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The research level of mathematics (what is done by professors and upper-level graduate students) tends to be heavily portrayed as focused on writing proofs to the exclusion of most anything else math-related. This was certainly the case for me as an undergraduate, and I pictured professors spending all of their time outside of teaching trying to write out formal proofs that X over blah is bounded by the gradient of Y in P-space, blah blah blah, rather than doing things I thought might have been more fun but still math-related. In fact, one of my textbooks started out with a "helpful" reminder that my childhood was over and that I was entering a world in which proofs were the most important thing. In other words, "knowing" math meant pretty much diddily-squat unless I could formally and rigorously write out proofs for everything I thought I knew.

I never really enjoyed proofs, and so was discouraged from following math any further than what was required for my non-math undergraduate degree. In fact, I've always been an intuitive, heuristic learner, combining past knowledge with intelligence and intuition to determine what a reasonable answer is even though I cannot "prove" it to be 100% correct.

To that end, are there research-level subfields of mathematics that can be used as examples of "There's more to math than proofs!"? In other words, these would be sub-fields that could be used to inspire students interested in research-level mathematics but who are weak in terms of proofs or do not find proofs interesting enough.

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    $\begingroup$ I do not have a PhD, so I'll be interested in the answers to this. My impression is that mathematical research means finding new mathematical relationships and mathematical objects, and that that involves proof. $\endgroup$
    – Sue VanHattum ♦
    Commented Sep 14, 2022 at 15:00
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    $\begingroup$ Is there a reason you need a research-level subfield of mathematics specifically? Mathematics-adjacent technical fields such as physics and engineering offer the opportunity to apply math to research and other problems without the same emphasis on proofs. $\endgroup$
    – Steve
    Commented Sep 14, 2022 at 17:25
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    $\begingroup$ It might also be good to emphasize to students interested in research that "there's more to proofs" than just sitting down with a blank piece of paper and a proposition handed down from a textbook author and cranking out a flawless stream of math-ese in a single pass. I think students get the wrong impression between reading textbooks and watching professors lecturing in courses. $\endgroup$
    – Steve
    Commented Sep 14, 2022 at 17:32
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    $\begingroup$ You seem to be operating under the misconception that proofs are some superfluous stricture arbitrarily enforced by the "elders of maths". They're not. They're what allows you to know whether a claim you make is true or not. Sure, in very simple cases you can "see it", but with more complicated claims your intuition may absolutely be mistaken. It's as if you said: 'I've always been an intuitive, heuristic architect, combining past knowledge with intelligence and intuition to determine whether a building will or will not fall down even though I cannot "prove" it to be 100% correct.' $\endgroup$
    – Adam Přenosil
    Commented Sep 15, 2022 at 9:55
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    $\begingroup$ A proof is just a watertight argument that something is true. Your dismissive quotation marks ('I cannot "prove" it to be 100% correct.') in fact dismiss the very idea that you need to provide convincing arguments for the claims that you make. "So and so is true because I can intuitively see it" is not a convincing argument. Of course, there is place in mathematics for other activitities in addition to writing proofs. But you should first get rid of what seem to be misconceptions about what the purpose of proofs is and why people demand them. $\endgroup$
    – Adam Přenosil
    Commented Sep 15, 2022 at 9:59

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I think a more correct view is that proof is the LAST of several stages involved in researching something in math. What follows is a quickly sketched out scenario of what is often the case.

Before proof, there is "playing around with some topic" enough to decide whether your background and ability might suffice to uncover something sufficiently useful to actually begin formulating conjectures. Or alternatively, reading over past attempts/results in the literature about a known problem enough to obtain sufficient background information to decide whether it might be worth while for you to pursue the problem.

Then comes some more focused investigations, maybe trying out the conjecture with some "toy examples" or seeing if you can prove weaker versions of the conjecture. For example, maybe you want to prove that every positive integer has a certain property. To vastly oversimply what usually is the case for the sake of illustrating what I mean, can you prove this for a specific integer? Two specific integers? Infinitely many integers? Can you prove it for even integers, or for odd integers, or for square free integers, or for all sufficiently large integers, etc.?

After this (or at the same time) comes a lot of attempts at heuristic arguments for why the result is true, with the hope that one such argument can be made sufficiently sound to ultimately result in a rigorous proof. Usually by the time one tries to write a rigorous proof you know whether the approach is likely to work (rigorously), so for a mathematician writing a proof (as opposed to a student still learning how to write proofs), it is somewhat like a judge writing a formal legal judgement after hearing a case and thinking over all the relevant facts and appropriate legal issues involved and coming to a decision about the case.

I'm not sure how much this will help, but this recent mathoverflow question gives an example of someone at the "playing around with a topic" stage, and in the comments I mention something that might be related and which is something I've never even gotten to the "playing around with a topic" stage despite having gathered some relevant literature on it over the years.

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  • $\begingroup$ But surely you have given up on a problem because you realized you did not know any methods to turn your heuristics or computations into a proof. $\endgroup$
    – Alexander Woo
    Commented Sep 15, 2022 at 13:46
  • $\begingroup$ @AlexanderWoo In that case, you call it a 'conjecture' rather than a 'theorem' and still get to put your name on it. Somebody else can come along a few years later and be lauded for proving the 'Renfro Conjecture of Prime Numbers' $\endgroup$
    – Silvio Mayolo
    Commented Sep 17, 2022 at 2:23
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    $\begingroup$ This is a good answer, but kind of skirts around the real thing being asked, which is if the OP can avoid being good at proofs. I feel like this kind of question (and answer) is like asking if you need to draw "classically" to be a great artist like Picasso. Obviously Picasso's art doesn't look "classical", but at the same time, he got there by mastering the skills of classical artists. $\endgroup$
    – Chan-Ho Suh
    Commented Sep 18, 2022 at 15:31
  • $\begingroup$ @Alexander Woo: I don't know what your comment is in reference to -- the mathoverflow OP's question or my research topic suggestion or something else. In my case, I haven't ever "sat down and carefully thought about" how Moore's notion of relatively uniformly convergent sequence (a notion provides a hierarchical classification of convergence notions lying between pointwise convergent and uniformly convergent) might allow for some type of generalized $\sigma$-porous strengthening of first Baire category for discontinuity sets of the limit functions. (continued) $\endgroup$
    – Dave L Renfro
    Commented Sep 18, 2022 at 18:16
  • $\begingroup$ Related to this, if anyone's interested, is that I did spend a few hours one afternoon in 2005 (after the main result in this paper was essentially completed) working on a generalized $\sigma$-porous strengthening of the fact that the complement of the Liouville numbers is a first Baire category set. Obviously a non-scaled porosity wouldn't work (every $\sigma$-porous set is both Lebesgue measure zero and first Baire category, and the Liouville numbers have full measure), but I managed to (continued) $\endgroup$
    – Dave L Renfro
    Commented Sep 18, 2022 at 18:20
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With the technological advances of the past couple decades, computational mathematics is now somewhat accessible to undergraduates. The wikipedia entry for computational mathematics lists out the various subfields. Basically, actually being able to compute something is, in and of itself, a worthwhile endeavor.

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    $\begingroup$ Yes - but you still want a proof that the computation actually computes what you think it does. $\endgroup$
    – Alexander Woo
    Commented Sep 14, 2022 at 17:01
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    $\begingroup$ At an undergraduate level, I'm happy if my students can actually implement the computation (in Mathematica, etc.) as that often involves understanding the main ideas thoroughly. There's often no need for them to give a proof (since we're usually starting with one in the literature). In cases where it's more of an exploratory project, the key elements are finding and describing the patterns in order to create conjectures. In those situations actual proofs (even of maybe a simple case) are bonus for an undergraduate. $\endgroup$
    – Aeryk
    Commented Sep 15, 2022 at 13:41
  • $\begingroup$ In many cases, when a computational mathematician writes a program, the purpose of the program is to find a proof, typically by doing some kind of (clever) exhaustive search and finding no counterexamples. $\endgroup$
    – kaya3
    Commented Sep 16, 2022 at 14:19
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I have taught Discrete & Computational Geometry to US undergraduates project-based, as opposed to assignment- and test-based. Some of the projects do involve proofs, but others are more experimental explorations, either computationally or physically. Here is one on "D-forms":

Gluing together two identical copies of a smooth convex shape, one rotated with respect to the other, produces an elegant 3D convex shape with one seam where the two perimeters are glued together. It is an unsolved problem to determine the 3D shape from the 2D shapes and how one is twisted with respect to the other. Explore some subclass of shapes and twists and formulate conjectures. Even studying doubly-covered rectangles is not entirely understood.

A student submitted an entire box of paper D-forms in support of her conjecture. :-)

surface formed by joining two ellipses

Image from CutOutFoldUp.

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  • $\begingroup$ In my experience it is possible "to inspire students interested in research-level mathematics but who are weak in terms of proofs ..." $\endgroup$
    – Joseph O'Rourke
    Commented Sep 16, 2022 at 20:51
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In other words, "knowing" math meant pretty much diddily-squat unless I could formally and rigorously write out proofs for everything I thought I knew.

You appear to believe that somebody can "know" mathematics without being able to prove things, but that their mathematics knowledge is devalued by stuffy gatekeepers who insist that all mathematicians should also be able to prove things. This is nonsense, akin to the idea that a skilled fiction writer should also be able to tell stories, or a skilled athlete should also be able to compete in sport.

When a textbook says "your childhood is over" and now proofs are important in mathematics, that is like saying to a writer that it is no longer sufficient to be able to spell words and construct grammatical sentences, because while those abilities are fundamental to writing, they don't make a writer. The next foundational skill the writer needs to learn is to tell stories, and the next foundational skill a mathematician needs to learn is to prove things.

I've always been an intuitive, heuristic learner, combining past knowledge with intelligence and intuition to determine what a reasonable answer is even though I cannot "prove" it to be 100% correct.

To me, what that means is that you frequently made guesses and you weren't generally able to justify why those guesses were correct, perhaps you weren't even always aware when one of your assumptions was a guess rather than something you really "knew" to be true. This approach may well have found you great success, but only because the mathematics you were doing was "too easy": easy to get intuition for and easy to make correct guesses about. But once "your childhood is over", being a good mathematician requires more than the ability to make guesses that are usually "reasonable".

I'm sorry if that sounds harsh, but I think it is a realistic assessment of this kind of attitude towards mathematical rigour. A student with this attitude is in what Terence Tao calls the "pre-rigorous stage".

To that end, are there [...] sub-fields that could be used to inspire students interested in research-level mathematics but who are weak in terms of proofs or do not find proofs interesting enough[?]

Yes, there are surely plenty of research areas in mathematics which don't "focus" on proofs; see the other answers for examples. But "not focusing on proofs" does not mean that one can become a good researcher in those subfields without learning how to prove things; rather it means that proofs are merely not the end product of that research. Good researchers in these areas (or in all areas, I should think) are at the "post-rigorous" stage, not the "pre-rigorous" stage.

To inspire students who struggle with proofs, I think "don't worry, you can still do mathematics if you're bad at proofs" is the wrong message. I would rather say: You can learn how to write proofs, and I will help you.

Likewise, if students find proofs uninteresting before undergraduate level, I think "don't worry, you can do all the other things and not focus on proofs" is the wrong message. Rather, I would say: The things you're proving now are not that interesting, sure, but that's because these are basic exercises to help you learn the basic proof techniques. But there's lots of interesting mathematics ahead of you, which you'll need proofs for, and it's much more fun once you're proving things that you do care about.

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When I describe undergraduate research to students majoring in mathematics, I ask them to browse the abstracts of the most recent MAA undergraduate poster session. Here is a link: Abstracts for the MAA Undergraduate Student Poster Session MAA MathFest 2022.

From this, one can easily become convinced that the proof-centric mode of research that you describe need not be the norm.

Another direction is to learn about what is called "experimental mathematics." There is a good Wikipedia page, and a highly respected publication title Experimental Mathematics. Their "aims and scopes" statement states that the journal publishes "papers on formal results inspired by experimentation, conjectures suggested by experiments and data supporting significant hypotheses in mathematics."

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Not quite an answer to your question, but if you have students who are interested in mathematics but not interested in (generating their own) proofs, encourage them to go into mathematics communication! Arguably, finding ways of effectively communicating mathematics to wide audiences is at least as important as discovering that mathematics in the first place.

Figuring out just the right way of communicating an idea requires (and creates!) a deep understanding of that idea, and communication skills will be valuable for your students no matter what they wind up doing.

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No. By definition, mathematics is the study of what can be formally proved, so if someone is not at all concerned with proofs, they are not doing mathematics (but possibly some math-related subject).

(I'm not sure I 100% believe this answer, but it's worth putting out there.)

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    $\begingroup$ I would say that the empirical discovery (by Oliver and Soundararajan) that the last digits of two consecutive primes are not uniformly random constitutes research, even in the absence of a proof: Unexpected biases in the distribution of consecutive primes. $\endgroup$
    – Joseph O'Rourke
    Commented Sep 14, 2022 at 17:30
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    $\begingroup$ But I would argue they definitely conducted their research with an eye towards the possibility of a future proof - they thought about how their conjectures might be proven given proofs of the Hardy-Littlewood conjectures, and they were guided in what to look for by these possibilities. $\endgroup$
    – Alexander Woo
    Commented Sep 15, 2022 at 0:15
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    $\begingroup$ @Kimball - it rules out most non-Greek ancient mathematics as well as most early modern and medieval mathematics. (But for your definition - surely it would be a big stretch to consider large cardinals or infinity categories or positive characteristic algebraic geometry quantifiable!) $\endgroup$
    – Alexander Woo
    Commented Sep 15, 2022 at 1:32
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    $\begingroup$ It feels very weird to me to define mathematics in a way that includes proofs of the Pythagorean theorem, but excludes ancient tables of Pythagorean triples; includes Hippasus of Metapontum's argument that the square root of two is irrational, but excludes the square root extraction algorithm from The Nine Chapters on the Mathematical Art; and includes Cauchy and Lagrange's proofs of Taylor's theorem, but excludes Madhava's power series for sine, cosine, and arctangent, Gregory's power series for various other functions, and maybe even Taylor's own statement of Taylor's theorem. (1/2) $\endgroup$
    – Vectornaut
    Commented Sep 16, 2022 at 3:17
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    $\begingroup$ And if we kick everything except formal proof out of mathematics, where does it go? Is a sine table astronomy when Āryabhaṭa computes it, navigation when Nathaniel Bowditch computes it, and both when Henry Briggs computes it? (2/2) $\endgroup$
    – Vectornaut
    Commented Sep 16, 2022 at 3:17
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Especially with the advent of relatively very fast numerical and symbolic manipulation software/computers, giving numerical evidence for various things is an eminently feasible, and interesting, heuristic ... apart from proving that the heuristic is correct. Indeed, in some cases, extensive numerical computation is useful, and perhaps necessary, in order to form conjectures about what should be proven (if possible!).

Yes, some sources take up a quite pompous attitude about "mathematics" being "only" things that have complete, rigorous proofs... and that everything else is irrelevant. Well, of course we'd like GOOD (=explanatory) proofs of things, if possible, rather than just numerical evidence, but sometimes we just don't get that. Just get numerical evidence... or other heuristics that are hard to justify.

As far as specifics: numerical simulation of n-body problems in celestial mechanics... numerical search for solutions to various diophantine equations ... are easy examples that come to mind. But/and I primarily wanted to make the point that math has many parts other than "formal proof"... so don't believe anyone who tries to tell you that "it's not math" unless it's formal proofs.

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IF you have kids that like math, but not proofs, I suggest modeling financial systems (e.g. actuarial issues) or the like (e.g. refinery operation) is a good activity for undergrad math majors. Data analysis or ops analysis projects. For sure, it will help them in the work world, just in case they don't end up being Andrew Wiles or a juco math teacher (which by the odds, most won't). If they do, it won't hurt them, either, to have a little wider perspective.

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Digital signal processing would seem like a good avenue.

The colors on this monitor, the audio and video, the compression, the encoding, the timing, everything digital is passing in and out of digital processing equations.

It can be very heavy for mathematics, and it's rather a factor of efficiency rather than proof, aiming for the most efficient way of processing digital signals and patterns.

Creatively there is a lot of research going into 3D equation based media that can be efficiently understood by AI... Actually, one of the biggest difficulties in AI is organizing data into compressed low data formats that can be processed by machine learning. It's good to be an expert in a particular field of media because the computer scientists often lack the mathematical knowledge of the data they process to inform the computers how to process efficiently. In 3D for example, it's all about quaternions, matrices, vectors, voxels, SDF's. In audio, it's all about waves, which have an intrinsic uncertainty in the time-frequency domain which resorts to quantum maths equations like Wigner distribution transforms.

The field of DSP and machine encoding-decoding of data is very mathematically heavy and reqires solutions demonstrating efficiency because the proof is already known.

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What are some research-level opportunities in mathematics that do not focus on proofs?

I'll give a common-sense answer without any particular knowledge:

Physics, Engineering, economics, some Investment Banking or market - related research, Cryptocurrency and online security (encryption), Chemistry, Computer Science, Machine Learning, Robotics, Statistics, Computer simulations, some computer programming/software engineering.... These all apply mathematical proofs and theorem without the need for knowing the proofs themselves, although of course you need to know the theorems.

If you want to study mathematics however, then you will need to read and write proofs in a comprehensive manner. There is no way around this. "Doing mathematics" requires theorem and proofs. Otherwise, it is not mathematics.

If you study an in-between ground - a math-heavy subject that is not 100% mathematics, like mathematical physics, or econometrics, then you will have to know (by heart) some important proofs, but not the unimportant proofs, for example you probably won't have to know the small lemmas leading up to a major theorem (although they might help you in your job/research).

Question answered.

"I cannot "prove" it to be 100% correct."

Absolute knowledge is unobtainable. Get over it.

If what you really meant is that you struggle with proofs because they are hard, then yes. So does everyone else. Comprehension of (reading and understanding) proofs, and proof-writing are advanced skills that take a lot of time and effort to develop and improve upon. Either you put the effort in - which for me is made easier by my enjoyment of studying mathematics, which motivates me to put the effort in myself to understand proofs - and you improve, or you don't put the effort in, and you stagnate in your abilities.

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