Proving Theorems with Computers
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Proving theorems with computers Kevin Buzzard ∗ May 1, 2020 Superhuman mathematics One might hence ask the following question: if one human had an understanding of all of modern How is breakthrough mathematics achieved? Here is pure mathematics simultaneously, how much further one example, from algebraic number theory. would they immediately be able to see? How many In his 1987 paper \Deforming Galois representa- insights are there which are needed to unblock one tions", Barry Mazur observes that the geometric con- field, but which have already been made in another? cept of a smoothly-varying complex family of rep- This question can of course be dismissed as a resentations of a group has an arithmetic analogue. thought experiment. Perhaps it is the kind of thing He sets up deformation theory in the purely alge- which philosophers might muse over, but it is the year braic setting of mod p and p-adic Galois representa- 2020 and pure mathematics is much too big for one tions, and makes some interesting observations about human to comprehend. the relationship between deformation rings and Ga- However, it is the year 2020 and hence we have lois cohomology. By 1990, Mazur and Tilouine have computer proof systems which are now in theory ca- raised a profound question about whether a certain pable of understanding all of modern pure mathe- universal deformation ring coming out of this the- matics. So why is our community not teaching it to ory is isomorphic to one of Hida's Hecke algebras. them? This is entirely within our grasp, and we have In 1993 Wiles uses new techniques in commutative absolutely no idea what will happen when we do. algebra to reduce a variant of this question to a nu- merical criterion, and a year later, aided by Taylor, he has pushed the strategy through. The semistable 1 Teaching mathematics to hu- Shimura{Taniyama conjecture (at that time often called the semistable Shimura{Taniyama{Weil con- mans jecture) follows, and hence, by earlier work of Ribet, Fermat's Last Theorem. Let us look at the way pure mathematics is learnt by This is but one of very many examples where humans, from undergraduate to PhD level. cross-fertilization has occurred in mathematics. The breadth of Mazur's mathematical knowledge (he was 1.1 The basics initially a topologist) played a key role here. In a 2014 article [Maz14] for the Math Intelligencer, Mazur Three key concepts in pure mathematics are the def- writes: \Reasoning by analogy is the keystone: it inition, the theorem statement, and the proof. We is present in much (perhaps all) daily mathematical will talk more about this trichotomy later on, but for thought, and is also often the inspiration behind some now let us focus on the concept of proof. A course of the major long-range projects in mathematics". introducing the formal notion of proof might cover concepts such as sets, functions and binary relations, ∗Kevin Buzzard is a professor of mathematics at Imperial College London. His email address is and basic theorems about these objects will be care- [email protected]. fully proved. For example, there might be a proof 1 that distinct equivalence classes for an equivalence 1.2 Developing intuition relation are disjoint. There are many ways that one can attempt to teach this to undergraduates. Re- After a while it becomes inconvenient to do mathe- cently I have become fond of a method where I bring matics in a purely axiomatic fashion. For example, a set of around 100 plastic shapes coloured red, yel- proving that if we remove a finite set of points from 2 low, green and blue into class. Two shapes are de- R then the resulting topological space is still path fined to be equivalent if they have the same colour, connected could of course in theory be done from and it is not hard to convince the students that this the axioms, but in practice, rather than attempting is an equivalence relation, and that the red shapes to write down the function defining a path between form an equivalence class, the blue shapes form an- two arbitrary points in the space, one would just other, and so on. The fact that distinct equivalence draw a picture. The same is also true when prov- classes are disjoint is now obvious. However this is an ing basic results about contour integrals in complex example, not a proof. The formal proof, which I then analysis { there are several \proofs by picture" in a go on to show the students, is a series of elementary typical development of the theory. The concept of steps, each of which follows from the rules of logic or a simple closed curve in the plane having an inside the axioms of an equivalence relation. and an outside will often be taken as read, although very few students will have seen a proof of the Jor- dan curve theorem at this point in their mathemat- ical education. Over time, students learning math- In proofs such as these, we are operating very close ematics begin to understand our unwritten rules of to the \machine" which drives formal mathematics { \what is allowed in practice". One is reminded of the machine which tells us that if P is true, and if P the apocryphal story of a student asking their pro- implies Q, then Q is true, and other such logical rules. fessor whether the fact just presented to the class as This axiom-based attitude continues to play an im- \obvious" was indeed obvious, and the professor go- portant role in subsequent classes such as first courses ing into deep thought to emerge 20 minutes later with in group theory, linear algebra, and real analysis. The the reply \yes". By this point in the development of a real numbers might be presented as a complete totally student's education, lecturers are expecting the stu- ordered archimedean field, that is, a structure satis- dents to \learn to fly”. Arguments in lectures may fying a list of axioms. Using only these axioms we can take place high above the axioms, with technical de- build a basic theory of real analysis from first princi- tails being dismissed as obvious or easy to verify, and ples. The theories of sequences, limits, infinite sums, left to the reader (perhaps with some hints). This is the beginning of what Terry Tao [Tao09] has called continuous functions R ! R, differentiation, integra- tion and so on can all be carefully built from the ax- the post-rigorous stage of mathematics. To borrow ioms for complete totally ordered archimedean fields. a phrase from computer science, students begin to Similarly, a lecturer presents the axioms of a group in learn the intuitive \front end" of mathematics. a first course on group theory, and from these axioms we can build the theory of subgroups, normal sub- 1.3 PhD Research groups, group homomorphisms, kernels, images and quotient groups, and prove the first isomorphism the- Those students who convince us that they can steer orem for groups. At this stage in the development of their mathematical arguments correctly are rewarded mathematics, every proof can be chased right down by being given PhD places. The prize for this \level- to the axioms of the system we are considering, and ling up" is that they are allowed access to the mathe- students are expected to learn from such courses that matical literature, and from now on they can assume mathematics can be done in this way. Much (but, as any result they like, as long as it is published in a we are about to see, not all) of undergraduate pure reasonably prestigious journal and their advisor be- mathematics is of this form. lieves it. A typical PhD thesis in pure mathemat- 2 ics will contain new proofs of results in a given the- typesetting system. Thirty years ago, only a small ory. In my personal case, this theory was the theory number of (typically young) mathematicians knew of p-adic Galois representations attached to modular how to write LATEX files, but now most of us do; this forms. By the time they graduate, a PhD student will program is now the standard typesetting system for typically know many theorem statements concerning mathematicians. If one runs the LATEX program on the objects they chose to study, and may well have a LATEX file, one of two things can happen. The file contributed to this list of theorem statements them- might contain errors, for example perhaps we acci- selves. Again to borrow a phrase from computer sci- dentally used a command in normal text mode which ence, the student knows the interface to each object is only valid in maths mode. In this case the LATEX in their area of expertise. The student is allowed to editor we are using will typically flag these errors and assume any results in the interface, and might well ask that we fix them. But when all the errors are know how to prove some of them { but possibly not gone, the LATEX program will compile the LATEX file, all of them. For example, when I was a PhD student, and the output will be a (hopefully) beautifully type- I had not read the details of the proof of the theo- set document. This document is typically a pdf file rem of Deligne which attached a p-adic Galois repre- nowadays, which can be read on a screen, printed out, sentation to a modular form, a key result from the or of course sent to another computer on the internet.