A simple argument for the existence of mathematical objects

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.

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.

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.

This is just a test, only a test!

-- Updated April 24th, 2012, 12:49 pm to add the following --

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

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

And what does saying that explain?

And what does saying that explain?

It asks why waste a lot of words that can be interpreted to mean nearly anything

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?

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.

Well I think if mathematical objects exist, they cannot be mental objects, since that would be to say they are dependent on thought. But, numbers, if they exist, would seem to exist necessarily. That is, they exist in all possible worlds. So, since the existence of minds is merely contingent, and they don't exist all possible worlds, the existence of numbers cannot be dependent upon them. (to be a dependent on minds would be to say that minds are a necessary condition for their existence. But, their being a necessary condition applies across possible worlds, and since there is a possible world w that contains the existence of numbers but not minds, it follows that minds are not a necessary condition for the existence of numbers).

About what I mean by numbers existing. Clearly what it means to exist is a rather difficult thing to define without avoiding circularity. 'There is a number 3' is just as unclear in meaning as 'There is a chair'. What does it mean to say there is a chair? that a chair exists? Berkeley says that all that means is that I can have sense experience of it. If I were to walk into the room, I could see it, touch it, and even taste and smell it if I wanted to, and that's what it means for the chair to exist. But this is clearly not what we mean, since no one would deny that the chair continues to exist without an observer. So, I think the best way to explain what it means to say numbers really exist is just that they are not fictions. In other words, numbers are not just invented by humans, nor are they just psychological projections of some kind. Polonious on the other hand was contrived, namely by the mind of William Shakespeare, and that's where the distinction lies (note that to say (∃x)(x is Polonious) is different from (∃x)(x is an idea & x is the idea of Polonious)). Existence, I believe, is not a category confined to just concrete objects.

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.Truth is quantifiable. Truth is relative to the sort of existent under consideration. Illogical fantasies exist, although the weight of truth carried by illogical fantasies is slight as judged by human reason. There are various sorts of existence, and the degree of truth carried by anything is relative to the sort of existence that it has, i.e. a possibility, a probability, a mad fantasy, a proven theory, an entity in spacetime or an abstraction of some property such as number or colour that the entity possesses.Each has a degree of truth however slight the degree of truth may be.

It is a matter of consensus how the degree of truth of any proposition is measured. Science seems to be the big winner in the consensus stakes, but we cannot be absolutely sure that scientific truths, whether concepts that exist or stuff that exists, correspond with overarching reality.

Personally I feel that if something exists in space and time it is more true than any abstraction from the tangible world of things . Therefore I believe that n(apples) is more true than n, especially if the apples are in front of me to be touched, smelled and eaten

in the hope people will take that for profundity?

No pris, only that it might stimulate some thinking in that direction. Profound I’m not, but Intuition strongly suggests that much of the apparent disagreement amongst us lies in that direction

Well I think if mathematical objects exist, they cannot be mental objects, since that would be to say they are dependent on thought

Brad does that mean that if there were no Universe numbers wouldn’t exist

When the Big Crunch occurs and there is only one of it all, is the number 1 the only one existing

In my brain, and yes it’s a relatively small one, Intuition suggests numbers have no reality beyond a process of electrons shifting around amongst cells

Pris in #24 expresses it far better than I

Each and every concept is true to some degree.Truth is quantifiable.

Aha! Bel my very favorite participant has nailed it. In other words, she is saying, as I understand her, and I concede I could be mistaken, as I have long maintained with virtually no support whatever, that nothing is entirely anything while everything is partly something else

Personally I feel that if something exists in space and time it is more true than any abstraction from the tangible world of things .

Einstein could not have better expressed it nor in fewer well-chosen words. Bel I sometimes wish I were single once more, much younger, and you lived nearby

Only however if I could take with me my present “smarts"

So nothing is entirely true while everything is partly so: 2 + 3 = 5 is partly false in the sense that numbers exist only as a figment of thought process and have no absolute “existence” like the rock sitting there

….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

Lo! the very idea of infinity itself might be invalid, merely a sound emanating as a result of interaction amongst brain cells transmitted by an evolutionary constriction of the humanoid breathing mechanism

While I conceded that her concept somewhat negates aspects of my own argument it is only because her thinking so far outweighs mine

I would also incidentally describe myself (and my No. 2 Son) as a Socialist if “Apodictical Existential Pantheist” didn’t sound more impressive

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.

humans have six legs.

I presume you mean entirely false. What if the fetus contains an extra set of dormant leg cells; how many legs does a woman have who is pregnant with twins

Perhaps not good examples but over the enormous range of possible examples you cannot draw a dividing line beween those good and those bad

Of course such thinking multiplies indefinitely the conditions required of the statement: “Humans of the world we know don't have six legs, assuming by “human” you mean …... while “leg”is defined as ……... until it requires an infinite number of words to bestow truth upon it. Indeed if the Universe is infinite forever and anything that can happen, will happen……….

the statement itself is paradoxical

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

Your problem here is that you are using a property of finite sets on infinite sets.

I’ll have to take your word for it, I’m not a mathematician, my reactions purely spontaneous

—but I'm not sure what you meant about the legitimacy of infinite universes.

Forgive me Pris but it’s hard to express some of these intuitional insights. It might prove for instance that an infinite Universe is impossible, calling into question the entire concept of infinity

…....just Intuition babbling on

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.

just change it to all humans have six legs.

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

I'm not prone to argue.

Good one Pris, I must remember it