mikeiavelli

Wow, Mister van Oortmerssen himself! I was a big fan of Bla's first-class environments! You're a legend!!

What syntax did you consider non-trivial? The summation sign? The transpose superscript? The use of greek letters? (beta, rho here) This is relatively basic notation for linear algebra, which I guess is the prerequisite for the reader.

I only skimmed it, but I want to point out to the author that in his slope equation, the slope is denoted by m, not x.

But yes, using the Laplacian seems natural as a metric for what he wants to measure. Coming from a physics background though, it felt weird being explained in terms of pixels in a picture what a laplacian is...

From the paper "Pólya’s conjecture for Euclidean balls" https://dms.umontreal.ca/~iossif/polya.pdf

"The celebrated Pólya’s conjecture (1954) in spectral geometry states that the eigenvalue counting functions of the Dirichlet and Neumann Laplacian on a bounded Euclidean domain can be estimated from above and below, respectively, by the leading term of Weyl’s asymptotics. Pólya’s conjecture is known to be true for domains which tile Euclidean space, and, in addition, for some special domains in higher dimensions. In this paper, we prove Pólya’s conjecture for the disk, making it the first non-tiling planar domain for which the conjecture is verified. We also confirm Pólya’s conjecture for arbitrary planar sectors, and, in the Dirichlet case, for balls of any dimension. Along the way, we develop the known links between the spectral problems in the disk and certain lattice counting problems. A key novel ingredient is the observation, made in recent work of the last named author, that the corresponding eigenvalue and lattice counting functions are related not only asymptotically, but in fact satisfy certain uniform bounds."

Maxwell was deeply influenced by Faraday's work but he was not his student: he first noticed Faraday’s work while he was at Cambridge in the 1850s. He found a way to translate Faraday’s experimental findings (expressed as the interaction of "lines of force") into a mathematical model (with fields), also taking into account Coulomb's and Ampère's laws. Maxwell then extended that model to take into account the conservation of charge / the continuity of currents.

While Maxwell’s work was inspired by Faraday's, it was also built upon the contributions of many other scientists (Coulomb, Ampère, Thomson, Neumann, Lenz, ...)

...but then, what material did you expect as a beginner?

"All you need are better titles!"

> Wolfram stumbled upon something quite profound that (from my perspective) is under appreciated by the majority of working physicists and mathematicians: Computation arises from very simple axiomatic systems

I disagree that his work offers profound new insights or that it has gone underappreciated.

Many mathematicians have explored the connections between computation, logic and formal systems for decades. I also take issue with the assertion that physicists have somehow "forgotten" or failed to appreciate these ideas within computer science. Many physicists are well versed in computational and information-theoretic concepts. The idea that simple rules can give rise to complex emergent behaviors is far from a new revelation.

Furthermore, Wolfram does not give due credit to this rich body of prior work. He presents his perspectives as groundbreaking discoveries, without properly acknowledging similar perspectives developed by others. This is not how science works.

(Emergent systems I guess.)

Have you seen the stuff he "works on" lately? What a pretentious use of pseudo-scientific jargon that means absolutely nothing. If someone piles on layers upon layers of concepts over many years, but nobody else uses those concepts, this is not science, not even a "new kind" of science.