unpopular opinion: imho there is no substitute for deep learning and doing problems. Math is unlike other subjects, where you have to actually do it head on; you cannot just passively consume it like listening to a history podcast or watching a video.
"Don't just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis?"
I heard the following nice metaphor from a professor: learning mathematics is like learning to ride a bike, at some point you just need to get on the bike, you don't just study some "theory of bicycles".
I don't think math is that different from other subjects, but it has this advantage that it is a whole world that you can carry with you and experiment, you don't need that many tools for your exploration.
> After scratching the itch of knowing that something's been written completely rigorously, it's easy to call a work done when maybe we shouldn't call it done yet. And there's contexts where this is maybe OK, research comes to mind...
It's funny, because all of the advice he gives in the rest of the video would be amazing to apply to research papers as well. I've had people discourage me from writing research papers pedagogically because it is "unusual," which really meant that it would be looked down upon. It's kind of tragic that the norm is to write research papers that strip away the human details of how a definition is motivated or how something was discovered. This kind of information is usually only shared through in-person discussions or occasionally research presentations, but it is often the core knowledge that enables one to do research in a given field.
It's a great video though. He gives a little checklist of ways to check if your work is clear:
- Do definitions have motivating examples?
- Do proofs [or other results] feel rediscoverable?
- Is there personality?
- Are core ideas illustrated with diagrams?
- Is relative importance highlighted?
...
He also gives a beautiful example of how powerful it is to give a motivating example for an abstraction before introducing it. I really got the sense from watching this of what makes him such a great math educator. Highly recommend his YouTube channel 3Blue1Brown for anyone who hasn't heard of him
I think most supervisors try to discourage students from writing in this way because there is a delicate art to it that you are unlikely to be able to meet at that career stage. When I read some of the texts from Martin Hairer and Cedric Villani in my field (or even Riemann's famous paper on the zeta function), you can see that they are motivational and informative but still concise at the same time. Full of personality too, especially Villani. That's a hard balance and it requires exceptional understanding of the topic and the reader.
At the same time, we really should be encouraging it more. I have found that in newer machine learning theory papers (strictly theory, not empirical work), there is something closer to a good balance that is expected even of students.
Including a motivating example is key to this, and should be considered mandatory.
> most supervisors try to discourage students from writing in this way because there is a delicate art to it that you are unlikely to be able to meet at that career stage.
And it seems to me they will continue to to unlikely to meet it if they don't get any practice at it. Which loops back to being another pedagogical problem.
Reviewer 1 will classify half or your analogies as misleading, reviewer 2 will complain about the other half, and anyway the editor will ask to reduce the number of pages before publication.
I'm pretty sure the vast majority of people understand linear algebra etc, if they were taught that the reason why linear algebra works is that algebra (algebra over a field) are an approximate isomorphism to physical reality.
From that perspective you do not need contrived examples like using function composition as alternative operator.
It would be simply enough to teach the student that any symbol system following the definition of an algebra over a field has the same properties as e.g. natural numbers with respect to the (possibly physical) composition operator.
If you don't understand this then it's because you don't understand linear algebra.
Consider this, you can write numbers as symbols or as words. There is no single universal language so a number can be represented by many words and mean the same thing.
Now let's ignore the concept of a number to begin with. Let's say numbers were never invented.
Everyone discovered that you can combine symbols a+a=b b+a=c and so on. They realize that the laws relating to how the words can be combined follow certain properties.
They will also notice that marbles under physical composition follow the same rules.
They then discover that the effect of having a marble in one hand and in another can be physically composed into one hand having two marbles.
They then think about giving the marble quantities a different representation, e.g. one marble = 1 finger
They invented an isomorphism.
Transforming the marbles into fingers then adding the fingers and then transforming back to marbles is the same as taking one and one marbles and putting them together.
And that is how linear algebra was invented, long before anyone even know what a mathematical definition was.
I don’t think that would help at all, to be honest. I think if you teach people that a vector is an arrow, and a matrix is a transformation of the arrow, you’d get much further. 3b1b has a great series on it in fact which I would consider to be the gold standard.
Sorry, long rant incoming. I think there are a lot of things wrong here.
I think you are right where you talk about the "one marble" and "one finger" isomorphism, that makes sense.
Everywhere else, you throw around a lot of terminology that gets very confusing.
> any symbol system following the definition of an algebra over a field has the same properties as e.g. natural numbers with respect to the (possibly physical) composition operator
What do you mean by symbol system? Are you suddenly in the discipline of logic and formal languages here? What are "natural numbers with respect to the composition operator"? what is _the_ composition operator? Function composition? I don't see any K-algebra structure here, please enlighten me. What does it mean to be physical here, referring to the composition operator? An algebra over a field is a pretty fundamental structure, intuitively polynomials over a field with some extra rules. But this is all still very abstract and I don't see how it is immediately apparent to be an "approximate isomorphism to physical reality", which I can only assume to mean what I think it means.
> Let's say numbers were never invented. [...] Everyone discovered that you can combine symbols a+a=b b+a=c and so on.
Whoah, slow down there, you are already assuming a lot. Infix notation, equality, and suggestively using the plus symbol. I would agree with your claim "everyone discovered [...]" when talking about concepts like addition, but you are talking about abstract "combining symbols", which in the way we do it now is a rather modern advancement, especially because you talk about the logic/language theory concept of words.
> Transforming the marbles into fingers then adding the fingers and then transforming back to marbles is the same as taking one and one marbles and putting them together. And that is how linear algebra was invented, long before anyone even know what a mathematical definition was
That is not linear algebra. That is simply group or ring theory, and really only restricted to the integers, really. The integers (or whatever you mean by "linear algebra") were exactly definitionally invented the moment they had a kind of mathematical definition of some sort. I think you are trying to say the concepts were discovered and/or used long before anyone knew about a mathematical definition.
> if it's not clear, any physical composition operator can form an algebra over a field
Please explain this better, I am not sure what you mean by a "physical composition operator". Do you mean that functions that are already K-linear operators over vector spaces form a (non commutative) K-algebra with the function composition operator as multiplication? Sure, that is true, but that already assumes so much structure. I don't think it is clear at all. I am not saying you are wrong here (smart physics concepts stuff around Noether theorem etc. come to mind, but I am not sure), but I don't think this is easy to see intuitively at all, as opposed to marbles and fingers etc.
> In a mechanical calculator, the gear movement is the composition operator. In a computer the adder is an electron based composition operator
Whoah, slow down again, please. What composition operator? There is no linear algebra here. Also, it is all over the integers, if anything, not over a field. Or if it is a field, then it is just approximated with floats, which mathematically amounts to things that are not even a true ring/algebra, because of rounding errors.
> Hence the reason why it is possible to build useful computers that perform useful calculations relates directly to the fact that there is an isomorphism between the real world state and the computer state
You make the conclusion sound so easy, yet you left out all of the important steps. And computers are famously not there as an "isomorphism between the real world state and the com...
1. This solution is how I solved it.
If you look at problem solving as A* search then the solution is not what was done to solve it. Learning mathematicians are supposed to learn to search trough the space of mathematics by examples of solutions. Each step presented on the blackboard is a step in the right direction, omitting steps in the wrong direction, and backtracking in the initial efforts to solve (or even define) a problem.
2. This idea is abstract and has nothing to do with reality.
Mathematics is a distillation process of abstraction. In this purification process all preconceptions of reality are filtered out, so the result is rigorous and universal. This often leads to a denial of any connection to reality. (A well known counterexamples is Conway's Analysis of go games, leading to the surreal numbers.)
3. This is the only and right way to do it.
You are presented with a single solutions, definitions or axioms, omitting alternative definitions. This get better in higher education (think of how many ways there are to define numbers).
This is not what it looks like doing math.
It leads to frustration in students who learn (wrongly) that math is about doing everything right the first time, intuitive thinking is wrong, and basically only about what you are not allowed to do (as the feedback they get are highlights of mistakes in exams).
Our task should be cleaning that up and reteach exploration, reteach that math - like any universal language - can describe everything however you want, finding solutions is trial and error, intuition is good and fast but fuzzy and can mislead you, to not stop with a solution but explore around it more for better understanding, and while we have good cause to define things as we do, it's not necessarily the only way to do things (looking at you, axiom of choice).
To use an analogy: Otherwise we stick to teaching orientation to scouts by fastest satnav routes.
i think your points 1 2 3 are true in the sense that this is perceived, and that sometimes, the way things are being taught lead to this. but in my mathematics career, as well as in school, i never had the impression that these sentiments were actually the beliefs of teachers.
i also think it is wrong to so confidently state in which way mathematics relates to reality. this is a very nuanced philosophical issue.
also, i don't see how your "counterexample" is one, let alone a well-known one, both things, the way you stated them, are very abstract and not "real world" at all, the way i see it. or are you here still talking about the fact that math gets inspired by things that happen in real life, as opposed to just inventing stuff just out of pure abstract inspiration? no mathematician i have ever met will deny this. and in fact, philosophically, it is an interesting question if it is even possible to separate a human's "purely abstract ideas" from how a human reasons in real life. that goes back to plato.
i also think one should be careful with these strong assignments of terms like "universal language" and "can describe everything however you want".
sorry for the rant. i agree with your sentiments about how we teach and how researches find out things.
He is right in a sense that a lot of abstract concepts, in e.g. combinatorics, can be explained even to middle school students if given as simple examples starting with trivial cases and showing the pattern. How many textbooks do that? Close to zero. How many research notebooks are full of these? I cannot say for sure but I suspect much greater than 0.
Interesting points. Regarding intuition, I highly recommend https://betterexplained.com as a great example of an alternative approach that aligns with your ideas.
I self-taught some math which gave me the benefit of no deadline, and I did not memorize derivations - I learned them. The typical experience is that the derivation given to you is that 'satnav route' and offers little intuition for what it does or how you would have even thought to do it.
The exercise for me was always the same: re-derive the proof by injecting context, motivation and intuition.
30 comments
[ 1.9 ms ] story [ 6.8 ms ] thread-The goat paul halmos
https://en.wikipedia.org/wiki/Deep_learning
I don't think math is that different from other subjects, but it has this advantage that it is a whole world that you can carry with you and experiment, you don't need that many tools for your exploration.
It's funny, because all of the advice he gives in the rest of the video would be amazing to apply to research papers as well. I've had people discourage me from writing research papers pedagogically because it is "unusual," which really meant that it would be looked down upon. It's kind of tragic that the norm is to write research papers that strip away the human details of how a definition is motivated or how something was discovered. This kind of information is usually only shared through in-person discussions or occasionally research presentations, but it is often the core knowledge that enables one to do research in a given field.
It's a great video though. He gives a little checklist of ways to check if your work is clear:
- Do definitions have motivating examples?
- Do proofs [or other results] feel rediscoverable?
- Is there personality?
- Are core ideas illustrated with diagrams?
- Is relative importance highlighted?
...
He also gives a beautiful example of how powerful it is to give a motivating example for an abstraction before introducing it. I really got the sense from watching this of what makes him such a great math educator. Highly recommend his YouTube channel 3Blue1Brown for anyone who hasn't heard of him
At the same time, we really should be encouraging it more. I have found that in newer machine learning theory papers (strictly theory, not empirical work), there is something closer to a good balance that is expected even of students.
Including a motivating example is key to this, and should be considered mandatory.
And it seems to me they will continue to to unlikely to meet it if they don't get any practice at it. Which loops back to being another pedagogical problem.
From that perspective you do not need contrived examples like using function composition as alternative operator.
It would be simply enough to teach the student that any symbol system following the definition of an algebra over a field has the same properties as e.g. natural numbers with respect to the (possibly physical) composition operator.
If you don't understand this then it's because you don't understand linear algebra.
Consider this, you can write numbers as symbols or as words. There is no single universal language so a number can be represented by many words and mean the same thing.
Now let's ignore the concept of a number to begin with. Let's say numbers were never invented.
Everyone discovered that you can combine symbols a+a=b b+a=c and so on. They realize that the laws relating to how the words can be combined follow certain properties.
They will also notice that marbles under physical composition follow the same rules.
They then discover that the effect of having a marble in one hand and in another can be physically composed into one hand having two marbles.
They then think about giving the marble quantities a different representation, e.g. one marble = 1 finger
They invented an isomorphism.
Transforming the marbles into fingers then adding the fingers and then transforming back to marbles is the same as taking one and one marbles and putting them together.
And that is how linear algebra was invented, long before anyone even know what a mathematical definition was.
I think you are right where you talk about the "one marble" and "one finger" isomorphism, that makes sense.
Everywhere else, you throw around a lot of terminology that gets very confusing.
> any symbol system following the definition of an algebra over a field has the same properties as e.g. natural numbers with respect to the (possibly physical) composition operator
What do you mean by symbol system? Are you suddenly in the discipline of logic and formal languages here? What are "natural numbers with respect to the composition operator"? what is _the_ composition operator? Function composition? I don't see any K-algebra structure here, please enlighten me. What does it mean to be physical here, referring to the composition operator? An algebra over a field is a pretty fundamental structure, intuitively polynomials over a field with some extra rules. But this is all still very abstract and I don't see how it is immediately apparent to be an "approximate isomorphism to physical reality", which I can only assume to mean what I think it means.
> Let's say numbers were never invented. [...] Everyone discovered that you can combine symbols a+a=b b+a=c and so on.
Whoah, slow down there, you are already assuming a lot. Infix notation, equality, and suggestively using the plus symbol. I would agree with your claim "everyone discovered [...]" when talking about concepts like addition, but you are talking about abstract "combining symbols", which in the way we do it now is a rather modern advancement, especially because you talk about the logic/language theory concept of words.
> Transforming the marbles into fingers then adding the fingers and then transforming back to marbles is the same as taking one and one marbles and putting them together. And that is how linear algebra was invented, long before anyone even know what a mathematical definition was
That is not linear algebra. That is simply group or ring theory, and really only restricted to the integers, really. The integers (or whatever you mean by "linear algebra") were exactly definitionally invented the moment they had a kind of mathematical definition of some sort. I think you are trying to say the concepts were discovered and/or used long before anyone knew about a mathematical definition.
> if it's not clear, any physical composition operator can form an algebra over a field
Please explain this better, I am not sure what you mean by a "physical composition operator". Do you mean that functions that are already K-linear operators over vector spaces form a (non commutative) K-algebra with the function composition operator as multiplication? Sure, that is true, but that already assumes so much structure. I don't think it is clear at all. I am not saying you are wrong here (smart physics concepts stuff around Noether theorem etc. come to mind, but I am not sure), but I don't think this is easy to see intuitively at all, as opposed to marbles and fingers etc.
> In a mechanical calculator, the gear movement is the composition operator. In a computer the adder is an electron based composition operator
Whoah, slow down again, please. What composition operator? There is no linear algebra here. Also, it is all over the integers, if anything, not over a field. Or if it is a field, then it is just approximated with floats, which mathematically amounts to things that are not even a true ring/algebra, because of rounding errors.
> Hence the reason why it is possible to build useful computers that perform useful calculations relates directly to the fact that there is an isomorphism between the real world state and the computer state
You make the conclusion sound so easy, yet you left out all of the important steps. And computers are famously not there as an "isomorphism between the real world state and the com...
1) Math Made Visual: Creating Images for Understanding Mathematics by Claudi Alsina and Roger Nelsen - https://bookstore.ams.org/CLRM/28
2) Roger Nelsen has also written other similar works like the 3-vol Proofs without Words, Nuggets of Number Theory: A Visual Approach, etc. - https://bookstore.ams.org/browse?Author=%22Roger%20B.%20Nels...
Visual Thinking in Mathematics: An epistemological study by Marcus Giaquinto - https://academic.oup.com/book/11260
The three lies (by omission) are:
1. This solution is how I solved it. If you look at problem solving as A* search then the solution is not what was done to solve it. Learning mathematicians are supposed to learn to search trough the space of mathematics by examples of solutions. Each step presented on the blackboard is a step in the right direction, omitting steps in the wrong direction, and backtracking in the initial efforts to solve (or even define) a problem.
2. This idea is abstract and has nothing to do with reality. Mathematics is a distillation process of abstraction. In this purification process all preconceptions of reality are filtered out, so the result is rigorous and universal. This often leads to a denial of any connection to reality. (A well known counterexamples is Conway's Analysis of go games, leading to the surreal numbers.)
3. This is the only and right way to do it. You are presented with a single solutions, definitions or axioms, omitting alternative definitions. This get better in higher education (think of how many ways there are to define numbers).
This is not what it looks like doing math. It leads to frustration in students who learn (wrongly) that math is about doing everything right the first time, intuitive thinking is wrong, and basically only about what you are not allowed to do (as the feedback they get are highlights of mistakes in exams).
Our task should be cleaning that up and reteach exploration, reteach that math - like any universal language - can describe everything however you want, finding solutions is trial and error, intuition is good and fast but fuzzy and can mislead you, to not stop with a solution but explore around it more for better understanding, and while we have good cause to define things as we do, it's not necessarily the only way to do things (looking at you, axiom of choice).
To use an analogy: Otherwise we stick to teaching orientation to scouts by fastest satnav routes.
i also think it is wrong to so confidently state in which way mathematics relates to reality. this is a very nuanced philosophical issue.
also, i don't see how your "counterexample" is one, let alone a well-known one, both things, the way you stated them, are very abstract and not "real world" at all, the way i see it. or are you here still talking about the fact that math gets inspired by things that happen in real life, as opposed to just inventing stuff just out of pure abstract inspiration? no mathematician i have ever met will deny this. and in fact, philosophically, it is an interesting question if it is even possible to separate a human's "purely abstract ideas" from how a human reasons in real life. that goes back to plato.
i also think one should be careful with these strong assignments of terms like "universal language" and "can describe everything however you want".
sorry for the rant. i agree with your sentiments about how we teach and how researches find out things.
I self-taught some math which gave me the benefit of no deadline, and I did not memorize derivations - I learned them. The typical experience is that the derivation given to you is that 'satnav route' and offers little intuition for what it does or how you would have even thought to do it.
The exercise for me was always the same: re-derive the proof by injecting context, motivation and intuition.