# What Is Your Ontology Of Mathematics?

What I am asking is: What is the ontological statis of mathematics? Do numbers exists? If so, what IS a number? How do numbers exist?

Explain your Philosophy Of Mathematics. For ideas, here is my Philosophy Of Mathematics:

numbers, their relations, and mathematical propositions are fundementally platonic in nature. What does this mean? This means, that a mathematical proposition such as "there cannot be finitely many prime numbers" exists immaterially and atemporally (not extended in space).

If you want me to defend this view, please ask. OR, if you have questions, please ask.

*waits for todangst to come...*

**#11**

1) Do you agree that all language/mathematics is rooted in rule following?

2) 'Following a rule' is a verb, an 'action' and that would make it potentially a physical phenomenon, right?

C) If you accept 1 and 2 then all concepts/numbers/relations could all be rooted in physical events. So they would exist as states in our physical universe.

**Chaoslord2004 wrote:**

But the physical universe COULD have been different. On this view, this means that mathematical facts COULD have been different. I mean, it is generally accepted in Physics that this is but one of many possible universes.

When we say 'possible world' we mean 'a world we can possibly conceptualise'.

I.e. a world we can construct using the concepts of our understanding (of which our mathematics is an essential part).

If this is so...and on your view, then this means that mathematics, in theory, could have been different. Thus, 2 + 2 could have equaled 5...it is just a matter of contingency that when 2 and 2 are added one gets 4.

I see it differently.

'Necessity' and 'Contingency' define limits on what's conceptually possible. So maths as we understand it is necessary for any conceivable picture of the world. There's potential for transcendent 'possibilities' of different mathematical/conceptual structures but any description we make pre-supposes our current ones so we couldn't say anything more than 'transcendent'.

However, assume you want to say that mathematical truths are necessary truths. Hence, when it was proved that there could be no greatest prime number, this was a truth that couldn't possibily be false. Even if the physical universe had wildly different constraints, this would still be true. Right? Do you accept this?

Any possible universe we can imagine/contemplate presupposes being conceptualised as we do. So mathematics is necessary for any possible universe we can imagine/contemplate/give constraints to. This allows for transcendent 'possibilities' that we can't conceptualise, but then we couldn't really talk of them full stop...

How does this answer your objection?

**#12**

**todangst wrote:**

Because the terms you've used have no ontological bearing. Immateriality and atemporality are negative terms devoid of any universe of discourse.

**chaos wrote:**

Why?

My good friend, I just explained why in what you've quoted here. Because concepts like 'immateriality' are necessarily defined only in negative terms, without ANY universe of discourse.

So there's nothing left over for them to be. So they can have no ontological status. I've made this point dozens of times, both here and on infidelguy. For a set of negative traits to be meaningful, there has to be a universe of discourse, a set of things left over, for it to 'be'.

If I show you a box with a comb and a penny in it, and say 'the object is not a comb" the negative definition guides you towards recognizing the penny as the object in question. So it is NOT the set of negative terms, but the the universe of discourse that provides the meaning!

So, the problem for terms like 'immateriality' is that they rule out EVERYTHING. There's nothing left over in the box, for something 'immaterial' to be... there isn't even a box.... so the term has NO ontological bearing.

It can only mean 'nothing'.

Oddly enough, I found a intelligent young man who has also expressed this point very well:

http://www.infidelguy.com/ftopicp-425732-.html#425732

His name on infidelguy was chaoslord.

And he wrote this:

"Defining something in the negative HELPS to illuminate what something is, but it needs more. Sure, negative definitions help, but positive definitions are needed. Here, let me show you:

X is defined as being a non-car. This is all I say. Now please, explain what the ontology of X is? What are X's properties?

A negative definition alone is insuffiecient. Even my one philosophy professor had to agree with this point to a degree."

So I'm perplexed as to why you ask me to explain this to you.

**chaos wrote:**

Furthermore, many things are defined in terms of the negation on something;

Yes, but withing a universe of discourse. A set of negative traits, devoid of anything left over for for the 'entity' to be can only refer to nothing.

**chaos wrote:**

take Quantification Theory. All that is really needed is one quantifier, and one can define either "existence" or "for all" in terms of one another. For example, without using the existential quantifier, here is on to define existence only using universal quantification: It is not the case, that for all x, not Px. Completely negative definition, and yet we know what it means.

Yes, we know that it means nothing! I'm really stunned to see this error after you yourself have helped to refute it in the past! You're confusing the fact that we know what the term 'non existence' or 'nothing' means as implying that non existence having ontological bearing. This is a common confusion. The fact that the term 'NOTHING" means something to us (not something) only comes from the fact that the term exists in contradistinction to existence. We know that it means 'not existence', i.e. NOTHING. But while we know 'what the the word means", "nothing" has no ontological bearing.

That's the actual point before you.

**chaos wrote:**

So your claim is demonstrably false.

No, your argument is based on a misunderstanding. You're confusing the fact that the term 'non existence' has linguistic meaning as a contradistinctive, with non existence having ontological status. You're also failing to appreciate that it is the universe of discouse that gives a set of negative traits its meaning, and not the negative terms themselves.

I agree, we don't have a great definition of immateriality.

The problem, again, is that there is no postive definition, and, furthermore, that creating a positive definition is impossible, because the concept is defined in contradistinction to matter. Yet this thread supposedly is about "ontology for numbers!

So do you see the problem now? How can you discuss ontology with terms that do not possess any ontological status?

**chaos wrote:**

All I was doing was saying in what domain numbers exist in. I still don't know what a number actually is. However, I was merely saying where these abstract entities exist: The platonic universe.

But you can't refer to it as a universe if it has no positive properties! A universe is a set of things. The only 'set' for the 'platonic realm' is the empty set. Null.

Platonism can only be an argument from ignorance. You'll never see a postive argument for it, you'll only see "materialism can't answer...."

So platonism is only incomplete materialism.

**chaos wrote:**

That is because you are treating "immateriality" as a predicate.

No. I am telling you that the term has NO ontological status. None. So how can you build an ontology upon terms with no ontological status?

**todangst wrote:**

But you don't believe that numbers are non existent. So they must be something. This leaves us with matter. Numbers, if they exist, must be matter. And one way they can exist as matter is as a relationship between sentient brains and the universe. The universe is the constant, the constraint upon the sentient brain. The brain is akin to paper, the universe, the pen. But there is more to this of course....the paper is not merely blank... it has properties, etc... I am not invoking tabla rasa here.... but whatever the specifics of how the brain works (we can leave that aside), we have a sentient brain and a constant universe, interacting.

**chaos wrote:**

fine. But the physical universe COULD have been different. On this view, this means that mathematical facts COULD have been different.

This is a non sequitur for two reasons.

It has never been demonstrated that the universe could have been different. It is only an assumption.

It has never been demonstrated that a 'different universe' would generate a fundamentally different type of mathematics.

So saying 'the universe could be different, so maybe 2+2=5" does not follow.

I mean, it is generally accepted in Physics

It is a quesiton many ask in physics, but no one has ever demonstrated that it is the case. I might even lean that way, but no one has ever shown that any basic law of the universe was flexible in a fashion that would allow for a fundamental change in mathematics!

that this is but one of many possible universes.

Can you show me any physicist who states that a change in matter would or could lead to a change in basic mathematics?

If this is so...and on your view, then this means that mathematics, in theory, could have been different.

Again, this is a non sequitur, and I am unaware of any physicist who has ever said such a thing.

**todangst wrote:**

I don't think you're not going to argue from inductive uncertainty, so I am unsure by what you mean here. How can it be shown otherwise, if immateriality is invisible, and incapable of having an effect on the material world?I really don't think that you want to rest your argument on the fallacy of arguing from inductive uncertainty, do you?

This has absolutely nothing to do with inductive reasoning...none.

Except that it does. You're arguing from the supposed limits of physics, an inductive science.

So, this is a red herring at best.

No, it's accurate at best.

There is no evidence AT ALL that the physical world is all there is.

Please be careful around walls and sharp things.

Seriously, arguing from any type of uncertainty doesn't help your cause, because your problem is that immateriality has no ontological status.

As far as I know, no one has given a probability calculation that makes "immateriality" unlikely.

That's an argument from inductive uncertainty as well as an argument from ignorance!

No one has to give any such "probability caculation". In order for you to give us your ontology for numbers, you have to give us terms with ontological bearing. If you can't demonstrate what 'immateriality' means, then that is YOUR problem.

If there has...good! Show it to me. To the best of my knowledge, however, there has not been.

Again, I'm going to point out the problem here.

Your term has NO ontological status.

So there's no way for you to 'rule it in' in the first place. There's no way to confirm it, there's no way to even form a hypothesis, as the term is empty.

Consider my infamous set of questions that usually pisses off any person I debate with:

1) Can you show that anything exists other than matter or energy? What are its "properties" - i.e. is it something natural? If not, how can we 'know" or "infer" anything about it. If we can't, what use is your 'hypothesis"? If it has no use, then why are we having this conversation?

2) How does something that is neither matter nor energy interact with our natural world?

3) How do you avoid violating the principle of conservation of energy? If no physical energy or mass is associated with "immaterial things", then there is a serious problem: a fundamental principle of physics is that any change in any physical entity is an acceleration requiring the expenditure of energy - but if these things have no matter or energy, where does the energy come from? what you have here is something akin to the impossibilty of perpetual motion - energy from nowhere. Dan Dennet states that these questions represent the fatal flaw in any dualistic argument (i.e. to immateriliaty) (- 1990 Consciousness Explained.)

**chaos wrote:**

Here is what we know (ruling out idealism): The material world exists. However, this fact does not rule out the ONLY possible mode of existence; this was my point. It had yet to be shown that this is the only possible mode of existence...even inductively.

Then you are conceding that you've argued to inductive uncertainty. Thank you.

**chaos wrote:**

Maybe im wrong, if so, show me an argument which demonstrates that the likely hood of their being anything above and beyond the material world is low. Go for it.

I've already made these points numerous times, both here and on infidelguy.

http://www.infidelguy.com/ftopicp-425447-.html#425447

http://www.infidelguy.com/ftopicp-425445-immateriality.html#425445

http://www.infidelguy.com/ftopicp-415990-.html#415990

http://www.infidelguy.com/ftopicp-415689-.html#415689

The term 'immateriality' has no ontological status, no meaning.

So you tell me, what hypothesis could you formulate to even get your investigation off the ground? How can you even begin if you don't have any way to conceptualize your hypothesis in positive terms?

What is immateriality? How does it 'work'? How can we even use the words "is" or 'work" in regards to something with no ontological status? How can we even call it 'something'?

**todangst wrote:**

No. You're forgetting that mathematics is a relationship between the brain and the universe.

I not sure what this means. Mathematics ITSELF is a relationship between the brain and the universe? What exactly are you saying?

Show me mathematics without a physical brain.

That's what I mean.

**todangst wrote:**

Remember your Dennett and the fact that our relations with the univerese are interactions.

**chaos wrote:**

Not that this has any baring on the matter, but Dennett is a platonist about mathematics...

Really? How so? He's rather clear about the nonsensical nature of immateriality.

**todangst wrote:**

Again, you're leaving out that the universe itself remains the same. The existence of the universe is an 'anchoring heuristic' that plays a part in determining what types of mathematics 'work' and which do not. This defeats your '2+2=5' argument.

Yes, so even if the universe was widely different, and if the laws of physics had been drastically different, 2 + 2 = 5 would be impossible. Why? Because there is something over and above the constitution of this universe.

"something" you say... well then, give me it's description, using terms with ontological status.

If you can't, then how can you call it 'something"

The universe itself is sufficient.

**todangst wrote:**

But still, different brains might come up different mathematics, right? But, even if different minds related to the same universe differently, this would not present a problem, because the differences would still have a relationship with each other, because both sets of numbers would be spawned from the interactions of the sentient brains with the universe. So, the differences would correlate, as they would be anchored by the fact that the same universe impresses itself upon these different minds in a consistent way, because again, the universe here is a constant.

**chaos wrote:**

When you say interaction, do you beings who empirically derive principles? Hence, mathematics is derived from the material world? Is this what you mean? If so...this is wrong, for we can prove things about mathematics that have no physical correlate:

Sorry, but you can't demonstrate ANYTHING without a physical correlate.

That's the real problem before you.

**chaos wrote:**

Take a mathematical sphere. A mathematical sphere cannot exist physically.

Actually, you mean to say it cannot have an extra-mental existence.

But it can exist abstractly, as a set of physical neurons in a physical brain.

Which means that it exists physically within a sentient brain.

If it did, it would have to consist of infinitely many points.

We can express the concept of infinity in a finite formula, which again, exists as a set of neurons in a physical brain.

In other words, a mathematical sphere cannot physically exist. Yet, we can prove theorems about it.

And these theorems are encoded in our neurons, in our physical brains.

Those who know the good, do the good. - Socrates

**#13**

I would like to preface this by agreeing with most of what you said todangst. I made an egregious error: I saw a potential short fall in Materialism as evidence for platonism. As a critical thinker, I must recognize the problems with platonism and suspend judgement on the ontological statis of mathematics. I must conceed that platonism is to strong of a claim as of right now. Rather, I will say that mathematical facts exist mind independent...but I suspend judgement, as of now, as to their ontological statis (be it material or platonic).

I thank you for bring this to my attention. I had a momentary lapse of judgement. However, I still disagree with your position. Below I indend to argue why it has problems.

**todangst wrote:**

A universe is a set of things.

This is logically impossible. This implies that there is a universal set. However, this is logically impossible. For, contrary to Cantor's Theorem, the power set of the universal set would be of the same cardnality as its power set. However, Cantor's Theorem states that the power-set of any set must be of greater cardnality then the original set. Hence, one can preform a reductio ad absurdum on the above claim to show it cannot be true.

Sorry for nitpicking. I couldn't ignore that.

**todangst wrote:**

It has never been demonstrated that the universe could have been different. It is only an assumption.

Im not up on Physics, all I know is that it is the expert opinion of many physicists that the universe could have been different. Im not sure if this is true.

I hope someone would demonstrate either the truth or falsity of this question. It would make things ALOT easier if physical and logical possiblity, physical and logical necessity, and so forth collapsed into one. If this is true, then one can see how Materialism could handle mathematical propositions. For one could consistently hold that mathematics is purely material, and yet necessary.

**todangst wrote:**

It has never been demonstrated that a 'different universe' would generate a fundamentally different type of mathematics.

This is a true point. This is an assumption I never thought to question. You are correct. However, it does follow that if mathematics is derived from the material world...and if enough of the material world is changed, then so would mathematics. Hence, while you are correct in one sense: that a change in the material world doesn't necessarily constitute a change in mathematics (for instance, very small changed). However, if enough changes...or changes that were pertenant to mathematics were changed, then so would mathematics. Hence, it would be possible for 2 + 2 = 4 to be false. I guess it isn't totally crazy to hold that mathematics is contingent, if im not wrong, John Stuart Mill held to this belief. However, I would contend that mathematical facts are facts which much obtain in all possible worlds (to use possible world semantics).

however, if pressed, I can give no good reason for holding mathematics as necessary, other than the strong intuition that it is.

**todangst wrote:**

Actually, you mean to say it cannot have an extra-mental existence.

But it can exist abstractly, as a set of physical neurons in a physical brain.Which means that it exists physically within a sentient brain.

here is the worry: What is the truth maker for mathematical propositions? What makes them either true or false? Since you agree that we do not decide these facts, what makes them true? in virtue of what are they either true or false?

**todangst wrote:**

We can express the concept of infinity in a finite formula, which again, exists as a set of neurons in a physical brain.

right, but makes the following proposition true: "there is no last prime number"? Since we don't, on your view, what does? Since I think you agree that statements about infinity are not nonsense, what makes statements about numbers not yet cognized true? Say the nth number of pie not calculated?

"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions

**#14**

**Numbers exist in material brains. They are encoded electrochemically, in neurons.**

By this reasoning certain numbers that have yet to be thought of do not exist. The number of 'numbers' between 0 and 1 is infinite 0.01, 0.011 etc.

**#15**

**qxdc wrote:**

Numbers exist in material brains. They are encoded electrochemically, in neurons.

By this reasoning certain numbers that have yet to be thought of do not exist.

Actually, they would exist potentially, we'd have the potential to formulate any specific number in these sets, because of the concept of sets. No one needs to sit down and think of .90128019010231923012901293012391203912093012391201202130129312 112 in order to substantiate it...

Now please, if you wish to continue this discussion, do not procede in this purely negative fashion. If you wish to argue for a non material basis for numbers, present your postive claim, along with a postive ontology that does not steal from materialism.

Those who know the good, do the good. - Socrates

**#16**

**Chaoslord2004 wrote:**

I would like to preface this by agreeing with most of what you said todangst. I made an egregious error: I saw a potential short fall in Materialism as evidence for platonism. As a critical thinker, I must recognize the problems with platonism and suspend judgement on the ontological statis of mathematics. I must conceed that platonism is to strong of a claim as of right now. Rather, I will say that mathematical facts exist mind independent...but I suspend judgement, as of now, as to their ontological statis (be it material or platonic). I thank you for bring this to my attention. I had a momentary lapse of judgement.

Good to hear it.

**todangst wrote:**

A universe is a set of things.

This is logically impossible.

No, it is not.

This implies that there is a universal set.

No, it does not.

Sorry for nitpicking. I couldn't ignore that.

You nitpicked incorrectly. A universe is a set of things, not a universal set of things. A penny and a comb sitting in a box is a universe of discourse. You even use the term 'platonic universe' in your discussion, so I just don't see why you're having a problem with my using the term "universe". If it continues to bother you, just rename my phrase "field of discourse"

**todangst wrote:**

It has never been demonstrated that the universe could have been different. It is only an assumption.

Im not up on Physics, all I know is that it is the expert opinion of many physicists that the universe could have been different.

Can you cite one who demonstrates that how it could have been different.

**todangst wrote:**

It has never been demonstrated that a 'different universe' would generate a fundamentally different type of mathematics.

This is a true point. This is an assumption I never thought to question. You are correct. However, it does follow that if mathematics is derived from the material world...and if enough of the material world is changed, then so would mathematics.

How does this follow? This does not follow unless there is a fundamental change to matter wherein matter is no longer anything at all like matter. Even then, I don't see how it necessarily follows.

**todangst wrote:**

Actually, you mean to say it cannot have an extra-mental existence. But it can exist abstractly, as a set of physical neurons in a physical brain. Which means that it exists physically within a sentient brain.

here is the worry: What is the truth maker for mathematical propositions? What makes them either true or false?

We do. We assign the values. Math is deductive, a set of equivalencies we create.

Since you agree that we do not decide these facts,

I do?

**todangst wrote:**

We can express the concept of infinity in a finite formula, which again, exists as a set of neurons in a physical brain.

right, but makes the following proposition true: "there is no last prime number"?

I'm not a mathematician, but I don't see any reason to appeal beyond materialism in order to answer this question.

Those who know the good, do the good. - Socrates

**#17**

**todangst wrote:**

Can you cite one who demonstrates that how it could have been different.

Any physicist who endorses the multiverse theory. Moreover, the Physicist Lee Smolin who wrote "Life of the Cosmos."

**todangst wrote:**

How does this follow? This does not follow unless there is a fundamental change to matter wherein matter is no longer anything at all like matter. Even then, I don't see how it necessarily follows.

It follows from the fact that if the material world had different physical constrants and laws, then matter would behave differently. If mathematics is derived from how the material universe worked, then it is possible that much of mathematics would be merely contingent upon the operation of the universe. Does this mean that mathematics would necessarily be different? No, but this is not the point. The point was that it was possible. This possibility...atleast prima facie, is absurd.

**todangst wrote:**

We do. We assign the values. Math is deductive, a set of equivalencies we create.

We assign the meanings to the symbols. However, after the meanings are fixed, it cannot be that we decide the truth of a mathematical proposition. given the meaning of "2" and the meaning of the addition operator, 2 + 2 cannot equal anything but 4. No matter how hard we try, we cannot make 2 + 2 = 5.

This whole thought is troubling. If we literally make the truth of abstract propositions, or of logical systems, then in virtue of what, can something be considered rational? If we decide that modus ponens is a valid inference rule, we could just as easily decide it wasn't. This seems highly dubious. I cannot accept this.

**todangst wrote:**

I'm not a mathematician, but I don't see any reason to appeal beyond materialism in order to answer this question.

Nor I. You can still be a mathematical realist and yet a a materialist. Penelope Maddy is a mathematical realist, but not an immaterial platonist.

"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions

**#18**

**Chaoslord2004 wrote:**

todangst wrote:Can you cite one who demonstrates that how it could have been different.Any physicist who endorses the multiverse theory.

You're missing my actual question. I am aware of the conjectures concering infinite universes with different basic laws, but that there could be fluctation in the basic physical laws of our universe is a hypothesis, not a theory. For all we know it may be impossible for there to be anything other than our basic laws.

So can you cite a physicist who moves past hypothesis and offers up a theory of how basic laws could be different, and who then how this would work. Again, I'm aware of Smolin and his Darwinian concept of universe generation, but I'm not aware of anyone who has ever demonstrated that our universe COULD actually be different, and how this would work.

**todangst wrote:**

How does this follow? This does not follow unless there is a fundamental change to matter wherein matter is no longer anything at all like matter. Even then, I don't see how it necessarily follows.

It follows from the fact that if the material world had different physical constrants and laws, then matter would behave differently. If mathematics is derived from how the material universe worked, then it is possible that much of mathematics would be merely contingent upon the operation of the universe. Does this mean that mathematics would necessarily be different? No.

And here's a second problem. First, it's merely a hypothesis that our basic laws could be different. Then, it's yet another hypothesis that a change in basic laws would somehow influence a priori propositions.

So we have groundless supposition X groundless supposition...

**todangst wrote:**

We do. We assign the values. Math is deductive, a set of equivalencies we create.

We assign the meanings to the symbols. However, after the meanings are fixed, it cannot be that we decide the truth of a mathematical proposition.

Why not? It follows from our definitions. Math is tautology.

given the meaning of "2" and the meaning of the addition operator, 2 + 2 cannot equal anything but 4. No matter how hard we try, we cannot make 2 + 2 = 5.

Again, that's because we identify '2' as being half of four, a priori. I don't follow your concern at all.

This whole thought is troubling.

But why?

If we literally make the truth of abstract propositions, or of logical systems, then in virtue of what, can something be considered rational?

I don't see any problem. I define X to be 'half' of Y, and then hold, that "half "is structurally the same thing as "add 2 to make a whole"... it's merely equalities, tautologies...

If we decide that modus ponens is a valid inference rule, we could just as easily decide it wasn't.

How so? How could we just as easily make that decision?

This seems highly dubious. I cannot accept this.

You've not given any reason why your dilemma could actually exist. If I define "2" as 'half of 4" then it follows that two '2's are equal to four, if by 'half' we mean "two are required to make a whole'

It's all deductive. Where is the problem?

**todangst wrote:**

I'm not a mathematician, but I don't see any reason to appeal beyond materialism in order to answer this question.

Nor I. You can still be a mathematical realist and yet a a materialist. Penelope Maddy is a mathematical realist, but not an immaterial platonist.

That's nice, but unless she has an ontology that makes sense, then I am unconcerned.

Those who know the good, do the good. - Socrates

**#19**

**todangst wrote:**

First, it's merely a hypothesis that our basic laws could be different.

Yes, but an intuitively plausible hypothesis.

**todangst wrote:**

Then, it's yet another hypothesis that a change in basic laws would somehow influence a priori propositions.

It's a necessary truth that it would. It follows from the assumption that logic and mathematics have their basis in the material world. If mathematics has as its foundation the material world, like John Stewart Mill thought, then it follows that if you change the foundation...you change the mathematics. i fail to see how it doesn't follow. Mathematics is no longer about discovering necessary truths, but it is merely contingent upon the current universe. This is the observation made by Husserl and Frege.

**todangst wrote:**

Why not? It follows from our definitions. Math is tautology.

Indeed, it does follow from the definitions. Given the definitions, there is a definitive answer. We don't decide the answer, we derive them from the definitions. See what im saying? Given the meaning of 2 and the meaning of addition, we discover that 2 plus 2 equals 4. In no way are we deciding that it equals 4. We couldn't decide that it equaled 5.

**todangst wrote:**

I don't see any problem. I define X to be 'half' of Y, and then hold, that "half "is structurally the same thing as "add 2 to make a whole"... it's merely equalities, tautologies...

that we discover. I could try as hard as I want, I cannot make a tautology false. This is my point. We may decide the rules, like in chess, but we don't determine the outcome. You seemed to be saying that we actually decided the truth value of a propositio. In a way, we do on your conception. Since we decide the definitions, to an extent we have control of the T-value. However, after the definitions are given, we cannot decide that, for example, 8 < 7. See what im saying?

**todangst wrote:**

How so? How could we just as easily make that decision?

It follows from your assumption that we determine the truth of a proposition. Hence, I could decide that Modus Ponens was invalid. Why not? If we decide the truth, I could claim that Modus Ponens was invalid. Of course, given your conception, all of rationality would break down, and logic and mathematics would be mere subjective opinion. How could Modus Ponens have any persuasive force if you can reject it? Don't like an argument I use in which I employ modus ponens? No problem, decide it isn't valid. Why wouldn't you be able to?

See the absurdity?

**todangst wrote:**

You've not given any reason why your dilemma could actually exist. If I define "2" as 'half of 4" then it follows that two '2's are equal to four, if by 'half' we mean "two are required to make a whole'

It's all deductive. Where is the problem?

Once again, illustrating that we may come up with the definitions, but after the definitions are given, the truth of a given proposition is out of our control.

"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions

**#20**

**Actually, they would exist potentially**

Exactally my point - potentially, not actually. You're not in dissagreement. Your position forces you to say that certain numbers 1) Don't exist untill thought of 2) Stop existing when no minds exist.

How do you respond?

**#21**

**axiom wrote:**

Actually, they would exist potentiallyExactally my point - potentially, not actually. You're not in dissagreement. Your position forces you to say that certain numbers 1) Don't exist untill thought of 2) Stop existing when no minds exist.

How do you respond?

This isn't that bold of a statement, this is basically the intuitionist view of mathematics. Mathematical objects have to be constructed.

"philosophy of mathematics, **intuitionism**, or **neointuitionism** (opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. That is, they are not analytic activities wherein deep properties of existence are revealed and applied. Instead, logic and mathematics are the application of internally consistent methods to realize more complex mental constructs." -Wikipedia

However, there are drawbacks to this. For example, you cannot show a mathematical object exists through disjunction or reductio. The object has to actually be constructed through a direct proof.

Moreover, it has a weird concept of truth. According to an intuitionist, the truth of a proposition is equivolant to there being a proof for that object. Hence, truth and provability are the same on the intuitionist view.

Interesting thought, however, I am afraid that this account of mathematics suffers the same fate as most philosophical theories: Its false.

"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions

**#22**

**Chaoslord2004 wrote:**

todangst wrote:First, it's merely a hypothesis that our basic laws could be different.Yes, but an intuitively plausible hypothesis.

Actually, that's the very point I'm challenging you on here. Show me how it is plausible. If you can't demonstrate this, then one what grounds can you assert it? Can you cite a physicist who moves past hypothesis and offers up a theory of how basic laws could be different, and who then how this would work?

**todangst wrote:**

Then, it's yet another hypothesis that a change in basic laws would somehow influence a priori propositions.

It's a necessary truth that it would. It follows from the assumption that logic and mathematics have their basis in the material world.

I've asked you to give me an argument to support this claim. I can't see how the axioms of existence, identity and sentience change no matter how a 'basic' law is changed.... You say "It follows from the fact that if the material world had different physical constrants and laws, then matter would behave differently." Fine, so matter 'behaves' differently, but how does this lead to a change in the law of identity?

If something exists, it exists as something. It has identity. A=A. How do different 'laws' unseat this? If different laws exist, then the metaphysical basis for the axiom of identity exists, provided there are sentient brains to generate it.

**todangst wrote:**

I don't see any problem. I define X to be 'half' of Y, and then hold, that "half "is structurally the same thing as "add 2 to make a whole"... it's merely equalities, tautologies...

that we discover. I could try as hard as I want, I cannot make a tautology false. This is my point.

Actually, I think you are missing that it is my point as well.

**todangst wrote:**

How so? How could we just as easily make that decision?

It follows from your assumption that we determine the truth of a proposition.

You're assuming something that I've not said and going, too, too far.. I'm not claiming that truth is merely a human declaration or creation, where on earth do you get that? I'm saying we assign the terms of a tautology arbitrarily. But that tautologies are necessarily true is not a mere human invention, it is necessarily so given basic ontology: To exist is to exist as something. So we are merely defining X and Y as equivalencies, based on the axiom of identity.

I say that "2" is half of 4. The terms are assigned however we like, but the process it relies on to give it truth, is not arbitrary. It is based on creating equalities, which in turn, is a recognition of the basic axioms of metaphysics.

So how does any change in physical law change the axiom of identity?

Hence, I could decide that Modus Ponens was invalid. Why not?

If we decide the truth, I could claim that Modus Ponens was invalid.

Again, you've got a strawman version of my post. I am not saying that we simply, willy nilly, 'decide truth' I am saying that we set up two terms to be equivalent.

XSDDFdsafls = asfdsafsdfds

The terms are arbitrary, but the process itself goes back to the law of identity.

See the absurdity?

Of your misreading of my point? Yes.

Now, I want to again find out how a change in physical laws could undo the axioms of existence and identity.... I can't see how it is possible. To exist is to exist as something, and right there we have the grounds for forming tautologies.

Those who know the good, do the good. - Socrates

**#23**

**axiom wrote:**

Actually, they would exist potentiallyExactally my point - potentially, not actually.

You're missing my point, and we are not quite in the agreement you think we are.... The potentiality exists as a finite method for generating any number. This finite process for creating a potential infinity exists in sentient brains.

You're not in dissagreement.

You're not following me.

Your position forces you to say that certain numbers 1) Don't exist until thought of

They exist as a potential -sentient brains can construct them as per a finite formula.

2) Stop existing when no minds exist.

The 'existence' of a number is neurochemical... No brains, no numbers. How else would numbers exist if there are no sentient brains?

How do you respond?

By saying that there's no dilemma. "Existing potentially" means that we already posses all we need to generate any such number: a finite method. The existence of numbers is contingent upon the existence of sentient brains that can generate them. 10 to the 234r234234234234 23423423423423432423432nd power didn't exist until just this moment, but the potential for any such number exists right this second.

Now please, if you wish to continue this discussion (I've already asked once), do not procede in this purely negative fashion. If you wish to argue for a non material basis for numbers, present your postive claim, along with a postive ontology that does not steal from materialism.

Those who know the good, do the good. - Socrates

**#24**

**Chaoslord2004 wrote:**

axiom wrote:

Actually, they would exist potentiallyExactally my point - potentially, not actually. You're not in dissagreement. Your position forces you to say that certain numbers 1) Don't exist untill thought of 2) Stop existing when no minds exist.

How do you respond?

This isn't that bold of a statement, this is basically the intuitionist view of mathematics. Mathematical objects have to be constructed.

Yes. Without a sentient brain to generate numbers, based on an initial a priori working out of the finite formula, no numbers would exist at all.

"philosophy of mathematics,

intuitionism, orneointuitionism(opposed to preintuitionism), is an approach to mathematics as the constructive mental activity of humans. That is, they are not analytic activities wherein deep properties of existence are revealed and applied. Instead, logic and mathematics are the application of internally consistent methods to realize more complex mental constructs." -Wikipedia

I don't see why there has to be a dichotomy here... why can't intuitionism begin analytically, or through a priori gleaing of 'deep properties of existence"?

However, there are drawbacks to this. For example, you cannot show a mathematical object exists through disjunction or reductio. The object has to actually be constructed through a direct proof.

Moreover, it has a weird concept of truth. According to an intuitionist, the truth of a proposition is equivolant to there being a proof for that object. Hence, truth and provability are the same on the intuitionist view.

Interesting thought, however, I am afraid that this account of mathematics suffers the same fate as most philosophical theories: Its false.

What account of mathematics avoids any problems?

Those who know the good, do the good. - Socrates

**#25**

Due to the context of this discussion I'm not going to be claiming my views as factual, just my views.

Numbers are a descriptive. Descriptives do not exist, just that which they describe exists. 4 doesn't exist, but there can be 4 of something.

Once-Proud Canadian, Enlightened Atheist, Gaming God.

**#26**

**todangst wrote:**

I've asked you to give me an argument to support this claim. I can't see how the axioms of existence, identity and sentience change no matter how a 'basic' law is changed.... You say "It follows from the fact that if the material world had different physical constrants and laws, then matter would behave differently." Fine, so matter 'behaves' differently, but how does this lead to a change in the law of identity?

I am not disputing these axioms...these axioms hold with metaphysical necessity. Unless you can derive all of mathematics from these principles...your critisism is bunk. Perhaps, communitivity would be different. For example, the following proposition might be true ~(X + Y = Y + X). Moreover, it might not be the case that 11 > 3.

**todangst wrote:**

If something exists, it exists as something. It has identity. A=A. How do different 'laws' unseat this? If different laws exist, then the metaphysical basis for the axiom of identity exists, provided there are sentient brains to generate it.

If you can show...not even formally, how these three axioms can give you all of mathematics, then by all means, go ahead. Unless you can, then it is not a necessary condition that I refute Identity in order to argue my case. So this is just a red herring

"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions

**#27**

**todangst wrote:**

I don't see why there has to be a dichotomy here... why can't intuitionism begin analytically, or through a priori gleaing of 'deep properties of existence"?

I am very interested in the claim that you can get all of mathematics from basic metaphysical axioms. I hope you flesh this out for me. I am intrigued

"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions

**#28**

Wait, I thought the Russellian and Fregean project of reducing mathematics to a single or few metaphysical logical axioms was thought to be philosophical lost cause all together, maybe I'm wrong though and someone could flesh out what the single or few metaphysical axioms are that mathematics reduces to. Further, I don't think mathematical truths are explicable to neurons firing off in the brain, and thus materialist in nature; 2+3=5 whether my neurons are firing off or not. Mathematics and logical truths apply to the world and are necessarily/universally true, and not reducable to any sort of physicallist's metaphysics (they are not reducible to material "stuff", thus my view of mathematics is anti reductive, but a relaxed naturalist approach. I think Wittgensteing was on to something when thinking about what it is use a language; since mathematics is said to be the "language of science", but I'm still thinking about the consequences of that position ( I still think that it is a whole lot better than a materialist approach to matthematics though).

**#29**

**drummermonkey wrote:**

Wait, I thought the Russellian and Fregean project of reducing mathematics to a single or few metaphysical logical axioms was thought to be philosophical lost cause all together

Close. Their goal was to reduce it to logic and set theory. The reason for this is simple: logic is consistent. Logic and set theory are the the firmest of firm foundations. If they could do this, they could show that mathematics was consistent. Their goal was really to reduce talk of the natural numbers to logic and set theory...for you could reduce talk of the reals, rationals, complex, and intergers to talk of the natural numbers. Moreover, it was shown that if Euclidian Geometry was consistent, then non-Eclidian Geometry was consistent. And, Geometry was consistent, if number theory was consistent. Hence, giving arithematic a firm foundation really gave all of mathematics a firm foundation.

Their project is generally thought to be a lost cause, however, the thesis supervisor of one of my Philosophy professors, is still trying to revive it.

"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions

**#30**

**Chaoslord2004 wrote:**

todangst wrote:I don't see why there has to be a dichotomy here... why can't intuitionism begin analytically, or through a priori gleaing of 'deep properties of existence"?I am very interested in the claim that you can get all of mathematics from basic metaphysical axioms.

I never made such a claim.

I hope you flesh this out for me. I am intrigued

It seems that every post I write to you is nothing more than a correction of your misperceptions of what I write...

Those who know the good, do the good. - Socrates

**#31**

**todangst wrote:**

I never made such a claim.

Ok, then what did you claim?

**todangst wrote:**

It seems that every post I write to you is nothing more than a correction of your misperceptions of what I write...

Then make you position more clear. I read your position, scratch my head asking myself: "what the hell can he mean?" then I give it a charitable interpretation. I apologise if I misunderstand you. Misunderstanding a position is common in Philosophy. Most of Philosophy is an attempt by people to understand just what in the hell is going on...its the nature of the beast.

"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions

**#32**

**Chaoslord2004 wrote:**

todangst wrote:I've asked you to give me an argument to support this claim. I can't see how the axioms of existence, identity and sentience change no matter how a 'basic' law is changed.... You say "It follows from the fact that if the material world had different physical constrants and laws, then matter would behave differently." Fine, so matter 'behaves' differently, but how does this lead to a change in the law of identity?I am not disputing these axioms...these axioms hold with metaphysical necessity. Unless you can derive all of mathematics from these principles...your critisism is bunk.

Sorry, but this has nothing to do with your claim. Will you EVER back up your claim by showing HOW changes in matter could lead to a change in metaphysics, or will you continue to trade in Raijo-ian assertions?

**todangst wrote:**

If something exists, it exists as something. It has identity. A=A. How do different 'laws' unseat this? If different laws exist, then the metaphysical basis for the axiom of identity exists, provided there are sentient brains to generate it.

If you can show...not even formally, how these three axioms can give you all of mathematics, then by all means, go ahead.

That's not my assertion. I've never said any such thing. What I've asked is this: how can a change in matter lead to a change in the law of identity? I didn't ask about all of mathematics, nor did I say that all mathematics can be derived from that law.

You said:

"It follows from the fact that if the material world had different physical constrants and laws, then matter would behave differently."

So again, can you show me how it could be different so that it changed 'mathematics' or is there nothing in your claim other than a naked assertion?

Those who know the good, do the good. - Socrates

**#33**

**Chaoslord2004 wrote:**

todangst wrote:I never made such a claim.Ok, then what did you claim?

todangst wrote:It seems that every post I write to you is nothing more than a correction of your misperceptions of what I write...Then make you position more clear. I read your position, scratch my head asking myself: "what the hell can he mean?" then I give it a charitable interpretation.

See below.

Those who know the good, do the good. - Socrates

**#34**

Let's review:

**todangst wrote:**

I don't see why there has to be a dichotomy here... why can't intuitionismbeginanalytically, orthrough a priori gleaing of 'deep properties of existence"?

Here's what you say

**Chaoslord2004 wrote:**

I am very interested in the claim that you can get

all of mathematics from basic metaphysical axioms.

I ask: why must there be a dichotomy. Why can't intutionism BEGIN analytically, OR through a priori gleaning of 'deep properties' of existence.

And you are off the to races having me claim that all mathematics comes from basic metaphysics.

If you read what I post, you see that your claim cannot be gleaned from my words.

So it's not a 'principle' of charity' but your rush to insert something that's not there.

And, furthermore, the entire point of the exchange is not for me to put forward any position at all, but to get you to answer my question concerning your own claim.

Those who know the good, do the good. - Socrates

**#35**

**Chaoslord2004 wrote:**

Does this say that mathematical facts discribe the workings of the human mind? If this is so, than their truth is a mere contingency. If the human mind had been different, than the facts would have been different. This is a hard pill the psychologist must swallow.

Gulp.

I have no problem swallowing this pill. The whole idea that the universe can be described in terms of little shredded up bits of itself may in fact be a very bizarre proposition unique to humans. Obviously this viewpoint has some utility, since we've been able to successfully manipulate our environment using math-based planning, but it's impossible to say how close or far away we are from a 1-1 correlation between information and its source. This is because we can't see the source, hidden as it is behind our perceptual limits. Certainly math is far from necessary. I myself go days or weeks without ever touching the stuff.

It was pointed out here that there is an infinite number of subdivisions between any two intergers. To me, that pretty much proves that the whole basic mathematical idea of a unit is just mental shorthand to help us compare quantities. It has no relation to anything outside of us.

I think many, many facts would indeed be different if our brains were different. What if we lacked the idea of mass because we refused to commit the error of seeing anything as unconnected from the rest of the universe?

Lazy is a word we use when someone isn't doing what we want them to do.

- Dr. Joy Brown

**#36**

**Tilberian wrote:**

Chaoslord2004 wrote:Does this say that mathematical facts discribe the workings of the human mind? If this is so, than their truth is a mere contingency. If the human mind had been different, than the facts would have been different. This is a hard pill the psychologist must swallow.Gulp.

I have no problem swallowing this pill.

There's no pill to swallow. It has not been demonstrated that a different material world would lead to a different set of basic metaphysics or a different set of mathematics... it is merely an assumption, an assertion without foundation.

It must first be demonstrated that it is possible. It must then be shown that a different 'type' of matter would lead to changes in mathematics. There's no reason why a change in a sub atomic particle would lead to a change in an a priori system!

In addition, the human 'mind' does not exist independently of the universe, it's a part of it. Things are not contingent solely on the 'mind' but on a 'brain working within the universe'.

Those who know the good, do the good. - Socrates

**#37**

**todangst wrote:**

Will you EVER back up your claim by showing HOW changes in matter could lead to a change in metaphysics

One example: Let us assume that matter behaved in such a way that non-quantom events were such that they could happen...but not have a cause. This would change the maxim: every non-quantom event must have a cause.

Ok, so let us assume mathematics has as its foundation, the material world. If this is true, then somewhere in the material world, we can derive from the material world, that 2 + 2 = 4. If that piece of the material world changed, 2 + 2 might not equal 4. It might equal 5...or 6, or something else. Will it necessarily change? No, but it is still a possibility.

**todangst wrote:**

or will you continue to trade in Raijo-ian assertions?

Show me some respect, ok?

**todangst wrote:**

What I've asked is this: how can a change in matter lead to a change in the law of identity?

Since I never disputed this, this is a red herring.

**todangst wrote:**

I didn't ask about all of mathematics, nor did I say that all mathematics can be derived from that law.

Fine, so why did you ask it? What possible purpose could that serve? I never disputed the law of identity. So if you question served no further function that to ask me if I disputed something, that I clearly never claimed to dispute, then I am left asking: What it is your point? I assumed your point was to build up mathematics from the basic metaphysical axioms.

"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions

**#38**

**todangst wrote:**

Why can't intutionism BEGIN analytically, OR through a priori gleaning of 'deep properties' of existence.

Ok, explain.

**todangst wrote:**

And you are off the to races having me claim that all mathematics comes from basic metaphysics.

After you asked me about the law of identity. I assumed you were going somewhere with the question. I assumed the question wasn't a dead-end.

**todangst wrote:**

And, furthermore, the entire point of the exchange is not for me to put forward any position at all, but to get you to answer my question concerning your own claim.

Fine, I gave the answer above. I had hoped to get your position on the matter. It appears you have no desire in giving it. Kinda sad, I wanted to hear it.

"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions

**#39**

**todangst wrote:**

Tilberian wrote:Chaoslord2004 wrote:Does this say that mathematical facts discribe the workings of the human mind? If this is so, than their truth is a mere contingency. If the human mind had been different, than the facts would have been different. This is a hard pill the psychologist must swallow.Gulp.

I have no problem swallowing this pill.

There's no pill to swallow. It has not been demonstrated that a different material world would lead to a different set of basic metaphysics or a different set of mathematics... it is merely an assumption, an assertion without foundation.

Heh, good point. Since our rules of math are somewhat independent of the actual nature of the universe, it does stand to reason that if our universe changed, math would not necessarily do so.

**todangst wrote:**

It must first be demonstrated that it is possible. It must then be shown that a different 'type' of matter would lead to changes in mathematics. There's no reason why a change in a sub atomic particle would lead to a change in an a priori system!

In addition, the human 'mind' does not exist independently of the universe, it's a part of it. Things are not contingent solely on the 'mind' but on a 'brain working within the universe'.

My likey.

Lazy is a word we use when someone isn't doing what we want them to do.

- Dr. Joy Brown

**#40**

It might be that I've misunderstood Chaoslord here, but I'll have a crack anyway:

Chaoslord says that mathematical truths are necessary but physical facts are contingent so it seems absurd that a necessary truth should depend on contingent. (correct me if I've strawmanned you)

First I'll distinguish between two kinds of dependence - causal and logical. The truth of a mathematical proposition is logically dependent on it following from axioms while the existence of mathematics causally depend on it being 'thought' by thinkers.

If I understand right, necessary truths are true by virtue of reason alone. So given that our reason is what it is and that mathematics is what it is, the physical conditions required for mathematics and reason to take the form that it does are transcendentally necessary. So the truths of mathematics are still a priori and necessary, but their existence (assuming ontological physicalism) requires physical conditions. This means that we can deduce these physical conditions transcendentally.

Most physical facts aren't determined by reason alone (we require experience as well) so are contingent in this respect.

**#41**

**Strafio wrote:**

It might be that I've misunderstood Chaoslord here, but I'll have a crack anyway:

Chaoslord says that mathematical truths are necessary but physical facts are contingent so it seems absurd that a necessary truth should depend on contingent. (correct me if I've strawmanned you)

First I'll distinguish between two kinds of dependence - causal and logical. The truth of a mathematical proposition is logically dependent on it following from axioms while the existence of mathematics causally depend on it being 'thought' by thinkers.

If I understand right, necessary truths are true by virtue of reason alone. So given that our reason is what it is and that mathematics is what it is, the physical conditions required for mathematics and reason to take the form that it does are transcendentally necessary. So the truths of mathematics are still a priori and necessary, but their existence (assuming ontological physicalism) requires physical conditions. This means that we can deduce these physical conditions transcendentally.

Most physical facts aren't determined by reason alone (we require experience as well) so are contingent in this respect.

Very nice.

Those who know the good, do the good. - Socrates

**#42**

**Strafio wrote:**

Chaoslord says that mathematical truths are necessary but physical facts are contingent so it seems absurd that a necessary truth should depend on contingent. (correct me if I've strawmanned you)

Correct.

**Strafio wrote:**

The truth of a mathematical proposition is logically dependent on it following from axioms

Really? So the truth of the proposition "7 + 3 = 10" was not true until it was deduced from the axioms of number theory? This seems highly suspect. It is identifing truth with probability.

**Strafio wrote:**

while the existence of mathematics causally depend on it being 'thought' by thinkers.

Perhaps. Prove it.

**Strafio wrote:**

If I understand right, necessary truths are true by virtue of reason alone. So given that our reason is what it is and that mathematics is what it is, the physical conditions required for mathematics and reason to take the form that it does are transcendentally necessary. So the truths of mathematics are still a priori and necessary, but their existence (assuming ontological physicalism) requires physical conditions.

This is only true if the physical constrants are logically necessary. Perhaps this is the case...I donno.

**Strafio wrote:**

This means that we can deduce these physical conditions transcendentally.

We can deduce empirical facts a priori? How is this not an inconsistency?

"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions

**#43**

**Chaoslord2004 wrote:**

Strafio wrote:The truth of a mathematical proposition is logically dependent on it following from axiomsReally? So the truth of the proposition "7 + 3 = 10" was not true until it was deduced from the axioms of number theory? This seems highly suspect. It is identifing truth with probability.

My understanding of mathematics is that it's a game with rules.

Discovering the axioms was a case of working out what rules we were following when we played the game 'mathematics'.

So 7+3=10 was still true as the rules had been informally established, just not formally established.

**Chaoslord2004 wrote:**

Strafio wrote:while the existence of mathematics causally depend on it being 'thought' by thinkers.Perhaps. Prove it.

That's a good demand. My wording was bad as I believe mathematics (like language) to be rooted in practice rather than thought. (Although the ontology of practice might be reducible to thought?)

My argument is that mathematics is a rule following activity.

The formal axioms we have discovered are the rules we have been subconsciously following when we did our mathematical practice. Correct maths was correct adherence to these rules and incorrect maths involved a breaking of the rules. So if maths exists as a human practice then its ontology must be the physical conditions of this human practice. (not that mathematics are limited to humans but creatures that did maths would have to have anthropomorphic characteristics, right?)

**Chaoslord2004 wrote:**

Strafio wrote:This means that we can deduce these physical conditions transcendentally.We can deduce empirical facts a priori? How is this not an inconsistency?

What it shows that physical conditions are not limited to being empirical facts. Some physical conditions can be a priori true. (a priori synthetic?)

Then again, it depends on whether a physicalist philosophy of mind is a priori. The transcendental argument would be that (a) Mathematical truths exist as thoughts and (b) thoughts exist as physical conditions.

If (a) and (b) are both a priori true then the physical conditions required for maths as it is would also be a priori true.

**#44**

**todangst wrote:**

Numbers exist in material brains. They are encoded electrochemically, in neurons. They are relational abstractions. Ideas that help sentient beings create relationships between entities, either real or hypothetical. Their existence depends upon the existence of sentient brains. No brains, no numbers. Why do numbers appear "Transcendant"? Because the universe itself transcends the individual brain, the universe impressess itself upon sentient brains in the same fashion.

Making the universe an active principle here seems like superstition. I think we relate to the universe, not it to us. If such a relationship between the universe and brain exists physically then it is by way of our perceptual access to the universe. This seems problematic to me in the following ways:

1) our perceptual access is limited. We don't get to perceive the enitre universe, so the idea of it providing this objective referential consistency seems wrong to me. That taken then, what is interesting is that: what our minds can concieve of is inherently bigger than what our senses perceive. That is our minds seem to inherently transcend the limits of the perceived universe. (NB I use inherently to mean a basic ability, I don't presuppose this to be physical or non-physical - how and why the brain has this function is still unknown - to me at least).

My hypothesis is that if a person grew up in a box, say 10m x 10m, he would inherently be able to 'think outside the box'. The mind seems to be able to transcend the limits imposed by its perceptual access - imagine 'elsewhere' if you will. This infact seems to be a basic theme in mythology - 'the land over the rainbow'. Furthermore if you put in 3 objects, say 3 apples in the room, he would again inherently be able to imagine more than 3 apples. He may not be able to count them, but he can imagine 'more'.

To conclude this point - I think we cannot get to the level of material sophistication to have perceptual access to a 'big universe', without first being able conceive of 'big numbers' - I don't think it can work the other way around.

(Aside: A type of equivalence of this 'spatial transcendence of the physical limits of perceptual access' is the mind's ability to imagine itself persisting after death (ie outside of the physical limits of time), but I'll go no further on that point beyond raising it.)

2) our perceptions and how we deal internally with them are subject to subjectivity (as you say the mind is not tabla rasa).

Even if the universe provides this 'constant' as a perspective why is it with numbers that the mind doesn't muck them up making them subjective things. You may have answered this but I can't see why the universe wouldn't give us the same objective basis with regards to other things like colours, touch and other sensations - thus exclude any notion of qualia? Why would it only do this with numbers?

Furthermore:

a) even if we had perceptual access to the whole universe, the universe isn't constant but expanding, does this mean mathematics changes with the expansion?

b) Is there any neuroscientific evidence to back up the idea of a number generating part of the brain? Are there any evolutionary precursors amongst other creatures? It seems on first appearence out of kilter with evolution...

Lastly I know this is an old thread and you may all have moved on from what is written here but I've just found this page, so cheers if you bother to read it.

**#45**

**RandomIdiot wrote:**

Lastly I know this is an old thread and you may all have moved on from what is written here but I've just found this page, so cheers if you bother to read it.

Funnily enough, I still hold the same position.

Whether the other guys have moved on from their position; they seem to have moved on from the entire site so I can't be sure.

It's a shame really... 3 years on and I still want to know what ChaosLord made of my last post!

**todangst wrote:**

Numbers exist in material brains. They are encoded electrochemically, in neurons. They are relational abstractions. Ideas that help sentient beings create relationships between entities, either real or hypothetical. Their existence depends upon the existence of sentient brains. No brains, no numbers. Why do numbers appear "Transcendant"? Because the universe itself transcends the individual brain, the universe impressess itself upon sentient brains in the same fashion.

**RandomIdiot wrote:**

Making the universe an active principle here seems like superstition. I think we relate to the universe, not it to us. If such a relationship between the universe and brain exists physically then it is by way of our perceptual access to the universe. This seems problematic to me in the following ways:

Seemed like a fair objection to me.

The bit in bold does seem quite strange; claiming that the expanse of the universe gives us the impression of transcendent numbers.

Perhaps if he returned he could elaborate more on what he meant by this.

**#46**

I don't see a problem with that last statement, unless you are reading more into 'transcendent' that was intended, as in some metaphysical sense. Using that word may have been asking for trouble.

The manifest existence of identifiable separate entities, as a fact of the nature of the perceivable universe, certainly is something that individual brains must handle, and so makes the emergence of the concept of the basic integers as a way of capturing an obvious aspect of collections of related entities, even to an extent in the brains of lower animals, pretty much inevitable, via evolutionary processes.

Favorite oxymorons: Gospel Truth, Rational Supernaturalist, Business Ethics, Christian Morality

*"Theology is now little more than a branch of human ignorance. Indeed, it is ignorance with wings." - *Sam Harris

*The path to Truth lies via careful study of reality, not the dreams of our fallible minds* - me

From the sublime to the ridiculous: Science -> Philosophy -> Theology

**#47**

Haha! I think the entire wording of that sentence was asking for trouble.

That said, when I read your interpretation (and I think it matches what he would have had in mind) it makes a lot more sense.

**#48**

This is great!! I was surfing the Internet looking for something on metaphysics of numbers and I found and atheist forum! If only my conservative friends knew!

I must be one of a kind because I subject myself to discussions on extreme opposite sides of a question! I must love a good argument! Or I just like messing with the extremes! Or maybe I try to look at both ends of a question and come out with something completely different

I any case:

I'm not an atheist (hope you dont kick me out) but I'm not a theist either. I just cant define "God"... whatever that is... I'm just a... hummm liberal?! kinda hippie, with a doomer touch... and a rational believer on a transcendent reality. My favorite philosopher is Bertrand Russell and my modern time hero is Jon Stewart from daily show

My conservative friends call me an atheist, my atheist friends call me a theist... my real friends just call me crazy

Enough of first time presentations! Onto the subject...

Ontology (metaphysics) is such a thought swamp!

First things first. I believe that Kalam Cosmological Argument is more or less correct. As such I think numbers have a reality outside space-time. In this Universe everything has a cause... there has been multiple causes for the existence of things since the Big Bang. The Big Bang origin is a mystery but is a must! So there has to exist a "platonic existence of something" that gave birth to the Universe. This thing has no cause because it exists outside space-time so it wont abide by the same rules. One of these things is clearly mathematics. The Universe itself falls apart without math whether there Humans in it or not. There can be no space-time reality without mathematics. So I tend to agree with Chaoslord2004. I have not many doubts about this so its unlikely you can convince me otherwise... As to meta-ethics... you should see the discussions I have on that other Christian conservative blog!

I know I know... this Kalam thing is being used by conservatives in our days but hey... they make an effort to their cause and sometimes comes something that makes some sense!

You know what amazes me the most about this atheist/theist clash is the desperation people have to protect their most beloved belief on the face of new facts without asking themselves: "Could I be wrong??".

For atheists I fear that that crap published in Lancet medical peer reviewed magazine turns out to be true... daunting... for theists I just fear the ET.

______________________________________________________________

"I once prayed to god for a bike, but quickly found out he didnt work that way...so I stole a bike and prayed for his forgiveness"

"All matter originates and exists only by virtue of a force... We must assume behind this force the existence of a conscious and intelligent Mind. This Mind is the matrix of all matter." (Max Planck)

"the existence of mind in some organism on some planet in the universe is surely a fact of fundamental significance. Through conscious beings the universe has generated self-awareness. This can be no trivial detail, no minor byproduct of mindless, purposeless forces. We are truly meant to be here." Paul Davies

**#49**

**Teralek wrote:**

I'm not an atheist (hope you dont kick me out) but I'm not a theist either. I just cant define "God"... whatever that is... I'm just a... hummm liberal?! kinda hippie, with a doomer touch... and a rational believer on a transcendent reality. My favorite philosopher is Bertrand Russell and my modern time hero is Jon Stewart from daily show

It sounds like you are an ignostic.

here is some information on the term:

http://en.wikipedia.org/wiki/Ignosticism

I don't understand why the Christians I meet find it so confusing that I care about the fact that they are wasting huge amounts of time and resources playing with their imaginary friend. Even non-confrontational religion hurts atheists because we live in a society which is constantly wasting resources and rejecting rational thinking.

**#50**

Hey! guess I am a Ignostic!!

Thank you!

I really have a semantical problem with the word "God" or "Deity"

______________________________________________________________

"I once prayed to god for a bike, but quickly found out he didnt work that way...so I stole a bike and prayed for his forgiveness"

"All matter originates and exists only by virtue of a force... We must assume behind this force the existence of a conscious and intelligent Mind. This Mind is the matrix of all matter." (Max Planck)

"the existence of mind in some organism on some planet in the universe is surely a fact of fundamental significance. Through conscious beings the universe has generated self-awareness. This can be no trivial detail, no minor byproduct of mindless, purposeless forces. We are truly meant to be here." Paul Davies

#1hmm numbers...

Well they would exist how ideas exist. Even if you have 4, lets say kitties, you don't have the number four you have kitties.

#2Chaoslord2004 wrote:And what does that mean?

You've answered a question concerning meaning, with two completely meaningless terms. Immateraility equates to nothingness.

Numbers are not nothing. They exist. And to exist is to exist as something. This is axiomatic. To exist is to exist as something.

Numbers exist in material brains. They are encoded electrochemically, in neurons. They are relational abstractions. Ideas that help sentient beings create relationships between entities, either real or hypothetical. Their existence depends upon the existence of sentient brains.

No brains, no numbers.

Why do numbers appear "Transcendant"? Because the universe itself transcends the individual brain, the universe impressess itself upon sentient brains in the same fashion.

If someone disagrees, they must first begin by defining immateriality. Have fun tilting at the windmills...

Those who know the good, do the good. - Socrates

Books on atheism.

#3todangst wrote:How so? Mathematical facts exist. Furthermore, they exist as something. However, it has yet to be shown that something is equivolent to being material. Furthermore, you need to demonstrate that immateriality is equivolant to "nothingness."

I may be a hard concept to define, but this doesn't mean it is "nothing."

todangst wrote:This was never in question. I agree.

todangst wrote:This is incoherent. What exactly are you arguing for? A version of Psychologism that Husserl use to subscribe too? Or the one that Kant subscribed to? Does this say that mathematical facts discribe the workings of the human mind?

If this is so, than their truth is a mere contingency. If the human mind had been different, than the facts would have been different. This is a hard pill the psychologist must swallow.

I assume that you would say mathematical facts are necessary, right? It is logically impossible for 2 + 2 to equal 5. I am pretty sure you would agree with this. Furthermore, I happen to know that you subscribe to a correspondence theory of truth. With these two beliefs in mind, you are almost forced to become a platonist. Here is why...

If you say "it is logically impossible for 2 + 2 = 5." You are using a declaritive sentence to assert a proposition regarding the natural numbers. You are saying, there is a fact that corresponds to this proposition. We don't decide the truth of the above proposition...its truth is determined by a fact. Did we create the fact? Of course not. Furthermore, it is hard to concieve of a mathematical proposition being made out of anything.

Here, let me make it more explicit: When the game of Chess was first invented, it consisted a finitely many man made rules. However, the results that this game produced were platonic in nature. Why? Because over the centuries people have discovered "superior stradegies." They didn't make them up...they discovered them.

Therefore, even if you subscribe to a kind of Formalism about Mathematics, the results they produce are hardly man made. You might agree with me, that they are not man made. Fine, then what determines their truth? What makes the proposition, 2 + 2 = 4 true? Presumably, because there is a mathematical fact that makes it true.

Also, numerals can exist in the brain...numbers cannot.

todangst wrote:Fine, what is your argument for this position? The more I thought about this position, the more it seemed incoherent.

todangst wrote:That which is not extended in space and time.

"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions

#4It is an abstract concept.

Let me take a stab at this.

It’s not exactly right, but I think you can see what I am trying to say.

Maybe someone else can improve on this idea.

The problem I have is my first sentence, where I have not defined what ‘number’ is:

It is a singularity sort of. Each thing in the universe is a single unit, in and of itself.

And so:

There is only one unit of each individual thing and a single unit is symbolized with the mark 1; numbers are symbols of the multiples of 1.

2 is the symbol for two 1’s; 5 is the symbol for five 1’s. In other words 5 is the symbol for 1 and 1 and 1 and 1 and 1 and 1. It would be very inconvenient to express how many things there are in that manner hence for simplification the symbolic term is 5.

A number is merely a symbol for how many items there are of things that fall into a class that is assigned a set of particular attributes.

There are many items in a room. Let’s say there are 100 items in the room. An item is any separate thing that is not of the room; for example a wall cannot be counted as an item.

After examining the items some might have similar attributes; for example, some items might be the color blue.

We then count how many items that are blue. Instead of saying, as we point to each item, 1 and 1 and 1 and 1 and 1, we refer to symbols that represent how many 1’s added to all the blue items we've pointed to. The first item is 1, but the second item is 1 and 1 together and we assign the symbol 2 to represent the count of 1 and 1. Now that we know 1 and 1 is a sequence of the count 1 we look for another blue item and refer to this set as 2 and 1, which we assign the symbol 3, etc

I tried.

People who think there is something they refer to as god don't ask enough questions.

#5AiiA wrote:This is similar to Frege's view of what a number was. Frege thought of numbers in terms of Set Theory. He defined what a number was, in terms of logic; as a side note, Frege did this so as to further his logicist project. Frege wanted to reduce all of mathematics, especially arithematic, to logic. For logic was, and still is, thought to be the firmest of firm foundations. Furthermore, it had been shown that if Arithematic was consistent, then so was Geometry, Analysis, and so forth. Hence, it was very important to put Arithematic on a firm foundation and to demonstrate its consistency. One way of doing this was through Logic. His project failed, for his approach gave rise to a contradiction: Russell's paradox. However, lets give the ole chap a round of applause for trying. Where was I going with this? I am pretty sure I forgot. Oh yeah, here is how Frege defines number:

He starts out by defining the number 0 as the set containing zero members. This makes sense. How many square-circles are there? zero. he then defines the number one as being the set of all singeltons. Hence:

1 = {{a}, {b}, {c}...}

he goes on to define the number two in a similar way. The set of all doubles:

2 = {{c, d}, {q, w}...}

and so on for all the other natural numbers.

Frege also argued against a number being a property of anything. Rather, he argued that a number simply indicated how many times a given concept (he used the word "concept" to mean property) instantiated. Thus, we say "how many red balls are there?" we would never say "how many..." One would obviously say, "how many WHAT"

"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions

#6Frege was using naive set theory.

Does the Zermelo–Fraenkel set theory solve the Frege inconsistensy?

lol my eyes are starting to hurt trying to read about it all.

People who think there is something they refer to as god don't ask enough questions.

#7AiiA wrote:to bad naive set theory was shown to be inconsistent; damn Russell's Paradox. Set Theories after Frege had to add special axioms for building sets. On Frege's conception, absolutly any collection was a set. Furthermore, Frege had what is called "axiom 5" which basically says that for any property P, that property picked out a set...this axiom cooked Frege's goose; it lead to Russell's paradox...which devistated Frege. Imagine poor Frege...his lifes work down the toilet, and the logicist project down the drain.

as far as I know, Zermelo–Fraenkel set theory avoids this particular inconsistency. Because special axioms were included to block sets from being members of themselves; the axiom was called the axiom of foundation. Because im a nerd, here is the formal axiom:

Hence, x and y are disjoined sets...meaning they do not have any members in common.

What ultimatly avoids Russell's Paradox is the Axiom Scheme Of Seperation:

This axiom lets one avoid Russell's Paradox because of the restriction on z.

As a side note, I think Zermelo–Fraenkel set theory is to restrictive. It is good for capturing well-founded sets, but I ultimatly think it is insufficient for capturing all possible sets. To capture all possible sets, one needs to work within non-wellfounded set theory. Non-wellfounded set theories capture all the well-founded sets, as well as the non-wellfounded sets. Do you know the difference between a wellfounded as opposed to a non-wellfounded set?

Back to the topic, the consistency of ZFC has yet to be shown. We hope it is consistent...but we don't know if it is. If ZFC is consistent, then as Godel demonstrated, a consistency proof cannot be constructed within ZFC. Oh well...

"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions

#8Chaoslord2004 wrote:Because the terms you've used have no ontological bearing. Immateriality and atemporality are negative terms devoid of any universe of discourse. Immateriality is a broken term, it can't refer to anything. In fact, it's defined that way!

So these terms do not tell you anything about numbers, or anything at all, for that matter. They merely tell you what something is not. I think that they are terms that belong in a Dr. Seuss book. They are merely calculatus eliminatus, devoid of a universe of discourse. So you've attempted to tell us something about numbers, and all you did was tell us what they aren't, without leaving over anything for them to be. If this completes your definition, then you are left with the fact that numbers are nothing at all.

Right. And who's disagreeing on that? I'm telling you that immateriality is a meaningless term. It refers to nothing.

Right! And that's your problem here. Numbers exist. But to exist is to exist as something, right? In fact, to 'exist as nothing' is contradictory to the axiom of existence.... So, we agree that to exist is to exist as something. But immateriality and atemporality refer to nothing. These terms are negatives. So to define something solely in negatives, devoid of any universe of discourse, is to say that it is nothing.

But you don't believe that numbers are non existent. So they must be something. This leaves us with matter. Numbers, if they exist, must be matter. And one way they can exist as matter is as a relationship between sentient brains and the universe. The universe is the constant, the constraint upon the sentient brain. The brain is akin to paper, the universe, the pen. But there is more to this of course....the paper is not merely blank... it has properties, etc... I am not invoking tabla rasa here.... but whatever the specifics of how the brain works (we can leave that aside), we have a sentient brain and a constant universe, interacting.

I don't think you're not going to argue from inductive uncertainty, so I am unsure by what you mean here. How can it be shown otherwise, if immateriality is invisible, and incapable of having an effect on the material world?

I really don't think that you want to rest your argument on the fallacy of arguing from inductive uncertainty, do you? Holding that "we can't rule out immateraility a priori, so this is a reason to hold to it" is raijo talk - embarassing irrationalism.

There are problems with your claim, both inductively and deductively:

1) Inductively: It has ONLY been shown that something is equivalent to matter. It has NEVER been shown that anything is immaterial.

But looking at this matter inductively is a fool's errand - for what are you looking for anyway?. What could possibly point to immateriality?

This leads us to the fact that this is a deductive matter:

2) Matter is a privileged concept, and as a privileged concept, it is positive, it is the grounds for which the binary - immaterial, is formed as a concept. You can't even talk about immateriality unless you first refer to matter.

Therefore, immateriality is a broken concept, the term can have no ontological bearing.

I don't need to demonstrate it, as it is definitional to the term. Immateriality is defined in negatives... it is NOT matter, it is NOT energy. And the definition rules out a universe of discourse, so there's nothing left over for it to 'be'. We can't even use the term

"be" in relation to it, because again, to exist is to exist as something, so we literally create an existential error by talking about immateriality.

So, seeing as the term is defined as a set of negatives, devoid of any universe of discourse, without any ontological bearing, there is no need for argument or demonstration. Its axiomatic that the term has no ontological bearing, because it is definitional to the term.

It is impossible to define in positive terms, because the term is defined as subordinate to 'matter' and therefore is a broken concept.

People may not believe it implies nothing, but itt cannot refer to anything, by definition, for it is a negative term, devoid of any universe of discoure.

todangst wrote:No, it is not. Please don't 'raijo' me.

The privileged aspects of materialism

No. You're forgetting that mathematics is a relationship between the brain and the universe. Remember your Dennett and the fact that our relations with the univerese are interactions.

Again, you're leaving out that the universe itself remains the same. The existence of the universe is an 'anchoring heuristic' that plays a part in determining what types of mathematics 'work' and which do not. This defeats your '2+2=5' argument.

But still, different brains might come up different mathematics, right? But, even if different minds related to the same universe differently, this would not present a problem, because the differences would still have a relationship with each other, because both sets of numbers would be spawned from the interactions of the sentient brains with the universe. So, the differences would correlate, as they would be anchored by the fact that the same universe impresses itself upon these different minds in a consistent way, because again, the universe here is a constant.

So if one set of sentient beings did mathematics 'differently' than another, the differerent beings could communicate through the use of a mathematical constant for a translation.

I've already answered this above.

Again, you're leaving out the universe. It's a pretty big thing to leave out.

todangst wrote:I think I've given it already. Numbers exist as a relational concept within sentient brains.

todangst wrote:I.e. - nothing.

You've define the term negatively.

Without any universe of discourse.

Without any ontological status.

So: Immateriality = nothing

So, you've just tilted at the windmill.

*********************

If numbers are immaterial, then they are nothing. Numbers are not nothing. So we need a rational, coherent way to talk about them. All we can say so far is that they must exist in sentient brains as a relational concept. We might be wrong. The fact that we might be wrong is not the basis for forming a a counter theory.

Those who know the good, do the good. - Socrates

Books on atheism.

#9At the moment, I subscribe to Wittgenstein's view.

Mathematics is best seen as a game where we follow rules.

He starts his description of maths with how we learn to count.

We learn the sequence 0-9.

We learn that after 9 comes 0 and count one step to the 'next digit'.

From there we can count the natural numbers to infinite... (i.e. there's no end).

The 'axioms of Mathematics' as a whole are a set of rules we have found that covers all the rules of all the 'games' under the name 'mathematics'. I think that makes me a nominalist... I think...

#10todangst wrote:Why? Furthermore, many things are defined in terms of the negation on something; take Quantification Theory. All that is really needed is one quantifier, and one can define either "existence" or "for all" in terms of one another. For example, without using the existential quantifier, here is on to define existence only using universal quantification: It is not the case, that for all x, not Px. Completely negative definition, and yet we know what it means. So your claim is demonstrably false.

I agree, we don't have a great definition of immateriality. However, why should this pervent us from using it? We are forced to use ill-defined terms all the time. I mean, hell, what exactly knowledge is is still debated. Does this mean we should consider Epistemology suspect because knowledge is ill-defined? In the Philosophy of Science, what constitues evidence is debated...should we give up on science until we can all agree as to what "evidence' is? perhaps...but this is a hefty price to pay.

It all boils down to this: We work with what we know, and go from there

todangst wrote:except it isn't defined this way. At best, it is a term we are still trying to figure out. It isn't really broken, but rather, it is a term we need to work at defining more regourously. Once again, this is no different that trying to define what constitutes a good scientific theory...this is still debated in the Philosophy Of Science.

todangst wrote:All I was doing was saying in what domain numbers exist in. I still don't know what a number actually is. However, I was merely saying where these abstract entities exist: The platonic universe. For example, I could tell you in what domain a car existed it, without really knowing much about cars: the physical universe. You are correct...I didn't define what exactly a number was: just where it was located.

todangst wrote:That is because you are treating "immateriality" as a predicate. This is as dubious as treating "the material world" as a predicate; both are domains of existence, and both pick out things with different properties. The material world picks out things with the properties of spacial extension and extention in space. The opposite of this is not being extended in space and not extended in time. Hence, if something is not extended in space and is not constricted by time, it is platonic.

todangst wrote:fine. But the physical universe COULD have been different. On this view, this means that mathematical facts COULD have been different. I mean, it is generally accepted in Physics that this is but one of many possible universes. If this is so...and on your view, then this means that mathematics, in theory, could have been different. Thus, 2 + 2 could have equaled 5...it is just a matter of contingency that when 2 and 2 are added one gets 4. If you accept this...I can't argue with you on this. Many have said mathematics is a contingent fact. Fine...this strikes me as crazy...but fine.

However, assume you want to say that mathematical truths are necessary truths. Hence, when it was proved that there could be no greatest prime number, this was a truth that couldn't possibily be false. Even if the physical universe had wildly different constraints, this would still be true. Right? Do you accept this?

If you accept this...then you pretty much have to accept platonism. For if the physical universe is organized in a contingent fasion, but the laws of mathematics are such that regardless of how the physical universe is constituted, these laws obtain, then you must necessarily posit something over and above the physical universe.

I really can't see any way around this.

todangst wrote:This has absolutely nothing to do with inductive reasoning...none. So, this is a red herring at best. There is no evidence AT ALL that the physical world is all there is. As far as I know, no one has given a probability calculation that makes "immateriality" unlikely. If there has...good! Show it to me. To the best of my knowledge, however, there has not been.

Here is what we know (ruling out idealism): The material world exists. However, this fact does not rule out the ONLY possible mode of existence; this was my point. It had yet to be shown that this is the only possible mode of existence...even inductively.

Maybe im wrong, if so, show me an argument which demonstrates that the likely hood of their being anything above and beyond the material world is low. Go for it.

todangst wrote:I not sure what this means. Mathematics ITSELF is a relationship between the brain and the universe? What exactly are you saying?

todangst wrote:Not that this has any baring on the matter, but Dennett is a platonist about mathematics...

todangst wrote:Yes, so even if the universe was widely different, and if the laws of physics had been drastically different, 2 + 2 = 5 would be impossible. Why? Because there is something over and above the constitution of this universe.

Or, are you saying that SOME aspect of the physical universe had to be necessary?

todangst wrote:When you say interaction, do you beings who empirically derive principles? Hence, mathematics is derived from the material world? Is this what you mean? If so...this is wrong, for we can prove things about mathematics that have no physical correlate: Take a mathematical sphere. A mathematical sphere cannot exist physically. If it did, it would have to consist of infinitely many points. In other words, a mathematical sphere cannot physically exist. Yet, we can prove theorems about it. Tarski, for instanced, proved some very weird things about mathematical spheres if the axiom of choice was assumed. Furthermore, we can prove things about the nature of infinity...even though we have never experienced an infinite set.

"In the high school halls, in the shopping malls, conform or be cast out" ~ Rush, from Subdivisions