Hawking's "End of Physics"

sophiecentaur

That's a very 'Optimistic' view. I believe the Victorians had a very similar idea at the end of the 19th Century and then along came QM, Relativity etc. etc.

Perhaps I should ask how big these "major steps" are and how "basic" did you want to go?

Just imagine taking a ruler and drawing a line under the final page in one's Physics notebook.
"Right, that's sorted out, now let's move on to something really difficult. How about todays Sudoku in the Guardian?"

Constantin

Those steps might need to be pretty big. As big as Quantum Mechanics or Relativity. At least in my optimistic point of view.

The basic rules have to deal with exactly what the Universe is. What the dimensions are. What the elementary particles are.
Right now it seems the Universe is all about geometry, information and probabilities. So at the basic level the rules should state exactly what the Universe's geometry is, the exact relationship between dimensions and the way the information exists, propagates and interacts.

At the very basic level, the amount of rules has to be small, very small.

Of course we'll still need a large number of complex theories for the more complex phenomena, and here active research may continue for very long, but finding the "basics" can't last for much longer. Just a few decades, at most, in my view.

Zarqon

I'm not sure I understand why you need to separate mathematical and physical axioms? Can't you just say that any potential GUT contains a certain set of axioms, both mathematical and physical, and that this total set would then be subject to Gödel's theorem. Then it follows that this set of axioms has to be consistent (because of the principle of explosion) and thus incomplete.

Constantin

"Grand Unified Theory" (GUT) and "Theory of Everything" (ToE) are two different things.
GUT doesn't have to explain everything, it's just an improvement over the existing Standard Model.

Zarqon

Ok, but it doesn't change much. Both a GUT and a TOE must be based on consistent sets of axioms, and as such, it appears that both will then necessarily be incomplete.

sophiecentaur

Ha - so GUT would really be just a MUT (Modest Unified Theory) and not that Grand after all.

Zarqon

... although I wouldn't recommend using the word "modest" anywhere near the front page of a grant proposal

sophiecentaur

J. Swift used it once with a tongue in HIS cheek - as in "A modest proposal".
Would 'Half arsed' go down better?

Jolb

My apologies for resurrecting an old thread, but it is a thread I really like a lot.

Anyway, I think there is a reason one needs to separate mathematical axioms from physical axioms: the world of math contains a lot of stuff that is impossible in the physical world. E.g., fractals. Fractals do not exist in our universe.

So mathematical axioms apply to the mathematical ("Platonic") universe, whereas physical axioms constrain the mathematical possibilites to only those things which apply to OUR universe, which certainly does not contain all the aspects of the mathematical universe.



[Note that in my previous post I used the word "universe" to refer to the PHYSICAL universe. Here I broaden the term "universe" to include both the physical and "platonic" universes.]

[[Second note: I do believe that there MAY be extant physical truths which cannot be mapped isomorphically to the mathematical "platonic" universe, so the physical universe is not necessarily a subset of the platonic universe. If this were the case, then a ToE/GUT would clearly be impossible, unless we expand our theories to include nonmathematical ones.]]

Reptillian

To paraphrase Gödel himself "Either the universe is incomprehensible, or the mind is more than a machine."

Jolb

Wow. That's an amazing quote. I never knew that Godel entertained "cognitive science" ideas. Do you have a source for that?

Godel is one of those mathematicians who definitely is underappreciated for his brilliance in physics, philosophy, etc. A true genius. No wonder Einstein and he were such good friends.

Zarqon

Really? The coastline would like a word with you :P

I think this doesn't change anything of what I wrote earlier. You may say you put up two sets of axioms (which may share some statements), but I'm simply saying that you can create a total set of axioms, containing all axioms from both sets defined on the combined space of your platonic + physical universe, and that this total set of axioms should still obey Gödel's theorem, i.e. be consistent and hence incomplete.

georgir

Ok, I am a bit confused.

In physics, we've already had theories which we thought were complete. They turned out to not match reality, so they got changed, but disregarding that, how are they "incomplete" in theoretical sense?

Assume reality were a simple physical model with just point-particles and infinite-speed gravity for example... What unprovable truths could there be in it to spell the "end of physics"?

After all, Goedel's theorem talks about "any" mathematical theory and is not concerned with reality... I can't see its physical equivalent.

Jolb

The coastline is not an example of a fractal, and the coastline paradox itself is somewhat nonsensical. I don't want to go on about debunking the coastline paradox, but could the coastline possibly be any longer than a line of n water molecules, where n is the total number of water molecules on earth (or the solar system, or the galaxy)?

The fact is that quantum physics dictates that once measurements are made on scales smaller than the Planck length, the only structure is randomness. A non-random fractal with infinite self-similarity (like the Mandelbrodt set) is an example of a non-random structure which goes down to infinitely small length scales, and as such cannot exist in a universe constrained by Heisenberg uncertainty on Planck-length scales.

I think I'll be repeating myself at this point. Physical axioms are things which constrain the wide world of mathematical possibilities to things that only apply to our universe. If we Godelize your "combined system" and it gives us a new axiom which has no specific reference to the physical universe, then it is a purely mathematical axiom and doesn't change what could be the complete set of physical axioms. The Godelization might only lead to new mathematical axioms and not any new physical axioms, and we could regard the complete set of physical axioms as the be-all-and-end-all theory of physics, despite mathematics being incomplete.

You're right. A simple enough physical theory could be complete, at least in the physical sense. There can be mathematical incompleteness without physical incompleteness.

What Hawking explained in the talk I linked earlier is that only now with the advent of M-theory does he think a physical kind of Godel's theorem could exist--it provides the needed complexity that wouldn't be there in your simple model.

Reptillian

The quote from Gödel is: "Either mathematics is too big for the human mind or the human mind is more than a machine"

I've been trying to figure out from whence this quote originates, but haven't had much luck.

I do agree that Gödel was one of the greatest, if not the greatest, mathematican/logician of the 20th century.

sophiecentaur

I don't find that quote very awe inspiring, as a matter of fact. Why should Mathematics NOT be too big for the human mind? Why should the human mind be more than a machine? The two alternatives are not mutually exclusive and there is too much of a hint of the 'spiritual' about the statement, playing to the vanity of the reader rather than to reason.

I also take issue with the idea of Science consisting of Axioms. It consists of hypotheses which can be verified of falsified. Mathematics happens to be a suitable tool for Science when its axioms happen to produce roughly parallel outcomes to what we observe in Science and that enables 'models' to be built and to allow predictions by extrapolation. There need be no more than that.

Jolb

The reason mathematics should not be too big for the human mind is because mathematics is a creation of the human mind. Each and every mathematical result is cooked up by a human using some understanding of the mathematics--thus everything in math is designed to be amenable to some kind of understanding. For that reason, I think that there is something 'spiritual' or at least 'transcendental' about saying that mathematics is beyond comprehension.

Well, if science were just about building model after model, why would theoretical physicists be so interested in a "unified theory"? The idea is to reduce the science to a minimal number of hypotheses--and these hypotheses are like the axioms of a formal system.