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It's both a toy and a tool. A tool for seeing the world as it is, beyond your immediate mental and physical capabilities. If you believe in god or something sim

It's both a toy and a tool. A tool for seeing the world as it is, beyond your immediate mental and physical capabilities. If you believe in god or something similar, it makes sense to see something divine in math.

PartiallyTyped 4y ago
What people find as "divine" is the "unreasonable" and "unexpected" effectiveness of mathematics in describing complexity and complex phenomena abstractly. Part of me thinks that we may be very close to reaching what the brain of the average mathematician can comprehend and grasp. After all, not all of us are Gausses, Newtons and Galois.
- auggierose 4y ago
  Math is not limited to what your brain can come up with on its own. The most important part of math, in my opinion, is that other people (or, other entities, if we include AI), can come up with insights (because of their superior brain power and/or expertise) you would never be capable of conceiving yourself, and package them up in a nice theory for you.
  - PartiallyTyped 4y ago
    While I do agree that mathematics are not limited by what either of us can conceive, there is a fundamental barrier in understanding that one may not be able to process due to sheer volume. A trivial example here, though a bit absurd, is 15000 page long proofs, or machine generated proofs that span hundreds of pages. It is one thing to conceive of something new, and another to fully grasp all that we have built as a species. Certain pieces of work required hundreds of mathematicians to verify, which to me suggests that we have hit the limit of what our brains can do.
    - auggierose 4y ago
      The proof of the Kepler conjecture has been formalised, because the original proof was just too large and contained computer calculations that mathematicians could obviously not verify just with their brains in any amount of time. So what? The high-level structure of the proof is quite understandable, and the statement of the Kepler conjecture is very easy to understand as well. What I am saying is that understanding something does not necessarily mean you understand it in all of its details down to the lowest level. And machines can help us understand the things we need to understand. But even with machines we will never understand everything in all of its detail down to the lowest level. We won't even know all the theorems of something as simple as Peano arithmetic. What you can do is not limited to what you can do without machine help, and just with your brain. I would say what we can do now will pale to what we will be able to do in the future, but of course this is speculation.
- qsort 4y ago
  I take the ESR view here: math is only unreasonably effective to describe your field because you have chosen to study a field that is effectively described by math. For example, consistent science-like theories of history are very suspicious to say the least. By and large, nature doesn't give a damn. E.g. the most natural and mathematically elegant way to describe space is euclidean geometry, which happens to be a "wrong" model. On some level, model selection is influenced by aesthetic considerations.
- bmacho 4y ago
  Hard problems has been estimated in math to be very hard, and they are falling far faster than ever.
7373737373 4y ago
With memory, seeing divinity in math probably requires active contemplation, since seeing simple formulas and geometric shapes repeatedly gets old fast > Many observers report they derive pleasure from discovering simple but novel patterns [...] The observer's learning process causes a reduction of the subjective complexity of the data, yielding a temporarily high derivative of subjective beauty: a temporarily steep learning curve. > Why are some musical pieces more interesting or aesthetically rewarding than others? Not the one the listener (composer) just heard (played) fifty times in a row without any noticable change. It became too subjectively predictable in the process. Not the weird one with completely unfamiliar rhythm and tonality. It seems too irregular and contain too much arbitrariness and subjective noise. The observer (creator) of the data is interested in melodies that are unfamiliar enough to contain somewhat unexpected harmonies or beats etc., but familiar enough to allow for quickly recognizing the presence of a new learnable regularity or compressibility in the sound stream: a novel pattern! Sure, it will get boring over time, but not yet. https://people.idsia.ch/~juergen/creativity.html https://people.idsia.ch/~juergen/creativity.html
- auggierose 4y ago
  Not sure I understand your comment, but I can assure you, math involves a lot of active contemplation.
  - 7373737373 4y ago
    Seeing a Tetrahedron, Euler's identity or a Klein bottle for the first time is much better than the 50th time, because you remember having seen it that many times by then and are not surprised by their meaning and properties anymore. Since you remember, having it remain meaningful requires actively exploring more properties of the object and its relations to other objects Some simple objects like the yin and yang symbol may remain more easily meaningful because they can have so many interpretations, compare this to say, a simple line, which is so ubiquitous its meanings are "too" unrestricted. Something like a Lie group may be too complex and too hard to understand for most to even consider it divine
    - 4y ago

PartiallyTyped 4y ago

What people find as "divine" is the "unreasonable" and "unexpected" effectiveness of mathematics in describing complexity and complex phenomena abstractly. Part of me thinks that we may be very close to reaching what the brain of the average mathematician can comprehend and grasp. After all, not all of us are Gausses, Newtons and Galois.

auggierose 4y ago
Math is not limited to what your brain can come up with on its own. The most important part of math, in my opinion, is that other people (or, other entities, if we include AI), can come up with insights (because of their superior brain power and/or expertise) you would never be capable of conceiving yourself, and package them up in a nice theory for you.
- PartiallyTyped 4y ago
  While I do agree that mathematics are not limited by what either of us can conceive, there is a fundamental barrier in understanding that one may not be able to process due to sheer volume. A trivial example here, though a bit absurd, is 15000 page long proofs, or machine generated proofs that span hundreds of pages. It is one thing to conceive of something new, and another to fully grasp all that we have built as a species. Certain pieces of work required hundreds of mathematicians to verify, which to me suggests that we have hit the limit of what our brains can do.
  - auggierose 4y ago
    The proof of the Kepler conjecture has been formalised, because the original proof was just too large and contained computer calculations that mathematicians could obviously not verify just with their brains in any amount of time. So what? The high-level structure of the proof is quite understandable, and the statement of the Kepler conjecture is very easy to understand as well. What I am saying is that understanding something does not necessarily mean you understand it in all of its details down to the lowest level. And machines can help us understand the things we need to understand. But even with machines we will never understand everything in all of its detail down to the lowest level. We won't even know all the theorems of something as simple as Peano arithmetic. What you can do is not limited to what you can do without machine help, and just with your brain. I would say what we can do now will pale to what we will be able to do in the future, but of course this is speculation.
qsort 4y ago
I take the ESR view here: math is only unreasonably effective to describe your field because you have chosen to study a field that is effectively described by math. For example, consistent science-like theories of history are very suspicious to say the least. By and large, nature doesn't give a damn. E.g. the most natural and mathematically elegant way to describe space is euclidean geometry, which happens to be a "wrong" model. On some level, model selection is influenced by aesthetic considerations.
bmacho 4y ago
Hard problems has been estimated in math to be very hard, and they are falling far faster than ever.

7373737373 4y ago

With memory, seeing divinity in math probably requires active contemplation, since seeing simple formulas and geometric shapes repeatedly gets old fast > Many observers report they derive pleasure from discovering simple but novel patterns [...] The observer's learning process causes a reduction of the subjective complexity of the data, yielding a temporarily high derivative of subjective beauty: a temporarily steep learning curve. > Why are some musical pieces more interesting or aesthetically rewarding than others? Not the one the listener (composer) just heard (played) fifty times in a row without any noticable change. It became too subjectively predictable in the process. Not the weird one with completely unfamiliar rhythm and tonality. It seems too irregular and contain too much arbitrariness and subjective noise. The observer (creator) of the data is interested in melodies that are unfamiliar enough to contain somewhat unexpected harmonies or beats etc., but familiar enough to allow for quickly recognizing the presence of a new learnable regularity or compressibility in the sound stream: a novel pattern! Sure, it will get boring over time, but not yet. https://people.idsia.ch/~juergen/creativity.html https://people.idsia.ch/~juergen/creativity.html

auggierose 4y ago
Not sure I understand your comment, but I can assure you, math involves a lot of active contemplation.
- 7373737373 4y ago
  Seeing a Tetrahedron, Euler's identity or a Klein bottle for the first time is much better than the 50th time, because you remember having seen it that many times by then and are not surprised by their meaning and properties anymore. Since you remember, having it remain meaningful requires actively exploring more properties of the object and its relations to other objects Some simple objects like the yin and yang symbol may remain more easily meaningful because they can have so many interpretations, compare this to say, a simple line, which is so ubiquitous its meanings are "too" unrestricted. Something like a Lie group may be too complex and too hard to understand for most to even consider it divine
  - 4y ago