Atla wrote: ↑Sat May 26, 2018 7:23 am
wtf wrote: ↑Wed May 23, 2018 7:00 pm
By assuming a weak form of the axiom of choice, one can show the existence of a gadget called a nonprincipal ultrafilter, which one can then use to produce a field (a mathematical system in which we can add, subtract, multiply, and divide) in which there are infinitesimals. This field is called the hyperreals. [Technically there are many such fields since different nonprincipal ultrafilters give rise to nonisomorphic fields of hyperreals. And it's a mathematical curiosity that the Continuum Hypothesis implies that there is only one unique field of hyperreals. As you can see we are in deep foundational waters].
Pardon me for my ignorant question (ignore it if it's just word salad): if we assume the Continuum hypothesis to be correct and so there is only one unique field of hyperreals, then would this uniqueness be somewhat similar to for example the uniqueness of the Monster group from group theory: similar in the sense that it would be a certain specific mathematical structure, found in higher mathematics, that's simply "there", for some reason?
(I just find it fascinating when mathematics seems to discover structures that are simply "there". I'm coming from the physical multiverse hypothesis angle; I have the impression that such structures may be somehow linked to the unique topology of our universe, but in ways we can't really understand yet. I also read somewhere that "set multiverse" ideas are a thing now in mathematics, and people who hold such views are more likely to think the the Continuum hypothesis may be correct.)
First, let's look at far simpler cases. Many abstract mathematical structures exist in the real world. For example take the set of permutations on three letters a, b, and c. We have: abc, acb, bac, bca, cab, and cba, six altogether. We can combine two permutations to get another (for ex. "swap the first two then swap the second two"), permutations are reversible, there's an identity permutation (abc) and composition of permutations is associative. So the six-element set of permutations on 3 letters is a
group, an abstract mathematical structure. Group theory finds physical application in quantum physics and crystallography. So groups have some sort of actual existence even though you can't pick one up and put it under a microscope or smash it with a hammer.
Likewise, the hyperreals, whether you regard all the distinct hyperreal fields in the absence of CH, or the unique one in the presence of CH, are "out there" in some sense. What sense that is, I can't say. Perhaps a philosopher can explain to me what kind of existence the group of permutations on 3 letters has, then I'll know.
However in the case of the hyperreals, this existence is more murky than it is for elementary examples like groups. To construct the hyperreals, we need a weak form of the
Axiom of Choice. So any such "construction" is actually nonconstructive. Nobody can visualize or write down any particular nonprincipal
ultrafilter or its corresponding field of hyperreals.
So I have two philosophical questions. One is, what kind of "existence" does an abstract group have? And, what kind of existence does a nonconstructive mathematical object have? Note that there is a philosophy of mathematics, namely
constructivism, that would accept the existence of the permutation group on 3 letters, but reject the existence of any nonconstructive object. So philosophers do discuss these issues.