A way to think about this process: "math" isn't quite a tower of abstract concepts; those only exist in any given mathematician's head, and die with them. Instead, "math" is a name for the novel pieces of language we invent, with the purpose of using them to talk about and (hopefully) share those internal abstract concepts, exposing to the rest of the world concepts that were entirely inaccessible (to everyone but the originator) before that language was introduced. Math is the attempt to communicate never-before-communicated epiphanies; to describe the shapes of never-before-pondered abstractions, with properties only describable using never-before-spoken words.
One thing that this implies, is that the first piece of language that lets us even vaguely point at some particular idea so that we can get to work on analyzing it further, might stick around long after we come up with some clearer or more coherent language, because the former one now has the momentum of historical use behind it.
Math, when done this way, then becomes a precarious pile of "just good enough to survive" conceptualizations, rather than a precise tower of "best-tool-for-the-job" conceptualizations. And that's no good for teaching.
I'm keen for etymologies and hence believe that math means the art of problemsolving or learning successfully. That includes language as a problem domain, and teaching tool for learning, but as modern development would have it, it's about structure and organization, not just in language.
I was just trying to highlight the fact that mathematics as an institution is a process of building up share-able symbolic abstractions; of inventing a "language" one new word at a time.
I'm using "language" here, and above, to refer not to the words used to discuss mathematics (the... "mathematology" of math), but rather to the thing that includes objects like mathematical operators (e.g. "+", "⨯", "∫", "⇒") as its "words." Not the language about mathematics, but rather the language that is mathematics: the ever-growing set of abstract tools with symbolic handles which we've constructed to allow us to manipulate other concepts inside our heads, in rigorous ways where you can trust that if you and another mathematician do the same named mental 'move' to the same source concept, you will both arrive at the same destination concept as a result.
For a cute analogy: you can think of a martial art as a vocabulary of known, precise body movements, that can be taught. You can think of mathematics as a vocabulary of known, precise mental movements, that can be taught. Yes, this makes mathematics an art; but, equivalently, this makes a martial art a language.
To sum up:
• Mathematics is itself a (formal) language. It doesn't really fit in the category of words like "biology"; it fits more in with words like "logic" or "C++".
• To say that someone is "doing" Mathematics just means that they are using that language to achieve a goal; it's about the same as saying that someone is "doing" Python.
• To say someone is a mathematician, is to say that that person works to explore and extend the language of Mathematics, to test its properties and its limits, and to invent new 'words' within it that may then be used by those "doing" Mathematics.
Math is where we invent language to refer to new forms of abstraction themselves: novel possible shapes for our thoughts to take when we think about other things. You can't talk about a new shape by analogy to existing shapes. Nor can you abstract an abstraction in a way that gives you anything more familiar. (Instead, you'll usually get more novel, ontologically-primitive abstractions, like going from monads to arrows, or going from numbers to fields to rings.)
Sometimes disciplines like physics will find a concept that we don't have any mental tools in our toolkit for yet. Then we build some. But we still refer to the process of doing that as doing mathematics—and then we apply that new mathematics back in the problem domain to talk about the new concept.
(For a good example, the formalization of quantum theory in physics, required the creation of infinite dimensional analysis in mathematics. Physicists did most of that work, but the work itself was still mathematics, not physics.)
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Now, other fields do still have a similar problem to mathematics, of historical momentum carrying forward old "things" (again: concepts, relationships, laws) when there are better, clearer "things" that could be used in their place. But when we're not working with pure abstractions—ways of thinking—we can make the effort to compare and contrast old and new "things", and decide that some might be more edifying than others.
It is possible for an especially-gifted physics teacher to write a very accessible physics textbook, because they need only pick all the clearest "things" to demonstrate. That teacher will still be stuck in a given paradigm—a way of thinking, a belief in the worth of some "things" over others, popular in the culture of their discipline at the time and place they worked. But they might be able to (barely) rise above it, if they think hard about the history of their discipline and the paradigms it has gone through in the past, compare-and-contrast those, and synthesize something that isn't quite just the paradigm they're immersed in.
Mathematics is uniquely problematic because it is entirely paradigm. It is a tree of paradigms—each new abstraction only making sense assuming the paradigm it was created in, and then becoming the paradigm for further abstractions still. Every mathematician, all the time, is trying to discover what a particular paradigm—their specialization—can be twisted to accomplish. Not one of us has the brain power to know the total space of things that one of these ways of thinking can be used to express—the problem domains the tool is applicable to—in order to know which tools show more or less promise at being "powerful." We know what we've discovered so far, but we have such an infinitesimal idea of the "space of all possible abstractions" that we could be totally missing some of the best, and using ones that are barely satisfactory.
Point a hypercomputer AI at "solving physics", and it'll spit out a description of the universe that will certainly have more "things" in it than we know about today—but which will still also contain a subset of the "things" we do have. (The most "carving nature at its joints"ing ones, presumably.) Those "things" that get carried over will, of course, be defined much more precisely; their concept-boundaries will be adjusted to include or exclude edge-cases we weren't aware of. But they'll be the same "things."
Point a hypercomputer AI at "solving mathematics"—of giving us the most powerful abstractions possible to solve problems with—and the result might be entirely unlike anything we've invented so far. It might not share any resemblance to current mathematics, beyond a shared fondness for sets. (For a silly hypothetical: it might turn out that there's a better abstraction for thinking about relationships between things than "functions", and that all the rest of mathematics and physics and whatever other problem domains simplify greatly if we think about the relationships in them in terms of that abstraction, instead of trying to specify relationships using the "function" abstraction.)
From "A Mathematician's Apology", G. H. Hardy:
> Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain.
From another angle: when developing new theories or models, your thoughts are all over the place and frankly it's boring to go over your own crappy notes afterwards and try to reconstruct them in a way that others can understand. And much of the time you forget exactly what happened along the way as well, so any story you reconstruct is going to have some hindsight bias, which defeats the purpose of trying to "teach the story".
Really, from the first answer:
> Based on my own experience as both a student and a teacher, I have come to the conclusion that the best way to learn is through "guided struggle".
This is the only way to "properly" learn mathematics or science. Anything else is only making you think you've learnt something.
> This is the only way to "properly" learn mathematics or science. Anything else is only making you think you've learnt something.
My point is not just that most textbooks make it too hard/inefficient to learn this stuff. My point is that most of it is never learned. The Legendre transform connects the Lagrangian and Hamiltonian mechanics, the two fundamental formulations of both quantum and classical physics, and yet most physicist cannot tell you why the transform is defined as it is. The reason is they don't take seriously the possibility that we'll find non-Lagrangian phenomena, and so they have not been forced to consider what observational and theoretical evidence led to it's identification in the first place.
Yes, Hardy was a great mathematician and he did say this --- but most mathematicians have tremendous respect for peers who strive for clear exposition in their lectures, their papers, and (if they write them) their books.
I am a professional mathematician, and Hardy's attitude is one I have never heard expressed by any of my peers.
"A Mathematician's Apology" is a fascinating read, but his description of mathematicians' attitudes is certainly not accurate today.
It indeed can be boring to reconstruct your thoughts in a way so that others can understand -- but many of us make the effort anyway, and doing so often leads to new insights.
> It is a melancholy experience for a professional mathematician to find himself writing about mathematics. The function of a mathematician is to do something, to prove new theorems, to add to mathematics, and not to talk about what he or other mathematicians have done. Statesmen despise publicists, painters despise art-critics, and physiologists, physicists, or mathematicians have usually similar feelings: there is no scorn more profound, or on the whole more justifiable, than that of the men who make for the men who explain. Exposition, criticism, appreciation, is work for second-rate minds.
I interpreted "men who explain" not as mathematicians that can explain their work well, but as people who try to explain mathematics in a "lay" way to cater for a large audience, whose explanation can very often become inaccurate, non-mathematical or just downright false, yet still get public credit for seeming to know the field very well, despite these inaccuracies, and even though they are not directly pushing the advancement of the field itself.
It's of course a good thing to try to reconstruct your own thoughts, but I wouldn't say it's unreasonable for a mathematician to omit doing that. Could you go into more detail on the examples you mention, where doing so led to new insights?
Good question. It's a bit hard to do so (especially without going into mind-numbing technical detail) -- in math you never quite know where insights really come from. "Fortune prepares the prepared mind."
But generally speaking, I would say that good exposition gets you thinking about: Why does the technique work? What is the key insight? What are its limitations? And if you think about such questions, you naturally get a better sense of for which other questions your techniques are also likely to work.
I agree with this statement but I think it misses the point entirely. Guided struggle is indeed necessary, but learning some theorem without seeing the impetus for its discovery is like learning how to play an instrument without understanding that the intent is to make music. Yeah, with enough time and struggle, you might be able to go through the motions and play some scales, but most people don't learn to play instruments that way, they learn to play a simple song or two, then go back and start with the scales and building musical theory.
Math textbooks sometimes try to do the same thing, but it seems like they always come up with the most inane and pointless exercises.
All of this is just to say, learning they why of math can help someone learn the how.
Do you believe that no new methods of learning will ever be invented?
Certainly, we can and do develop newer and simpler ways of understanding previous theories. And teaching the historical sequence of events can help with understanding; I myself experienced that with [1] for modern analysis. However, these understanding-aids don't teach you how to do mathematics, and only marginally improve your ability to apply those models and theories to existing real-world problems. To improve your ability to do mathematics, active exercises are necessary. Really, it's the same with many other fields, you don't get to be a good musician merely by reading about music and music theory.
I assumed they used the term "struggle" poetically, it certainly doesn't have to be unpleasant. But you have to put in some active mental exploratory effort. I found this post [2] a good summary of the skill set. But it's very abstract and likely won't make much sense unless you've been through the experience yourself.
These understanding-aids are also sometimes unnecessary. If you've done enough of the right kinds of exercises, they are of themselves an aid to understanding. For example, I could understand category theory better, not by learning about how this theory was developed historically, but by writing lots and lots of similar programs, and having a natural tendency to syntactically (and without much thought) refactor my code to be less repetitive, eventually leading me to various "category theory aimed at programmers" blog posts and papers. This one [3] of course deserves a mention, but there are many more.
To further emphasise this point, very brilliant mathematicians can just "pick up" models and concepts and work creatively and productively on them, without needing these aids.
My other point was that, the understanding-aids are very rarely what actually happened in the head of the people that developed a theory. Even historical narratives have distortions, and they are rarely detailed or precise enough to describe the rejected options, nor why they were options in the first place. (This fact, is also why they are not useful for teaching how to do mathematics.) There are exceptions, but reconstructing them is a boring process with little reward, especially since new developments 10 years later might explain it in even simpler terms.
That said, I would disagree with this part (from the top answer to the OP):
> a) The goal is to learn how to do mathematics, not to "know" it.
Modern mathematics has so much damn material these days that it's impossible to learn everything you need in order to solve modern-level problems, merely by teaching yourself all models and all theories "the hard way". Understanding-aids are certainly needed, and I use them very often myself, and I certainly prefer resources that teach using good analogies, proper context, descriptions of the motivations behind a theory, step-by-step "n/n+1" exercises, and everything else that other people mentioned here.
[1] "Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus" [2] https://medium.com/@jeremyjkun/habits-of-highly-mathematical... (discussed on HN here https://news.ycombinator.com/item?id=12187469) [3] https://bartoszmilewski.com/2014/10/28/category-theory-for-p...
I've noticed this too. I've tried to learn some quantum physics for fun, and it seems to me that most textbooks have surprisingly little description of the actual physical experiments behind the physics. Maybe this is something they usually cover more in lectures?
Like, I haven't found much details about the actual physical experiments around the spin of electrons (and certainly not of other particles). Sometimes there's an abstract description of a basic experimental setup.
I read just a few days ago that Feynman would redo old experiments that other physicist weren't doing anymore because they were established science. This is a great learning experience if you have access to the equipment, but most people don't. There should be books that describe how physics was developed, experiment by experiment, with lots of details.
If you read a 1000 page textbook on experimental physics, thats just the cliffnotes. The very highest level "here are the results of 400 years of serious research". There can't be any detail.
If you want the details on the stern gerlach experiment, for example, read the research papers. They will contain all the detail necessary to reproduce the experiment.
Lectures are just like those books. They just teach a very broad overview of what has been discovered.
The books you think should exist do exist. They're just not written for casuals.
Furthermore, it's not necessary to communicate the actual, usually circuitous, route taken historically by scientists. Instead, you could just describe the series of hypothetical simplified experiments whose results would lead one to quantum mechanics and rejecting alternatives. This is never attempted in a serious way (trivial uses of the Stern-Gerlach experiment as a model of quantum mechanics not withstanding).
"Principles of Quantum Mechanics"
Some of those flaws:
(1) Dirac postulates, as most people do, that measurable observables are to be identified with Hermitian operators. This is a mistake that can be traced back to von Neumann. In reality, the larger class of normal operators are perfectly fine as measurable observables. Indeed, measurements are properly associated with only an orthonormal basis, and it is completely unnecessary to label them with eigenvalues, real or otherwise. (To see this, just observe that there is no physical difference between experiments that measure x and x^3 for the position of a particle.) Dirac's discussion on page 35 is just wrong.
(2) Unless my memory fails me, Dirac gives little to no justification for why we use tensor products to build up the state space of a many-body system from the state space of a single-body system, a tectonic shift from classical mechanics at the heart of the weirdness and power of quantum mechanics.
(3) Dirac was written before Bell's inequality. I mean, just look at the pithy discussion by Dirac (p. 4-7) to justify fundamental indeterminacy, one of the most profound things we know about the universe. Do you think this would have convinced Newton? Or Dirac in 1925? (We know it didn't convince Einstein.) This sort of thought experiment is lovely for an article in Scientific American to give laymen a sense of where things come from, but it's nowhere near the rigor with which we should teach physicists.
Much of science is really just subtle modifications or tests of major theories, but I feel like you could write up... say 12 keystone experiments in 100 page summaries and publish it in 2-3 volumes, (eg, QM and relativity volumes).
The problem I have with only presenting "polished" results is that we lose the context of our modeling -- eg, QM seems to have included non-determinism in an effort to preserve locality, but locality couldn't be preserved in the face of other results, so is non-determinism merely an extra assumption included for legacy reasons (eg, technical debt because no one wants to clean up the model/people forgot why we included it)?
You'd never even think to ask that question if you only saw the cleaned up model of QM divorced from its (philosophical) roots.
its just that someone who is doing environmental physics doesn't necessarily understand papers on string theory. science is specialized, yea. if you want to understand research papers, get an education.
or just do something else. cutting edge research is not for everyone. not everyone is supposed to understand it. whats next? people demanding research papers be dumbed down to adhere to the freedom of information act?
science is hard. big deal. you cant have all the things at the same time. read the hawking books written for casuals if you want a cute little story about science.
Not necessarily. The classic example is Shinichi Mochizuki's work.
He's done some incredible work but he's basically invented his entire field out of thin air, he doesn't publish frequently, and the papers he does publish are essentially impenetrable.
He likes it this way and doesn't want to "dumb down" his work for the mere mortals who need to review it. That's essentially the end state of your argument - if you can't comprehend it you're not "in his field" and you're obviously not qualified to discuss it. And he's certainly not going to waste his time teaching some dunces the basics of his field.
https://www.newscientist.com/article/dn26753-mathematicians-...
https://www.newscientist.com/article/2099534-mathematicians-...
Writing in a clear way, without drammatic grammatical errors, is not "dumbing things down".
However, this knowledge is only communicated to other experts in the same - often very narrow - subfield. Often the definitions that give you a hard time when reading a paper have been refined over several years and are basically known to all other people working in the same field.
This is not ideal, but there is simply not enough manpower to produce good and generally accessible summaries of current research topics every few months...
No, that's the point I'm trying to make. They do not have enough training in making themselves understood. Hell, a good portion of them don't even have the language in which they are writing as their primary language.
Including some of the enlightening historical path towards that result is not "dumbing things down".
Unfortunately, many academics receive little if any training in good presentation before being expected to lecture or write at undergraduate or graduate level, and while some are naturally gifted presenters anyway, most inevitably are not. Consequently, many career academics have no idea how poor their presentation skills are, how ineffective their presentation is as a direct result, or how much better they could be. They just get stuck at a very low level, but without the kind of introductory/remedial training that would be given to someone whose career involved presentation skills in the professional world. And of course if anyone with broader experience dares to suggest that there might be room for improvement, the instinctive reaction is denial.
A little irony is that some of the most engaging and informative presenters I have ever seen or read come from that same community, but despite the emphasis on peer review in their research work, when it comes to soft skills the weaker presenters typically have no idea how bad they are and therefore make no attempt to learn from their stronger colleagues and improve.
I once sat in a review meeting at the end of a year with members of the faculty responsible for teaching collecting feedback from many of the undergraduate students. When challenged about the poor quality of many lectures, the response was essentially that they can't make the lecturers go and learn how to lecture competently because the lecturers wouldn't stand for it.
Try that attitude in the professional world and you'd be in a remedial programme on your way to getting fired.
Fortunately, I expect that with the advances in modern technology and changes in modern careers, the old-school universities that think a famous name and charging high fees mean they can get away with anything will soon be obsolete, and so will the incompetent parts of the academic community sheltering within them. They will need to find new ways to offer dramatically more value than interested people can find on their own with all the modern resources we have available, or they won't be able to justify people taking several years out of their lives and paying a fortune in fees to attend any more.
Try the free and new course "Quantum Mechanics for Everyone" that despite the title is certainly not dumbed down at all, on edX.
https://www.edx.org/course/quantum-mechanics-everyone-george...
The Stern-Gerlach experiment (1922) is what you're looking for. "Modern Quantum Mechanics" by Sakurai opens with an explanation of this experiment and its consequences.
What's missing for a lot of people is "What does this theorem apply to? Why does it apply? How do I break this theorem down into its component pieces to understand it?"
Indeed if your math text started off with a concept of a real world thing you're trying to figure out and then break it down to a series of paradigms that the student understands before applying each of the mathematical components that applies to each of these paradigms before finally combining them all to the final theorem, people would get it far more easily and be much less intimidated by it. Then over time students will spot paradigms that combine more elegantly to the final theorem and understand how to substitute them.
I think Math is being taught backwards in schools and this is why so many people are intimidated by and shy away from math... and I was a student who both struggled and enjoyed Math... and I still struggle with it because we were taught to think about it backwards.
We started out well with the basic building blocks, but somewhere between basic algebra and calculus, this all got turn around ass backwards. Where does the motivation to understand derivatives and integration come from if one doesn't understand the implications of their application?
Starting off with the general concept and breaking down into paradigms is not always possible. For example, say you want to teach people the Stokes' theorem of integration of manifolds [1]. It is the generalization of the divergence, Kelvin-Stokes and Green theorems that are very useful in electromagnetis, fluid mechanics and probably more. The theorem says, broadly, that if you are measuring some quantity in a certain space, you only need to know how does it "accumulate" on the boundaries of that space. But that will not be useful to an engineer or physicist. To really understand that theorem and break it down you need an entire course on differential geometry, with requisites on topology and calculus. Therefore, in the analysis courses for engineers/physicists the most probable situation is that either they present you the theorem and tell you how to perform the required operations on differential forms (which, if you don't know what they are, is a complete mistery/dark magic), or they go to the specific theorems (Gauss, Kelvin-Stokes, Green) which seem clunky and completely magic again.
The summary is that to be able to know what does a theorem apply to, why does it and how to break it, you literally need to study mathematics: that's what mathematics is. And the way to study it is always backwards, it's not like physics or programming where they can show you the high-level stuff, you can see what's happening and then you start to understand it. In math, most of the time, if you learn some subject starting from the high level stuff, it is difficult to even know what on earth are they talking about.
Better analogy:
You cannot code a Python script without knowing how to call the APIs you need to implement the functionality you want.
Knowing how an operating operating system works is more like knowing how foundational mathematics is axiomatized. Interesting and occasionally useful, but not immediately necessary for the theory of differential equations.
I think you might even be able to deal with differential equations without involving limits, by simply asserting all necessary theorems about derivatives as axioms. This corresponds to relying on a robust library (= the theorems other mathematicians have proved), without worrying about implementation details (= the proofs of those theorems, which might be much more difficult to understand than the results can be used).
Computing theory would have a word or two about this idea.
You can not code a Python script without knowing Python any less than you can use a mathematical theorem without knowing the definitions.
And yes, you can know what problem a piece of mathematics solve without knowing the math. How else would the math be invented in the first place?
> How else would the math be invented in the first place?
Usually the first step for problems to be solved is their expression in a mathematical way. See the heat equation: Newton (and probably others, history is not my strong point) already captured the core idea of heat transfer (transfer rate is proportional to the temperature difference) but it could only be really solved by Fourier once all of those ideas were put in mathematical expressions.
Yes, I agree this MO answer (and several others) seem to be rationalizing the situation rather than acknowledging how suboptimal it is.
I think I'm a good example of the system's failure. Coming in to college, I was something of an ideal candidate for becoming a mathematician. I had some success in Olympiads and already had decided I wanted to study math. My goal in life was to be a math professor. I enrolled in a top college and took many graduate-level classes. However, by my senior year, when it was time for me to decide the next step in life, grad school or industry, I had become somewhat disenchanted with theoretical math. Math was so abstract I started losing interest: all this commutative algrebra (for example) I learned wasn't making me feel like I had any new insights into solving math problems, outside commutative algebra problem sets.
And so I went into industry.
However, I can't help but think that if I had more knowledge of the motivation behind all the abstract math, I wouldn't have lost interest. All that machinery of commutative algebra was invented for specific reasons, such as solving polynomial equations in the rationals through algebraic geometry. Years later, through casually reading math on the internet, I've been getting hints as to what power these highly abstract frameworks give you for solving concrete problems. But without seeing the end goal, and having some idea why I should be learning this in the first place, I felt like I was just getting lost in abstract nonsense.
>Most of that foundational information is lost when it's not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject.
This is exactly right. I personally failed to reconstruct enough to keep myself interested in the subject.
Contrary to what you say, practical applications are easy to be found where they are simple and helpful. They make for fun math hobbyist like to play with and that works nice in terms of making kids interested in math. Unfortunately they do not lead to cutting edge science nor math needed for physics and engineering.
1) Search JSTOR: https://www.jstor.org/action/doBasicSearch?Query=proofs+and+...
2) Use Sci-Hub to download it (handy bookmarklet): javascript:document.location.hostname=document.location.hostname + ".sci-hub.cc";
Please use absolute references (a link to the specific answer) instead of fuzzy references that change over time.
Currently, your answer is the top-rated, so which one did you mean? (And even if your answer was not at the top, the top will almost certainly change over time, leading to all kinds of misunderstandings.)
EDIT: It seems that this referred to the top-rated answer on MO, not HN:
https://mathoverflow.net/a/13149/66043
This demonstrates my point even more: Please use absolute references, as fuzzy references make for misunderstandings! (And as my comment received quit a lot of upvotes in the beginning, I'm pretty sure I'm not the only one who thought this was about the top-rated HN answer.)
From my own experience (which, I'll admit, is not much to extrapolate from), the former was great for its "insight" and the latter just had too much unnecessary "information". Until I got stuck in an actual maths problem. Then suddenly my opinion switched, and the latter became a trusted map through strange lands.
You're right about the amount of information that's encoded in a definition or theorem. And it is very difficult to portray why that information is important (never mind how to access it, see other's comments about scaffolding and cathedrals and the like). I fear that trying to describe why groups are defined like they are would just lead to more impenetrable tomes - but instead of being maps to the lost traveler, they'd be more like tourist adverts. But the questions the source was asking is "why are books like this?" and "how do others [learn maths] in this situation?"
Well, maths books are like this because they are guides to help people doing maths. And you learn maths by doing it. I don't think its defeatist to say that no book or lecture about riding a bike would compare to the experience of actually riding a bike. To paraphrase the top-rated answer, why would you expect anything different from maths?
But to answer the underlying question of both yourself and the source: how do we improve this? I think pedagogy should focus on getting to "do maths" faster. You really learn by getting stuck in a problem.
This it the wrong analogy and does not capture the MO answer. The correct analogy would be training vehicle designers by teaching them all the parts necessary to build a modern bicycle without teaching them any of the ways early bicycles were designed and why those designs were discarded.
Some of us like it and don't see value in the long winded motivational style. Things like group theory are targeted on people who are interested in math itself and like the mathematics for what it really is.
It is interesting to learn history once in a while and there are good books about that. However, most of the time you want to move on faster.
There is a huge amount of information encoded in the choice of exactly how to define thing, and which theorems people care about. This reflects a long process of trial-and-error as the field was constructed. For a famous philosophical treatise on this using the Euler characteristic as an example, see "Proofs and Refutations” by Imre Lakatos
https://en.wikipedia.org/wiki/Proofs_and_Refutations
Part 1: https://math.berkeley.edu/~kpmann/Lakatos.pdf
Most of that foundational information is lost when it's not written down somewhere accessible; contrary to the answerer, only a small fraction is reconstructed by students as they learn the subject.
It's a huge problem, it applies to physics textbooks too, and it doesn't have to be this way. Unfortunately, the problem has been known for decades and there's not much reason to expect things to change. (Lakatos wrote the above in 1976.)