The n-Category Café

Thanks! My ideas on the background of “platonism” in mathematics, with some art and poetry. Barry Mazur remarked on that: “somehow I would have balanced that list (of “platonic” ideas in math, TR) with “things sought that turned out not to exist” (e.g., the squaring of the circle, trisection of a general angle by Euclidean construction methods) as further hints that a platonic mentality seems to be very energetically at work—in the minds of many mathematicians.” Maybe one could wonder about “platonic mirages” like Hilberts proof theory program.

Nice piece, David! Anyone who wants to go straight to it should click here.

Ulf needs a more intelligent spell-checker. He speaks of “rightest political views”, and the “a priory”.

The link to the EMS newsletter was a good idea!

S Y M M E T RY A Journey into the Patterns of Nature by Marcus du Sautoy
“The universe is built on a plan the profound symmetry of which is somehow present in the inner structure of our intellect.” by Paul Valery

Why, though, is symmetry so pervasive in nature? It is not just a matter of aesthetics. Just as it is for me and mathematics, symmetry in nature is about language. It provides a way for animals and plants to convey a multitude of messages, from genetic superiority to nutritional information. Symmetry is often a sign of meaning, and can therefore be interpreted as a very basic, almost primeval form of communication.

plato.stanford.edu/entries/pythagoras/
“Some of these earlier sources were heavily contaminated by the Neopythagorean view of Pythagoras as the source of all true philosophy, whose ideas Plato, Aristotle and all later Greek philosophers plagiarized. Nicomachus (ca. 50 – ca. 150 CE) assigns Pythagoras a metaphysics that is patently Platonic and Aristotelian and that employs distinctive Platonic and Aristotelian terminology … In the Pythagorean Memoirs, Pythagoras is said to have adopted the Monad and the Indefinite Dyad as incorporeal
principles, from which arise first the numbers, then plane and solid figures and finally the bodies of the sensible world”

“The tendency of people to see purpose in symmetry suggests at least one reason why symmetries are often an integral part of the symbols of world religions.”

SH: The ancient Greeks perceived patterns and attributed them to a Divine provenance, as also in the case of the divine proportion. Symmetry in nature and mathematics accounts for patterns ranging from the hexagonal honeybee cells to the way in which “Moonshine forms a way of explaining the mysterious connection between the monster finite group and modular functions from classical number theory.”

Can you hear the sound of Platonism’s drumbeat vanishing just as the mythological gods have perished when scientific knowledge has replaced primitive belief?!

Thomas wrote: “The sketch of the “platonic” mindset seems to fit esp. that of number theorists. An interesting suggestion I received is about Wilfried Owen’s poem “The naming of the parts” as illustration of emotions related to “naming” and the short stories of Tim OBrien “The things they carried” on that and a complex concept of “reality” similar to those of the ancient greeks.”

The idea of the Muses has been hear since at least the days of the ancient Greeks. The eye receives something like two million bits of information per second. Only about two thousand bits of information are passed onto consciousness as a basis for decisions. Perhaps the unconscious can process other conscious ideas and provide unusual creativity. Some people report creativity being passed on in dreams. I think this Muse idea (unconscious) makes a lot more sense than creative input received from an eternal realm of ideas existing beyond space and time. Muses don’t have to be actual personas, just functions of the unconscious. So perhaps everyone has that potential. But I don’t think one has to buy into the notion that everyone is networked into a Jungian Collective Unconscious. If there are archetypes, it is because they are part of the normal evolved brain information that babies are born with that allows them to learn without needing to be taught to learn, early, not language. And because someone writes a historical perspective doesn’t mean that they don’t reveal their own belief inclinations within that primarily historical account. No matter whether they had the intention to do so or not.

Here’s what Alain Connes says about mathematical realism. I find it quite congenial to my own way of thinking.

Q: In your conversations with Changeux you discussed mathematics and reality. Have you advanced in your thinking about this?

I have no doubt that mathematical reality is something which exists, that it exists independently of my own brain trying to see it, and has exactly the same properties of resistance as external reality. When you want to prove something, or when you examine if a proof is correct or not, you feel the same anguish, the same external resistance as you do with external reality. Some people will tell you that this reality does not exist because it is not “localized” somewhere in space and time. I just find this absurd and I adopt a diametrically opposed point of view: for me even a human being is better described by an abstract scheme than by a material collection of cells – which in any case are totally renewed and replaced over a relatively short period of time and hence possess less meaning or permanence than the scheme itself, which might eventually be reproduced in several identical copies……

If one wants to reduce everything to “matter localized somewhere” one soon meets a wall which comes from quantum mechanics and one finds that this reduction of the outer reality to matter is an illusion that only makes sense at intermediate scales but by no means at a fundamental level. Thus I have no doubt on the subtleness and existence of a reality which can be neither reduced to “matter” nor “localized”.

Here is a very nice article by Laurent Lafforgue sketching the “platonic mindset”. Philosopher’s ideas differ (that link explains Connes’ harsh remarks on that).

News on the “Plato code”…