3 ms·

If something can be proven by any method, that means that it is true. An elegant proof is often more illuminating, but it isn't precluded by a direct method. I

by ISL 9y ago

If something can be proven by any method, that means that it is true. An elegant proof is often more illuminating, but it isn't precluded by a direct method. If the proven theorem is useful, isn't it better to be able to make use of it earlier, rather than wait for the elegant proof? An example: If the AdS/CFT conjecture is demonstrably proven, it will elevate it from "useful computational technique" to "law of mathematics, probably how nature does it, too". I'll take any proof, as early as I can get it, in order to guide our experimental work.

eric_h 9y ago
https://en.wikipedia.org/wiki/AdS/CFT_correspondence https://en.wikipedia.org/wiki/AdS/CFT_correspondence In case anyone else had no idea what the parent was talking about.
- groby_b 9y ago
  I still have no idea :) Wikipedia on Quantum Physics is usually pretty atrocious, and this page is no exception. It reads like... well, nerds showing off knowledge. (There's nothing wrong with that, but unless you're already pretty steeped in things quantum, the page is jargon without meaning) A more readable summary: http://whystringtheory.com/toolbox/ads-cft/ http://whystringtheory.com/toolbox/ads-cft/
  - aisofteng 9y ago
    That's because it's an encyclopedia, not a romance novel.
    - dzamo_norton 9y ago
      There's a point there that is obscured by the overreach to "romance novel": it's because it's a one-page encyclopedia article on a highly-technical and esoteric subject not a 300-page introductory textbook to be studied over the course of a year. Then again "If someone cannot explain something in plain English, then we should question whether they really do themselves understand what they profess." - Richard Feynmann
      - wiz21c 9y ago
        >>> If someone cannot explain something in plain English This does not mean : "in plain English and in five minutes".
      - coldtea 9y ago
        >Then again "If someone cannot explain something in plain English, then we should question whether they really do themselves understand what they profess." - Richard Feynmann Being able to explain it doesn't mean the recipient will be able to understand the explanation. Richard Feynman has all sorts of explanations in his books and lectures that are still beyond most people's understanding.
  - PhasmaFelis 9y ago
    > Wikipedia on Quantum Physics is usually pretty atrocious, and this page is no exception. It reads like... well, nerds showing off knowledge. Indeed. Wikipedia on math in general is useless unless you already understand it.
    - alimw 9y ago
      It's probably not something I should advertise, but a lot of my maths I learned from Wikipedia.
    - tmearnest 9y ago
      I view it as a reference mainly. It's usually not very helpful for learning, unless the subject is trivial.
raattgift 9y ago
> If the AdS/CFT conjecture is demonstrably proven, it will elevate it from "useful computational technique" to "law of mathematics, probably how nature does it, too". Gauge/gravity duality under particular circumstances is certainly interesting, but unfortunately our universe is not AdS and in particular does not have a boundary on which the conformal field theory can be constrained. So being able to define string theory nonperturbatively on a Minkowski space defined on the boundary of spacetime is certainly interesting, but as far as we can tell our spacetime does not have that boundary and moreover it would be much nicer if we had a background independent formulation for a nonperturbative string theory, and the gauge/gravity duality does not really offer that. One might draw the analogy to being able to define a patch of Minkowski space around a point on a geodesic through a general curved spacetime: General Relativity explicitly promises only that a smooth coordinatization (e.g. Fermi coordinates) is possible in an infinitesimal neighbourhood around the point although in weak gravity the patch can be much bigger (and the system of coordinates more "Cartesian"). The worry (by this analogy) should be that gauge/gravity correspondence falls apart in strong gravity. Moreover, it is an enormous stretch to argue that if a large class of bulk theories have the same symmetries as certain boundary theories (or are related in some other way) then therefore there background independent physics are recovered. Rovelli enjoys pointing out that in GR you can always select a preferred frame and map an SR theory over that and do useful computations, but that doing so gives no insights into proper Lorentz-invariant physics, so how does choosing a preferred spacetime and mapping a field theory over that providing insights into quantum gravity? However, some string theorists (notably Verlinde recently) have been trying to get out of the AdS "box" and think about other chosen spacetimes (e.g. asymptotic de Sitter) to see if some gauge/gravity ideas are even workable for those spacetimes. Maybe they are. > I'll take any proof, as early as I can get it, in order to guide our experimental work. If you assume -- because General Relativity survives all observational tests and is expected to be completely correct in weak gravity -- that the string theory has to reduce to (or at least fully reproduce) classical GR in the appropriate limit then you can't expect any experiments which could distinguish between the two in the near future. But if you have a string theory that departs from GR in a testable limit, then sure, compare that with existing and future evidence. However, string theory might testable outside the gravitational sector if the bulk-boundary correspondence covers more than that sector. Given that string theory tends to come with infinite fields, extra dimensions, supersymmetry and other theoretical objects that are not in the Standard Model, then BTSM experiments may have something to say about such string theory models.
noobermin 9y ago
But people often write papers anyway assuming X and Y and Z are true just to see what happens because of it, it's not like you really need it to be true to at least publish things. And for your example that touches on applications of math, it approaches scientific research from the wrong perspective. If some math follows reality, (agrees with experiment and even better makes predictions that turn out to be right), there is has to be some backing to it, even if the rigor isn't fully there. Physics often runs with ideas before Math has caught up (see QM in the early days, HEP in general, etc). So, if we back off from applications who, quite frankly, only care about whether X is true in only a few, really important number of conjectures I can think of--big questions like P=NP for example--then math itself I imagine is, well, a study for its own sake. So, what above exhaustive proofs of your run-of-the-mill conjecture? Who would want to read a paper from a math person that merely iterates through a 200TB cache of data? People forget that science is a social phenomenon as much as it is systematic process.
- rrobukef 9y ago
  I don't want a math paper detailing 200TB of data. I want a paper telling me the generation of the digital input, the method that was used to actually prove it if it's novel. I want the authors to show how they verified the result and if it's important how they verified the automatic verification. The paper will end with "Oh and by the way, we used this conjecture because it was difficult, it's true. You can verify it yourself." I want this all so I can do this myself, so I can prove a run-of-the-mill conjecture and not waste my time with it. I want this because proving is difficult and there is more to SAT than mathematical proofs, it is used in the industry. I want this because I trust a verified proof that the airplane won't crash more than a human proof. The availability of computer proofs does not devalue the study of mathematics.

by ISL 9y ago

eric_h 9y ago

https://en.wikipedia.org/wiki/AdS/CFT_correspondence https://en.wikipedia.org/wiki/AdS/CFT_correspondence In case anyone else had no idea what the parent was talking about.

groby_b 9y ago
I still have no idea :) Wikipedia on Quantum Physics is usually pretty atrocious, and this page is no exception. It reads like... well, nerds showing off knowledge. (There's nothing wrong with that, but unless you're already pretty steeped in things quantum, the page is jargon without meaning) A more readable summary: http://whystringtheory.com/toolbox/ads-cft/ http://whystringtheory.com/toolbox/ads-cft/
- aisofteng 9y ago
  That's because it's an encyclopedia, not a romance novel.
  - dzamo_norton 9y ago
    There's a point there that is obscured by the overreach to "romance novel": it's because it's a one-page encyclopedia article on a highly-technical and esoteric subject not a 300-page introductory textbook to be studied over the course of a year. Then again "If someone cannot explain something in plain English, then we should question whether they really do themselves understand what they profess." - Richard Feynmann
    - wiz21c 9y ago
      >>> If someone cannot explain something in plain English This does not mean : "in plain English and in five minutes".
    - coldtea 9y ago
      >Then again "If someone cannot explain something in plain English, then we should question whether they really do themselves understand what they profess." - Richard Feynmann Being able to explain it doesn't mean the recipient will be able to understand the explanation. Richard Feynman has all sorts of explanations in his books and lectures that are still beyond most people's understanding.
- PhasmaFelis 9y ago
  > Wikipedia on Quantum Physics is usually pretty atrocious, and this page is no exception. It reads like... well, nerds showing off knowledge. Indeed. Wikipedia on math in general is useless unless you already understand it.
  - alimw 9y ago
    It's probably not something I should advertise, but a lot of my maths I learned from Wikipedia.
  - tmearnest 9y ago
    I view it as a reference mainly. It's usually not very helpful for learning, unless the subject is trivial.

raattgift 9y ago

> If the AdS/CFT conjecture is demonstrably proven, it will elevate it from "useful computational technique" to "law of mathematics, probably how nature does it, too". Gauge/gravity duality under particular circumstances is certainly interesting, but unfortunately our universe is not AdS and in particular does not have a boundary on which the conformal field theory can be constrained. So being able to define string theory nonperturbatively on a Minkowski space defined on the boundary of spacetime is certainly interesting, but as far as we can tell our spacetime does not have that boundary and moreover it would be much nicer if we had a background independent formulation for a nonperturbative string theory, and the gauge/gravity duality does not really offer that. One might draw the analogy to being able to define a patch of Minkowski space around a point on a geodesic through a general curved spacetime: General Relativity explicitly promises only that a smooth coordinatization (e.g. Fermi coordinates) is possible in an infinitesimal neighbourhood around the point although in weak gravity the patch can be much bigger (and the system of coordinates more "Cartesian"). The worry (by this analogy) should be that gauge/gravity correspondence falls apart in strong gravity. Moreover, it is an enormous stretch to argue that if a large class of bulk theories have the same symmetries as certain boundary theories (or are related in some other way) then therefore there background independent physics are recovered. Rovelli enjoys pointing out that in GR you can always select a preferred frame and map an SR theory over that and do useful computations, but that doing so gives no insights into proper Lorentz-invariant physics, so how does choosing a preferred spacetime and mapping a field theory over that providing insights into quantum gravity? However, some string theorists (notably Verlinde recently) have been trying to get out of the AdS "box" and think about other chosen spacetimes (e.g. asymptotic de Sitter) to see if some gauge/gravity ideas are even workable for those spacetimes. Maybe they are. > I'll take any proof, as early as I can get it, in order to guide our experimental work. If you assume -- because General Relativity survives all observational tests and is expected to be completely correct in weak gravity -- that the string theory has to reduce to (or at least fully reproduce) classical GR in the appropriate limit then you can't expect any experiments which could distinguish between the two in the near future. But if you have a string theory that departs from GR in a testable limit, then sure, compare that with existing and future evidence. However, string theory might testable outside the gravitational sector if the bulk-boundary correspondence covers more than that sector. Given that string theory tends to come with infinite fields, extra dimensions, supersymmetry and other theoretical objects that are not in the Standard Model, then BTSM experiments may have something to say about such string theory models.

noobermin 9y ago

But people often write papers anyway assuming X and Y and Z are true just to see what happens because of it, it's not like you really need it to be true to at least publish things. And for your example that touches on applications of math, it approaches scientific research from the wrong perspective. If some math follows reality, (agrees with experiment and even better makes predictions that turn out to be right), there is has to be some backing to it, even if the rigor isn't fully there. Physics often runs with ideas before Math has caught up (see QM in the early days, HEP in general, etc). So, if we back off from applications who, quite frankly, only care about whether X is true in only a few, really important number of conjectures I can think of--big questions like P=NP for example--then math itself I imagine is, well, a study for its own sake. So, what above exhaustive proofs of your run-of-the-mill conjecture? Who would want to read a paper from a math person that merely iterates through a 200TB cache of data? People forget that science is a social phenomenon as much as it is systematic process.

rrobukef 9y ago
I don't want a math paper detailing 200TB of data. I want a paper telling me the generation of the digital input, the method that was used to actually prove it if it's novel. I want the authors to show how they verified the result and if it's important how they verified the automatic verification. The paper will end with "Oh and by the way, we used this conjecture because it was difficult, it's true. You can verify it yourself." I want this all so I can do this myself, so I can prove a run-of-the-mill conjecture and not waste my time with it. I want this because proving is difficult and there is more to SAT than mathematical proofs, it is used in the industry. I want this because I trust a verified proof that the airplane won't crash more than a human proof. The availability of computer proofs does not devalue the study of mathematics.