I made a public bet with Norman in early 2020 (late Feb) about whether the university we were at would hold classes until the end of the term (he believed they would not due to covid). Norman was the only one with the guts to publicly point out the obvious (anybody with the Wikipedia page for China case numbers and a spreadsheet could have figured out what was going to happen in late Feb). Among other things he was a great lecturer and a talented go player, although I never really thought that rational trig was worth all the effort - regular math seems to have plenty of utility even if you think it is built on shaky philosophical foundations.
> I never really thought that rational trig was worth all the effort - regular math seems to have plenty of utility even if you think it is built on shaky philosophical foundations
For those familiar with "normal" (irrational?) trig, it's certainly an extra effort to learn, and may hinder communication, which gives a poor cost/benefit ratio.
For those unfamiliar with "normal" trig (e.g. not reaching that stage in school, or having later forgot it all), then rational trig certainly seems easier to learn and understand. I also like the way it generalises, e.g. to hyperbolic space (relativity), by simply changing the dot-product.
Part of the utility of rational trig seems to be that it can work in any field, while functions like sin and cos only work for reals. I.e. you can use rational trig describe the "geometry" of a "triangle" whose points have coordinates in a finite field
> Norman was the only one with the guts to publicly point out...
Sorry; what context was this in? UNSW maths department or some other social circle? I know I was anticipating lockdowns some time around then for exactly the same reasons as you list, but I assumed that all the mathematicians would have figured it out around the same moment.
Yes UNSW maths. Maybe everybody else had it figured out, but certainly Norman was the first to publicly acknowledge that all the learning would have to be online within a few months.
NJW posits that Real Numbers and Set Theory are based on the notion that it is possible to do an infinite number of calculations which he considers disingenuous.
He highlights in some of his Youtube videos that in respected Math textbooks the definition of real numbers is left vauge.
In his opinion set theory has the same kind of holes the we are expected to accept that we can add an infinite quantity of things to a Set by describing a function or simply having a desciption of the elements of the Set.
The gist of the argument is that addition + other operations on non-computable numbers (which the real numbers contain) require infinite algorithms or something similar (unlike addition on computable irrational numbers which may require infinite work, but the algorithms are finite). You can therefore get situations where, say, the tenths digit in a sum of non-computable numbers is not defined because of potentially infinite carries, and there's no way to determine if the sequence of carries terminates or not. He discusses the problem in the context of different representations of real numbers, including infinite decimals, cauchy sequences, and dedekind cuts etc. This is just the gist of it.
This is different than saying the definition of the reals is “vague”. The models and various definitions are not vague. They are as precise as the axioms of Euclidean geometry or any other axiomatic system. His objections are reasons why he doesn’t like the axioms. One can either accept or reject the axiomatic system but it isn’t vague.
ZFC models the set of real numbers, but only provides a model for a measure-zero amount of specific individual real numbers. It just says "yeah they exist".
People like Wildberger believe that anything that exists in math should have some way of determining its exact value, otherwise, what is "it"?
That I understand. I objected to the idea that the theory was vague rather than that the theory contains objects that are vague. Even so, if one thinks the theory contains vague objects then we run into considerations such as the following:
