This will be a series of lectures on the philosophy of mathematics, given at Oxford University, Michaelmas term 2018. The lectures are mainly intended for undergraduate students preparing for exam paper 122, although all interested parties are welcome.

My approach to the philosophy of mathematics tends to be grounded in mathematical arguments and ideas, treating philosophical issues as they arise organically. The lectures will accordingly be organized around mathematical themes, in such a way that naturally brings various philosophical issues to light.

Here is a tentative list of topics, which may be updated as the term approaches.

**Lecture 1. Numbers.** Numbers are perhaps the essential mathematical idea, but what are numbers? We have many kinds of numbers—natural numbers, integers, rational numbers, real numbers, complex numbers, hyperreal numbers, surreal numbers, ordinal numbers, and more—and these number systems provide a fruitful background for classical arguments on incommensurability, the irrationality of $\sqrt{2}$, transcendental numbers, the infinitude of primes, and lead naturally to discussions of platonism, Frege’s number concept, Peano’s numbers, Dedekind’s categoricity arguments, and the philosophy of structuralism.

**Lecture 2. Rigour.** Let us consider the problem of mathematical rigour in the development of the calculus. Informal continuity concepts and the use of infinitesimals ultimately gave way to formal epsilon-delta limit concepts, which provided a capacity for refined notions, such as uniform continuity, equicontinuity and uniform convergence. Nonstandard analysis resurrected the infinitesimal concept on a more secure foundation, providing a parallel development of the subject, which can be understood from various sweeping perspectives. Meanwhile, increasing abstraction emerged in the function concept, which we shall illustrate with the Devil’s staircase, space-filling curves and the Conway base 13 function.

**Lecture 3. Infinity.** We shall follow the allegory of Hilbert’s hotel and the paradox of Galileo to the equinumerosity relation and the notion of countability. Cantor’s diagonal arguments, meanwhile, reveal uncountability and a vast hierarchy of different orders of infinity; some arguments give rise to the distinction between constructive and non-constructive proof. Zeno’s paradox highlights classical ideas on potential versus actual infinity. Time permitting, we shall count into the transfinite ordinals.

**Lecture 4. Geometry.** Classical Euclidean geometry, accompanied by its ideal of straightedge and compass construction and the Euclidean concept of proof, is an ageless paragon of deductive mathematical reasoning. Yet, the impossibility of certain constructions, such as doubling the cube, trisecting the angle or squaring the circle, hints at geometric realms beyond Euclid, and leads one to the concept of constructible and non-constructible numbers. The rise of non-Euclidean geometry, especially in light of scientific observations and theories suggesting that physical reality may not be Euclidean, challenges previous accounts of what geometry is about and changes our understanding of the nature of geometric and indeed mathematical ontology. New formalizations, such as those of Hilbert and Tarski, replace the old axiomatizations, augmenting and correcting Euclid with axioms on completeness and betweenness. Ultimately, Tarski’s decision procedure hints at the tantalizing possibility of automation in our geometrical reasoning.

**Lecture 5. Proof.** What is proof? What is the relation between proof and truth? Is every mathematical truth, true for a reason? After clarifying the distinction between syntax and semantics, we shall discuss formal proof systems and highlight the importance of soundness, completeness and verifiability in any such system, outlining the central ideas used in proving the completeness theorem. The compactness theorem distills the finiteness of proofs into an independent purely semantic consequence. Computer-verified proof promises increasing significance; it’s role is well illustrated by the history of the four-color theorem. Nonclassical logics, such as intuitionistic logic, arise naturally from formal systems by weakenings of the logical rules.

**Lecture 6. Computability.** What is computability? Gödel’s primitive recursive functions were a robust class, yet he gave reasons to despair of a fully satisfactory answer. Nevertheless, Turing’s machine concept, growing out of his careful philosophical analysis of computability, laid a foundation for the contemporary computer era, and the widely accepted Church-Turing thesis asserts that Turing has the right notion. Meanwhile, the distinction between computable decidability and computable enumerability, highlighted by the undecidability of the halting problem, shows that not all mathematical problems can be solved by machine, and a vast hierarchy looms in the Turing degrees, an infinitary information theory. Complexity theory refocuses this on the realm of feasible computation, with the still-unsolved P vs. NP problem standing in the background of nearly every serious issue in theoretical computer science.

**Lecture 7. Incompleteness.** The Hilbert program, seeking to secure the consistency of higher mathematics by finitary reasoning about the formal system underlying it, was dashed by Gödel’s incompleteness theorems, which show that no consistent formal system can prove even its own consistency, let alone the consistency of a higher system. We shall describe several proofs of the first incompleteness theorem, via the halting problem, via self-reference, and via definability. After this, we’ll discuss the Rosser variation, the second incompleteness theorem, and Tarski on the non-definability of truth. Ultimately, one is led to the inherent hierarchy of consistency strength underlying all mathematical theories.

**Lecture 8. Set theory.** We shall discuss the emergence of set theory as a foundation of mathematics. An initially naive theory, challenged fundamentally by the Russell paradox, grew into Zermelo’s formal set theory, founded on the idea of a cumulative universe of sets and providing a robust general context in which to undertake mathematics, while also enabling the clarification of fundamentally set-theoretic issues surrounding the axiom of choice, the continuum hypothesis and an increasingly diverse hierarchy of large cardinal concepts. The development of forcing solved many stubborn questions and illuminated a ubiquitous independence phenomenon, feeding into philosophical issues concerning the criteria by which one should add new axioms to mathematics and the question of pluralism in mathematical foundations.

This class sounds great! Those are some lucky kids!

Joel, this sounds wonderful! Unfortunately I won’t be crossing the pond. Will notes, texts, or videos eventually become available? I am particularly intrigued by and interested in your lecture descriptions on Number, Infinity, and Set Theory. (I once taught a semester elective on The Mathematics of Infinity at The Beekman School in NYC).

All the best,

Charlie Sitler

I’m not sure about videos, although I shall inquire about it. Meanwhile, I am preparing lecture notes, which with some work, may eventually become a book.

The structure of the course is quite different from many introductory courses of philosophy of mathematics that I know. It will be a very interesting book. Looking forward to it.

Yes, indeed. I was a little worried about that, but decided to organize things in the way that I understand the subject, in a way that makes sense to me, leading to the philosophical issues from the mathematics. How is your syllabus organized? I would be interested to learn.

Sounds like a well-rounded introduction that can be very useful. How much time will you have to explain that set theory didn’t just spring into existence and that it took some convincing (by Skolem I think) that first-order logic with set theory was an option. Many people (including Hilbert) expected that first-order logic was insufficient (which it is, but that’s a debate for another day).

Oh yes, this is a very interesting part of the history. Zermelo’s original formulation was essentially second-order, and this was important in his categoricity proof, showing that the models of his theory are precisely the $V_\kappa$ for inaccessible cardinals $\kappa$. But ultimately, he lost the battle and the theory now bearing his name is nearly universally understood as a first-order theory.

Put me down as another eager customer for the book version!

Glad to hear it! Thanks for the vote of confidence.

The purpose of all mathematics (including large cardinals) is to build taller skyscrapers, longer bridges, and better I-phones; any philosophy of mathematics needs to reflect this purpose.

I suppose we’ll discuss that view. What do you think are the strongest arguments in its favor? Meanwhile, it seems to me that many mathematicians also study mathematics for reasons that seem to have little to do with applications, and are rather closer to values of art and beauty. Or perhaps it is similar to the reasons why some scientists find scientific investigation of nature to have value, even when it is divorced from practical application. Do you think they are wrong to do so? I suppose they could stop, and become taxi drivers or software engineers, but from my perspective, we would be worse off if they did so.

Mathematicians should study the areas of mathematics that they deem elegant, but these areas of mathematics are more often than not applicable to the universe we live in.

The applicable areas of mathematics are often much deeper than the unapplied areas of mathematics. For example, the mathematical objects used in cryptography need to be ‘structured’,’combinatorially complicated’, and ‘easily computable’ objects. These characteristics that make a certain mathematical object suitable for cryptography are the same characteristics that make a mathematical object interesting for purely aesthetic reasons and also make it possible to develop a deep theory behind such a mathematical object. Furthermore, any ‘structured’,’combinatorially complicated’, and ‘easily computable’ mathematical structure has a fairly good chance of being applicable to cryptography.

The creator of the universe has decided to make the laws of nature based upon the most useful and beautiful mathematics. Mathematicians who are interested in mathematics for its own sake and for the sake of beauty should do more applied mathematics than pure mathematics. After all, applied mathematics is the mathematics that the creator likes enough to actually use, and we should trust that the creator has good mathematical taste.

As a case study of the elegance of applied mathematics, let’s look at the braid groups. The braid groups have diverse applications including cryptography (braid based cryptography has suffered from attacks recently though; it is a work in progress), quantum computation, and several other areas that I do not know anything about. On the other hand, the theory of braid groups is one of the most profound mathematical theories encompassing diverse areas of mathematics including set theory (the relation between braids and very large cardinals is currently undeveloped and unclear but as time goes these two areas will be unified). A good objective way to test the elegance of a mathematical structure is to extract a countable ordinal from the structure in a natural, computable, and unexpected manner. The Dehornoy ordering on the positive braids (this linear ordering appears in many different contexts) has order type w^(w^w) which exceeds the proof theoretic ordinal of theories such as PRA. The fact that a reasonably large computable ordinal could be extracted from the positive braids suggests that braid groups have a combinatorial depth beyond what one would find in pure mathematics.

The process of doing applied mathematics is also more satisfying than the process of doing pure mathematics since with applied mathematics since applied mathematics often requires a more well-rounded skill set. Applied mathematics is more interdisciplinary than pure mathematics, and the interdisciplinary nature of applied mathematics fosters development instead of stagnation. For instance, with applied mathematics, instead of simply proving theorems and making definitions, one will also make algorithms, perform computer experiments to gain knowledge that cannot easily be gained simply by writing theorems and proofs, and use the knowledge gained from

computer experiments to prove more theorems.

Any good area of mathematics including set theory should be broad enough to have practical applications. I believe that in the future, large cardinals due in-part to their high consistency strength, will have a prominent place in more traditional areas of mathematics and will be used to construct new mathematical structures that will have real world applications in areas such as cryptography.

Pure mathematics is ok, but even pure mathematicians need to give applied mathematics a chance. A pure mathematician would be much better off by investigating some applied areas.

In which room/time are you giving these lectures? I might want to drop in on the 3 forthcoming ones if there’s space.

The lectures are in the Oxford Faculty of Philosophy building, the Radcliffe Humanities building, in the Lecture Room on the 2nd floor, Thursdays 12-1.