Thread: Mathematical Numbers are Theoretical; Not Proven

Indeed, from the axioms of addition, which are really just based upon common experience.

There are other modes of addition, for example addition modulo 4, in which that wouldn't necessarily be true.

And it all becomes a lot less clear when you ask still very simple questions like,

Does x^n + y^n ever = z^n, for n>2?

Obviously the answer is either yes or no, but whether or not you can prove it with mathematics is a different matter (in this case you actually can determine the answer is no, but there are other problems which you can't).

The world of math is exactly the same. There is only one world of mathematics, and it exists without language. You can use different languages to describe it (decimal, hex, binary, etc...) but they only make sense if they are coherent throughout the whole world of mathematics. You say you can throw random symbols together and they would make a coherent system...it doesn't work like that. I can say "asjdh jkahsldkah jvhkjlhl" and tell you it makes sense in another language, but it doesn't mean it does. Just like you can spout gibberish in language, you can spout gibberish in math. Just like a dog is a dog no matter what the name, adding the idea of one to the idea of one will create the idea of two. Changing the symbols used to represent that just changes how it looks on paper.

I'm afraid this isn't true.

Mathematics is essentially the 'language'. To speak of langauge describing the world of mathematics is not true; mathematics is the language which describes various aspects of our reality. Whatever rules you come up with for what you are allowed to do in mathematics, there will always be some statements which are true, but unprovable. Mathematics is not only a language; it is also a limited language.

I didn't say anything about new systems of maths being 'random'. There are still precisely defined rules; it's just that the rules are different.

For example, one consistent mathematical system is that of Euclidean geometry, which has six (I think) axioms, and then builds upon them. This system, as with all mathematical systems, will be limited. There will be some isolated geometric facts which are true but you can't prove with Euclid's rules. Another consistent system is hyperbolic geometry, which uses different, and mutually exclusive axioms, and will establish new facts within that system which are completely wrong in the other.

Originally Posted by Xei

For example, one consistent mathematical system is that of Euclidean geometry, which has six (I think) axioms, and then builds upon them. This system, as with all mathematical systems, will be limited. There will be some isolated geometric facts which are true but you can't prove with Euclid's rules. Another consistent system is hyperbolic geometry, which uses different, and mutually exclusive axioms, and will establish new facts within that system which are completely wrong in the other.

Don't these systems of geometry fit what I said? Different languages describing the same world?

But the systems describe mutually contradictory facts..?

How can any world exist if it negates its own existence? It simply makes no sense.

Originally Posted by Xei

But the systems describe mutually contradictory facts..?

How can any world exist if it negates its own existence? It simply makes no sense.

The languages are imperfect ways of describing reality. If they are in fact describing the same aspect of reality and make contradictory claims, wouldn't that be a fault of the language? Even if the systems work within themselves, one would have to be wrong when it comes to reality, because the universe doesn't change depending on whether or not you are using euclidean or hyperbolic geometry. They are just different ways of describing what is.

I don't think you're being very consistent. You were saying that there is a mathematical reality. Now you're saying that physical reality is real and mathematics just describes various models which may apply to it, which is what I was saying...

Mathematics is a just series of logical systems in which you start with a small number of facts and rules which you can apply to them, and build up from there. Whether or not they have any bearing upon reality depends on how closely your facts and rules model those of reality. Some will appear quite similar, others will have no relevance at all, but all will only be true in themselves, not describing some single coherent otherworld, and all of them will be limited in the fact that they will be unable to determine various truths in that system.

Originally Posted by Xei

Indeed, from the axioms of addition, which are really just based upon common experience.

They are factual whether they are experienced or not. This many *** ** and this many ***** are the same. A thing is the thing that it is. That is pure logic. It might even be the most fundamental principle of existence.