On this topic I recommend the "learn X in 20 minutes a day" series of math books for folks trying to get started in subjects they've had difficulty with in the past. I always struggled with Calculus until that book. The math is taught almost entirely by visuals and intuition hints. I came away understanding Calc, not just how to perform the steps. It helped me divine the purpose of calculus, not just the process. I've used them for Geometry and stats to similar effect. Highly recommend for anyone who share the sentiments of the OP.
In the top answer one of the last paragraph sounds like the real answer:
> When I was a graduate student, we had a wonderful working seminar on Sunday mornings with bagels and cream cheese, where I learned a lot about differential geometry and Lie groups with my classmates.
At the end, the students needed context and enlightment. We cannot underestimate other dimensions of the learning experience.
Because they are written by mathematicians. In my case, when I have learned a mathematical topic, the intuition becomes obvious and the derivations/proofs seem to be much more important for gaining a complete understanding. I have gone up against texts with complete bewilderment, only to come back after gaining the intuition and found the extensions of the core premises and proofs provided by the text to be highly enlightening.
Great math teachers understand the need to teach intuition. He wasn't a math teacher, but I think Richard Feynman is the pinnacle of this. See [1] to see how he expresses intuition about physics, and his Red Books[2] for how he teaches mathematical physics with all the qualities I believe makes a great maths text for students.
Also, there's a linear algebra MOOC which also teaches great intuition before delving into proofs and heavy detail [3]. I mention these examples because they are exemplars of this idea of teaching intuition.
There really needs to be a version of the Feynman lectures for mathematics.
Although, this is what Arnol'd has to say [1]:
"Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap... In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences."
Feynman learned mathematics from a series of self teaching books published in the 1940's suffixed "...for the Practical Man" and prefixed with Arithmetic, Algebra and Calculus. I have the full set and this is a rather good solution to the problem. They teach you insight and how to think about things as well as the mechanical aspects. This is IMHO a well solved problem if you don't mind skipping more modern abstractions such as limits.
From there he was given a calculus book, the title of which I cannot remember. I never got that far.
I suspect you have to at least follow the same path to have the same intuition.
I sometimes get the feeling that we seem to have taken a huge step backwards in math books over the past 50 years. Back when I was in college and studying multivariate calculus I happened to find a small, ~100 page, book called something like "Introduction to Multivariate Calculus" from the 50s in a used book store. This tiny books not only covered basically the whole curriculum of my course, but did it in much greater clarity then the 500+ page that was our textbook. I can basically thank that book for me passing that course. I find on the whole that especially introductory mathbooks have gotten harder to follow and less clear (and a lot longer) over the past few decades.
I've taken the liberty of taking a quick snap of a random page in "Arithmetic for the Practical Man" to include below for those poor people poisoned by modern textbooks:
I see horrible modern behemoths of over a 1000 pages that leave you dazed, confused and full of facts but nowhere to go with them. EE textbooks are even worse on this front than your average mathematics text book. I've seen one proudly promoting over 1500 pages and 1000 illustrations, but doesn't even get as far as an opamp or discuss anything at system level.
bought the series as well, love it. It's my daughter's favorite math series. One interesting thing I noticed in this regard is textbooks from the 30-60s have way more textual descriptions. They seem to spend more time looking at the problem or concept in a literary way and that might have helped to build a better understanding for the student.
I think it's a bit misleading to say he learned math from those books. He got his start there, but surely the bulk of his mathematical knowledge was more advanced. However, it's quite possible that he retained the attitude from those early books. It seems to me though that he already had that attitude prior to reading the "practical man" books, and it is more that they particularly resonated with him because of it.
In search for books like these in the past, I found "Understanding Analysis" (Stephen Abbott). I went through the first two chapters and I liked it. It is written in a narrative which is both entertaining and instructive. He explains the problem, why is it relevant, ways of approaching it, etc. "It is designed to capture the intellectual imagination."
From the preface:
"This book is an introductory text. The only prerequisite is a robust understand-
ing of the results from single-variable calculus. The theorems of linear algebra
are not needed, but the exposure to abstract arguments and proof writing that
usually comes with this course would be a valuable asset. Complex numbers are
never used.
The proofs in Understanding Analysis are written with the beginning student
firmly in mind. Brevity and other stylistic concerns are postponed in favor
of including a significant level of detail. Most proofs come with a generous
amount of discussion about the context of the argument. What should the
proof entail? Which definitions are relevant? What is the overall strategy?
Whenever there is a choice, efficiency is traded for an opportunity to reinforce
some previously learned technique. Especially familiar or predictable arguments
are often deferred to the exercises.
The search for recurring ideas exists at the proof-writing level and also on
the larger expository level. I have tried to give the course a narrative tone by
picking up on the unifying themes of approximation and the transition from the
finite to the infinite. Often when we ask a question in analysis the answer is
“sometimes.” Can the order of a double summation be exchanged? Is term-by-
term differentiation of an infinite series allowed? By focusing on this recurring
pattern, each successive topic builds on the intuition of the previous one. The
questions seem more natural, and a coherent story emerges from what might
otherwise appear as a long list of theorems and proofs."
That rant is a bit ridiculous to post here - the foundations of computer science is largely the result of mathematicians who weren't particularly interested in physics.
Well for one Turing was certainly somewhat interested in physics. From his WikiP page:
In 1928, aged 16, Turing encountered Albert Einstein's work; not only did he grasp it, but it is possible that he managed to deduce Einstein's questioning of Newton's laws of motion from a text in which this was never made explicit.
Then there was von Neumann and several others. If not interested then at least well educated in physics.
That's the most egotistical thing I've ever read. It's pretty typical of physicists though. Whenever you see a 'scientist' on television pontificating about things they don't really understand (from psychology to economics to theology to whatever happens to be relevant) they're usually physicists.
I don't think it's the intuition. I think it's the part where people are explicitly and implicitly taught to avoid metaphors, since they are considered bad analogues and "window dressing on top of objective literal truths".
The sad part it, Lakoff and Johnson already provided a good counter-argument that thesis in the eighties with their landmark 'Metaphors We Live By,' suggesting that metaphors are the main way humans make sense of the world, almost as if they are the fundamental intuition you refer to. Since then then the proof for this case has only been piling up.
Especially in the field of machine learning we're finding more hard evidence that metaphor are not decoration, but fundamental parts of how to transfer information. Using rich metaphors to pass on implicit information between teacher and student is known as "privileged information":
> When Vladimir Vapnik teaches his computers to recognize handwriting, he [he harnesses] the power of “privileged information.” Passed from student to teacher, parent to child, or colleague to colleague, privileged information encodes knowledge derived from experience. That is what Vapnik was after when he asked Natalia Pavlovich, a professor of Russian poetry, to write poems describing the numbers 5 and 8, for consumption by his learning algorithms. (...)
[After coming up with a simple way to "quantify" the poetry], Vapnik’s computer was able to recognize handwritten numbers with far less training than is conventionally required. A learning process that might have required 100,000 samples might now require only 300. The speedup was also independent of the style of the poetry used.
Now, of course, knowing how to come up with a good metaphor is a skill in itself, and bad metaphors do lead people astray. But they do so precisely because they are so good at transferring information - wrong information, in the case of bad metaphors.
He's talking about a fairly advanced math topic, abstract algebra.
At some point, you graduate from "being taught" to "teaching yourself." By the time you get PhD, you need to teach yourself, because you're studying things no one has ever studied before.
I'm not a mathematician, but it's my understanding from what I've read that part of it is becoming used to groping around in the dark, sometimes getting stuck, and accepting that as normal. Solving the really tough puzzles means getting stuck a lot.
This might explain the relative lack of attention towards the user experience of non-mathematicians?
I find it amusing that you call it "user experience", a very computery/hackery term for mathematics. There are so many computer-like expectations for mathematics that I see on HN. :-)
Anyway, yes. Every mathematician I know acknowledges that frustration is the natural state of affairs. If you're not frustrated that's because you haven't been doing enough mathematics yet. There's always a bigger problem, a new concept to master, a new way to look at an old idea.
Yes, to extend the metaphor, writers often pride themselves in offering a good "user experience." Bad writing does not flow smoothly and is unnecessarily difficult to read.
Good games try to provide an optimal learning experience, providing just enough challenge to be interesting without players getting stuck. Play-testing is vital; if your players commonly get stuck in ways you didn't intend, it's a bug.
There's a lot to be said for designing a learning experience to flow smoothly. We can admire the work that goes into making that happen. (It then seems strange that, by contrast, writers of math books often don't seem to be playing the same game.)
One thing a well-designed experience doesn't do, though, is prepare you for being stuck and overcoming difficulty when you're not on an artificially smoothed path.
> It then seems strange that, by contrast, writers of math books don't seem to be playing the same game.
Oh, but they are playing that game. It's just a very difficult game. Everyone wants easy math books and lots of people are trying to write easy math books (for example, my buddy Ivan and his No Bullshit guides: https://minireference.com/ ). It's just a very difficult game, and very few have come close to success. When they do succeed, it's usually only for one kind of audience and not another. For example, Spivak's Calculus is widely admired in the mathematical community for its presentation, but I wouldn't be surprised if HN derided it for being stuffy, too mathematical, elitist, and full of itself.
> He's talking about a fairly advanced math topic, abstract algebra.
That's actually a remedial-level topic for MO, which focuses on research mathematics. It's meant to be an easy example that everyone can relate to in this discussion, like talking about how children learn how to add. You'll see they say, "simple example". I'm not trying to be elitist, just trying to explain MO.
As an aside, I found the conclusion of this simple example about how normal subgroups are about being kernels of homomorphisms hard to relate to, how would you know that without even knowing what a homomorphism is, which needs knowing what a group is? These a-ha moments that come after learning the material and make the learner assume that the teacher is an absolute idiot by not starting from the a-ha moment are very frustrating. It's kind of impossible to start from a-ha without the learner first bumping into all the dark corners and hard work that light the way to the a-ha. I myself have had topics ranging from (simplicial, singular) homology to the completeness of the reals and elliptic curves explained in so many ways and nothing ever made sense to me until I sat down and struggled through all the explanations offered to me. I don't think there's a way to convey insights to a mind that hasn't struggled towards those insights.
Here I am going to quote Michael D Alder, from the introduction to an old edition of some lecture notes of his that I can no longer find online:
I try to start every course of lectures with an overview or
outline of the material in the course. Before I do that, a (true)
cautionary tale. While I was an undergraduate doing Mathematics,
I told my Calculus lecturer, a celebrated mathematician called
Ian Porteous:
“I have been getting the strong sensation in your course of being
dragged at high speed down a narrow track in a jungle. There have
been dimly sensed paths off to the sides but we have been
galloping after you and have no time to explore these
possibilities.”
I went on: “What I would like, is to be taken up to a high place
and given a view over the country through which we have been and
that where we are headed, an overview of the subject.”
He replied with an amused grin, “Wouldn’t we all.”
He elaborated the point, which was that in order to have an
overview of a subject in Mathematics, you have to crawl all over
it on your hands and knees, and then you move next door and do
the same with another mathematical subject, and then, if the
subjects are related, you may get a higher level view of the two
bits you have sorted out in detail. And then you can do some more
detailed understanding of some other bits and link those, and
maybe get a higher order insight linking the links. But a higher
order view from a height without doing the detailed work is not
possible.
The reason it is not possible is that you are looking at ideas.
You can only develop a higher level idea by understanding the
base level. We could tell you the words for the higher level
ideas, but they wouldn’t mean anything. For this reason, course
outlines in Mathematics are intelligible only after you have done
the course, not before. This is extremely maddening, particularly
to philosophers, journalists, post-modernists and others with
similar intellectual handicaps, but that’s the way things are.
It follows that explaining what the course is about in a general
way is a waste of time and can’t really be done, but I shall do
it anyway.
Edit: I just serendipitously found a passage of his in a very similar vein. Read section 5.1, "Cultural Anthropology": ftp://www.biophysics.uwa.edu.au/pub/Mathematics/Alder/DiffGeom.pdf
By the Curry–Howard correspondence, programming is in fact (formal) theorem proving.
Mathematics got a lot less interesting after I realized it amounted to a giant, informally-specified, mostly undocumented body of code designed to run on the human brain... from that perspective it's hard to see why one should prefer mathematics to a well-written software program that does the same thing.
So it's legacy code that runs on the oldest hardware that we have.. Plenty of people are interested in updating this code so that it runs on more modern platforms. Actually, this is my favorite way to drill into a piece of mathematics: port it to python.
Is Curry-Howard interesting if you're not programming with advanced type systems? It's my understanding that writing a function that returns an integer (for example) corresponds to proving that an integer exists. Whoop-de-do? This would imply that unless you're constructing a sophisticated return type, you're not proving anything interesting.
My first programmation book was the K&R book.
I hated reading it and was pretty sure that was nothing I would ever understand (or use).
25 years later, I now understand all the details of the book.
And now I hate ... the C language itself (with passion!).
Fortunately, nowadays, there is so much learning material that you have the opportunity to choose the one that suits you.
For example, I know that I cannot apply a theory upon reality. I have to start with examples and build my own vision by abstracting from a lot of examples.
So I choose my learning material accordingly.
My current interest, at the moment, is monad. Believe me, there is a lot of abstract articles about that.
But my entry point was a tutorial about implementing some stuff in Javascript, that happened to be functors, monads, etc.
Believe it or not, but I used that knowledge the week after in one of my Java projects. And my colleagues considered it was "a nice trick" (which is true!).
In one word, the learning process must be fluid. Don't try to force anything. If the book medium does not suit you, don't blame the book. Or yourself. Try to find another resource and keep learning.
There exist purely theoretical computer science topics. Ullman's automata theory lectures are freely (legally?) available online. There are also authors like Knuth who are surprisingly practical.
This problem goes way beyond math and programming and into physics and general engineering. There are a surprising number of engineers who can manipulate the linear algebra tools but can't actually engineer structures or figure out linear circuits. EE filter design is another good example.
If you look at higher ed vocational training as kind of a prep-school or maybe a qualification filter for an apprenticeship it makes more sense. Once you figure out linear algebra you're qualified to apprentice to someone to teach you how to actually use it.
The top-rated answer is either defeatist, or just rationalization for the sentiment "I had to go through this and figure out everything myself, so you should too".
There is a huge amount of information encoded in the choice of exactly how to define thing, and which theorems people care about. This reflects a long process of trial-and-error as the field was constructed. For a famous philosophical treatise on this using the Euler characteristic as an example, see "Proofs and Refutations” by Imre Lakatos
Most of that foundational information is lost when it's not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject.
It's a huge problem, it applies to physics textbooks too, and it doesn't have to be this way. Unfortunately, the problem has been known for decades and there's not much reason to expect things to change. (Lakatos wrote the above in 1976.)
> This reflects a long process of trial-and-error as the field was constructed.
A way to think about this process: "math" isn't quite a tower of abstract concepts; those only exist in any given mathematician's head, and die with them. Instead, "math" is a name for the novel pieces of language we invent, with the purpose of using them to talk about and (hopefully) share those internal abstract concepts, exposing to the rest of the world concepts that were entirely inaccessible (to everyone but the originator) before that language was introduced. Math is the attempt to communicate never-before-communicated epiphanies; to describe the shapes of never-before-pondered abstractions, with properties only describable using never-before-spoken words.
One thing that this implies, is that the first piece of language that lets us even vaguely point at some particular idea so that we can get to work on analyzing it further, might stick around long after we come up with some clearer or more coherent language, because the former one now has the momentum of historical use behind it.
Math, when done this way, then becomes a precarious pile of "just good enough to survive" conceptualizations, rather than a precise tower of "best-tool-for-the-job" conceptualizations. And that's no good for teaching.
I'm not so sure. In most fields, we certainly discover new things (concepts, relationships, laws.) But we can usually describe those new things by analogy to existing things, because—given enough abstraction and analogy—they have recognizable "shapes." There are existing mental tools in our civilization that fit the concepts, and let us toss them around and look at them from all the angles.
Math is where we invent language to refer to new forms of abstraction themselves: novel possible shapes for our thoughts to take when we think about other things. You can't talk about a new shape by analogy to existing shapes. Nor can you abstract an abstraction in a way that gives you anything more familiar. (Instead, you'll usually get more novel, ontologically-primitive abstractions, like going from monads to arrows, or going from numbers to fields to rings.)
Sometimes disciplines like physics will find a concept that we don't have any mental tools in our toolkit for yet. Then we build some. But we still refer to the process of doing that as doing mathematics—and then we apply that new mathematics back in the problem domain to talk about the new concept.
(For a good example, the formalization of quantum theory in physics, required the creation of infinite dimensional analysis in mathematics. Physicists did most of that work, but the work itself was still mathematics, not physics.)
---
Now, other fields do still have a similar problem to mathematics, of historical momentum carrying forward old "things" (again: concepts, relationships, laws) when there are better, clearer "things" that could be used in their place. But when we're not working with pure abstractions—ways of thinking—we can make the effort to compare and contrast old and new "things", and decide that some might be more edifying than others.
It is possible for an especially-gifted physics teacher to write a very accessible physics textbook, because they need only pick all the clearest "things" to demonstrate. That teacher will still be stuck in a given paradigm—a way of thinking, a belief in the worth of some "things" over others, popular in the culture of their discipline at the time and place they worked. But they might be able to (barely) rise above it, if they think hard about the history of their discipline and the paradigms it has gone through in the past, compare-and-contrast those, and synthesize something that isn't quite just the paradigm they're immersed in.
Mathematics is uniquely problematic because it is entirely paradigm. It is a tree of paradigms—each new abstraction only making sense assuming the paradigm it was created in, and then becoming the paradigm for further abstractions still. Every mathematician, all the time, is trying to discover what a particular paradigm—their specialization—can be twisted to accomplish. Not one of us has the brain power to know the total space of things that one of these ways of thinking can be used to express—the problem domains the tool is applicable to—in order to know which tools show more or less promise at being "powerful." We know what we've discovered so far, but we have such an infinitesimal idea of the "space of all possible abstractions" that we could be totally missing some of the best, and using ones that are barely satisfactory.
Point a hypercomputer AI at "solving physics", and it'll spit out a description of the universe that will certainly have more "things" in it than we know about today—but which will still also contain a subset of the "things" we do have. (The most "carving nature at its joints"ing ones, presumably.) Those "things" that get carried over will, of course, be defined much more precisely; their concept-boundaries will be adjusted to includ...
> "math" is a name for the novel pieces of language ...
I'm keen for etymologies and hence believe that math means the art of problemsolving or learning successfully. That includes language as a problem domain, and teaching tool for learning, but as modern development would have it, it's about structure and organization, not just in language.
Sure, literally, yes, the word "mathematics" is not an -ology; it isn't a discipline concerned chiefly with carving up reality and giving words to the results.
I was just trying to highlight the fact that mathematics as an institution is a process of building up share-able symbolic abstractions; of inventing a "language" one new word at a time.
I'm using "language" here, and above, to refer not to the words used to discuss mathematics (the... "mathematology" of math), but rather to the thing that includes objects like mathematical operators (e.g. "+", "⨯", "∫", "⇒") as its "words." Not the language about mathematics, but rather the language that is mathematics: the ever-growing set of abstract tools with symbolic handles which we've constructed to allow us to manipulate other concepts inside our heads, in rigorous ways where you can trust that if you and another mathematician do the same named mental 'move' to the same source concept, you will both arrive at the same destination concept as a result.
For a cute analogy: you can think of a martial art as a vocabulary of known, precise body movements, that can be taught. You can think of mathematics as a vocabulary of known, precise mental movements, that can be taught. Yes, this makes mathematics an art; but, equivalently, this makes a martial art a language.
To sum up:
• Mathematics is itself a (formal) language. It doesn't really fit in the category of words like "biology"; it fits more in with words like "logic" or "C++".
• To say that someone is "doing" Mathematics just means that they are using that language to achieve a goal; it's about the same as saying that someone is "doing" Python.
• To say someone is a mathematician, is to say that that person works to explore and extend the language of Mathematics, to test its properties and its limits, and to invent new 'words' within it that may then be used by those "doing" Mathematics.
Yes, I might have missed that you used "language" as an allegory rather than a metaphor. I find ironic how this reaffirms the stereotype of the divide between language and mathematics, ie. mathematicians being bad at maths and the other way around, when really the combination of both creates a synergistic effect that helps each to surpass the effectiveness of each on its own.
Highly recommended book by Lakatos. The history and meta-mathematical aspects of proof-theoretic constructs are so often neglected that one can only imagine that there's a latent desire to attain a priestly purity to the proof-theoretic process that serves only to hamper true mathematical maturity in the field and individual learning.
"I had to go through this and figure out everything myself, so you should too".
Some of us like it and don't see value in the long winded motivational style. Things like group theory are targeted on people who are interested in math itself and like the mathematics for what it really is.
It is interesting to learn history once in a while and there are good books about that. However, most of the time you want to move on faster.
Please use absolute references (a link to the specific answer) instead of fuzzy references that change over time.
Currently, your answer is the top-rated, so which one did you mean? (And even if your answer was not at the top, the top will almost certainly change over time, leading to all kinds of misunderstandings.)
EDIT: It seems that this referred to the top-rated answer on MO, not HN:
This demonstrates my point even more: Please use absolute references, as fuzzy references make for misunderstandings! (And as my comment received quit a lot of upvotes in the beginning, I'm pretty sure I'm not the only one who thought this was about the top-rated HN answer.)
I took their post to be referring to the top reply on the math exchange site, which may actually be a stable thing at this point as the question is not fresh. I may be wrong though.
Humans are strange. Perhaps, though it's not certain, if your original comment were couched as a softer request, then that softer approach might not pique certain small subsets so much?
I can remember feeling really intimidated by the proofs of theorems. What I now think is that at least some proofs are analogous to code you struggle when writing to get something working. Later, once you'd gotten the thing done you'd realised that there were steps that could be eliminated and still others that could be cleverly combined. Then later again when you wanted to show it off you tidied it up some more and what you then presented to others is much more elegant but perhaps harder to understand.
It's becoming more obvious to me with time that math and its corresponding proofs are very similar in concept to our code and unit tests. Theorems like our software are built from rudimentary building blocks, each needing to be understood in order to build the software and then the tests are the proofs that each of the building blocks are correct...
What's missing for a lot of people is "What does this theorem apply to? Why does it apply? How do I break this theorem down into its component pieces to understand it?"
Indeed if your math text started off with a concept of a real world thing you're trying to figure out and then break it down to a series of paradigms that the student understands before applying each of the mathematical components that applies to each of these paradigms before finally combining them all to the final theorem, people would get it far more easily and be much less intimidated by it. Then over time students will spot paradigms that combine more elegantly to the final theorem and understand how to substitute them.
I think Math is being taught backwards in schools and this is why so many people are intimidated by and shy away from math... and I was a student who both struggled and enjoyed Math... and I still struggle with it because we were taught to think about it backwards.
We started out well with the basic building blocks, but somewhere between basic algebra and calculus, this all got turn around ass backwards. Where does the motivation to understand derivatives and integration come from if one doesn't understand the implications of their application?
That's a somewhat naive way to say it. Math is not code, it's not programming. You can code a Python script without knowing how does a operating system work. You cannot understand nor use a theorem without knowing the definitions involved. I can tell people, for example, that the concept of derivatives and integration comes very handily when solving differential equations that model real world phenomena (for example, fluid motion). But how can I explain what is a differential equation in a manner that's useful to someone who doesn't know what a derivative is? Math, for better or worse, tends to not have direct applications (on the other side, much like other subjects in their basic stages) and you need to advance before getting to a point where you learn something you really need and then see the usefulness of it all. It's like everything: you don't know what you need and how strongly you need it (despite how many times people will tell you that you need it) until you really have that necessity.
Starting off with the general concept and breaking down into paradigms is not always possible. For example, say you want to teach people the Stokes' theorem of integration of manifolds [1]. It is the generalization of the divergence, Kelvin-Stokes and Green theorems that are very useful in electromagnetis, fluid mechanics and probably more. The theorem says, broadly, that if you are measuring some quantity in a certain space, you only need to know how does it "accumulate" on the boundaries of that space. But that will not be useful to an engineer or physicist. To really understand that theorem and break it down you need an entire course on differential geometry, with requisites on topology and calculus. Therefore, in the analysis courses for engineers/physicists the most probable situation is that either they present you the theorem and tell you how to perform the required operations on differential forms (which, if you don't know what they are, is a complete mistery/dark magic), or they go to the specific theorems (Gauss, Kelvin-Stokes, Green) which seem clunky and completely magic again.
The summary is that to be able to know what does a theorem apply to, why does it and how to break it, you literally need to study mathematics: that's what mathematics is. And the way to study it is always backwards, it's not like physics or programming where they can show you the high-level stuff, you can see what's happening and then you start to understand it. In math, most of the time, if you learn some subject starting from the high level stuff, it is difficult to even know what on earth are they talking about.
The existence of fields that merge math and programming does not mean that "doing math" and "programming" are similar actions/similar fields. I've studied both and there is a world of difference.
> How else would the math be invented in the first place?
Usually the first step for problems to be solved is their expression in a mathematical way. See the heat equation: Newton (and probably others, history is not my strong point) already captured the core idea of heat transfer (transfer rate is proportional to the temperature difference) but it could only be really solved by Fourier once all of those ideas were put in mathematical expressions.
The very act of putting all those ideas into a mathematical expression is in essence programming. You take a whole bunch of ideas, codify them and put them together to either explain something else or produce a product - a theory or formula. Which in essence is what programming is.
> You can code a Python script without knowing how does a operating system work. You cannot understand nor use a theorem without knowing the definitions involved.
Better analogy:
You cannot code a Python script without knowing how to call the APIs you need to implement the functionality you want.
Knowing how an operating operating system works is more like knowing how foundational mathematics is axiomatized. Interesting and occasionally useful, but not immediately necessary for the theory of differential equations.
I think you might even be able to deal with differential equations without involving limits, by simply asserting all necessary theorems about derivatives as axioms. This corresponds to relying on a robust library (= the theorems other mathematicians have proved), without worrying about implementation details (= the proofs of those theorems, which might be much more difficult to understand than the results can be used).
You might be able to teach someone how to solve differential equations mechanically the same way you can teach someone how to press keys on the keyboard to code a Python script. Yes, it works, but it's specific and the moment something is slightly different it will not work. For someone to be able to work with DEs they need to know how to work with derivatives and integration, and I think it will be worse to not introduce the formal definition of a derivative. Giving just the rules of calculation will appear as magic or arbitrary, and even worse, it may lead people to wrong conclusions (such as operating differentials as if they were fractions because that worked with ODEs). And that's not even entering into all the theory of differential equations, which usually in engineering it does not get much attention (existence and unicity is not that worrying for an engineer, you tend to already know that a solution exists as you did an experiment/saw something/etc).
> It's a huge problem, it applies to physics textbooks too, and it doesn't have to be this way.
I've noticed this too. I've tried to learn some quantum physics for fun, and it seems to me that most textbooks have surprisingly little description of the actual physical experiments behind the physics. Maybe this is something they usually cover more in lectures?
Like, I haven't found much details about the actual physical experiments around the spin of electrons (and certainly not of other particles). Sometimes there's an abstract description of a basic experimental setup.
I read just a few days ago that Feynman would redo old experiments that other physicist weren't doing anymore because they were established science. This is a great learning experience if you have access to the equipment, but most people don't. There should be books that describe how physics was developed, experiment by experiment, with lots of details.
People severely underestimate how much detail is going into science.
If you read a 1000 page textbook on experimental physics, thats just the cliffnotes. The very highest level "here are the results of 400 years of serious research". There can't be any detail.
If you want the details on the stern gerlach experiment, for example, read the research papers. They will contain all the detail necessary to reproduce the experiment.
Lectures are just like those books. They just teach a very broad overview of what has been discovered.
The books you think should exist do exist. They're just not written for casuals.
research papers can all be read by "people in the field".
its just that someone who is doing environmental physics doesn't necessarily understand papers on string theory. science is specialized, yea. if you want to understand research papers, get an education.
or just do something else. cutting edge research is not for everyone. not everyone is supposed to understand it. whats next? people demanding research papers be dumbed down to adhere to the freedom of information act?
science is hard. big deal. you cant have all the things at the same time. read the hawking books written for casuals if you want a cute little story about science.
show me one paper that you cant understand because of the grammar but would be perfectly fine reading it if only the lyrical style was a bit more up your alley.
Effective communication is about far more than whether the reader/viewer can eventually understand the material. Even basic presentation skills can make a huge difference both to the speed at which someone can receive and understand new information and to how well they will retain that information later.
Unfortunately, many academics receive little if any training in good presentation before being expected to lecture or write at undergraduate or graduate level, and while some are naturally gifted presenters anyway, most inevitably are not. Consequently, many career academics have no idea how poor their presentation skills are, how ineffective their presentation is as a direct result, or how much better they could be. They just get stuck at a very low level, but without the kind of introductory/remedial training that would be given to someone whose career involved presentation skills in the professional world. And of course if anyone with broader experience dares to suggest that there might be room for improvement, the instinctive reaction is denial.
A little irony is that some of the most engaging and informative presenters I have ever seen or read come from that same community, but despite the emphasis on peer review in their research work, when it comes to soft skills the weaker presenters typically have no idea how bad they are and therefore make no attempt to learn from their stronger colleagues and improve.
Unfortunately, as your own choice of word "exceptional" implies, most are not so lucky.
I once sat in a review meeting at the end of a year with members of the faculty responsible for teaching collecting feedback from many of the undergraduate students. When challenged about the poor quality of many lectures, the response was essentially that they can't make the lecturers go and learn how to lecture competently because the lecturers wouldn't stand for it.
Try that attitude in the professional world and you'd be in a remedial programme on your way to getting fired.
Fortunately, I expect that with the advances in modern technology and changes in modern careers, the old-school universities that think a famous name and charging high fees mean they can get away with anything will soon be obsolete, and so will the incompetent parts of the academic community sheltering within them. They will need to find new ways to offer dramatically more value than interested people can find on their own with all the modern resources we have available, or they won't be able to justify people taking several years out of their lives and paying a fortune in fees to attend any more.
For the casual reader, the "bad writing" aspect of the difficulty is severely outweighed by the "lack of domain knowledge" aspect, by several orders of magnitude. If you take a random modern research paper, and put it through a handful of editors to make it superbly written in a clear, convincing language that gets its points across very easily, the casual reader uneducated in the background knowledge will barely be able to tell the difference.
Plus, it seems that authors sometimes feel compelled to appear smarter by leaving out the intuition and motivation behind a result, and in particular the easier and more intuitive simple cases (which might have been the inspiration for the result in the first place), and instead present only the most abstract & most general version of the result that they managed to prove.
Including some of the enlightening historical path towards that result is not "dumbing things down".
A large part of any scientists job is communicating results. The people you criticize may not be novelists, but they are certainly professionals when it comes to communicating technical knowledge.
However, this knowledge is only communicated to other experts in the same - often very narrow - subfield. Often the definitions that give you a hard time when reading a paper have been refined over several years and are basically known to all other people working in the same field.
This is not ideal, but there is simply not enough manpower to produce good and generally accessible summaries of current research topics every few months...
> The people you criticize may not be novelists, but they are certainly professionals when it comes to communicating technical knowledge.
No, that's the point I'm trying to make. They do not have enough training in making themselves understood. Hell, a good portion of them don't even have the language in which they are writing as their primary language.
> research papers can all be read by "people in the field".
Not necessarily. The classic example is Shinichi Mochizuki's work.
He's done some incredible work but he's basically invented his entire field out of thin air, he doesn't publish frequently, and the papers he does publish are essentially impenetrable.
He likes it this way and doesn't want to "dumb down" his work for the mere mortals who need to review it. That's essentially the end state of your argument - if you can't comprehend it you're not "in his field" and you're obviously not qualified to discuss it. And he's certainly not going to waste his time teaching some dunces the basics of his field.
As someone who works on the foundations of quantum mechanics (how to formulate, why we think it's right, etc.), I'll disagree with this. The fact that much of the detail from many experiments is available somewhere is not much more useful than being told that code is available from the authors if only you email them and wait 3 months. The barrier to figuring out this stuff are immense.
Furthermore, it's not necessary to communicate the actual, usually circuitous, route taken historically by scientists. Instead, you could just describe the series of hypothetical simplified experiments whose results would lead one to quantum mechanics and rejecting alternatives. This is never attempted in a serious way (trivial uses of the Stern-Gerlach experiment as a model of quantum mechanics not withstanding).
Dirac derives quantum physics from first principles, including the relevant thought experiments, in one of his books. that was good enough for me. made a lot of sense.
Dirac's book is a perfect example of what I'm talking about. Just think about the fact that this book from 1958, which makes a valiant attempt at justifying a few parts of the formalism (it's on my bookshelf), is basically a high-water mark even though it's very flawed, especially given what we know today. What does it tell us about the ability of physicist to transmit ideas to the next generation with high-fidelity if, to understand why the quantum formalism is how it is, you recommend reading a 60 year old book by a guy who was alive when it was being formulated!
Some of those flaws:
(1) Dirac postulates, as most people do, that measurable observables are to be identified with Hermitian operators. This is a mistake that can be traced back to von Neumann. In reality, the larger class of normal operators are perfectly fine as measurable observables. Indeed, measurements are properly associated with only an orthonormal basis, and it is completely unnecessary to label them with eigenvalues, real or otherwise. (To see this, just observe that there is no physical difference between experiments that measure x and x^3 for the position of a particle.) Dirac's discussion on page 35 is just wrong.
(2) Unless my memory fails me, Dirac gives little to no justification for why we use tensor products to build up the state space of a many-body system from the state space of a single-body system, a tectonic shift from classical mechanics at the heart of the weirdness and power of quantum mechanics.
(3) Dirac was written before Bell's inequality. I mean, just look at the pithy discussion by Dirac (p. 4-7) to justify fundamental indeterminacy, one of the most profound things we know about the universe. Do you think this would have convinced Newton? Or Dirac in 1925? (We know it didn't convince Einstein.) This sort of thought experiment is lovely for an article in Scientific American to give laymen a sense of where things come from, but it's nowhere near the rigor with which we should teach physicists.
I really wish I could find a book (or series) that started with explaining the (non-)results of the Michelson-Morley experiment, and traced through the major experiments that underpin modern theory. Just picking out the major quintessential experiment (types) -- things like the Bell test.
Much of science is really just subtle modifications or tests of major theories, but I feel like you could write up... say 12 keystone experiments in 100 page summaries and publish it in 2-3 volumes, (eg, QM and relativity volumes).
The problem I have with only presenting "polished" results is that we lose the context of our modeling -- eg, QM seems to have included non-determinism in an effort to preserve locality, but locality couldn't be preserved in the face of other results, so is non-determinism merely an extra assumption included for legacy reasons (eg, technical debt because no one wants to clean up the model/people forgot why we included it)?
You'd never even think to ask that question if you only saw the cleaned up model of QM divorced from its (philosophical) roots.
> Like, I haven't found much details about the actual physical experiments around the spin of electrons (and certainly not of other particles).
The Stern-Gerlach experiment (1922) is what you're looking for. "Modern Quantum Mechanics" by Sakurai opens with an explanation of this experiment and its consequences.
> I've tried to learn some quantum physics for fun, and it seems to me that most textbooks have surprisingly little description of the actual physical experiments behind the physics.
Try the free and new course "Quantum Mechanics for Everyone" that despite the title is certainly not dumbed down at all, on edX.
Look for books on experikental physics! Pysics bachelors measure many natural constants in practice labs. Many compendiate text books will have rich sources. I'd look for ones from the time the of the discovery to get the gory details which many textbooks avoid to achieve didactic reduction
> Most of that foundational information is lost when it's not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject.
From "A Mathematician's Apology", G. H. Hardy:
> Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain.
From another angle: when developing new theories or models, your thoughts are all over the place and frankly it's boring to go over your own crappy notes afterwards and try to reconstruct them in a way that others can understand. And much of the time you forget exactly what happened along the way as well, so any story you reconstruct is going to have some hindsight bias, which defeats the purpose of trying to "teach the story".
Really, from the first answer:
> Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through "guided struggle".
This is the only way to "properly" learn mathematics or science. Anything else is only making you think you've learnt something.
>> Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through "guided struggle".
> This is the only way to "properly" learn mathematics or science. Anything else is only making you think you've learnt something.
My point is not just that most textbooks make it too hard/inefficient to learn this stuff. My point is that most of it is never learned. The Legendre transform connects the Lagrangian and Hamiltonian mechanics, the two fundamental formulations of both quantum and classical physics, and yet most physicist cannot tell you why the transform is defined as it is. The reason is they don't take seriously the possibility that we'll find non-Lagrangian phenomena, and so they have not been forced to consider what observational and theoretical evidence led to it's identification in the first place.
I learned math through guided struggle. My high school had a kind of macho attitude about it, we were all about math competitions and pushing through university level stuff etc. Today I feel that was a good way to study, but much of the struggle could've been avoided. You must solve problems to progress, but they don't have to be hard problems. They just need to be formulated at level n but require solutions at level n+1. Devising such problems is hard and many teachers don't bother, instead they give you definitions at level n+1 right away and make you solve problems about those. That's the root of the problem IMO.
No, definitely not, and I agree with the other sibling replies. I was more specifically responding to the part that was dismissive of the "so you should too" point.
Certainly, we can and do develop newer and simpler ways of understanding previous theories. And teaching the historical sequence of events can help with understanding; I myself experienced that with [1] for modern analysis. However, these understanding-aids don't teach you how to do mathematics, and only marginally improve your ability to apply those models and theories to existing real-world problems. To improve your ability to do mathematics, active exercises are necessary. Really, it's the same with many other fields, you don't get to be a good musician merely by reading about music and music theory.
I assumed they used the term "struggle" poetically, it certainly doesn't have to be unpleasant. But you have to put in some active mental exploratory effort. I found this post [2] a good summary of the skill set. But it's very abstract and likely won't make much sense unless you've been through the experience yourself.
These understanding-aids are also sometimes unnecessary. If you've done enough of the right kinds of exercises, they are of themselves an aid to understanding. For example, I could understand category theory better, not by learning about how this theory was developed historically, but by writing lots and lots of similar programs, and having a natural tendency to syntactically (and without much thought) refactor my code to be less repetitive, eventually leading me to various "category theory aimed at programmers" blog posts and papers. This one [3] of course deserves a mention, but there are many more.
To further emphasise this point, very brilliant mathematicians can just "pick up" models and concepts and work creatively and productively on them, without needing these aids.
My other point was that, the understanding-aids are very rarely what actually happened in the head of the people that developed a theory. Even historical narratives have distortions, and they are rarely detailed or precise enough to describe the rejected options, nor why they were options in the first place. (This fact, is also why they are not useful for teaching how to do mathematics.) There are exceptions, but reconstructing them is a boring process with little reward, especially since new developments 10 years later might explain it in even simpler terms.
That said, I would disagree with this part (from the top answer to the OP):
> a) The goal is to learn how to do mathematics, not to "know" it.
Modern mathematics has so much damn material these days that it's impossible to learn everything you need in order to solve modern-level problems, merely by teaching yourself all models and all theories "the hard way". Understanding-aids are certainly needed, and I use them very often myself, and I certainly prefer resources that teach using good analogies, proper context, descriptions of the motivations behind a theory, step-by-step "n/n+1" exercises, and everything else that other people mentioned here.
>> Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through "guided struggle".
>This is the only way to "properly" learn mathematics or science. Anything else is only making you think you've learnt something.
I agree with this statement but I think it misses the point entirely. Guided struggle is indeed necessary, but learning some theorem without seeing the impetus for its discovery is like learning how to play an instrument without understanding that the intent is to make music. Yeah, with enough time and struggle, you might be able to go through the motions and play some scales, but most people don't learn to play instruments that way, they learn to play a simple song or two, then go back and start with the scales and building musical theory.
Math textbooks sometimes try to do the same thing, but it seems like they always come up with the most inane and pointless exercises.
All of this is just to say, learning they why of math can help someone learn the how.
> mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain.
Yes, Hardy was a great mathematician and he did say this --- but most mathematicians have tremendous respect for peers who strive for clear exposition in their lectures, their papers, and (if they write them) their books.
I am a professional mathematician, and Hardy's attitude is one I have never heard expressed by any of my peers.
"A Mathematician's Apology" is a fascinating read, but his description of mathematicians' attitudes is certainly not accurate today.
It indeed can be boring to reconstruct your thoughts in a way so that others can understand -- but many of us make the effort anyway, and doing so often leads to new insights.
The full paragraph (in fact, the very first paragraph of the essay) reads:
> It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.
I interpreted "men who explain" not as mathematicians that can explain their work well, but as people who try to explain mathematics in a "lay" way to cater for a large audience, whose explanation can very often become inaccurate, non-mathematical or just downright false, yet still get public credit for seeming to know the field very well, despite these inaccuracies, and even though they are not directly pushing the advancement of the field itself.
It's of course a good thing to try to reconstruct your own thoughts, but I wouldn't say it's unreasonable for a mathematician to omit doing that. Could you go into more detail on the examples you mention, where doing so led to new insights?
> Could you go into more detail on the examples you mention, where doing so led to new insights?
Good question. It's a bit hard to do so (especially without going into mind-numbing technical detail) -- in math you never quite know where insights really come from. "Fortune prepares the prepared mind."
But generally speaking, I would say that good exposition gets you thinking about: Why does the technique work? What is the key insight? What are its limitations? And if you think about such questions, you naturally get a better sense of for which other questions your techniques are also likely to work.
How's about a half way step here: I agree that guided struggle is probably the most likely path to learning, but giving the student a concrete context they can apply does wonders. Back when I was first learning Calc, I had a professor notice I was also writing game software, at the time I was making a "Missile Command" clone. He pointed out that I could use calc to create "guided missiles" and more calc for various other things I had solvers. The light dawned in my head, and suddenly just the realization of the concrete application caused several calc concepts I'd been struggling with evaporate. Additionally, that created a reason, a justification for interest and context, and math almost instantly became peer to my interest in software, practically erasing any conceptual difference they had in my head. I cite that memory as the moment I became a mathematician.
Or maybe, just maybe, some things really are complicated and there is no quick and easy way to learn them. You really have to struggle and hit your head several times until enlightment comes to you. Books and teachers can only do so much, they can help you and guide you, but often there simply does not exist any magical way to convey stuff to make you instantly understand it.
While that is true, there is really no need to make it (deliberately?) harder by withholding relevant examples, applications, and historical background.
No one is deliberately withholding them out of malice. Practical applications are often convoluted and complicated or too forced. It is like wanting to do triple flips on trampoline before you can jump to sit on it. The sport analogy hold for historical background too - it sometimes help or makes it more interesting, but most often helps as much as knowing history of flips helps to learn flips.
Contrary to what you say, practical applications are easy to be found where they are simple and helpful. They make for fun math hobbyist like to play with and that works nice in terms of making kids interested in math. Unfortunately they do not lead to cutting edge science nor math needed for physics and engineering.
The top-rated answer states that "the goal is to learn how to do mathematics, not to 'know' it". I don't think that's a defeatist sentiment or a rationalization for the state-of-things. It's certainly true that one would gain more insight that shows how modern-maths-as-we-know-it evolved from a centuries' long struggle to understand basic patterns (exempli gratia, Cox's "Primes of the Form x^2 + ny^2"). But it is also true that, when you are doing mathematics, such a book is not nearly as helpful as an impenetrable tome like Neukirch's Algebraic Number Theory.
From my own experience (which, I'll admit, is not much to extrapolate from), the former was great for its "insight" and the latter just had too much unnecessary "information". Until I got stuck in an actual maths problem. Then suddenly my opinion switched, and the latter became a trusted map through strange lands.
You're right about the amount of information that's encoded in a definition or theorem. And it is very difficult to portray why that information is important (never mind how to access it, see other's comments about scaffolding and cathedrals and the like). I fear that trying to describe why groups are defined like they are would just lead to more impenetrable tomes - but instead of being maps to the lost traveler, they'd be more like tourist adverts. But the questions the source was asking is "why are books like this?" and "how do others [learn maths] in this situation?"
Well, maths books are like this because they are guides to help people doing maths. And you learn maths by doing it. I don't think its defeatist to say that no book or lecture about riding a bike would compare to the experience of actually riding a bike. To paraphrase the top-rated answer, why would you expect anything different from maths?
But to answer the underlying question of both yourself and the source: how do we improve this? I think pedagogy should focus on getting to "do maths" faster. You really learn by getting stuck in a problem.
> I don't think its defeatist to say that no book or lecture about riding a bike would compare to the experience of actually riding a bike. To paraphrase the top-rated answer, why would you expect anything different from maths?]
This it the wrong analogy and does not capture the MO answer. The correct analogy would be training vehicle designers by teaching them all the parts necessary to build a modern bicycle without teaching them any of the ways early bicycles were designed and why those designs were discarded.
> The top-rated answer is either defeatist, or just rationalization for the sentiment "I had to go through this and figure out everything myself, so you should too".
Yes, I agree this MO answer (and several others) seem to be rationalizing the situation rather than acknowledging how suboptimal it is.
I think I'm a good example of the system's failure. Coming in to college, I was something of an ideal candidate for becoming a mathematician. I had some success in Olympiads and already had decided I wanted to study math. My goal in life was to be a math professor.
I enrolled in a top college and took many graduate-level classes. However, by my senior year, when it was time for me to decide the next step in life, grad school or industry, I had become somewhat disenchanted with theoretical math. Math was so abstract I started losing interest: all this commutative algrebra (for example) I learned wasn't making me feel like I had any new insights into solving math problems, outside commutative algebra problem sets.
And so I went into industry.
However, I can't help but think that if I had more knowledge of the motivation behind all the abstract math, I wouldn't have lost interest. All that machinery of commutative algebra was invented for specific reasons, such as solving polynomial equations in the rationals through algebraic geometry. Years later, through casually reading math on the internet, I've been getting hints as to what power these highly abstract frameworks give you for solving concrete problems. But without seeing the end goal, and having some idea why I should be learning this in the first place, I felt like I was just getting lost in abstract nonsense.
>Most of that foundational information is lost when it's not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject.
This is exactly right. I personally failed to reconstruct enough to keep myself interested in the subject.
"closed as no longer relevant ", I find that is a growing problem with StackOverflow. My colleagues and I personally asked several relevant questions (like this one, that has many upvotes on HN), that were shut down or deleted by StackOverflow heavy-handed mods as "off topic" , "not relevant" etc. This made me give up on contributing and treat it as a read-only resource.
I'm very curious about this: is there a reason the reductionists seem to win in communities like this? Wikipedia has a similar culture. Are there systemic forces that point this direction, or is it just an accident of the history of the specific people involved?
There may be so much spam that they practice a shotgun banning approach without comprehension or real content evaluation. Somehow the best "spam banners" get elevated perhaps?
There should be a niwdoG's law: the longer an argument goes on, on the internet, the greater the likelihood that someone will accuse someone else of being a nerd that spends all their time indoors, while arguing on the internet themselves.
It's worth noting that SE and Wikipedia are two of the most valuable reference resources human civilization has produced, ever in history. Maybe the reductionism contributes to their high level of signal.
My problem is that it feels like they could have been much more useful.
Annoyingly often it is also the most useful questions that are closed while things I consider trivia style/karma-farming operations like: "what is the reason for x" seems to be totally OK.
I too find those trivia questions and their answers interesting but IMO they are a distraction.
SE aims to answer every relevant question, even basic questions. Karma is a reward for the users which do activities to generate the relevant Q&A entries.
As a member of both, no, they're really not. The information on SE is very useful but highly transient (and redundant, but at least we constantly try to rectify that). It'd be extremely more valuable if it could be condensed into a guide or textbook or what have you to teach idioms to beginners.
As for Wikipedia, I'm almost convinced most information on it is erroneous, unverifiable, or useless in the forms it is presented. I've stopped making contributions and try not to ever reference it at all.
This gives no indication one way or the other of which direction causality points.
ie. I could just as accurately say: maybe their value as a reference has led to reductionism. We can't go back and do an experiment, so we'll never really know.
probably because both have the goals of being encyclopedic sources of knowledge, and successfully complete this goal following reductionist policies, and its a hell of a lot easier to follow a strict philosophy than a lenient one (because the latter invites heavy legal discussion by armchair lawyers, while the former minimizes it).
In short, likely because its efficient and apparently sufficient
I don't really agree with this, but I upvoted it because it's the best charitable explanation I've seen, even if I'm frustrated by what I perceive to be a missed opportunity.
Which part do you disagree with? That its easier to enforce, or that its sufficiently effective?
The former is intuitively true; finding good rules to enforce strictly is difficult, but enforcing is simple (is the act explicitly allowed/banned or not?). Find broad rules is trivial, but then you spend all the enforcement time arguing edge cases (see: US gov)
The latter is defended by the assumption that wikipedia/overflow do indeed follow such a policy, and its difficult to argue that they are not successful. There might be something better out there, but at least currently the market seems to have converged on this policy; ala "democracy is the worst form of government, except for the all other forms we've tried.
I disagree with the part about it achieving the goal of being an encyclopedic source of knowledge. Except it largely comes down to how you define "encyclopedic". I would like it to mean "broad and inclusive", but it often means "highly curated", which tends to translates to "conservative".
I don't think the convergence of the market on this sort of policy provides any indication one way or the other, because of network effects. Both of those sites won their space before the culture we're discussing here became entrenched, and because of network effects, it is not realistic to unseat them.
Im not sure you can easily argue either is "highly curated"; they both cover an extremely large domain and cover at least a substantial subset of it. It's hardly the case that any one/group handpicked particular subjects of interest, rather they denied anything not of interest (and accepted the rest)
I imagine rather than "broad and inclusive", it should be "broad and authoritative", which then tends towards conservative when contraversial (which makes sense: how can you be authoritative when a general agreement can't be reached?)
And ofc, wikipedia was preceded by c2wiki, and stackoverflow by innumerable Q&A sites, both predecessors being substantially more liberal in what they accepted. And both beat out their predecessors, presumably largely because quality control was made much more difficult, and often absent, in the face of liberal acceptance.
> And both beat out their predecessors, presumably largely because quality control was made much more difficult, and often absent, in the face of liberal acceptance.
That doesn't mesh with my experience of what made the sites popular at the time of their growth. Wikipedia was the only wiki that anyone had ever heard of, and SO was the only software focused Q&A site besides expertsexchange which had a freemium model that made it unusable (not to mention a funny domain name!).
"Are there systemic forces that point this direction,"
When you can "contribute" to the community simply by clicking a delete button, rather than actually adding new and useful information, you're bound to get this sort of situation.
StackOverflow mods are appointed to police content. The more content they remove, the more it looks like they're doing their jobs.
It's the same with Wikipedia. Deleting someone else's hard work counts as a "contribution". It shows up in your profile just like writing a new multi-page article would.
When the same credit is awarded for for destroying as for creating, the destroyers are always going to win.
I recognized a few of the names who deleted the post to be fine mathematicians who have contributed quite a bit to mathoverflow, so I don't think that applies. Mathoverflow is a Q&A site for professional mathematicians, you may have it confused with math.se.
StackOverflow just need to lighten up. There's been times where it looks like the Moderator isn't even reading the responses, or shut it down because he/she personally didn't find it interesting? I sometimes wonder if some the Moderators have a complete grasp of the English language?
And the rewrites at Wikipedia, sometimes it seem like there's a financial motive to the rewrite. Years ago I bought a Domain name. It was the all to familiar neologism. Someone tried to buy it. I said no. The very next day the singular of a made up word was registered and his website was online. And Wikipedia was rewritten to "see the singular." that day too.
MathOverflow is not StackOverflow. For that matter, StackExchange mods in general are not 'appointed'. A very few are employees of the StackExchange company, but the vast majority are just normal people that have contributed a lot.
MathOverflow is intended for professional mathematicians. It isn't a general purpose discussion forum for mathematics. That such a thing doesn't really exist in a satisfactory form can't really be helped.
> StackOverflow mods are appointed to police content. The more content they remove, the more it looks like they're doing their jobs.
I can safely say you're wrong in this regard. Content removal is a big fraction of what the mods do because the contributors filter editing, closing, pointing out duplicates, migrating, etc. so that mods have to deal with only important stuff.
In fairness, it was closed after being open for more than two and a half years. Although the point remains that the "no longer relevant" message is needlessly grating.
Most of the SEs, in my experience, dislike questions, like this one, that boil down to subjective opinion. This makes them hard to answer, as answers no longer depend on fact or statistics, but someone's opinion of something. Frankly — while I feel for the questioner, and agree w/ him — this one seems very close to that edge.
It isn't immediately clear to me that this question is relevant like you claim; the FAQ isn't really clear on it either:
> The site works best for well-defined questions: math questions that actually have a specific answer. You'll notice that there is the occasional question making a list of something, asking about the workings of the mathematical community, or something else which isn't really a math question. Such questions can be helpful to the community, but it is extremely tricky to ask them in a way that produces a useful response.
On MathOverflow? Every StackExchange site has completely different moderators, who are all just users that have used the site a great deal, contributed great answers and great questions.
I asked a question on an electronics stack, but it was shut down by mods because they apparently couldn't answer the question, and decided it wasn't clear/valid. Clear case of mod ego.
Having participated frequently in SO and others, I understand why it is necessary. But, I feel that so much is wasted with such a qualified audience. There is so much potential to be explored there.
I don't have a solution either. I just wish I had.
Agree this is a problem; there are a huge number of worthwhile questions that get closed on stack exchange. This is because they prioritize the reputation of the community over answering all worthwhile questions (and to be fair maybe they wouldn't still be with us if they hadn't). Still most of the mods would do well to remind themselves of the official advice
I've historically had the opposite problem. I find attempts in both science and mathematics textbooks to provide some sort of real world context to be distracting and wasteful. I remember that impression going all the way back to high school, but the most recent memory comes from early in college. My discrete math book had some chapter that started with a page and a half about volcanoes.
Volcanoes.
Why are you wasting my precious time with this nonsense before getting to the meat? There are other things I need to read and study. There are problems I need to use these ideas to solve... and not only the ones in the textbook.
But then, I can also recognize that such winding introductions to a subject or an idea might be helpful to people who "get bored" as the author describes. Though I suspect the remedy might be to acquire and deliberately practice study skills. It would be nice if publishers sold versions of their textbooks both with and without what the individual posing the original question would identify as "motivation." I tend to think of it as fatty prose.
Half-assed physical analogs can be a drag, especially when the analogy breaks down after any bit of scrutiny. Geometric intuition, on the other hand, I often find worth the weight of a thousand words. There have been times I haven't been able to make my way through a dense piece of mathematical text, only to pop open another book on the same topic to find a single picture that makes all of the pieces click.
The conversation is about very advanced math topics (advanced undergraduate or graduate level). At some point in the levels of abstraction it becomes hard to show concrete applications of the ideas, or rather the applications of math is to do ever more advanced math.
I find more deplorable the fact that even basic math topics are often covered in the same dry way, without discussing practical applications or introducing topics through real-world scenarios. Many math books take the attitude "You have to know this, because I say so." Whenever I write about math, I try to start with a concrete example or a useful application of the theoretical result—it's always possible to come up with something for most of first year stuff. Seriously, you'd be surprised how much better reading UX is if you start each chapter/section with a motivating example.
I find that almost all books I read do discuss "applications". No mathematics is an island. But as you say, the problem is that a lot of those applications don't really count, because the domain of the application is as foreign as the original concept.
For example, knowing that the snake lemma, a purely categorical statement, is most useful in homological algebra (such as, for example, simplicial or singular homology), is utterly useless if you don't already have an interest in algebraic topology. There really is almost no other motivation for the snake lemma, so now we're faced with the problem of trying to convince you that it's interesting by trying to convince you that algebraic topology is interesting. It can be done, and maybe we'll eventually bottom out in something like financial statements or bridge-building, or another topic that is widely recognised as "useful" and very far-removed from the snake lemma. Either way, it will be a long and arduous path, and I hardly think mathematicians can be blamed for this or be dismissed as elitists for the inherent difficulties of the subject.
But even for "first year stuff", the applications are kind of pointless. Do you really want to learn calculus because of physics? That's the most obvious and most historical application, but calculus is so foundational that you might as well motivate addition of real numbers by saying that numbers are added in physics too. More likely for the HN crowd, you want to learn calculus because you want to know how a neural network's backprop algorithm works, but how is the first year teacher going to anticipate that this is your particular interest in calculus?
At some point, I think there has to be a little "trust me, this is useful" and you just struggle through the subject until you can see on your own, after the fact, what the struggle was about. First wax on, wax off, Daniel-san. Then you will learn how you really were learning how to block karate blows.
I had a math teacher that explained that beyond simple and clear explanations, the other way to learn math was to do example questions. Lots of examples, to see all the patterns and subleties. The subject he'd most often quote was permutations and combinations - either you got it pretty much instantly or it would just be a struggle that you could only improve at so much by doing a lot of practice questions.
The best math class I ever took was probability, where the professor worked history of the field and famous historical examples into his lectures. It helped that probability has a colorful past, but the right motivation really brings things to life.
Also, for self-teaching, I've found some math books that are setup to be in the form "Topic A with Applications to Topic B." If I care about B, such a book will typically does a good job motivating A, even if it isn't the purest introduction to the area. I can always read a book that is a more canonical intro to A later.
When I was teaching calculus, I made a fundamental decision of how to define limits:
I used limits of sequences.
In our university, we encounter limits in either a handwavy way in Calc I or an epsilon-delta way in Advanced Calculus. Sequences and series are introduced in Calculus II.
Continuity is then described in the same terms. Or perhaps in terms of open sets if there is some topological topic.
I think this is pretty terrible for intuition. Not just that, but it's not even general enough - it requires normed spaces, so then you have to generalize again.
Much simpler to talk about ancient Greeks and Zeno's paradox. And then rigorously define limits of sequences, and define limits of real valued functions in terms of "for any sequence x_n that approaches x, y_n = f(x_n) approaches y". Simple and right away lets me show counterexamples beyond "left and right limits", like cos(1/x), and show students how to produce two sequences that converge to different numbers.
There are similar ah-ha moments when discussing fundamental concepts of linear algebra, number theory, complex numbers etc.
The two best books I have ever found on teaching Complex Numbers are:
1) Bak and Newman
2) Schaum's Outlines
They actually give you the understanding and ceeling behind WHY analytic functions are the way they are, and derive holomorphic functions from that. Imho a more terrible approach is that of Serge Lang and proving everything the other way, with Taylor power series.
Bottom line - make a directed graph of how you will teach your subject and then figure out the best entry point an direction for the greatest cohesion, as you would when telling a story.
If you are curious, now I teach a course on "Thinking Matematically" and here are this semester's results of that approach:
I feel when it comes to motivation in math, I just want to know why people got so excited about a particular theorem in the first place. I'm ok with the answer "before this theorem we assumed all these different things we proved in similar ways were different. Now we know about their commonality and it allows us to borrow mathematical machinery and use it here".
For example, take something like measure theory. A reasonable motivation for measure theory to me is remembering your introductory probability class how you had to learn a probability mass function for discrete spaces and a probability density function for continuous spaces. Now these two ideas are obviously nearly the same idea. I mean the notation used is a pretty big hint. But you need measure theory to have the right concepts to describe how they are the same.
Where are all those kinds of motivation for things like topology and cohomology?
I asked the ##math chat room on Freenode about cohomology recently, more precisely I expressed that it was for me the scariest math-word-that-I-don't-know-yet, and I received the explanation "it isn't that scary once you get used to it; it's just a way to repair exactness of sequences." Maybe that helps?
For topology, I feel like there's a sort of meeting-of-two-different-things; one starts with being very frustrated with the delta-epsilon-definition of "limit" and its one-dimensional nature, and the limit-definition of "continuous" and its clumsiness. The other starts from wanting to play with spheres and Möbius strips and knots and the like. When you're playing with these shapes a bijective mapping between two surfaces is not a fine-grained-enough idea because it is not continuous; adding continuity gives "homeomorphisms" which also aren't a fine-grained-enough idea because they do not make reference to the space an object is embedded in; wrap a torus about itself in a pretzel knot and you have something which is homeomorphic to a torus but in 3D you can't get there without tearing part of the surface through the other one, but in 4D you can. So finally we come to the idea of an isotopy, which bumps "continuous" to the next level by saying "Just like you can have a continuous path of points in space, you can have a continuous path of homeomorphisms from one to another," and that's where the pretzel knot becomes finally distinct from the torus in 3D, there is no continuous path from the homeomorphism of the pretzel knot to the torus, to the identity homeomorphism of the torus to itself. Or something like that. So this path is then an "isotopy" and then certain things are nicely isotopy-invariant and so forth.
The topology explanation is about what I had in mind for motivation. I think to appreciate cohomology, I need appreciate what problems in homology it makes easy to solve. To appreciate that I need have the vocabulary of algebraic topology.
One way to maybe then to describe homology is it gives you way to take about shapes and surfaces in the language of algebra.
I like quanta for general purpose math articles. they typically take a fairly recent paper and explain it qualitatively but still pretty accurately. almost like war stories.
> And of course it is historically backwards; groups arose as people tried to solve problems they were independently interested in.
Reminds me of the joke that philosophy students university learn the work of people who didn't go to university.
Would a text written by the discoverer of a field be more interesting?
I've found important papers easier to read than textbooks covering them... OTOH the original paper can be too close to their own motivation, and not treat - or yet be aware - of its general significance.
A motivating problem doesn't have to be practical or relatable... it just must lack obvious solution. A puzzle.
Good example of something (although it's perhaps a bit trivial) that is a puzzle that seems to make even non-mathematicians curious is the 'justification' for greco-latin squares.
The idea is that you have 6 files of 6 men. Each is of a different regiment, each is of a different rank. Is it possible to arrange them so that no row and no file have two men of the same rank or from the same regiment?
You can mention this in your first lecture on algebraic geometry (or, for that matter, combinatorial geometry i.e. matroid theory) and then come back to it when you talk about projective geometries.
If a dry presentation of groups doesn't excite you, then there's plenty of books that are full of examples and motivations.
Personally, I prefer the dry stuff (e.g., Herstein) because that's the abstract-est abstraction. Rotations and matrices are groups? Don't care. I just wanna see the strange hidden properties of abstract structures reveal themselves.
I think this, as much of higher mathematics, will change soon.
Today's technology in natural language processing is reaching the point where it will be possible to marry a natural language processing system to an automated theorem prover and have it generate formally verified proofs from math prose proofs.
Once this technology can readily process the textbooks making up the PhD curriculum, I think there will be a culture shift. Quickly there will be a new standard that math results should be formally verified. The hallmark of math, after all, is that it can be proven correct!
But with an increased role for computers will come an increased appreciation for the things that only humans can provide. Motivation and explication will be more valued when the technical aspects of theorem-proving are automated.
If what you are saying is indeed going to happen, then the "problem" will become even worse. The formally verified proofs tend to be much more unintelligible than the human generated ones, and when they stop being so, the humans will become deprecated in general.
Practically speaking though, working mathematicians do not care that much about formally verified proofs. Working mathematicians are more interested in insight and understanding, and not necessarily in being completely sure of every detail. Formally and automatically verified proofs are much better suited for programming, as the automatic verification of the correctness of the program is after all _the_ best regression test.
So, while interesting in principle, I doubt formal methods will change much in how we do mathematics. Hopefully they'll change how we do software engineering though.
Perhaps it would mean that theorems are sometimes known to be true or false before anyone understands why. Digging through automatically generated proofs looking for interesting insights seems like a rather different experience than groping in the dark, not knowing whether a proof is possible?
Yeah and wait a second, isn't that the problem people are talking about in this thread? Aren't people basically complaining that things are discussed and proven without being justified?
I don't really see how some fantastical process of formally verifying all of mathematics would have any relevance to teaching, anyway.
If you have a statement that humans doesn't know how to prove, finding a proof via automatically generated proof is kinda like trying to decrypt RSA by factoring the key. In both cases, you're looking for a specific key in the search space is extremely large. You can put some insights into the tool that search for the solution, from simpler ones (e.g. using fast multiplication algorithm to verify the candidate quicker), or more sophisticated ones (e.g. using generalized number field sieve instead of trial division), but in the end, they don't help you much in practice -- the search space is still too large to expect to find a key in the lifetime.
It's an interesting analogy but I think it proves too much. You could argue that computers will never win at Chess or Go because the search space is too large, and look what happened.
Although it's not proven, we have fairly good reason to believe there are no sufficiently efficient shortcuts for factoring large prime numbers, while there are shortcuts for proving many difficult mathematical theorems. After all, humans can do one but not the other.
You make a good point. I agree that the analogy is not perfect, and that if you assume that breaking RSA is computationally hard for some intrinsic reasons[1], then the theorem proving is more like chess and go, rather than RSA. However, theorem proving is still much more difficult than chess and go, if you consider time spent on a theorem vs. on a single game, and on the number of good mathematicians vs. the number of good chess/go players. I think we'll have human-level theorem proving solved by machines at some point in future, though not very soon. Either way, humans will be well deprecated by then.
[1] - Practically speaking though, the biggest reason we believe that factoring is hard is that we haven't really figured out how to do this, so our belief that it's hard is really build upon our feeling on hardness of theorem proving. :) I think we have more intrinsic reasons to believe than P != NP than that factoring is hard.
I don't think so. I'm not really competent to say if the NLP part will be solved soon, but (fully) automated theorem proving is very hard, and I don't see deep learning applying to it very well.
What could happen, though (hopefully), is that mathematicians start using more and more the various proof assistants (Coq, Isabelle/HOL, lean, …). These allow to write definitions, statements, theorems in a more programming-like way, with structured proofs; to develop modular libraries; and even in some cases to use automated provers to wave away details of the proofs. The problem of adoption of such provers, however, is a culture problem, not really a technology problem: as said elsewhere, mathematicians do not seem to care too much about formal correctness. It's only if automated tools help gaining insights (automatic search for (counter)examples, for example; function plotting; builtin algebra solving system… ?) that they will convince more people to use them. Modularity might also be a selling point, but not sufficient to overcome the steep learning curve of these tools.
I have some hopes in [lean](https://github.com/leanprover/lean), a new proof assistant inspired from Coq, which puts the focus on good UI and automation.
I don't really think that makes any sense. Why would textbooks, which are already written informally and aren't intended to be computer-verified, be written less formally for the purpose of being fed into a computer?
Formally-verified proofs are a great idea, it's my main area of interest in mathematics, but they have essentially zero relevance to a textbook. Textbooks don't need to be formally verified.
Mathematics traditionally has a macho ethos. This goes all the way back to Euclid, and his "Let no one ignorant of geometry enter" sign. The training process primarily involves solving puzzle-like problems. Cambridge University still has Wranglers and Senior Wranglers [1], chosen for success at solving puzzle-like problems, not originality. (Hardy once wrote that this set English mathematics back a hundred years.)
This history infects mathematics books.
The other traditional problem with mathematics is terse and obscure notation. There are many implicit assumptions embedded in published mathematical papers and books. This is not helpful. It's like reading code snippets without the declarations.
The other traditional problem with mathematics is terse and obscure notation.
It's only a problem if you are unfamiliar with the subject area, and if you are, you have very little chance to understand the paper or book anyway. Mathematics is just difficult, way more difficult than most other stuff people might be doing.
Here's an example: consider this classical paper[1], "Vector Bundles Over An Elliptic Curve" by M. Atiyah. The author is a well known and regarded mathematician, and the paper itself has around 1000 citations. Sections 1. and 2., "Generalities" and "Theorems A and B", just recall the basic notions and theorems that are absolutely necessary to have any grasp whatsoever as to what is going on in the paper.
If you know nothing about algebraic geometry, the classical way to learn these is Robin Hartshorne's famous textbook "Algebraic Geometry". It is famous for being both good, self-contained to a large degree, and having very well chosen exercises, but also for being quite terse and often difficult to follow. Here's an Amazon link[2]. You can look at the table of contents. To really fully understand these two sections from Atiyah's paper, you need to have very good understanding of Chapter 2. "Schemes", and at least first 5-6 sections of Chapter 3. "Cohomology". This is 200 pages of pretty terse mathematics. At a fast understanding pace of 4 pages a day, it will take you two months to even have some basic toolset to understand the Atiyah's paper.
But, if you try to understand the Harthshorne's textbook, you'll quickly find out that it also has some prerequisites of its own. Also, the "4 pages a day" pace is only possible if you've already spent 2-3 years learning how to learn mathematics.
I encourage anyone to try to understand even first 2-3 sentences of the "Generalities" section. Google and Wikipedia the unfamiliar terms, you can also try to look it up in Harthorne's textbook, or Vakil's lecture notes, or any other source. The notions used in these first 2-3 sentences are basic to anybody working in the field, and yet one needs to spend hours to fully understand these when starting from scratch.
Compare this to other famous paper from other field, "The Market for Lemons" by G. Akerlof[3]. This is also a very famous paper by a well regarded economist, who received a Nobel Memorial Prize In Economic Sciences for it. It is a much easier read, precisely because the economic sciences do not operate on nearly the same level of complexity as mathematics. Once you know what are some common sense notions like, supply, demand, utility etc., and some very basic calculus, you can easily follow the argument without too much training.
My point here is that it's not that it's difficult to read mathematics just because it uses terse and obscure notation. It all is just genuinely difficult and complex, and it is impossible to invent better notation that will transfer days and weeks worth of understanding straight to the reader's brain. I would love it to be the case, but then it would cease to be as fun and rewarding to really understand.
No, mathematics is [naturally] hard to understand and this is why it's fun to understand: it's very challenging and immensely gratifying when you get it.
No, it is fun and rewarding because it is genuinely hard to understand. Once you really understand it, it tends to become more obvious in hindsight, but good luck getting your understanding across to someone else who hasn't spent as much time thinking about this as you.
Economics would also be really hard to understand if it didn't operate on the real world: something very familiar to us.
I can imagine a mathematical version of economics operating on some abstract constructs designed to emulate an economic system. It would reuse no terminology from the real world for the sake of producing an abstract notion, completely (or at least artificially) decoupled from the system it was designed to emulate.
I would imagine that Akerlofs paper, encoded into this form, would be at least as hard and involved to understand as the one from Atiyah.
As long as the as the "abstract constructs" themselves weren't much more complicated than the current real-world economic concepts, it would only be slightly more difficult. You would need some background reading, but not any more than skimming relevant Wikipedia articles for definitions. And in that case, you'd have every reason to be pissed at economists who don't present their findings in much simpler and obvious way.
If, however, the abstract constructs were much more complicated then the current economic concepts, and if you were trying to solve problems on much higher level of abstraction than economists currently are, then it would just be mathematics, and indeed it would be more difficult.
I, personally, cannot imagine how you could rewrite Akerlof's paper to be as hard to understand as Atiyah's. I can, with great difficulty, follow Atiyah's paper only because I spent literally _years_ learning the necessary background material. I am completely unable to relay my understanding to someone who hasn't spent years doing the same. I wish I was -- I'd revolutionize algebraic geometry then, just like Alexander Grothendieck revolutionized it around when the Atiyah's paper was written. On the other hand, if someone rewrote Akerlof's paper in an intentionally obscure way, you could easily rewrite it back in a clear way, once you spend the effort to understand the obscure version yourself.
> Here's an example: consider this classical paper[1], "Vector Bundles Over An Elliptic Curve" by M. Atiyah. The author is a well known and regarded mathematician, and the paper itself has around 1000 citations.
Sir Michael Atiyah is an Abel prize and Fields medal winner. Apart from that, he has an excellent sense of humor. I had a chance to grab lunch with him last year and it was quite entertaining to say the least.
Have you actually hung around with some mathematicians? I find them to be on the whole quite a humble bunch of people. My feeling is that any hubris is burnt up in all of the great suffering that it takes to do good mathematics.
I think the machos all migrated to physics in the first half of the 20th century. Those guys are full of "big talkers". Have you ever heard of a physicist refusing the Nobel prize? No, but it has happened (more than once i think) for the Fields medal.
Machos go to whatever gets them social status. They will move from math to physics when physics becomes important and then to programming when that one becomes cool. If art would be cool, machos would go back to writing poems.
It was Plato not Euclid who's said to have had that sign ("...no one ignorant of geometry..."). I agree that Euclid was a bad influence in this way, though we know so little about his time, maybe he'd have hated that part of his influence. We have a bit more variety from Archimedes: The Method, aka "here's how I really figured that stuff out". It was long lost until pretty recently, and so didn't get to influence anything post-Hellenistic.
Also agreed about the notation. I hope something better evolves out of code.
People always say that math students don't learn about applications of math.
This was never the case for me. When I learned trig ratios, I always understood some basic things that trig ratios could be used for. The teacher always introduced some applications, we always had a lot of word problems, and I could fill in the gaps myself.
Same for calculus. When I learned calculus, I always understood some things that calculus could be used for.
So I understood how those things could be applied to general, everyday sorts of problems. What was missing, though, was that I had nothing to which I could apply those techniques, besides homework.
Learning math (and reading STEM papers) has become easier for me since I now have actual problems to solve. Don't get me wrong: I'm not solving particularly challenging problems or using particularly advanced math. Nothing that tens of thousands of people haven't done before me. But I do need to understand the problems, solutions, and some of the context in order to successfully implement them. This provides a motivation that was always missing before.
I suspect this general narrative is true for a lot of people: that having an actual problem to solve is almost necessary to get a student to really learn the material, instead of just coasting along for a grade.
High school trigonometry and introductory differential & integral calculus are not the kind of books being described in this discussion.
The example in the original post is books about group theory (or the group theory sections of abstract algebra books more generally). I can attest that this subject is very rarely described in textbooks with clear examples shown before definitions and theorems; usually the presentation is entirely abstract, following a pure definition–theorem–proof kind of structure. But many other areas of pure mathematics at the undergraduate level and above are presented in a similar fashion.
(I recommend Nathan Carter’s book Visual Group Theory for a lovely counter-example to the prevailing trend, which starts with the concrete, and is very accessible. http://web.bentley.edu/empl/c/ncarter/vgt/)
We used to "run through" books like that. Their reasoning was to prove a theorem you only need the definitions/axioms. They really wanted us to be able to grasp the truths of a logical system from just its theorems and definitions. It was horrifically difficult. (Not all professors taught that way there.)
I feel that a lot of blame lies at modern academias curriculums. They feel every student needs to graduate in X number of years with a pretty long list of courses. It leaves little time for students who need or want more time with topics.
A lot of trig and geometry never really clicked for me until I had to use them in shop class. For instance, planning out the dimensions and cuts that you need to make in a rafter to get the desired roof pitch for a shed of certain dimensions. Or laying out the stringers for a set of stairs.
- e; The story of a number (Eli Moar)
- An imaginary tale; The story of -1 (Paul Nahin)
- The Poincare Conjecture (Daniel O'Shea)
- The man who loved only numbers (Paul Hoffman)
- Prime Obsession (John Derbyshire)
They're part biography, part history, and give a little colour to the subject that isn't available from your typical college textbook.
Its because math is used as test material. Thats all it is in american educations and many others i might add. Its a fucking crime. All of science and almost everying else is treated similarly. They build tests into the material you are to learn. At the end of the day you get a monstrosity. I say that if we are going to try to hide iq tests inside the material then we should have no qualms about simply giving everyone iq tests separately from learning material. Yes there are flaws with that proposal but it would still be better than what we have now where all of the material is molested. Imagine if people actually got a thorough education. No more grtting to the job only to learn that they didnt teach you what you need to know. In computer science this problem is fucking atrocious.
For an answer, there have been various influences:
(1) Whatever math was before 1900 or so, by the time of the Russell paradox and its fix with axiomatic set theory, the style of the fix was to be close to Russell-Whitehead (if I have that right) idea that proofs could be checked essentially mechanically by just symbol substitution and manipulation. E.g., in those days there was a book on the natural numbers, that is, 1, 2, 3, ... that apologized for numbering the pages before the natural numbers had been carefully defined!
There was even a name given to this style of writing, telegraph style.
(2) Of course, the books written on axiomatic set theory easily fell into the telegraph style. Even there the writers were getting into trouble: They gave names for the various axioms; they didn't explain why the names they gave were appropriate, and I never could discover why. But I was eager to get out of the sub-sub-basement of axiomatic set theory ASAP so just did a f'get about it.
(3) When books were written on abstract algebra, e.g., basic set theory, construction of the main number systems -- the naturals, integers, rationals, reals, complex -- and then went on to the main algebraic systems defined with axioms -- groups, rings, fields, vector spaces -- it was easy to stay with the telegraph style. E.g., it was tough to find a book on abstract algebra that also discussed group representations and its applications to quantum mechanics and molecular spectroscopy.
(4) Long calculus was often done with a lot of intuition and nothing like some carefully done definitions, theorems, and proofs as in, say, W. Rudin, Principles of Mathematical Analysis. And physics and engineering kept drawing diagrams with "little interval dx" etc. So, when abstract algebra was proving theorems, the calculus authors also wanted to be careful at least about delta-epsilon arguments. Then including a lot of physics, mechanical engineering, touching on the heat equation or fluid flow, was considered off-topic. Bummer.
(5) The series of astoundingly carefully written books, close to telegraph style, by the team Bourbaki was influential.
(6) During the Cold War and the Space Race, US math was awash in grant money and essentially turned its back on the physical science motivations and applications. Some of the funding people started to get angry about that, and we got the Tom Lehrer song and joke about abstract math being about "the analytic algebraic topology of locally Euclidean metrization of infinitely differentiable Riemannian manifolds" or some such.
But, sure, especially in analysis, for a good proof, there's often a good picture and if can see the picture then can construct the proof easily. E.g., for positive integer n, the set of real numbers R, and convex f: R^n --> R, that f is continuous has a really cute picture. Same for Jensen's inequality. In linear algebra, the polar decomposition says that each square matrix is just (A) a rigid motion, rotation and/or reflections followed by (B) moving a sphere into an ellipsoid by stretching and/or shrinking on mutually orthogonal axes. One or more of the axes goes to zero if and only if the matrix is singular. Etc. Nice picture.
Currently, then, there is an opportunity for math authors to include motivations for their subject, definitions, and theorems, intuitive descriptions and helpful pictures, applications, issues, open questions, etc. Uh, when reading a proof, for each of the assumptions, check off where it was used in the proof! The definitions, theorems, and proofs can still be fully precise and solid.
Some students of, say, analysis have long tried to find and draw pictures that would clarify what was going on in some definitions, theorems, and proofs. I would advise new students to do that also.
> E.g., in those days there was a book on the natural numbers, that is, 1, 2, 3, ... that apologized for numbering the pages before the natural numbers had been carefully defined!
This was almost certainly a joke, although it must be said that it was probably more of a parody than a haha-joke.
The reality is that mathematics is written in a rigorous style because rigour is part of mathematics. You can waffle on for as long as you like about the intuition and the applications and whatever else, but you need to eventually get to the rigour.
The physicists and engineers might get away with 'drawing diagrams with little interval dx', but mathematics simply cannot.
Physics and mechanical engineering are not mathematics. Maybe don't try to learn mathematics if you want to actually learn physics?
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[ 3.1 ms ] story [ 170 ms ] thread> When I was a graduate student, we had a wonderful working seminar on Sunday mornings with bagels and cream cheese, where I learned a lot about differential geometry and Lie groups with my classmates.
At the end, the students needed context and enlightment. We cannot underestimate other dimensions of the learning experience.
Great math teachers understand the need to teach intuition. He wasn't a math teacher, but I think Richard Feynman is the pinnacle of this. See [1] to see how he expresses intuition about physics, and his Red Books[2] for how he teaches mathematical physics with all the qualities I believe makes a great maths text for students.
Also, there's a linear algebra MOOC which also teaches great intuition before delving into proofs and heavy detail [3]. I mention these examples because they are exemplars of this idea of teaching intuition.
[1] https://www.youtube.com/watch?v=4zZbX_9ru9U
[2] https://www.amazon.com/Feynman-Lectures-Physics-Vol-Mechanic...
[3]https://www.edx.org/course/linear-algebra-foundations-fronti...
There really needs to be a version of the Feynman lectures for mathematics.
Although, this is what Arnol'd has to say [1]:
"Mathematics is a part of physics. Physics is an experimental science, a part of natural science. Mathematics is the part of physics where experiments are cheap... In the middle of the twentieth century it was attempted to divide physics and mathematics. The consequences turned out to be catastrophic. Whole generations of mathematicians grew up without knowing half of their science and, of course, in total ignorance of any other sciences."
[1] https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html
I asked the question on math.SE:
https://math.stackexchange.com/questions/62190/mathematical-...
From there he was given a calculus book, the title of which I cannot remember. I never got that far.
I suspect you have to at least follow the same path to have the same intuition.
I've taken the liberty of taking a quick snap of a random page in "Arithmetic for the Practical Man" to include below for those poor people poisoned by modern textbooks:
http://i.imgur.com/Bg9OiiK.jpg (926KiB)
I see horrible modern behemoths of over a 1000 pages that leave you dazed, confused and full of facts but nowhere to go with them. EE textbooks are even worse on this front than your average mathematics text book. I've seen one proudly promoting over 1500 pages and 1000 illustrations, but doesn't even get as far as an opamp or discuss anything at system level.
From the preface: "This book is an introductory text. The only prerequisite is a robust understand- ing of the results from single-variable calculus. The theorems of linear algebra are not needed, but the exposure to abstract arguments and proof writing that usually comes with this course would be a valuable asset. Complex numbers are never used.
The proofs in Understanding Analysis are written with the beginning student firmly in mind. Brevity and other stylistic concerns are postponed in favor of including a significant level of detail. Most proofs come with a generous amount of discussion about the context of the argument. What should the proof entail? Which definitions are relevant? What is the overall strategy? Whenever there is a choice, efficiency is traded for an opportunity to reinforce some previously learned technique. Especially familiar or predictable arguments are often deferred to the exercises.
The search for recurring ideas exists at the proof-writing level and also on the larger expository level. I have tried to give the course a narrative tone by picking up on the unifying themes of approximation and the transition from the finite to the infinite. Often when we ask a question in analysis the answer is “sometimes.” Can the order of a double summation be exchanged? Is term-by- term differentiation of an infinite series allowed? By focusing on this recurring pattern, each successive topic builds on the intuition of the previous one. The questions seem more natural, and a coherent story emerges from what might otherwise appear as a long list of theorems and proofs."
In 1928, aged 16, Turing encountered Albert Einstein's work; not only did he grasp it, but it is possible that he managed to deduce Einstein's questioning of Newton's laws of motion from a text in which this was never made explicit.
Then there was von Neumann and several others. If not interested then at least well educated in physics.
Especially in the field of machine learning we're finding more hard evidence that metaphor are not decoration, but fundamental parts of how to transfer information. Using rich metaphors to pass on implicit information between teacher and student is known as "privileged information":
> When Vladimir Vapnik teaches his computers to recognize handwriting, he [he harnesses] the power of “privileged information.” Passed from student to teacher, parent to child, or colleague to colleague, privileged information encodes knowledge derived from experience. That is what Vapnik was after when he asked Natalia Pavlovich, a professor of Russian poetry, to write poems describing the numbers 5 and 8, for consumption by his learning algorithms. (...) [After coming up with a simple way to "quantify" the poetry], Vapnik’s computer was able to recognize handwritten numbers with far less training than is conventionally required. A learning process that might have required 100,000 samples might now require only 300. The speedup was also independent of the style of the poetry used.
http://nautil.us/issue/6/secret-codes/teaching-me-softly
Now, of course, knowing how to come up with a good metaphor is a skill in itself, and bad metaphors do lead people astray. But they do so precisely because they are so good at transferring information - wrong information, in the case of bad metaphors.
At some point, you graduate from "being taught" to "teaching yourself." By the time you get PhD, you need to teach yourself, because you're studying things no one has ever studied before.
It's kind of sobering to realize that even the pro's have this problem.
This might explain the relative lack of attention towards the user experience of non-mathematicians?
Anyway, yes. Every mathematician I know acknowledges that frustration is the natural state of affairs. If you're not frustrated that's because you haven't been doing enough mathematics yet. There's always a bigger problem, a new concept to master, a new way to look at an old idea.
Good games try to provide an optimal learning experience, providing just enough challenge to be interesting without players getting stuck. Play-testing is vital; if your players commonly get stuck in ways you didn't intend, it's a bug.
There's a lot to be said for designing a learning experience to flow smoothly. We can admire the work that goes into making that happen. (It then seems strange that, by contrast, writers of math books often don't seem to be playing the same game.)
One thing a well-designed experience doesn't do, though, is prepare you for being stuck and overcoming difficulty when you're not on an artificially smoothed path.
Oh, but they are playing that game. It's just a very difficult game. Everyone wants easy math books and lots of people are trying to write easy math books (for example, my buddy Ivan and his No Bullshit guides: https://minireference.com/ ). It's just a very difficult game, and very few have come close to success. When they do succeed, it's usually only for one kind of audience and not another. For example, Spivak's Calculus is widely admired in the mathematical community for its presentation, but I wouldn't be surprised if HN derided it for being stuffy, too mathematical, elitist, and full of itself.
That's actually a remedial-level topic for MO, which focuses on research mathematics. It's meant to be an easy example that everyone can relate to in this discussion, like talking about how children learn how to add. You'll see they say, "simple example". I'm not trying to be elitist, just trying to explain MO.
As an aside, I found the conclusion of this simple example about how normal subgroups are about being kernels of homomorphisms hard to relate to, how would you know that without even knowing what a homomorphism is, which needs knowing what a group is? These a-ha moments that come after learning the material and make the learner assume that the teacher is an absolute idiot by not starting from the a-ha moment are very frustrating. It's kind of impossible to start from a-ha without the learner first bumping into all the dark corners and hard work that light the way to the a-ha. I myself have had topics ranging from (simplicial, singular) homology to the completeness of the reals and elliptic curves explained in so many ways and nothing ever made sense to me until I sat down and struggled through all the explanations offered to me. I don't think there's a way to convey insights to a mind that hasn't struggled towards those insights.
Here I am going to quote Michael D Alder, from the introduction to an old edition of some lecture notes of his that I can no longer find online:
Edit: I just serendipitously found a passage of his in a very similar vein. Read section 5.1, "Cultural Anthropology": ftp://www.biophysics.uwa.edu.au/pub/Mathematics/Alder/DiffGeom.pdfReference and theory are very useful, but context is how they are tied together in implementation.
Mathematics got a lot less interesting after I realized it amounted to a giant, informally-specified, mostly undocumented body of code designed to run on the human brain... from that perspective it's hard to see why one should prefer mathematics to a well-written software program that does the same thing.
So it's legacy code that runs on the oldest hardware that we have.. Plenty of people are interested in updating this code so that it runs on more modern platforms. Actually, this is my favorite way to drill into a piece of mathematics: port it to python.
Fortunately, nowadays, there is so much learning material that you have the opportunity to choose the one that suits you.
For example, I know that I cannot apply a theory upon reality. I have to start with examples and build my own vision by abstracting from a lot of examples. So I choose my learning material accordingly.
My current interest, at the moment, is monad. Believe me, there is a lot of abstract articles about that. But my entry point was a tutorial about implementing some stuff in Javascript, that happened to be functors, monads, etc.
Believe it or not, but I used that knowledge the week after in one of my Java projects. And my colleagues considered it was "a nice trick" (which is true!).
In one word, the learning process must be fluid. Don't try to force anything. If the book medium does not suit you, don't blame the book. Or yourself. Try to find another resource and keep learning.
This problem goes way beyond math and programming and into physics and general engineering. There are a surprising number of engineers who can manipulate the linear algebra tools but can't actually engineer structures or figure out linear circuits. EE filter design is another good example.
If you look at higher ed vocational training as kind of a prep-school or maybe a qualification filter for an apprenticeship it makes more sense. Once you figure out linear algebra you're qualified to apprentice to someone to teach you how to actually use it.
There is a huge amount of information encoded in the choice of exactly how to define thing, and which theorems people care about. This reflects a long process of trial-and-error as the field was constructed. For a famous philosophical treatise on this using the Euler characteristic as an example, see "Proofs and Refutations” by Imre Lakatos
https://en.wikipedia.org/wiki/Proofs_and_Refutations
Part 1: https://math.berkeley.edu/~kpmann/Lakatos.pdf
Most of that foundational information is lost when it's not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject.
It's a huge problem, it applies to physics textbooks too, and it doesn't have to be this way. Unfortunately, the problem has been known for decades and there's not much reason to expect things to change. (Lakatos wrote the above in 1976.)
A way to think about this process: "math" isn't quite a tower of abstract concepts; those only exist in any given mathematician's head, and die with them. Instead, "math" is a name for the novel pieces of language we invent, with the purpose of using them to talk about and (hopefully) share those internal abstract concepts, exposing to the rest of the world concepts that were entirely inaccessible (to everyone but the originator) before that language was introduced. Math is the attempt to communicate never-before-communicated epiphanies; to describe the shapes of never-before-pondered abstractions, with properties only describable using never-before-spoken words.
One thing that this implies, is that the first piece of language that lets us even vaguely point at some particular idea so that we can get to work on analyzing it further, might stick around long after we come up with some clearer or more coherent language, because the former one now has the momentum of historical use behind it.
Math, when done this way, then becomes a precarious pile of "just good enough to survive" conceptualizations, rather than a precise tower of "best-tool-for-the-job" conceptualizations. And that's no good for teaching.
Math is where we invent language to refer to new forms of abstraction themselves: novel possible shapes for our thoughts to take when we think about other things. You can't talk about a new shape by analogy to existing shapes. Nor can you abstract an abstraction in a way that gives you anything more familiar. (Instead, you'll usually get more novel, ontologically-primitive abstractions, like going from monads to arrows, or going from numbers to fields to rings.)
Sometimes disciplines like physics will find a concept that we don't have any mental tools in our toolkit for yet. Then we build some. But we still refer to the process of doing that as doing mathematics—and then we apply that new mathematics back in the problem domain to talk about the new concept.
(For a good example, the formalization of quantum theory in physics, required the creation of infinite dimensional analysis in mathematics. Physicists did most of that work, but the work itself was still mathematics, not physics.)
---
Now, other fields do still have a similar problem to mathematics, of historical momentum carrying forward old "things" (again: concepts, relationships, laws) when there are better, clearer "things" that could be used in their place. But when we're not working with pure abstractions—ways of thinking—we can make the effort to compare and contrast old and new "things", and decide that some might be more edifying than others.
It is possible for an especially-gifted physics teacher to write a very accessible physics textbook, because they need only pick all the clearest "things" to demonstrate. That teacher will still be stuck in a given paradigm—a way of thinking, a belief in the worth of some "things" over others, popular in the culture of their discipline at the time and place they worked. But they might be able to (barely) rise above it, if they think hard about the history of their discipline and the paradigms it has gone through in the past, compare-and-contrast those, and synthesize something that isn't quite just the paradigm they're immersed in.
Mathematics is uniquely problematic because it is entirely paradigm. It is a tree of paradigms—each new abstraction only making sense assuming the paradigm it was created in, and then becoming the paradigm for further abstractions still. Every mathematician, all the time, is trying to discover what a particular paradigm—their specialization—can be twisted to accomplish. Not one of us has the brain power to know the total space of things that one of these ways of thinking can be used to express—the problem domains the tool is applicable to—in order to know which tools show more or less promise at being "powerful." We know what we've discovered so far, but we have such an infinitesimal idea of the "space of all possible abstractions" that we could be totally missing some of the best, and using ones that are barely satisfactory.
Point a hypercomputer AI at "solving physics", and it'll spit out a description of the universe that will certainly have more "things" in it than we know about today—but which will still also contain a subset of the "things" we do have. (The most "carving nature at its joints"ing ones, presumably.) Those "things" that get carried over will, of course, be defined much more precisely; their concept-boundaries will be adjusted to includ...
I'm keen for etymologies and hence believe that math means the art of problemsolving or learning successfully. That includes language as a problem domain, and teaching tool for learning, but as modern development would have it, it's about structure and organization, not just in language.
I was just trying to highlight the fact that mathematics as an institution is a process of building up share-able symbolic abstractions; of inventing a "language" one new word at a time.
I'm using "language" here, and above, to refer not to the words used to discuss mathematics (the... "mathematology" of math), but rather to the thing that includes objects like mathematical operators (e.g. "+", "⨯", "∫", "⇒") as its "words." Not the language about mathematics, but rather the language that is mathematics: the ever-growing set of abstract tools with symbolic handles which we've constructed to allow us to manipulate other concepts inside our heads, in rigorous ways where you can trust that if you and another mathematician do the same named mental 'move' to the same source concept, you will both arrive at the same destination concept as a result.
For a cute analogy: you can think of a martial art as a vocabulary of known, precise body movements, that can be taught. You can think of mathematics as a vocabulary of known, precise mental movements, that can be taught. Yes, this makes mathematics an art; but, equivalently, this makes a martial art a language.
To sum up:
• Mathematics is itself a (formal) language. It doesn't really fit in the category of words like "biology"; it fits more in with words like "logic" or "C++".
• To say that someone is "doing" Mathematics just means that they are using that language to achieve a goal; it's about the same as saying that someone is "doing" Python.
• To say someone is a mathematician, is to say that that person works to explore and extend the language of Mathematics, to test its properties and its limits, and to invent new 'words' within it that may then be used by those "doing" Mathematics.
Some of us like it and don't see value in the long winded motivational style. Things like group theory are targeted on people who are interested in math itself and like the mathematics for what it really is.
It is interesting to learn history once in a while and there are good books about that. However, most of the time you want to move on faster.
Please use absolute references (a link to the specific answer) instead of fuzzy references that change over time.
Currently, your answer is the top-rated, so which one did you mean? (And even if your answer was not at the top, the top will almost certainly change over time, leading to all kinds of misunderstandings.)
EDIT: It seems that this referred to the top-rated answer on MO, not HN:
https://mathoverflow.net/a/13149/66043
This demonstrates my point even more: Please use absolute references, as fuzzy references make for misunderstandings! (And as my comment received quit a lot of upvotes in the beginning, I'm pretty sure I'm not the only one who thought this was about the top-rated HN answer.)
What's missing for a lot of people is "What does this theorem apply to? Why does it apply? How do I break this theorem down into its component pieces to understand it?"
Indeed if your math text started off with a concept of a real world thing you're trying to figure out and then break it down to a series of paradigms that the student understands before applying each of the mathematical components that applies to each of these paradigms before finally combining them all to the final theorem, people would get it far more easily and be much less intimidated by it. Then over time students will spot paradigms that combine more elegantly to the final theorem and understand how to substitute them.
I think Math is being taught backwards in schools and this is why so many people are intimidated by and shy away from math... and I was a student who both struggled and enjoyed Math... and I still struggle with it because we were taught to think about it backwards.
We started out well with the basic building blocks, but somewhere between basic algebra and calculus, this all got turn around ass backwards. Where does the motivation to understand derivatives and integration come from if one doesn't understand the implications of their application?
Starting off with the general concept and breaking down into paradigms is not always possible. For example, say you want to teach people the Stokes' theorem of integration of manifolds [1]. It is the generalization of the divergence, Kelvin-Stokes and Green theorems that are very useful in electromagnetis, fluid mechanics and probably more. The theorem says, broadly, that if you are measuring some quantity in a certain space, you only need to know how does it "accumulate" on the boundaries of that space. But that will not be useful to an engineer or physicist. To really understand that theorem and break it down you need an entire course on differential geometry, with requisites on topology and calculus. Therefore, in the analysis courses for engineers/physicists the most probable situation is that either they present you the theorem and tell you how to perform the required operations on differential forms (which, if you don't know what they are, is a complete mistery/dark magic), or they go to the specific theorems (Gauss, Kelvin-Stokes, Green) which seem clunky and completely magic again.
The summary is that to be able to know what does a theorem apply to, why does it and how to break it, you literally need to study mathematics: that's what mathematics is. And the way to study it is always backwards, it's not like physics or programming where they can show you the high-level stuff, you can see what's happening and then you start to understand it. In math, most of the time, if you learn some subject starting from the high level stuff, it is difficult to even know what on earth are they talking about.
1: https://en.wikipedia.org/wiki/Stokes%27_theorem
Computing theory would have a word or two about this idea.
You can not code a Python script without knowing Python any less than you can use a mathematical theorem without knowing the definitions.
And yes, you can know what problem a piece of mathematics solve without knowing the math. How else would the math be invented in the first place?
> How else would the math be invented in the first place?
Usually the first step for problems to be solved is their expression in a mathematical way. See the heat equation: Newton (and probably others, history is not my strong point) already captured the core idea of heat transfer (transfer rate is proportional to the temperature difference) but it could only be really solved by Fourier once all of those ideas were put in mathematical expressions.
Better analogy:
You cannot code a Python script without knowing how to call the APIs you need to implement the functionality you want.
Knowing how an operating operating system works is more like knowing how foundational mathematics is axiomatized. Interesting and occasionally useful, but not immediately necessary for the theory of differential equations.
I think you might even be able to deal with differential equations without involving limits, by simply asserting all necessary theorems about derivatives as axioms. This corresponds to relying on a robust library (= the theorems other mathematicians have proved), without worrying about implementation details (= the proofs of those theorems, which might be much more difficult to understand than the results can be used).
I've noticed this too. I've tried to learn some quantum physics for fun, and it seems to me that most textbooks have surprisingly little description of the actual physical experiments behind the physics. Maybe this is something they usually cover more in lectures?
Like, I haven't found much details about the actual physical experiments around the spin of electrons (and certainly not of other particles). Sometimes there's an abstract description of a basic experimental setup.
I read just a few days ago that Feynman would redo old experiments that other physicist weren't doing anymore because they were established science. This is a great learning experience if you have access to the equipment, but most people don't. There should be books that describe how physics was developed, experiment by experiment, with lots of details.
If you read a 1000 page textbook on experimental physics, thats just the cliffnotes. The very highest level "here are the results of 400 years of serious research". There can't be any detail.
If you want the details on the stern gerlach experiment, for example, read the research papers. They will contain all the detail necessary to reproduce the experiment.
Lectures are just like those books. They just teach a very broad overview of what has been discovered.
The books you think should exist do exist. They're just not written for casuals.
its just that someone who is doing environmental physics doesn't necessarily understand papers on string theory. science is specialized, yea. if you want to understand research papers, get an education.
or just do something else. cutting edge research is not for everyone. not everyone is supposed to understand it. whats next? people demanding research papers be dumbed down to adhere to the freedom of information act?
science is hard. big deal. you cant have all the things at the same time. read the hawking books written for casuals if you want a cute little story about science.
Writing in a clear way, without drammatic grammatical errors, is not "dumbing things down".
Unfortunately, many academics receive little if any training in good presentation before being expected to lecture or write at undergraduate or graduate level, and while some are naturally gifted presenters anyway, most inevitably are not. Consequently, many career academics have no idea how poor their presentation skills are, how ineffective their presentation is as a direct result, or how much better they could be. They just get stuck at a very low level, but without the kind of introductory/remedial training that would be given to someone whose career involved presentation skills in the professional world. And of course if anyone with broader experience dares to suggest that there might be room for improvement, the instinctive reaction is denial.
A little irony is that some of the most engaging and informative presenters I have ever seen or read come from that same community, but despite the emphasis on peer review in their research work, when it comes to soft skills the weaker presenters typically have no idea how bad they are and therefore make no attempt to learn from their stronger colleagues and improve.
I once sat in a review meeting at the end of a year with members of the faculty responsible for teaching collecting feedback from many of the undergraduate students. When challenged about the poor quality of many lectures, the response was essentially that they can't make the lecturers go and learn how to lecture competently because the lecturers wouldn't stand for it.
Try that attitude in the professional world and you'd be in a remedial programme on your way to getting fired.
Fortunately, I expect that with the advances in modern technology and changes in modern careers, the old-school universities that think a famous name and charging high fees mean they can get away with anything will soon be obsolete, and so will the incompetent parts of the academic community sheltering within them. They will need to find new ways to offer dramatically more value than interested people can find on their own with all the modern resources we have available, or they won't be able to justify people taking several years out of their lives and paying a fortune in fees to attend any more.
Including some of the enlightening historical path towards that result is not "dumbing things down".
However, this knowledge is only communicated to other experts in the same - often very narrow - subfield. Often the definitions that give you a hard time when reading a paper have been refined over several years and are basically known to all other people working in the same field.
This is not ideal, but there is simply not enough manpower to produce good and generally accessible summaries of current research topics every few months...
No, that's the point I'm trying to make. They do not have enough training in making themselves understood. Hell, a good portion of them don't even have the language in which they are writing as their primary language.
Not necessarily. The classic example is Shinichi Mochizuki's work.
He's done some incredible work but he's basically invented his entire field out of thin air, he doesn't publish frequently, and the papers he does publish are essentially impenetrable.
He likes it this way and doesn't want to "dumb down" his work for the mere mortals who need to review it. That's essentially the end state of your argument - if you can't comprehend it you're not "in his field" and you're obviously not qualified to discuss it. And he's certainly not going to waste his time teaching some dunces the basics of his field.
https://www.newscientist.com/article/dn26753-mathematicians-...
https://www.newscientist.com/article/2099534-mathematicians-...
Furthermore, it's not necessary to communicate the actual, usually circuitous, route taken historically by scientists. Instead, you could just describe the series of hypothetical simplified experiments whose results would lead one to quantum mechanics and rejecting alternatives. This is never attempted in a serious way (trivial uses of the Stern-Gerlach experiment as a model of quantum mechanics not withstanding).
"Principles of Quantum Mechanics"
Some of those flaws:
(1) Dirac postulates, as most people do, that measurable observables are to be identified with Hermitian operators. This is a mistake that can be traced back to von Neumann. In reality, the larger class of normal operators are perfectly fine as measurable observables. Indeed, measurements are properly associated with only an orthonormal basis, and it is completely unnecessary to label them with eigenvalues, real or otherwise. (To see this, just observe that there is no physical difference between experiments that measure x and x^3 for the position of a particle.) Dirac's discussion on page 35 is just wrong.
(2) Unless my memory fails me, Dirac gives little to no justification for why we use tensor products to build up the state space of a many-body system from the state space of a single-body system, a tectonic shift from classical mechanics at the heart of the weirdness and power of quantum mechanics.
(3) Dirac was written before Bell's inequality. I mean, just look at the pithy discussion by Dirac (p. 4-7) to justify fundamental indeterminacy, one of the most profound things we know about the universe. Do you think this would have convinced Newton? Or Dirac in 1925? (We know it didn't convince Einstein.) This sort of thought experiment is lovely for an article in Scientific American to give laymen a sense of where things come from, but it's nowhere near the rigor with which we should teach physicists.
Much of science is really just subtle modifications or tests of major theories, but I feel like you could write up... say 12 keystone experiments in 100 page summaries and publish it in 2-3 volumes, (eg, QM and relativity volumes).
The problem I have with only presenting "polished" results is that we lose the context of our modeling -- eg, QM seems to have included non-determinism in an effort to preserve locality, but locality couldn't be preserved in the face of other results, so is non-determinism merely an extra assumption included for legacy reasons (eg, technical debt because no one wants to clean up the model/people forgot why we included it)?
You'd never even think to ask that question if you only saw the cleaned up model of QM divorced from its (philosophical) roots.
The Stern-Gerlach experiment (1922) is what you're looking for. "Modern Quantum Mechanics" by Sakurai opens with an explanation of this experiment and its consequences.
Try the free and new course "Quantum Mechanics for Everyone" that despite the title is certainly not dumbed down at all, on edX.
https://www.edx.org/course/quantum-mechanics-everyone-george...
From "A Mathematician's Apology", G. H. Hardy:
> Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain.
From another angle: when developing new theories or models, your thoughts are all over the place and frankly it's boring to go over your own crappy notes afterwards and try to reconstruct them in a way that others can understand. And much of the time you forget exactly what happened along the way as well, so any story you reconstruct is going to have some hindsight bias, which defeats the purpose of trying to "teach the story".
Really, from the first answer:
> Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through "guided struggle".
This is the only way to "properly" learn mathematics or science. Anything else is only making you think you've learnt something.
> This is the only way to "properly" learn mathematics or science. Anything else is only making you think you've learnt something.
My point is not just that most textbooks make it too hard/inefficient to learn this stuff. My point is that most of it is never learned. The Legendre transform connects the Lagrangian and Hamiltonian mechanics, the two fundamental formulations of both quantum and classical physics, and yet most physicist cannot tell you why the transform is defined as it is. The reason is they don't take seriously the possibility that we'll find non-Lagrangian phenomena, and so they have not been forced to consider what observational and theoretical evidence led to it's identification in the first place.
Do you believe that no new methods of learning will ever be invented?
Certainly, we can and do develop newer and simpler ways of understanding previous theories. And teaching the historical sequence of events can help with understanding; I myself experienced that with [1] for modern analysis. However, these understanding-aids don't teach you how to do mathematics, and only marginally improve your ability to apply those models and theories to existing real-world problems. To improve your ability to do mathematics, active exercises are necessary. Really, it's the same with many other fields, you don't get to be a good musician merely by reading about music and music theory.
I assumed they used the term "struggle" poetically, it certainly doesn't have to be unpleasant. But you have to put in some active mental exploratory effort. I found this post [2] a good summary of the skill set. But it's very abstract and likely won't make much sense unless you've been through the experience yourself.
These understanding-aids are also sometimes unnecessary. If you've done enough of the right kinds of exercises, they are of themselves an aid to understanding. For example, I could understand category theory better, not by learning about how this theory was developed historically, but by writing lots and lots of similar programs, and having a natural tendency to syntactically (and without much thought) refactor my code to be less repetitive, eventually leading me to various "category theory aimed at programmers" blog posts and papers. This one [3] of course deserves a mention, but there are many more.
To further emphasise this point, very brilliant mathematicians can just "pick up" models and concepts and work creatively and productively on them, without needing these aids.
My other point was that, the understanding-aids are very rarely what actually happened in the head of the people that developed a theory. Even historical narratives have distortions, and they are rarely detailed or precise enough to describe the rejected options, nor why they were options in the first place. (This fact, is also why they are not useful for teaching how to do mathematics.) There are exceptions, but reconstructing them is a boring process with little reward, especially since new developments 10 years later might explain it in even simpler terms.
That said, I would disagree with this part (from the top answer to the OP):
> a) The goal is to learn how to do mathematics, not to "know" it.
Modern mathematics has so much damn material these days that it's impossible to learn everything you need in order to solve modern-level problems, merely by teaching yourself all models and all theories "the hard way". Understanding-aids are certainly needed, and I use them very often myself, and I certainly prefer resources that teach using good analogies, proper context, descriptions of the motivations behind a theory, step-by-step "n/n+1" exercises, and everything else that other people mentioned here.
[1] "Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus" [2] https://medium.com/@jeremyjkun/habits-of-highly-mathematical... (discussed on HN here https://news.ycombinator.com/item?id=12187469) [3] https://bartoszmilewski.com/2014/10/28/category-theory-for-p...
I agree with this statement but I think it misses the point entirely. Guided struggle is indeed necessary, but learning some theorem without seeing the impetus for its discovery is like learning how to play an instrument without understanding that the intent is to make music. Yeah, with enough time and struggle, you might be able to go through the motions and play some scales, but most people don't learn to play instruments that way, they learn to play a simple song or two, then go back and start with the scales and building musical theory.
Math textbooks sometimes try to do the same thing, but it seems like they always come up with the most inane and pointless exercises.
All of this is just to say, learning they why of math can help someone learn the how.
Yes, Hardy was a great mathematician and he did say this --- but most mathematicians have tremendous respect for peers who strive for clear exposition in their lectures, their papers, and (if they write them) their books.
I am a professional mathematician, and Hardy's attitude is one I have never heard expressed by any of my peers.
"A Mathematician's Apology" is a fascinating read, but his description of mathematicians' attitudes is certainly not accurate today.
It indeed can be boring to reconstruct your thoughts in a way so that others can understand -- but many of us make the effort anyway, and doing so often leads to new insights.
> It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.
I interpreted "men who explain" not as mathematicians that can explain their work well, but as people who try to explain mathematics in a "lay" way to cater for a large audience, whose explanation can very often become inaccurate, non-mathematical or just downright false, yet still get public credit for seeming to know the field very well, despite these inaccuracies, and even though they are not directly pushing the advancement of the field itself.
It's of course a good thing to try to reconstruct your own thoughts, but I wouldn't say it's unreasonable for a mathematician to omit doing that. Could you go into more detail on the examples you mention, where doing so led to new insights?
Good question. It's a bit hard to do so (especially without going into mind-numbing technical detail) -- in math you never quite know where insights really come from. "Fortune prepares the prepared mind."
But generally speaking, I would say that good exposition gets you thinking about: Why does the technique work? What is the key insight? What are its limitations? And if you think about such questions, you naturally get a better sense of for which other questions your techniques are also likely to work.
1) Search JSTOR: https://www.jstor.org/action/doBasicSearch?Query=proofs+and+...
2) Use Sci-Hub to download it (handy bookmarklet): javascript:document.location.hostname=document.location.hostname + ".sci-hub.cc";
Contrary to what you say, practical applications are easy to be found where they are simple and helpful. They make for fun math hobbyist like to play with and that works nice in terms of making kids interested in math. Unfortunately they do not lead to cutting edge science nor math needed for physics and engineering.
From my own experience (which, I'll admit, is not much to extrapolate from), the former was great for its "insight" and the latter just had too much unnecessary "information". Until I got stuck in an actual maths problem. Then suddenly my opinion switched, and the latter became a trusted map through strange lands.
You're right about the amount of information that's encoded in a definition or theorem. And it is very difficult to portray why that information is important (never mind how to access it, see other's comments about scaffolding and cathedrals and the like). I fear that trying to describe why groups are defined like they are would just lead to more impenetrable tomes - but instead of being maps to the lost traveler, they'd be more like tourist adverts. But the questions the source was asking is "why are books like this?" and "how do others [learn maths] in this situation?"
Well, maths books are like this because they are guides to help people doing maths. And you learn maths by doing it. I don't think its defeatist to say that no book or lecture about riding a bike would compare to the experience of actually riding a bike. To paraphrase the top-rated answer, why would you expect anything different from maths?
But to answer the underlying question of both yourself and the source: how do we improve this? I think pedagogy should focus on getting to "do maths" faster. You really learn by getting stuck in a problem.
This it the wrong analogy and does not capture the MO answer. The correct analogy would be training vehicle designers by teaching them all the parts necessary to build a modern bicycle without teaching them any of the ways early bicycles were designed and why those designs were discarded.
Yes, I agree this MO answer (and several others) seem to be rationalizing the situation rather than acknowledging how suboptimal it is.
I think I'm a good example of the system's failure. Coming in to college, I was something of an ideal candidate for becoming a mathematician. I had some success in Olympiads and already had decided I wanted to study math. My goal in life was to be a math professor. I enrolled in a top college and took many graduate-level classes. However, by my senior year, when it was time for me to decide the next step in life, grad school or industry, I had become somewhat disenchanted with theoretical math. Math was so abstract I started losing interest: all this commutative algrebra (for example) I learned wasn't making me feel like I had any new insights into solving math problems, outside commutative algebra problem sets.
And so I went into industry.
However, I can't help but think that if I had more knowledge of the motivation behind all the abstract math, I wouldn't have lost interest. All that machinery of commutative algebra was invented for specific reasons, such as solving polynomial equations in the rationals through algebraic geometry. Years later, through casually reading math on the internet, I've been getting hints as to what power these highly abstract frameworks give you for solving concrete problems. But without seeing the end goal, and having some idea why I should be learning this in the first place, I felt like I was just getting lost in abstract nonsense.
>Most of that foundational information is lost when it's not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject.
This is exactly right. I personally failed to reconstruct enough to keep myself interested in the subject.
Annoyingly often it is also the most useful questions that are closed while things I consider trivia style/karma-farming operations like: "what is the reason for x" seems to be totally OK.
I too find those trivia questions and their answers interesting but IMO they are a distraction.
Again, this a pull quote that couldn't even be asked to be written. Pure gold!
Wikipedia is entertaining, valuable ?
https://en.wikipedia.org/wiki/CRC_Handbook_of_Chemistry_and_...
As for Wikipedia, I'm almost convinced most information on it is erroneous, unverifiable, or useless in the forms it is presented. I've stopped making contributions and try not to ever reference it at all.
ie. I could just as accurately say: maybe their value as a reference has led to reductionism. We can't go back and do an experiment, so we'll never really know.
In short, likely because its efficient and apparently sufficient
The former is intuitively true; finding good rules to enforce strictly is difficult, but enforcing is simple (is the act explicitly allowed/banned or not?). Find broad rules is trivial, but then you spend all the enforcement time arguing edge cases (see: US gov)
The latter is defended by the assumption that wikipedia/overflow do indeed follow such a policy, and its difficult to argue that they are not successful. There might be something better out there, but at least currently the market seems to have converged on this policy; ala "democracy is the worst form of government, except for the all other forms we've tried.
I don't think the convergence of the market on this sort of policy provides any indication one way or the other, because of network effects. Both of those sites won their space before the culture we're discussing here became entrenched, and because of network effects, it is not realistic to unseat them.
I imagine rather than "broad and inclusive", it should be "broad and authoritative", which then tends towards conservative when contraversial (which makes sense: how can you be authoritative when a general agreement can't be reached?)
And ofc, wikipedia was preceded by c2wiki, and stackoverflow by innumerable Q&A sites, both predecessors being substantially more liberal in what they accepted. And both beat out their predecessors, presumably largely because quality control was made much more difficult, and often absent, in the face of liberal acceptance.
That doesn't mesh with my experience of what made the sites popular at the time of their growth. Wikipedia was the only wiki that anyone had ever heard of, and SO was the only software focused Q&A site besides expertsexchange which had a freemium model that made it unusable (not to mention a funny domain name!).
When you can "contribute" to the community simply by clicking a delete button, rather than actually adding new and useful information, you're bound to get this sort of situation.
StackOverflow mods are appointed to police content. The more content they remove, the more it looks like they're doing their jobs.
It's the same with Wikipedia. Deleting someone else's hard work counts as a "contribution". It shows up in your profile just like writing a new multi-page article would.
When the same credit is awarded for for destroying as for creating, the destroyers are always going to win.
StackOverflow just need to lighten up. There's been times where it looks like the Moderator isn't even reading the responses, or shut it down because he/she personally didn't find it interesting? I sometimes wonder if some the Moderators have a complete grasp of the English language?
And the rewrites at Wikipedia, sometimes it seem like there's a financial motive to the rewrite. Years ago I bought a Domain name. It was the all to familiar neologism. Someone tried to buy it. I said no. The very next day the singular of a made up word was registered and his website was online. And Wikipedia was rewritten to "see the singular." that day too.
MathOverflow is intended for professional mathematicians. It isn't a general purpose discussion forum for mathematics. That such a thing doesn't really exist in a satisfactory form can't really be helped.
I can safely say you're wrong in this regard. Content removal is a big fraction of what the mods do because the contributors filter editing, closing, pointing out duplicates, migrating, etc. so that mods have to deal with only important stuff.
[1] https://en.wikipedia.org/wiki/Deletionism_and_inclusionism_i...
It isn't immediately clear to me that this question is relevant like you claim; the FAQ isn't really clear on it either:
> The site works best for well-defined questions: math questions that actually have a specific answer. You'll notice that there is the occasional question making a list of something, asking about the workings of the mathematical community, or something else which isn't really a math question. Such questions can be helpful to the community, but it is extremely tricky to ask them in a way that produces a useful response.
I don't have a solution either. I just wish I had.
https://stackoverflow.blog/2010/09/29/good-subjective-bad-su...
and be a bit more lenient.
Volcanoes.
Why are you wasting my precious time with this nonsense before getting to the meat? There are other things I need to read and study. There are problems I need to use these ideas to solve... and not only the ones in the textbook.
But then, I can also recognize that such winding introductions to a subject or an idea might be helpful to people who "get bored" as the author describes. Though I suspect the remedy might be to acquire and deliberately practice study skills. It would be nice if publishers sold versions of their textbooks both with and without what the individual posing the original question would identify as "motivation." I tend to think of it as fatty prose.
That's why there are two books. :)
I find more deplorable the fact that even basic math topics are often covered in the same dry way, without discussing practical applications or introducing topics through real-world scenarios. Many math books take the attitude "You have to know this, because I say so." Whenever I write about math, I try to start with a concrete example or a useful application of the theoretical result—it's always possible to come up with something for most of first year stuff. Seriously, you'd be surprised how much better reading UX is if you start each chapter/section with a motivating example.
For example, knowing that the snake lemma, a purely categorical statement, is most useful in homological algebra (such as, for example, simplicial or singular homology), is utterly useless if you don't already have an interest in algebraic topology. There really is almost no other motivation for the snake lemma, so now we're faced with the problem of trying to convince you that it's interesting by trying to convince you that algebraic topology is interesting. It can be done, and maybe we'll eventually bottom out in something like financial statements or bridge-building, or another topic that is widely recognised as "useful" and very far-removed from the snake lemma. Either way, it will be a long and arduous path, and I hardly think mathematicians can be blamed for this or be dismissed as elitists for the inherent difficulties of the subject.
But even for "first year stuff", the applications are kind of pointless. Do you really want to learn calculus because of physics? That's the most obvious and most historical application, but calculus is so foundational that you might as well motivate addition of real numbers by saying that numbers are added in physics too. More likely for the HN crowd, you want to learn calculus because you want to know how a neural network's backprop algorithm works, but how is the first year teacher going to anticipate that this is your particular interest in calculus?
At some point, I think there has to be a little "trust me, this is useful" and you just struggle through the subject until you can see on your own, after the fact, what the struggle was about. First wax on, wax off, Daniel-san. Then you will learn how you really were learning how to block karate blows.
If you can no longer "see" a thing, you can not usefully explain it.
Also, for self-teaching, I've found some math books that are setup to be in the form "Topic A with Applications to Topic B." If I care about B, such a book will typically does a good job motivating A, even if it isn't the purest introduction to the area. I can always read a book that is a more canonical intro to A later.
I used limits of sequences.
In our university, we encounter limits in either a handwavy way in Calc I or an epsilon-delta way in Advanced Calculus. Sequences and series are introduced in Calculus II.
Continuity is then described in the same terms. Or perhaps in terms of open sets if there is some topological topic.
I think this is pretty terrible for intuition. Not just that, but it's not even general enough - it requires normed spaces, so then you have to generalize again.
Much simpler to talk about ancient Greeks and Zeno's paradox. And then rigorously define limits of sequences, and define limits of real valued functions in terms of "for any sequence x_n that approaches x, y_n = f(x_n) approaches y". Simple and right away lets me show counterexamples beyond "left and right limits", like cos(1/x), and show students how to produce two sequences that converge to different numbers.
There are similar ah-ha moments when discussing fundamental concepts of linear algebra, number theory, complex numbers etc.
The two best books I have ever found on teaching Complex Numbers are:
1) Bak and Newman
2) Schaum's Outlines
They actually give you the understanding and ceeling behind WHY analytic functions are the way they are, and derive holomorphic functions from that. Imho a more terrible approach is that of Serge Lang and proving everything the other way, with Taylor power series.
Bottom line - make a directed graph of how you will teach your subject and then figure out the best entry point an direction for the greatest cohesion, as you would when telling a story.
If you are curious, now I teach a course on "Thinking Matematically" and here are this semester's results of that approach:
Numbers and Algebra: https://qbix.com/docs/mathematically1.pdf (https://vimeo.com/215335666)
Sets and Infinity: https://qbix.com/docs/mathematically2.pdf (https://vimeo.com/210500111)
Boolean Algebra: https://qbix.com/docs/mathematically3.pdf
Logic and Probability (Coming up)
The above videos were made with help from video exditors on upwork.com
For example, take something like measure theory. A reasonable motivation for measure theory to me is remembering your introductory probability class how you had to learn a probability mass function for discrete spaces and a probability density function for continuous spaces. Now these two ideas are obviously nearly the same idea. I mean the notation used is a pretty big hint. But you need measure theory to have the right concepts to describe how they are the same.
Where are all those kinds of motivation for things like topology and cohomology?
For topology, I feel like there's a sort of meeting-of-two-different-things; one starts with being very frustrated with the delta-epsilon-definition of "limit" and its one-dimensional nature, and the limit-definition of "continuous" and its clumsiness. The other starts from wanting to play with spheres and Möbius strips and knots and the like. When you're playing with these shapes a bijective mapping between two surfaces is not a fine-grained-enough idea because it is not continuous; adding continuity gives "homeomorphisms" which also aren't a fine-grained-enough idea because they do not make reference to the space an object is embedded in; wrap a torus about itself in a pretzel knot and you have something which is homeomorphic to a torus but in 3D you can't get there without tearing part of the surface through the other one, but in 4D you can. So finally we come to the idea of an isotopy, which bumps "continuous" to the next level by saying "Just like you can have a continuous path of points in space, you can have a continuous path of homeomorphisms from one to another," and that's where the pretzel knot becomes finally distinct from the torus in 3D, there is no continuous path from the homeomorphism of the pretzel knot to the torus, to the identity homeomorphism of the torus to itself. Or something like that. So this path is then an "isotopy" and then certain things are nicely isotopy-invariant and so forth.
One way to maybe then to describe homology is it gives you way to take about shapes and surfaces in the language of algebra.
https://www.quantamagazine.org/
Reminds me of the joke that philosophy students university learn the work of people who didn't go to university.
Would a text written by the discoverer of a field be more interesting?
I've found important papers easier to read than textbooks covering them... OTOH the original paper can be too close to their own motivation, and not treat - or yet be aware - of its general significance.
A motivating problem doesn't have to be practical or relatable... it just must lack obvious solution. A puzzle.
The idea is that you have 6 files of 6 men. Each is of a different regiment, each is of a different rank. Is it possible to arrange them so that no row and no file have two men of the same rank or from the same regiment?
You can mention this in your first lecture on algebraic geometry (or, for that matter, combinatorial geometry i.e. matroid theory) and then come back to it when you talk about projective geometries.
Personally, I prefer the dry stuff (e.g., Herstein) because that's the abstract-est abstraction. Rotations and matrices are groups? Don't care. I just wanna see the strange hidden properties of abstract structures reveal themselves.
... for example by noting that rotations and matrices form a group?
Today's technology in natural language processing is reaching the point where it will be possible to marry a natural language processing system to an automated theorem prover and have it generate formally verified proofs from math prose proofs.
Once this technology can readily process the textbooks making up the PhD curriculum, I think there will be a culture shift. Quickly there will be a new standard that math results should be formally verified. The hallmark of math, after all, is that it can be proven correct!
But with an increased role for computers will come an increased appreciation for the things that only humans can provide. Motivation and explication will be more valued when the technical aspects of theorem-proving are automated.
Practically speaking though, working mathematicians do not care that much about formally verified proofs. Working mathematicians are more interested in insight and understanding, and not necessarily in being completely sure of every detail. Formally and automatically verified proofs are much better suited for programming, as the automatic verification of the correctness of the program is after all _the_ best regression test.
So, while interesting in principle, I doubt formal methods will change much in how we do mathematics. Hopefully they'll change how we do software engineering though.
I don't really see how some fantastical process of formally verifying all of mathematics would have any relevance to teaching, anyway.
Although it's not proven, we have fairly good reason to believe there are no sufficiently efficient shortcuts for factoring large prime numbers, while there are shortcuts for proving many difficult mathematical theorems. After all, humans can do one but not the other.
[1] - Practically speaking though, the biggest reason we believe that factoring is hard is that we haven't really figured out how to do this, so our belief that it's hard is really build upon our feeling on hardness of theorem proving. :) I think we have more intrinsic reasons to believe than P != NP than that factoring is hard.
What could happen, though (hopefully), is that mathematicians start using more and more the various proof assistants (Coq, Isabelle/HOL, lean, …). These allow to write definitions, statements, theorems in a more programming-like way, with structured proofs; to develop modular libraries; and even in some cases to use automated provers to wave away details of the proofs. The problem of adoption of such provers, however, is a culture problem, not really a technology problem: as said elsewhere, mathematicians do not seem to care too much about formal correctness. It's only if automated tools help gaining insights (automatic search for (counter)examples, for example; function plotting; builtin algebra solving system… ?) that they will convince more people to use them. Modularity might also be a selling point, but not sufficient to overcome the steep learning curve of these tools.
I have some hopes in [lean](https://github.com/leanprover/lean), a new proof assistant inspired from Coq, which puts the focus on good UI and automation.
Formally-verified proofs are a great idea, it's my main area of interest in mathematics, but they have essentially zero relevance to a textbook. Textbooks don't need to be formally verified.
This history infects mathematics books.
The other traditional problem with mathematics is terse and obscure notation. There are many implicit assumptions embedded in published mathematical papers and books. This is not helpful. It's like reading code snippets without the declarations.
[1] https://en.wikipedia.org/wiki/Wrangler_(University_of_Cambri...
It's only a problem if you are unfamiliar with the subject area, and if you are, you have very little chance to understand the paper or book anyway. Mathematics is just difficult, way more difficult than most other stuff people might be doing.
Here's an example: consider this classical paper[1], "Vector Bundles Over An Elliptic Curve" by M. Atiyah. The author is a well known and regarded mathematician, and the paper itself has around 1000 citations. Sections 1. and 2., "Generalities" and "Theorems A and B", just recall the basic notions and theorems that are absolutely necessary to have any grasp whatsoever as to what is going on in the paper.
If you know nothing about algebraic geometry, the classical way to learn these is Robin Hartshorne's famous textbook "Algebraic Geometry". It is famous for being both good, self-contained to a large degree, and having very well chosen exercises, but also for being quite terse and often difficult to follow. Here's an Amazon link[2]. You can look at the table of contents. To really fully understand these two sections from Atiyah's paper, you need to have very good understanding of Chapter 2. "Schemes", and at least first 5-6 sections of Chapter 3. "Cohomology". This is 200 pages of pretty terse mathematics. At a fast understanding pace of 4 pages a day, it will take you two months to even have some basic toolset to understand the Atiyah's paper.
But, if you try to understand the Harthshorne's textbook, you'll quickly find out that it also has some prerequisites of its own. Also, the "4 pages a day" pace is only possible if you've already spent 2-3 years learning how to learn mathematics.
I encourage anyone to try to understand even first 2-3 sentences of the "Generalities" section. Google and Wikipedia the unfamiliar terms, you can also try to look it up in Harthorne's textbook, or Vakil's lecture notes, or any other source. The notions used in these first 2-3 sentences are basic to anybody working in the field, and yet one needs to spend hours to fully understand these when starting from scratch.
Compare this to other famous paper from other field, "The Market for Lemons" by G. Akerlof[3]. This is also a very famous paper by a well regarded economist, who received a Nobel Memorial Prize In Economic Sciences for it. It is a much easier read, precisely because the economic sciences do not operate on nearly the same level of complexity as mathematics. Once you know what are some common sense notions like, supply, demand, utility etc., and some very basic calculus, you can easily follow the argument without too much training.
My point here is that it's not that it's difficult to read mathematics just because it uses terse and obscure notation. It all is just genuinely difficult and complex, and it is impossible to invent better notation that will transfer days and weeks worth of understanding straight to the reader's brain. I would love it to be the case, but then it would cease to be as fun and rewarding to really understand.
[1] - https://math.berkeley.edu/~nadler/atiyah.classification.pdf [2] - https://www.amazon.com/Algebraic-Geometry-Graduate-Texts-Mat... [3] - https://www.iei.liu.se/nek/730g83/artiklar/1.328833/AkerlofM...
Mathematics is fun to understand because it is [artificially] hard to understand?
I can imagine a mathematical version of economics operating on some abstract constructs designed to emulate an economic system. It would reuse no terminology from the real world for the sake of producing an abstract notion, completely (or at least artificially) decoupled from the system it was designed to emulate.
I would imagine that Akerlofs paper, encoded into this form, would be at least as hard and involved to understand as the one from Atiyah.
If, however, the abstract constructs were much more complicated then the current economic concepts, and if you were trying to solve problems on much higher level of abstraction than economists currently are, then it would just be mathematics, and indeed it would be more difficult.
I, personally, cannot imagine how you could rewrite Akerlof's paper to be as hard to understand as Atiyah's. I can, with great difficulty, follow Atiyah's paper only because I spent literally _years_ learning the necessary background material. I am completely unable to relay my understanding to someone who hasn't spent years doing the same. I wish I was -- I'd revolutionize algebraic geometry then, just like Alexander Grothendieck revolutionized it around when the Atiyah's paper was written. On the other hand, if someone rewrote Akerlof's paper in an intentionally obscure way, you could easily rewrite it back in a clear way, once you spend the effort to understand the obscure version yourself.
Sir Michael Atiyah is an Abel prize and Fields medal winner. Apart from that, he has an excellent sense of humor. I had a chance to grab lunch with him last year and it was quite entertaining to say the least.
Have you actually hung around with some mathematicians? I find them to be on the whole quite a humble bunch of people. My feeling is that any hubris is burnt up in all of the great suffering that it takes to do good mathematics.
I think the machos all migrated to physics in the first half of the 20th century. Those guys are full of "big talkers". Have you ever heard of a physicist refusing the Nobel prize? No, but it has happened (more than once i think) for the Fields medal.
Also agreed about the notation. I hope something better evolves out of code.
This was never the case for me. When I learned trig ratios, I always understood some basic things that trig ratios could be used for. The teacher always introduced some applications, we always had a lot of word problems, and I could fill in the gaps myself.
Same for calculus. When I learned calculus, I always understood some things that calculus could be used for.
So I understood how those things could be applied to general, everyday sorts of problems. What was missing, though, was that I had nothing to which I could apply those techniques, besides homework.
Learning math (and reading STEM papers) has become easier for me since I now have actual problems to solve. Don't get me wrong: I'm not solving particularly challenging problems or using particularly advanced math. Nothing that tens of thousands of people haven't done before me. But I do need to understand the problems, solutions, and some of the context in order to successfully implement them. This provides a motivation that was always missing before.
I suspect this general narrative is true for a lot of people: that having an actual problem to solve is almost necessary to get a student to really learn the material, instead of just coasting along for a grade.
The example in the original post is books about group theory (or the group theory sections of abstract algebra books more generally). I can attest that this subject is very rarely described in textbooks with clear examples shown before definitions and theorems; usually the presentation is entirely abstract, following a pure definition–theorem–proof kind of structure. But many other areas of pure mathematics at the undergraduate level and above are presented in a similar fashion.
(I recommend Nathan Carter’s book Visual Group Theory for a lovely counter-example to the prevailing trend, which starts with the concrete, and is very accessible. http://web.bentley.edu/empl/c/ncarter/vgt/)
I feel that a lot of blame lies at modern academias curriculums. They feel every student needs to graduate in X number of years with a pretty long list of courses. It leaves little time for students who need or want more time with topics.
- e; The story of a number (Eli Moar) - An imaginary tale; The story of -1 (Paul Nahin) - The Poincare Conjecture (Daniel O'Shea) - The man who loved only numbers (Paul Hoffman) - Prime Obsession (John Derbyshire)
They're part biography, part history, and give a little colour to the subject that isn't available from your typical college textbook.
Throwing on you dull definition of matrix multiplication is completely useless unless you know where this definition comes from.
For an answer, there have been various influences:
(1) Whatever math was before 1900 or so, by the time of the Russell paradox and its fix with axiomatic set theory, the style of the fix was to be close to Russell-Whitehead (if I have that right) idea that proofs could be checked essentially mechanically by just symbol substitution and manipulation. E.g., in those days there was a book on the natural numbers, that is, 1, 2, 3, ... that apologized for numbering the pages before the natural numbers had been carefully defined!
There was even a name given to this style of writing, telegraph style.
(2) Of course, the books written on axiomatic set theory easily fell into the telegraph style. Even there the writers were getting into trouble: They gave names for the various axioms; they didn't explain why the names they gave were appropriate, and I never could discover why. But I was eager to get out of the sub-sub-basement of axiomatic set theory ASAP so just did a f'get about it.
(3) When books were written on abstract algebra, e.g., basic set theory, construction of the main number systems -- the naturals, integers, rationals, reals, complex -- and then went on to the main algebraic systems defined with axioms -- groups, rings, fields, vector spaces -- it was easy to stay with the telegraph style. E.g., it was tough to find a book on abstract algebra that also discussed group representations and its applications to quantum mechanics and molecular spectroscopy.
(4) Long calculus was often done with a lot of intuition and nothing like some carefully done definitions, theorems, and proofs as in, say, W. Rudin, Principles of Mathematical Analysis. And physics and engineering kept drawing diagrams with "little interval dx" etc. So, when abstract algebra was proving theorems, the calculus authors also wanted to be careful at least about delta-epsilon arguments. Then including a lot of physics, mechanical engineering, touching on the heat equation or fluid flow, was considered off-topic. Bummer.
(5) The series of astoundingly carefully written books, close to telegraph style, by the team Bourbaki was influential.
(6) During the Cold War and the Space Race, US math was awash in grant money and essentially turned its back on the physical science motivations and applications. Some of the funding people started to get angry about that, and we got the Tom Lehrer song and joke about abstract math being about "the analytic algebraic topology of locally Euclidean metrization of infinitely differentiable Riemannian manifolds" or some such.
But, sure, especially in analysis, for a good proof, there's often a good picture and if can see the picture then can construct the proof easily. E.g., for positive integer n, the set of real numbers R, and convex f: R^n --> R, that f is continuous has a really cute picture. Same for Jensen's inequality. In linear algebra, the polar decomposition says that each square matrix is just (A) a rigid motion, rotation and/or reflections followed by (B) moving a sphere into an ellipsoid by stretching and/or shrinking on mutually orthogonal axes. One or more of the axes goes to zero if and only if the matrix is singular. Etc. Nice picture.
Currently, then, there is an opportunity for math authors to include motivations for their subject, definitions, and theorems, intuitive descriptions and helpful pictures, applications, issues, open questions, etc. Uh, when reading a proof, for each of the assumptions, check off where it was used in the proof! The definitions, theorems, and proofs can still be fully precise and solid.
Some students of, say, analysis have long tried to find and draw pictures that would clarify what was going on in some definitions, theorems, and proofs. I would advise new students to do that also.
This was almost certainly a joke, although it must be said that it was probably more of a parody than a haha-joke.
The reality is that mathematics is written in a rigorous style because rigour is part of mathematics. You can waffle on for as long as you like about the intuition and the applications and whatever else, but you need to eventually get to the rigour.
The physicists and engineers might get away with 'drawing diagrams with little interval dx', but mathematics simply cannot.
Physics and mechanical engineering are not mathematics. Maybe don't try to learn mathematics if you want to actually learn physics?