Something that kind of interests me; to the Pythagoreans mathematics was magic, involving mystical insight into the ultimate nature of reality. Similar ideas continued in Plato and those he influenced, with echoes down to the early modern rationalists. But Pythagorean mathematics was exceedingly primitive and limited. Modern logic and mathematics are vastly more powerful and sophisticated, and yet the Pythagorean mysticism has almost completely disappeared. Why were people more impressed with mathematics when it was more limited? I imagine novelty was a factor, and they say familiarity breeds contempt, but I have a couple of other hypotheses. First, there seem to be obvious places here and there in modern logic and mathematics where we seem to face choices, where it looks like a matter of “doing this is convenient for this purpose” rather than “this is the only way things could be.” This tends to weaken the idea that the fundamental nature of reality is being revealed. Second, there are parts of modern mathematics that are deeply weird (e.g. many things about how infinite cardinals work). I imagine there are people who find it hard to accept that those parts of mathematics at least could be describing any real world.
I’m certainly not arguing for returning to Pythagorean mysticism, but of the two reasons I propose why people might be ruling that out, I’m inclined to think the first reason looks fairly good while the second strikes me as highly suspect. I’m quite curious as to which has been most influential (or whether it’s some other factor or factors, perhaps familiarity really is the driving force, or perhaps external influences, say from hostility to mysticism in other fields, are important).
Tegmark level IV is the modern version of “Pythagorean mysticism” (it states that “the ultimate nature of reality” is math). So, there is no need to return to anything, we are already there. Well, some of us are, even on this forum. As for “the fundamental nature of reality”, there is no indication for it to be a real thing, the deeper we dig, the more we find. But yes, in the map/territory model, math helps build better maps of the territory, whether or not it is the “ultimate” territory. The “weird” parts may or may not be useful in building better maps, it’s hard to tell in advance. After all, the (freshly revealed) territory ls also weird.
Cantor who first did the first work on infinite cardinals and ordinals seemed to have a somewhat mystic point of view some times. He thought his ideas about transfinite numbers were communicated to him from god, whom he also identified with the absolute infinite (the cardinality of the cardinals which is too big to itself be a cardinal). This was during the 19th century so quite recently.
I’d say that much mysticism about foundational issues like what numbers really are, or what these possible infinities actually mean, have been abandoned by mathematicians in favour of actually doing real mathematics. We also have quite good formal foundations in terms of ZF and formal logic nowadays, so discussions like that do not help in the process of doing mathematics (unlike, say, discussions about the nature of real numbers before we had them formalised in terms of Cauchy sequences or Dedekind cuts).
I don’t think those attitudes are quite as gone as you seem to think although that might be the mind projection fallacy. It’s better understood, to the mystery and superstition are gone, but the senses of transcendent awe and sacred value are still there. Certainly, with Tegmarkian cosmology (that I tend to hold as axiomatic, in the literal “assumption++” sense), it’s very much the “one ultimate nature of reality”.
Something that kind of interests me; to the Pythagoreans mathematics was magic, involving mystical insight into the ultimate nature of reality. Similar ideas continued in Plato and those he influenced, with echoes down to the early modern rationalists. But Pythagorean mathematics was exceedingly primitive and limited. Modern logic and mathematics are vastly more powerful and sophisticated, and yet the Pythagorean mysticism has almost completely disappeared. Why were people more impressed with mathematics when it was more limited? I imagine novelty was a factor, and they say familiarity breeds contempt, but I have a couple of other hypotheses. First, there seem to be obvious places here and there in modern logic and mathematics where we seem to face choices, where it looks like a matter of “doing this is convenient for this purpose” rather than “this is the only way things could be.” This tends to weaken the idea that the fundamental nature of reality is being revealed. Second, there are parts of modern mathematics that are deeply weird (e.g. many things about how infinite cardinals work). I imagine there are people who find it hard to accept that those parts of mathematics at least could be describing any real world.
I’m certainly not arguing for returning to Pythagorean mysticism, but of the two reasons I propose why people might be ruling that out, I’m inclined to think the first reason looks fairly good while the second strikes me as highly suspect. I’m quite curious as to which has been most influential (or whether it’s some other factor or factors, perhaps familiarity really is the driving force, or perhaps external influences, say from hostility to mysticism in other fields, are important).
Tegmark level IV is the modern version of “Pythagorean mysticism” (it states that “the ultimate nature of reality” is math). So, there is no need to return to anything, we are already there. Well, some of us are, even on this forum. As for “the fundamental nature of reality”, there is no indication for it to be a real thing, the deeper we dig, the more we find. But yes, in the map/territory model, math helps build better maps of the territory, whether or not it is the “ultimate” territory. The “weird” parts may or may not be useful in building better maps, it’s hard to tell in advance. After all, the (freshly revealed) territory ls also weird.
Cantor who first did the first work on infinite cardinals and ordinals seemed to have a somewhat mystic point of view some times. He thought his ideas about transfinite numbers were communicated to him from god, whom he also identified with the absolute infinite (the cardinality of the cardinals which is too big to itself be a cardinal). This was during the 19th century so quite recently.
I’d say that much mysticism about foundational issues like what numbers really are, or what these possible infinities actually mean, have been abandoned by mathematicians in favour of actually doing real mathematics. We also have quite good formal foundations in terms of ZF and formal logic nowadays, so discussions like that do not help in the process of doing mathematics (unlike, say, discussions about the nature of real numbers before we had them formalised in terms of Cauchy sequences or Dedekind cuts).
I don’t think those attitudes are quite as gone as you seem to think although that might be the mind projection fallacy. It’s better understood, to the mystery and superstition are gone, but the senses of transcendent awe and sacred value are still there. Certainly, with Tegmarkian cosmology (that I tend to hold as axiomatic, in the literal “assumption++” sense), it’s very much the “one ultimate nature of reality”.