What is mathematical existence?

Question

When I make a claim in a proof that a mathematical entity exists, is this no more than saying that the theory I'm working within is consistent, and that all the steps upto that point in the proof are allowed moves in the theory?

By mathematical entity I mean a number, say 3; or an algebraic system such as the group of rotations of the square or the ring of natural numbers; or the category of varieties (a variety is an algebraic system with a fixed number of operations). It could also be a circle, or the category of all manifolds. It could be set theory axiomatized in ZFC.

Not only do these entities relate to each other (for example, after 2 is 3; the group of rotations of a square is related to that of a cube) but also they have structure (3=2+1 in the ring of natural numbers; or the circle, a 1-dimensional manifold is made up of 0-dimensional points)

Existence is a loaded word, we experience in our everyday lives with the existence of chairs and tables or the colour red. It seems very odd that the same word is used for entities that are unavailable directly to the senses, but at the same time it is 'oddly' appropriate.

Taking the platonic perspective - that these entities exist in their own world - and that we access this world through an effort of will. Then supposing that our own physical universe does have a complete mathematical theory. Then the entities in this theory, as well as the theory itself appear to exist in this other separate world, and our own physical world (with us in it as we are physical beings) appears to be incarnated in someway in this other world. This seems odd. Perhaps this suggests that the universe cannot have an underlying mathematical theory?

On the other hand, taking an entirely formal point of view seems (to me) deeply unsatisying. Its an interesting/entertaining point of view, granted.

For one thing, a formalisation appears to be only a perspective or an approximation. For example I have a circle in mind, I could choose to formalise this as a figure of plane geometry as Euclid originally did or a 1-dimensional smooth manifold. It is approximate, as formalisations have changed over time, and are likely to keep on doing so.

Answer 1

There are a number of different opinions on that; a brief overview can be found here.

Answer 2

Let's be concrete. Take a specific existential statement:

There exists n such that n^2 - 2 n + 1 = 0

This is a sentence related with some "natural number theory" (whathever it could mean), but

1. we are not necessarily considering a specific number theory, the claim could be related to any such theory, so which one should be assumed to be consistent? Which steps should we consider to be correct?
2. given a specific number theory the claim needs not be provable in this theory in order to allows us to make that claim, it could be evident from outside that theory (it could have been proved in a theory which is not "number theory" but is (for ex.) geomety or set theory), so again: which are the "steps" that should be considered correct? Which is the theory that is supposed to be consistent?

Answer 3

At some point I had read Stanford's Principia Metaphysica, in one of its firsts more draft forms (so, what I say may be totally outdated right now).

It stated that we can define existence as the action of a certain element encoding (or complying with) certain properties.

Taking Marcos example, but simplifying it further, let's take an example of:

There exists n such that n + 1 = 0

We can come up with a solution that exists and will codify such properties. That's how imaginary numbers made their way into mathematics.

There is, however, a situation where certain objects do not exist.

There exists n such that n + 1 = 0 AND n belongs to N.

(Asuming "N" refers to the set of natural numbers)

However, you can prove that both conditions are contradictory because of the definition of the properties themselves and not the properties that their different elements encode. In such a way, then no element will exist that complies with those, because the condition itself is absurd, but not because elements exist or do not exist by their own.

I know that the implication of this is that we will only know of existence by defining new properties that mathematical objects comply with, but that is, in a certain way, phenomenology: is what we can perceive.

Answer 4

If you assert that a mathematical entity exists, then it's up to you to know what you mean by that. What I mean when I assert, for example, that the natural numbers exist is that they exist in what I understand to be the ordinary everyday meaning of the word "exist", which is roughly captured by Quine's prescription that "to exist is to be a value of a bound variable". (Of course, what I mean is, in and of itself uninteresting, but I'm pretty sure this is also what most people mean.)

In that sense, both the natural numbers and my living room sofa exist, even though the natural numbers and my living room sofa are very different sorts of things. You appear to suggest that this is some sort of a problem. But we don't ordinarily reject concepts just because they are broadly applicable. My blood is very different from a fire truck, but it doesn't bother me in the least that the concept of "red" applies to both of them.

Answer 5

To my mind, only the intuitionist/constructivist definition actually works: Mathematical existence is constructability -- the ability to generate the object or to generate as close an approximation to the object as you wish in imagination, given enough time. This is what is required to actually imagine the object. Things in the imagination are revealed as you explore them. Contrary to a lot of imaginations of how imagining ought to work, things do not pop into your head fully formed. Thinking, even very basic conceiving, takes time. And these are imaginary objects.

Classical mathematical existence is, implicitly, the expectation that something should be constructable, without the requirement that enough information exists to render it actually possible to construct or imagine the object. Otherwise, the negative result of later logic would not rankle us so much. It would not bother us to know that there are true-and-unproveable statements, unless we believed that the intention of being true in mathematics was to be proveable. We expected the constructability of all Platonic objects we can predict the existence of, and we cannot have it.

But I think those results really shoot down the Classical expectation. So we need to adapt, and the right direction to move is toward Intuitionism or farther into Construcivism. An object exists if you can construct approximations of it. It is reasonable to continue arguing in a classical fashion, but you need to realize that you are only talking about what ought to exist if the classical Platonic worldview actually held water, as a loose wrapper around the actual reality of the mind.