In philosophical debates about absolute truth, people cite “the truths of pure mathematics” as beyond reproach—eternal and universal things discovered/invented by us fallible mortals. But the more deeply I look into these issues myself, the more I see evidence that mathematics is not as stable as I’d supposed:

  • constructivists and intuitionists argue that the foundations of mathematics don’t make sense
  • logicians accuse mathematicians of not being rigorous enough
  • mathematicians themselves admit they totally ignore foundational issues and just concentrate on getting interesting results that make sense within their set of assumptions and could probably be “straightened up” to satisfy the logicians
  • Bill Thurston referred to mathematics itself as a social entity — it is the dynamical creation of a community, it lives inside the heads of the people who prove these things and not on paper.
  • John L Bell and Geoffrey Hellman: “Contrary to the popular (mis)conception of mathematics as a cut-and-dried body of universally agreed upon truths and methods, as soon as one examines the foundations of mathematics, one encounters divergences of viewpoint and failures of communication that can easily remind one of religious, schismatic controversy.

Norman Wildberger thinks real numbers have been a wrong turning in mathematics. He also claims, here in the video above, that angles θ are illogical. (Or maybe I should say, certain angles are used illogically.) Some angles, like 60°, can be constructed via ruler and compass. But other angles like 34° and 26° are not constructible.

So although “I know what you mean” when you talk about a real number or an angle that measures 90.1°, maybe we should both recognise that they don’t really make sense and speak in air quotes.


Related but different. On the topic of left-brains, right-brains, closed-minds, and open-minds in science. You can see youtube user njwildberger being beaten up on the XKCD forums for suggesting such unconventional and—ick!—philosophical ideas. Listen to these self-satisfied, smarter-than-thou sabelotodos savaging the “ridiculousness” of someone who would undercut this Well Established Knowledge.

I find that incredible because XKCD’s vision of science seems to be about open-mindedness, learning from data, and accepting the truth based on logic rather than tradition or popularity.

The Data So Far


OK, “data” needs to be replaced with something else in theoretical maths. But you could at least listen to what the guy’s saying rather than his credentials or his sweatpants. (Conversely: if John Conway says it, does that make it true? He gives talks in sweatpants as well.)

I bet ≥ some of these know-it-alls have lauded Galileo for smashing the accepted wisdoms handed down from Aristotle with cold, hard logic. What’s the difference to making fun of njwildberger because he’s suggesting something weird or unconventional? Prima facie it makes sense to me.

Maybe you don’t care about the foundational issues (isn’t that called hand-waving elsewhere?), or maybe you can disprove what he’s saying—but this PageRank 7 site is just attacking him rather than his idea. (For example they look at his publication record to see if he’s “someone we should take seriously”.)

You want to know why people aren’t interested in science? I think it’s in part because science and maths is associated with such stuck-up, judgmental people—putting down everyone who’s less “intelligent” than they are.

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