A simple argument for the existence of mathematical objects

(p1) The statement '3+2=5' is true.

(p2) If the statement '3+2=5' is true, then the number 3 exists and the number 2 exists.

(c1) Therefore, the number 3 exists and the number 2 exists.

(p3) If the number 3 exists and the number 2 exists, then mathematical objects exist.

(c2) Mathematical objects exist (from c1 & p3)

Let's take a look at some of the reasoning behind premise 2. Clearly we want to say that '3+2=5' is a true mathematical statement. But in virtue of what? What needs to be the case in order from '3+2-5' to be true? Well, it needs to be the case that there exists a number 3 and that there exists a number 2, such that their conjunction is 5. This is the case with any true statement. If I say, "the cat is on the mat", there needs to exist both the cat and the mat, such that the cat stands in an 'is on' relation to the mat, in order for it to be true. Therefore, since the statement is in fact true, it follows that such numbers really exist, and therefore that mathematical objects in general exist. Of course, my entire argument rests on the assumption that the statement is in fact true, but I'm sure most would concur that it is.

“2 + 3 = 5” is a sort of tautology. Numbers are a figment of our reasoning process; they don’t exist in the same sense as a cat or a mat

Sorry Brad but your assertion is a kind of futile semantic exercise

“2 + 3 = 5” is a sort of tautology. Numbers are a figment of our reasoning process; they don’t exist in the same sense as a cat or a mat

Sorry Brad but your assertion is a kind of futile semantic exercise

Yes, it is a tautology as all mathematic propositions are, but that does nothing to refute my contention. For instance, just think of the tautology, "All bachelors are unmarried". This is tautological, yet bachelors clearly exist in the same sense as cats and mats. So, its being a tautology is irrelevant to whether or not such objects exist.

As Dalehileman says, it's true by definition. it's a tautology, unlike "the cat is on the mat" which is an empirical observation.

I think the mistake you making, by implication, is that '2', '3' and '5' exist as objects in the same sense that cats and mats exist. This does not follow. All you can say is that they exist as concepts. And you can say this because the statement is either true or false.

[Scott]

What about the tautology Santa Claus is Santa Claus? Does the claim that the tautology is 'true' mean that Santa Claus exists?

Beat me to that one.

'true' therefore 'exists', doesn't follow.

So, its being a tautology is irrelevant to whether or not such objects exist.

Point swell taken Brad. However in no way does the equation confer objectivity

What about the tautology Santa Claus is Santa Claus? Does the claim that the tautology is 'true' mean that Santa Claus exists?

Actually, I think that's a great question. Something that I've been pondering over lately too. Just like Parmenides, I think it doesn't make sense to say 'x is x' if x doesn't exist in any sense (whether as a physical object, or as a mental object, or as an abstract object). So, 'Santa Claus is Santa Claus' seems to have a hidden existential quantifier in it. That is, it says (∃x)(x=x) rather than just (x)(x=x).

-- Updated April 23rd, 2012, 6:37 pm to add the following --

However in no way does the equation confer objectivity

I'm not sure what you mean by this.

I'm not sure what you mean by this.

The equation 2 + 2 = 5 or for that matter any mathematical “tautology”.

To make an argument of this type, you first have to dispose of Kant's claim that existence is not a predicate, that is, existence is not a property of things that can be adjoined by any argument.

What does your statement 3+2 = 5 mean if the terms 3, 2, and 5 do not exist? Would they not exist at least by virtue of their definitions? And then after your argument, what have you achieved beyond mental existence for them? In other words, what kind of existence do these terms have in order for the statement to be true? And what kind do they have after your proof?

To put it another way, 2,3 and 5 must exist (as mental entities) before you can make the statement 2+3 = 5. If not, that statement would be meaningless. However, when you are finished with your argument, you have nothing new added to their ontological status.

There are all kinds of magic you might do with this kind of argument:

If Venusians and Martians both have six legs, then it follows that Martian have six legs. (True) Therefore Venusians and Martians exist.

What does your statement 3+2 = 5 mean if the terms 3, 2, and 5 do not exist? Would they not exist at least by virtue of their definitions?

Pris it’s purely a semantic matter.

Pris it’s purely a semantic matter.

So you think the '3' does not refer to the number three or anything else?

When you declare that the statement 3+2 = 5 is true, you really mean that the (existing) number 3 added to the (existing) number 2 is the (existing) number 5—whatever sense you give to existing. Unless its terms refer to something definite, the statement means nothing. The something definite can be merely an idea, but if the terms refer to nothing, the statement cannot be deemed true or false.

His argument is backwards, it's the existence of 2,3, and 5 as numbers that are a prerequisite for the truth of the statement 3 + 2 = 5, not the other way around. If they refer to nothing or are merely names, then 3 + 2 = 5 is just the same as a + b = c where a,b and c are undefined.

-- Updated April 23rd, 2012, 10:49 pm to add the following --

In fact, he says it himself in the very first post:

What needs to be the case in order from '3+2-5' to be true? Well, it needs to be the case that there exists a number 3 and that there exists a number 2, such that their conjunction is 5. This is the case with any true statement.

Pris it’s purely a semantic matter.

So you think the '3' does not refer to the number three or anything else?

When you declare that the statement 3+2 = 5 is true, you really mean that the (existing) number 3 added to the (existing) number 2 is the (existing) number 5—whatever sense you give to existing. Unless its terms refer to something definite, the statement means nothing. The something definite can be merely an idea, but if the terms refer to nothing, the statement cannot be deemed true or false.

His argument is backwards, it's the existence of 2,3, and 5 as numbers that are a prerequisite for the truth of the statement 3 + 2 = 5, not the other way around. If they refer to nothing or are merely names, then 3 + 2 = 5 is just the same as a + b = c where a,b and c are undefined.

-- Updated April 23rd, 2012, 10:49 pm to add the following --

In fact, he says it himself in the very first post:

What needs to be the case in order from '3+2-5' to be true? Well, it needs to be the case that there exists a number 3 and that there exists a number 2, such that their conjunction is 5. This is the case with any true statement.

My argument isn't backwards. I'm not saying that '3+2+5' is true, and because of that it means 3 and 2 exist. The 'if-then' in (p2), which is where I think your accusation comes from, only serves as the logical form of the argument (namely modus ponens).

It's the first statement that requires the existence of 2,3,and 5.

(p1) The statement '3+2=5' is true.

You can't justify (p1) as true unless its terms refer to existing numbers.

It's the first statement that requires the existence of 2,3,and 5.

(p1) The statement '3+2=5' is true.

You can't justify (p1) as true unless its terms refer to existing numbers.

Well, I've assumed it to be true. So I do agree with you. My concern here is to show that holding the statement as true then requires an ontological commitment to mathematical objects as real existent things (though not concrete). That's the point of (p2). If it is the case that '3+2=5' is true, then it is the case that such numbers exist.

-- Updated April 23rd, 2012, 10:52 pm to add the following --

To make an argument of this type, you first have to dispose of Kant's claim that existence is not a predicate, that is, existence is not a property of things that can be adjoined by any argument.

What does your statement 3+2 = 5 mean if the terms 3, 2, and 5 do not exist? Would they not exist at least by virtue of their definitions? And then after your argument, what have you achieved beyond mental existence for them? In other words, what kind of existence do these terms have in order for the statement to be true? And what kind do they have after your proof?

To put it another way, 2,3 and 5 must exist (as mental entities) before you can make the statement 2+3 = 5. If not, that statement would be meaningless. However, when you are finished with your argument, you have nothing new added to their ontological status.

There are all kinds of magic you might do with this kind of argument:

If Venusians and Martians both have six legs, then it follows that Martian have six legs. (True) Therefore Venusians and Martians exist.

Not necessarily. I agree with Kant that existence is not a predicate; it is a quantifier. Secondly, your 'analogous' argument is not structured as mine is. Where is the premise that says if p, then x exists? That is, the conclusion doesn't follow from the premises. Lastly, If you don't like the grammar of 'x exists', then I could have simply said '(∃x)(x...)'. I just figured everyone would have understood what I meant.