Logical difficulties in modern mathematics (2012)

I am curious as to what his view would be on, say, the positive solution to the 3n+1 problem (i.e. every 3n+1/2 sequence eventually becomes a cycle). Some of these sequences may be profoundly long, so long that in his view, they may not exist. Or what if someone proves that 3 appears only finitely many times in the decimal expansion of sqrt{2} (God forbid!). How is the proof of something simple, say the solution of the Koenigsberg bridge problem by Euler any different: after all he did not enumerate all the possible paths! Even if he could, it is easy to find a manageably sized graph with too many possible paths.

Yeah, pretty confident. He doesn't like the computable or uncomputable reals; although I suspect he would like the uncomputables less.

Eg, "These phoney real numbers that most of my colleagues pretend to deal with on a daily basis ... such as sqrt(2), and pi, and Euler’s number e." [0]

Even in the article you cite, the irony of a pure mathematician of all people complaining that a concept has no tangible link to reality is a bit of a give away that he is speaking from the heart rather than the head. That isn't a valid complaint about pure mathematics; the point is patterns for patterns sake. So what if there are no known examples of your pattern? Study it anyway!

Great case of the flaw maketh the masterpiece; apart from that one little quirk with infinite things he is a lovely character and a force to be reckoned with. And I expect his personality motivates a lot of interesting research from him regardless.

[0] https://njwildberger.com/2014/10/06/the-infinitely-real-delu...

"Measure theory a la Bourbaki" is probably not what you meant if you did probability theory and statistics. Bourbaki famously sidestepped classical measure theory with sigma algebras by constructing Radon measures as functionals via functional analysis. This is sufficient for many purposes but not for probability theory. First and foremost, sigma algebras in probability theory are not just a technical device to avoid paradoxes with unmeasurable sets. Sigma algebras play a starring role in conditional expectation, stochastic processes and martingale theory.

The best way to appreciate the information-theoretic role of sigma algebras is to look at them in the simplest case, where you have a discrete-time, finite-valued process. Then a sigma algebra is equivalent to a partition of the state space and it represents the information that can be gained from an observation; it's like a random variable without specific values, just the discriminating information from different outcomes. To say that a random variable is measurable with respect to the sigma algebra is to say that its value may only depend on information that can be gained from an observation. A filtration of sigma algebras corresponds to a causal series of observations where the observer learns more information over time.

The conditional expectation of a random variable with respect to a sigma algebra (or partition or other random variable) is another random variable that tells you the expectation over the states consistent with a given observation; this new random variable is measurable with respect to the sigma algebra you conditioned on, which as mentioned earlier means it only depends on the information gained from an observation. The conditional expectation is the best least-squares estimator given the information from an observation in the same way that the usual expectation is the best least-squares estimator given no information.

I would start with identity, that's all everything boils down to, and that math is as much a language, a shorthand to express ideas and communicate across the world like some state department, as it is computational.

Then I would explain that imaginary numbers mean harmonics. exponential functions give you a derivative, one function can serve as input to another, PID control, ... ... and that's about it.

I'm not sure if he's a finitist or ultrafinitist [1] but the uncountable reals are clearly a problem for him. I'd like to argue for their usage from a "soft" point of view as opposed to just giving axioms.

Mathematics and physics both began with the advent of astronomy: Babylonians and others were curious to trace star patterns and from there both physics and math developed in tandem and influenced each other greatly. Really, neither would have developed without the other.

Calculus was invented for calculations in physics. This gave rise to differential equations which we use to model so many nontrivial things. The differential eqns describe flow and continuity and arguably reality. Differentiation and smoothness can't be defined over finite sets in the same way. My philosophical counter-argument to finitists is that clearly we're on the right path to understanding the universe and nature when using the reals. It seems foolish to shy from this because computers have trouble computing some functions. Statements like "there are only finitely many atoms in the universe" don't improve our understanding of much but PDEs explain.

[1] https://en.wikipedia.org/wiki/Ultrafinitism