Did God create numbers?

[Rauffenburg]

It seems like yesterday's database error has somewhat messed up this thread.

 

[-Vincent-]

Well, this has been interesting reading. I have more primative ideas with regard to numbers.

Numbers are words and exist as much as any other word. All words name something and as names they are symbolic.

The Austrailian aborigines only have three number words which are: one, two, and many. So they do not have a system of mathematics. They actually participate in higher forms of communication which are visual. However, that goes beyond this question about numbers.

Perhaps numbers were originally developed as a way to signify birth order: first, second, third, fourth, born etc.. The next development would be the consideration possible groups like two present and four absent. This would lead to arithmetic.

But numbers are names which are helpful in managing our perceptions of any group of things. Ordinality or quantiy can be addressed with the names which are numbers.

Is ordinality real. It is at least real as an idea. But anyway numbers address the properties of our perceptions, but not properties of the things percieved.

We may percieve a group of five apples but five is not a physical property of the apples it is only a property of that temporal perception of five apples. If you eat one apple you may say you percive four remaning apples, so you see number does refer to the perception.

In his yeshua,

Vincent

 

[JonF]

If God created everything, does that mean he created numbers? Are numbers necessary things or contingent?

Numbers exist as an abstract construct. Did God create abstract constructs? I would say "yes" at least in this case. Numbers come directly from the concepts of set theory and formal logic. Both of which come from the concept of "consistency." I personally believe the universe is only consistent because God is consistent and the universe was created to reflect his nature.

 

[JonF]

Although numbers are isomorphic to certain sets, Paul Benacerraf in "What numbers could not be" (The Philosophical Review, Vol. 74, Jan 1965, pp. 47–73) points out the problem with this as a definition. When you learned in kindergarten that 2+3=5, did you use that set-theoretic definition of 'five', or another one, such as {{{{{{}}}}}}?

I think you are missing the point of the classic set theoretic construction of the natural numbers. You prove their mathematical existence by that construction, then you define a measure with standard cardinality. How this measure is defined for finite sets is such a way that bijections are equivalent. Then you give the measures names, i.e. 1, 2, 3, 4...

Little kids don't do the formal math, or the construction. What they do learn to do is look at two groups of objects and "pair" them up and realize that gives them some property of sameness.

And even if kids do some other process, what pray tell does that have to do with math? Intuition is often wrong in math and logic. And i'm pretty sure you would agree, even if what they conceptionalize as "5" and what I conceptualize as "5" are different in conception, they are the same object in a different, yet equivalent form. And as far as math is concerend that makes them the same.

 

[-Vincent-]

There are cultures which have number words and no formal logic and consequently no mathematics or set theory. What numbers mean to logicians and mathematicians is a modern question which only applies to some groups.

Numbering is ancient and has been has been developed differently by different cultures. Seek and you will find, and some simply have not sought mathematics...

 

[Radagast]

I think you are missing the point of the classic set theoretic construction of the natural numbers.

I do know a little about foundations of mathematics, actually. Personally, I believe numbers to be eternally existent abstract objects described by axioms. There are other viewpoints, but I think they fall over on close examination.

You prove their mathematical existence by that construction...

The existence of numbers was never in doubt. Nor the truth of the Peano axioms, which I think are on more solid footing than, say, the Axiom of Choice.

 

[StrugglingSceptic]

You seem to be mixing two incompatible philosophies of mathematics here -- empiricist ('Five' only exists insofar as I have five apples or five atoms, which means mathematics is essentially a subset of physics) and logicist (five is defined to be the set you described, which is an abstract object).

A big thumbs up from me, Radagast, but I fear that Wiccan_child was craftier in his use of language:

More generally, 'five' is defined as the quantity of things in the set:

This isn't the logicist definition of five. The logicist says that five is the constructed set, not the quantity of things in the set. Of course, the problem is that Wiccan_child hasn't really offered a reasonable definition if he's using bulky phrases like "quantity of things", but at least we can't fault him for being a logicist.

think you are missing the point of the classic set theoretic construction of the natural numbers. You prove their mathematical existence by that construction, then you define a measure with standard cardinality. How this measure is defined for finite sets is such a way that bijections are equivalent. Then you give the measures names, i.e. 1, 2, 3, 4...

The existence of numbers was never in doubt. Nor the truth of the Peano axioms, which I think are on more solid footing than, say, the Axiom of Choice.

Another thumbs up.

In some sense, though, the set-theoretic construction does prove the existence of natural numbers. Well, it proves their existence within the formal system of set theory! Furthermore, it shows that arithmetic can be embedded within set theory, so that we don't need the one set of axioms so long as we have the other. And lastly, in big formalisation projects for mathematics, we will generally start with set theory (or type theory) and construct the natural numbers just as described, and then claim to have proven the existence of the naturals.

But the point here is whether this construction gives us any confidence in the axioms of PA or whether it has any ontological import for natural numbers. I would certainly agree with Benacerraf that, when we talk about numbers, we are not talking about (say) the Von Neumann Hierarchy. And as for confidence, it seems clear to me that the axioms of PA are much more obviously true of the natural numbers than the axioms of ZF are true of these strange objects we call "sets", so that the ZF construction does nothing whatsoever for our confidence in PA.

I'd love to chat a bit with you about the Axiom of Choice, if you fancy doing so by private messaging. I wonder if AC gets a bad rep, and that it's things like the axiom of infinity together with the axiom of powerset that are the real culprits. But anyway, it's great having what I assume is a professional logician/mathematician here at the philosophy forums!

 

[Radagast]

And as for confidence, it seems clear to me that the axioms of PA are much more obviously true of the natural numbers than the axioms of ZF are true of these strange objects we call "sets", so that the ZF construction does nothing whatsoever for our confidence in PA.

I would certainly agree.

... I wonder if AC gets a bad rep...

The difficulty with the Axiom of Choice is that, although it's intuitively true for countable sets, intuition is a poor guide when it comes to uncountable sets. Pretty much everyone accepts it, but sometimes with a vague uneasiness.

 

[JonF]

The difficulty with the Axiom of Choice is that, although it's intuitively true for countable sets, intuition is a poor guide when it comes to uncountable sets. Pretty much everyone accepts it, but sometimes with a vague uneasiness.

So much is dependant on it, what choice do we have?

 

[Radagast]

So much is dependant on it, what choice do we have?

The sensible thing is to distinguish carefully between what depends on the Axiom of Choice and what doesn't. It could perhaps lead to as-yet-undiscovered inconsistencies.

But getting back to the OP, the numbers are on more solid grounds than the sets, which is why Platonists like me think they have an eternal existence.

 

[JonF]

But getting back to the OP, the numbers are on more solid grounds than the .

I disagree, i think sets and the axioms of logic give rise to numbers.

 

[Radagast]

I disagree, i think sets and the axioms of logic give rise to numbers.

Well, as a logicist (which is how I interpret your comment) you're in good company with Russell, Whitehead, and co. And that would also make numbers eternal, as a sort of epiphenomenon of logic.

But I think Paul Benacerraf ("What numbers could not be", The Philosophical Review, Vol. 74, Jan 1965, pp. 47–73) demolishes that viewpoint completely. Is the number 5 really identical to the set {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}}? Or perhaps to {{{{{{}}}}}}? I don't think so.

 

[JonF]

Well, as a logicist (which is how I interpret your comment) you're in good company with Russell, Whitehead, and co. And that would also make numbers eternal, as a sort of epiphenomenon of logic.

But I think Paul Benacerraf ("What numbers could not be", The Philosophical Review, Vol. 74, Jan 1965, pp. 47–73) demolishes that viewpoint completely. Is the number 5 really identical to the set {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}}? Or perhaps to {{{{{{}}}}}}? I don't think so.

i do, and so are all other bijective sets. That and bolded part is rather important for my view.

 

[Radagast]

i do, and so are all other bijective sets. That and bolded part is rather important for my view.

What exactly do you think the number 5 really is?

 

[JonF]

What exactly do you think the number 5 really is?

a property that is describing all sets bijective to {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}}

 

[Radagast]

What kind of property, though?

 

[StrugglingSceptic]

a property that is describing all sets bijective to {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}}

Do you mean the property of being in one-one correspondence with {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}}?

Minor issue: I understand what you mean, but just for us pedants, can we use the more standard terminology? That is, can we say that functions may be bijective, while sets may be in one-one correspondence?

 

[StrugglingSceptic]

I disagree, i think sets and the axioms of logic give rise to numbers.

Which axioms of logic? There are many different systems to choose from.

And this is one reason that natural numbers are on more solid footing than sets: nobody disputes the Peano axioms. Whatever natural numbers are, the axioms of PA are true of them. This cannot be said for logic and set theory. For instance, in one version of set theory, namely type theory, it is not possible to construct the Von Neumann numbers, let alone talk about being in one-one correspondence with the set representing 5.

 

[JonF]

Do you mean the property of being in one-one correspondence with {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}}?

Minor issue: I understand what you mean, but just for us pedants, can we use the more standard terminology? That is, can we say that functions may be bijective, while sets may be in one-one correspondence?

They mean the same thing, so sure

Which axioms of logic? There are many different systems to choose from.

Almost all systems are built around the principle of non contradiction, and almost any of these you can construct numbers with. FOL+ seems rather universal.

And this is one reason that natural numbers are on more solid footing than sets: nobody disputes the Peano axioms. Whatever natural numbers are, the axioms of PA are true of them. This cannot be said for logic and set theory. For instance, in one version of set theory, namely type theory, it is not possible to construct the Von Neumann numbers, let alone talk about being in one-one correspondence with the set representing 5.

This is a good point, but i'd love to see you define PA without the notion of sets. Where as i can sure define set's with out natural numbers. IMHO different sets of axioms about sets and logic give rise to different systems, all of which are necessary in the sense that they are systems of non-contradiction, thus having some sense of entrensice truth. Numbers happen to be one of these things that describe the real world very well.

What kind of property, though?

intrinsic

 

[StrugglingSceptic]

Almost all systems are built around the principle of non contradiction, and almost any of these you can construct numbers with. FOL+ seems rather universal.

My guess is that typed theories are actually used more widely than first-order set theories for formalising mathematics. Certainly, the biggest formalisation project currently going (the Flyspeck Project) is all done in type theory (specifically, a variant of the simply-typed lambda calculus called HOL).

This is a good point, but i'd love to see you define PA without the notion of sets.

In simple type theory, a complete logic for mathematics which trades in functions rather than sets, the Church numerals are a fair start. You can define the number 5, for instance, as the lambda abstraction

\f. \x. f (f (f (f (f x))))

That is, 5 is a function which maps f to a function which applies f five times to its argument. The successor operation is then

\n. \f. \x. f (n f x)

But you can't get the whole of PA like this in type theory. Eventually, we need a predicate which is true just of the natural numbers. The axiom which gives us this is "the axiom of infinity", analogous to the one in ZF. In ZF, it gives us a set closed under the Von Neumann successor function x |-> x u {x}. In type theory, it gives us a function on some definite type which is one-one but not onto.

This is just simple-type theory. In theories with more expressive types, you can do a lot more. For instance, if you have algebraic data types, you can encode the natural numbers with a type definition:

type nat = Zero | Successor nat

This before anyone has even mentioned sets or functions. And you get systems for logic which are even more powerful than this, such as the one which forms the basis of the theorem prover Coq. Again, there is no consensus on what axioms characterise logic. We have ZF set theory, NBG set theory, NF set theory, simple-type theory, polymorphic type theory, dependent type theory, and none of these theories is isomorphic to another: they all see logic very differently. Simple type theory, as I remarked, does not recognise a set such as {{}, {{}}}, while ZF does. So does this set really exist or not? It would seem that not everyone would agree on the matter. On the other hand, just about everyone agrees that there are these things called natural numbers and that they are carved out by the Peano axioms. So to reiterate, I would say that the natural numbers are on much more solid footing.