118118 said:
If these three criteria are objective, then could we not just say that there is a perfect mathematics, just that we have not found it yet.
I was thinking in terms of mathematics as an educational/scientific institution and the subtle tensions within it.
I don't think there is a perfect anything. There is always imperfection, conflict and compromise.
118118 said:
So some kind of numbers do exist independent of the mind, its just that we have some evidence that the number system we do use now is not those numbers. Not that I would agree with that, as I can't see 1+1=1 describes the world literally, as well as 1+1=2, unless the the things being counted are poorly defined. So that mathematics is only ambiguous if the thinsg being counted are.
I'm still going to say that direct representation of the world is unimportant - except for educational purposes. I think 1+1=1 always works whereas 1+1=2 only usually works. The moral is not that the former is superior but that neither statement is a literal statement about the physical world.
118118 said:
Knotted: I thinki that the fictional concept of maths is suppsoed to be tenable. But, how do you explain that something that is only true in the story, is true in science. 1+1=2 is not only true in the story of mathematics, but also in scientific work. If you claim that it is not true that 1+1=2 in science, then my encyclopedia disagrees with you. And, I would add that 1+1=3 cannot be true in scientific study, as that would be incompatible with what is.
Hartry Field wrote a paper describing Newtonian mechanics without reference to arithmetic. I haven't read this myself, but there seems to be a consensus that it has demolished the view that arithmetic is essential to understand science.
Plus as I have shown very simply that an arithmetic with 1+1=3 is just as good a tool for scientific study.
Maybe number and standard arithmetic always creeps in disguise, and I tend to believe it does, but it needn't be a starting point and can be represented with all sorts of formalities with all sorts of physical interpretations.
So the idea that the formalities of mathematics are real is a nonsense. If there is a reality to mathematics we do not have the language to even refer to it. Outrageous as it sounds, this is pretty much what I feel. I see the formality of mathematics as being like an artist's impression of the underlying 'mathematical reality'. Just as the impressionists showed that art does not need to be literal in order to depict its subject well, I don't think mathematics need be literal in order to describe reality well.
118118 said:
Eta: What I quoted above is also true of phillsophy. Does that mean that no philosophy is a fiction
No, but I'd be hard pressed to prove philosophy is not just fiction in any case.