The Search for Foundations

An argument for a particular philosophy of mathematics.

“Alert readers recognized the book as a work of philosophy – a humanist philosophy of mathematics.” Now, “humanist” is a bad word in some religious circles, since

one brand of humanism, secular humanism, rejects the need for a deity. I’m not sure about Hersh’s use here, but it’s clear he does reject the need for some “ideal” mathematical world, and mathematics as “the mind of God.” I believe it can be read to just mean “arising from human values and concerns.” In this sense, medicine, political science, art, science, and sports are all “humanist.”

So what is the “humanist philosophy of mathematics” that they espouse?

1. Mathematics is human. It’s part of and fits into human culture.

2. Mathematical knowledge isn’t infallible. Like science, mathematics can advance by making mistakes, correcting and recorrecting them. (This fallibilism is brilliantly argued in Lakatos’s Proofs and Refutations.)

3. There are different versions of proof or rigor, depending on time, place, and other things. The use of computers in proofs is a nontraditional rigor. Empirical evidence, numerical experimentation, probabilistic proof all help us decide what to believe in mathematics. Aristotelian logic isn’t the only way to decide.

4. Mathematical objects are a distinct variety of social-historic objects. They’re a special part of culture. Literature, religion, and banking are also special parts of culture. Each is radically different from the others.

There’s no need to look for a hidden meaning or definition of mathematics beyond its social-historic-cultural meaning. Social-historic is all it needs to be. Forget foundations, forget immaterial, inhuman “reality.”

The Euclid Myth – the belief that the books of Euclid contain truths about the universe that are clear and indubitable. Starting from self-evident truths, and proceeding by rigorous proof, Euclid arrives at knowledge which is certain, objective, and eternal.

Indubitable: “Impossible to doubt; unquestionable.”

Rationalism: The view that reason is the source of knowledge or justification. Reason is privileged over all other sources of knowledge.

“Reason was the faculty that allowed man to know the Good and to know the Divine. The existence of this faculty was best seen in mathematics.”

Empiricism: The view that knowledge or justification comes from sensory experience (or amplifications of sensory experience, e.g. radiotelescopes). It emphasizes the role of evidence and experience.

Materialism: The view that physical matter is the only reality, and that all phenomena are the result of physical interactions.

Empiricism and materialism are the basis of modern science.

Rationalists lost, and empiricists won.◦ “The accepted assumptions in science are now. . .

those of materialism with respect to ontology, and empiricism with respect to epistemology.”

Except for mathematics:◦ “For the rationalists, mathematics was the best

example to confirm their view of the world. For empiricists it was an embarrassing counterexample which had to be ignored or somehow explained away.”◦ “The mathematician-in-the-street, with his

common-sense belief in mathematics as knowledge, is the last vestige of rationalism.”

Kant attempted to unify rationalism and empiricism.

A Critique of Pure Reason

Concern for a priori knowledge – knowledge which can be timeless and independent of experience. ◦ Analytic a priori – known to be true by logical

analysis◦ Synthetic a priori – inborn, like intuition of space

and time.

Kant: Knowledge of space is systematized in geometry. Euclidean geometry is forced upon us by the way our mind works, our synthetic a priori knowledge.

The Euclid Myth is alive and well in Kantian philosophy. Indeed, it was alive and well for nearly everyone – mathematicians, philosophers (including, reluctantly, empiricists) until well into the 1800’s. Then disaster struck our happy little village.

Non-Euclidean Geometry Examples of space-filling curves and

continuous, nowhere-differentiable functions brought into question our geometric intuitions.

Solutions:◦ Base real numbers on positive integers◦ Base positive integers on sets.

Peano Axioms:◦ 1. Zero is a number. ◦ 2. If a is a number, the successor of a is a number. ◦ 3. Zero is not the successor of a number. ◦ 4. Two numbers of which the successors are equal

are themselves equal. ◦ 5. (Induction axiom.) If a set X of numbers contains

zero and also the successor of every number in X, then every number is in X.

From these you can build the Whole Numbers, or prove that some object you create behaves like the whole numbers.

From the Whole Numbers, you can build:◦ Integers as equivalence classes of ordered pairs of

whole numbers: , , iff .◦ Rational numbers as equivalence classes of ordered

pairs of integers (whose second elements are never 0): , , iff .◦ Reals as carefully defined infinite sets of rational

numbers (such as Dedekind cuts or Cauchy sequences).◦ And so on to complex numbers, quaternions, etc.

From set theory, you can build the counting numbers:

and so on. . . .

Actually, to do this you need the Axiom of Infinity in your Axioms for Set Theory:

and

This gives you an infinite set that is also “inductive” – you can define and prove things by induction.

Anyway, after the discovery of non-Euclidean geometry and “monsterous” functions in calculus, the foundations of mathematics –the bedrock, immutable truth – moved from geometry to arithmetic and eventually to set theory.

If we could just prove set theory, which was very close to being logic, was a good foundation, we’d be back to having a strong foundation for all of mathematics.

Gottlob Fregeattempted to show that arithmetic (i.e. the natural numbers) could be built from set theory.

Bertrand Russell discovered “Russell’s Paradox”.

Suppose Is This showed that

you can’t just go creating any old sets you want.

| .A x x x

Russell’s paradox, and other similar problems that came to light in “informal” set theory, caused the Crisis in Foundations. There were three responses, which have come to dominate historical accounts of the philosophy of mathematics.◦ Logicism (Russell and Whitehead◦ Constructivism or Intuitionism (Brouwer)◦ Formalism (Hilbert)

Attempted to reduce mathematics to logic. Principia Mathematica , by Russell and

Whitehead. Two Problems:◦ By the time they were done, the foundational logic

was a holy mess – you couldn’t really claim it was just “rules of correct reasoning.”◦ The same thing that got Hilbert eventually also

applied to this work. Stay tuned.

Founded by LuitzenEgbertus Jan Brouwer.

Sort of. Also Kronecker,

Poincaré, Weyl. Elaborated on by

Brouwer’s student, Heyting.

1. Took the counting numbers as foundational and intuitive.

2. Felt mathematics was about what human minds can construct, and not about language.

3. Rejected the Law of the Excluded Middle (LEM), i.e. “either P or not P”. Claimed you didn’t know P unless you could check every case.

A proof Brouwer wouldn’t believe: Theorem: There are irrational numbers x and

y such that is rational.

Proof: We know that is irrational. If is rational, we have the result. If it is

irrational, then is rational, since

What’s wrong with this proof from an Intuitionistic viewpoint?

It assumes is either rational or irrational, which uses the LEM.

The proof gives no way of determining which of the two alternatives is true.

Brouwer saw LEM as implying that every proposition could be proved or disproved, and he rejected this implication.

Brouwer felt numbers and indeed all things that mathematics is concerned with are mental constructions and only exist in our minds.

Heyting developed an “intuitionistic” logic.

The major problem with Intuitionism was thatmuch of mathematics was then, and even more is now, based on logical deductive methods that were rejected by Brouwer and his followers. Thus, for example, many existence proofs were not acceptable. Thus, much of modern mathematics can’t be proved constructively.

It is now mostly an historical artifact and a study for a few mathematical logicians and philosophers.

Hilbert was peeved with the set-theoretic paradoxes.

“The present state of affairs where we run up against the paradoxes is intolerable. Just think, the definitions and deductive methods which everyone learns, teaches and uses in mathematics, the paragon of truth and certitude, lead to absurdities! If mathematical thinking is defective, where are we to find truth and certitude?”

Provide a proof, from within mathematics, that classical mathematics was consistent (notice there is no mention of truth here), and do so with proofs acceptable to everyone (including Brouwer).

He was willing to give up “truth” in favor of consistency – getting rid of those darn paradoxes.

Eventually, this plan was brought down, along with the more general “Formalism” that grew out of it, and along with Russell and Whitehead, by the work of Kurt Gödel.

But before we talk about that, we need to talk about Formalism as a broader movement.

Mathematics is a game of logical deduction, or “the science of rigorous proof.”

It begins with axioms and produces theorems.

It rejects the notion of truth and rejects the Euclid Myth. Euclidean and Non-Euclidean geometries are neither true nor false, but just the result of correct logical deduction from different axiom sets.

Mathematical statements aren’t about anything and don’t mean anything.

“From the formalist point of view, we haven’t really started doing mathematics until we have stated some hypotheses and begun a proof. Once we have reached our conclusions, the mathematics is over.”

From the viewpoint of science (especially the logical positivists), mathematics is a language.

The important thing is consistency – lack of any contradictions.

Nicolas Bourbaki, a group of mathematicians writing under a pen name, became the strongest proponents of mathematical formalism in the 1950’s and 1960’s, creating a series of graduate texts in mathematics using axiomatic developments, with very little or no diagrams or applications.

I’m pretty much infected with it, as are you.

It doesn’t square with what working mathematicians actually do.

I doesn’t square with what most of us think about mathematical and arithmetical statements.

And then there’s Gödel.

In 1930 – 1931, Kurt Gödel proved that any consistent formal system with enough power to develop elementary arithmetic would have statements that were true, but unproveable.

Thus, the system would be incomplete – unable to decide the truth of some statements.

This is Gödel’s First Incompleteness Theorem. Gödel’s Second Incompleteness Theorem showed

that you cannot prove the consistency of arithmetic from within arithmetic.

In other words, Hilbert’s plan was doomed to failure.

It was a depressing time for many mathematicians, apparently.

The particular unproveable statement that Gödel created was a sneaky, underhanded, self-referential monster that was created just for the purposes of the proof.

But: we have actually come upon some more or less “regular” mathematical statements that are undecidable in that neither they nor their negations can be proved from our set-theory axioms.

The first four axioms of Euclidean Geometry (properly updated and formalized) are incomplete in something like this sense.

A simple way to think of this is that the first four axioms are true in Euclidean Geometry, but are also true in Hyperbolic Geometry, so that the axioms and any theorems that follow from them aren’t strong enough to decide how parallel lines should behave.

One way to show a system is incomplete to create a model in which some statement is true, and another model in which the statement is false.

This is easy in geometry, but it is much harder when your axiom set describes most of mathematics – like the axioms of set theory.

Nevertheless, in our usual set theory, there are many such statements, assuming set theory is itself consistent: CH, GCH, the Axiom of Choice, the Well Ordering Principle, Zorn’s Lemma, the Hausdorff Maximal Principle, Martin’s Axiom, Suslin’s Conjecture, ◊, V=L, etc. etc.

Suslin Conjecture: every countable-chain-condition dense complete linear order without endpoints is isomorphic to the real line.

Martin’s Axiom: MA(k) is the statement that for any partial order P satisfying the countable chain condition (hereafter ccc) and any family D of dense sets in P, with | D | at most k, there is a filter F on P such that F ∩ d is non-empty for every d in D. Martin’s Axiom says that MA(k) holds for every k less than the continuum.

In modern set theory there are two methods used to show a statement is unproveable.

Gödel developed a model of set theory by carefully constructing only those sets which absolutely need to exist to satisfy the axioms. This is called the Constructible Universe, and is denoted by L. When we say V=L, we are invoking the model. It’s the minimal model for set theory, in some sense, and in it the Continuum Hypothesis is true, because it has only those infinite sets that absolutely need to be there.

The second method was developed by Paul Cohen at Stanford in the 1960’s. It is called “forcing” and is a method for adding a lot of sets to a model in a way that doesn’t specify too much about them – they are called “generic” sets. He proved that there was a model of set theory with an infinite set of size between that of the integers and that of their power set.

Together, they proved both CH and AC independent of ZFC!

We are doomed to have to make decisions on these things “outside” of the axiom systems in which they appear, since the axioms systems can’t prove or disprove them.

The axiom systems just ain’t strong enough to decide everything.

So formalism fails as a foundation for mathematics.

That doesn’t mean it isn’t alive and well, of course, just that it didn’t do what it was originally intended to do.

Also, most working mathematicians couldn’t care less about this issue, really.

Remember, they tend to be Platonists on weekdays, and formalists on Sundays.

But mostly they want to be left alone to do some mathematics.