A maths question for a physicist

AlexPontik · 2021-05-12 23:57:15

Does the below argument make sense?

1. Why say that the following phrase is nonsense?
“If a logical system is consistent, it cannot be complete.”
Because:
The phrase “if a logical system is consistent, it cannot be complete”, is itself a logical system, it is consistent with what it says, and if that is so, something is missing from this phrase, according to what the phrase says. And so this bring us to the second phrase.

2. Why say that the following phrase is nonsense?
“The consistency of axioms cannot be proved within their own system.”
Because:
A system which has axioms for itself, in order for the system to call them axioms for itself, the system has to have a consistent behavior around those axioms and so when it behaves inconsistently with regard to those axioms, the inconsistency between those axioms and the system’s behavior the system can prove to itself.
If what is written above is false, then when a system behaves inconsistently with regard to some axioms it has for itself, that inconsistency it cannot prove to itself, and it keeps behaving inconsistently with regard to those axioms…but…
if the system keeps behaving inconsistently with regard to some axioms and cannot prove to itself that it does so with regard to those axioms, then it doesn’t seem to me it can consistently keep regarding them as axioms for the system, and then something else replaces them, and that something else is what the system calls axioms for itself.

Kind regards

Last edited by AlexPontik (2021-05-13 00:06:16)

Bob · 2021-05-13 03:25:17

hi AlexPontik

Arhh! The good old Gödel Incompleteness Theorems.

You've posted this before I think.

At University level and above, there's a group of mathematicians that think this sort of thing is important. For the rest of us, life goes on, safe in our ignorance about what it is and why it is important (if indeed it is).

Mathematical modelling requires a set of axioms, so we can be sure what the originator is talking about and a set of proofs so we can get something useful out of the model.

Ever since the Russell paradox: "If the barber shaves everyone who doesn't shave themselves, then who shaves the barber?"

mathematicians have worried about axiom systems and whether there's some mighty flaw in the way models work. It's important because, if there is a flaw, then thermodynamics, astronomy, electromagnetism etc etc comes tumbling down and nothing can be safely modelled. If we are going to use maths then we want it to work.

What Gödel did was to show that axioms on their own aren't enough. He showed there exists theorems of this kind:

"This theorem cannot be proved".

Because if it can, then we have a logical inconsistency.

As a physicist, you are used to theories becoming popular and then being overtaken by new knowledge. eg. Newtonian mechanics worked well for centuries and even allowed the discovery of new planets, but got overtaken by Einstein's relativity. And now that's in trouble because of quantum theory. And as they smash particles with ever increasing energy, new stuff pops into the Universe.

But, in the world of physics (and several dozen other disciplines that use maths), we can always do the experiment to find out if the model is any good. For a brief while during my undergraduate years I did a course that 'proved' Gödel's theory. It was hard work and I couldn't re-hash any of it now. But I'm prepared to accept it, within the narrow world of logic. Have I been able to live with this shattering knowledge? Funnily enough, yes I have. The World doesn't know about Gödel, so it keeps turning anyway.

I recommend you use your brain power for something else and stop worrying about unprovable theorems. The World (and this forum) needs you!

Best wishes,

Bob

AlexPontik · 2021-05-14 00:16:40

Hi Bob,

thanks for you reply, let me comment on some of your phrases, which grabbed my attention, and you can guide me according to your view on these, so that I fully get your thinking.

Bob wrote:

Mathematical modelling requires a set of axioms, so we can be sure what the originator is talking about and a set of proofs so we can get something useful out of the model.

I would put it as, in mathematical modelling axioms seem to exist, which commonly make sense to people other than the model's originator.

The ability of humans to make sense of what seems to be happening around them, is not unique in humans, other animals do so as well, but humans seem to be the only ones in this planet using their senses commonly with written language, meaning that they can write things down, and what is written down can be common sense to many others.

When words in some order make sense to humans regardless of their background, time or place, then this is usually what humans call common sense, for example:

A dog defecating on the street is probably an unpleasant site, if you are not the dog's owner, and even then things don't look all that bright for the viewer, but most people wouldn't say that what makes sense for the dog is to go to a toilet and make sure it doesn't make a mess, as it is a dog, and most people get that.

A human defecating on the street however, is a completely different story and most humans get that, some humans don't get that, and there are also the one who like to pretend that they don't get that, and what in the end applies commonly for humans, is what humans commonly get (pretty straightforward argument if you ask me, but I may be wrong in my thinking).

Bob wrote:

Ever since the Russell paradox: "If the barber shaves everyone who doesn't shave themselves, then who shaves the barber?"

The barber shave himself once and after that commits suicide, is what seems to be happening here to me, and that was all the story about this magical barber...

Bob wrote:

What Gödel did was to show that axioms on their own aren't enough. He showed there exists theorems of this kind:
"This theorem cannot be proved".
Because if it can, then we have a logical inconsistency.

If this theorem cannot be proved, then it doesn't make sense to humans to write that theorem down, or it doesn't make sense to keep it written down as it is unchanged, because...

otherwise if this theorem cannot be proved, then it does make sense to humans to write that theorem down, and it does make sense to keep it written down as it is unchanged, but...

if in the end this theorem cannot be proved, and it does make sense to humans to write that theorem down, and it does make sense to keep it written down as it is unchanged, it doesn't seem to me that what humans write down ends up making sense to them, nor that that making sense was their intention of writing that theorem down.

Bob wrote:

As a physicist, you are used to theories becoming popular and then being overtaken by new knowledge. eg. Newtonian mechanics worked well for centuries and even allowed the discovery of new planets, but got overtaken by Einstein's relativity. And now that's in trouble because of quantum theory. And as they smash particles with ever increasing energy, new stuff pops into the Universe.
But, in the world of physics (and several dozen other disciplines that use maths), we can always do the experiment to find out if the model is any good. For a brief while during my undergraduate years I did a course that 'proved' Gödel's theory. It was hard work and I couldn't re-hash any of it now. But I'm prepared to accept it, within the narrow world of logic. Have I been able to live with this shattering knowledge? Funnily enough, yes I have. The World doesn't know about Gödel, so it keeps turning anyway.

As a physicist, this is what seems to me to be happening:
1) Humans have a current theory of physics, which describes what seems to be happening to some degree.
2) A new theory is proposed, which described what seems to be happening to a degree closer to reality, and usually can be simplified to the old theory (the old theory wasn't untrue, it was close enough to reality, but now something closer is proposed).
3) Experiments have to be done, in order to verify that the new theory seems to be happening in reality.

Humans are not imagining stuff, and stuff happens around them.
Stuff happens around humans, and humans can imagine how that is.
And what seems to be happening in the end is something else than anything a human can imagine, even though any human can say it.

Bob wrote:

I recommend you use your brain power for something else and stop worrying about unprovable theorems. The World (and this forum) needs you!

if you make a theory with unprovable theorems, you don't have to worry about a lot, as you are already are not making much sense, it seems to me.
If you are trying to make sense however, what you theorize about, has to be common sense for others, and the way this is common sense for others, is their proof, not yours in the end.
What I mean is I welcome any input, however I didn't decide to spend my time and effort writing here, so that I prove something to myself, I am doing so as I cannot disprove to me what I wrote, and what I wrote is pretty simply written for anyone who wants to discuss that exactly, it seems to me.

(and arguments of the type, Godel's language is too advanced for the common mind, is too advanced for the common minds in this forum)

Bob · 2021-05-14 01:09:56

Theorems can exist which cannot be proved. Most of physics starts out with empirical laws which are maybe proved later. eg Kepler came before Newton.

The point about the barber is you cannot say who shaves him without getting tied up in a paradox. The same can be expressed in terms of library catalogues. I could create a catalogue listing all the books with a red cover. If I give it a red cover then it contains itself ie one of the books listed is the catalogue.

I could also create a catalogue of all the catalogues that don't contain themselves. The catalogue in the last paragraph isn't in it.

But should I put the catalogue in itself. If it's in then it shouldn't be as it's only supposed to list those that don't contain themselves. But if it isn't in itself then it's one of those that don't contain itself so it should be in.

This creates a whole confusion surrounding set theory and that was being used as the basis of number theory. and so the debate about an axiomatic approach to maths began. (Euclid never had this problem! )

But I am prepared to leave all that to those who want to think about it and stick to simpler topics.

Bob

AlexPontik · 2021-06-05 03:11:59

…reality for mathematicians, either is inconsistent, or incomplete…and the opposite they cannot prove to them-selves within the system…before they lose their balance in reality…they have no idea…and let’s say ok with all of these but…

Argument

Physics as a science, progresses as follows:
1.There is a current theory, at any given time.
2.A candidate theory, which is more exact regarding what really is happening appears from research as a proposed new theory.
3. Experiments have to be conducted to verify the new theory.
4. When experiments are conducted, they can have the following results.
5. Nothing happens, the experiments fail to show any results, which has happened in the past.
6. Something happens, the experiments had the expected results, which has hap-pened in the past, and science keeps following its path.
7. Something else happens...which was the case with some previous experiments...or else we wouldn't be looking for a new theory, as then all experiments would point only to something, and nothing else...but up to now, this isn't the case, and the future still happens next, and not before next happens.
8. What seems to be happening, is that before people actually make things in their lives that do something...they make things that don't do something exactly...and they find that early at best, or late at worst...but the complete story they all know from the be-ginning, pretty consistently, it seems to me...as it could be the case with the argument I am making here and below.

And all the above in summary is

AXIOM: In any experiment conducted in reality, nothing can happen as a result, some-thing can happen as a result, or...something else can happen as a result.

This is an axiom that seems consistent and complete to me, and I dare say...logical.

Isn't it?

Math Is Fun Forum

#1 2021-05-12 23:57:15

A maths question for a physicist

#2 2021-05-13 03:25:17

Re: A maths question for a physicist

#3 2021-05-14 00:16:40

Re: A maths question for a physicist

#4 2021-05-14 01:09:56

Re: A maths question for a physicist

#5 2021-06-05 03:11:59

Re: A maths question for a physicist

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