A simple argument for the existence of mathematical objects

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.

The statement [all humans have six legs] is false if there is a single human who doesn't have six legs.

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


Naturally I’d have to specify that Marty doesn’t consider one of his kind who has lost a leg, as human.

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

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.

Alas, well, the fuzzy object with indefinite outline does after have a faint stripe here and there and maybe even a small touch of color

seems to be that no definite statements can be made about anything

I’ll have to concede however that some seem more definite than others

-- Updated April 27th, 2012, 11:05 am to add the following --

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,

Pris while I don’t know any of them personally their existence is almost a certainty while surely some of them are six-footed

The idea we’re alone is absurd. A very conservative estimate places our number at six sextillion but it must surely be much higher than that as just recently it’s been suggested that there are more planets than stars

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.

1. I agree with you that possible worlds are fictions, but I don't see why that matters. Possible world semantics is just a tool for thinking about modality, so for it to 'work' and provide true conclusions there does not actually need to be real possible worlds.

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)).

4. I don't see what's the problem here. 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.

5. I'm not seeing how that follows. Why do we need to perceive them just as we perceive material objects?

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.

Some of our uncertainty about this issue seems to stems from the fact that there are at least two approaches commonly used to establish truth: truth from definitional logic, and truth through observation. Of these, truth by definitional logic is easier to establish and, though it can be highly complex, it can with reason be described as tautological because at some level the conclusion is already present in the premises. Truth through observation requires much more care, and because it is based on probability it is never quite as certain – but it is more basic when trying to establish the validity and ‘reality’ of an object or idea. If a truth statement is not at some level based on observations, I think it cannot be viewed as the real truth.
Let’s start with the discussion of six legged humans. Many people would agree that having two legs is a part of the definition of being human – if this is accepted, then a six-legged human is ruled out by definition. (Real definitions usually allow for special contingencies – for example, amputees are not disqualified from being human because the reason for their one-leggedness can be explained.) For those who consider two legs to be irrelevant to the definition of what is human – perhaps language and empathy are all that is needed – then the weaker, but still powerful argument, must be used that no-one throughout history has ever observed a six legged human, and we think it likely given our knowledge of genetics (as a science, this is also based on observation) that nobody ever will. We are now using a probabilistic argument rather than a logical one.
The statement ‘3 + 2 = 5’ can be examined for its truth in both these ways. It is unquestionably true by a logic derived from definitions: 3 means 1 + 1 + 1 and 2 means 1 + 1; 5 is the only numeral defined as 1 + 1 + 1 + 1 + 1 so by definition it is the sum of the two other numerals. (Note that it is not hard to define a mathematics in which this would not be true – for example, ‘family math’ might argue that one human plus one human can equal three or more, depending how many children they have!)
The deeper way of viewing the truth of ‘3 + 2= 5’ is to explore the numerals not with reference to their definitions (which like the numerals themselves are products of human thought), but in relation to the aspects of the ‘real world’ to which we have related them. Numerals (and mathematics too) have qualified as valid concepts because they have proved over history to effectively help us describe what we perceive – like other mental concepts they have seemed to correspond very consistently to our observations. Numerals are a characteristic of groups of objects (any objects) in the same way that color, for example, can be considered a characteristic of any single object. (I am using the terms object and group of objects to refer to actual concrete phenomena like cats or groups of cats on mats. The term characteristic means a noteworthy aspect of an object (e.g. black) or group (e.g. count of 3).
To relate these thoughts to the original issue under discussion: if

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

simply asserts the logical truth of '3+2+5, then I do not consider that it has anything to do with the real world existence or not of mathematical objects; but if it asserts the deeper truth that this is a consistent pattern that has been observed though history in the world of specific concrete objects, then I think it does have bearing on the issue (though I would perhaps argue that the term should not be 'mathematical objects' -- can't think what to call them, however.

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.

They're close but not necessarily equivalent, because (c) is not incompatible with the following disjunction:

Numbers exist necessarily or it is not possible that numbers exist.

But since premise (a) rejects their being impossible, (e) follows. 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.

As far as how we become aware of their existence, I would say we acquire knowledge of them through the intellect. The intellect just being a non-sensuous form of cognition.

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.

Existence of the abstract is intermittent depending entirely upon apprehension by a living person. This characteristic of the knowledge base is most troublesome for those who believe that the rules of the physical domain should apply to the abstract, particularly the notion of persistence of existence. So they reify and take joy in the non-existent things they create. Which do not exist.

Existence of the abstract is intermittent depending entirely upon apprehension by a living person.

Well put Eric, according to the general principle that nothing is entirely anything while everything is partly something else so there’s no distinct dividing line between the abstract and the concrete. Thus we have a rock at one extreme and God at the other with The Catholic Church and Costco somewhere between

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.

Prismatic

Not true. 2 3 and 5 ontologically have different shapes when printed. [and 2+3 =5 is the fingers and thumb] snicker

Prill

Not true. 2 3 and 5 ontologically have different shapes when printed. [and 2+3 =5 is the fingers and thumb] snicker

Prill

Perhaps you need to try that one again and see if you can make some sense.