Bernado's Mathematical Universe

[SanteriSatama]

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.

We don't need real number proof of Fourier's theorem to listen digitally stored and rationally computed computational music. I'm very happy that this tech works and makes so much music available. Maybe and probably something analogical is required to actualize the experience, as well as the whole universe and duration.

[Squidgers]

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.

We don't need real number proof of Fourier's theorem to listen digitally stored and rationally computed computational music. I'm very happy that this tech works and makes so much music available. Maybe and probably something analogical is required to actualize the experience, as well as the whole universe and duration.

Whatever the fundamental reality is, it can be said to have both experiential parts, and measurable parts - and both of these are isomorphisms of what is ontological.

Do you agree with this?

[SanteriSatama]

Whatever the fundamental reality is, it can be said to have both experiential parts, and measurable parts - and both of these are isomorphisms of what is ontological.

Do you agree with this?

I agree until "Isomorphism", which is a challenging technical term. The foundational creative algorithm that generates uniqueness can't be easily defined in terms of isomorphism. Computational irreducibility that generates forms like rule 30 is not isomorphic to anything else, no deterministic pattern has been found that could predict ahead of actual computation.

[Eugene I]

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.

Well, no, because analytical math is based on the theory of real numbers. As an engineer I do use computed Fourier transforms obtained by simulators. But I also need Fourier and Laplace transforms in analytical form because that gives me much more insight into the system behavior and properties. And analytical forms are only obtainable though calculus and algebra which is all based on real number theory.

This situation is similar to physics: everyone admits that QM is inconsistent and incompatible with GR/SR, and it is a big problem from the point of view of the foundations of physics. But QM practically works very well and physicists use it everyday. If one day in the future a unified and consistent physical theory becomes available, everyone will be happy to adopt it, yet QM equations will still be widely used as a practical approximation. Mechanics engineers still use Newton laws because they practically work well enough, even though they are inconsistent from the point of view of foundations of physics and QM/SR.

Similarly, the real number theory is inconsistent and I applaud your (and other mathematicians) search for a more consistent foundation of mathematics. However, it has not happened yet, so we engineers still need to use math tools based on real numbers in the meantime. And even if times come when a more consistent foundation will be developed, real number theory and its calculus will still be widely used because it is simple and it practically works very well in spite of its foundational inconsistencies.

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.

I definitely agree, but with the reservations stated above. The most consistent mathematical theory is not necessarily the one which is the most practical and easy to use, and vice versa. So, it is possible that even if a cleaner foundation of math would be developed, people will still use the real numbers calculus because it is easier to use for practical purposes.

[Eugene I]

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.

That's a good point, and I admit that I do not have an answer. My position is rather agnostic here. From the point of view of analytical philosophy, I don't even know what the term "ontological" exactly means, so I use this term with a disclaimer that "I do not actually know what I'm talking about".

From the experiential/phenomenological perspective, all we know is that there is conscious experience of phenomena and forms, including mathematical meanings and ideas, that's a fact of our direct experience. Whether these ideas/meanings have any "ontological" status - I have no idea and have no way to either prove or disprove it.

[AshvinP]

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.

That's a good point, and I admit that I do not have an answer. My position is rather agnostic here. From the point of view of analytical philosophy, I don't even know what the term "ontological" exactly means, so I use this term with a disclaimer that "I do not actually know what I'm talking about".

From the experiential/phenomenological perspective, all we know is that there is conscious experience of phenomena and forms, including mathematical meanings and ideas, that's a fact of our direct experience. Whether these ideas/meanings have any "ontological" status - I have no idea and have no way to either prove or disprove it.

Needless to say, I disagree. What we know for certain is that there is ideating activity, and we can come to know the essence of that activity by considering our own ideating activity and its living transformations. We can then conclude that meanings are ontic and universal. Meaning is the fundamental essence of the Cosmos. It is the inward reality which the outward creation is always pointing to. I see no need for dual-aspect monism here, because idealism works just fine as a metaphysical framework which encompasses this living essence of ideating activity and meaning.

[SanteriSatama]

Well, no, because analytical math is based on the theory of real numbers. As an engineer I do use computed Fourier transforms obtained by simulators. But I also need Fourier and Laplace transforms in analytical form because that gives me much more insight into the system behavior and properties. And analytical forms are only obtainable though calculus and algebra which is all based on real number theory.

This situation is similar to physics: everyone admits that QM is inconsistent and incompatible with GR/SR, and it is a big problem from the point of view of the foundations of physics. But QM practically works very well and physicists use it everyday. If one day in the future a unified and consistent physical theory becomes available, everyone will be happy to adopt it, yet QM equations will still be widely used as a practical approximation. Mechanics engineers still use Newton laws because they practically work well enough, even though they are inconsistent from the point of view of foundations of physics and QM/SR.

Similarly, the real number theory is inconsistent and I applaud your (and other mathematicians) search for a more consistent foundation of mathematics. However, it has not happened yet, so we engineers still need to use math tools based on real numbers in the meantime. And even if times come when a more consistent foundation will be developed, real number theory and its calculus will still be widely used because it is simple and it practically works very well in spite of its foundational inconsistencies.

Many quantum physicists consider the current mathematical and logical foundation of the theory highly problematic, and are looking for a different approach, a better theory of math.

When I say coherent, I mean coherent, not consistent. Posing questions based on bivalent logic leads to foundational undecidablity in many contexts, so LNC can't be a global foundational axiom of coherent foundations.

I agree that most practical applied math does not necessarily correlate with most coherent foundations, but math is also an open and evolving system, so who knows what we manage to cook up. Fragmentation into computation theory, continuous geometry and set theory is not a healthy situation, and coherent and communicative foundation could prove also very productive in terms of applied math.

People were doing calculus also before analytic geometry aka coordinate geometry. Coordinate geometry has its benefits, but also costs - e.g. equilateral triangle does not exist in rationally valued Cartesian plane. Einstein was aware of the philosophical problems of "coordinate ontology", when he spoke about necessity of coordinate invariance, but with standard formulations of standard theories being highly dependent from real complex plane, the situation has not exactly improved since Einstein's philosophical doubts against coordinate ontology of coordinate geometry.

Taking holistic philosophical distance to analysis, the etymological meaning of the word has semantic relation with partition. Partition in the mereological sense, not the modern definition based on natural numbers. Continuous geometry and mereological partition (and fractions in that sense!) predate metaphysical theory of natural numbers. Egyptian fractions are called such because they thought mereologically, the "one" in the numerator as continuum and denominator values as mereological partitions of continuum.

In our era we are - however slowly - coming back to our senses from the modernist trip into reductionism - materialistic reductionism which was based on reduction to metaphysical quantification ("There exists unit of measurement!") which became absurd point-reductionism about century ago when Cantor and Hilbert made their post-modern linguistic turn of formalism and logicism (the latter failing almost immediately thanks to Gödel and Turing).

Foundational thinking in mathematics has been dragging behind variety of holistic philosophies that have emerge after the failure of modernist metanarrative of materialistic reductionism. There's also loads of cumulative evolution of mathematics, which does not foundationally depend from heuristic methods of formalism or coordinate geometry analysis. Wolfram's key finding, IMHO, is that only repetition and nesting are computationally reducible. Nesting is by definition a mereological part-whole relation, and does not depend from metaphysical postulation of existential quantification. Thus it seems very natural that a more coherent foundation could be built from the mereological perspective.

Maybe a mereological theory of Fourier-partition could be developed based on Stern-Brocot type structures and Ford circles?

[Squidgers]

Whatever the fundamental reality is, it can be said to have both experiential parts, and measurable parts - and both of these are isomorphisms of what is ontological.

Do you agree with this?

I agree until "Isomorphism", which is a challenging technical term. The foundational creative algorithm that generates uniqueness can't be easily defined in terms of isomorphism. Computational irreducibility that generates forms like rule 30 is not isomorphic to anything else, no deterministic pattern has been found that could predict ahead of actual computation.

I'm using "Isomorphic" here in a similar way to saying something is "the same thing represented (or interpreted) in a different way."

Our experiences are representations of reality, as our models are representations of reality. Yet neither is reducable to the other, and our knowledge and understanding of reality is more complete with both.

[Squidgers]

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.

That's a good point, and I admit that I do not have an answer. My position is rather agnostic here. From the point of view of analytical philosophy, I don't even know what the term "ontological" exactly means, so I use this term with a disclaimer that "I do not actually know what I'm talking about".

From the experiential/phenomenological perspective, all we know is that there is conscious experience of phenomena and forms, including mathematical meanings and ideas, that's a fact of our direct experience. Whether these ideas/meanings have any "ontological" status - I have no idea and have no way to either prove or disprove it.

Needless to say, I disagree. What we know for certain is that there is ideating activity, and we can come to know the essence of that activity by considering our own ideating activity and its living transformations. We can then conclude that meanings are ontic and universal. Meaning is the fundamental essence of the Cosmos. It is the inward reality which the outward creation is always pointing to. I see no need for dual-aspect monism here, because idealism works just fine as a metaphysical framework which encompasses this living essence of ideating activity and meaning.

How are you defining "meaning" here? Is it a process (whitehead) or some kind of substance?

I always saw "idealism" as more of an umbrella term under which a "dual-aspect monism" would fall under. Is what you are talking about a monism, dualism, or something else?

[SanteriSatama]

Our experiences are representations of reality, as our models are representations of reality. Yet neither is reducable to the other, and our knowledge and understanding of reality is more complete with both.

Ah. I don't believe in representation theory, as I don't believe in objective realism.