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.

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.

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.

Your assumption of the truth of p1 is unwarranted unless 2,3, and 5 exist and the signs + and = have their accustomed meaning.

To put it another way, in order for the statement 2 + 3 = 5 to be true presupposes that 2, 3, and 5 refer to existing objects of thought.

Okay, so if you don't think mathematical objects really exist, that would imply (by modus tollens) that the statement '3+2=5' is false. But do you really want to concede to that? Mathematics is in-dispensable, since it is the basis of most of our strongest sciences. That would entail an extremely anti-realist view and destroy knowledge all together.

Not what I am saying at all. You had it right in your first post when you wrote:

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.

That statement is exactly right and it says that for 3 + 2 = 5 to be true, there are pre-conditions, including that the numbers exist (in some sense). My criticism is that you try to deduce the truth of the preconditions from the (assumed) truth of the statement whose truth requires them. The better statement in place of your p2, would be: 3 + 2 = 5 only if the numbers 2,3,and 5 exist.

The existence of mathematical objects has been a problem of ontology for a long time. I believe the better question is how should we think of the existence of mathematical objects? Clearly they are not material objects and while they have the character of mental objects, their behavior is far more rigid than is usual with mental objects. We think that everyone who understands it correctly has the same notion of the number 2 and computes with it in the same way.

When you say the sentence numbers exist, what do you mean by the word exist since you cannot mean they exist as material objects? Consider for a moment an entirely different example, the true sentence There exists in the play Hamlet by William Shakespeare a character named Polonius. You would not want to conclude just from that true sentence that Polonius ever existed as a real person. When you discuss what happens in a play, you do not draw conclusions about material reality, but confine yourself to the universe of the play.

Possibly that is the way to think of the existence of mathematical objects—in a universe of their own so that when we speak of their existence, it always means only just within the universe of discourse of mathematics. That means that of the existence of mathematical objects can only be given within that universe, which assumes general existence to begin with. Or you could begin with the universe of sets and construct mathematics out of that or any other formal system with axioms. However among the axioms there will (one suspects) always need to be an assumption of existence to get you started.

Perhaps you have another idea of what it means to say that mathematical objects really exist.

I repeat, it’s all a matter of semantics--------

And what does saying that explain?

Rather than attempt to resolve questions of philosophy through careful thinking, you just wave your hands in the air and gravely announce, "It's all a matter of semantics" in the hope people will take that for profundity?

Some comments on your argument.

1. The difficulty I see with the notion of possible worlds is that while the idea offers a convenient and pleasant semantics for modal logic and clarifies ideas, it is in fact a fiction itself, that is, a mental construct and, for all we know, nothing more than that. We know only one world by experience. We do not know that any others exist at all, that they have any reality whatsoever beyond our imagination.

2. Consequently the claim that mathematical objects are not mental objects because they exist in all possible worlds, which worlds are themselves an invention of the mind, does not work for me. We think that mathematical objects are necessary, but that is beyond verification.

3. Even if we assume that there are possible worlds, that assumption itself seems to me far more extensive than the mere assumption that numbers exist in some extra-mental sense. It erects a huge stage of inaccessible possibilities that we can only imagine, but not explore.

4. The existence of material objects is fairly clear—their existence can be experienced directly through senses or their extension by scientific instruments, not to say that their existence depends on the senses, only that it is known and recognized by senses.

5. In order to prove things about numbers in mathematics you either have to assume their existence or you have to construct them out of simpler systems of numbers—for example the construction of real numbers from the rationals as given by Dedekind in the nineteenth century as a way to make mathematical analysis rigorous. To get mathematics off the ground and investigate deeper properties, you need to assume existence of a sufficiency of numbers or sets. If numbers have an existence as real entities of some kind, it seems you ought to be able to encounter them as directly as you do material objects in science.

6. That numbers are not fictions may very well be true, but it fails to satisfy as an answer to what kind of existence they do have. That is to say, it is hard to see what conclusions you could draw out of that non-fictional hat.

Each and every concept is true to some degree.

I think this is nonsense and quickly leads to contradictions. For example, the concepts that 1 = 0 or that √2 is a rational number would have to be true to some degree. What would that mean? We would have to allow that to some degree Fridays are always holidays in every country or that humans have six legs.

Perhaps worse still, the statement itself is paradoxical since it requires that each and every concept is absolutely false must be true to some degree.

Presumably I have not correctly understood what you meant to say.

-- Updated April 26th, 2012, 3:47 pm to add the following --

….while I’ve long wondered whether our concept of numbers might fall flat in the face of infinity. For instance if there are an infinite number of numbers then how can there also be an infinite number of even ones, and odd too, etc, and somehow doesn’t this call into question the legitimacy of infinite universes

Not at all. Your problem here is that you are using a property of finite sets on infinite sets. It is characteristic of an infinite set that there can exist a 1-1 correspondence of itself with a proper subset. The following sets all have the same infinite cardinality:

1) the natural numbers

2) the even natural numbers

3) the natural numbers which are perfect squares

4) the natural numbers which are primes

You can stop worrying about the status of the natural numbers—they are doing just fine—but I'm not sure what you meant about the legitimacy of infinite universes.

Try again Statements such as humans have six legs are usually interpreted as predicating a characteristic of the species, but since you want to quibble, just change it to all humans have six legs. That is an example of a statement which cannot be true to some extent.

Yes but only if you maintain a distinct dividing line between paradoxical and not

As Cleopatra said to Marc Anthony, I'm not prone to argue.

I imagine a race of six-legged humanoids in a distant galaxy who don’t consider us “human” so you still have to attach conditions. Lawsuits have been conducted and won on the basis of such apparent absurdities.

If you wish to justify that the statement all humans have six legs is "true to some extent," you need to do better than that. The statement is false if there is a single human who doesn't have six legs. Given your penchant for spontaneity, I am reluctant to suggest you yourself as a useful exemplar of standard human anatomy, but count your lower limbs immediately without fail and report back to us.

Let me reiterate that six-legged Marty who considers himself human but doesn’t consider us so sees the statement as true. So you have to specify at least that what you mean by “human” excludes, say, anything not earthbound. You will have to continue adding conditions until what you’re saying is, “ 'All humans have six legs' is false” provided that which I callI call human has fewer than six legs” which is of course a tautology

No, I don't. It may be true that my circle of acquaintances is not quite as extensive as yours, which seems to extend to alien life forms, but I don't accept your Humpty-Dumpty argument that words can mean anything you wish them too and that I must rule out any crazy interpretations you might put forth, which would apparently be an endless task.

Once again, the statement all humans have six legs is false if any one single human fails to have six legs. It cannot be "true to some extent." Perhaps you need to refresh your understanding of negations of statements with existential and universal quantifiers.

Pris we simply view the Megillah differently. Yours is black and white with definite outline while mine is fuzzy gray in all diretions

I agree your thinking is fuzzy. Sorry, logical mistakes do not count as just a difference of opinion. Forgive me if I misrepresent your position, but your fuzzy point of view seems to be that no definite statements can be made about anything because words can be interpreted anyway you please. That's an easy way of ensuring you never lose an argument—you can't be pinned down to anything definite—but it makes discussions with you useless.

2. It does not matter if it is beyond verification. I believe the matter can resolved using a priori reasoning, or at least reasoning to the best conclusion.

(a) The existence of numbers is not impossible.
(b) If the existence of numbers is not impossible, then their existence is either necessary or contingent.
(c) The existence of numbers is not contingent.
(d) Therefore, if the existence of numbers is not impossible, then their existence is necessary.
(e) Therefore, the existence of numbers is necessary (from (a) & (b) & (c)).

We agree on using the possible worlds semantics as simply that and nothing more. That's good.

1. Isn't there a problem in making statements about the existence of numbers in the absence of any clear notion of what that expression means—assuming you intend it to mean some kind of existence beyond existence as mental constructs. We ought to know the meaning of the existence of numbers before asserting anything about it. There is no problem with numbers existing as ideas which (almost) everyone holds in the same way and uses in the same way, but if there is any existence beyond that, its context is not clear.

2. I assume that by contingent here you simply mean not necessary . If that is correct, then statement (c) seems to be equivalent to (e) and looks even more dubious. What can be the justification for saying that the existence of numbers is not contingent?

Certainly, the existence of physical objects is much more certain since we can perceive them via our senses, but from that it does not follow that supra-sensible objects don't exist.

I wouldn't claim that "supra-sensible" objects don't exist, but the question is: how do we become aware of their existence? The answer seems to be that we become aware of them as abstractions from sensible objects. A child learns to count using fingers, etc. and gradually forms the idea of number as an abstraction distinct from objects counted, something that exists in his mind. Then later he learns to do arithmetic, memorizes the times tables, etc. When does he encounter numbers directly?

If all you are claiming is that numbers are abstractions of a certain kind, I see no problem, but I think you want something more.

What justification is their for thinking their existence cannot be contingent? My reason for thinking that would be that they are abstract objects, and abstract objects, if they exist, seem to exist necessarily. For how else could an abstract object exist? Contingent things are typically concrete particulars, such as tables, chairs, and persons.

I take contingent existence to mean existing only under conditions. Wouldn't you consider an abstract object to have contingent existence if it existed only in the mind of one person? One person might have a unique idea of a particular abstraction.