What is "Pure Math"?
Understanding humanity's oldest academic discipline
Alice shows up to her first calculus class in college. Halfway through, the professor is emphatically gesturing at an ancient chalkboard covered with white chicken scratch, proclaiming that the ‘symbols’ on the board explain everything. Alice is reminded of endearing characters like Gandalf and Dumbledore often speaking in riddles whose meaning is not clear until much later. Alice decides these characters were much more endearing in books than actually present in her life and having the power to determine her grade.
It should be no surprise then that the subject of math has assumed a vaguely mystical aura. Not only does it have some inherently alien characteristics (e.g., communication happens through a non-verbal language with little-to-no contextual structure), but the point that it is “really important” and “enabled [insert awesome technology]” is hammered non-stop since elementary school. However, perhaps the most mystifying characteristic is that it is usually described completely wrong.
Somehow the perceived ‘importance’ of math leads people (including professors!) to describe math as if it was a science. They present the fact that objective truth exists in a mathematical framework as analogous to (or even ‘more scientific’ than) how experimental truth exists in physics. They point to all the ‘scientific’ impacts of math: how physicists use math to calculate properties of the universe, or how error-correcting codes enabled modern computers, or how Archimedes discovered leverage which led to basically all of modern construction capabilities. But this ‘scientific’ image is all wrong!1
(Post?-)Modern pure math is really an art, and this phrase is not meant as some intellectual preening. I mean this in the most boring way possible: there are no objectively ‘important’ subjects in today’s pure math, trends emerge and die down at approximately the same rate that Leonardo DiCaprio gets new girlfriends, and most if not almost entirely all of today’s pure math is completely useless. To be absolutely clear: none of this is a critique; indeed, mathematicians themselves will tell you that “real mathematics must be justified as art if it can be justified at all.2”–G.H. Hardy. This perhaps shocking testimony from a famous mathematician presents two confusions: (1) the seemingly stark contradiction between what we are told all our lives about math (that it is extremely useful, that it is the reason we have all this fancy technology, etc.) versus mathematicians’ perspective that they actively don’t care about applications of their work, and even purposefully try to avoid them. And (2) “then what the heck is real math?!” To clear up both of these points we need to go a little into the history of math.
What Math Was
I would like to preface this section with two disclaimers: (1) I am not a historian and for the purposes of this article, am not interested in giving equal weight to different historical periods, and (2) much more could be said about the history of mathematics. I would also like to emphasize the following terminology (which is commonly used by mathematicians to delineate different eras of math): “modern” mathematics means math with axiomatic foundations and rigorous standards for proof, which began in the 19th century. I will also be delineating this math from “today’s” math because (as I will argue) it is very different in character. So “modern” here does not mean the same thing as when a historian (who would refer to this period as the “early modern period”) or layperson says “modern”, while “today’s math” does refer to math as it is studied today.
From the beginning of history until around Medieval times, math was mostly used for calculation: Pythagoras’s infamous theorem, Archimedes’s studies on leverage (and the rest of his research), Ptolemy’s astrological predictions, The Nine Chapters of the Mathematical Art, etc. The math that was studied could more-or-less be broken up into two groups, both of which focused on finding numbers that tell you something:
Counting.3
Calculating the length/area of some simple shape (circle, rectangle, triangle, etc.).
Even within the narrow scope of ancient mathematics, some quite deep math and complicated calculations were carried out. Astronomy was what emerged when you took the second camp of study and ran with it very far, and then looked up at the sky at night.
Next, the math of the Renaissance era, through the Scientific Revolution, and up until the mid/late eighteenth century laid the motivation for the abstractions that would later define modern math (and all the math that followed hence). Great mathematicians like Euler, Newton, Leibniz, and Descartes published extensive works, including the theory of calculus. Math of this era is characterized as having much more complicated motivations and goals as opposed to the previous era, and also branching into new directions in terms of the calculations that people cared about. No longer was it satisfactory to approximate a trajectory with triangles and calculate the hypotenuses; mathematicians now wanted to know exactly how long that trajectory was. At the same time, there was a fundamental lack of rigor in the work of these (great) mathematicians.4 It was not until the controversies that arose from different interpretations of their work (that were only possible due to the lack of rigor and technicality of the subjects) that a more precise mathematics emerged. Today’s mathematicians (and students) would assess that the original works of these titans were largely ‘intuitionistic’ and lacked proper rigor. With all that being said, there was still some improvement in the rigor from previous eras. Another point of note is that math at this time was more or less synonymous with “physics”; the subjects had yet to split.
Now we have arrived to modern (recall this does not mean “today’s”) mathematics. The character of this math can be summarized as arising from disputes between famous mathematicians (of this era) on subtle points of calculus. See, even if Newton and Leibniz had already written their treatises and many mathematicians had studied them since, when one got too far away from the base theory there seemed to start arising contradictions. The notions of “infinitesimals,” “limits,” and even “function” had yet to be precisely defined, leading mathematicians to have their own (sometimes changing) interpretations, and thus leading to ill-stated theorems and proofs. There were different instances of famous mathematicians rejecting other famous mathematicians’ works for seemingly arbitrary5 reasons. Joseph Fourier’s paper on Fourier series was rejected by a committee that included Laplace and Lagrange due to “lack of rigor.” The greatest mathematician to exist up to this point was Augustine Cauchy, and even he fell into this trap when he published an article on the convergence of series of continuous functions that claimed to prove something false6 (ironically, the counterexample was provided by Fourier). So was all of math a lie? What was the point of the entire field if two different calculations of the same phenomenon could be different? These dramatic questions caused a revolution within mathematics in the late 18th and all of the 19th century. Infamous mathematicians like Cauchy7 and Weierstrass established a more rigorous calculus, eventually resolving earlier contradictions.8 Abstract algebra (which is somewhat naturally axiomatic because the objects it studies are quite explicitly collections of axioms) made its debut through the works of Grassmann, Hamilton, Abel, Galois, and others.
Following these figures were the logicians, Cantor9, Peano, Russell, and Whitehead, who sought an axiomatic ‘maximally rigorous’ mathematics. More modern mathematicians followed, continuously seeking to make mathematics unquestionably rigorous. Notable in this quest (beyond the already-mentioned logicians) are Hilbert, Zermelo, and Frankel. Zermelo and Frankel established a completely rigorous theory of sets that could encapsulate most (if not all) of math at the time, giving more hope to the notion that all of mathematics could be established on top of unquestionably rigorous footing. Hilbert, along with being perhaps the greatest mathematician that ever lived, was a particular fanatic in the pursuit of proving that mathematics was logically sound10. Hilbert assigned the mathematicians of the world an infamous list of problems that they should focus on in August 1900, and the second problem of this list was proving that you could not have contradictions in arithmetic (this sounds easy but it is not). You can imagine his (and every other mathematician’s) surprise when later on Kurt Gödel (at the young age of 25) came along and proved that you could not (internally) prove that arithmetic has no contradictions. Gödel essentially killed Hilbert’s program for a self-contained fully-rigorous mathematics, but so much progress had been made in axiomatization—and the standards for rigor had been elevated so much higher within the community—that the culture of writing complete formal proofs stuck and remains venerated to this day.11
What Math Is
Another consequence of the modern period of mathematics was that rigorous foundations had been struck in a lot of different fields. Many of these fields arose directly from applications, but required serious abstraction in order to make rigorous. A good example of this is what is called a manifold; we can all draw a torus (donut) quite easily, but it takes a textbook several pages of abstract jargon in order to define one rigorously. Suddenly, every mathematical object had axioms attached to it, and researchers could study just the axioms and their interrelations. The abstraction of the working with these systems of axioms meant that every result was rather powerful; by working with axioms instead of specific objects, a simple connection could be formed between entire classes of objects at once. Mathematics quickly became untethered from context. It also split from physics as an academic discipline.
It is this decoupling from practical applications that characterizes today’s math, which I shall also call “post-modern” math. During the modern era of math, a mathematician did not have to “go far down” from the level of abstraction they were working at in order to reach an application; it was just a matter of remembering one real-world example of a mathematical object (e.g., a torus as a specific example of a manifold). Suddenly the math had context again. But what happens when many of the ‘first-level’ connections are already made? Well then you abstract the abstraction! This takes many different forms and is also not a well-defined concept, but the gist of it is true nevertheless. Apart from obvious abstractions like category theory, clever researchers defined new abstract structures on top of (or relating to) existing abstract structures, and pretty soon none of the math that they were studying has a direct interpretation in terms of applications. And to be clear: there was some useful mathematics that fell out of this progress. Whatever the case may be with the motivation for the math, some results were found to have very important implications in applied fields (e.g., cryptography, algorithms, robotics, guiding spaceships, etc.).
This is why the idea that “math is useful” and “makes things work” is both true and false. For most of math’s history, this was true. In particular, most of the math that is taught in high school and “makes things work” was discovered during periods in mathematical history where mathematicians focused on applications, or just after the end of this period (at the start of the post-modern period of math). During the post-modern period of math, the task fell to engineers and ‘applied mathematicians’ to look at real-world problems and try to tackle them with either new math or by interpreting the complicated math that has been established by pure mathematicians. Mathematicians themselves do not (in general) consider applications for their work.
Moreover, the fundamental driver for mathematical progress completely changed. Some other valuation metric had to take the place of “relevance to applications,” and this metric is today a sense of “aesthetic taste.” One example of a type of “beautiful result” is when two distant parts of math that are otherwise not commonly studied together are connected in a new way, e.g., the Langland’s program. Another type of “aesthetic” result is one stemming from a simple question but requiring very deep math to actually solve, e.g., the Geometrization/Poincare conjecture, which states essentially that three-dimensional shapes are “nice” (of course one then needs to define “nice” and then you’re dealing with manifolds and so you’ve got pages and pages of definitions…). A deep problem stemming back to ancient times is the distribution of the prime numbers, which is also closely related to the famous Riemann hypothesis, and both remain unsolved and closely tied to what is considered “important” in post-modern mathematics.
The shift in valuation metric is publicly admitted to different extents in different fields. While number theory has some of the most explicit potential for application (cryptography) within pure math, a relatively small fraction of researchers in the field actually consider cryptography when searching for research questions12. Algebraists are—in my experience—even less concerned with applications. Their ideal research problem is more-or-less starting with a set of axioms and wondering whether this set implies some other statement. I’m not trying to diminish their work, as I formed this opinion speaking to some of the most brilliant algebraists in the country, but it’s difficult to describe their work in any other way. Analysts—especially in Partial Differential Equation (PDE) theory—like to advertise that their research has many applications. I would not say that this is exactly false advertising, but it is not exactly honest either. What is definitely true is that PDEs arise very frequently in applications from engineering, to finance, to physics, to weather. Furthermore, it’s certainly true that familiarity with PDEs and knowing how to analyze them can be useful in some practical scenarios. What is not-so-certain is how relevant certain central parts of PDE theory are to applications. In practice, PDEs are frequently passed to numerical PDE solvers13 and the result is used as the real solution14. Outside of some very specific research roles in industry, it is unlikely that, e.g., existence (note: this is not about what the solution is, but if it even exists) and uniqueness of the solution of a PDE will be relevant to the engineer, sell-side quant, or weather forecaster,15 but these subjects do comprise a supermajority of the coursework in PDE theory and maintain an important status within research. Of course, different PDE theorists will study different types of questions, so the applicability really depends on what you’re actually studying. But reader be warned: just because a mathematician claims16 what they are studying “has applications to X” does not actually mean that someone doing “X” would ever think their work is remotely important or useful. In fact, the mathematician writing the paper likely does not know what the application is17!
The most difficult-to-parse field is mathematical physics. First of all, the field is so big that it encompasses too much for it to be a reasonable hope that the level of applicability would remain somewhat steady. What makes mathematical physics so much more confusing is that the name and aura of the field emphasizes applications to physics, but it is perhaps the most guilty of having researchers using the “this has applications to X” phrase without knowing what those applications are or (more importantly) explaining why a certain theoretical contribution aids those applications to begin with. I will discuss this point further in a future article, but the gist of it is this: if your article requires a PhD in pure mathematics and years of study in some niche part of mathematical physics for me to understand it, and if I am a physicists who does not know those things, I’m not going to commit to that massive startup cost in the hope that your article actually helps me. Instead, I’ll develop my own methods for solving my problems. So your “application to physics” is now buried so deep within pure mathematics that no one in physics will ever see it. Effectively, you do not have an application to physics.
And now we have arrived to the critical tension within the community of pure mathematics, and between the pure math community and the rest of the world. Pure mathematicians cannot justify their work through applications today (or when they can, their culture de-emphasizes connecting their work back down to the application and demonstrating what new insight was revealed by their theoretical result). A lot of the math community follows G.H. Hardy’s philosophy of “math as art.” At the same time, for the practical purposes of getting funding via grants and prizes, mathematicians borrow against their historical reputation that their math will eventually tie back to reality in a useful way. Mathematicians unabashedly cite vague connections to different applications but never actually do the work of connecting the theoretical results they develop to the application. This causes or will cause a major tension with the external world, especially with funding organizations.
Going Forward
Eventually, that historical credit of applicability will start (or arguably already has started) to run dry, and then suddenly grants will be significantly harder to obtain. The terrible financial model of academia that is overly reliant on grants and cheap student labor will cause pain in math departments, as it does for the humanities. At the same time, the few sub-disciplines within pure math that do find applications will continue to receive funding, mirroring the funding situation in the humanities where projects that can be connected to politically relevant themes have an easier time getting funding. Today, this is happening with mathematics related to AI; topics that can be related (at least in grant proposals) to theoretical advancements in understanding the training process for AI (which is not hard to formulate as a purely theoretical question) have seen an influx of funding. The result of this trend may be that certain fields of math stagnate and recede, and the well-funded subfields managing to demonstrate applications begin to dominate, leading to a much more homogenous set of subjects studied within math departments.
None of the previous paragraph is meant as normative statements, though I do think that—as a point of honesty—mathematicians should explore the contributions of their work to applications if they are going to cite said applications as a justification of their work. I.e., if you’re going to get funded to prove some super technical theorem, and in your grant proposal (and eventually the abstract of your paper) you say that this work “has applications to physics,” then you should have a section detailing exactly what the contribution within physics is (written in a way that physicists will understand). I want to emphasize that the dangers the math community face are, in some sense, fundamentally tied to the terrible funding model of universities, and not morally “the fault of” moving towards abstract mathematics. Pure math is wonderful and beautiful and very much worth studying. But it should not be surprising that electing to reject applications will put your academic discipline in a similar situation to humanity departments, whose work has great but intangible value. Moreover, the value of pure mathematics is even harder to understand than the humanities; when I read about the history of early modern French salons I can reflect about the real world and my own life, even for this relatively niche topic. But if I spend a few days understanding a theorem on the stable homotopy groups of spheres, I am unlikely to understand my place in the world any better than beforehand.
The last aspect of today’s pure mathematics I’ll mention is the small but growing movement within math to transition to computer-checked proofs. Before AI, there was already considerable effort within the mathematical community to essentially code up all of mathematics and have a computer check that everything is logically valid (by running the code). With the arrival of AI, this has in some sense sped up due to an unrelated motivation. I’ll talk more about this in a future blog post, but essentially the idea some AI companies are having is: “math is very rigorous and has very clearly-defined logical arguments. Maybe if we can train an AI model to prove new math, we can both: discover new results useful for applications (thus making a boatload of money), and maybe the AI model will pick up on the logical thinking required for pure math and generalize it outside of mathematics, thus tackling the AI hallucination problem (and probably making a second boatload of money).” There are other practical reasons for seeking to train a model to write mathematically rigorous (and verifiable) proofs, but I’ll save them for an article more focused on this subject. What is notable for the mathematical community’s future direction is that the most famous mathematician alive today (Terrence Tao) has gotten pretty deeply involved with both the computer verification of established mathematics and the potential for AI to write proofs. My personal hope is that from this movement there will at least arise a new community standard for all proofs to be verified by a computer; I feel if math’s claim to fame is its rigor, then mathematicians have a moral obligation to verify their work if technology allows for that.
If the whole AI formalization movement works out to at least a small degree, there is hope of automatically converting existing written proofs into computer programs that are verifiable, thus making the labor cost of formalizing mathematics very small. With that said, anyone who is familiar with today’s journal review process will tell you that it is deeply flawed, and that it is almost certain that there exist published mathematical papers with logical flaws. The main question then is: how significant are the flaws that exist? Due to important work being read by many different mathematicians in the course of building off of it, it is most likely that the flaws in established math will be relatively minor. Also, since mathematicians work by using a sense of intuition for what is true and then formalizing that intuition, it is likely that many of the results with problematic proofs can be recovered. On that note, I’ll leave you with the following interaction I had with a tenured computer science professor while I was a math major (circa 2019):
Me: “Hey have you heard about these formalization languages for checking math?”
Prof: “No, what are they?”
Me: [explanation]
Prof: “I bet they’ll find a lot of mistakes!”
And wrong on many different levels too!
This quote is by G.H. Hardy in the afore-linked book “A Mathematician’s Apology”. The book is an argument for the justification of math as an academic discipline, and from the get-go it rejects any and all justifications related to applications. G.H. Hardy strongly believed that the applications are happy coincidences, but that to a mathematician they shouldn’t really matter for their work and/or mathematical interests, at one point in the book proclaiming that the “uselessness” of “real math” is one of its virtues.
The reason that I am citing this book (as opposed to a collection of sources) is that it is the quintessential argument for the justification for pure math. Students of pure math around the world come across this book as a matter of culture in their field, and it is the source that is always cited in online forums discussing either “why math"?” or “what is math?”.
“Counting” as a field in modern math and today’s math is one of the hardest fields that exists, and also means a completely different thing than counting as the ancients were doing it. To put it crudely, ancients were interested in how many of [real thing] will I have after doing Y. Or, how many revolutions of Earth does it take before the seasons repeat?
So reader be warned: if your professor at university says they study counting, understand that they are likely one of the smartest people you will ever meet.
Source (which is also an interesting read going more specifically into the development of rigor in math).
Sometimes the reason given was a “lack of rigor,” but given that everyone at this time period was not being rigorous enough, I’m classifying this as “arbitrary.” In particular, note that Fourier’s works ended up being completely formalized later, similar to other great mathematicians’ works of this time, so the rejection of Fourier’s works due to “lack of rigor” should be taken in context.
“What is amazing here is Cauchy's attitude. He totally disregarded Fourier's counterexample and did not admit having made a mistake: not only did he "prove" his theorem, but he repeated it in a paper read to the Academie des Sciences as late as 1853.” (page 233 in Segre, Michael. Peano's axioms in their historical context. Arch. Hist. Exact Sci. 48 (1994), no. 3-4, 201-342)
Cauchy both made an error due to lack of rigor and was one of the mathematicians leading the charge for a more rigorous mathematics.
p. 208 in Bottazzini, Umberto. The higher calculus: a history of real and complex analysis from Euler to Weierstrass. Translated from the Italian by Warren Van Egmond. Springer-Verlag, New York, 1986
I’m morally obligated to say here that Cantor did some of the most beautiful math ever. If you’ve ever asked yourself “can one infinity be larger than another,” then you might find it interesting to know that Cantor definitively answered that question with a ‘yes’.
There was a debate during this time on the philosophy of mathematics, and Hilbert led the so-called “formalist” camp.
It is worth describing a little more what Gödel proved and the cultural and epistemological ramifications of his (second) incompleteness theorem (though this also could be its own blog post or an entire semester-long course). What Gödel proved was that you cannot use just the axioms of arithmetic to prove that arithmetic does not have contradictions (“is consistent”). You need to use a stronger set of axioms containing those of arithmetic. But then, this stronger set of axioms cannot prove its own consistency either, and you need to embed those axioms in a still-stronger set, and so on. This creates a fundamental epistemological problem that you can never know if the full axiomatic framework you have established is consistent; you can only ever prove that a strict sub-collection of the axioms are consistent. Thus to get some proof of consistency for the theory you are working in, you always have to make an assumption outside of that theory.
The cultural ramifications of this “incompleteness” is that mathematicians essentially make the aesthetic judgement that arithmetic is consistent and move on with their lives: “Arithmetic is simple and well-understood, and if it is inconsistent (has a contradiction), then everything else in our world is more-or-less wrong anyways, so let’s just accept that it’s true and focus on other questions.” In particular, mathematicians can prove that the theory resulting in adding certain sets of axioms to arithmetic result in a consistent theory if and only if arithmetic is consistent (known as a conservative extension), and so you can work in a fairly complicated theory and only rely on the assumption that arithmetic is consistent to know that the full theory is consistent as well. A similar second option is to instead work in ZFC (“Zermelo-Frankel set theory with Choice”), which is also well-understood and “seems” consistent, and within which you can prove that arithmetic is consistent (but still ZFC cannot prove its own consistency due to Gödel!). All of the math the average pure math PhD student studies is contained within ZFC.
For what it’s worth, I’ve heard from friends studying number theory that the NSA still loves number theory PhDs.
It should then be noted that numerical PDEs (the field related to solving PDEs using algorithms) lies within the domain of applied mathematics, and that these numerical PDE solvers have theoretical guarantees bounding the error from the true solution.
An interesting nascent field is the training of neural networks to solve PDEs. This is potentially useful in higher dimensions where ‘classical schemes’ for numerically solving a PDE become too computationally costly to run. However, unlike classical schemes, there are no convergence guarantees for neural networks solving PDEs, and so the user is basically hoping that the solution they get is close to the true solution.
There are probably some physicists that care about existence of solutions to PDEs. At the same time, some of the most famous PDEs arising out of physics tormented mathematicians for decades because it was not possible to show the existence or uniqueness of these PDEs in the “usual sense” (while physicists kept using them as if they were interpretable in the “usual sense”) and so Schwartz and other mathematicians (most recently Hairer) had to come up with a suitable way to re-interpret the PDE so that in some new context it did admit a solution.
I am guilty of this.
I am also guilty of this.