Did God create numbers?

[Nooj]

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

 

[Wiccan_Child]

Numbers are not things, but an emergent property of groups of things. 'Five' only exists insofar as I have five apples. More generally, 'five' is defined as the quantity of things in the set:

5 = | {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}} |.

 

[Nooj]

Numbers are not things, but an emergent property of groups of things. 'Five' only exists insofar as I have five apples. More generally, 'five' is defined as the quantity of things in the set:

So before he created 'things', [the emergent property of] numbers would not exist?

The other alternative as I see it is that the property of numbers existed before he started creating things. But that implies something (or a property of something) existing independently of God's creating process.

 

[Washington]

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

Yes he did, because if he didn't we'd all be stuck with going, "One, one, one, one, one, one, one. . . . . . . " And, "Yup I've got one leg, one ear and one eye. What you got? Ah, One leg, one ear and one eye also. Ain't life grand."

 

[Wiccan_Child]

So before he created 'things', [the emergent property of] numbers would not exist?

No, because numbers don't exist. They're an emergent property, not things in and of themselves. It's like velocity, or height: they doesn't exist, but rather are properties of things which do exist.

The other alternative as I see it is that the property of numbers existed before he started creating things. But that implies something (or a property of something) existing independently of God's creating process.

Well, Christianity posits that God created everything that was made. It says nothing about those things which weren't made. Numbers could fall into that category (depending on what one considers to be 'numbers').

 

[AdrocK48]

Numbers is one of those things where it is hard to date exactly, like time. These things, as some would argue, have always been there, and seemed to come into existance when they started to be developed into recognizable digits or symbols. I would be one who falls into this category of arguement. just because something is not measured for the entirety of the known creation, that does not mean it does not exist for the entirety of that time. God has created everything, but he chooses to reveal things at different times to different people (see deuteronomy 29:29)

 

[PhilosophicalBluster]

G-d created us, and in creating us he gave us the ability to create and comprehend a number system, which we did. Depending on your conclusion of this statement, numbers can be made by either us or Him.

 

[PantsMcFist]

WC is right on the money - numbers are rational, but there's no way to ostensively define them. Like most scientific concepts, they're conceptual tools in an interpretive framework. Good evidence for numbers not being a universal concept does exist in some anthropological studies. There are cultures which exist in the South America jungles that have no concept of sequential numbers, and these tribals cannot tell the difference between groups of items which exceed 3 or 4. They were getting ripped off pretty hardcore in trading, and some missionaries decided to teach them a base 10 number system. After two years of trying, the tribe decided it was unable of learning the difference between 7 or 9 or w/e arbitrary number.

 

[Penumbra]

That reminds me of the quote,

"God created the integers, all the rest is the work of man."
Leopold Kronecker, mathematician

I've always been stumped when trying to consider whether logic/math is something that had to come into existence at one point, or is just completely "there" as unchanging rules of how the universe operates. I can't think of a proof that could be used either way.

 

[SaintPhotios]

I think, first of all, we have to concede that numbers have some mode of existence in so far as they have meaning. They don't have physical properties, but they exist as much as concepts can be said to exist. If you disqualify numbers from having any degree of existence whatsoever, you also have to discount such items as words, concepts, etc. So let's not say that because numbers are not physical, they must have no existence. If you're a nominalist, then you don't buy that and you will say that abstract objects have no existence. But this thread is assuming that there is a God and He created things. So naturalism can't really apply here.

If God created everything, then it naturally has to be admitted that God created numbers. If prior to God there was nothing, there would be no necessary or contingent truths. In fact, to even talk about what is would imply existence and therefore proceed creation. This is why many eastern theologians, who refused to attribute properties to God, included existence. They rather spoke of God as transcending existence -- as it is a concept, and as such, was included in the act of creation.

I would think even if you view numbers as mere conceptions and having no specific created nature in themselves, you would still have to say that God, as the creator of mind, still created numbers in so far as He created minds with the operative ability to conceive of them.

.... and are numbers necessary or contingent? That really depends entirely on how you define numbers, and also on a number of other things. For instance, I would say that reliability of perception is a necessary truth. I'm basing this one the presumption that basically, we can knows things. If there is no order to our perception, then knowledge to any degree is impossible. But if any sort of knowledge is possible, then reliability of perception is a necessary truth. If we assume perception, then we much assume the object of perception. This alone requires a plurality of objects. Even if the only objects that exist are the perceiver and the object of the perceiver's perception, then there is a plurality of things. This alone is a basic confirmation of numbers. There are in existence, at least, a plurality of objects which can be assigned numerical values. If nothing else, this alone would constitute an analytic truth.

 

[Radagast]

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

Personally, I think numbers are (as St Augustine said) eternal, not created. In the words of American scholar and politician Edward Everett:

“In the pure mathematics we contemplate absolute truths, which existed in the Divine Mind before the morning stars sang together, and which will continue to exist there, when the last of their radiant host shall have fallen from heaven.”

As part of the Mind of God, mathematical truths are necessary, not contingent. This is consistent with the Platonist view of mathematics held by many great mathematicians, and works for me as a professional mathematician.

But I'm sure you'd enjoy the alternate viewpoints presented in any good book on philosophy of mathematics.

Numbers are not things, but an emergent property of groups of things. 'Five' only exists insofar as I have five apples. More generally, 'five' is defined as the quantity of things in the set:

5 = | {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}} |.

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).

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 {{{{{{}}}}}}?

 

[Wiccan_Child]

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).

I don't see the difference. '5' is still the number of things I have. In the first case, I have five physical objects. In the latter, I have five abstract objects. Either way, the principle is the same.

And I vehemently oppose the suggestion that mathematics is subordinate to physics .

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 {{{{{{}}}}}}?

Kindergarten ?

While we initially learn about numbers by way of analogy (the discreteness of apples and cakes is most useful, and delicious), a thoughtful child will realise that the 'five' in 'five cakes' and the 'five' in 'five apples' is the same thing: though the thing itself may differ, the quantity you have is the same. Thus, numbers are properties of things (as evident in the set definition), not things unto themselves.

 

[Radagast]

I don't see the difference. '5' is still the number of things I have. In the first case, I have five physical objects. In the latter, I have five abstract objects. Either way, the principle is the same.

I think you agree with me then: 'five' refers to a number:

It doesn't refer to that specific complex set, because in kindergarten you understood 'five' without getting into that particular set (in fact, I misread your post: you didn't define 5 to be that set, as logicists do, you defined 5 to be the cardinality of that set ).

And numbers are logically prior to things, so that 2+3=5 is an abstract truth, not an experimental fact of physics.

Thus, numbers are properties of things (as evident in the set definition), not things unto themselves.

Or maybe you don't? If numbers are properties of physical things, then mathematics is subordinate to physics. I'm still having trouble placing your philosophical position.

Let's try something else. The number pi = 3.14159265358979323846... is the ratio of the circumference to the diameter of an infinitely perfect circle. Is pi a property of physical things, or an abstract thing unto itself?

 

[Wiccan_Child]

I think you agree with me then: 'five' refers to a number:

Yes. Five isn't a thing in and of itself, but rather, it's a property of (among other things) the set {, , , , }.

It doesn't refer to that specific complex set, because in kindergarten you understood 'five' without getting into that particular set

Because it is a useful way of introducing the more general concept of 'five'.

And numbers are logically prior to things, so that 2+3=5 is an abstract truth, not an experimental fact of physics.

Exactly. That's kinda why mathematics isn't subordinate to physics.

Or maybe you don't? If numbers are properties of physical things, then mathematics is subordinate to physics. I'm still having trouble placing your philosophical position.

I said they're properties of things, but not specifically physical things.

Let's try something else. The number pi = 3.14159265358979323846... is the ratio of the circumference to the diameter of an infinitely perfect circle. Is pi a property of physical things, or an abstract thing unto itself?

Neither: it's a property of an abstract thing. And, indeed, though perfect circles don't exist in real life, π crops us all the time in physical equations.

 

[Radagast]

Yes. Five isn't a thing in and of itself, but rather, it's a property of (among other things) the set {, , , , }.

The question is: what kind of property? You do seem to think that set theory is logically prior to numbers, and many mathematicians with that approach wind up defining a number to be a specific set, which I'm not really happy with (largely because I'm much more comfortable with the Peano axioms than the ZF axioms).

Neither: it's a property of an abstract thing. And, indeed, though perfect circles don't exist in real life, π crops us all the time in physical equations.

I would go further and say that π itself is an abstract thing, though also useful to physicists.

 

[Wiccan_Child]

The question is: what kind of property?

That question goes beyond the scope of this thread. Suffice to say, they are a property.

You do seem to think that set theory is logically prior to numbers, and many mathematicians with that approach wind up defining a number to be a specific set, which I'm not really happy with (largely because I'm much more comfortable with the Peano axioms than the ZF axioms).

I do not believe that numbers are sets, abstract or otherwise. Properties of sets, yes, but not sets in and of themselves.

I would go further and say that π itself is an abstract thing, though also useful to physicists.

I disagree, though we might be quibbling over semantical minutiae.

And then there's Logic waaaay over to the right.

 

[Radagast]

That question goes beyond the scope of this thread. Suffice to say, they are a property.

With respect, the question of what numbers actually are is the whole point of the thread.

Personally, I think natural numbers are eternally existing abstract objects (I don't think they're sets either) understood by intuition, and formalised by the Peano axioms. This gives them the same level of existence as propositions in pure logic (and indeed, there are some deep relationships between the two).

I disagree, though we might be quibbling over semantical minutiae.

Hey, I've run whole lecture courses on semantic minutiae!

 

[Rauffenburg]

@Wiccan Child

No, because numbers don't exist. They're an emergent property, not things in and of themselves. It's like velocity, or height: they doesn't exist, but rather are properties of things which do exist.

Whether numbers exist or not does first of all depend on your notion of existence. If existence simply means to be the vaule of a quantifier, then numbers do exist. Because I can speak in such a way: There exists an x which is a number and fulfills some property P (e.g. being greater than 2). Since there is a number greater than two, the sentence it true, and therefore numbers do exist.

The next point to emphasize is: It actually isn't important whether numbers may be properties of sets in the first place, as you suggested (even though I do not understand what that should mean right now). When we speak of the natural numbers, the real numbers and so on, we speak of sets of objects. We treat numbers as elements of these sets. And once we do, we can quantify over these elements since those sets can be domains about which we speak. And consequently we can assign properties to natural numbers and so on - and of course we can say that numbers with such and such properties exist or do not exist. Even if numbers are properties of sets they may still exist independently because properties may exist independently once we quantify over them - their instances do depend on the existence of sets of which they are properties - but their existence itself may be independent of the existence of their instances.

So to put it less complicated: Your idea seems to be a case of begging the question. You already assume that properties do not exist independent of their instances. But you have not brought forth anything to motivate this premiss. I on the contrary say: If you accept the logical notion of existence formalized by the existence quantifier, then it is a clearcut case that numbers do exist, no matter whether they are properties of anything else or not.

@SaintPhotius

If you're a nominalist, then you don't buy that and you will say that abstract objects have no existence. But this thread is assuming that there is a God and He created things. So naturalism can't really apply here.

Let me please remind you that nominalism and naturalism are not the same.

 

[Rauffenburg]

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[Rauffenburg]

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