Have you guys heard of Professor Wildberger? He’s a mathematics professor at the University of New South Wales. I stumbled upon this video a couple years ago and thought it was really interesting. It is called, “The mostly absent theory of real numbers.” He goes through some standard textbooks and shows how most do not define real numbers. Of the few that do, they do not do so in a way he finds satisfactory. He says near the end that “the real numbers cannot be constructed in the usual kind of fashion. So next time it will be dedekind cuts that will be on the chopping block.”
Then I started watching his other videos, and I like his teaching style a lot. I’m not a math major, but I do enjoy these videos. Have you seen them? What do you think?
https://www.youtube.com/watch?v=fXdFGbuAoF0
I skipped around this video so I just have a few random thoughts.
1. Elementary textbooks like Stewart are not going to deal with axiomatic questions that Prof. Wildberger is interested in. Even Courant, who is more interested in fundamental issues purposely dodges this one because it is a distraction from the main purpose of his book. These books are trying to teach students how do do calculus, not how to build an axiomatic mathematics.
2. Fundamental problems like this in mathematics have been bugging people for well over a century. Given that Kurt Gödel proved that arithmetic is incomplete way back in 1931 (about the same time as Courant’s book) shows that these kinds of questions will *never* be solved to the satisfaction of Wildberger. I don’t know if he mentions Gödel in any of his lectures but he certainly should.
https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems
3. There have been some attempts at axiomatic definitions. Wildberger alludes to one in his review of Spivak’s book. It’s probably something along these lines:
https://en.wikipedia.org/wiki/Real_number#Axiomatic_approach
He would probably not approve of this either.
My impression is that Wildberger is on a fool’s errand. But I have to admit this is well outside of my expertise so he might be on to something for all I know.
Perhaps he is on a fool’s errand. I don’t know enough to say one way or the other. I didn’t understand how much we rely on axioms in mathematics, until coming across his video here, then watching others of his.
I am no expert on the subject matter, but I was not impressed. To me, he comes off as an old kook with an obscure axe to grind. The camera-in-face presentation, tight focus on his portion of the laundry room, lecture given to no students, agonizingly slow development and the ever-present preambles, re-ambles, defenses and re-setting of the stage just give the impression of a guy who didn’t get tenure soldiering on to answer a question that nobody is asking — nobody in the audience, anyway.
He seems like a perfectly nice man, and he gets dressed up to come to work in the basement — he’s not offensive or combative in his presentation. It seems that he contrives strawmen such as the textbook definition thing. As Doc Lorentz says, some of these texts would not likely cover that anyway, and at any rate, the approach with the texts seems a bit cargo-cult.
I recall as hearing as a child that there was a missing number (meaning integer) between 6 and 7. It has always intrigued me, not for the answer, but for the origin of the question. There may be something to it. There may be something to an infinite number of handy conjectures, each just waiting to be ridden into batte by a champion such as this guy.
But I can see that your time is nearly up here, so Good Luck! to you, Professor Wildberger! My hat is off to you and I bow low to usher you and your fine steed on your way, off to the frontlines in another town, where there is also no fighting going on, but the people there will receive you well and listen to your stories, so long as you pay in coin at the tavern.
ON THE OTHER HAND, let’s assume that he is correct. He should be able to explain his point crisply. Is there a video under five minutes in which he does so? This video is obviously one of many, and pretty far gone along on a road I’ve never seen. SO it all looks strange to me, and I don’t recognize anybody here, and I don’t trust the driver. I’d be more willing to sit for the ride if I could see a map. I get where he says we’re going, but I got no landmarks.
I don’t think there is a video under five minutes from the professor. He is, after all, a professor! He likes to teach, which means he likes to talk.
I guess it all looks a bit strange to me as well, which is one reason I enjoyed it. Never even considered the idea that we rely on axioms as much as we do for math. I just found the whole idea fascinating. In my next life, I will be a math major so I can learn more about all this stuff. For now, I am content to find somebody who can explain things in a way I find interesting.
I don’t quite get the reference to cargo cult? I had to look it up. Not sure I see what you meant there.
Cargo cult is probably an allusion to Feynman’s Caltech commencement address.
http://calteches.library.caltech.edu/51/2/CargoCult.htm
Not sure that’s applicable to this guy but maybe.
Thanks. That looks interesting. I will read it this evening.
Earlier on another thread I had to look up mopey.
This one is out of my league.
Have fun y’all.