reply to post by IrnBruFiend
I read your thread during the afternoon, local hour and have been thinking on & off about the questions you posed since. Here are some of my opinions
(I can't stress opinions
To the best of my understanding numbers in general are a prioritic in nature, or maybe it's accurate to say that numbers precede experience that tell
us of their existence. I believe this is true of mathematics in general; hence, its status as the purest science. Besides that mathematics is not
about numbers, but axioms and theorems derived from axioms. From the standpoint of pure mathematics I speculate
that numbers are simply a
consequence of axioms. That said, it's probably true that ideas about numbers preceded axiomatic methods. I don't know whether this is certainly
true, but my stab-in-the-dark intuition compels me to believe this.
The one wrinkle I'm familiar with in terms of pure mathematics centers on Kurt Godel's Incompleteness Theorem (IT). I'm by no means an expert on
this subject matter, but to the best of my understanding Godel's IT showed that "it is impossible to establish the internal logical consistency of a
very large class of deductive systems unless one adopts principles of reasoning so complex that their internal consistency is as open to doubt as that
of the systems themselves." (Nagal, Newman, "Godel's Proof").
IT applies to Zermelo-Fraenkel Set Theory; Z-F Set Theory is the language of mathematics. I doubt there is a man or woman alive today capable of
explaining or realizing the ... full
implications of this theorem, and by full I mean those implications that step outside of pure mathematics
& into philosophy. At any rate, that's my opinion.
Mention of IT is intended to suggest that pure mathematics is not broken per se, but its pureness of truth is not consistent. So to ask, "Is math
always correct?" is, I think, context dependent. Math's logical rigor is a boon, but at a fundamental level (i.e. a level that matters little to
the majority of practitioners of mathematics) it is in some sense a bane as well. From my perpsective that's odd, but so is the fact that you can
divide a circle's circumference by its diameter and arrive at pi regardless the size of various circles (sidenote: I doubt this was discovered a
priori, but by experimentation alone). In any event I mean odd in cool way, but it's still damn odd nonetheless.
Does zero exist? My mind refuses to believe that there is such thing as "nothing", presumably representative of zero. My reasoning is that from a
reference standpoint of meaning it's not possible to definitively define "nothing", but allude to its sense meaning. In other words I'm not, hmm,
confident that someone can prove a prior or positivistically that "nothing" exists. For that matter I reckon it could be argued that integers
don't really exist either, but I tend to lean toward the opinion that in some weird metaphysical way they might. As for zero itself I think of it in
terms of limits and infinity. That's formal education for you I guess, but the ideas about the limit is powerful and compelling.
Do I think zero and infinity exist? I don't think zero exists. It's certainly a useful idea, but I don't think it exists. As for infinity I
purposely choose to say nothing (pun intended).