[goldenkey]

It's equivalent to the measures you mentioned.

Here's a paper which discusses fractional degrees of freedom: Effective degrees of freedom of a random walk on a fractal by AS Balankin · 2015 · Cited by 42 — This allows us to define the fractional dimensional space allied ... number of effective dynamical degrees of freedom on the fractal

https://pubmed.ncbi.nlm.nih.gov/26764671/

To hammer the relationship home, consider the holographic principle, which is based on observation of black holes, and states that our reality only needs 2 spatial dimensions instead of 3. Both Hawking and Susskind have eluded to this being the case solely because of the symmetries in the laws of physics. The symmetries cause a massive redundancy/pattern in the field values over the 3d space, such that we should theoretically be able to predict the state of the entire field given only the values at two-thirds of the volume.

Therefore, you can imagine a 2d surface which contains the state of our universe, and some kind of computational (possibly geometric/algebraic) projector, which understands the redundancies, reads the 2d surface, and renders a sparse 3d volume. In the case of our universe, the projection operation might be extraordinary complex, requiring a deep understanding of the laws of physics and the redundancies they induce into the underlying state that they operate on.

https://en.wikipedia.org/wiki/Holographic_principle

[hellbannedguy]

I looked at you information, and your quote interesting.

If you happen to come back, what do you mean by this partial quote,

"Hardy couldn't have been more wrong about the innocence of pure mathematics."

I'm just getting more interested in math the older I get.

Thanks in advance.

[greyhair]

Math is more interesting, purely as math, as I get older. I found the same is true for history as well. The trick is to find good history authors for the particular time you are interested in. Same with math.

I graded very well in math through college, but later in life, I went back and explored more how all the concepts relate. In college, you are sort of fed calculus through a fire hose, and you just have to 'accept it' and move on. And you are left wondering, how were these ideas, these conclusions, reached? Until you go back through and see the long history of infinite series and see the various attempts to codify solutions. The problem is, as a student, you cannot possibly spend that much time deriving the whole solution from scratch and still hope to finish a degree in four years. As Carl Sagan said “If you wish to make an apple pie from scratch, you must first invent the universe.”

Which is why you can never stop learning. I am in my mid 60s, and I still learn something new regularly. Something big at least once a year, something smaller at least once a month. Never stop learning.

[goldenkey]

I meant that mathematics has been used for nuclear bombs, breaking cryptography, and other nefarious purposes. Physics is a sort of applied mathematics, where reality is the testing ground for models created on paper.

It was originally thought that Einstein's theory of special relativity was just a cool mathematical idea. But it was much much more. Mathematics has consequences that are so deep that even now it is rearing its head in AI..differentiable loss functions.