# Archaic mathematical reality as referred to by mathematician Alain Connes

Reading Alain Connes' less-technical books (with coauthors) Conversations on Mind, Matter, and Mathematics (1995) and Triangle of Thoughts (2001) left me especially impressed with his platonist view on Mathematics that assumes existence of an archaic mathematical reality outside space-time yet as inexhaustible as normal physical reality.

I am wondering whether his specific arguments build on top of an established school within philosophy (not just platonism in the philosophy of mathematics in general) or whether they triggered follow-up work by philosophers. If any such school or work exists (in normal physical reality, i.e. :), who are the leading exponents and what articles or books could I consult for a more detailed presentation?

Here are some relevant quotes from the two books:

Take prime numbers, for example, which as far as I’m concerned, constitute a more stable reality than the material reality that surrounds us [...]

[If we consider Gödel’s incompleteness theorem] from a different angle – namely, as asserting that true propositions about positive integers can’t be reduced, by means of logical inference, to a finite number of axioms – it can be seen to imply that the quantity of information contained in the set of all such true propositions about positive integers is infinite. I ask you: isn’t that the distinguishing characteristic of a reality independent of all human creation? [...]

What I find fascinating about mathematical reality, and about the effort made by human being to try to understand the objects that populate it, is that it’s often possible to characterize a particular object up to isomorphism by its properties [...] I would be very difficult, I think, to make similar statements about external physical reality. How could even the earth, for instance, our own planet, be defined protectively? One could say that it’s the third planet in a system revolving around a star situated in the spiral arm of a galaxy, but obviously this doesn’t single it out from a great man other planets [...]

All the logical conclusions that we arrive at be deductive reasoning belong, in my opinion, to a projective system of thought [...] The true or not-true properties of integers, by contract, belong to archaic reality [...] These amount to experimentally verified facts about archaic reality [...]

It does seem to me that one indisputable feature of external reality is that it constitutes a constant source of information that, while not, of course, immediately comprehensible to the brain, isn’t reducible to the past [...] Each second that passes in a give volume is going to produce a certain number of new bits of information that are irreducible to the past [...] To me this is one of the basic attributes of external reality [...]

Reality is a source of information in the sense that there are things constantly coming into existence that can’t be reduced to past events – things that are really new [...]

To me, our ability to comprehend the external physical world implies that there exists an archaic mathematical reality that exists on the same footing as external physical reality […] The thing that’s extraordinary about mathematical conceptualization is that we’re able to conceive of a universe as a four-dimensional object, we don’t have to situate it within a larger dimensional space. We can picture it intrinsically, as it were – on its own terms […] There’s no longer any causality, because there’s no longer any time! Once you adopt this way of looking at the outside world, the notion of causality is simply one of the features of the mathematical model of the universe. There’s no longer any obstacle to conceiving of this archaic mathematical reality as something that exists alongside the universe.

(Connes' ideas around subjective time also seemed quite inspiring and potentially deep.)

• I've read only his Triangle of Thoughts; I didn't think that there was anything particularly innovative in his philosophy of mathematics there. The cohesiveness of mathematics as a subject and its ever-expanding horizons is fairly standard. Commented Sep 23, 2013 at 12:40
• @MoziburUllah The ever-expanding horizons of mathematics may be fairly standard in mathematical discourse (around Gödel's incompleteness theorem, say), but are they also in philosophical discourse on mathematical platonism? For instance, the article in SEP concentrates on three claims: reality, abstractness, and independence; inexhaustability does not seem to play a direct role there.
– Drux
Commented Oct 5, 2013 at 1:53
• Now that you point it out, I suppose not. Reality & abstractness appears to be what we mean by the ontological status of numbers, which is a fairly standard subject. Independence comes from an analysis of axiomatic systems & logic. The inexhaustibility of mathematics and its cohesiveness doesn't appear to be standard topics, so perhaps I was too hasty there! But certainly platonism is fairly standard, and commonly held by most mathematicians; and those that don't act as if it is true. Commented Oct 5, 2013 at 19:00
• @MoziburUllah I've added some relevant quotes and noted in the process that Connes relates inexhaustibility (perhaps specifically) to reality, so maybe this is how his (and the sought school's) views fit into a common one.
– Drux
Commented Oct 6, 2013 at 2:38

Most of this seems to be fairly generic mathematical platonism, but one thing which stood out to me, and seems worth giving a specific answer on, is the following:

What I find fascinating about mathematical reality, and about the effort made by human being to try to understand the objects that populate it, is that it’s often possible to characterize a particular object up to isomorphism by its properties [...] I would be very difficult, I think, to make similar statements about external physical reality. How could even the earth, for instance, our own planet, be defined protectively? One could say that it’s the third planet in a system revolving around a star situated in the spiral arm of a galaxy, but obviously this doesn’t single it out from a great man other planets [...]

This seems to suggest a kind of structuralism. Connes seems fairly clear that, once we've characterised a mathematical object up to isomorphism, then we've identified it uniquely (in contrast to the case with the earth). That is, and roughly speaking there are no distinct, yet isomorphic mathematical objects; isomorphic objects are identical. And this, or something like this, is one of the key tenets of structuralisms of various kinds. The article linked above is a good start on this. A more recent paper developing this core idea (in quite a technical direction – be warned!) is 'Structuralism, Invariance, and Univalence' by Steve Awodey (link)

• Yes that passage jumped out at me as well as taking a step beyond garden-variety platonism. Commented Apr 25, 2014 at 16:34