Posted by: Gary Ernest Davis on: February 14, 2010

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Alain

“*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)

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?

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.”

From what I’ve read thus far, my perspective is much closer to Changeux’s than Connes’. Mathematical ideas exist within the human mind which is ultimately a physical system. Mathematical reality exists inside physical reality, not prior to or alongside it.

I tend to view mathematics as a problem of information compression. Mathematicians seek to maximize the amount of knowledge that can be obtained from a minimal number of assumptions. Our experiences as human beings in a physical reality help determine what information is important to include in this compression.

This question of mathematical reality v. physical reality is much more complicated than this article suggests.

For example, quantum mechanics teaches us that “physical reality” is a combination of unrealized quantum probabilities combined with an observer collapsing that quantum wave into a particular state. Different observers or observations result in different physical realities.

However, that isn’t the crux of this discussion. It really comes down to a comparison of the physical universe with the mathematical universe.

In favor of the mathematical universe being “more real” is that the mathematical universe doesn’t depend on the physical universe. You could change one of the basic constants of the physical universe and it would change completely. But the mathematical universe would remain unchanged. It has, in that sense, its own reality that applies to all universes. Neal Stephenson’s Anathem explores this in an enjoyable (to me) novel.

You can also argue that we know nothing of physical reality, as any Philosophy student knows. We only know what we sense and what models we build in our brains.

In favor of the physical world is the tangibility. Many of us use the term “real” to mean “something I can sense”. Using that approach, you get the chemist who only uses mathematics as a tool. I can’t “touch” mathematics, so that makes it something less real than the physical universe, like God or love or stories about dragons. Plus, as referenced in the article, Godel’s incompleteness theorem can easily be used against mathematical reality.

Me? I have to come down on the side that mathematics applies across all kinds of physical realities, and therefore it has a “more real” reality than the collections of strings or quarks that exist in a quantum cloud until we observe them (and “oh by the way”, what we observe isn’t the same as what is out there).

Thanks.

His view makes sense even if he doesn’t have the neurobiological evidence to back it up. Physical reality is much more subjective than people might believe and it is colored by experience and belief so in that sense he is right. The reality a trained mathematician sees is much different from the one that anyone else sees and much of mathematics is about defining physical experience so maybe the mathematician can rightfully claim his external reality is indeed part of his mathematical world.

David,

you seem to have expressed very nicely what I am beginning to understand about Connes’ point of view. Many moons ago I pretty much rejected Platonist ideas in favor of a more constructivist point of view. But Connes has a point, I feel, that we may well know aspects of mathematical reality better than we know physical reality. We certainly seem to know them differently.

I still do not think it’s a case of either/or, and I think we have a long way to go to know what “reality” really is (if it’s anything at all).

I like very much your phrase “The reality a trained mathematician sees is much different from the one that anyone else sees.. “

Regarding Connes’ use of ‘archaic’, I think it works in the sense of ‘arche’ as beginning, first cause, origin – http://en.wikipedia.org/wiki/Arche .

We tend to think of ‘archaic’ as ‘old’, but there’s another sense in the Greek that corresponds more to what we mean by ‘archetype’.

Does it make sense to think of mathematical objects as ‘archetypical’?

If you go with platonic shadows, you eventually end up with an ontological argument, and define God into existence. For examples:

Nothing is better than life in heaven.

A ham sandwich is better than nothing.

Therefore, a ham sandwich is better than life in heaven.

or

First, we define God as the best ham sandwich, ever.

Now, a ham sandwich that exists is clearly superior to a ham sandwich that does not.

Therefore, a ham sandwich exits!

1 | Jameson Graber

February 14, 2010 at 1:12 pm

Language is a funny thing. Somehow we humans can get awfully worked up about the names we give to things. No one can deny that mathematics actually produces “facts,” which in some sense seem more permanent than (hence prior to) physical reality. But some people use the term “reality” only to mean whatever is in the physical world, and it seems bizarre to them to use the word “reality” to refer to something abstract that doesn’t exist in time and space (like the laws of logic).

You’re right that most of the time a pragmatic approach is best; if people want to reserve the word “reality” for physical reality, let them have it. It is, after all, just a word. I sense that underneath all this metaphysical discussion of what is “reality” is really just a need to assign importance to what we do. If scientists study “reality” but mathematicians just push symbols around all day, then I suppose mathematics must be subservient to science. But there are legitimate reasons not to think this way, and my feeling is that’s what mathematical Platonists are really trying to say.