1. It's much more interesting to study something when you have a historical perspective that you can relate to later on in the course.
2. It makes you comfortable about the idea that such developments in mathematics are made by fellow humans only and they can also do such things if they put in the efforts. This might sound like a small addition but it's detrimental in developing such sense in young kids.
There is a difference in history and thought process.
My view is that school should help kids find their thought process. Will likely be similar to successful thought processes. Didn't have to be, though.
The most instructive thing I have ever heard, was seeing that Feynman made his own notations for learning math tricks in high school. Fur some reason, this really made me regret not trying new ways of things.
There was an article recently on the importance of notation. Making your own goes a long way to understanding others.
Do math education specialists need to be taught the history of the debate over whether mathematics education should include the history of mathematics?
Certainly yes (within limits), and as many people said above that applies to all fields.
Sometimes knowing how/by whom/when something was discovered or entered into practice gives some perspective on what you are doing today.
Absolutely no! 90% of educated population does not handle basic math (such as rule of proportions). History would dilute math even more. But it would be easier for teachers and students to 'pass'.
I agree. The fact that high school students are struggling with epsilon-delta notation of limits is not because they lack historical context. It is just far beyond what most people can grasp.
Most of my school mates had zero understanding of the mathematics we learned. They memorised how to compute the distance between a point and a sphere, a line and a sphere, a plane and a sphere, and then they'd solve one of these problems for the exam, get a good grade, and forget everything immediately afterwards. At no point have they understood what a scalar product is and why they are using it.
If you'd really want to improve mathematics education, you should focus a lot more on the basics, and teach advanced subjects only to the handful of students that are interested in them.
I don't know about American conditions, but I remember that many struggled with epsilon delta because it was their first encounter with formal logic. "For every... there exists a..." took some getting used to for them.
Probably not. It's interesting to plow through Newman's four volume "The World of Mathematics" as a grad student, but inflicting mathematical history on ordinary school kids is cruel. They don't need that much math.
Unless you're a mathematician, math should be viewed as a tool, like a lathe. You don't need to know the history of the lathe, and how Maudslay made it a precision machine tool. (His original lathe is in the Kensington Science Museum. It's one of those historic artifacts which looks very different from its predecessors, and it's successors look a lot like it.)
Few people need to know how to build up mathematics from minimal axioms. Nobody should have to struggle through Whitehead and Russell below the PhD level. We have power tools for that now. The original Boyer-Moore theorem prover from 1992 can build up constructive number theory from the axioms in under a minute.[1] I fixed it up recently to run on GNU Common Lisp and put it on Github, so it's runnable on modern machines.
There's certainly no excuse for inflicting Newton's notation for calculus on kids. It's not even clear that classical geometry proof approaches are that useful.
Of course it's very important to study the history, but probably not in the first presentation. But at the same time, it's definitely true, for most subjects, that historical references often help to motivate less committed students and enliven the experience.
On Maudslay's role in the development of machine tools, this paper looks interesting:
On the other hand, often our tools have somewhat peculiar forms. For example, “trigonometry” as commonly taught in high schools and used in science/engineering/mathematics uses bizarre names, terrible notation conventions, and a giant pile of unmotivated formulas to memorize.
Teaching a bit of history alongside helps students understand why it takes that particular form.
The word “sine” comes from a weird Latinization of an Indian word for “half a bowstring”. Draw a picture of a circle with a vertically oriented chord (the word “chord” also implies a bowstring), and a student will have a much easier time remembering what the sine is. Likewise tangent (Latin for touching) and secant (Latin for cutting) make more sense if you think about the meanings of the words. Cosine means the sine of the complementary angle, etc.
The reason we call inverses “arcsine”, etc. is because originally these were written as quasi-sentences, and the concept of a mathematical function was not well developed. So sin⁻¹ x would be expressed as something like: arc (sin. = x). That is, the length of the arc whose sine is x. This form was cumbersome so later got shortened to arcsin x.
The origins of trigonometry are in astronomical measurement, which is why we have 360° in a circle (each degree is roughly one day of movement (365 days/year), rounded to a nearby highly composite number), and come from the Sumerian/Babylonian numerical tradition which used a base sixty number system. Hence “first minutes”, “second minutes”, “third minutes”, etc. of a degree. “Minute” (Latin for small) implies 1/60 of the larger unit.
The reason trigonometry focuses on learning a big pile of formulas is because before the era of electronic calculators, people needed to do all computations by hand, or by interpolating in pre-computed lookup tables. The goal of “trigonometry” is to take a given problem and convert it to a form with the easiest hand computation and the fewest table lookups possible, so that the mechanical work can be handed off to a team of human computers who can go through the laborious arithmetic. Memorizing trigonometry formulas is a way to cut the work done by the human computers to a small fraction of what it might take for the original problem as posed.
Trigonometry was important in science/engineering because until recently the abstract vector concept and idea of combining simple single-number parts into “complexes” were not well developed. People solved problems by breaking them into coordinates and discrete lengths and angles. Solving triangles and converting between polar/cartesian coordinates were important steps in almost any 2-dimensional problem.
If we really wanted to avoid the “ontogeny recapitulates phylogeny” model, we would scrap the current form of trigonometry (certainly not spend 4+ months exclusively focusing on it) and set the high-level ideas on a more logical foundation which was easier to learn and reason about, ditching the parts now anachronistic in an electronic computer age. We would give students harder problems to solve and fewer formulas to memorize. But that could leave students unfamiliar with the existing language commonly used in the existing literature, so to some extent we’re stuck by our history. http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
> The reason we call inverses “arcsine”, etc. is because originally these were written as quasi-sentences, and the concept of a mathematical function was not well developed. So sin⁻¹ x would be expressed as something like: arc (sin. = x).
Any pointers to resources about those "quasi-sentences"? I'm interested in language, broadly speaking, so info about how mathematical notation evolved is interesting to me. The rest of your comment is great to, I'd love to read a book about stuff like this if there is one!
The most comprehensive source about this kind of thing is Cajori’s History of Mathematical Notations. You should be able to find a used copy of both volumes for a reasonable price.
- Get a clear notion of what you desire to accomplish, then you will probably get it.
- Keep a sharp look-out upon your materials: Get rid of every pound of material you can do without. Put yourself to the question, ‘What business has it there?’
- Avoid complexities and make everything as simple as possible.
- Remember the get-ability of parts.
“It was a pleasure to see him handle a tool of any kind, but he was quite splendid with an 18-inch file.”
Bringing up Whitehead and Russell indicates you may have somewhat missed the argument here. The point is to be guided by the history, not build up from axioms. Mathematics was not historically built up from axioms! Whitehead and Russell are very recent historically, so you would certainly not start with them (or even get to it at all) by following a historical approach.
Essentially this is saying that you can use the sequence of historical discovery as a guide to the appropriate sequence of lies-to-children [0].
You don't think that a little historical perspective helps bring the subject to life? I think it's great for both the mathematically inclined students and their not so interested counterparts. Context helps those who care see the bigger picture and those who don't to at least understand why this might be important.
I work as an applied mathematician and in my opinion trying to work though the historical context can sometimes hurt. Basically, there are two issues at play. One, everyone has a finite amount of time to learn things, so working through historical methods to add context can take away from gaining time and experience with flat out better, modern methods. Two, it often confuses people as to what methods to use. I work a lot in optimization theory and solvers. In modern optimization theory, you essentially never formulate, use, or look at the Euler-Lagrange equations. However, whenever I work with engineers, they was to use these equations because they learned them in school and think that it's necessary to use them to optimize. It's not. Just discretize everything and use the KKT conditions. Hell, don't discretize everything and use the KKT conditions because they work just fine in Banach spaces. It's not that the E-L equations are wrong, but they're covered for historical context and it cheats people out of the time they need to learn modern methodology. As another example, when people learn iterative linear system solvers, they often start with the Jacobi method or Gauss-Seidel. However, these methods are almost never used because they're absolutely terrible compared to modern Krylov methods. Again, though, I often work with engineers who learned about them in school and want to use them when they should have been learning the differences between things like MINRES and CG and what needs to be done to precondition the system.
For full mastery of a field, sure it helps to know the historical record. Certainly, I think there are little historical tidbits that can keep a class more interesting and put some things in context. However, by in large, I often see people use completely terrible algorithms because they were taught in what I consider a historical manner and they did not have the time, nor perhaps the instruction, to learn more modern methods, which can often be formulated in an independent way without regard to older algorithms. Fields evolve and sometimes we realize there's just flat out better ways of doing things that don't need earlier results.
Reading the above, I wonder if the people who are writing the linear equality/inequality parts of SAT solvers for program proof know that. Internally, numerical equalities and inequalities become rows in a sparse matrix. The columns represent program variables. Solution is done with rational arithmetic. The original Oppen-Nelson prover of the 1970s solved those with a Gauss-Jordan approach, and most follow-on work through at least the 1990s was basically the same machinery.
In this context, your idea of history is of a "Historian's history". The article on the other hand is about "more historical context" in the progression of mathematical tools and ideas.
Worth considering, is that if this practice gained widespread appeal, the quality of how it is done (by the best and probably by the majority) would improve.
For my opinion, I think the idea "incorporate history" is just the most obvious first attempt at more story telling in mathematics. If these "stories" are not a part of the classroom then they may come too late or never for the majority.
I do agree that math is a tool, but it should be taught with more rhetorical tools. To trade a little into your knowledge space, think about the fondness many have for "The Little Schemer".
When I learned calculus in school our teacher took an hour to describe what the problem was and how people over history had solved these problems before calculus came around. This made the topic much more interesting to me. You probably don't need detailed math history but a few minutes showing that things like calculus have not always been around and how they were discovered is pretty useful.
My math teacher taught me not only calculus, but also when and who got up to what. It doesn't have to take a long time, but a bit of context helps a lot. Euler went to Russia and yada yada bridges graphs etc.
Same goes for all science disciplines. You need to have a rough idea that Darwin worked in the 19th century, that much of thermo came about in the late 19th century, that quantum is a 20th century thing. You need to know what people were wondering about, and what experiments they came up with.
I've been listening to a lot of audio courses lately, and those little nuggets really help to understand things.
The point of the little stories such as how Watson and Crick came up with the double helix is to help recall. It's hard to remember dry facts, much easier to remember stories. People are kinda built that way.
I agree that historical context should be taught in classes. I wrote a Master's adding plenty of historical notes about "why" this and that approximation was thought of, and proposed. Sometimes it doesn't take much - just reading the early Schrödinger papers can give you an idea of his train of thought. It really helps make sense out of theory, for me.
"Euler went to Russia and yada yada bridges graphs"
I know little of this history, but is there any link between him going to Russia and him solving the seven bridges of Königsberg problem?
I'm asking because I don't know whether Euler ever visited Königsberg and because Königsberg is in Russia now (renamed to Kaliningrad after WWII), but was in Prussia at the time.
He went to work in St Petersberg, which is just up the coast. No idea if he sailed or drove, but it seems quite likely to have been on his way regardless of where in Europe he was before (Switzerland?).
Someone probably drove the vehicle in question for Euler but:
b.2.b One who drives a vehicle or the animal that draws it; a charioteer, coachman, cabman, etc.; also, one who drives a locomotive engine. (Often with defining word prefixed, as cab-driver, engine-driver, etc., for which see the first element.) [...]
c 1450 St. Cuthbert (Surtees) 6016 All þe dryuers ware agaste þat þe sledd suld ga our faste. 1581 Savile Tacitus 93 (R.) Buffons, stage-players, and charet drivers. 1725 Pope Odyss. xiii. 99 Fiery coursers in the rapid race Urg'd by fierce drivers thro' the dusty space. [...]
"Euler left Basel on 5 April 1727. He travelled down the Rhine by boat, crossed the German states by post wagon, then by boat from Lübeck arriving in St Petersburg on 17 May 1727."
Still inconclusive, but given that we know this, I would say chances are non-zero that the sources this was derived from spell out where that ship made stops.
One could also look at his journeys from Leningrad to Berlin and, years later, back, but both were after his publication on the 7 bridges problem.
> how Watson and Crick came up with the double helix
s/came up with/stole the idea from looking at Rosalind Franklin's X-ray data/
To be fair, both Watson and Crick strongly insisted she also get the Nobel, but she'd tragically passed a way shortly before the award was announced, and the rules are very clear about no posthumous awards. Still, it's sad that we don't teach Franklin's role.
I have never, not once, come across a discussion of Watson and Crick that did not also talk about Franklin's role.
Even when I was in high school back in the 80's, her contribution was front and center, along with the other two. Can we please let the myth that somehow her contribution is forgotten and not taught to die?
Always the headline is Watson and Crick. Not Franklin with some minor assistance from ...
Franklin is being brought into the "Watson and Crick" discussion. Rather than it being Franklin first, foremost and clearly most important. Let's not airbrush that her research was provided to Watson and Crick who claimed the glory without her consent.
No one forgets that her data was taken/stolen. The fact of the matter is that she didn't come up with the final structure given her brilliant experimental work and results. The history of science is littered with forgotten people who were scooped of the glory just short of the line. Franklin is very well recognised given that she didn't discover the structure of DNA.
Yes Yes Yes! One of the best math books I ever read, "Math for the Millions" by Lancelot Hogben taught each math topic as something that was discovered as civilization grew to have more complex problems.
Hearing their stories communicates to you the existence of the millenia-wide fellowship of truthseekers, and that what you are receiving is not just fodder for tests---it is your inheritance.
I'll be the devils advocate. Why? I can understand calculus perfectly fine without understanding how people got there. In fact, I'd argue much of what you learn before call--in the order of discovery--actively hinders understanding.
Because advancing science and mathematics is not a straight line but full frustrations inside a cloud of uncertainty and anxiety. It is important to appreciate what the world looked like before the discovery and how someone tackled a problem to change this view of the world because when you yourself make discoveries you will be in the exact same situation.
However if all you care is to use what has been discovered, which is becoming less and less valuable, then you don't need to learn history of mathematics and science.
Although this is not good evidence but rather an anecdote, I cannot remember any significant person who has made fundamental contributions to mathematics or science that was completely ignorant of the history of the field.
Let me give an example from calculus - a continuous function. How do you define the concept? The definiton of the concept changed quite a lot in the past 250 years or so. (Please take the following explanation with a grain of salt, I am not writing a thesis on the topic, just pointing out stuff.)
For Euler, continuous function was pretty much intuitive notion. He only composed functions with only occasional point discontinuities, so it wasn't a big deal for him to even not have a proper definition.
Then people like Bolzano and Dirichlet came along and realized they need a better definition, because there can be some really weird cases. So they formalized the continuity with limits (which is typical way how to define it in basic calculus).
Later yet, people understood better what it means to be a real number by looking at notions such as countability and measurability. While this doesn't affect continuity itself, it does affect understanding of what is a real fuction.
Then came more abstraction, to metric spaces and eventually topological spaces, which redefined "continuous function" yet again as a morphism between topological spaces.
Another shift in thinking about continuity happened when theory of distributions was invented. This actually completely reverses the intuition - instead of properly definining reals and then on top of real function define what it means to be continuous, you define the "function" itself in an entirely different way, in which the continuity becomes somewhat irrelevant.
Finally, modern mathematics is quite obsessed with category theory and various ways to make everything into some algebra. In a way, we care less what reals really are, only what we can do with them (or their sets).
So I think to understand the intuitive relation of all these different definitions, you need to understand a little bit of history.
Many people will find it easier to follow maths lessons if there are a few bits of history sprinkled in here and there. It adds a human element and honoring the great ones hundreds of years (or even millennia) after they are dead serves as an implicit demonstration of how important their discoveries are to us. Not a terribly powerful demonstration, but less futile than repeatedly yelling "hey, this is important!"
Also, mathematical concepts don't come with natural names attached. But we need consistent labels for successful communication. It's much easier to not confuse those labels if you know a bit about the history that led to the naming.
I strongly believe math should be taught with the science that motivated its development- calculus should be taught with classical mechanics. But I'm less sure the history of science is worthwhile.
The problem is opportunity cost. Schools (highschool, college intro physics) spend a great deal of time discussing previous models of the atom. What's the value of teaching Thomson's plum pudding model of the atom, really? They could start with the current model and list all of the observations/experiments that have shown the model to be useful. Previous models could be relegated to an appendix, or a history class.
This would free up time to more comprehensively discuss 20th and 21st century physics, which are sorely neglected at this level.
Yeah, calculus seems really abstract and pointless until you realise that it's 90% of physics. At school they shy away from calculus so much that you just do a version of physics where they've simplified away the calculus, which is extremely confusing.
I think there is value in teaching that there have been prior models (but more energy should be spent on the current thinking) because it shows that good scientists are constantly researching and revising their ideas as they get more information. Science is never done.
To be clear I agree there's value, I just don't think the value is as high as the other things that could be taught in that time.
I think appendices are really a good solution here. Those who are curious can read them- I know I would have. Those who aren't, can stick to focusing on the current model.
I'm not sure I agree. I think that understanding that knowledge is never done, and that science is about exploring new ideas and questioning our current understanding is much more important to the layperson than knowing how atoms are structured.
I also think that couching science as a journey--a mystery to be solved, with clues and red herrings along the way--will help to get students interested in learning the details. (This is all just my gut instinct; I have no experience in science education, so I might not know what I'm talking about).
The value of teaching Thomson's plum pudding model of the atom is to understand the evolution of ideas. It also helps the young minds understand that solving problems is often messy and we have to revisit our understanding of scientific models as new facts emerge. It might be obvious to you but the young students would feel that solutions would come in a single shot.
Ideally if the students have strong basics 20th and 21st century physics should be tried by the students themselves.
“Need”? Depends on the level, and maybe not. It can certainly be helpful though. I particularly recommend in primary school teaching about the history of counting boards / abacuses, the history of trigonometry, and most importantly the history of logarithms (every high school student should learn rudimentary use of a slide rule). All three will help clarify why particular tools and notations became standard, and give insight into pre-electronic-computer science and engineering more generally.
For anyone interested in the history of mathematics per se at the undergraduate level, I recommend Stillwell’s book, https://amzn.com/144196052X
More important than “mathematical history” is to teach students some measure of physics/engineering alongside the mathematics, to help motivate concepts.
I think the history of anything is important. It helps the student learn why things are significant, what they changed, and how things were before it occurred.
Does anyone have a good ref that reviews the history of dividing by zero? The current way of dealing with it just seems so ad hoc and dissatisfying... there has to be a long controversial history but searching around I find very little that isn't repeating the same stuff.
You mean that it's undefined? It actually makes perfect sense that it is left this way.
Consider addition and subtraction as functions:
f(x) = x + 42
g(x) = x - 42
Now, realize that these functions are inverses of each other. Each one undoes the operation of the former:
f(g(x)) = 42 + x - 42 = x
g(f(x)) = x + 42 - 42 = x
If you change 42 to any other real number, this relationship still holds. This is because f and g are functions that relate a number uniquely to another number. The inverse then, is just flipping that relationship (7 -> 49 has the inverse 49 -> 7).
You can define similar functions for multiplication and division, and things will work out in the same fashion until you try to replace 42 with 0. What happens then is that the multiplication relationship is no longer unique. For
f(x) = 0 * x
f(x) is 0 for any choice of x! This is a problem when we try to flip the relationship as before to invert the operation. Since f(7) = 0 and f(49) = 0, should g(0) be 7 or 49?
Moreover, the fact that the right and and left hand limit of 1/x as x -> infinity does not converge means that there's no obvious way to "plug the hole" so to speak and define 1/0 as some value.
I'm in the same boat as only finding repeating stuff; but for other's reference, here is the repeating stuff:
The earliest known attempt at dealing with division by zero comes from Brahmagupta in 628 (in the Brāhmasphuṭasiddhānta). His definition is as follows (non division-by-zero axioms ommited):
1) A non-zero number divided by 0 is a fraction with zero as the denominator.
2) Zero divided by a non-zero number is a fraction with zero as the numerator and the finite quantity as the denominator
Zero: The Biography of a Dangerous Idea by Charles Seife is one of the most engaging books I've read about mathematics. In its Amazon reviews, I also see The Nothing That Is: A Natural History of Zero by Robert Kaplan recommended.
I'm a mathematician and I'm not aware that there was a "long controversial history". (I'll ask a colleague who teaches history of math and see if she knows anything.) I have a lot of students who plan to teach, and I tell them that their students (and other people) will probably ask why "you can't divide by zero". It's important to explain what is meant by "can't". (Does it mean no one knows how to do it? Or that some mathematical authorities issued a decree? Of course, it's nothing like that. Those kinds of misunderstandings can arise because people don't understand how math is done by mathematicians, because it is done in ways that are different from the way we do things in everyday life.)
Anyway, the explanation is simple and quick enough that I can do it in any class where I'm discussing number systems (e.g. linear algebra, number theory, abstract algebra). The "tl;dr" is that if you want to "divide by 0" you will have to give up something else, and none of the things you have to choose from are things you'd want to give up.
In more detail, suppose you could "divide by 0". Division is defined as multiplying by the multiplicative inverse. (If you don't like that definition, you have to explain what you'll substitute as the definition of division - and note that, mathematicians want a definition that extends smoothly to "number systems" that may be very unfamiliar.)
So saying you can divide by 0 is the same as saying that 0 has a multiplicative inverse - call it 0^(-1). By definition of multiplicative inverse, 0 * 0^(-1) = 1.
On the other hand, in any reasonable number system (specifically, in any ring), 0 * x = 0 for any x. The proof is easy - it uses the definition of "0", the definition of additive inverse, the distributive law, and associativity of addition. (Try it!) Therefore, 0 * 0^(-1) = 0, so 0 = 1.
If this doesn't seem enough of a contradiction, just note that it follows from this (and the definition of "1") that x = 0 for all x. So the only "number" in the whole world is 0. Well, that makes life simple, but not very interesting.
So: If you want to "divide by 0", you're going to have to give up one of those algebraic axioms I mentioned. Which one would you give up? Associativity of addition? The distributive law?
I think it's really important to explain (particularly to kids learning math) that math is not a bunch of arbitrary rules. "Not dividing by 0" is not an arbitrary rule - it's a matter of making a trade-off.
>So: If you want to "divide by 0", you're going to have to give up one of those algebraic axioms I mentioned. Which one would you give up? Associativity of addition? The distributive law?
Just to expand on this, mathmaticians have done this; in several ways. For example, the Projetivly Extended real line, and Riemann sphere add "∞" to the Real or Complex numbers respectivly, such that 1/0=∞. Note that 0/0 remains undefined, ∞=-∞, 0∞ is undefined, and ∞ + ∞ is undefined (I am probably missing other "oddities" of these constructions).
There is also a more general way of defining division by 0, that avoid undefined instances: wheels.
As bikenaga mentions, the standard definition of a "reasonable" number system that involves addition and multiplication is a ring. In general, division by anything is not defined because elements are not guaranteed to have multiplicative inverses [0]. For example, the integers form a ring, but 5/3 is not defined in the integers.
If you add the following two properties to a ring, you get an integral domain: 1) Commutativity: xy = yx 2) if xy=0 then x=0 or y=0. Again, the integers are an example. For a non integral domain ring, consider the integers mod 4, where 2 * 2 = 0.
Once you have an integral domain, there is a standard way of defining division by any non-0 element: fractions. Informally, we that x^(-1) is the fraction 1/x, and a/b = x/y iff ay=bx. Addition and multiplication of fractions are defined as you learned in grade school. [1] As you would expect, applying this approach to the integers gives you the rational numbers. More formally, we defined the fraction x/y as the ordered pair (x,y).
To define a wheel, we modify the above construction slightly. Specifically, we say that a/b = x/y iff there exists an s,s' such that (sa,sb) = (s'x,s'y) or sa/sb = s'x/s'y [2].
Addition and multiplication remain unchanged, but we define a new operation for taking inverses: /(x,y) = (y,x). That is to say that, to take the "inverse" of an element, you swap the numerator and denominator.
In this system, we define 0 = (0,1) = 0/1 and 1 = (1,1) = 1/1.
Division by 0 is now a simple matter: 0/0 = (0,1)/(0,1) = (0,1)(1,0) = (0,0)
Notice that, under this construction, (0,0) is not the zero element; (0,1) is. Further, the equation
(0,0) + x = (0,1) has no solution.
If you keep poking at this structure, I am sure that you can find other bad things that happen.
[0] In fact, depending on who you ask, a ring is not even required to have a multiplicative identity (eg. 1).
[1] This construction gives you a structure known as a field; which is, in my opinion, the point where most non-mathematicians would start consider the algebraic structure to be a reasonable number system.
[2] Under this construction, we can also loosen the requirements of the underlying ring. Specifically, any commutative ring will do. We do not require that xy=0 implies x=0 or y=0.
The stuff on wheels is interesting! I found the paper by Carlstrom (http://www2.math.su.se/reports/2001/11/) and it alludes to applications, though I didn't see any specifics about applications (to computer science, at least). Thank you for the pointer.
If you avoid division by 0, wheels revert back to normal fractions [0]. This should mean that you can use them as a drop in replacement for a rational number datatype. Doing so should allow you to defer checking for division by zero; possible moving the check outside of a tight loop. Granted, this should also be doable as an optimization of normal rational arithmetic.
The only use cases I can think of for wheels amount to them being a principled way of adding NaN to the number system. Of course, if history is anything to go by, a hundred years from now someone may look back on this comment the same way we look back on people calling sqrt(i) "imaginary".
[0] At least under the explicit construction presented.
This conversation may have moved well above my head, but I mean start with 0/0 = "undefined". I don't see why that should be the case.
x = 0/0
x*0 = 0
1*0 = 0
2*0 = 0
...
From this we see that it isn't really that x is undefinable, rather it can be any value at all. There is apparently no issue with an equation having two equally valid solutions (eg quadratic formula), so at what point are there too many?
I think I see what you're asking. I think the answer is: Before we start talking, you have to tell me what all the rules are. So when you say "start with 0/0 = 'undefined'", what is the ambient number system? The proof I gave earlier showed the number system can't be a ring - so it's not the real numbers, the integers, the rationals ... at least not by the standard definitions of those number systems. When you do math, you don't make up the rules as you go.
As far as the example you gave goes, you start by assuming 0/0 is defined. But if you're trying to show 0/0 is defined you're assuming what you want to prove. The logic isn't correct.
Note that giving 0/0 the name "x" doesn't do anything.
Simply naming something doesn't establish any fact. It just makes "x" shorthand for "0/0".
Anyway, observing that 0/0 * 0 = 0 but also 1 * 0 = 0, 2 * 0 = 0, and so on doesn't establish any necessary connection between "0/0" and 1, 2, ... You wouldn't conclude from "1 * 0 = 0" and "2 * 0 = 0" that "1 = 2", or that "1 could be 2", for instance. So nothing has happened. But what could happen? Remember that you started by assuming that "0/0" was defined. "Assume" in math means you've assumed it's true. In that case, you're done, right? Its "definedness" isn't probabilistic. And if starting with that assumption you did find out something true, it doesn't follow that the assumption is "independently" true. (The truth of "if P, then Q" and the truth of "Q" do not together imply the truth of "P".)
You might want to look at the post on wheels higher up this thread. It shows what you could do - namely, use a different set of rules.
Wow, this was a really interesting explanation. I think this is perfectly appropriate for junior school while the explanation about non-converging limits offerred above by someone is well suited to college students.
no. the math curriculum is too crowded already. i liked reading dirk struik, but the selfishness of mathematicians and the patrons they served is only entertainment in the end. want to update the curriculum? then replace some geometry and calculus with statistics.
no. the math curriculum is too crowded already. i liked reading dirk struik, but the selfishness of mathematicians and the patrons they served is only entertainment in the end. want to update the curriculum? then replace some geometry and calculus with statistics.
Maybe not for math, but I think a case could definitely be made for programming. If I had understood the motivation behind Unicode and its full implications, I would have spent much less time suffering in confusion about encoding errors.
Unicode is a terrible mess. I remember reading about it with enthusiasm in the nineties, but it didn't take long to realize the committee of designers simply punted on every difficult decision instead of making a stand. They dumped everything on the programmers. At this point, I think it would be less work to convert 7 billion people to using ascii than it would to enable 10 million programmers to use that pile of a standard robustly. History lessons won't help us here.
> committee of designers simply punted on every difficult decision instead of making a stand
Are you sure that it was a bad thing? If they made a stand, we would now had to use (arguably worse) UCS2 or UCS4 encoding instead of UTF-8 (which de facto won).
UTF-8 is a nifty compression scheme - like a Huffman encoding that doesn't require a table to implement, but the committee doesn't deserve any credit for that (unless Thompson and Pike were on the committee, which I doubt).
Besides, the reason most of us actually like UTF-8 is because it leaves ascii alone (which is all I ever use) while pretending to handle the general case. It doesn't help end users or programmers deal with any of the nonsense around multiple ways to encode glyphs (combining codes vs accented codes), deal with surrogates (yes, people encode surrogates in UTF-8), lexical sorting, or anything else. I'll bet there are dozens of incompatible ways strings are UTF-8 encoded in the real world, each of them a bug for interoperability, and all of that blame falls on Unicode being a terrible standard.
I didn't develop a love for mathematics until I picked up Mathematics for the Nonmathematician at my local library. Having some historical context to what we were learning would have had a more profound impact on me during high school.
Yes. Probably the most inspiring feature of my first number theory course was a healthy dose of history: about Gauss, Sophie Germain, Riemann, Hardy, Ramanujan. It was helpful, psychologically, to see the process that gave rise to number theory, instead of viewing its ideas as God-given.
Sometimes in math the history provides helpful context and motivation, and when people leave it out it makes things confusing. I'm not going to elaborate on this because I assume most people here already agree with this!
But sometimes the modern way is so much cleaner and better, so that even if you do want to learn the history, everything will probably be easier to understand if you learn the modern way first and know in advance what truth it is that they were working their way towards. Sometimes the historical way is just awful.
(When I took representation theory in college, the professor thought it would be funny to at one point show us the original definition of an irreducible character. Nobody should ever have to learn representation theory in such a way!)
This website does a really good job explaining complex calculus ideas to me. One of which is 'e', where the history and its origins will help you understand why it behaves the way it does. Without this context, 'e' is just some arbitrary number that you have to memorize. This is solely why I believe that history of math should definitely be taught.
Finance is also another area where history should be taught since ideas like continuously compounded interest was a recent development, and it's history explains why anyone wanted to compound continously.
Kalid from BetterExplained here, glad you're enjoying it!
I often go to the history of a concept for both historical appreciation and, practically, to understand it better. We often study Calculus without looking at what Archimedes was able to do -- break shapes into smaller parts -- and without really seeing how the notion of infinitesimals come into play. (Epsilon/delta definitions don't give the same insight.)
The natural log was discovered before e -- why would that be? Sine, cosine, trig functions -- they started as measurements of triangles, evolved into analytic definitions of their own -- why does this progression make sense?
In my mind, truly understanding a concept means you understood the path it took to the current state.
Oh wow, this is why I love Hacker News - thank you for helping me overcome my mathematical anxieties in college as I was working on my Computer Science major. Your site really helped me gain an appreciation for math since I had terrible teachers as a kid and led me to believe that I could never 'get it'. Would love to see some more linear algebra materials as well
> One of which is 'e', where the history and its origins will help you understand why it behaves the way it does. Without this context, 'e' is just some arbitrary number that you have to memorize.
What? I have no idea about the history of `e` (though I have a good idea who "found" it) but it is incredibly important and it's pretty easy to explain why -- if you have a dollar in the bank, earning interest at an instantaneous rate of 100% per annum, but continuously compounded, `e` is the amount of money you'd have at the end of one year.
Very, very simple. When the rate of change is equal to the quantity itself, `e` is the thing you exponentiate.
Absolutely yes!
Without history students are given an often misleading view of how math constructs evolve. Not only does historical context help some students learn and accept certain math concepts (helping answer the 'why?'), but it also provides insight as to how math research is actually carried-out.
Absolutely yes!
Without history students are given an often misleading view of how math constructs evolve. Not only does historical context help some students learn and accept certain math concepts (helping answer the 'why?'), but it also provides insight as to how math research is actually carried-out.
The fact that mathematics definition are a construction and not something that's absolutely obvious, and also that notations have evolved and so are a bit arbitrary as well is absolutely crucial to demystify mathematics. I think the reason mathematicians still wonder if it's necessary is because it would only show that maths are a human construction and not a heaven's revelation, and so makes math more mundane.
The danger is that incorporating history is very easily to do badly. I remember various science lessons at school which were made duller and more confusing by the teacher taking a historical approach.
We have better mathematical notation, explanations etc these days that make various topics far easier to understand. Try reading most old mathematics books and it's a tough experience. It's certainly also possible to have a good understanding of mathematics without really knowing much about the history e.g. I understand Galois Theory but I really don't know much at all about the history of trying to solve polynomials or indeed about Galois himself other than he went and got himself killed in a duel. In fact when I was younger, too much history might have put me off the subject altogether. I'm old enough now to appreciate history but I wasn't when I was at school.
Instead, I think a better approach is to identify parts of mathematics that teachers struggle to teach either in terms of concepts or motivation. Then one can look more carefully at those and see whether examples from history (or indeed other contexts) might help, rather than necessarily using history as the starting point. I think the article may be in accord here, but it is so easy for the message to get interpreted differently by teachers.
I'm skeptical that adding more history to mathematics education would increase motivation. If mathematics has a contender as the most boring and dreaded subject, it's history. For motivation, tie in pop cultural references and show applications that students actually find interesting. But historical context could be useful on other fronts. E.g., discussing the arguments from historical debates probably helps to solidify certain ideas.
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[ 3.5 ms ] story [ 180 ms ] thread1. It's much more interesting to study something when you have a historical perspective that you can relate to later on in the course.
2. It makes you comfortable about the idea that such developments in mathematics are made by fellow humans only and they can also do such things if they put in the efforts. This might sound like a small addition but it's detrimental in developing such sense in young kids.
I think that explaining the thought process is even more important than the results.
My view is that school should help kids find their thought process. Will likely be similar to successful thought processes. Didn't have to be, though.
The most instructive thing I have ever heard, was seeing that Feynman made his own notations for learning math tricks in high school. Fur some reason, this really made me regret not trying new ways of things.
There was an article recently on the importance of notation. Making your own goes a long way to understanding others.
Do Doctors Need to be Taught the History of Medicine?
Do math education specialists need to be taught the history of the debate over whether mathematics education should include the history of mathematics?
Most of my school mates had zero understanding of the mathematics we learned. They memorised how to compute the distance between a point and a sphere, a line and a sphere, a plane and a sphere, and then they'd solve one of these problems for the exam, get a good grade, and forget everything immediately afterwards. At no point have they understood what a scalar product is and why they are using it.
If you'd really want to improve mathematics education, you should focus a lot more on the basics, and teach advanced subjects only to the handful of students that are interested in them.
Unless you're a mathematician, math should be viewed as a tool, like a lathe. You don't need to know the history of the lathe, and how Maudslay made it a precision machine tool. (His original lathe is in the Kensington Science Museum. It's one of those historic artifacts which looks very different from its predecessors, and it's successors look a lot like it.)
Few people need to know how to build up mathematics from minimal axioms. Nobody should have to struggle through Whitehead and Russell below the PhD level. We have power tools for that now. The original Boyer-Moore theorem prover from 1992 can build up constructive number theory from the axioms in under a minute.[1] I fixed it up recently to run on GNU Common Lisp and put it on Github, so it's runnable on modern machines.
There's certainly no excuse for inflicting Newton's notation for calculus on kids. It's not even clear that classical geometry proof approaches are that useful.
[1] https://github.com/John-Nagle/nqthm
Of course it's very important to study the history, but probably not in the first presentation. But at the same time, it's definitely true, for most subjects, that historical references often help to motivate less committed students and enliven the experience.
On Maudslay's role in the development of machine tools, this paper looks interesting:
FT Evans "The Maudslay Touch: Henry Maudslay, Product of the Past and Maker of the Future" http://www.tandfonline.com/doi/abs/10.1179/tns.1994.007?jour...
Teaching a bit of history alongside helps students understand why it takes that particular form.
The word “sine” comes from a weird Latinization of an Indian word for “half a bowstring”. Draw a picture of a circle with a vertically oriented chord (the word “chord” also implies a bowstring), and a student will have a much easier time remembering what the sine is. Likewise tangent (Latin for touching) and secant (Latin for cutting) make more sense if you think about the meanings of the words. Cosine means the sine of the complementary angle, etc.
The reason we call inverses “arcsine”, etc. is because originally these were written as quasi-sentences, and the concept of a mathematical function was not well developed. So sin⁻¹ x would be expressed as something like: arc (sin. = x). That is, the length of the arc whose sine is x. This form was cumbersome so later got shortened to arcsin x.
The origins of trigonometry are in astronomical measurement, which is why we have 360° in a circle (each degree is roughly one day of movement (365 days/year), rounded to a nearby highly composite number), and come from the Sumerian/Babylonian numerical tradition which used a base sixty number system. Hence “first minutes”, “second minutes”, “third minutes”, etc. of a degree. “Minute” (Latin for small) implies 1/60 of the larger unit.
The reason trigonometry focuses on learning a big pile of formulas is because before the era of electronic calculators, people needed to do all computations by hand, or by interpolating in pre-computed lookup tables. The goal of “trigonometry” is to take a given problem and convert it to a form with the easiest hand computation and the fewest table lookups possible, so that the mechanical work can be handed off to a team of human computers who can go through the laborious arithmetic. Memorizing trigonometry formulas is a way to cut the work done by the human computers to a small fraction of what it might take for the original problem as posed.
Trigonometry was important in science/engineering because until recently the abstract vector concept and idea of combining simple single-number parts into “complexes” were not well developed. People solved problems by breaking them into coordinates and discrete lengths and angles. Solving triangles and converting between polar/cartesian coordinates were important steps in almost any 2-dimensional problem.
If we really wanted to avoid the “ontogeny recapitulates phylogeny” model, we would scrap the current form of trigonometry (certainly not spend 4+ months exclusively focusing on it) and set the high-level ideas on a more logical foundation which was easier to learn and reason about, ditching the parts now anachronistic in an electronic computer age. We would give students harder problems to solve and fewer formulas to memorize. But that could leave students unfamiliar with the existing language commonly used in the existing literature, so to some extent we’re stuck by our history. http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf
Any pointers to resources about those "quasi-sentences"? I'm interested in language, broadly speaking, so info about how mathematical notation evolved is interesting to me. The rest of your comment is great to, I'd love to read a book about stuff like this if there is one!
https://smile.amazon.com/History-Mathematical-Notations-Dove...
- Get a clear notion of what you desire to accomplish, then you will probably get it.
- Keep a sharp look-out upon your materials: Get rid of every pound of material you can do without. Put yourself to the question, ‘What business has it there?’
- Avoid complexities and make everything as simple as possible.
- Remember the get-ability of parts.
“It was a pleasure to see him handle a tool of any kind, but he was quite splendid with an 18-inch file.”
Essentially this is saying that you can use the sequence of historical discovery as a guide to the appropriate sequence of lies-to-children [0].
[0] https://en.wikipedia.org/wiki/Lie-to-children
For full mastery of a field, sure it helps to know the historical record. Certainly, I think there are little historical tidbits that can keep a class more interesting and put some things in context. However, by in large, I often see people use completely terrible algorithms because they were taught in what I consider a historical manner and they did not have the time, nor perhaps the instruction, to learn more modern methods, which can often be formulated in an independent way without regard to older algorithms. Fields evolve and sometimes we realize there's just flat out better ways of doing things that don't need earlier results.
Worth considering, is that if this practice gained widespread appeal, the quality of how it is done (by the best and probably by the majority) would improve.
For my opinion, I think the idea "incorporate history" is just the most obvious first attempt at more story telling in mathematics. If these "stories" are not a part of the classroom then they may come too late or never for the majority.
I do agree that math is a tool, but it should be taught with more rhetorical tools. To trade a little into your knowledge space, think about the fondness many have for "The Little Schemer".
My math teacher taught me not only calculus, but also when and who got up to what. It doesn't have to take a long time, but a bit of context helps a lot. Euler went to Russia and yada yada bridges graphs etc.
Same goes for all science disciplines. You need to have a rough idea that Darwin worked in the 19th century, that much of thermo came about in the late 19th century, that quantum is a 20th century thing. You need to know what people were wondering about, and what experiments they came up with.
I've been listening to a lot of audio courses lately, and those little nuggets really help to understand things.
The point of the little stories such as how Watson and Crick came up with the double helix is to help recall. It's hard to remember dry facts, much easier to remember stories. People are kinda built that way.
I know little of this history, but is there any link between him going to Russia and him solving the seven bridges of Königsberg problem?
I'm asking because I don't know whether Euler ever visited Königsberg and because Königsberg is in Russia now (renamed to Kaliningrad after WWII), but was in Prussia at the time.
b.2.b One who drives a vehicle or the animal that draws it; a charioteer, coachman, cabman, etc.; also, one who drives a locomotive engine. (Often with defining word prefixed, as cab-driver, engine-driver, etc., for which see the first element.) [...]
c 1450 St. Cuthbert (Surtees) 6016 All þe dryuers ware agaste þat þe sledd suld ga our faste. 1581 Savile Tacitus 93 (R.) Buffons, stage-players, and charet drivers. 1725 Pope Odyss. xiii. 99 Fiery coursers in the rapid race Urg'd by fierce drivers thro' the dusty space. [...]
Yes, he was in Switzerland, in Basel. Googled a bit. http://www-groups.dcs.st-and.ac.uk/history/Biographies/Euler...:
"Euler left Basel on 5 April 1727. He travelled down the Rhine by boat, crossed the German states by post wagon, then by boat from Lübeck arriving in St Petersburg on 17 May 1727."
Still inconclusive, but given that we know this, I would say chances are non-zero that the sources this was derived from spell out where that ship made stops.
One could also look at his journeys from Leningrad to Berlin and, years later, back, but both were after his publication on the 7 bridges problem.
s/came up with/stole the idea from looking at Rosalind Franklin's X-ray data/
To be fair, both Watson and Crick strongly insisted she also get the Nobel, but she'd tragically passed a way shortly before the award was announced, and the rules are very clear about no posthumous awards. Still, it's sad that we don't teach Franklin's role.
Even when I was in high school back in the 80's, her contribution was front and center, along with the other two. Can we please let the myth that somehow her contribution is forgotten and not taught to die?
Franklin is being brought into the "Watson and Crick" discussion. Rather than it being Franklin first, foremost and clearly most important. Let's not airbrush that her research was provided to Watson and Crick who claimed the glory without her consent.
So yeah, Franklin. Franklin is double helix.
Edit: name typo
Oddly, Wilkins is always left out too. That trio won the Nobel.
Maybe Watson and Crick were just better at selling themselves.
Each episode tends to discuss historical, political, and economic conditions in addition to the primary topic.
At some point I stumbled onto stories about newton, leibniz, and everything that drove them. It was riveting and brought the magic back.
https://www.youtube.com/watch?v=dW8Cy6WrO94&index=1&list=PL3...
However if all you care is to use what has been discovered, which is becoming less and less valuable, then you don't need to learn history of mathematics and science.
Although this is not good evidence but rather an anecdote, I cannot remember any significant person who has made fundamental contributions to mathematics or science that was completely ignorant of the history of the field.
For Euler, continuous function was pretty much intuitive notion. He only composed functions with only occasional point discontinuities, so it wasn't a big deal for him to even not have a proper definition.
Then people like Bolzano and Dirichlet came along and realized they need a better definition, because there can be some really weird cases. So they formalized the continuity with limits (which is typical way how to define it in basic calculus).
Later yet, people understood better what it means to be a real number by looking at notions such as countability and measurability. While this doesn't affect continuity itself, it does affect understanding of what is a real fuction.
Then came more abstraction, to metric spaces and eventually topological spaces, which redefined "continuous function" yet again as a morphism between topological spaces.
Another shift in thinking about continuity happened when theory of distributions was invented. This actually completely reverses the intuition - instead of properly definining reals and then on top of real function define what it means to be continuous, you define the "function" itself in an entirely different way, in which the continuity becomes somewhat irrelevant.
Finally, modern mathematics is quite obsessed with category theory and various ways to make everything into some algebra. In a way, we care less what reals really are, only what we can do with them (or their sets).
So I think to understand the intuitive relation of all these different definitions, you need to understand a little bit of history.
Also, mathematical concepts don't come with natural names attached. But we need consistent labels for successful communication. It's much easier to not confuse those labels if you know a bit about the history that led to the naming.
If you know that people didn't know some science/math at a time, it makes it clearer why they acted in some way.
The problem is opportunity cost. Schools (highschool, college intro physics) spend a great deal of time discussing previous models of the atom. What's the value of teaching Thomson's plum pudding model of the atom, really? They could start with the current model and list all of the observations/experiments that have shown the model to be useful. Previous models could be relegated to an appendix, or a history class.
This would free up time to more comprehensively discuss 20th and 21st century physics, which are sorely neglected at this level.
I think appendices are really a good solution here. Those who are curious can read them- I know I would have. Those who aren't, can stick to focusing on the current model.
I also think that couching science as a journey--a mystery to be solved, with clues and red herrings along the way--will help to get students interested in learning the details. (This is all just my gut instinct; I have no experience in science education, so I might not know what I'm talking about).
Ideally if the students have strong basics 20th and 21st century physics should be tried by the students themselves.
For anyone interested in the history of mathematics per se at the undergraduate level, I recommend Stillwell’s book, https://amzn.com/144196052X
More important than “mathematical history” is to teach students some measure of physics/engineering alongside the mathematics, to help motivate concepts.
Consider addition and subtraction as functions:
Now, realize that these functions are inverses of each other. Each one undoes the operation of the former:f(g(x)) = 42 + x - 42 = x g(f(x)) = x + 42 - 42 = x
If you change 42 to any other real number, this relationship still holds. This is because f and g are functions that relate a number uniquely to another number. The inverse then, is just flipping that relationship (7 -> 49 has the inverse 49 -> 7).
You can define similar functions for multiplication and division, and things will work out in the same fashion until you try to replace 42 with 0. What happens then is that the multiplication relationship is no longer unique. For
f(x) is 0 for any choice of x! This is a problem when we try to flip the relationship as before to invert the operation. Since f(7) = 0 and f(49) = 0, should g(0) be 7 or 49?Moreover, the fact that the right and and left hand limit of 1/x as x -> infinity does not converge means that there's no obvious way to "plug the hole" so to speak and define 1/0 as some value.
The earliest known attempt at dealing with division by zero comes from Brahmagupta in 628 (in the Brāhmasphuṭasiddhānta). His definition is as follows (non division-by-zero axioms ommited):
1) A non-zero number divided by 0 is a fraction with zero as the denominator.
2) Zero divided by a non-zero number is a fraction with zero as the numerator and the finite quantity as the denominator
3) Zero divided by zero is zero.
Anyway, the explanation is simple and quick enough that I can do it in any class where I'm discussing number systems (e.g. linear algebra, number theory, abstract algebra). The "tl;dr" is that if you want to "divide by 0" you will have to give up something else, and none of the things you have to choose from are things you'd want to give up.
In more detail, suppose you could "divide by 0". Division is defined as multiplying by the multiplicative inverse. (If you don't like that definition, you have to explain what you'll substitute as the definition of division - and note that, mathematicians want a definition that extends smoothly to "number systems" that may be very unfamiliar.)
So saying you can divide by 0 is the same as saying that 0 has a multiplicative inverse - call it 0^(-1). By definition of multiplicative inverse, 0 * 0^(-1) = 1.
On the other hand, in any reasonable number system (specifically, in any ring), 0 * x = 0 for any x. The proof is easy - it uses the definition of "0", the definition of additive inverse, the distributive law, and associativity of addition. (Try it!) Therefore, 0 * 0^(-1) = 0, so 0 = 1.
If this doesn't seem enough of a contradiction, just note that it follows from this (and the definition of "1") that x = 0 for all x. So the only "number" in the whole world is 0. Well, that makes life simple, but not very interesting.
So: If you want to "divide by 0", you're going to have to give up one of those algebraic axioms I mentioned. Which one would you give up? Associativity of addition? The distributive law?
I think it's really important to explain (particularly to kids learning math) that math is not a bunch of arbitrary rules. "Not dividing by 0" is not an arbitrary rule - it's a matter of making a trade-off.
Just to expand on this, mathmaticians have done this; in several ways. For example, the Projetivly Extended real line, and Riemann sphere add "∞" to the Real or Complex numbers respectivly, such that 1/0=∞. Note that 0/0 remains undefined, ∞=-∞, 0∞ is undefined, and ∞ + ∞ is undefined (I am probably missing other "oddities" of these constructions).
There is also a more general way of defining division by 0, that avoid undefined instances: wheels.
As bikenaga mentions, the standard definition of a "reasonable" number system that involves addition and multiplication is a ring. In general, division by anything is not defined because elements are not guaranteed to have multiplicative inverses [0]. For example, the integers form a ring, but 5/3 is not defined in the integers.
If you add the following two properties to a ring, you get an integral domain: 1) Commutativity: xy = yx 2) if xy=0 then x=0 or y=0. Again, the integers are an example. For a non integral domain ring, consider the integers mod 4, where 2 * 2 = 0.
Once you have an integral domain, there is a standard way of defining division by any non-0 element: fractions. Informally, we that x^(-1) is the fraction 1/x, and a/b = x/y iff ay=bx. Addition and multiplication of fractions are defined as you learned in grade school. [1] As you would expect, applying this approach to the integers gives you the rational numbers. More formally, we defined the fraction x/y as the ordered pair (x,y).
To define a wheel, we modify the above construction slightly. Specifically, we say that a/b = x/y iff there exists an s,s' such that (sa,sb) = (s'x,s'y) or sa/sb = s'x/s'y [2].
Addition and multiplication remain unchanged, but we define a new operation for taking inverses: /(x,y) = (y,x). That is to say that, to take the "inverse" of an element, you swap the numerator and denominator.
In this system, we define 0 = (0,1) = 0/1 and 1 = (1,1) = 1/1.
Division by 0 is now a simple matter: 0/0 = (0,1)/(0,1) = (0,1)(1,0) = (0,0)
Notice that, under this construction, (0,0) is not the zero element; (0,1) is. Further, the equation
(0,0) + x = (0,1) has no solution.
If you keep poking at this structure, I am sure that you can find other bad things that happen.
[0] In fact, depending on who you ask, a ring is not even required to have a multiplicative identity (eg. 1).
[1] This construction gives you a structure known as a field; which is, in my opinion, the point where most non-mathematicians would start consider the algebraic structure to be a reasonable number system.
[2] Under this construction, we can also loosen the requirements of the underlying ring. Specifically, any commutative ring will do. We do not require that xy=0 implies x=0 or y=0.
The only use cases I can think of for wheels amount to them being a principled way of adding NaN to the number system. Of course, if history is anything to go by, a hundred years from now someone may look back on this comment the same way we look back on people calling sqrt(i) "imaginary".
[0] At least under the explicit construction presented.
As far as the example you gave goes, you start by assuming 0/0 is defined. But if you're trying to show 0/0 is defined you're assuming what you want to prove. The logic isn't correct.
Note that giving 0/0 the name "x" doesn't do anything. Simply naming something doesn't establish any fact. It just makes "x" shorthand for "0/0".
Anyway, observing that 0/0 * 0 = 0 but also 1 * 0 = 0, 2 * 0 = 0, and so on doesn't establish any necessary connection between "0/0" and 1, 2, ... You wouldn't conclude from "1 * 0 = 0" and "2 * 0 = 0" that "1 = 2", or that "1 could be 2", for instance. So nothing has happened. But what could happen? Remember that you started by assuming that "0/0" was defined. "Assume" in math means you've assumed it's true. In that case, you're done, right? Its "definedness" isn't probabilistic. And if starting with that assumption you did find out something true, it doesn't follow that the assumption is "independently" true. (The truth of "if P, then Q" and the truth of "Q" do not together imply the truth of "P".)
You might want to look at the post on wheels higher up this thread. It shows what you could do - namely, use a different set of rules.
This may be more than you wanted to know ...
Are you sure that it was a bad thing? If they made a stand, we would now had to use (arguably worse) UCS2 or UCS4 encoding instead of UTF-8 (which de facto won).
Besides, the reason most of us actually like UTF-8 is because it leaves ascii alone (which is all I ever use) while pretending to handle the general case. It doesn't help end users or programmers deal with any of the nonsense around multiple ways to encode glyphs (combining codes vs accented codes), deal with surrogates (yes, people encode surrogates in UTF-8), lexical sorting, or anything else. I'll bet there are dozens of incompatible ways strings are UTF-8 encoded in the real world, each of them a bug for interoperability, and all of that blame falls on Unicode being a terrible standard.
So yes, I'm sure.
Sometimes in math the history provides helpful context and motivation, and when people leave it out it makes things confusing. I'm not going to elaborate on this because I assume most people here already agree with this!
But sometimes the modern way is so much cleaner and better, so that even if you do want to learn the history, everything will probably be easier to understand if you learn the modern way first and know in advance what truth it is that they were working their way towards. Sometimes the historical way is just awful.
(When I took representation theory in college, the professor thought it would be funny to at one point show us the original definition of an irreducible character. Nobody should ever have to learn representation theory in such a way!)
This website does a really good job explaining complex calculus ideas to me. One of which is 'e', where the history and its origins will help you understand why it behaves the way it does. Without this context, 'e' is just some arbitrary number that you have to memorize. This is solely why I believe that history of math should definitely be taught.
Finance is also another area where history should be taught since ideas like continuously compounded interest was a recent development, and it's history explains why anyone wanted to compound continously.
I often go to the history of a concept for both historical appreciation and, practically, to understand it better. We often study Calculus without looking at what Archimedes was able to do -- break shapes into smaller parts -- and without really seeing how the notion of infinitesimals come into play. (Epsilon/delta definitions don't give the same insight.)
The natural log was discovered before e -- why would that be? Sine, cosine, trig functions -- they started as measurements of triangles, evolved into analytic definitions of their own -- why does this progression make sense?
In my mind, truly understanding a concept means you understood the path it took to the current state.
Yeah, my only linear algebra content is:
General Overview:
https://betterexplained.com/articles/linear-algebra-guide/
Matrix Multiplication for Programmers:
https://betterexplained.com/articles/matrix-multiplication/
Hoping to flesh it out over time.
I've always thought that betterexplained was a great resource, and was wondering whether there was any way I could contribute.
https://aha.betterexplained.com - forum to discuss ideas. It's in the background currently but I'd like to make it into more of a public place.
If you like, shoot me an email, I'm putting together a list of people who might want to be contributors. Thanks!
https://betterexplained.com/contact
What? I have no idea about the history of `e` (though I have a good idea who "found" it) but it is incredibly important and it's pretty easy to explain why -- if you have a dollar in the bank, earning interest at an instantaneous rate of 100% per annum, but continuously compounded, `e` is the amount of money you'd have at the end of one year.
Very, very simple. When the rate of change is equal to the quantity itself, `e` is the thing you exponentiate.
We have better mathematical notation, explanations etc these days that make various topics far easier to understand. Try reading most old mathematics books and it's a tough experience. It's certainly also possible to have a good understanding of mathematics without really knowing much about the history e.g. I understand Galois Theory but I really don't know much at all about the history of trying to solve polynomials or indeed about Galois himself other than he went and got himself killed in a duel. In fact when I was younger, too much history might have put me off the subject altogether. I'm old enough now to appreciate history but I wasn't when I was at school.
Instead, I think a better approach is to identify parts of mathematics that teachers struggle to teach either in terms of concepts or motivation. Then one can look more carefully at those and see whether examples from history (or indeed other contexts) might help, rather than necessarily using history as the starting point. I think the article may be in accord here, but it is so easy for the message to get interpreted differently by teachers.