Simple question about the mathematics of theoretical physics!

[Speakpigeon]

Simple question about the mathematics of theoretical physics:

How much of non-trivial formal logic is necessary in the mathematics used in, or implicitly underpinning, theoretical physics?

I am looking for actual examples of any explicit logical proofs used. I need the actual logical formulae that make up the logical ground on which may rest recent theoretical physics such as Quantum Mechanics or String Theory (or just links to such), if at all possible.

I pray to God this question makes sense not just to me!
EB

 

[steve_bank]

Science, applied science, and engineering share the same core matgh

Algebra-trig
Calculus
Differential Equations
Linear Algebra
Probability-Statistics
Multivariable Calculus

Proofs od methodologies are left to realm of mathematicians. Scientists and engineers are generally applied mathematicians. A foundation text on feedback control systems was done y a mathematician at Bell in the 30s. It contains all the foundational proofs electrical engineers rely on today. We generally relu on mathematicians for underlying validity of techniques.

For example LaPlace and Fourier Transforms, a foundational math technique in technology. There is a proof that quarantines the uniqueness of a Laplace-Fourier Transform pair. Any exception found would bring down a lot of technology.

Beyond simple proofs there are no logical rules on developing a new proof and technique. It is not a linear logical process.

Read the book How To Read And Do Proofs. Written initially for the non mathematician. Experience it first hand.

 

[Phil Scott]

Simple question about the mathematics of theoretical physics:

How much of non-trivial formal logic is necessary in the mathematics used in, or implicitly underpinning, theoretical physics?

I am looking for actual examples of any explicit logical proofs used. I need the actual logical formulae that make up the logical ground on which may rest recent theoretical physics such as Quantum Mechanics or String Theory (or just links to such), if at all possible.

I pray to God this question makes sense not just to me!
EB

If by "logical formulae", you mean the formulas of formal logic, then there are probably zero examples. Even in pure mathematics, where rigour has much higher demand, there are basically zero examples outside of mathematical logic. Hilbert's proposed 6th problem for 20th century mathematics was an axiomatisation of physics, which would get someway towards formal logic, but as wikipedia currently has it:

The 6th problem concerns the axiomatization of physics, a goal that twentieth century developments of physics (including its recognition as a discipline independent from mathematics) seem to render both more remote and less important than in Hilbert's time.

 

[steve_bank]

Hilbert asked 'Are all true mathematical propositions provable?'. It led in part to the Turing Machine. Did a paper on it in a CS class.

 

[Phil Scott]

Hilbert asked 'Are all true mathematical propositions provable?'. It led in part to the Turing Machine. Did a paper on it in a CS class.

Nice. It's the Entscheidungsproblem. The page there talks about it having its genesis in Leibniz, which was how it was taught to me when we did Goedel's theorem at uni, but you can drill down further into history to read scholastic philosophers hinting at similar ideas.

 

[Speakpigeon]

Simple question about the mathematics of theoretical physics:

How much of non-trivial formal logic is necessary in the mathematics used in, or implicitly underpinning, theoretical physics?

I am looking for actual examples of any explicit logical proofs used. I need the actual logical formulae that make up the logical ground on which may rest recent theoretical physics such as Quantum Mechanics or String Theory (or just links to such), if at all possible.

I pray to God this question makes sense not just to me!
EB

If by "logical formulae", you mean the formulas of formal logic, then there are probably zero examples. Even in pure mathematics, where rigour has much higher demand, there are basically zero examples outside of mathematical logic. Hilbert's proposed 6th problem for 20th century mathematics was an axiomatisation of physics, which would get someway towards formal logic, but as wikipedia currently has it:

The 6th problem concerns the axiomatization of physics, a goal that twentieth century developments of physics (including its recognition as a discipline independent from mathematics) seem to render both more remote and less important than in Hilbert's time.

So, how do we even know whether what we're doing is at all correct?

Just because what we do all seems to work so very well so far? Or something like that?

Do we just have to trust physicists that they really understand what they are doing?
EB

 

[Phil Scott]

If by "logical formulae", you mean the formulas of formal logic, then there are probably zero examples. Even in pure mathematics, where rigour has much higher demand, there are basically zero examples outside of mathematical logic. Hilbert's proposed 6th problem for 20th century mathematics was an axiomatisation of physics, which would get someway towards formal logic, but as wikipedia currently has it:

So, how do we even know whether what we're doing is at all correct?

Just because what we do all seems to work so very well so far? Or something like that?

Do we just have to trust physicists that they really understand what they are doing?
EB

When I see rocket launches set to fling probes onto comets, I tend to assume these folk know what they're doing

 

[Speakpigeon]

Good, I'm fully reassured now!

So, how much use would be a method for doing logic proofs that would be an extension of our sense of logic? Not needed?
EB

 

[steve_bank]

You never know. A bump from a probe today and 1000 years later Earth is clobbered by an asteroid. Keeps me awake at night.

 

[barbos]

If by "logical formulae", you mean the formulas of formal logic, then there are probably zero examples. Even in pure mathematics, where rigour has much higher demand, there are basically zero examples outside of mathematical logic. Hilbert's proposed 6th problem for 20th century mathematics was an axiomatisation of physics, which would get someway towards formal logic, but as wikipedia currently has it:

So, how do we even know whether what we're doing is at all correct?

Just because what we do all seems to work so very well so far? Or something like that?

Do we just have to trust physicists that they really understand what they are doing?
EB

When I see rocket launches set to fling probes onto comets, I tend to assume these folk know what they're doing

Yeah, but then NASA tries to build EM drive...

 

[Phil Scott]

Good, I'm fully reassured now!

So, how much use would be a method for doing logic proofs that would be an extension of our sense of logic? Not needed?
EB

I'm not sure I understand the proposal. But then, it might be that I've been mucking about with formal logic for so long that I've completely forgotten any sort of innate sense of logic.

Computers are excellent at the symbol crunching, and in some domains, kick the crap out of humans. In other domains, they're just a bit stupid. So a big part of doing formal verification and formal methods is coding up algorithms that somehow impart our intuition about the domains we're good at to the machine. It sometimes works.

As for the original post, I'd like to eventually see formal logic and formal verification get into physics. What I'm thinking of, for a start, is seeing formal verifiers used as part of computer-aided design, where you've got formal axiomatic systems for Newtonian mechanics and you can state and verify properties of the machine you're modelling. We can go from there.

 

[Speakpigeon]

Good, I'm fully reassured now!

So, how much use would be a method for doing logic proofs that would be an extension of our sense of logic? Not needed?
EB

I'm not sure I understand the proposal. But then, it might be that I've been mucking about with formal logic for so long that I've completely forgotten any sort of innate sense of logic.

I think it's more general than that. I think we tend to take for granted a lot of whatever is going on inside our own conscious mind. As I understand it, our sense of logic is basically some unconscious brain process delivering logical evaluations to our conscious mind. When that happen, we may be tempted to see it as something we just did consciously from start to finish. That misconception is probably a necessary one. Our brain wouldn't want to burden the conscious mind with irrelevant considerations of who is actually doing all the hard work! So, we just see the result and take that to be the whole of it.

Also, the conscious mind is really bombarded all the time with all sorts of "evaluations", most of them not of a logic nature. See my thread on "impressions" in the Science section. So, again, it has to be done in a reasonable manner. All these evaluations have to be as unobtrusive as possible. Basically, it's "displayed" to our conscious mind, like on a train departure board, and that's it. If you're too busy elsewhere and don't pay attention to them, that's just too bad.

There's maybe also a side effect, but I'm not sure. Impressions are few whenever you're seriously concentrated on something, i.e. intellectual work. They very nearly disappear, as if to not disturb you when you're doing serious work. So, I would say, it may be that even logical evaluation from our sense of logic are somewhat affected by this restriction. Difficult to say, just a possibility. So, if you want to have logical intuitions, first work hard to train your brain to abstract the subject-matter you're interested in, and once your brain has a good handle on those abstraction, go for a walk and look at the beautiful things around you. Let those impressions come to you.

I think Newton did something like that!

Computers are excellent at the symbol crunching, and in some domains, kick the crap out of humans. In other domains, they're just a bit stupid. So a big part of doing formal verification and formal methods is coding up algorithms that somehow impart our intuition about the domains we're good at to the machine. It sometimes works.

Yes.

As for the original post, I'd like to eventually see formal logic and formal verification get into physics. What I'm thinking of, for a start, is seeing formal verifiers used as part of computer-aided design, where you've got formal axiomatic systems for Newtonian mechanics and you can state and verify properties of the machine you're modelling. We can go from there.

I had the impression the "axiomatic" method had been shown terminally defective by Gôdel?

What about something more analytical?
EB

 

[Phil Scott]

I couldn't comment on the stuff about our sense of logic. It's not really something I give any thought.

I had the impression the "axiomatic" method had been shown terminally defective by Gôdel?

No. The consequences of Goedel are largely overblown. The axiomatic method is still the basis of all formal verification, such as with the examples I gave in the other thread.

 

[steve_bank]

Good, I'm fully reassured now!

So, how much use would be a method for doing logic proofs that would be an extension of our sense of logic? Not needed?
EB

I'm not sure I understand the proposal. But then, it might be that I've been mucking about with formal logic for so long that I've completely forgotten any sort of innate sense of logic.

Computers are excellent at the symbol crunching, and in some domains, kick the crap out of humans. In other domains, they're just a bit stupid. So a big part of doing formal verification and formal methods is coding up algorithms that somehow impart our intuition about the domains we're good at to the machine. It sometimes works.

As for the original post, I'd like to eventually see formal logic and formal verification get into physics. What I'm thinking of, for a start, is seeing formal verifiers used as part of computer-aided design, where you've got formal axiomatic systems for Newtonian mechanics and you can state and verify properties of the machine you're modelling. We can go from there.

Back in the 80s Artificial Intelligence was bee touted as the end all of engineering. Engineering experience would be reduced to a software system It never fully materialized. What it ended up being was large arrays of rules in CAD tools that could be checked quickly by computer beyond practical human ability.

It did have an impact in mechanical and electrical engineering under the heading Design Rule Checking. In printed circuit board design especially in electronics it could allow a less skilled designer to produce quality error free designs that previously required a high level of experience. Same with electronic circuit design. Same in mechanical design. A mechanical design comprised of multiple parts like an engine can be completely evaluated for mechanical fit and tolerances. Not something humans can easily do with high accuracy.

As to reducing science to a formal proof, that is not how modern science works. Science is math model based. A math model is mathematically simulated and the results compared to observation. The BB Theory was simulated. Objects like galaxies appeared. The creative process of generating a model can not be reduced to linear Aristotelian Logic, so to speak.You are not going to create a human mind through a set o rules. Software is in the end applying rules as algorithms.

I read a book on Goedel. He considered it possible a brain/mind analog could be raised and taught like a human. He thought if the brain was logically consistent the Incompleteness Theorem would apply.

 

[Speakpigeon]

He thought if the brain was logically consistent the Incompleteness Theorem would apply.

I think the idea would be to understand why it doesn't matter one bit.

One answer is obvious. Our brain is really all we have to make sense of the world, Gôdel or not.

But, maybe, that's just the very short answer.
EB