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Did anyone see Terrence Howard on Joe Rogan? Would love to see what the motte thinks.
Personally I'm withholding judgment until he goes up against someone credible.
I watched it, I liked it.
If you want to listen to someone who doesn't say 1x1 = 2, then much of what he was talking about reminded me of Eric P Dollard. If you can stomach a two hour podcast, maybe consider this three-and-a-half hour lecture, History and Theory of Electricity.
The real meat and potatoes of his views are not new. He's deep into the waves/frequencies/spin is everything method of analysis, which makes sense. This is also part of his criticism of 1x1=1, in that it assumes axiomatically a rectilinear universe, which is not the way the actual universe exists. The most interesting parts were when he was talking about the periodic table and the solar system.
For the periodic table, he claims that elements are essentially harmonics of one another, the same frequencies (spin) doubled can turn one element into another.
As for the 1x1=2, I'm very sympathetic. I distinctly remember my complex algebra course, where we rederived addition, subtraction, multiplication, and division such that they are consistent with what we see in a number line. It reminds me that these operators are axioms themselves, and that axioms are taken as true, not proved. Ultimately, I think he gets lost in the weeds of units. For example, he asks what is $1 x $1, and reports that people say $2, and don't want to accept $1. But of course, $1 x $1 is $^2 1 (one dollar-squared, or one square dollar). What the hell is a square dollar? The same thing as a square second, but with currency.
The other reason I dislike it is because you can derive multiplication from your fingers, in a similar manner to addition. Put one finger up on each hand, and count them, and you get to 2 (addition). Two fingers on each hand gets you 6, and three fingers gets you 9. Now, instead of counting them, cross the fingers of one hand against the other and count the overlaps. One finger and one finger, overlapped, gets one intersection. Two fingers on each hand, 4 intersections, and three fingers gets 9 intersections.
His problem is root 2. For our finger example, what combination of fingers gets you two intersections? You have to use different numbers on each hand.
But notice, we have changed units! We are counting intersections, not fingers! So again, I'm sympathetic in general, but in particular it breaks down.
It is possible to build up a thorough and comprehensive axiomatic theory starting from a geometry without the parallel postulate. But what's described here seems like an extremely painful, sloppy, and intentionally confusing usage of notation. Possibly just wrong, probably not even wrong.
In terms of foundational mathematics, building up from geometric definitions like crossing lines is an extremely cumbersome method of defining your axioms. Even if you do not insist on using intentionally confusing notation like 1x1=2. As you said, you immediacy run into annoyances in terms of defining basic things like the irrationals in sqrt(2).
If you wanted to make a serious attempt at analyzing alternatives to the conventional axiomatic assumptions, it would be much more clear to begin with variations on Zermelo–Fraenkel set theory, with or without the axiom of choice and the continuum hypothesis. This would be a much more rigorous and clear way of showing how your systems produces a non-Peano arithmetic. If someone is unwilling to go through that work, it seems extraordinarily unlikely that they are producing anything interesting, correct, and non-trivial.
Though the foundational crisis may non be resolvable, the generally accepted formalism provides the necessary mathematical tools to do an extraordinarily good job of describing reality. If someone wants to propose a different formalism, it better provide a better or more useful description of reality. Saying that the current formalism does not perfectly describe reality so we should adopt a formalism that is less useful and more confusing, is pure nonsense.
To quote Hilbert (1927 The Foundations of Mathematics):
Laying out a formalism with overlapping but ill-defined versions of "spin" and "product," is not cleverness or some deep philosophical insight, it's an expression of sloppy thinking.
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