Bernado's Mathematical Universe

[Squidgers]

Form and content:

[Squidgers]

And a link to Euler

https://www.deviantart.com/woodmath/art ... -268936785

[SanteriSatama]

And a link to Euler

https://www.deviantart.com/woodmath/art ... -268936785

etc.

[Squidgers]

And a link to Euler

https://www.deviantart.com/woodmath/art ... -268936785

etc.

You might get a kick out of this (or annoyed)

https://www.academia.edu/35451881/The_L ... nit_Sphere

[SanteriSatama]

You might get a kick out of this (or annoyed)

https://www.academia.edu/35451881/The_L ... nit_Sphere

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.

If you are interested, try comparing the article you linked, and Norman's presentations of Famous Math Problems 21a-d and 22a-d. What differences do you find, what benefits and caveats in the respective approaches?

https://www.youtube.com/c/njwildberger/playlists

There's lots to digest and it's not easy stuff (at least for poor me), so I have no definitive opinion on whether Norman's approach works and how well. Take a look, maybe you'll find it fun.

[Eugene I]

Interesting discussion, although I could not read all 9 pages of it.

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, tedious and ugly. Most of the advances of the modern mathematics can be attributed to the achievements of the analytical and formal mathematics. In a way, continuous/analytical method is a compromise: it is more powerful because it uses certain mathematical tools that are more problematic (less logically consistent - I agree with Santeri here), yet practically more powerful and efficient. Demanding the absolute consistency (like in computational mathematics) is a purist approach that would leave us with much smaller and weaker set of tools and would make mathematical progress much slower. Yet, the discrete and computational mathematics has their own practical advantages in certain areas where they are better applicable and more efficient. So, from the practical point of view, the discrete/computational and analytical/continuous approaches are simply different practical methods (or thinking algorithms) for achieving certain mathematical results. The question is not which one is right and which is wrong, but which one is more appropriate and more efficient in every particular practical case.

The second aspect is philosophical. IMO the issues discussed here are very similar to issues in physicalism when scientists ascribe ontological realities to the abstractions of scientific physical models (such as fields, particles etc). In other words, issues arise when we believe that the scientific models tell us what reality IS, not only what reality DOES.

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.

So, the problem here is actually not in the mathematical ideas of real numbers, infinitesimal points or uncountable infinities per se, but in our belief that they represent the reality in a certain ontological way, in our belief that they tell us are what reality IS. Once we say goodbye to such naive ontological beliefs, we can use all the practical power of the analytical and formalist mathematics without running into any philosophical problems or paradoxes.

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.

[SanteriSatama]

Nice to hear from you and good to have you back!

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.

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...

Proof theory shifted gear after Gödel and Turing debunked Hilbert's program, and the Curry-Howard correspendence (which has classical, intuitionist and category theory formalisms) became a game changer. Wolfram's hypergraphs seem to be another major step linking deep structures of computational proof theories to various areas of mathematics and physics - the hypergraph presentation of Euclid's Elementa was a marvel to see, a whole new level of mathematics.

Based on fairly simple and comprehensible notions of computational reducibility and irreducibility, we seem to be moving towards the kind of proof theory, where actual computation simply proves itself, and we can deconstruct Wittgenstein style the arbitrary language games of mathematics and questions and conjectures based on those. And as computation self-actualizes as computation happens, whether by human computers or machines, and not excluging other spiritual math beings, computational languages very much act as they compute, in accordance with Wittgenstein's philosophy of language as action instead of theoretical aboutness-relation.

The big questions are Wigner's question, empirism of quantum potential, and empirical continua of time and causality when (not where!!!) measuring and computation occurs. The Wild Fox Koan of Computation expands Buddhist "now" into Bergson-duration.

Demanding the absolute consistency (like in computational mathematics)

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.

It's formalism which historically demands both LNC and LEM, but fails to satisfy either. Formalism is not a very Nagarjuna style catuskoti philosophy of mathematics... should mathematical philosophy of coherence exclude and deny Buddhist logic? Should these fields be kept separate and dissassociated and mutually incoherent? Why would such compartmentalization be preferable to coherence?

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.

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?

our belief that they represent the reality in a certain ontological way, in our belief that they tell us are what reality IS.

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?

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.

Now we talk. I agree and rest my case.

[Eugene I]

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?

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.

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.

[Squidgers]

Demanding the absolute consistency (like in computational mathematics) is a purist approach that would leave us with much smaller and weaker set of tools and would make mathematical progress much slower.

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
as ontological.

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.

I agree, but also think that there are other options to consider.

I think it is probably the wrong question to ask if mathematics is ontologically real or not. Perhaps a better question (which I think you answer below) is: which parts of ontological reality could be anything like mathematics?

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.

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?

Perhaps music is a good analogy here. There is an "inside" to music - the experience of it. And a structure to the music - the waveform. You can't reduce music to the waveform, but the waveform is isomorphic to the experience.

This form and content, structure and experience or objective and subjective aspect isn't some duality but a dual aspect monism - both representations of the one.

[SanteriSatama]

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.

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.

Engineers don't use "real numbers" of set theory. Every approximation ever computed by an engineer is a rational number, a decimal number with few digits in the form n+n/10+n/100+n/1000 etc. Stevin, who invented decimal numbers, was an engineer.
So, this discussion is not about trying to take away rational approximations and floating point computations from engineers. Engineers have no use for non-computational math, the computable and practical version of Fourier analysis does as computation does. This is about foundations of pure mathematics, trying to clean up a horrible conceptual mess and think philosophy of mathematics as well as we can. Proof theory is highly theoretical beast and mainly a philosophical problem, not the primary worry of engineering.

When people call rational numbers of applied math "real numbers" of set theory, and think they are same thing, math as whole becomes incomprehensible and does not make any sense. People get turned away from the absurd jargon of standard math, thinking they are too dumb to comprehend. No, people are not dumb, the monk Latin of formalism and set theory is absurd. Do you know is the official formalist definition of "set" or "point"? Neither do I, nobody does, those are given as undefined and semantically empty "primitive notions". Euclid's definitions of 'point' make perfect sense: 1) Point has no part (merelologic definition) and 3) Point is end of a line. A discontinuity.

Making pure math foundations coherent and comprehensible is very important, because in our technological and mathematical sociology, we can't afford to be naive about math en masse.

PS: there are attempts to develop more coherent pure math theories of calclulus - which BTW goes back to Archimedes, who of course knew nothing of "real numbers". I'm aware of at least non-standard analysis and Norman Wildberger's algebraic calculus.