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This is Norman Wildberger's personal website where he expresses his unusual opinions: https://njwildberger.com/
There is a lot to unpack here, so I'll highlight a few important things:
- Debate with Daniel Rubin: He links to this on the homepage of his website. Rubin was incredibly respectful of Wildberger's ideas, ideas which I suspect most mathematicians would dismiss as nonsense pretty quickly. I appreciate that Rubin was willing to do this, because while I do not agree with Wildberger, it only grants more credibility to conspiratorial quacks when they are ignored by experts. One issue with this "debate", however, is that Rubin gave Wildberger most of the talking time and did not push back very much. He did, however, outline his objections more clearly in another video, which Wildberger neglected to link to or mention on his website. (At least that I could find. Feel free to correct me on this.) It can be found at this link: https://www.youtube.com/watch?v=GnepxZ-ZZOI
- Modern math compared to religion: Wildberger asserts that modern math is in some sense religious, believing in things on the grounds of faith alone. I might respect his objections a bit more if he didn't do this, since many of his other objections are at least understandable, but this assertion that mathematicians so desperately want their beliefs to be true, and that they don't have any real arguments is absurd, and potentially harmful. The assertion is baseless, and it paints mathematicians as complete fools, rather than the geniuses that many of them are.
- Conspiratorial wording: Wildberger uses a lot of conspiratorial wording such as "delusion" and "blindly accept." In this way, he appeals to people who are conspiratorially minded, and who want to feel like they are smarter than the experts. This idea of a widespread delusion is simply nonsense. Mathematicians do not blindly accept statements such as "...and then taking this to infinity..." and they frequently question the meaning of this sort of statement when applied to a context in which it has no formal definition or where its application cannot be justified. They don't simply search for evidence which agrees with their preconceived notions (as Wildberger would suggest) either. Take, for example this paper: https://vixra.org/pdf/1208.0009v4.pdf. A mathematician as described by Wildberger would blindly accept its conclusions, nodding their heads every time "as n goes to infinity" is mentioned. In the real world, however, any credible source will tell you that the problems that this article claims to solve remain unsolved. This is because a real mathematicians questions the use of limits in this paper, and recognizes it as invalid.
- The impact of Wildberger's conspiratorial wording: This is what really caused me to lose any remaining respect I had for Wildberger. Many of his followers hold the belief that modern math is a complete waste of time that does nothing for society. A trip to fantasy land that mathematicians get paid to take. This is problematic for two reasons. Firstly, it makes it seem as though mathematicians don't actually do anything, negating the sheer amount of work and effort that math takes. Secondly, it has lead many of his followers to believe that if mathematicians could only wake up, our technology would be drastically better, and millions of lives could be saved. (One look at the comments on one of his posts reveals just how many of them believe all of these things.) The irony in this is that without the concepts that Wildberger rejects, he wouldn't be making these blog posts on a computer, nor would we understand nearly anything of what we do today about the universe. Switching to Wildberger's ultrafinist math would kill, not save millions.
- "Are mathematicians scientists?": The short answer is no. They aren't supposed to be. Science uses inductive reasoning. Math uses deductive reasoning. Science can change with new evidence. Math is not evidence based, and proofs are set in stone. Science uses experimentation to draw conclusions. Math uses abstract deductive proofs. Science is observation based. Math is done in the abstract, and you can't observe abstract objects in the same way that you can observe physical ones. According to Wildberger, however, the approach of science is the only valid one. This completely misses the point of math, which brings us to my next point.
- Model vs. match: Mathematicians do not assert their axioms as objective truths. Math is not intended to be part of the physical world. Math, like any field of study, should be judged by its usefulness, regardless of how that usefulness arises. Math allows us to model things in reality, but it is not itself part of physical reality. It is a model, not a match, and that is the way it is supposed to be. This is because the physical world can be somewhat of an enigma. In theory, we shouldn't be able to make any predictions at all, because we don't know, for example, that just because F = ma this one time, that F will equal ma the next time we apply force to an object. The equation "F = ma" wasn't found in some deep dark cave signed "Creator of the Universe," we just observed that this equation is consistent with our observations. This is the beauty of math: Our mathematical models can make predictions about something without us actually needing to see it. That is why math is not observation based. Because that would defeat the purpose. Sure, the fact that math exists separately from the physical world means that it doesn't always match the physical world, but that is okay. No one is claiming that everything in math has a counterpart in reality.
- The law of (logical) honesty: Wildberger's law of honesty is a good one. The issue is that it is a moral principle, not a logical one. Not pretending to do something you can't is good life advice, but for the sake of logic, considering theoreticals is incredibly important and useful, and there is no problem with it. Wildberger says that this law of honesty invalidates a question such as "If you could jump to the moon, would it hurt?" I have no issue answering this question: Yes, it would. In fact, you would definitely die. You would accelerate incredibly quickly through the earth's atmosphere and into space, and if you weren't already dead, you would find yourself in the vacuum of space where your blood would boil. Wildberger would suggest that this wouldn't happen, because no one can jump to the moon anyway. My response to this is that it is possible to talk about what would happen if one were to jump to the moon, even if that won't happen. Why is it important to be able to use theoreticals though? No one really cares about what would happen if they could jump to the moon, but considering theoreticals can be very important. Wildberger agrees that it has been proven that there is no rational number equal to the square-root of 2. What exactly is this proof? Well, feel free to look it up if you want the details, but to summarize, it begins by assuming that the square-root of 2 can be written as a fraction, and demonstrates that this leads to a logical contradiction. That's right, we are not only imagining that we can do something that we can't, but we are using that assumption to prove that we can't by showing that it leads to a logical contradiction. It is undeniable that if the assumption that something can be done leads to a logical contradiction, it cannot be done, and yet under Wildberger's "law of honesty" (at least as he applies it) this sort of proof by contradiction is invalid.
Obviously, this does not address everything that Wildberger has to say, as he has said a lot, so if there is something specific that you want me to address, feel free to point it out.
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