RE: Are Numbers Real?
October 17, 2018 at 8:08 am
(This post was last modified: October 17, 2018 at 8:42 am by vulcanlogician.)
(October 16, 2018 at 9:31 pm)polymath257 Wrote: It's more that it is true because we say certain axioms are true. No, it is not a fact about all right triangles. As an example, there is a triangle on a sphere which has three right angles: pole to equator, 90 degrees along the equator, then back up to the pole. That is a perfectly legitimate triangle in spherical geometry. And the sum of its angles is 270 degrees, not 180. In such a triangle, even defining which is the hypothesis is problematic (it is a equilateral triangle after all). And the Pythagorean equality fails badly.
Thank you for your post. This sheds quite a bit of light on things. It is like Rob was saying before: math is only true because its underlying axioms are assumed to be true. But I don't think this presents a problem from mathematical Platonism or moral objectivism (not that I endorse mathematical Platonism or anything, just pointing this out).
What is an axiom? Let's define it as a basic assumption for the purposes of this discussion. I'm sure there is a more precise definition to be had among mathematicians, but that's pretty much what it means in philosophy and (I'm guessing) that's pretty much what it means in math. But a basic assumption can be correct, can it not?
I'm going a bit out of my wheelhouse here (and correct me if I'm wrong) but Euclidean geometry assumes that all its calculations transpire in flat 2D or 3D space. Those basic assumptions were so successful because, by and large, a great deal of the physical world conforms to those assumptions. My point (in the other thread) was that morality is an objective enterprise, just like math. In math, we didn't "choose" the axioms upon which any given system is based out of thin air. There was good reason for our assumptions... at the time when those systems were formulated, their assumptions were considered to be universal (of course they they aren't technically.... but pretty close).
In regards to Jorm's post above, the success of mathematics seems to indicate that (at least some of) our basic assumptions (axioms) are correct. Otherwise, we got quite lucky in selecting them.
What I would disagree with concerning Rob is this:
(October 16, 2018 at 5:09 am)robvalue Wrote: It’s more like every single person draws up their own moral (mathematical) system, and so what is true in one system is not true in another. It just so happens that certain mathematical systems are so incredibly useful that it’s highly practical to all use the same one in most applied tasks.
Maths applied through science can give us data and predict outcomes, but it can’t tell us which outcomes are preferable without also including exact criteria for what "preferable" means. It can’t do the ethics for you.
This seems to say
A) People are given a wide berth in selecting axioms, as if they can just pick whichever ones they want. I mean... they can... and the math will still work (I get that). But given the success of certain systems of mathematics in describing the physical world, this seems to suggest that we selected the "right" ones concerning those systems--ie. some of our basic assumptions were correct.
B) That math is "useful" in physics in the same way that the myth of Santa Clause is useful in our ethical discourses with our children. That is, it's a useful story on a practical level, but it is otherwise made up. Rob seems to say, as mathematical fictionalists propose, math does not make truth statements. What is your take on Rob's post?
I would posit that (in ethics as well as math) a truth statement based upon certain axioms is a truth statement nonetheless... provided the basic assumptions (axioms) are correct. But I'm learning quite a bit here, so I'm going to stand back and listen a bit more before advancing any new claims.
