There’s a growing list of facts in mathematics, physics and philosophy that worry me. I want to start keeping track of them. So this post will be dynamic in the sense that I’ll add worries to it as they come up.
Post 1: Jan. 23rd, 2014
(1) Summing the natural numbers (all of them) and obtaining -1/12. That’s worrisome enough, it’s further perplexing that it’s used in quantum field theory (so I’ve heard).
(2) All functions from the reals to the reals are continuous if you use the intuitionist logic of Brouwer. I know very little about this but the multiplicity of logics in which different results are obtained is worrisome.
(3) Toposes: I really know nothing of this other than what I read in McLarty’s book on categories (but I am working my way forward). A topos sounds like a category equipped to do mathematics, it sounds very big and it includes, to my understanding, a deductive system. Some of these toposes don’t all for non-not=true or the excluded middle. So this concern is related to concern #2
(4) What’s a mathematical object and what’s a mathematical tool. Alain Connes makes this distinction in “Triangle of Thoughts.” It has caused me to think a lot about the things I study and their existence/utility. If Nonstandard Analysis, for example, yields no additional theorems that couldn’t be proved in Standard Analysis, what does that imply about the infinitesimals and infinitely large elements in the theory?
(5) (added 1/25/14) I’m a platonist and so believe that mathematical objects (whatever they are) are independent of me, my culture etc. What of mathematical tools? To what extent do they bear the mark of the species using those tools? This partially revolves around the privileged position that low dimensions, such as 3 and 4, seem to have in topology and geometry (there’ll be posts on that eventually). It is a coincidence due to the fact that the species studying mathematical objects have imaginations rooted in three dimensions with the ability to generalize just a bit? Do we run out of vision as the dimensions increase? A good counter-argument would be infinite dimensional spaces such as Banach and Hilbert spaces but I have increasing reservations about infinity. Is there something profound about low dimensional topology/geometry that all species in all times would find and wonder at or is this a matter of only sensing information that our sensory organs are equipped to receive.
More to come, feel free to share (I know I’ve left some out).