Re: Bernado's Mathematical Universe
Posted: Tue Jul 20, 2021 6:40 am
Squidgers wrote: ↑Tue Jul 20, 2021 9:05 am And a link to Euler
https://www.deviantart.com/woodmath/art ... -268936785
You might get a kick out of this (or annoyed)SanteriSatama wrote: ↑Tue Jul 20, 2021 11:49 amSquidgers wrote: ↑Tue Jul 20, 2021 9:05 am And a link to Euler
https://www.deviantart.com/woodmath/art ... -268936785
etc.
Same basic intuitions from a quick look, but if and when you can do it fully computationally, of course that is preferable to transcendental functions. Norman was studying Lie-theory (continuous groups) before he lost faith in axiomatic set theory and realized that to make genuine progress, we need radical rethinking of foundations.Squidgers wrote: ↑Tue Jul 20, 2021 12:11 pm You might get a kick out of this (or annoyed)
https://www.academia.edu/35451881/The_L ... nit_Sphere
I can't agree that current standards of analytical/formal satisfy criteria of continuous. That's a major philosophical issue where also Buddhist philosophy and empirical insights are highly relevant - do we follow coherence theory of truth, and concatenate also empirical continuity in the truth conditions? Or do we take the formalist position and exclude empirical reality and truth conditions from foundations mathematics, and reduce mathematics to arbitrary language games, which leads to fragmentation and any hope of coherence and intuitivity and common sense communicativity of mathematics? I wonder what Nagarjuna would say on the issue...Eugene I wrote: ↑Wed Jul 21, 2021 8:23 pm There are two aspects in this issue. The first one is practical. If we look at discrete/computational mathematics vs continuous/analytical/formal, in most cases the latter has practical power by far exceeding the former. I encourage to take any advanced theorem proof of analytical mathematics and demonstrate that the same result can be achieved as elegantly and efficiently using discrete or computational approach. As an example, try to prove the Fourier theorem using only the tools of discrete/computational mathematics. Even if it might be possible, it would be exhaustingly long and ugly.
Does computational mathematics really demand consistency? Formal languages - as simple as Schönfinkel's Turing complete S and K combinators - are just simple rules of string manipulation. Just writing with a syntax. Various formal logics and their proof theories can be derived from combinators, but there's no necessity to do so. There are no logical axioms which formal languages should follow, in that sense they are not attached to any horn of the tetralemma.Demanding the absolute consistency (like in computational mathematics)
The real issue is that by their own axiomatic logical criteria, real numbers are absurd. The claim that real numbers can do basics arithmetics is blatant violation of the most basic syllogism:Similarly, we run in these endless arguments whether "real numbers are real" because we only consider two options: ether they exist as some ontological realities, or (if not) they should be dumped as nonsense.
It's much more serious. The physicalist-materialist belief in objective reality is highly unplausible, hence their representation theory of mathematics is worthless. Wigner's question and empirism of quantum measurement theory imply that mathematics does not represent reality, measurement decoheres and actualizes reality, or at least some aspects of reality. Does the materialist-formalist paradigm of mathematical physics decohere a good reality? Could we do better?our belief that they represent the reality in a certain ontological way, in our belief that they tell us are what reality IS.
Now we talk. I agree and rest my case.IMO it is not because the reality is ontologically mathematical, but because it is "functionally" mathematical: at least in idealism, Consciousness has fundamental ability to think and to manipulate meanings, and that includes mathematical meanings. Meanings and thoughts is what Consciousness (the Mathematician- the Creator - the Thinker) "does", while the meanings/thoughts are not something that ontologically exist independent of Consciousness, but are only aspects (forms) of Consciousness, while Consciousness itself is irreducible to its set of meanings/thoughts/forms and also includes formless aspects. So, speaking specifically about mathematical meanings, the Reality (as Consciousness) is irreducible to mathematical objects only, but they still do exist as forms/thoughts/ideas that Consciousness can create and manipulate.
I would call it "matehmaticalism". Physicalism is a belief that the ideas used in physical models ("fields", "particles") represent some "material" ontological realities. Likewise, "mathematicalism" is a belief that mathematical ideas, and specifically real numbers and uncountable infinities, represent some "ideal" ontological realities. And this is definitely absurd.SanteriSatama wrote: ↑Wed Jul 21, 2021 10:10 pm The real issue is that by their own axiomatic logical criteria, real numbers are absurd. The claim that real numbers can do basics arithmetics is blatant violation of the most basic syllogism:
A: Real numbers can do arithmetic.
B: Variables a and b represent real numbers.
So, according to the syllogism, referents of a and b should be able to do arithmetic, you'd think? But the probability that a and b are non-computable and non-demonstrable is 1!
Should philosophy of mathematics kept apart from rest of philosophy, apart from ethics which generally does not consider dishonesty a virtue?
...Does the materialist-formalist paradigm of mathematical physics decohere a good reality?
Considering what is ontological about mathematics doesn't have to subsume other formal approaches to mathematics. It would be a separate/new category. Having a fully tautological, complete and consistent system of mathematics wouldn't "look" like anything that has come before. E.g. in Eidomorphism numbers are considered to be thoughts, which are modeled around sinusoidal waves. So in a sense, it's more like a mathematical language put over what is ontological, rather than interpreting our known mathematics
I agree, but also think that there are other options to consider.Similarly, we run into these endless arguments whether "real numbers are real" because we only consider two options: ether they exist as some ontological realities, or (if not) they should be dumped as nonsense. In other words, ether the reality IS ontologically real numbers, or it is NOT, and in latter case real numbers has no relevance or usefulness at all.
Would you not still say that these forms/meanings/thoughts exist ontologically? Or does it have to exist independent of consciousness to be considered ontologically real for you?As to the original question of why the reality seems to be mathematical, IMO it is not because the reality is ontologically mathematical, but because it is "functionally" mathematical: at least in idealism, Consciousness has fundamental ability to think and to manipulate meanings, and that includes mathematical meanings. Meanings and thoughts is what Consciousness (the Mathematician- the Creator - the Thinker) "does", while the meanings/thoughts are not something that ontologically exist independent of Consciousness, but are only aspects (forms) of Consciousness, while Consciousness itself is irreducible to its set of meanings/thoughts/forms and also includes formless aspects. So, speaking specifically about mathematical meanings, the Reality (as Consciousness) is irreducible to mathematical objects only, but they still do exist as forms/thoughts/ideas that Consciousness can create and manipulate.
Pure math perspective is different from applied math, the sad history is that set theory has been an attempt to justify applied calculus with absurd theory of pure math. The result is horrible mess.Eugene I wrote: ↑Thu Jul 22, 2021 1:51 am To me, as an engineer, math is simply a set of thinking algorithms that help solving practical problems but do not need to have any relevance to ontological reality. Engineering is based on calculus and calculus is based on real numbers. No real numbers theory = no technology, so without real numbers theory we would still live in the medieval times. All electronics is based on the theory of Fourier and Laplace transforms that could not be developed would the idea of real numbers. So I don't care whether real numbers represent any ontological realities, or whether they are clean from paradoxes and inconsistencies, or whether they are computable or not. As long as they help to solve engineering problems they are good enough. They make no sense but they practically work. So, I am still an advocate of the real number theory simply because of its huge practical usefulness, as long as we don't take real numbers "ontologically" and "religiously".
But if computational mathematics can give me thinking algorithms and ideas that can be as useful and as easy to apply to practical problems as calculus and Fourier/Laplace transforms, I will be happy to dump the calculus with all its real number theory and switch to the alternative tools. But I just do not see it happening.