Is Mathematics Truly Limited by Physics?

[Hornbein]

And I can show you many aspects of the principles used in Economics that have no resemblance with the "physical laws of the universe". Yet, they also use mathematics!

Zz.

I suggest you take a break and look at this again in an hour or day or so.

 

[ZapperZ]

I suggest you take a break and look at this again in an hour or day or so.

Why? It tastes better after it has been fermented?

Zz.

 

[TeethWhitener]

It used to be that way.

I think the OP's argument is:

Mathematics is limited by the capabilities of mathematicians.
Mathematicians are physical objects.
Physical objects are limited by physics.

-> Mathematicians are limited by physics.
-> Mathematics is limited by physics.

I find nothing wrong with this. It is impossible to conceive of that which is impossible to conceive. Note also that replacing mathematicians with computers or mystics or whatever makes no difference to the argument. It applies to everything within our Universe.

This is similar to the argument put forth by early detractors of computer-aided proofs in math. The thinking goes thus:

A valid proof of X requires verification of X.
Let's say we use a computer to prove X.
Computers are physical objects.
The behavior of physical objects is governed by the inductive (in the common non-mathematical sense of the word) laws of physics, rather than the deductive laws of logic.
The laws of physics cannot be verified absolutely like the laws of logic can.
Therefore, the computer proof of X cannot be verified absolutely.
Therefore the computer proof of X is not a proof of X.

Essentially, the detractors claimed that a computer can never prove a statement because we have no way of knowing in general with absolute confidence that the computer didn't go wrong (i.e., behaved in a way contrary to what we expected). Of course, this problem extends to the fact that our brains (the classic theorem proving tool) are also physical objects, which has led into discussion in the philosophy of mathematics about what exactly a proof is.

 

[Hornbein]

Essentially, the detractors claimed that a computer can never prove a statement because we have no way of knowing in general with absolute confidence that the computer didn't go wrong (i.e., behaved in a way contrary to what we expected). Of course, this problem extends to the fact that our brains (the classic theorem proving tool) are also physical objects, which has led into discussion in the philosophy of mathematics about what exactly a proof is.

IMO a proof is an exercise in persuasion. There is a definite element of psychology and (dare I say it) marketing. I'm not being critical, the proof acceptance process is pragmatic, a system to make best use of finite resources.

About twenty years ago there was a movement toward automated verification of proofs. It was abruptly abandoned. I think automated proof is a worthy goal, but beyond what can be done at this time. I guess what they discovered is that it was far too much work to prove anything to a computer. If proving to a computer were a requirement, all mathematical progress would halt. First it would be necessary to enter all the basic proofs, a Herculean and entirely impractical effort. So the current methods of proof must suffice until artificial intelligence technology improves.

Kurt Godel proved that there is no limit to the minimum length of a proof. Starting with a simple proof it is possible to construct a proposition with a proof that must be longer than the first, and so on. So it is practically impossible to prove certain things.

There is a proposed proof of the abc conjecture. The claimant has excellent credentials and reputation. But the proof is so long and complicated that no movement has arisen to verify the alleged proof. If no group will make the effort then we may never know whether or not this is a proof. If a computer ever verifies a proof like that, or a proof with that complexity raised to a power of one thousand, then humans would have no other criteria than trust or lack thereof.

I think that quite often the proposition to be proved is of little importance. What really matters is the discovery of a fresh approach or insight in the course of the proof. It would seem to me that a proof of the Goldbach conjecture that occupies a centillion terabytes would be utterly useless.

There was the controversial computerized proof of the 4-color conjecture. It was useful in the sense that mathematicians (and cranks) could then stop wasting their time on attempted proofs. An expert (Gerhard Ringel) told me that he felt there was no other way to prove it, so good riddance. But it could very well be in some cases that an impenetrable computerized proof would discourage the discovery of a simple, worthwhile proof. In that case the computerized proof would serve the function of inhibiting progress. But probably not. I bet there are a thousand cranks right now trying to come up with a one-page proof of Fermat's last theorem.

IMO automated and natural methods of proof verification are complementary. It would be nice to have both. What's the problem?

 

[Gjmdp]

The rules of Physics, and the rules for studying Physics are different. You are mixing this terms.
To study Physics or whatever we need Mathematics, and this is like a computer program: there is syntax (logical and organized methods) but not semantics (rules or concepts of Physics).
The rules of Physics, the concepts, or the "semantics" are expressed with help of Mathematics, but NOT only with Mathematics (speed is not a mathematical concept).
All this concepts are limited by Physics.

With Mathematics:
The rules of Mathematics are expressed by itself, Mathematics. You can't explain 1+1=2 by anything that isn't Mathematics.
Now, are the rules for studying Mathematics limited by Physics?
Well, the environment for studying Mathematics in this universe is completely physical, not mathematical.
The speed of light is limited on this universe, and for some of you this seems to be a random rule. But how do we know is a random rule? How do we know there could be an universe with no limit on the speed of light? We don't know, we haven't been on that universe and we can't have a proof of the possibility of such universe never.

Maybe (I don't know), the speed of light has to be limited Mathematically.
So this makes me ask: May Physics be as limited as Mathematics? Is that a possibility? I'm not saying anything is a branch of anything, there are things equal limited (Physics and Chemistry are limited by the universe) but there aren't the same thing (Physics and Chemistry aren't the same thing by definition)

 

[gleem]

It seem this thread is bordering on philosophy and therefore imperiled. But let me add my opinion

From the OP

So, does this branch have a name?

No. why should it. Any subset of a discipline must still satisfy the tenets of that discipline. Mathematics is not bound by physical constraints.

May Physics be as limited as Mathematics? Is that a possibility?

Physics as tediously discussed heretofore is not a branch of mathematics. Insofar as physics uses mathematics as a tool it will be limited in its goals of establishing how our universe works if Physics does not have the right tool for the project at hand it will be handicapped. Physics is limited by constraints beyond our control. Mathematics is limited by our imagination.

Maybe (I don't know), the speed of light has to be limited Mathematically.

Mathematics is a system of logic, rules and objects that pertain to them. Light is generated by physical processes and in condensed matter propagated by physical interactions. At this time its propagation in a vacuum is a subject of investigation Mathematics will play a role in the eventual explanation but will not be the cause of it just as it is not the cause of any physical phenomenon unless you think it causes 1 +1 to equal 2.

If mathematics has bounds it is only in our ability to express the concepts.

For a good read on the interaction math and physics read "Mathematics Queen and Servant of Science" by Eric Temple Bell.

 

[Gjmdp]

It seem this thread is bordering on philosophy and therefore imperiled. But let me add my opinion

From the OPNo. why should it. Any subset of a discipline must still satisfy the tenets of that discipline. Mathematics is not bound by physical constraints.
Physics as tediously discussed heretofore is not a branch of mathematics. Insofar as physics uses mathematics as a tool it will be limited in its goals of establishing how our universe works if Physics does not have the right tool for the project at hand it will be handicapped. Physics is limited by constraints beyond our control. Mathematics is limited by our imagination.
Mathematics is a system of logic, rules and objects that pertain to them. Light is generated by physical processes and in condensed matter propagated by physical interactions. At this time its propagation in a vacuum is a subject of investigation Mathematics will play a role in the eventual explanation but will not be the cause of it just as it is not the cause of any physical phenomenon unless you think it causes 1 +1 to equal 2.

If mathematics has bounds it is only in our ability to express the concepts.

For a good read on the interaction math and physics read "Mathematics Queen and Servant of Science" by Eric Temple Bell.

The intention of this thread is to know more about the nature of Physics, Mathematics and the difference of its limits. Well, when I said if Mathematics would be equal limited as Physics I was saying if it was also limited by the universe, like Chemistry.
Of course I know Mathematics is limited by our imagination: Well wait, only by our imagination? Let's look that again:

We can imagine 1+1=3 but that isn't true! Mathematics it's limited by our imagination and by Logic!
We can imagine as far as we can but always with logic (and not always we imagine with logic), because without it, Mathematics wouldn't have sense.

Then, the limits of logic are now the limits of Mathematics: Is logic limited by the universe? Because then Mathematics will be limited by the universe, as Physics!
So that's the question.

 

[Orodruin]

The thread is a philosophical discussion rather than a scientific one. It will remain closed.