Dr. Norman Wildberger is a mathematician born in Canada, living in Australia. Dr. Wildberger has an impressive series of educational YouTube videos where he explores rational and irrational numbers.
However, if you watch this guys videos, he has a problem with all irrational numbers (pi, e, sqt2, phi, etc). He thinks its okay to approximate them for practical use. I mean, isn't that what Calculus does? But calculus usually ends up converging toward a real solution except when it doesn't.
However, this guy says that irrational numbers are not numbers and they are something else (meta-numbers, number galaxies?). He has a problem with analytical proofs that use infinite series as well. He thinks those don't logically exist either.
So, the question is.... when it comes to proofs and logic and analytical solutions involving irrational numbers or infinite series, etc, what is math to do with these derivations that are not real proofs but are approximations.
How does this affect the so-called proof that there are more irrational numbers than rational because irrationals are uncountable like all real numbers while rational numbers are theoretically countable?
Ultimately, I think his comparison of rational numbers and irrational numbers to stars and galaxies was insightful. They may appear the same from a distance as points of light in the sky until you look closer with a telescope.
However, he gives away his philosophical bias in another video where he says something like 'irrational numbers and infinite sequences are illogical because they are not tied to reality, we cant even imagine them, "we are not gods".'
So, his premise was that all math should be worked out analytically and logically without these expressions.
However, he was rather inconsistent in his arguments in my view because he then went on to give multiple theorems, expressions, and proofs for rational numbers showing infinite rational numbers between points. Infinity is very necessary and useful in Calculus and in limit testing for convergence. Therefore, I think his comments were more a refection of his problems with and rejection of God. Sad.
Despite this man's denial of faith, I came away from his lectures stronger.
1. if man can at least imagine that we can imagine infinity
2. and if infinity is a real, logical, and useful expression beyond mortal comprehension.
3. then the reality of infinity in mathematics is a strong evidence for God (infinite mind) in the Universe.
"And the Lord God spake unto Moses, saying: The heavens, they are many, and they cannot be numbered unto man; but they are numbered unto me, for they are mine." (Pearl of Great Price, Moses 1:37)
At one point that talking about infinity was a manifestation of human arrogance. In reality, I think the converse is true. Math is not the language of man. Math is the language of the Universe, the language of creation and the language of God. Therefore, man shouldn't get caught up in rejecting those parts that his mortal and finite mind can't comprehend.
P.S. He asked everyone to come up with a name for and define 0/0. He said if 1/0 = infinity (8), then we should give a name to 0/0 = (*). Some suggested "infinitesimal". I suggested "Zot".