A way to improve thinking is to imagine more, and be more rigorous. What a better place to exert these skills than in mathematics and logic? Things are clearer there.
The crucial Axiom Of Choice (AC) in mathematics has crazy consequences. After describing what it is, and evoking some of its insufferable consequences, I will expose why it ought to be rejected, and why the lack of a similar rejection, at the time, in a somewhat similar situation, may have help in the decay of Greco-Roman antiquity.
This is part of my general, Non-Aristotelian campaign against infinity in mathematics and beyond. The nature of mathematics, long pondered, is touched upon. A 25 centuries old “proof” is mauled, and not just because it’s fun. There is deep philosophy behind. Call it the philosophy of sustainability, or of finite energy.
The Axiom of Choice makes you believe you can multiply not just wine, fish and bread, but space itself: AC corresponds, one can say, to a wasteful mentality.
The Axiom of Choice says that, given a collection C of subsets inside a set S, one can consider that a set exists, made of elements, each one of them is an element in exactly one of the subsets. That sounds innocuous enough, and obvious. And obvious it is, if one thinks of finite sets. However, if C is infinite, it gets boringly complicated.
Moreover, AC has a consequence: given a unit sphere, one can cut it in disjoint pieces, and reassemble those pieces to build two unit spheres. Banach and Tarski, both Polish mathematicians working in what’s now Western Ukraine, the object of Putin’s envy and greed, demonstrated this Banach-Tarski paradox. It’s viewed as an object of wonder in General Topology.
I prefer to view it as an object of horror. (The pieces are not Lebesgue measurable, that means not physical objects. Such non measurable objects had been found earlier by Vitali and Hausdorff)
Punch line? The Axiom Of Choice (AC) is central to all of modern mathematics. Position of conventional mathematicians? The fact that AC is so useful, all over mathematics, proves that AC can be fruitfully considered to be true.
My retort? Maybe what you view as fruitful mathematics is just resting on a false axiom, or, at least one against nature, and thus, is just plain false, or against nature. One may be better off, studying mathematics that is not against nature..
As I showed earlier, calculus survives the outlawing of infinity in mathematics. That pretty much means that useful mathematics survives.
You see a problem with mathematics, even the simplest arithmetic, is that, once one has admitted the infinity postulate, thanks to the Cantor Diagonal process, one can always find undecidable propositions (this is part of the Incompleteness Theorems of mathematical logic: Gödel, etc.).
That means a field such as Euclidean geometry is infinite, in the sense that it has an infinite number of non-provable theorems. Each can be decided both ways: false, or true. Each gives rise to two mathematics.
Yet, even modern mathematicians will admit that studying Euclidean geometry for an infinite amount of time is of little interest. Proof? They don’t do it.
Yet, what’s the difference with what they are doing?
Mathematics is neurology, and neurology can be anything, but infinite. Think about what it means. Yes, mathematics is even cephalopod neurology, with the octopus’ nine brains. Fractals, for example, are part of math, but far from the tradition of equating angles or algebraic expressions.
It’s a big universe out there. The number one consequence to draw from the history of science, is that scientists make tribes. Quite often those tribes go astray… for more than 1,000 years (see notes). Worse: my making science, and, or mathematics, uninteresting, they may lead to a weakening of public intelligence.
I would suggest that effect, making science, and mathematics priestly and narrow minded, contributed to the powerful anti-intellectual tsunami that struck the Roman empire.
Greek mathematicians had excluded all mathematics as unworthy of consideration, but for a strict subset of “Euclid’s Elements” (some of the present Euclid Elements were added later). The implementation of those discoveries were made by others (Indians, and to some extent, Iranians and Arabs).
It turned out that these more practical mathematics, excluded by Euclid, because they were viewed as non rigorous and primitive, led to deeper and more powerful insights.
The irony was that Euclid’s Elements, in the guise of rigor, were using an axiom that was not needed, in general, the parallel axiom. That axiom, by supposing too much, killed the imagination.
I suggest nothing less happening nowadays, with the Axiom of Choice: it’s one axiom too far.
Up to a recent time, if one was not a Supersymmetric (SUSY) physicist, it was impossible to find a job, except as a taxi cab driver. There was a practical axiom ruling physics: the world had got to be supersymmetric.
Now the whole SUSY business seems to be imploding as the CERN’s LHC came up empty, and it dawned on participants that there was no reason for an experimental confrontation in the imaginable future… I have studied SUSY, and I have a competitive theory, where there are two hints of experimental proofs imaginable (namely Dark Energy and Dark Matter).
I said the AC was one axiom too far, but actually I think infinity itself is an axiom too far. I exposed earlier what’s wrong with the 25 centuries old proof of infinity (it assumes one can use a symbol one cannot actually evoke, because there is no energy to do so!).
The geocentric astronomy ruled from Aristarchus of Samos (who proposed the heliocentric system, 3C BCE) until Buridan (who used inertia, that he had discovered to make the heliocentric system more reasonable; ~1320 CE; Copernic learned Buridan in Cracow, Poland). It could be viewed as an axiom.
Hidden axioms are found even in arithmetic, for example the Archimedean Axiom was used by all mathematicians implicitly, before Model Theory logicians detected it around 1950 (it says, given two integers, A and B, a third one can be found, D, such that: AD > B; if not fulfilled one gets non-standard integers).