RE: Math and Reality
March 18, 2012 at 9:11 am
(This post was last modified: March 18, 2012 at 10:56 am by Categories+Sheaves.)
(March 17, 2012 at 8:01 pm)mannaka Wrote: I'm fascinated by your idea. This is an old topic now, but was any consensus formed on this matter? Like Vaeolet Lilly Blossom I'm not entirely sure how necessary stability is, nor was I aware of how wide-spread symmetry is.Hooray! An excuse to rant about math!
As far as the issue about falsifiablility goes (admittedly, it's not the point being necrobumped) I'm with this guy's thinking: the 'core' statements of a paradigm (e.g. matter is indivisible) are too general to be falsifiable. It's only once we start adding auxiliary hypotheses/concepts (what other properties of matter allow us to see that it is, in some sense, indivisible?) that we can get falsifiable statements about reality, and this thing we call science.
With that out of the way: The stability or invariance of of a thing (under certain operations) is a pretty basic result of having any sort of equivalence. How do I come to say, "I have four textbooks in my backpack"? Four things in my backpack satisfy the definition of 'textbook'. In regard to the question "Is this object a textbook?", these four objects produce equivalent results, and the invariance of this result under textbook-swapping forms a "symmetry." Within the operations we can perform on the books in the world, some of these operations are "symmetries" and preserve that earlier statement about my backpack (e.g. I take Spivak's Calculus book out of my backpack and replace it with Munkres' Topology book) while some of these operations don't preserve that statement (e.g. I swap Spivak's book for the collected poems of Robert Creeley). This is all an armchair concern for now... but when we start talking about atoms or electrons instead of textbooks, this becomes a practical matter. It's not like nature and math are part of a larger conspiracy to make everything symmetric. It's just that any equivalence of things (e.g. any two electrons will behave the same way in the same circumstances) necessarily yields a sense of symmetry. In the same way a definition of "up" (or really any continuous function f: R^3 -> R) yields a sense of a "level surface."
