# What is mathematical existence?

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.

• It's not clear what you mean when you say a "mathematical entity exists". If you could expand the question a bit by adding context or defining what you mean by mathematical entity, it would be easier to see what you are trying to ask. – stoicfury Oct 8 '11 at 20:48
• Is there any chance I can persuade you to develop your question a bit further? What led you to ask this community for an explanation about this? What have you found out so far? – Joseph Weissman Oct 8 '11 at 21:26
• Was there an alternative you were thinking of? Or is 'consistency' not satisfying for you? 'no more' comes with a lot of emotional baggage, so it'd be good to know what direction you'd like an explanation to take. – Mitch Oct 11 '11 at 13:49
• A mathematical entity could be a number, say 2; or it could an entire algebraic system, say the group of rotations of a square or the ring of reals, or the entire category of algebraic varieties (a variety is any algebraic system with a fixed number of operations) etc. – Mozibur Ullah Oct 19 '11 at 3:36

## 4 Answers

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

• Since I see the warning below about one-line answers, I'll simply add here: the question is so broad and vague as to indicate that no preliminary research has been done by the questioner. I linked to the Wikipedia page that gives an overview of some of the many different theories of mathematical existence; to do more (in the absence of a more refined question) seems futile. In other words: sometimes a one line/one link answer is enough, if you ask me. – Michael Dorfman Oct 10 '11 at 13:48

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?

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.

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.

• So by defining we create new objects? For example when we define determinant we have already shown that there exists a unique number that can be obtained by multiplying and adding the entries of the matrix. Can we do the same with function? Can we prove that there exists a mapping and we call that mapping function? Or we just create that entity and discover some results using the axioms and other definitions? In other words is every definition we make just an abbrevation for entities that can be proved to exist by the axioms or we can define whatever we want? – ado sar Jul 29 '20 at 11:17