18 October 2008

On the Platonic existence of numbers

Yesterday I came across an introductory paper by Olle Häggström, Objective truth versus human understanding in mathematics and in chess (2007), The Montana Math. Enthusiast. In it, Olle supports two ideas, both of which I have always held firmly. First that numbers exist independently of humans, and second that the human way of doing mathematics will always play a role in it and, no matter how advanced our computational machines turn out to be, they will never substitute the way we think, we prove and, more importantly, we understand mathematics (and science more generally).

Olle questions whether the first philosophical idea (the platonic ecistence of numbers) can be as dangerous as the following argument for the existence of God: " To anyone who has met God, His existence can no longer be in doubt. " I don't think so. Of course, accepting the existence of numbers as independent of humans goes beyond the realm of science and mathematics, but it is not dangerous. In fact, what Olle does not mention in his article is that we did have a whole system, a religion if you wish, that was based, precisely, on the concept of number as a divine object: Why, Pythagoras himself established his school which lasted for 600 years before it was destroyed by early Christians (they stoned to death the last of the Pythagorean mathematicians, Hypatia of Alexandria). I maintain that we would have ended up with a much better society had numbers still been the object of divinity, rather than a merciless God. I would definitely go to church every Sunday (or Saturday or Friday...) if we were to discuss numbers, and discuss I mean, rather than listen (without the possibility of asking any deep questions) to a boring priest, minister, rabbi, etc, and performal silly rituals.

Anyway, I'm digressing. Back to Olle's article, I would like to add that if we accept that numbers are independent of the human experience then we quickly reach the fact that it is merely the empty set that is the only thing that exists. Indeed, all numbers can be constructed from the integers which themselves can be constructed from the empty set, which I here denote by o. Indeed, zero is defined as o. One is defined as the set that contains o. Two is defined as the set that contains o and 1, and so on: Integer n is defined as the set containing all the previous integers. This model is, arguably, the best we have: It leads naturally to the construction of transfinite ordinals (Cantor) as well as surreal numbers (Conway). The latter ones are representations of two-player strategic games, just as the game of chess that Olle discusses in his paper.

So, the empty set is all there is then. Do you hear some reverberations of Zen Buddhism or Ancient Taoism? Hm, yes, indeed. Not that I will take these systems too seriously, but they do rely on the concepts of emptiness and nothingness, respectively.

As for whether Maths is human or not, I do agree again with Olle in that it is not the building stones of Mathematics that are human, but it is the way we do it that is. We, as humans, have to decide what to study, what to accept, what to prove, how to prove it, how to interpret and understand a proof, and how to use a certain result. Suppose we had reached a stage where we could solve all ordinary differential equations of the form
P(D)y = z
where P is a polynomial with real coefficients of degree less than 40, z=z(x) is a given function of one real variable x, y=y(x) is the unknown function, while D is the derivative operator. But say that the explicit solution required a few thousand of pages to write down. So? Would we accept it as an answer? Certainly not! Who says so? Why, everyone, mathematicians, engineers, physicists, practitioners... Only a robot would seriously maintain that a 1000-page formula is an answer. The reason for our dislike of such a formula is, precisely, because of our physical (and therefore mental) limitations. One could, possibly, imagine another world, where "humans" were 4 times as tall, with eyes 3 times as big, brain 5 times as large, and so on. Then mathematics would have been different. This is a bit of a naive explanation, but does convey my point.

Take another example. In Maths, we are interested in rates of convergence. What does this mean? It means to find out how fast a certain sequence converges. But the answer, i.e. the rate of convergence, must be given in terms of an "elementary" function, i.e. a polynomial, an exponential, and so on--only a handful of them, or in terms of another function we understand. Again, this is because of our human understanding of what constitutes elementary.

Let me also mention that (many) mathematicians are often faced with a choice: should we study this or that? Why should we accept or reject the axiom of choice? (Axioms are, roughly speaking, statements that cannot be proved or disproved, based on previously accepted statements.) If we do, we get a certain kind of Maths. If we don't we get another. What is better? Again, the answer is beyond the field of Mathematics. It has to do with us, humans, with the way we want to interpret the world, with the machines we need to construct, the tools to use to cook better food, etc.

I find the quotation, in Olle's paper, of a statement of Gowers interesting:

Namely, that we can live without the idea that an ordered pair (x,y) really is a funny set of the form {{x},{x,y}}, and that undergraduates would be confused by it.

Why should we need to take one point of view? I do agree with Gowers that, esp. nowadays, most undergraduates would get confused. And that when we introduce an ordered pair we do have to say the obvious thing, instead of reducing it to set theory. At least not immediately. But I do maintain (and have often found useful) to have the ability to reduce intuitively understood concepts to its fundamentals, to the axioms and objects of set theory, say. I take no sides. I maintain that both rigour and intuition are absolutely necessary. I am not surprised that Gowers seems to be taking on one side only. His colleague, Alan Baker, writes in the preface of his wonderful book, A Concise Introduction to the Theory of Numbers , Cambridge U. Press (1985), that "[t]there is no need to enter here into philosophical questions concerning the existence of [the integers]". He is right: his book is a wonderful speedy introduction to those aspects of number theory that lead to the solvability of Diophantine equations. Diophantus (from the works of whom--destroyed by early Christians, preserved by Arabs--all modern Algebra stems) has discovered algorithms for solving (systems) of classes of polynomial equations in integers. He didn't care about the existence of integers: Integers did exist and if you dared accept that square root of 2 was irrational you might lose your head. But Baker adds the adverb "here" in his sentence. He means, I hope, that elsewhere, at some other time, the student might wish to question the existence of integers, wonder why they should exist and convince herself or himself that they do (or do not!).

This is why Science and Mathematics is much more desirable than blind faith: we discuss, we question, we argue, we come to a conclusion, we revise, we discuss again, all along based on proofs and physical evidence.

Olle says (and I find this amazing!) that Freeman Dyson maintains that the statement "there exists a power of two, 2^n, such that, when written in decimal and read backwards it is a power of 5" is not provable!

It is quite rare that we encounter statements that are unprovable. Although, from a counting point of view, most statements within a mathematical system should be independent of its axiomatic foundation, it is very hard to bump into one. Is that not then another evidence supporting the idea that the way we do Mathematics, the way we seek to discover (for discoverers we are) truth within its vast archipelago, is, indeed, very human-based?


  1. Pantelis Loupos2 Nov 2011, 02:05:00

    Sorry in advance if my following questions proved to be naive.

    First of all, I have recently been told by one of my Professors that there exist different theories of probability based on the axioms we accept. That made me think that Mathematics are like religion, i.e you have a choice on which mathematical system you want to adhere to (you don't have uniqueness).. Could you please share your view on this analogy?

    Secondly (there is a correlation with my first question), I was always wondering what if our axioms do not tell the whole truth? As you have nicely stated "We, as humans, have to decide what to study, what to accept, what to prove...", so our mathematics is based on our human senses. But what if our "human mathematics" is just a projection of some "Universal mathematics"?
    Then we should stop think that we can describe everything by mathematics, and just restrict ourselves to the understanding of the world we can sense? After all, Godel's theorems are a proof of our limitations...

    Best Regards

  2. Quite the opposite: mathematics is not at all like religion. In mathematics, we have theorems which are proved on the basis of axioms (which do not contradict one another). At all times, we make sure that we do in mathematics makes sense. There is no proof concept in religion. Religion is full of contradictions. There is nothing, absolutely anything, which makes religion be anywhere close to mathematics.

    The professor who told you there different theories of probability based on the axioms accepted, probably has not had much experience in applying probability or seeing how and why the commonly accepted probability theory makes a lot of sense, especially because it has a vast explanatory power.

    Yes, by changing one axiom, e.g., by replacing countable additivity by finite additivity, we create a different theory, but, so far, we have not seen any advantage of this. This is experience-based statement.

    The "laws" in mathematics, the axioms, are never given. We use the word "given" for convenience. It is one of the hardest tasks of mathematics to come up with axioms for a theory. This comes after years, decades, centuries of experience. It took several thousand years to write down a consistent system of axioms of Euclidean geometry (Hilbert).

    Religion starts with rules that nobody should question. Mathematics creates its axioms in the end, not in the beginning. And then, this end becomes a new beginning.

    If, in the future, we see that probability needs another axiom, we will add it, but, so far, it is not needed.



What measure theory is about

It's about counting, but when things get too large.
Put otherwise, it's about addition of positive numbers, but when these numbers are far too many.

The principle of dynamic programming

max_{x,y} [f(x) + g(x,y)] = max_x [f(x) + max_y g(x,y)]

The bottom line

Nuestras horas son minutos cuando esperamos saber y siglos cuando sabemos lo que se puede aprender.
(Our hours are minutes when we wait to learn and centuries when we know what is to be learnt.) --António Machado

Αγεωμέτρητος μηδείς εισίτω.
(Those who do not know geometry may not enter.) --Plato

Sapere Aude! Habe Muth, dich deines eigenen Verstandes zu bedienen!
(Dare to know! Have courage to use your own reason!) --Kant