Could God have created Mathematics.

Son Goku

Could Mathematics have taken any other form, but what we know.

Could 1 + 1 ever equal 3. Could x^n + y^n = z^n ever be true for n>2.

God could have chosen another character for the physical Universe or the physical universe could have had another character, but would that character have always been mathematical.

Gone West

1 is an abstract term that people use.
Perhaps asking something like "could pi ever equal something other than 22/7"
I reckon yes.

SkepticOne

Son Goku wrote:

Could Mathematics have taken any other form, but what we know.

Could 1 + 1 ever equal 3.

Words like 3 or 2 are only created in the first place because they have some use in communities and part of the meaning of 2 is 1+1. If 1 + 1 was 3 then what would the use of 2 be?

I think for mathematics to lose its meaning we have to do away with the idea of things being part of classes (e.g. an individual lion being part of the general class of lions). Only throught this does number have any meaning. But individual things being part of general classes is a fundamental requirement of language.

Son Goku

Words like 3 or 2 are only created in the first place because they have some use in communities and part of the meaning of 2 is 1+1. If 1 + 1 was 3 then what would the use of 2 be?

I just choose 1 + 1 = 3, because it is common everyday maths.
I mean could the operation symbolized by '+' take "the number symbolized by 1" and "the number symbolized by 1" as its arguments and return "the number symbolized by 3".


I'm more asking if mathematics is prior to or more primary than the universe.
Does Maths exists without the laws which it describes? For instance F = ma is a specific tensor mapping of the vector a into the vector F through the scalar 'm'. Despite the fact that the Universe could have taken any arbitrary vector mapping as its force law or no vector mapping at all, vector mapping would still have been a concept in these other worlds that never came to be.
Complex Vector Spaces would be one example. All our dimensions are Real as opposed to complex, so although we have Complex Vector space as an idea they do not pertain to physical reality.
Do these mathematical ideas exist "before" the physical world.

Perhaps asking something like "could pi ever equal something other than 22/7"

I would argue that Pi must always equal "3.14......" in all domains. It simply is the ratio of circumference to diameter in Euclidean space.

Gone West

Now were getting somewhere

...." in euclidean space"
Thats exactly my point.
Could space exsist in a non-euclidean manner?
I think this is a more "xen" or "to the point" question than could god have created mathematics. Could there be a universe which is not comprehendable(sp?) to us? Could the diameter of a circle fit the circumference 12 times? Or could it change with the circle? These are all unanswered questions, but are very hot topics in philosophy. However, that is not to say that having the argument is pointless. The wisdom is gained from the discussion, not the answer.

or something like that.

Son Goku

FuzzyLogic wrote:

Now were getting somewhere
...." in Euclidean space"
That’s exactly my point.
Could space exist in a non-Euclidean manner?

Yeah, flat physical(real) space-time isn't Euclidean it is Minkowskian. And curved spaces can take any Lorentzian form.
Euclidean space doesn't exist in the real world.

FuzzyLogic wrote:

Could there be a universe which is not comprehendible(sp?) to us? Could the diameter of a circle fit the circumference 12 times? Or could it change with the circle? These are all unanswered questions, but are very hot topics in philosophy. However, that is not to say that having the argument is pointless. The wisdom is gained from the discussion, not the answer.
or something like that.

Well there is "stretched affine space" where the diameter of the circle can fit into the circumference 12 times.

When I say could God have created mathematics I mean would he have had any choice. Would geometric entities and there relations be so axiomatic that God had no degrees of freedom from Mathematics.

That the universe must have been constructed from mathematical principles and in this sense they must be prior.

There is an non-theistic version of this obviously.

Gone West

But in this affine space, wouldn't the basic principals of mathematics not be altered?

I just used the term euclidian to "dumb-down" the argument slightly :P
Some people mightn't understand this curved-space malarkey.

Obligatory eistein quote
"What really interests me is whether God had any choice in the creation of the world."

(dude, im on Irc if you want to chat about this, its pretty interesting stuff)

Son Goku

FuzzyLogic wrote:

But in this affine space, wouldn't the basic principals of mathematics not be altered?

Not really. They have different relations from Euclidean geometry, thats about it.

Obligatory eistein quote:"What really interests me is whether God had any choice in the creation of the world."

I'd argue that "God" did have a choice in what mathematical character the physical world would take, but it had to be mathematical character.

SkepticOne

I don't think there's any need to get into advanced mathematics really.

Son Goku wrote:

I just choose 1 + 1 = 3, because it is common everyday maths.
I mean could the operation symbolized by '+' take "the number symbolized by 1" and "the number symbolized by 1" as its arguments and return "the number symbolized by 3".

But "1" and "the number symbolised by '1'". Nothing is added by the phrase "the symbol represented by". It is like someone saying "the object represented by the word 'bus' is coming." This conveys the same meaning as simply "the bus is coming".

My earlier remarks remain valid. If one views mathematics as a set of symbols and the rules governing their use then, in whatever universe we can imagine, provided we use those symbols and rules we get the same results.

Earlier you asked:

Could x^n + y^n = z^n ever be true for n>2?

Yes, if we give to the symbols 'x', '^', '+' and so on the meanings that we attach to them in this universe.

Of course, some alien species in this other universe may use different symbols. If there is a one-to-one mapping between the alien system and our own (I believe this is called a homomorphism), then the two systems are the same mathematically for the purposes of this discussion. These symbols don't need to point to anything outside the formal system itself.

I think this is where the problem with the original question occurs. The assumption is that '1', '+', '3' refer to things; that they refer to something 'out there' in the universe. I think the confusion arises because they fit into our language in the form of nouns. Although many nouns refer to things in the world, they don't always. In the case of mathematics we can define the meaning of symbols in terms of other symbols in the system with no reference to anything outside the system.

Son Goku

SkepticOne wrote:

I think this is where the problem with the original question occurs. The assumption is that '1', '+', '3' refer to things; that they refer to something 'out there' in the universe. I think the confusion arises because they fit into our language in the form of nouns. Although many nouns refer to things in the world, they don't always. In the case of mathematics we can define the meaning of symbols in terms of other symbols in the system with no reference to anything outside the system.

I'm more so arguing that Mathematics can only have the form we know of.
Either a domain is non-mathematical or mathematical and that mathematics has no other form.

That there are no "degrees of freedom" in mathematics construction.

I don't think there's any need to get into advanced mathematics really.

More complicated maths elucidates (ironically) the question, in my opinion, because it isn't as tied to language. (And I know maths is a language, but it isn't an object language)

SkepticOne

Son Goku wrote:

I'm more so arguing that Mathematics can only have the form we know of.
Either a domain is non-mathematical or mathematical and that mathematics has no other form.

That there are no "degrees of freedom" in mathematics construction.

What do you mean by "degrees of freedom"? Can you not choose your axioms?

Son Goku

SkepticOne wrote:

What do you mean by "degrees of freedom"? Can you not choose your axioms?

I'm go to limit maths to geometry just for simplicity.
Yes, it is possible to choose your axioms, but I believe that you will still end up with the same construct.
For example Euclidean space can have euclid's five axioms or alternative ones proposed by different geometers at different times, however you will still have flat space as a conclusion.
From this you will generalise to Complex Euclidean space independant of your complex number axioms.

To take another example from geometry, a discussion of curved surfaces will always lead to the metric tensor.

SkepticOne

Son Goku wrote:

I'm go to limit maths to geometry just for simplicity.
Yes, it is possible to choose your axioms, but I believe that you will still end up with the same construct.
For example Euclidean space can have euclid's five axioms or alternative ones proposed by different geometers at different times, however you will still have flat space as a conclusion.

But not every combination of axioms will lead to Euclidean space. We have to choose the approprate axioms. If we leave out or modify some of them, we get something other than Euclidean geometry.

You were asking if mathematics was in some way prior to the universe. I think the fact that we have to choose the axioms means that this need not be so. We choose the axioms and the results follow tautologically. If we want specific results we choose the axioms accordingly.

The motive for choosing some forms (e.g. Euclidian geometry) over other forms could be historical or aesthetical or might be built in to our minds through evolution. We live on the Earth which is close to flat locally. Space-time is also flat locally. Would there have been an evolutionary advantage to taking into account curvatures for our hunter-gatherer ancestors?

Of course, it is always possible to imagine that a space of mathematical forms somehow exists. We then discover these forms. This would be the Platonist view. I have heard that mathematicians often take this view in their working lives. The logician Godel was a Platonist of this type.

While there is no way to prove the Platonist wrong, it is not the only view. If we can explain things without recourse to an unseen realm of forms then this might be better according to Occam's razor.

I have explained it in terms of a human-chosen system of axioms and logical deduction. Things 'must be' a certain way (e.g. 2+2=4) because we have set the system up in such fashion that that is simply the logical outcome.

Gurgle

You're all insane!

Maths invented God.