On foundations

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.

Peter Cameron and Jack Edmonds

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.

Any thoughts?

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About Peter Cameron

I count all the things that need to be counted.
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3 Responses to On foundations

  1. A question. Several years ago I visited a seminar at the local philosophy department, which was mostly about the philosophy of mathematics by Paul Lorenzen. My following question might misrepresent Lorenzen’s ideas as I did not read him for 7 or 8 years. I just looked into some of my notes from that time.

    Paul Lorenzen had strange constructivism. He started by writing down words and concatenating these words. So he gets arithmetic in the same way as you are suggesting. Then he goes on and on and reconstructs with these basic operations large parts of mathematics (including ordinal numbers such as $$\omega^{\omega^{\omega}}$$ or all “useful” analysis). The goal of his work was to justify the usual mathematics with these simple operations on words. Are Lorenzen’s ideas similar to the things you/Jack want? Or am I missing some crucial difference? Unfortunately, I do not know how much of his work is available in English.

  2. It’s difficult for me to answer. It sounds not too far from what Jack is advocating, but I think the philosophy is as important to him as the mathematical details, and knowing nothing of Lorenzen’s philosophy I can’t say. Certainly Jack is prepared to do without large parts of mathematics; I think he doesn’t believe in real numbers beyond the computable ones. Also, since this isn’t my philosophy (though I would be happy to help develop it as an interesting exercise), I don’t know what counts as a solution.
    Thanks for the reference.

    • Lorenzen gave a John Locke Lecture in 1967/68. You missed him by one or two years I suppose, but then he did not even talk about mathematics there. I think that I read the German version of the following book, which I would recommend (as long as the translation is good).

      Differential and Integral: A constructive introduction to classical analysis, The University of Texas Press, Austin, 1971.

      The only constructivism I know, where you actual keep the intermediate value theorem in its usual formulation (and do not add epsilons and deltas into the statement itself).

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