Philosophy of Mathematics Formalisation

Quick Recap In the last session we looked at several familiar number systems: the naturals, integers and rationals. We also briefly introduced the idea of the real numbers, which are designed to represent the points on a continuous line. These are at least the rationals plus all the irrationals we get by taking square roots, cube roots and so on; maybe others, too, who knows? But the real numbers are a lot stranger than the more familiar ones. We can easily make mistakes if we rely on our intuition to guide us.

Know Your Number System!

PART 1: MOTIVATION What was wrong with maths around 1800?

Irrationality of √2 It had been known since the time of the ancient Greeks that √2 can’t be expressed as a fraction – at least not exactly. In modern terms we say it’s “irrational” – meaning it isn’t found among the rational numbers. The real numbers are supposed to contain “all the numbers we need” – rational and irrational alike.

Dirichlet Function Even worse is the Dirichlet function (1829): f(x) = 1 if x is rational, 0 otherwise. At first blush it’s not clear what’s going on with this function at all. We can’t even draw a graph of it. In fact we can prove (with a suitable theory) that the Dirichlet function is nowhere continuous. This is indeed a strange property.

Analysis in the C19 Early analysts were concerned with finding technical solutions to practical problems arising from the rapid development of the calculus: Augustin Louis Cauchy (1789–1857), Bernhard Bolzano (1781–1848), Niels Henrik Abel (1802–1829), Peter Lejeune Dirichlet ( ) Karl Weierstrass (1815–1897) Bernhard Riemann (1826–1866) A little later, in response to their work, focus fell on the nature of number and of the continuum and the project became philosophically deeper: Leopold Kronecker ( ) Richard Dedekind ( ) Georg Cantor ( ) Giuseppe Peano ( )

Axiom of Completeness

Cauchy’s Version of Convergence Cauchy’s Cours d’Analyse of 1821 can be thought of as the first attempt to write a textbook in rigorous analysis. It was hugely influential. Note the date: this is just the time when Dirichlet is pointing to problems in the foundations of Fourier’s use of calculus to solve the heat and wave equations. We’ll focus on the idea of a continuum of points, not on continuity of functions. The ideas are basically the same, but clearer in the former case. Here is a modern definition of continuity of a sequence in the spirit of Cauchy (notice: no mention of limits!) :

A Definition of the Reals

The Real Numbers

Density

PART 2: FORMALIZATION The proposed solution to all these problems

The Need for Rigour Actual infinities pose a serious problem: they’re difficult to think about and it’s easy to be led into absurdities (i.e. contradictions). Kant’s antinomies warm us about this: for him, there are limits beyond which reason can go, but knowledge can’t. Yet infinities lurk at the heart of much mathematics, including the calculus that underpins almost all of modern physics. Can we find a way to think about them that’s robust enough to save us from both absurdity and ignorance? Can questions like the Continuum Hypothesis be decided ion a principled way, or do they simply lie beyond what anyone’s capable of knowing?

The Axiomatic Method In modern terminology, every statement in Euclid’s Elements belongs to one of the following: An axiom – that is, a definition or something taken to be basic. A rule of reasoning that allows you to make deductions. A theorem – that is, something claimed to be true. A proof – that is, an argument showing that the theorem necessarily follows from the axioms. The axioms are as minimal as possible. If you don’t agree with them, nothing in the Elements works for you. A valid proof shows why you’re forced to accept the theorem if you accept the axioms and follow the rules of reasoning. The axioms and rules should therefore be “intuitively true” in some sense. A lot of high-powered C18 and C19 mathematics has abandoned this approach. It sets a very high standard but appears to be too difficult (or at least too slow) for advanced problems.

Formal Languages A formal language isn’t much like a “natural” language like English or French. Nor does it look like the kind of mathematics human beings do. It looks more like the kind of code a computer might use. It consists of: a restricted set of symbols; A set of rules for combining them to make expressions; Another set of rules for combining expressions to make proofs.

Syntax vs Semantics What we’ve just described is syntax: mere symbol-shuffling. We like to think there’s another layer: the meaning of the symbols. This is their semantics. Without this, a formal language is just a kind of abstract game. To matter to us, it must be about something. Still, we can use it to state (purely abstract) axioms and proofs of theorems. Given a set of formally-encoded axioms, the set of all the theorems we can prove from them is called their theory. This is captured more formally by the idea of an interpretation of the pure symbolic syntax. We find a model – something outside the theory that it’s suppposed to be “about”. Then we map the symbols of the language, one by one, onto features of the model.

Contradiction A theory is consistent if it doesn’t prove any contradictions. If you can produce a proof of “P” and of “not P”, the theory is inconsistent. Assumption: if a theory is inconsistent, it has no model. That is, there are no “real” contradictions. It follows that if a theory demonstrably has a model, it’s consistent. Ex falso quodlibet It’s a feature of classical logic that if you can prove a contradiction, you can use that contradiction to prove any other statement you like, even if the two have nothing to do with each other.

Suppose we, who are working in a formalized language, have a theory that we interpret as being about sea creatures. Suppose this theory can prove “Whales are fish” and also, by a different line of argument, “Whales are not fish”. We may now prove “Humans are fish” as follows: “Whales are fish” is true, so “Either whales are fish or humans are fish” is also true. This is because “either X or Y” is true if X is true, regardless of whether Y is true. But “Whales are not fish” Well, “Either whales are fish or humans are fish” and “Whales are not fish”. So it must be that humans are fish! This is because when “either X or Y” is true and X is false, Y must be true. Nobody thinks this is a good proof that humans are fish. But it’s not so easy to point to one step that causes the trouble.

Peano Arithmetic If we’re going to express mathematics formally, that had better include the basic system of whole-number arithmetic before trying to do all the fancy bits. This already includes non-trivial things like Goldbach’s Conjecture. First-Order Peano Arithmetic gives a simple formal theory that “obviously” has the arithmetic of the natural numbers as a model. That is, its axioms are explicitly chosen for that purpose. Most of it is uncontroversial, but we do need to add some machinery to do induction, which is how we get to prove a statement like “every even number is the successor of an odd number”. Without induction, we can prove for any particular even number, that it’s the successor of an odd number; we just can’t prove it for all of them at once. This is a direct consequence of the fact that there are infinitely many of them, and it’s where the trouble starts.

Russell’s Paradox Naïve set theory includes the Principle of Comprehension. Anything is a set that can be described by a suitable formal language that can be interpreted as making consistent statements about sets and their elements. This was, broadly speaking, Frege’s approach in his attempts to express arithmetic purely in terms of set theory (actually it’s more complicated than that, but we’re close enough). The PoC licenses us to declare: “S is the set of all sets that don’t contain themselves” S is a contradictory object; any theory that allows this sentence to be proved is absurd: ex falso quodlibet. Yet we don’t need much “machinery” to do it: just the ability to refer to sets and what they include.

A Version of the ZF Axioms Extensionality: Two sets that have exactly the same elements are the same set. Regularity: All sets are well-founded. Replacement: This is an axiom schema that allows us to “build up sets from other sets” by saying things like “X is the set of all x such that…” without falling into Russell’s Paradox. You’ll sometimes see an “Axiom of Separation” as an alternative. Null set: there exists a set of which nothing is an element Unordered pairs: given sets x and y, there exists a set {x, y} Union: given sets x and y, x union y exists. Power set: given x, its power set exists. Infinity: A specific infinite set exists.

The Axiom of Infinity This axiom guarantees the existence of a set that can be interpreted as “the set of natural numbers”. There are various ways to build sets that can be interpreted as natural numbers – the details aren’t important. This axiom plays a similar role to induction in Peano Arithmetic. Without it we have finite set theory, which would appear to be a theory of the potential infinite only. It can say “this even number is the successor of an odd number” for any even number you choose. It can say “in this set of even numbers, every one is the successor of an odd number” for any set of numbers you like, however large. But it can’t say “every even number is the successor of an odd number”. The point to note is that we can’t get our first infinite set just by juggling with finite sets. We must “import” or “assert” it.

Hilbert’s Programme “Real” mathematics is given to us in (Kantian) intuition. Facts like 2+2=4 are really facts and we can know and prove that they’re true from first principles (a priori). Is all mathematics “real” in this sense? In particular, what about calculus (which relies on the continuum) and Cantor’s set theory with its multiple sizes of infinity? Perhaps this lies outside the bounds of knowledge, in the pure play of reason where antinomies are possible. Hilbert hopes not, though! The way to draw this boundary is to reconstruct as much mathematics as possible formally, with great rigor and care, out of elements like basic arithmetic that are undoubtedly part of our immediate intuition. To proceed rigorously we need axioms for each field of maths. But in the more exotic fields, how can we tell our axioms are well- formed, i.e. that they don’t give rise to a contradictory theory? We must prove it. Mathematics owes us a proof of its own consistency.

Hilbert’s Programme 1.Express statements about infinite objects as finite strings of symbols in a precisely-defined formal language. 2.Express rules of inference as precisely-defined operations on symbol strings. 3.Derive theorems from the axiom strings by finitely many applications of rules of inference. 4.Prove that the rules of inference produce no contradictory sentence, by purely finite reasoning about strings of symbols.

Hilbert wrote these words in 1922, responding to the intuitionist programme of Weyl, Brouwer and others. His aim, then, is to preserve the achievements of the C19 against skepticism and paradox.

Propositional Logic This formal language uses letters to represent propositions, such as “Whales are fish”. The built-in symbols, besides letters, are ¬ (“not”), | (“or”) and & (“and”), along with parentheses “(“ and “)”. The rules for forming expressions are: Every letter on its own is an expression If X is an expression, so is ¬(X) If X and Y are expressions, so is (X | Y) If X and Y are expressions, so is (X & Y) Nothing else is an expression We then need a set of rules for combining expressions into proofs. These are usually pretty short but in the interests of space we won’t set this out here. Here is the proof we just did on the previous slide: P (P | Q) ¬(P) (¬(P) & (P | Q)) Q

Predicate Logic