Axiomatics and the least upper bound property (I) | Real numbers and limits Math Foundations 120

The role of axiomatics in mathematics is a highly contentious one. Originally the term always referred to Euclid, and his use of the term to mean `a self-evident truth that requires no proof '. However in modern times the meaning of the term has shifted dramatically, to the idea that an Axiom is `a convenient fact that we assume'.

This casts considerable doubt on the validity of the usual claim that `Mathematics is built on Axioms", which these days appears more and more as a religious position rather than a scientific one. Is that what we want our subject to be?? Your belief system as opposed to my belief system??

In this video we discuss this shift in meaning and its consequences when trying to set up a theory of real numbers. We will be discussing this important issue further when we get around to critiquing `modern set theory'.

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Here are the Insights into Mathematics Playlists:

[robharwood]

"The nice thing about axiomatic systems is that you have so many to choose from."

The above is 'translated' from one of my favourite software engineering jokes (attributed to Andrew S. Tanenbaum, according to WikiQuote):
"The nice thing about standards is that you have so many to choose from."

[maxpercer]

Watching many of his videos it occurs to me that the uploader (normal wildberger) is not presenting all the sides of this issue, either out of ignorance, dishonesty, or just doesn't care (or trolling). I will assume he is not trolling or dishonest or doesn't care.
So what is the issue? _Mathematical realism versus mathematical anti-realism_ . The issues get nuanced, and he drops a few names here and there. Don't be led astray by someone who is not presenting all sides of the issue. It is misleading. I hate it when people do that. It is better to say this is my opinion versus 'all these mathematicians who are using infinite sets exist are wrong'. In fact many mathematicians are formalists (math is a syntax game) so this is a non-issue - saying they are wrong would be like telling chess players they are wrong about the rules of chess. But for non-formalists, the 'completed infinity' versus potential infinity issue does come up (typically people use the term 'infinity' without qualification to mean potential or unfinished infinity). It comes up when you assume a bijection exists between Natural Numbers and the Real numbers in a proof by contradiction. If bijections with N and other sets cannot exist at all, then the proof by contradiction would not be a contradiction, and the conclusion would not be reached (that the real numbers are uncountable).
Can you get away with Cantorian mathematics by replacing so called completed infinities with potential infinities? Maybe. That is an interesting challenge. I don't think that many people dispute that Cantorian mathematics has been useful for mathematicians, whether it is fictitious or not. There are other examples of this in daily life. I can enjoy the Simpsons cartoon even though I know it is fictitious ( I am not a platonist about cartoons).
Though I will go out on a limb and say that norman wildberger is claiming aspects of cantor's mathematics, e.g. ordinal arithmetic, is not merely fictitious but meaningless or self contradictory- it is like talking about square circles (at least that is the accusation being posed at Cantorian mathematics which has now been adopted by mainstream mathematicians - Cantor received a lot of criticism in his day; notably from the mathematician Kronecker).
Anyways, I will leave this issue to a philosopher who is trained in modern mathematics to soberly evaluate these arguments and present all viewpoints. There are many since the issue becomes nuanced. One ought to be transparent.
If you can construct all of mathematics without infinity in a manner that is also pedagogically secure, all the best. I am not throwing out my set theory book, aptly titled 'sets', and i have a book called 'mathematics of infinity' (the author straight up reveals in chapter 0 he is a mathematical platonist).
The bottom line, this uploader knows that mathematically inexperienced students are watching his video. He should be honest and say, well this is debated by mathematically trained philosophers and this is my viewpoint. Again he is presenting one hybrid view, and these viewpoints have been classified by philosophers, see . Be warned.