Some Funky Math Systems
June 19, 2022•653 words
Just thought I'd share some ideas that show how math isn't strictly the truth (I know I said I planned on posting updates, but my life isn't very exciting and I'd seems bad to email the subscribers of this blog just mundane updates considering my precedent).
It seems quite intuitive that if something is not false, then it must be true. However, some mathematicians don't think this is true. There are compelling examples supporting their view, like the liar paradox [1] (e.g. statements like "this sentence is false") and fuzzy logic [2] (i.e. there are degrees of truth). This is the view of the constructivist [3]. I read this introduction [4] to it, and I found it fascinating some of the proofs they reject. There are the more obvious stuff like how not not true = true isn't guaranteed, but also less obvious stuff like the Axiom of Choice [5] (this basically posits that there is also a way to choose objects from sets, i.e. choice) {note that rejecting the Axiom of Choice for things like the Banach-Tarski paradox [6] doesn't necessarily imply constructivism¹}. Although, the rejection of not not truths has big ramifications for a lot of proofs in math. Quite a number of proofs assume that if they prove an object can't be nonexistent, then it must exist. However, constructivists demand an actual example of the object existing for it to exist. They also don't necessarily accept standard theorems like the Intermediate Value Theorem in calculus (which is a very important theorem for all the other theorems in a standard calculus class).
Most people in the modern day are quite used to the concept of infinities, but some mathematicians like Norman Wildberger [7] reject the use of infinities in math. His main criticism is that sets are poorly defined and that infinity doesn't exist in the real world. This is the concept of finitism [8]. I don't know enough about set theory to comment on his first critique, but from consuming his content he simply asks whether sets can be made of real objects (like people) but assumes the viewer finds this idea obviously false (i.e. he doesn't say why). In the latter, I currently agree. The recent success of quantum physics suggests that reality is quantized (i.e. not infinitely small) as the once-considered-continuous light wave was quantized into photos. Hence the idea of a continuum may not actually exist. Consider what a lot of us think is continuous, like a line. If we draw a line, it is impossible to have it completely connected as there is always space between atoms, space inside atoms, space inside quarks, etc. He also raises points along the lines of "how can we be sure numbers like 10101010101010101010101010101010 exist or have factors?". He argues that since this number is larger than the current estimate of the number of atoms in the universe, it can't exist or describe something physical nonetheless have mathematical theorems guarantee a prime factorization. Thus this perspective takes quite a conservative perspective to mathematics. And indeed, he does seem to have some more conservative views, suggesting a pessimism on intellectualism's ability. I think he is wrong though. In my opinion, math is an attempt to describe an ideal, something more platonic then science and reality can ever achieve. Furthermore, basing math off of physics means that if some idea shakes physics' core, then math is affected as well. Also, limiting the future of math based on present day physical approximations seem to be purposefully limiting yourself. Tying math to physics suggests math is less true in an absolute sense.
¹Law of excluded middle [9] (basically that not not true = true) is a law of thought, ZF [10] (a lot of modern math) uses the laws of thought
[1] https://en.wikipedia.org/wiki/Liar_paradox
[2] https://en.wikipedia.org/wiki/Fuzzy_logic
[3] https://en.wikipedia.org/wiki/Intuitionistic_logic
[4] https://www.ams.org/journals/bull/2017-54-03/S0273-0979-2016-01556-4/S0273-0979-2016-01556-4.pdf
[5] https://en.wikipedia.org/wiki/Axiom_of_choice
[6] https://en.wikipedia.org/wiki/Banach%E2%80%93Tarski_paradox or VSauce: https://www.youtube.com/watch?v=s86-Z-CbaHA
[7] https://njwildberger.com/
[8] https://en.wikipedia.org/wiki/Finitism
[9] https://en.wikipedia.org/wiki/Law_of_excluded_middle
[10] https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory