Republic of Mathematics blog

Alain Connes thinks so, if I have understood his point of view correctly:

“You conceive of my external mathematical reality as  part of the external physical world. What I am about to tell you is going to come as a surprise, then. For me, it’s just the opposite: external physical reality is a part of archaic mathematical reality.” (Conversations on Mind, Matter, and Mathematics, Jean-Pierre Changeux and Alain Connes, p. 206)

Alain Connes

Despite Connes’ use of the word “archaic” – a term he does not explain in the book with Changeux – his point of view seems to be expressed fairly clearly: mathematical reality is not, for  Connes, a part of physical reality, including the physical reality of the brain.  The very notion of physical reality, for him, is derived from mathematical reality.

Now this is pretty strong stuff, especially for that great cohort of scientists and engineers who see mathematics as just a language for describing reality.

I addressed the issue of whether mathematics is just a language in a previous post. My answer is that mathematics is not just a language because it contains empirically discovered and logically proven facts, such as the prime number theorem.

Nonetheless, a pragmatic scientist might argue that she is more interested in computational cell biology and network analysis than she is in the prime number theorem, and that, for her, mathematics provides a convenient set of language terms and concepts with which to frame her theoretical and experimental work on protein-protein interactions. I find it hard, and somewhat pointless, to argue against such a pragmatic point of view.

So is there some simple way for a pragmatic scientist to understand what Connes is getting at?

I think there might be, and I think such a path comes from the very human activity of counting.

Kids love to count. They usually learn to count from their mothers, sometimes fathers, as they walk along, counting steps, counting cookies, and many other things they encounter in daily life.

I see human activity from a fairly constructivist point of view:  it seems to correspond best to the reality of human activity, for me. So from that perspective you might think I would be quite opposed to Connes’ very Platonist view of mathematical objects.

Let’s continue with counting. As kids count they learn the number names, and learn to match those one-to-one with actions – such as stepping – and with objects – such as steps.  Kids undergo a development with regard to their understanding of numbers and eventually come to terms with arithmetical operations on numbers, and even move on to fractions and decimals.

Many kids, as young as 5 years of age, will discuss the potential infinity of numbers.  Their insight is that they can , at least in imagination, always count one more, so there is no end to counting numbers.

There is a point of view, generally called strict finitism, which asserts there is a biggest counting number, because the finite constraints of the “real” world prohibit us form counting to arbitrarily large numbers.

Of course, 5 year old children know they begin to tire as they count further and further, yet in their imagination they can see a tireless person always able to count one more.

So the vision of the counting numbers as without bound is an imaginative act.

Many people, teachers included, happily use the everyday word “set” and, in their imaginations,  bundle all possible counting numbers into an object – the set of all counting numbers.

But mathematics and and logicians have long discovered that there are serious problems in just treating set formation as a naive notion, free of constraints. The most celebrated problems is that posed by Bertrand Russell: the set of all sets that do not belong to themselves. (Does this set belong to itself or not? If it does, then it doesn’t. If it doesn’t, then it does.)

To overcome these logical difficulties mathematicians devised formal properties – axioms – for set formation that they hoped would place mathematics on a from logical footing.

Henri Poincaré was quite opposed to these efforts, because, for him, the counting numbers were part of our primal (= “archaic”?) experience, and subjecting them to the strictures of set formation was a perversion of our fundamental human knowledge.

Nevertheless. the human experience of  always being able, in imagination, to count one more, has ben encapsulated in the Peano axioms for the set of natural numbers.

This set of natural numbers – a well described set in mathematics – is different, for me, from the counting numbers of our human experience. Why? Because nothing in our imagination of counting one more tells us that the counting numbers can be encapsulated as a single object – the collection of all counting numbers. That is a different act of imagination.

The formalization of the act of imagining all the counting numbers as a single collection is what leads directly to the formation of the set of all natural numbers, in set theory, satisfying Peano’s axioms.

So, to sum up my perspective: the human activity of counting, leads to an act of imagination that we can always count one more. Another act of imagination  gathers, in imagination, all the counting numbers into a single object. The foundational problems in set theory say, “Whoa! you need to be careful about doing that, because there are hidden paradoxes lurking“. So set theory is formalized in axioms, and so are the natural numbers, along with the notion of mathematical induction. Now we have a formally described set of natural numbers and we can discover and prove deep things about it, such as the prime number theorem

We have discovered and proved, deep aspects of mathematical reality which seems to be prior to, or at least alongside, physical reality.

Connes’ point of view, I believe (and not to put words in his mouth) is that we have, through a process of reflecting on our acts of human activity, and human imagination, discovered aspects of archaic mathematical reality. In this sense, the mathematical reality is prior to physical reality.

What do you think?

Addenda

1. Michael Atiyah wrote a nice review of Changeux and Connes’ book, way back in 1995.

2. To be even more extreme about the nature of archaic mathematical reality, suppose human beings never came into being on this planet. Suppose the dinosaurs were still around. In fact, let’s imagine we are in the dinosaur era. No humans to carry out counting. I’m guessing dinosaurs didn’t count.

Was the prime number theorem true then?

I feel that Connes’ point of view would be yes (but, again, I may be putting words in his mouth).

To me the question ” Was the prime number theorem true then?” is a  misapplication of the human notion of “truth”, extrapolated, without justifcation, from everyday affairs, as in: “Is it true that you loaned Henry  $500?”

Yet, I find Connes’  Platonist view intriguing: maybe, just maybe, through successively reflecting upon, and refining, our human activities, such as counting, we hit upon, (stumble upon ?) deep mathematical truths that were in fact, eternally true.

It certainly feels that way when you make a mathematical discovery and prove a theorem.

3.  Leibniz was supposed to have written:

Sans les mathématiques on ne pénètre point au fond de la philosophie.

Sans la philosophie on ne pénètre point au fond des mathématiques.

Sans les deux on ne pénètre au fond de rien.

[Without mathematics we cannot penetrate deeply into philosophy.

Without philosophy we cannot penetrate deeply into mathematics.

Without both we cannot penetrate deeply into anything.]

4. Niels Bohr is supposed to have said, in relation to the reality of the quantum world:

“There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature.”