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Epistemology - m a t h 4 w i s d o m - g m a i l
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Introduction E9F5FC Questions FFFFC0 Software |
See: Math 美 - How is beauty related to symmetry?
- How is beauty related to entropy?
- How does mathematical beauty express that there is no sense of inside?
读物 - John Baez. A Quest for Beauty and Clear Thinking
- Source of mathematical beauty in physics - principle of least action - God's will: showing as little good will as possible. Whereas people show more good will. Jesus: "Give to the one who asks" and not more.
Christopher Alexander's principles of life and different kinds of opposites are relevant. Main points: - Centers arise from the "center".
- Centers (and structures) are preserved.
Look for: - The role of duality...
- The role of opening up ever new dimensions...
- Where we imagine the center - beyond us or among us. In our plane or beyond it?
- The sense of time - Mandelbrot set is given by the time for each point to settle down.
- The interaction with us - the simplex engages us, our imagination, in a way that the Mandelbrot set does not.
- Relate Mandelbrot set to the kneading process.
Matematikos grožis
One milestone (or miracle) in knowing everything is to generate all mathematical objects and truths in a unified way, but especially, generate all mathematical insights in a way that builds our intuition. The relevant outlook will, I imagine, be in terms of beauty. I mean that we may think of mathematical insight as guided by the wish for beauty. Beauty - wholeness preserving transformations - natural generalizations
- coordinate free
- Wikipedia: In the 1990s, Jürgen Schmidhuber formulated a mathematical theory of observer-dependent subjective beauty based on algorithmic information theory: the most beautiful objects among subjectively comparable objects have short algorithmic descriptions (i.e., Kolmogorov complexity) relative to what the observer already knows.[17][18][19] Schmidhuber explicitly distinguishes between beautiful and interesting. The latter corresponds to the first derivative of subjectively perceived beauty: the observer continually tries to improve the predictability and compressibility of the observations by discovering regularities such as repetitions and symmetries and fractal self-similarity. Whenever the observer's learning process (possibly a predictive artificial neural network) leads to improved data compression such that the observation sequence can be described by fewer bits than before, the temporary interestingness of the data corresponds to the compression progress, and is proportional to the observer's internal curiosity reward.
- Math Beauty blog and Readings
- What is Good Mathematics? Terrence Tao.
- Alexander Bogomolny, Cut the Knot Manifesto "The peculiar beauty of Mathematics lies in deduction, in the dependency of one fact upon another. The less expected a dependency is, the simpler the facts on which the deduction is based -- the more beautiful is the result."
- Beautiful, simple proofs
Mathematicians are often guided by their sense of beauty. Beauty is a key to the big picture in mathematics as well as physics and philosophy. Questions about beauty - What is meant by beauty?
- What principles determine it?
- To what extent is beauty objective and subjective?
- How does beauty lead to mathematical insight?
Investigation: Collect examples In this section, we collect examples of beauty in mathematics. Richard Stanley’s post at Math Overflow, Most intricate and most beautiful structures in mathematics, produced the following list, ordered by votes received: - The absolute Galois group of the rationals
- The natural numbers (and variations)
- Homotopy groups of spheres
- The Mandelbrot set
- The Littlewood Richardson coefficients (representations of Sn etc.)
- The class of ordinals
- The monster vertex algebra
- Classical Hopf fibration
- Exotic Lie groups (See the Scientific American article about A.Garrett Lisi’s application of E8 to physics: A Geometric Theory of Everything)
- The Cantor set
- The 24 dimensional packing of unit spheres with kissing number 196560 (related to the monster vertex algebra).
- The simplicial symmetric sphere spectrum
- F_un (whatever it is)
- The Grothendiek-Teichmuller tower.
- Riemann’s zeta function
- Schwartz space of functions
Below we gather more examples, both basic and advanced: - e^2pii + 1 = 0
- more generally, Euler’s formula: e^2piix = cos(x) + i sin(x)
- the unique decomposition of natural numbers into prime numbers
- Euler’s polyhedron formula V - E + F = 2
- the classification of the Platonic solids
- the relationship between a polynomial and its graph
- binomial theorem and Pascal's triangle
- elementary proofs of the Pythagorean theorem (such as: "four times a right triangle is the difference of two squares" - the basis for infinitesimal rotation).
Investigation: Analyze examples In this section, we analyze the examples collected above to consider: - In what sense are they beautiful?
- What makes them beautiful?
- What are the simplest examples of beauty?
- Which examples yield the most beauty for the least drudgery?
Investigation: Look for unifying principles or contexts. - Urs Schreiber notes that many of the beautiful structures relate to string theory.
- Relation between two completely different domains, especially dual, complementary domains.
Investigation: Compare with beauty in chess. Matematikos grožis žadina vaizduotę Tęsiu prieš dvejus metus skaitytą pranešimą apie kūrybos prasmę ir meno taisykles. Žavesio jausmą nusakysiu kaip paneigimą bjauresio sąlygų - vidinio gyvenimo. Užtat grožis yra tiesos atskleidimas išoriniame gyvenime. Dviem pavyzdžiais iš matematikos parodysiu kaip architekto Kristoferio Aleksandro gyvybės dėsniai įvairiais kampais nuduoda arba išsako kūrybinę prasmę. Neišsemiamas fraktalas Mandelbrot aibė magina akis, tačiau matematiko vaizduotę žadina simpleksas. Paprastojo simplekso paveika grįsiu dėsnius, kaip grožis žadina vaizduotę. Paaiškinsiu, kaip Mandelbrot aibė galėtų labiau sužavėti matematiką. 1) Gražuolė tampa vis gražesnė. 2) Nebylė vs. tūkstantis šypsenų. Grožis: išsisakymas: Dvasios trilypė meilė visiems: išsakytojas, išdėstymas, klausytojas. Akis (1 taškas); šypsena (1 tiesinė atkarpa) - tai Mona Lisa šypsena. Supažindinti su dviem pavyzdžiais Mandelbrot ir simplex, ir parodyti kaip jie tampa gražūs. Abiems pavyzdžiams pritaikyti Christopher Alexander gyvybės dėsnių lentelę, kurią prieš dvejus metus pristačiau. Simplex yra: - total order
- decomposition
- all six pertvarkymai?
Opposites: - square roots of -1
- 1 and -1
- point and center
Simplekse ryšį tarp taško ir židinio galime ištrinti - galime ištrinti tašką - ir tuo suglaudinti sandarą - ir tai yra gražu. Gražu, kad židinys nepažymėtas. Sieti su Christopher Alexander mintimis. Mandelbrot aibė nėra matematikui graži nes tai yra gražu akims o ne vaizduotei. Mandelbrot eiga irgi nėra graži nes tai tarsi atsitiktinė. Tačiau ji tampa graži, kada paaiškėja - jog tai išplaukia iš begalinės sekos
- jog tai nusakoma Catalan skaičių
- jog tai skaičiuoja kompiuterių generuojamą kalbą
- jog tai gali būti užkoduota kiekviename taške kompleksinių skaičių plokštumoje
Harvey Friedman nurodė kokie ypatingi yra įrodymai, kad kažkas neįmanoma: - impossible to square the circle
- the real numbers are not countable
- Goedel's incompleteness theorem
But these are not beautiful... because they are negative results, not constructive results. There are other results which are amazing because they go beyond contemporary axiomatics: - imaginary numbers
- Euler's tricks with divergent infinite series, the Riemann function
- infinitesimals
- sphere of radius i
- Ramanujan
These are pretty in some sense, but not beautiful. There is a sense of a method at work, but it is not grounded, not systematic. So what is beautiful is something between these extremes. It is a framework between the two which can ground the two types of results. What is beautiful is when the imagination can "see" the gist in either case. http://www.kurzweilai.net/how-bio-inspired-deep-learning-keeps-winning-competitions Sąmoningumas - savasties ženklas - nes savastis kartojasi - nuo kurios gali būti skirtumai būti paskaičiuoti. Suglaustinumas - compressibility - means a pattern exists - Christopher Alexander's recurrent activity evokes structure. (+2) Search for new patterns. Self-compression - want to compress oneself as a pattern compressor. (Consciousness +3) "Interesting" = derivative of beauty. Juokai remiasi glaudinimu. P.S. I forgot to say in my other letter that I'll be suggesting a talk at a philosophy (aesthetics) conference here in Vilnius, Lithuania, most likely to be accepted, about mathematical beauty. I appreciate thoughts on mathematical beauty (I suppose through a new thread). My main thought so far is that mathematicians (at least me) would typically not consider the Mandelbrot set as beautiful because you can only see it, you can't imagine it. Whereas Galois theory is beautiful because it empowers the imagination. So I want to explain what it takes for the Mandelbrot set to become beautiful for a mathematician. Also, I want to link in with architect Christopher Alexander's 15 principles of life. I'll bring up some thoughts that may be relevant for you and your art teachers and art students. I'm thinking through a talk that I will propose for an aesthetics conference here in Vilnius, Lithuania. I want to talk about mathematical beauty. So I'm wondering what I can say about that. But I'm thinking it would be good to use my work with the Mandelbrot set as an example. I think my main idea is that, from a mathematician's point of view, (at least my own), the Mandelbrot set is not beautiful. It's not beautiful because in math beauty is not what you see, but what you imagine. And I can't imagine the Mandelbrot set. It's just lots of noise. There's no melody. Whereas I can imagine an equilateral triangle and so it stands out amongst all of the triangles. I can play with it in my mind, watch it dance around. I can hum that tune. What's truly beautiful is Galois theory where you have this assurance that you can play around with a group of dynamic actions and that will correspond to a polynomial and its solutions. That is amazingly beautiful. There is order in the universe. Or the dualities that you notice with the Platonic solids. Or the fact that there can only be those solids and no others, that what matters are the number of edges and vertices and faces and whether they satisfy Euler's formula. That makes it seem like we have signs of a Why out there some where. But I will go through steps to show what can make the Mandelbrot set beautiful mathematically, step by step. The kinds of steps I wrote about that this is not some accidental set. And that it relates to lots of key things in math. And that (perhaps) every single point is actually encoding something meaningful, so that the whole complex plane is an analysis of all of the possible behaviors of automata. And furthermore if the intricate structure of the Mandelbrot set was visually displaying how those behaviors are related. Something like that would be awesomely beautiful. |

This page was last changed on September 07, 2023, at 12:44 PM