Robbert Dijkgraaf - How is Mathematics Truth and Beauty?
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- Опубликовано: 16 сен 2024
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What is it about mathematics that mathematicians employ the language of philosophy to speak about “truth” and the language of art to speak about “beauty”? What makes mathematical propositions true? What makes them beautiful. Conversely, can mathematical propositions be true without being beautiful and/or be beautiful without being true?
Robbert Dijkgraaf is a theoretical physicist and string theorist. He is professor at and director of the Institute for Advanced Study in Princeton, New Jersey, and a tenured professor at the University of Amsterdam.
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Best discussion on the topic so far
Dear Robert Lawrence Kuhn,
In mathematics, there is excellent consensus (there is not unanimity, but I would put less than 1% the number of mathematician “dissidents”) about the assumptions that must be assumed as starting point. Those assumptions are called Zermelo Fraenkel set theory with Choice, and it is abbreviated as ZFC. That theory has about 100 years. Until now, any physical phenomenon or experiment force to require taking, even one different assumption than those 9 axioms that conform ZFC. And I firmly believe will never be such a physical requirement. What have those 9 axioms that fit so good with the reality? It is a deep and very interesting mystery.
The opposite would be: There are mathematical facts logically based in ZFC that are not true in the physical word? The answer is YES! One of those facts is one of those 9 axioms that goes, “There is a set with infinity elements.” However, there is another interesting mathematical fact that is not true in the physical world: Given a point c (the center) and a number r (the radius), a sphere S in mathematics is the set of all points of the space that are at a distance from c, equal or less than r. In the physical world, a sphere is something such as billiard pool balls. There is a theorem (it is a well-proven fact) in mathematics called Banach-Tarski paradox that says that “S can be decomposed into a finite number of point sets (a finite number of pieces) and reassemble with those pieces two balls identical to the original.
Regards,
Carlos E. Freites
Well that depends if you accept the axiom of choice
I feel in this discussion too much weight is given to the universe we happen to inhabit. Mathematics does not exist "in" a universe. It exists on a deeper more fundamental level. It is intriguing that our minds can fathom this level.
I believe that in order for something to be natural there should be mathematics that describe that process. The real question is whether we have discovered those mathematics or not, and if we are sufficiently intelligent and creative to understand that.
This channel is beauty;)
(perceived) reality and its mathematical structure are what they are because of INEVITABILITY.
how could you possibly have any existence at all, in any Universe/realm/domain, without any consistent logical structure?
Logically consistent structure is a fundamental ingredient of existence.
and could the laws of that reality be 'chosen' any different? not sure ... because of the restrictions of our locality and our limited cognition we only see a part of reality of course, who knows.
Dijkgraaf is awesome by the way.
Well why can there not just be nothing at all? That seems mathematically consistent. Yet here we are. I feel a strange kind of satisfaction from the fact that anything exists. Feels like a big victory over nothingness. Surely that must mean something?
@@reneahn5908 absolute nothingness has no structure, no time, no dimension, nothing ... i would call it UNexistence, or UNreality.
Perhaps more interestingly: absolute nothingness lacks any and all laws, so a whole Universe could spawn form it, as long as the parts of the spawned Universe describe each-other.
again the spawning of 1 or multiple Universe(s) from a theoretical absolute nothingness, seems INEVITABLE to me.
(nothingness is inherently unstable)
@@rjwelsinga Interesting approach. So does this mean you assume that the apparent regularity is emergent? From something akin to chaos?
@@reneahn5908 hmm not sure if i mean emergent and/or chaos, good question :)
absolute nothingness is not equal to chaos, nor does it 'spawn' true chaos ... nothingness inevitably 'spawns' a structured Universe, from 'the get-go'.
(when thinking about these matters, language is a super tricky thing: there would be no time 'in' nothingness, nor would there be an 'in', nor would there be 'being'.)
I am a illogical unfortunate who doesn't see truth and beauty in mathematics. From my earliest school days, math (and related subjects) were presented in a manner that caused tension, confusion, and nervousness in me. If I asked a question, I would immediately be confronted with an attitude of "Well, let's back up for the slow poke". I noticed some of my school mates had similar experiences. None of my math/physics/chemistry teachers could ever inspire the interest that I had for history, literature, or languages (or even biology). I AM however, extremely thankful for the scientists and physicists who keep progress rolling along with their expertise.
For mathematics to function we need three conditions, open space, quantum building blocks and balance of energy between them. By open i mean space that doesn't lean on any specific direction, like here on Earth where we're in a constant free fall to the center of massive sphere. Quantized mean all same parts must be equal, quarks or electrons doesn't have any individual distinctions, they are all exact copies as far as we understand them. And balance means things constantly exchange energy potentials, there can be only so much charge between particles since energy can't come from nowhere.
Truth is, our solar system is in a free fall towards center of galaxy and it feels attraction from galaxy cluster. Quantum particles exist in different reference frames and contain potentials like acceleration or other forms of energy. Third one is the worst, every particle is coupled with every other particle in entire universe, everything is universally connected.
Just to remind us what is the scale we're talking about debating philosophy and aesthetics of mathematics, it's a giant mess, but it works without a single glitch. Because mathematics.
Is there something deeper than space...
@@rubiks6 Wouldn't like to argue with Mr. Feynman, I'm not worthy, but same could be said for all massless particles in the universe. Everything with no mass travel at light speed, where time supposedly stop and can't every change. This is why we can see never changing past looking at some distance.
Kudos -- 444 Gematria -- 🗽
Beauty? Try this..... e^(i pi) + 1 = 0 !! Euler’s equation. Now THAT.... is BEAUTIFUL my friends! It suggests an infinite intelligence behind our reality. Maybe not..... but .... DAMN...... that is beautiful!
second
"Mathematics is true" is not Mathematics at all..
first
"Nature is deeply logical"... how did nature acquire this logic??
It's tautologically true that nature has that logic
Conservation of energy.
Mathematics from subconscious mind of cosmos which is part of God divine being.