Is a GUT Impossible Due to Godel's Incompleteness Theorem?

[Jolb]

"Why should Mathematics NOT be too big for the human mind? Why should the human mind be more than a machine? "

I agree with you. The human mind *evolved* for certain purposes. Why should it necessarily be good at things it did not evolve for?

Humans needed to be able to deal, at least intuitively, Newtonian mechanics because our ancestors had to hunt by throwing spears while hunting, to throw rocks at enemies etc. There was no need to understand multi-particle processes, so it is not surprising that things like phase transitions which are simple and neat empirically, have not been fully understood theoretically by humans.

Well, your examples here are not mathematical examples. You are stating physical examples that are impossible to understand, not mathematical examples. As I said before, all mathematics has to be understandable because it is a creation of humans all using an understanding of the mathematics.

Incidentally, the only way we can understand multi-particle processes and phase transitions theoretically is by using some sort of mathematics. Just remember that a lot of what we're discussing is the fact that physics≠mathematics.

However, for those ideas/concepts to ever make it into our math and brains, it must have some real meaning, even as purely information in our brains and communicated through physical mediums. ... My reasoning tells me that all math that we know of or can imagine has to be grounded in the physical universe as reality, or else it would not be possible to manifest in our ideas/thoughts/communication - it would not exist.

I disagree with the idea that mathematical entities do not exist unless they are manifested physically as some informational process in a human brain. That would be to say that any time a new mathematical thought is conjured up by a mathematician, he is creating it out of nowhere. (This seems to violate some sort of "conservation law" to me.) As an example, you are saying that the Mandelbrodt set did not exist until the instant that Mandelbrot thought of it. That is nonsense. The Mandelbrot set always did and always will exist in the abstract mathematical ("platonic") reality.
As a second argument for the independent existence of mathematical entities, take for example a universal Turing machine. A universal Turing machine is capable of running any deterministic algorithm, of which there are infinitely many. Assuming each algorithm takes a finite amount of processing time, then it would be impossible to execute all of them. So there are always infinitely many algorithms which have never been executed.

Your argument says that a given algorithm does not exist until a Turing machine executes it. I would argue that as long as a Turing machine exists with the potential to run that algorithm, then that algorithm already exists without it already having been executed.

You don't even really need a universal Turing machine to make this example work. Imagine you had a super good calculator that can multiply any two numbers. For simplicity, use the natural numbers 1, 2, 3, ..., which of course there are infinitely many of. The algorithm in this case would be the calculator actually performing the multiplication. Your argument would say that the algorithm of multiplying two numbers A and B does not exist until one such calculator actually performs the multiplication of A and B.One thing I should clarify before someone points out that my two responses are ostensibly contradictory: "Mathematics" is not the same as the "abstract mathematical reality". The abstract mathematical reality always did and always will exist. Mathematics is a human activity that explores certain elements of the mathematical reality. Therefore it's totally consistent to say the following two things:
1. Mathematical discoveries must be understandable because they are derived by a human.
2. The mathematical entities these discoveries describe are not a creation of the human mind--rather, the mathematical result is a description of a discovered property of the mathematical universe.

 

[DragonPetter]

I disagree with the idea that mathematical entities do not exist unless they are manifested physically as some informational process in a human brain. That would be to say that any time a new mathematical thought is conjured up by a mathematician, he is creating it out of nowhere. (This seems to violate some sort of "conservation law" to me.) As an example, you are saying that the Mandelbrodt set did not exist until the instant that Mandelbrot thought of it. That is nonsense. The Mandelbrot set always did and always will exist in the abstract mathematical ("platonic") reality.

Hmm. . maybe what I wrote out was really confusing, because what you just said is basically part of the point I was trying to make. So, I think you misinterpreted what I meant as the opposite of my meaning haha. I typed a response to SophieCentaur, but my login timed out and I lost it :( I'm going to re-respond to him, and maybe that will clarify what I meant for you too.

 

[DragonPetter]

That is a pretty arbitrary assumption - relying on your own take on the meaning of the word "real", I think. It's easy to make up total nonsense which, apart from the fact that it may consist of 'real' brain impulses, can hardly be thought of as real. (my definition and not yours, perhaps)

Well, I guess I should have been more rigorous in what I meant by "real" in the context of math. First, what I said is just as much a question as it is a vague conjecture, so hopefully someone can point out a flaw in my reasoning or give a meaningful response to improve my thoughts on these subjects. I am way out of my league when I try to think about things related to the ideas of Gödel, Turing, Shannon, etc.

Anyway, I won't be able to give you a rigorous or official definition of what I meant by real. Real to me is existence in physical reality, or in other words, some physical manifestation or part inside the universe. If something is removed from the universe (which I think is impossible), and the universe looks the same as it did before it was removed, then it wasn't real. Likewise, this definition applies to math too. If we added a new concept of some math that does not apply to the universe (ie something that could never have been thought of from existence in that universe), and the universe did not change, then I would say that math concept is not real. If the only way to add that concept was to alter the universe in some way, then I would say its real (even if the concept does not apply to anything else that we observe in physical reality). When others -casually- talk about math being real, and the distinction I'm trying to make from, they say math is "real" when it either applies to the universe in a way that it can A) describes or B) represents some part of the universe. If we can come up with a mathematical structure or concept that does not do A or B, we will say it is not part of physical reality. Lots of high school examples exist where we say "that has no physical meaning", like the negative root used as a solution to an equation that is describing an only positive physical quantity. The questions I have call that statement "that has no physical meaning" into question, and I'm not trying to imply some ground breaking new concepts or matrix altering reality with this. I'm just simply raising questions from my own confusion. That negative root solution exists, even if it doesn't fulfill A or B. It came from a system comprised of the student's body and their pen and paper and has been figuratively burned into the universe as some pattern that is physically there. The only way it can fulfill A or B is to describe itself as just a concept physically imprinted in the universe in some way (neurons, bits, etc). In that way, something we say of math that is outside of physics, still ultimately sits physically as a part of our universe in some way, which is why I quoted your post originally. The universe almost seems to physically represent the negative root as useless information because that's the only place the information can go as the events and systems interact and the solver's brain comes to the solutions of the equation, and we just disregard it as "not physically real" because it has no further use to us, even though the positive root is not much different physically (in its imprint in the universe) from the negative root as information.

Say a mathematician creates an abstract algebra structure, and we might say that structure is neat and interesting, but it only belongs to the realm of mathematics. We might say it neither A) describes reality nor B) can represent any part of the universe as we observe it (including approximately). But the mere fact that the mathematicians brain described the structure precisely, thought of the concept, processed the concept (importantly: developed a proof, which also must exist/obey the laws of physical reality for it to actually be completed), etc. implies that the structure exists in some manifestation. If it didn't, his brain would be processing something that is not a part of reality - does not exist - and I find that impossible. I just think that the structure can exist while still not fulfilling A or B, except to describe itself. The only exception/flaw with this train of thought that I can think of would be that his thoughts, concepts, processing, etc. only approximate that ideal mathematical structure that exists regardless of a universe. I don't know which would be more incorrect, however if the laws of the universe allow the mathematician to complete a proof, then it seems that the idea has to exist physically. I could just be swimming in circles because of some logical fallacies that I have :P

I actually prefer the idea that math exists as a framework independent of our universe, but while I was enjoying that thought, other thoughts tried to challenge it. That's the only reason I am asking about this. I realize this post is kind of out there now, so I'm going to stop.

 

[DragonPetter]

Your argument says that a given algorithm does not exist until a Turing machine executes it. I would argue that as long as a Turing machine exists with the potential to run that algorithm, then that algorithm already exists without it already having been executed.

Well, I did not imply that from my thoughts that I shared. I'm not trying to argue one way or another really, just trying to sort through the thoughts which are largely unestablished.

That is one possible conclusion you could get from my thoughts though. Mathematical concepts, algorithms, etc. only exist when they become represented in the universe physically.

Another possible conclusion seems to bump shoulders with what you just said, which would be that the mathematics is built into the universe, or in other words that the only mathematics that exists are mathematics that exist in the universe. The reason I say this bumps shoulders with what you said is because you say that the algorithm exists implicitly if a turing machine exists to execute it (which itself is an unestablished statement). Well, the turing machine only exists if a universe can allow it to exist under its natural laws, and so you would say the algorithm exists without being executed if the universe's laws allow it to exist, which would imply that the mathematics is built into the universe's laws. This of course seems to apply to all kinds of ideas outside of just math ideas, such as dragons can exist if the laws allow them to, but we have never seen a dragon. Dragons, along with any other information that does not describe physical reality, are only self referencing information; not physical entities beyond the information that encodes this idea. Similarly, we can come up with all kinds of math that may be similar to reality, but does not exist other than its self-reference (meaning it still has SOME existence as a part of our universe). This still does not get anyone very far because there could always be math that exists outside of the universe, and this is math that we would never even be able to access, and turing machines that we cannot access from within our universe. I don't know, I think I have ruined this discussion by bringing dragons into the equation.

 

[ApplePion]

Me: "Humans needed to be able to deal, at least intuitively, Newtonian mechanics because our ancestors had to hunt by throwing spears while hunting, to throw rocks at enemies etc. There was no need to understand multi-particle processes, so it is not surprising that things like phase transitions which are simple and neat empirically, have not been fully understood theoretically by humans."

Jolb: "Well, your examples here are not mathematical examples. You are stating physical examples that are impossible to understand, not mathematical examples."

All the actual physics regarding multi-particle systems is pretty much known. It is the same physics as the physics for low-particle number systems (of course taking into effects like Fermi-Dirac and/or Bose-Einstein, effects whose physics is understood.) So humans not being able to explain the simple phase transition properties is due to math insufficiency.

Consider the three body problem. We know the basic physics just as well for that as for the two body problem. But we have effective mathematical techniques to handle the two body problem, but not to handle the three body problem.

Jolb: "As I said before, all mathematics has to be understandable because it is a creation of humans all using an understanding of the mathematics."

For that to be true it would, for example, have to be the case that if the Goldbach conjecture has a proof that some human would necessarily figure out the proof. How can you preclude the possibility that it has a proof but not one that a human will come up with or be able understand? As I stated earlier there are some concepts dogs will never understand. How do you assume that there are no numerical concepts that humans will never understand?

 

[Jolb]

All the actual physics regarding multi-particle systems is pretty much known. It is the same physics as the physics for low-particle number systems (of course taking into effects like Fermi-Dirac and/or Bose-Einstein, effects whose physics is understood.) So humans not being able to explain the simple phase transition properties is due to math insufficiency.

Consider the three body problem. We know the basic physics just as well for that as for the two body problem. But we have effective mathematical techniques to handle the two body problem, but not to handle the three body problem.

The fact that not all mathematical problems can be solved with our current techniques does not mean that the math is not understandable. In fact, there exist mathematical proofs that certain problems do not have analytical solutions--take for example the Abel-Ruffini theorem. Just because certain mathematical problems are not possible to solve analytically, or aren't possible to solve with our current techniques, doesn't imply that the math is not understandable. Each and every symbol in every mathematical text has a clear definition, and you can look up each and every symbol to interpret and understand what it means.

On the other hand, in physics, writing down the equations for a model which cannot be solved to make predictions doesn't mean that the physics is completely understood. Was all of quantum mechanics understood the day Schrodinger wrote his equation? If the physical model dictates some physically measurable quantities, and if we fully understand the model, then we should be able to use it to generate accurate predictions. If the model is not usable, we can't make predictions, and there is a huge gap in the understanding. Even though we can write down the Navier-Stokes equations that dictate how our atmosphere evolves over time, we still don't understand the model enough to accurately predict the weather more than a day or two in advance.

For that to be true it would, for example, have to be the case that if the Goldbach conjecture has a proof that some human would necessarily figure out the proof. How can you preclude the possibility that it has a proof but not one that a human will come up with or be able understand? As I stated earlier there are some concepts dogs will never understand. How do you assume that there are no numerical concepts that humans will never understand?

Let me make a distinction: it is possible that the Goldbach conjecture follows from the axioms of mathematics, and that proving it might be impossible for humans. Statements that are generated by Godelization are known to be true but unprovable (inside the axiomatic system that was Godelized). But a mathematical proof by definition must have each and every symbol clearly defined. Each symbol in the proof must have a definition that is understandable by humans, since all these definitions have been created by humans.

Furthermore, there are plenty of fairly basic mathematical ideas which cannot be proved, and even these have a definite meaning. Take for example the axiom of choice and the axiom of determinacy. Nonetheless, anything that can be written down mathematically must have an understandable meaning based on the elementary definitions, because the mathematics is endowed with an understandable meaning ab initio.

Just to restate my point more succinctly: there may be extant mathematical objects in the mathematical universe which are beyond human understanding, but the mathematics that humans do must be understandable.

 

[ApplePion]

Jolb: "The fact that not all mathematical problems can be solved with our current techniques does not mean that the math is not understandable. In fact, there exist mathematical proofs that certain problems do not have analytical solutions--take for example the Abel-Ruffini theorem. Just because certain mathematical problems are not possible to solve analytically, or aren't possible to solve with our current techniques, doesn't imply that the math is not understandable. Each and every symbol in every mathematical text has a clear definition, and you can look up each and every symbol to interpret and understand what it means...etc. etc."

You should re-read my post and try to respond to what I actually said.

 

[Jolb]

Well, you should re-read my post and try to see how what I said IS a response to what you said. I think everything in my post after the second quote is quite obviously a response to what you said.

Just to spell out why what I said after your first quote constitutes a response, I could add the following qualifier at the beginning of my response.

You seem to be arguing that your examples of non-understandable mathematics indeed qualify as mathematics and not physics (despite all being physics examples) because these physical problems have been reduced to purely mathematical problems which cannot be solved. (According to you, writing down the basic equations of a physical model constitutes a full understanding of the physics, even if the equations do not give physical predictions since they are unsolvable.) So your argument is basically that math must not be understandable because there exist mathematical problems which are not yet or not in principle solvable.The fact that not all mathematical problems...

Is my interpretation of your argument off? I feel that my response was pretty germane to what you said, but let me know if I'm missing your point.