Jack Edmonds stayed in my house for two and a half weeks, while giving his two courses on polyhedral combinatorics in London (I reported on the second one here).
Whenever we spend time together, we have a robust discussion about the foundations of mathematics, from which we both gain something. The picture below shows an earlier discussion in a Chinese restaurant after Jack spoke in the Combinatorics Study Group at Queen Mary. The picture was taken by Carrie Rutherford.
I want to report on our discussion last week. I am aware of the risk of misrepresenting Jack’s views; but I will trust to our friendship and carry on anyway. But of course, what I say here is my own take on things.
Jack is scathing in his dismissal of ZFC (Zermelo–Fraenkel set theory with the Axiom of Choice) as a foundation for mathematics. In his view, the basic problem is the Power Set axiom, since it is not specified what a subset of a set is. It is true that many vagaries slip in through this open window; but varying some of the other axioms (e.g. by assuming Gödel’s axiom V=L, or by replacing the Axiom of Infinity with its negation) makes the power set axiom harmless. I think part of Jack’s objection is the non-effectiveness; even if we understand what the subsets of a set are, there is no efficient way to list them all.
I think he is fairly content with Peano’s axioms. But the problem really goes deeper. One of our discussions was about the commutative law for multiplication. I discussed this here. To summarise what I said: the commutative law is obvious to everyone who has thought about a rectangle of dots; proving it in PA is a nightmare, involving a double induction; proving it in ZFC is much closer to the intuitive argument, since there is a natural bijection between A×B and B×A (just turn around all the ordered pairs).
Jack’s view, as I understand it, is that the foundations of mathematics should be much closer to our psychological picture than it is at present. I will discuss a bit what he thinks should be done after a short digression.
Alan Turing, in a famous extended passage, invited us to watch a mathematician at work. She is equipped with a large notebook, a pencil, and an eraser. She writes or erases something on the page, or turns over some pages to find the result of a previous calculation, or turns over some pages to find a blank sheet to continue working on. These moves depend both on what she reads on the current page and on her thought processes (the state of her brain, if you like). Turing’s conclusion is that a mathematician is a Turing machine, since these actions are exactly what the eponymous machine does.
Not so long ago, I described how Ron Aharoni also invited us to watch a mathematician at work, in order to contrast with a poet at work. Rather than the extreme busyness of Turing’s mathematician, Aharoni’s spends most of his time staring into space.
Many eminent mathematicians, including Gauss and Poincaré, have left accounts of how they made their discoveries. I have collected some of these accounts. They are much closer to Aharoni’s mathematician than to Turing’s.
Anyway, Jack believes that mathematics is done in the manner Turing suggests, and consists of manipulations of words. So he would like a foundation of mathematics based on words and related directly to actual mathematical practice.
Natural numbers are the lengths of words, and so can be regarded as equivalence classes of words (two words being equivalent if their letters in natural order match up). Now addition can be defined by concatenation, and multiplication by substitution. (To multiply m by n, take a word of length m, and substitute a word of length n for each of its symbols.)
Now we have to choose what the basic assumptions are. Though I haven’t tried very hard, I suspect that proving the commutative law in this formulation would not be much easier than proving it in Peano arithmetic.