So I'm reading Penrose, and all of sudden he explodes into excitement like a schoolgirl, fawning over complex numbers because they are "magical" and perform "miraculous" things, further spilling exclamation marks in the surrounding paragraphs about how he's only scratched the surface of "number magic!"
https://www.physicsforums.com/threads/are-complex-numbers-magical.68277/
So I'm reading Penrose, and all of sudden he explodes into excitement like a schoolgirl, fawning over complex numbers because they are "magical" and perform "miraculous" things, further spilling exclamation marks in the surrounding paragraphs about how he's only scratched the surface of "number magic!"
There's more magic in complex analysis than in complex arithmetic. — jgill
Penrose’s [sees complex numbers structures] as the primary fabric of the Universe. … Penrose uses this designation in regards to the mathematical structure of his main research focus in quantum gravity and twistor theory (Penrose, 1967).
The main idea that Penrose articulates concerning the complex structures is that the results that require arduous computations with the use of the of the real structures are obtained “for free” as the complex structures come into play.
Penrose does not hesitate to push this magic to the extreme as he openly states:
Nature herself is as impressed by the scope and the consistency of the complex-number system as we ourselves, and has entrusted to these numbers the precise operations of her world at its minutest scales (Penrose, 2005, p. 73).
I have, in fact, done a fair amount of research involving Moebius (or linear fractional) transformations (that crop up in these physics discussions), but going the pure analysis direction in the complex plane rather than the geometries coming off the Riemann sphere. — jgill
Gisin has traced the problem of the block universe to an unexpected source: mathematics itself. He notes that a century ago, mathematicians were split about how to describe numbers whose decimal digits trail off into infinity.
On one side, led by David Hilbert, were those who thought every such real number was a “completed object” that exists in its entirety, timelessly, even though it has infinite digits. On the other side were ... “Intuitionistic mathematics.” In intuitionistic mathematics, numbers are created over time, with digits materializing in succession.
Spoiler: Hilbert’s side won. “Time was expulsed from mathematics” and as a byproduct, from physics, too, writes Gisin (Gisin, 2020a). But, he wondered, what would happen if physics were re-written in the language of intuitionistic mathematics? Would time become “real” again?
Gisin asks us to consider “chaotic” systems, in which two almost-but-not-quite-identical starting points evolve to wildly different end points. A classic example is the weather. ... whether you are talking about the weather, the evolution of the entire universe, or just your choice of what to have for dinner tonight, it is distressing to think that the future is already fully determined.
Of course, physics is not required to make us comfortable. But Gisin points out that intuitionistic mathematics could offer a natural way out of the deterministic lockup. In the intuitionistic view, numbers—like the values of pressure, wind speed, humidity, and so on—do not have definite values from the get-go, but rather develop over time, with randomly-generated digits unscrolling as time passes. This mathematical treatment allows for a universe in which time actually flows, events truly happen, and randomness and chance are injected moment by moment.
https://www.templeton.org/wp-content/uploads/2021/11/Time-by-Kate-Becker-1.pdf
... from the very beginning of their investigations, the pathways of Cantor and Peirce are opposite to one another; while Cantor and many of his successors in the 20th century try systematically to delimit the continuum, Peirce tries to unlimit it—to bring it nearer to a supermultitudinous continuum, not limited in size, truly generic in the transfinite, never totally actualizable.
Spoiler: Hilbert’s side won. “Time was expulsed from mathematics” and as a byproduct, from physics, too, writes Gisin (Gisin, 2020a). But, he wondered, what would happen if physics were re-written in the language of intuitionistic mathematics? Would time become “real” again?
Einsteinian relativity is what expulses time from physics. — Metaphysician Undercover
Relativity united time and space in a way that made more general sense. — apokrisis
So what are your thoughts here when one direction looks to track the "deep maths" of Nature and the other choice may be just unphysical pattern spinning? What do we learn if this is the case? — apokrisis
But am I right that you argue the complex plane has lessons in terms of the physics of chaos - patterns of convergence~divergence? — apokrisis
what would happen if physics were re-written in the language of intuitionistic mathematics? Would time become “real” again?
But Gisin points out that intuitionistic mathematics could offer a natural way out of the deterministic lockup.
Zoom in on your complex plane with its pattern of curl, and do you start to lose any sense of whether some infinitesimal part is diverging or converging? — apokrisis
One can ask again whether maths made the right pragmatic choice even if Peirce is the metaphysically correct choice? — apokrisis
Some thinkers assume that Nature is Mathematical (abstract values, sans meaning) while others believe that Nature is Mental (logic plus meaning). So, I suspect that the replies to your topic will divide along those lines. A purely mathematical universe just is, and must be taken for granted. That's why many scientists assume that Energy (cause) & Laws (logic) exist eternally, and need no explanation. But some scientists observe that the universe, that began from abstract Cause & Laws, has evolved animated self-aware beings with personal values & meanings. How is that possible?If somehow evolution has equipped us with mathematical minds, it is fair to hypothesize that the "book of nature is written in the language of mathematics" just because we see it that way. — Doru B
Most of the proposed answers I've seen, to the "invented vs discovered" question, seem to conclude that it's a little of both. The "unreasonable effectiveness of mathematics" in science indicates that Nature is in some sense fundamentally mathematical. But the history of math shows that humans using abstract "pure" mathematical principles, eventually find concrete practical applications for many of them.The question seems a correspondent of the most popular question “Was mathematics invented or discovered?” and relates to the nature of mathematics as well as to the philosophical problem of applicability of mathematics. However, there are anthropocentric and evolutionary features that the philosophical investigations on this topic have not focused on much: — Doru B
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