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Update from since I first posted this question: just keep sticking with it. I think that I am through the thick of it now and am finally doing ok. I really want to add an answer to this question myself though when I get the time.

TLDR; To summarize, the emphasis in most real math courses seems to be on proof-based things about seemingly esoteric or abstract structures which is, sadly, making me lose my love for math and motivation to do it. Moreover, this exercise doesn't seem to even have bearing on mathematical discovers yet it is all I can see I look deeper into math classes. I have not had the opportunity, to grasp the functional connection between axioms and their use in math. And finally, it may be helpful to know that I know solidly up the basics of matrices, multivariable calculus, and the basic ideas in differential equations.

I have heard that most mathematicians have discredited "game formalism". As per here, it seems like in the working world they don't really care much about foundations and axioms either: Why did we settle for ZFC? . I love math and am interested in studying some more advanced topics, yet this seems to be all that we learn about. If mathematicians do not use axioms to solve problems, then why does every class seem to be about knowing the definitions of structures and then proving things about them? I have not taken a course in real analysis or abstract algebra, but I have dipped my toes in enough to know what it's about.

When I hear people say things like "in math we invent rules and see what is true" or "it's true because that's what we define it to be", I just lose morale. I do not understand how manipulating symbols according to pointless rules is any sort of "game", because it is not fun in any way. It does not reveal truths about the universe as the type of math I know and love.

If this is what advanced math is, then I don't want to become a mathematician. Please, what is the purpose of learning so much axiomatic math if the working mathematicians does not use it? Isn't that an indicator that it shouldn't be taught?

And then there is the conflicting trio of reasons for axiomatic math which I have never resolved:

  • It seems like the axiomatization of things has come about through a messy process of taking into account and adjusting for many esoteric examples instead of one crystallized concept. I don't see how I can learn what these concepts are without a ridiculous search through historical documents which would take a lifetime of work. And it feels like you have to know literally everything about a subject too to include all those little examples, which is hopeless. Having to blindly take by faith that something is useful takes all the fun out of math, and it turns into grunt memorization.
  • If axiomatization of something is not just a more compact and reasonable way of saying what you know, as I have talked about above, but instead a way to capture abstraction that can apply to literally everything, then why do the definitions of things generally not transfer smoothly into the real world? Why do they not have a concise "essence" or "concept" in any lingo beyond math? My interest in this was first sparked when a friend explained that anything, not just "mathematical' things but things like colors could be viewed as vectors. The idea of the inverse of a function is very nice for example. I spent a year pursuing crystallized, applicable English definitions of mathematical objects, but to no avail. I wasted an entire year of my life, when I could have been learning other math, and do not want wish to do it again. Here is one such question I asked during that period of my life: What do Monic and epic morphisms imply?
  • If axioms are not made for everything, but just a few specific mathematical objects, then once we see the abstract connection between between those few structures it would seem simpler to solve them in each individually. If it is only 2 or 3 things that satisfy a given set of axioms, why do we talk about it otherwise? Is 50% of advanced math really just learning how to save time in one or two examples? I don't think so.

I believe I have the wrong core philosophy on axioms and their purpose, but being a young isolated self-learner, I lack the subtle ways axioms are utilized which can only be picked up through a classroom environment. If a teacher says "let this be true" vs "we arrive at this because" vs "we'll give this a name because it comes up a lot" vs "this equations represents the geometric object X" vs "we say this because the notation is efficient", are saying very different things with very different implications despite introducing the same object, with the same definition. But again, I lack the correct contextual attitude towards axioms as the core of math.

To summarize, the emphasis in most real math courses seems to be on proof-based things about seemingly esoteric or abstract structures which is, sadly, making me lose my love for math and motivation to do it. Moreover, this exercise doesn't seem to even have bearing on mathematical discovers yet it is all I can see I look deeper into math classes. I have not had the opportunity, to grasp the functional connection between axioms and their use in math. And finally, it may be helpful to know that I know solidly up the basics of matrices, multivariable calculus, and the basic ideas in differential equations.

I have tried so hard for so long, but I think I may not carry on with math without this.

Thank you, so, so, much

Pineapple Fish
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    "it is all I can see I look deeper into math classes" Look in a different direction! Try applied math for example! – littleO May 30 '20 at 06:36
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    You may be focusing too much on logic, set theory, and foundational type books/literature. There is a huge amount of mathematics where these issues are pretty much never raised (except in some very technical and tangential corner of the field). Perhaps you may want to consider working through one of the standard complex variables/analysis texts, as much of the theory is fairly straightforward and there are a huge number of applications and connections to other mathematical topics. – Dave L. Renfro May 30 '20 at 06:56
  • What is the question? – lcv May 30 '20 at 07:21
  • Why do you think math is about inventing axioms? Most mathematicians are content with ZFC and do their mathematics in this system. – J. De Ro May 30 '20 at 07:42

1 Answers1

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There is a lot there, and I don't have a good answer for all of it, so I'll just make a few points.

Axiomatization does often turn out to be effective in attacking problems in different fields. For example, groups appear in many places in math. Your argument seems to be that one might as well just study each particular case on its own merits. The fact is that any particular case of a group (say, a group of matrices, or a group of permutations) will have many special properties. It may have so many in fact, that without a unifying concept such as that of a group, it may become hard to see the forest for the trees. Realizing that they are groups, that certain maps between them are morphisms, that certain subsets are subgroups may (in favorable cases) strip away many of their extraneous features and focus attention on what is most important. This is an advantage from a psychological standpoint when trying to solve a problem.

It is true, however, that a lot of people do derive enjoyment from playing around with abstract axiom systems, so it is pleasant for them to study abstract theories. They may not feel the same impatience you do to find applications. The same way one can enjoy number puzzles or games like chess, one can have a taste for abstract mathematics. (In fairness, it must be said that many abstract theories have arisen because of their applications - for example functional analysis for integral equations and PDEs.)

Does that mean they'll be any less adept at applying the abstract knowledge to more concrete problems when they turn their attention to them? In most cases, probably not. I don't think I'd be wrong in saying that the same people who have enjoyed pointless number puzzles since they were children also turn out to be the best at applying math to concrete problems - epidemic curves, computational genetics, many fields in computer science, etc. The more you enjoy thinking about numbers and other abstract systems, the more intimately familiar you become with them and, ultimately, the better you become at using them to solve concrete problems.

When you consider the very advanced math that is used regularly by theoretical physicists, it's hard to draw a definite line between what pure math will likely be useful in applications and what won't. But the difference in ability between someone who knows basic undergraduate analysis and abstract algebra and someone who doesn't is likely to be substantial. For one, the actual content is very likely to be used heavily in applications. Secondly, the habits of thought one develops in solving problems in these fields are the same ones that prove useful in solving problems across most disciplines in which mathematical methods are fruitful. Lastly, in the case of analysis in particular, one develops very concrete skills, such as proficiency with inequalities, that are useful in applied problems.

So where does this leave you, given that you don't believe you'll enjoy the material if it's taught in too abstract a way? If you want to continue far enough with pure math that you can be a competent applied mathematician, you probably need to find high-quality textbooks that teach the material with a constant eye to applications, in order to retain your interest.

In analysis, a couple of books come to mind, namely Calculus with Applications by Peter Lax, and Mathematical Analysis by Zorich. Most probability books mix theory and interesting examples constantly.

In algebra, I'm afraid I can't name any books offhand, but I believe there are books that teach abstract algebra in conjunction with computer applications such as cryptography. In any case, number theory is a natural concrete domain of application that comes up constantly in even the "purest" abstract algebra textbooks.

Whatever your opinion of axiomatization, there is one thing that I think is beyond question. Proof is central to mathematics - not just to the theory, but also in solving concrete problems. While there are often real-world cases in which heuristic arguments are more applicable than airtight proofs, there's little doubt that the best problem-solvers in quantitative fields are those who have honed their skills at solving problems in a logically flawless way.

Edit. Let me add that while mathematicians may vary in their attitudes towards axiomatization per se, they all enjoy proofs, or else they wouldn't be in this field. In perhaps 50-75% of the problems you do in a basic analysis course, you will be proving facts about sets of real numbers, derivatives, integrals, etc., even though abstract structures like metric spaces form part of the course and may sometimes be helpful in solving a proportion of the more concrete problems.

If you don't enjoy these problems, the problem you're experiencing may be not just an aversion to axiomatics and abstraction, but a reaction to the inherent difficulty of the problems themselves. If that is the case, then you may be right to be asking yourself if math is for you.

So apart from the suggestion of trying algebra and analysis books that emphasize applications (while teaching the theory competently), you could also consider going into an area which would make use of multivariable calculus, differential equations, etc., but where the demands in terms of rigorous math would be relatively smaller (especially at the undergraduate level). Some examples:

  • statistics (with an emphasis on applications as opposed to theory)
  • physics (oriented towards the experimental side)
  • computer science
  • economics

An example of a brilliant mathematician who famously disliked the trend towards math that he considered excessively abstract was V.I. Arnold. If you feel that your problem may be with abstract math rather than math in and of itself, you could look into reading works of his, especially in the fields of differential equations, PDEs and mechanics. A warning, though - this may be a good deal harder than the abstract math you've looked at and found offputting. (Arnold had no problems using and developing abstract mathematics when it was useful; one of his papers was titled (translation) "On the differential geometry of infinite-dimensional Lie groups and its applications to the hydrodynamics of perfect fluids.")

Anonymous
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  • +1. Especially for the point where you mention that along the way one acquires certain patterns of thought helpful in different fields of study. I used to believe mathematical thinking is very different from and has absolutely nothing to do with the almost brute force memorizing found in fields like medicine and biology, but really, one can almost say that you simply end up with an overall better brain as you make progress studying maths. From one perspective, maths can be seen as a strenuous exercise in logic, which is definitely useful for learning sciences in general. – Km356 May 30 '20 at 08:34
  • Hhm yeah. I also forgot to mention I'm not really an "applied" kind of person in the sense of physics or other fields beyond math.I like pure math like geometry or calculus, for example, if you would call it that. I actually enjoy abstract concepts, so long as they have a clear abstract purpose. My litmus test for understanding something is "could I have derived this myself?" But I accept anyways because it's my fault for not putting that in the question. Thank you for taking the time to write such a detailed answer, I really appreciate it. +1 I think there was something else but I forgot it.. – Pineapple Fish May 31 '20 at 05:24
  • Can you expound a little more on the machinery of how focusing on the abstract structures takes away the extraneous features? Since I have posted this question, I have found a very compelling quote by a topologist and close friend of Emmy Noether's about her legacy: "It was she who taught us to think in terms of general and simple algebraic concepts - homomorphic mappings, groups and rings with operators, ideals - and not in cumbersome algebraic computations; and [she] thereby opened up the path to finding algebraic principles in places where such principles had been obscured by some comp... – Pineapple Fish Jun 24 '20 at 17:30
  • licated special situation..." – Pineapple Fish Jun 24 '20 at 17:30
  • I don't know that I can elaborate on this in general. I can give you an example where I think abstract thinking is useful. Say you want to prove that for a fixed integer $c \geq 0$, there is a polynomial formula giving the sum of the first $n$ $c$th powers, i.e. $P(n) = \sum_{k = 1}^n k^c$. Really what you are looking for is a polynomial $P$ with the property that $P(0) = 0$ and $P(n) - P(n-1) = n^c$. Now for each $e$, consider the vector space $V_e$ of polynomials $P$ with coefficients in $\mathbb{Q}$ and degree $\leq e$. Now let $\Phi \colon V_{c + 1} \to V_c, P(X) \mapsto P(X)- P(X - 1)$... – Anonymous Jun 26 '20 at 20:23
  • We have $\dim V_{c + 1} = c + 2 = \dim V_c + 1$, $\Phi$ is linear, and it is easy to see that $\ker \Phi$ has dimension $1$ since it consists of the constant polynomials. These facts alone prove that $\Phi$ is surjective, so a polynomial of the required kind exists. The first few polynomials of this kind (for $c = 0, 1, 2, 3$) are $X$, $X(X+1)/2$, $X(X+1)(2X+1)/6$, $X^2 (X+1)^2/4$. When I was in high school I worked out these formulas up to $c = 6$ or $7$, but had no inkling how to prove that one could be found for any $c$, much less that a simple argument was enough. – Anonymous Jun 26 '20 at 20:30