Posted by: Gary Ernest Davis on: January 24, 2010

My ex-wife used to wonder what was the difficulty in forming, or talking about, the set of all natural numbers. This was not a reason for our eventual separation. Â It did, however, cause her some angst in wondering why we can’tÂ just imagine all the counting numbers bundled together in a set:

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My explanation to her involved three things:

1. Bertrand Russell’s apparent paradox of the set S of all sets that do Â not belong to themselves (does S belong to itself or not?). This is a so-called paradox of naive set theory in which there are few restrictions on the formation of sets.

2. A consequent need to formalize the process of set formation, for example, as in the axioms specified in Zermelo-Fraenkel set theory.

3. The perhaps surprising fact that Zermelo-Fraenkel set theory without the axiom of infinity has models with only finite sets.Â So to even assert that there is a formally constructed set of natural numbers we need to assert, as an axiom, that there is an infinite set.

This formal axiom is based on our informal, or naive, experience that for any number to which we have counted we can, at least in imagination, count one more.

There is no doubt that we count and that, unless we are too tired, can always count one more. But this experiential fact does not, in itself, justify our assertion that there exists a set of all natural numbers.

What we are doing in asserting the existence of the set of natural numbers is making real something that is an idea. This process is usually called reification, in whichÂ an abstraction is treated as if it were a concrete, real physical entity.

So at the heart of mathematics lies an act of reification, a taking as real some thing – the set of natural numbers – that is an abstraction from our human activity of counting.

The process of reification is explicitly addressed in Buddhist thought, where it is generally thought to be not a good thing because it leads to the delusion of permanence for mental constructions that are bound to decay:

“*All things and events, whether â€˜materialâ€™, mental or even abstract concepts like time, are devoid of objective, independent existence. … things and events are ’empty’ in that they can never possess any immutable essence, intrinsic reality or absolute â€˜beingâ€™ that affords independence*.” Â Dalai Lama (2005). The Universe in a Single Atom: The Convergence of Science and Spirituality. Broadway. ISBN 076792066X & ISBN 978-0767920667

Yet, to a mathematician, at least all those I have met or read, the natural numbers do indeed have an immutable essence. The set of natural numbers is as real and permanent to most mathematicians as any physical reality.

Once mathematicians have accepted the reality of the set of natural numbers they can begin to ask questions. A very early observation about the set of natural numbers is that every natural number Â has a decomposition, in an essentially unique way (apart from rearrangement), as a product of natural numbers that themselves are ‘prime’ – that is, Â can only can only be expressed as a product of the number 1 and themselves. Examples of prime numbers are 2, 3, 5, 7, 11, and 13.

It is a fact about natural numbers Â that 53 is prime number.

It is a deeper fact that there are infinitely many prime numbers.

It is an even deeper fact that theÂ the prime number is approximately equal to , with the relative error of this approximation approaching 0 as Â increases without bound. Â Here is the natural logarithm of : Â the area under the curve from .

Many mathematicians believe, but no-one has yet proven, that an even deeper fact is true about natural numbers (Riemann’s zeta hypothesis).

To me this is a very odd and puzzling issue. I have considerable respect for the Buddhist point of view on impermanence. Heraclitus’ statement about not being able to step into the same river twice also relates to the impermanence of a supposed object which we have reified through the word “river”.

We might, daily, look up the likely low temperature estimate, yet there is no real thing called “temperature’ – just readings, for example, of the height of a mercury column. The “temperature” is a mental construct to be found nowhere in the physical world.

Yet physics, and science in general, thrive on constructs such as “temperature”, “pressure”, speed”, and “force” , which can be defined operationally, but are taken by most scientists to refer to an objective reality.

So in mathematics we treat the abstract construct of the set of natural numbers as a real object and then discover deep properties of this abstracted set.

On such myths mathematics and science thrive!

I remember a conversation/argument I had with a fellow grad student years ago (before the proof of Fermat’s Conjecture) regarding whether that conjecture might be independent of the Peano Axioms.

He argued that if it were independent, then there would exist a model of the Axioms in which the conjecture was false. And if that were the case then, in that model, there would be a solution to x^n + y^n = z^n with n >= 3. But if that statement held in one model, then it would have to hold in all models and be verifiable via the Axioms, hence the conjecture would be provably false and not independent.

Something like that (I’ve been away from math for a long time).

I bring this up because it was during that time that I think I rejected completely the idea of “the set of natural numbers” as being a real thing. The whole idea of different “models” of the Peano Axioms and statements which are true but not provable in some model of those Axioms made me question the very concept itself.

Now certainly, we can develop the language Set Theory, and then “define” the set of natural numbers in that language, and then everything feels all warm and cozy for awhile. But its really just postponing the problem because you end up with the same problem regarding models of Set Theory. Which model is the “real” one?

Anyway, I feel I’m rambling so I’ll stop now.

Nice article,

John

1 | Ryan R

January 24, 2010 at 10:53 pm

“So in mathematics we treat the abstract construct of the set of natural numbers as a real object and â€“ as if by magic â€“ discover deep properties of this set.”

These discoveries were not made through magic, but rather diligent scientific study. Mathematicians generate hypotheses about the natural numbers and test them. Some turn out to be true, some turn out to be false, and some of them turn out to be inconclusive. Magic is about exploiting the weaknesses of the human mind; mathematics is about exploiting its strengths.

I’m also a little skeptical about the necessity of reification in pure math. I’m quite content to leave the natural numbers as an abstract concept and explore what kinds of hypothetical properties they might have, without making any reference to “reality”. Reification only seems necessary in the field of applied math, where one wishes to take conclusions from the realm of the abstract and apply them to a real world situation.

Gary Davis

January 25, 2010 at 6:22 am

Ryan,

the phrase “as if by magic” did not refer to the process of discovering facts about natural numbers, but that we seem to discover a deep reality about an object that many would regard as not real.

I changed the phrase so as not to confuse. Thanks.