Mathematical realism (a tought experiment)

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Telmo Menezes

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Jun 12, 2020, 5:56:36 AM6/12/20
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Hello all,

I've been reading here often the claim that physics is about the "real stuff" and math is a human construction that helps us make sense of the real stuff, but it is just an approximation of reality. So here's a thought experiment on this topic.

Let us imagine I program a digital computer to keep iterating through all possible integer values greater than 2 of the variables a, b, c and n. If the following condition is satisfied:

a^n + b^n = c^n

then the computer turns on a light. I let it run for one year. Will the light turn on during that year?

So my questions are:

(1) Can you use theoretical physics to make a correct prediction?
(2) Can you use math to make a correct prediction?

Notice that I am asking a question that is as hard-nosed as it can be. No metaphysics, just a question about an observable event in a physical system during a well-defined time period. Will the light turn on?

What gives?

Best,
Telmo

Brent Meeker

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Jun 12, 2020, 2:40:08 PM6/12/20
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On 6/12/2020 2:55 AM, Telmo Menezes wrote:
Hello all,

I've been reading here often the claim that physics is about the "real stuff" and math is a human construction that helps us make sense of the real stuff, but it is just an approximation of reality. So here's a thought experiment on this topic.

Let us imagine I program a digital computer to keep iterating through all possible integer values greater than 2 of the variables a, b, c and n. If the following condition is satisfied:

a^n + b^n = c^n

then the computer turns on a light. I let it run for one year. Will the light turn on during that year?

So my questions are:

(1) Can you use theoretical physics to make a correct prediction?

Yes.  Theory of theoretical physics includes arithmetic and in fact your question assumes it.


(2) Can you use math to make a correct prediction?

Not unless the math can predict how fast the computer runs and how reliable it is.

Brent


Notice that I am asking a question that is as hard-nosed as it can be. No metaphysics, just a question about an observable event in a physical system during a well-defined time period. Will the light turn on?

What gives?

Best,
Telmo
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Telmo Menezes

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Jun 12, 2020, 11:13:01 PM6/12/20
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Am Fr, 12. Jun 2020, um 18:39, schrieb 'Brent Meeker' via Everything List:


On 6/12/2020 2:55 AM, Telmo Menezes wrote:
Hello all,

I've been reading here often the claim that physics is about the "real stuff" and math is a human construction that helps us make sense of the real stuff, but it is just an approximation of reality. So here's a thought experiment on this topic.

Let us imagine I program a digital computer to keep iterating through all possible integer values greater than 2 of the variables a, b, c and n. If the following condition is satisfied:

a^n + b^n = c^n

then the computer turns on a light. I let it run for one year. Will the light turn on during that year?

So my questions are:

(1) Can you use theoretical physics to make a correct prediction?

Yes.  Theory of theoretical physics includes arithmetic and in fact your question assumes it.

So we can conclude that arithmetic is part of physical reality, at least as much as any other thing that physics talks about?

(2) Can you use math to make a correct prediction?

Not unless the math can predict how fast the computer runs

It doesn't matter how fast the computer runs, and we know this thanks to a mathematical proof, not a theory in physics. And that's how we know how this particular physical system will behave.

and how reliable it is.

If we use Newton's laws to predict the movement of a ball, we assume that someone will not show up and kick it around, that the ball is not unbalanced, etc. Maybe I can suggest a system with an uneven number of redundant computers and such a simple voting mechanism that a probability of failure is infinitesimal, like NASA used to do. It can even be geographically distributed. The voting mechanism itself can become decentralized, something like the bitcoin network. Or I can become more creative and look for some Turing-complete phenomena in nature. You get my point.

Telmo.

Brent



Notice that I am asking a question that is as hard-nosed as it can be. No metaphysics, just a question about an observable event in a physical system during a well-defined time period. Will the light turn on?

What gives?

Best,
Telmo
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Jason Resch

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Jun 13, 2020, 4:08:04 AM6/13/20
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Excellent question Telmo!  I arrived at a very similar thought-experiment in the past, writing:

In fact, incompleteness is not limited to mathematics and mathematical problems, but extends into physical systems too. Consider an intricate arrangement of dominoes. The question of how long it takes for the last domino to fall after the first is toppled is a purely physical question having some definite answer.

Likewise, a physical computer is a physical system, and questions about its future behavior can be framed as a physical problem. For example, we could ask how long after pushing the power button will it take for the screen to light up. But things get murky in the case the computer runs a computation before turning the screen on.

Let’s say the computer runs a search for a proof of some unproven statement when it is turned on, and only when it completes does it light up the screen.  Now the physical question of how long it takes for this physical light to switch on is reduced to a mathematical problem. Where things become very unclear is when due to incompleteness, the computer might never find such a proof.

It turns out that some physical questions cannot be answered without solving fundamental problems in the foundation of mathematics.
 
So where things get hairy is when the computer is not only looking for some example which may or may not exist, but a proof which may not doesn't exist in the generally assumed/accepted system of axioms. Then if we want to answer this purely physical question of "will this light ever turn on?", we need to delve into foundations of mathematics. We get dragged into the mathematical debate of what system of axioms allows a proof to be found, and is that system of axioms consistent?

There's no way of escaping it that I see.

Jason

Telmo Menezes

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Jun 13, 2020, 4:22:39 PM6/13/20
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Am Sa, 13. Jun 2020, um 05:01, schrieb Brent Meeker:


On 6/12/2020 9:25 PM, Telmo Menezes wrote:


Am Sa, 13. Jun 2020, um 04:08, schrieb Brent Meeker:


On 6/12/2020 8:12 PM, Telmo Menezes wrote:


Am Fr, 12. Jun 2020, um 18:39, schrieb 'Brent Meeker' via Everything List:


On 6/12/2020 2:55 AM, Telmo Menezes wrote:
Hello all,

I've been reading here often the claim that physics is about the "real stuff" and math is a human construction that helps us make sense of the real stuff, but it is just an approximation of reality. So here's a thought experiment on this topic.

Let us imagine I program a digital computer to keep iterating through all possible integer values greater than 2 of the variables a, b, c and n. If the following condition is satisfied:

a^n + b^n = c^n

then the computer turns on a light. I let it run for one year. Will the light turn on during that year?

So my questions are:

(1) Can you use theoretical physics to make a correct prediction?

Yes.  Theory of theoretical physics includes arithmetic and in fact your question assumes it.

So we can conclude that arithmetic is part of physical reality,

No, you can conclude it's part of theories of physics.


It points to underlying reality at least as much as a physical theory does, that's my point.

I agree.  But what points is distinct from the thing pointed to.




at least as much as any other thing that physics talks about?

(2) Can you use math to make a correct prediction?

Not unless the math can predict how fast the computer runs

It doesn't matter how fast the computer runs, and we know this thanks to a mathematical proof, not a theory in physics. And that's how we know how this particular physical system will behave.

No we don't.  What happens when you runs out of registers to contain the numbers?


In that case an exception is triggered and nothing happens. The light doesn't turn on. Will it turn on before exhausting whatever memory space is available?

Not if it perfectly reliable.  But then why not just postulate a computer whose light is burned out?   Is there something special about Fermat's last theorem, now that we know the answer?  You've made it seem profound, but it's logically equivalent to a program that says, "Don't turn on the light."

I'm not trying to sound profound. What I am trying to do is to confront the idea that empiricism is the only way to figure out a world where the only real things are the ones that "kick back". As far as I can tell, this very real question can only be solved in the platonic realm. No actual experimentation will help settle it -- although I concede that it will help adjust your bayesian priors. I think this is interesting.

When Andrew Wiles proved Fermat's last theorem, was he doing physics?

- If yes, then he provided an answer for a question about systems that "kick back" without any empirical grounding whatsoever.

- If no, then physics has to share the stage with math.

Do you believe I am missing an option?





and how reliable it is.

If we use Newton's laws to predict the movement of a ball, we assume that someone will not show up and kick it around, that the ball is not unbalanced, etc.

Newton also assumed physics was deterministic.


What's your point?

Newton was wrong.  As far as we know now, nothing can be perfectly reliable because all physical processes include some randomness.

Are you sure? I don't possess your level of sophistication in theoretical physics, but as far as I understand, there are two types of randomness:

(1) Non-linear dynamics. In such cases, it's not that we cannot write laws that perfectly describe the system, but in practice we would need extremely high to infinite precision to be sure about the outcome (e.g. weather prediction, throwing dice, etc). I assume we all agree on this, and it doesn't make Newton wrong -- perhaps only a bit ignorant, but we can forgive him given that he lived a long time ago.

(2) Fundamental / primary randomness as a brute fact of reality. This is kind of this topic of this mailing list, right? If MWI is correct, then this sort of randomness is, in a sense, an illusion created by our limited perception of all there is. There is no definite answer to this question, correct?

So, if we agree that we only care about (2) here, I would say that I do not share your certainty.




Maybe I can suggest a system with an uneven number of redundant computers and such a simple voting mechanism that a probability of failure is infinitesimal, like NASA used to do.

An idealization.


Language itself is an idealization. This sort of refutation is applicable to anything one can say.

Exactly so. Which is why you should no more confuse arithmetic with reality than you do Sherlock Holmes.


The only reality that you and me have access to is idealized. Is there such thing as a non-idealized reality? This is a metaphysical question. I won't bother you with discussion on the ontological status of Sherlock Holmes.

Telmo

Brent
As far as the laws of mathematics refer to reality, they are not
certain, and as far as they are certain, they do not refer to
reality.
        -- Albert Einstein

Telmo Menezes

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Jun 13, 2020, 4:29:57 PM6/13/20
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Am Sa, 13. Jun 2020, um 08:07, schrieb Jason Resch:


On Fri, Jun 12, 2020 at 4:56 AM Telmo Menezes <te...@telmomenezes.net> wrote:

Hello all,

I've been reading here often the claim that physics is about the "real stuff" and math is a human construction that helps us make sense of the real stuff, but it is just an approximation of reality. So here's a thought experiment on this topic.

Let us imagine I program a digital computer to keep iterating through all possible integer values greater than 2 of the variables a, b, c and n. If the following condition is satisfied:

a^n + b^n = c^n

then the computer turns on a light. I let it run for one year. Will the light turn on during that year?

So my questions are:

(1) Can you use theoretical physics to make a correct prediction?
(2) Can you use math to make a correct prediction?

Notice that I am asking a question that is as hard-nosed as it can be. No metaphysics, just a question about an observable event in a physical system during a well-defined time period. Will the light turn on?

What gives?



Excellent question Telmo!  I arrived at a very similar thought-experiment in the past, writing:

In fact, incompleteness is not limited to mathematics and mathematical problems, but extends into physical systems too. Consider an intricate arrangement of dominoes. The question of how long it takes for the last domino to fall after the first is toppled is a purely physical question having some definite answer.

Dominos is an an excellent idea for this, much better than mine.

Likewise, a physical computer is a physical system, and questions about its future behavior can be framed as a physical problem. For example, we could ask how long after pushing the power button will it take for the screen to light up. But things get murky in the case the computer runs a computation before turning the screen on.

Let’s say the computer runs a search for a proof of some unproven statement when it is turned on, and only when it completes does it light up the screen.  Now the physical question of how long it takes for this physical light to switch on is reduced to a mathematical problem. Where things become very unclear is when due to incompleteness, the computer might never find such a proof.

It turns out that some physical questions cannot be answered without solving fundamental problems in the foundation of mathematics.
 
So where things get hairy is when the computer is not only looking for some example which may or may not exist, but a proof which may not doesn't exist in the generally assumed/accepted system of axioms. Then if we want to answer this purely physical question of "will this light ever turn on?", we need to delve into foundations of mathematics. We get dragged into the mathematical debate of what system of axioms allows a proof to be found, and is that system of axioms consistent?

There's no way of escaping it that I see.

Exactly. There is something truly bizarre and at the same time unescapable about this.

Telmo

Jason


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Brent Meeker

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Jun 13, 2020, 5:18:05 PM6/13/20
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The empirical grounding was implicitly assumed in supposing that the computer implemented the computation as mathematicians intended so that Wiles proof could be translated into some conclusion about the physical process.  It's really no different than any other testing of mathematics by running a computer.  On the math-fun list, to which I subscribe, there is a discussion right now of whether one should define 0^0=1.  I pointed out that in Lisp (expt 0 0) returns 1.  Does that prove that Lisp is doing physics?  or that the computer programmer made some assumption about mathematical axioms?

Brent


- If no, then physics has to share the stage with math.

Do you believe I am missing an option?





and how reliable it is.

If we use Newton's laws to predict the movement of a ball, we assume that someone will not show up and kick it around, that the ball is not unbalanced, etc.

Newton also assumed physics was deterministic.


What's your point?

Newton was wrong.  As far as we know now, nothing can be perfectly reliable because all physical processes include some randomness.

Are you sure? I don't possess your level of sophistication in theoretical physics, but as far as I understand, there are two types of randomness:

(1) Non-linear dynamics. In such cases, it's not that we cannot write laws that perfectly describe the system, but in practice we would need extremely high to infinite precision to be sure about the outcome (e.g. weather prediction, throwing dice, etc). I assume we all agree on this, and it doesn't make Newton wrong -- perhaps only a bit ignorant, but we can forgive him given that he lived a long time ago.

(2) Fundamental / primary randomness as a brute fact of reality. This is kind of this topic of this mailing list, right? If MWI is correct, then this sort of randomness is, in a sense, an illusion created by our limited perception of all there is. There is no definite answer to this question, correct?

So, if we agree that we only care about (2) here, I would say that I do not share your certainty.




Maybe I can suggest a system with an uneven number of redundant computers and such a simple voting mechanism that a probability of failure is infinitesimal, like NASA used to do.

An idealization.


Language itself is an idealization. This sort of refutation is applicable to anything one can say.

Exactly so. Which is why you should no more confuse arithmetic with reality than you do Sherlock Holmes.


The only reality that you and me have access to is idealized. Is there such thing as a non-idealized reality? This is a metaphysical question. I won't bother you with discussion on the ontological status of Sherlock Holmes.

Telmo

Brent
As far as the laws of mathematics refer to reality, they are not
certain, and as far as they are certain, they do not refer to
reality.
        -- Albert Einstein

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Brent Meeker

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Jun 13, 2020, 5:25:26 PM6/13/20
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On 6/13/2020 1:29 PM, Telmo Menezes wrote:


Am Sa, 13. Jun 2020, um 08:07, schrieb Jason Resch:


On Fri, Jun 12, 2020 at 4:56 AM Telmo Menezes <te...@telmomenezes.net> wrote:

Hello all,

I've been reading here often the claim that physics is about the "real stuff" and math is a human construction that helps us make sense of the real stuff, but it is just an approximation of reality. So here's a thought experiment on this topic.

Let us imagine I program a digital computer to keep iterating through all possible integer values greater than 2 of the variables a, b, c and n. If the following condition is satisfied:

a^n + b^n = c^n

then the computer turns on a light. I let it run for one year. Will the light turn on during that year?

So my questions are:

(1) Can you use theoretical physics to make a correct prediction?
(2) Can you use math to make a correct prediction?

Notice that I am asking a question that is as hard-nosed as it can be. No metaphysics, just a question about an observable event in a physical system during a well-defined time period. Will the light turn on?

What gives?



Excellent question Telmo!  I arrived at a very similar thought-experiment in the past, writing:

In fact, incompleteness is not limited to mathematics and mathematical problems, but extends into physical systems too. Consider an intricate arrangement of dominoes. The question of how long it takes for the last domino to fall after the first is toppled is a purely physical question having some definite answer.

Dominos is an an excellent idea for this, much better than mine.

Likewise, a physical computer is a physical system, and questions about its future behavior can be framed as a physical problem. For example, we could ask how long after pushing the power button will it take for the screen to light up. But things get murky in the case the computer runs a computation before turning the screen on.

Let’s say the computer runs a search for a proof of some unproven statement when it is turned on, and only when it completes does it light up the screen.  Now the physical question of how long it takes for this physical light to switch on is reduced to a mathematical problem.

Only by (the sometimes wrong) assumption that the computer will implement the mathematics you assume.

Brent

Where things become very unclear is when due to incompleteness, the computer might never find such a proof.

It turns out that some physical questions cannot be answered without solving fundamental problems in the foundation of mathematics.
 
So where things get hairy is when the computer is not only looking for some example which may or may not exist, but a proof which may not doesn't exist in the generally assumed/accepted system of axioms. Then if we want to answer this purely physical question of "will this light ever turn on?", we need to delve into foundations of mathematics. We get dragged into the mathematical debate of what system of axioms allows a proof to be found, and is that system of axioms consistent?

There's no way of escaping it that I see.

Exactly. There is something truly bizarre and at the same time unescapable about this.

Telmo

Jason


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Bruno Marchal

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Jun 14, 2020, 5:23:05 AM6/14/20
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On 12 Jun 2020, at 11:55, Telmo Menezes <te...@telmomenezes.net> wrote:

Hello all,

I've been reading here often the claim that physics is about the "real stuff" and math is a human construction that helps us make sense of the real stuff, but it is just an approximation of reality. So here's a thought experiment on this topic.

Let us imagine I program a digital computer to keep iterating through all possible integer values greater than 2 of the variables a, b, c and n. If the following condition is satisfied:

a^n + b^n = c^n

then the computer turns on a light. I let it run for one year. Will the light turn on during that year?

So my questions are:

(1) Can you use theoretical physics to make a correct prediction?

Yes, by using the idea (in theoretical physics) that the physical reality cannot contradict elementary arithmetic.



(2) Can you use math to make a correct prediction?


Yes, by assuming that the physical reality will not contradict Wiles’ proof of Fermat.




Notice that I am asking a question that is as hard-nosed as it can be. No metaphysics, just a question about an observable event in a physical system during a well-defined time period. Will the light turn on?

What gives?

I have done the experience, and the light turned on during that year. My prediction was false, and I thought that an experiment suggest something went wrong in Wiles proof. Then I woke up :) 

Bruno





Best,
Telmo

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Bruno Marchal

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Jun 14, 2020, 5:41:01 AM6/14/20
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No. Nice argument.








and how reliable it is.

If we use Newton's laws to predict the movement of a ball, we assume that someone will not show up and kick it around, that the ball is not unbalanced, etc.

Newton also assumed physics was deterministic.


What's your point?

Newton was wrong.  As far as we know now, nothing can be perfectly reliable because all physical processes include some randomness.

Are you sure? I don't possess your level of sophistication in theoretical physics, but as far as I understand, there are two types of randomness:

(1) Non-linear dynamics. In such cases, it's not that we cannot write laws that perfectly describe the system, but in practice we would need extremely high to infinite precision to be sure about the outcome (e.g. weather prediction, throwing dice, etc). I assume we all agree on this, and it doesn't make Newton wrong -- perhaps only a bit ignorant, but we can forgive him given that he lived a long time ago.

(2) Fundamental / primary randomness as a brute fact of reality. This is kind of this topic of this mailing list, right? If MWI is correct, then this sort of randomness is, in a sense, an illusion created by our limited perception of all there is. There is no definite answer to this question, correct?

So, if we agree that we only care about (2) here, I would say that I do not share your certainty.




Maybe I can suggest a system with an uneven number of redundant computers and such a simple voting mechanism that a probability of failure is infinitesimal, like NASA used to do.

An idealization.


Language itself is an idealization. This sort of refutation is applicable to anything one can say.

Exactly so. Which is why you should no more confuse arithmetic with reality than you do Sherlock Holmes.


The only reality that you and me have access to is idealized. Is there such thing as a non-idealized reality? This is a metaphysical question. I won't bother you with discussion on the ontological status of Sherlock Holmes.

Your thought experience is actually by itself a good answer to Brent. If Fermat’s mathematical truth was of the type of Sherlock Homes sort of reality, it would not be possible to use it to make any physical prediction.

Mathematics is always done when doing physics, and indeed, that is why we have a computer in the head, it computes all the time.
In fact when we look at what the physicists do, what we see are people who bet on some measurable numbers, and infer or extrapolate mathematical relation between those measurable numbers. Such relations are never proved, only inferred, but they might become theorem, in some metaphysics (and that is necessarily the case in Mechanist Metaphysics).

Then some “mystic” people infer that there is a non mathematical origin to those mathematical relations, that they might called God, or Universe, or Matter, but that is the part which looks like Sherlock Holmes … 
We can test Mechanism/physicalism, but we cannot really test mathematicalism, because a machine cannot distinguish a non computable reality from a (mathematical) oracle. The dream argument strikes again.

Assuming some non-mechanism, all positions remains open.

Bruno







Telmo

Brent
As far as the laws of mathematics refer to reality, they are not
certain, and as far as they are certain, they do not refer to
reality.
        -- Albert Einstein


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Bruno Marchal

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Jun 14, 2020, 5:54:15 AM6/14/20
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Incompleteness explains why the mathematical reality will look empirical by the internal self-aware creature of the arithmetical reality. Physics becomes necessarily the product of a sort of mathematical experience.
In particular, the physical reality is not a mathematical reality among others. It is much more fundamental than that, and can be described a universal digital-machine (mathematical) experience made by almost all universal machine in any Turing universal/complete realm.

The conceptually hard problem of consciousness is reduced to a conceptually simple, but mathematically complicated mathematical problem: to derive physics from the mathematical psychology of the universal numbers, which are dreaming, because the arithmetical reality is “build that way” (to talk like Hardy).

Science is born from the discovery of mathematics, behind the psychological, and physical, appearances. 

Bruno



Telmo Menezes

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Jun 14, 2020, 1:54:24 PM6/14/20
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Am So, 14. Jun 2020, um 09:23, schrieb Bruno Marchal:

On 12 Jun 2020, at 11:55, Telmo Menezes <te...@telmomenezes.net> wrote:

Hello all,

I've been reading here often the claim that physics is about the "real stuff" and math is a human construction that helps us make sense of the real stuff, but it is just an approximation of reality. So here's a thought experiment on this topic.

Let us imagine I program a digital computer to keep iterating through all possible integer values greater than 2 of the variables a, b, c and n. If the following condition is satisfied:

a^n + b^n = c^n

then the computer turns on a light. I let it run for one year. Will the light turn on during that year?

So my questions are:

(1) Can you use theoretical physics to make a correct prediction?

Yes, by using the idea (in theoretical physics) that the physical reality cannot contradict elementary arithmetic.

We agree.

(2) Can you use math to make a correct prediction?


Yes, by assuming that the physical reality will not contradict Wiles’ proof of Fermat.




Notice that I am asking a question that is as hard-nosed as it can be. No metaphysics, just a question about an observable event in a physical system during a well-defined time period. Will the light turn on?

What gives?

I have done the experience, and the light turned on during that year. My prediction was false, and I thought that an experiment suggest something went wrong in Wiles proof. Then I woke up :) 

This is the problem with putting too much faith in numbers, it may always be that one can still wake up :) (a little provocation)

Telmo

Bruno





Best,
Telmo

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Telmo Menezes

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Jun 14, 2020, 2:02:29 PM6/14/20
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So my previous joke is not so bad... But I'm not sure I understand correctly what you mean here with "mathematical oracle", can you clarify?

Telmo

Assuming some non-mechanism, all positions remains open.

Bruno







Telmo

Brent
As far as the laws of mathematics refer to reality, they are not
certain, and as far as they are certain, they do not refer to
reality.
        -- Albert Einstein


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Bruno Marchal

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Jun 15, 2020, 5:42:44 AM6/15/20
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On 14 Jun 2020, at 19:53, Telmo Menezes <te...@telmomenezes.net> wrote:



Am So, 14. Jun 2020, um 09:23, schrieb Bruno Marchal:

On 12 Jun 2020, at 11:55, Telmo Menezes <te...@telmomenezes.net> wrote:

Hello all,

I've been reading here often the claim that physics is about the "real stuff" and math is a human construction that helps us make sense of the real stuff, but it is just an approximation of reality. So here's a thought experiment on this topic.

Let us imagine I program a digital computer to keep iterating through all possible integer values greater than 2 of the variables a, b, c and n. If the following condition is satisfied:

a^n + b^n = c^n

then the computer turns on a light. I let it run for one year. Will the light turn on during that year?

So my questions are:

(1) Can you use theoretical physics to make a correct prediction?

Yes, by using the idea (in theoretical physics) that the physical reality cannot contradict elementary arithmetic.

We agree.

(2) Can you use math to make a correct prediction?


Yes, by assuming that the physical reality will not contradict Wiles’ proof of Fermat.




Notice that I am asking a question that is as hard-nosed as it can be. No metaphysics, just a question about an observable event in a physical system during a well-defined time period. Will the light turn on?

What gives?

I have done the experience, and the light turned on during that year. My prediction was false, and I thought that an experiment suggest something went wrong in Wiles proof. Then I woke up :) 

This is the problem with putting too much faith in numbers, it may always be that one can still wake up :) (a little provocation)

You point on a difficult question where I (and you) might not completely agree with Descartes, but I am not sure (right now). I will reread the passage in Descartes to be sure. The question is can we dream that circle are triangular, or that p -> p is not a tautology. From what you say, it looks you think that this is possible, and we agree on this … up to some point (I cannot dream that I am not conscious, for example).

Bruno




Telmo

Bruno





Best,
Telmo

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Bruno Marchal

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Jun 15, 2020, 6:04:11 AM6/15/20
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So my previous joke is not so bad…

I agree :)



But I'm not sure I understand correctly what you mean here with "mathematical oracle", can you clarify?


Take a set S of number. An oracle is some entity that can answer the question “does n belong to S” and give the answer to the machine which will act accordingly.

You can see it as a supplementary control structure like “If x belongs to S do this else do that”.

Of course, if S is a computable/recursive set, the oracle is equivalent with a subroutine. The interest is when S is a non computable set, like the halting set H (the set of all I such that ph_i stop), for example.

A natural question was: can we solve all problem in arithmetic if we give H as oracle to a Turing machine? The answer is no, for example the set TOT (the set of all I such that phi_i(x) is a total function) is still not computable. That means that TOT is more non solvable than H. A machine with TOT as oracle can solve “is n in H”, but a machine with H as oracle can still not solve all question of the type “is n in Tot”. This has led Turing, Post, Kleene to the study of the degrees of unsolvability, which classifies the degree of complexity of the non computable set of numbers definable in arithmetic.

For example, the Turing machine with oracle will contain a new type of quadruplet like q_i S_j q_k q_r, and it means “if the number of one on my tape belong to the set O (the oracle) then I will be in the state q_k, else I will be in the state q_r. 

People note phi_i^O the phi_i that are partial computable with the oracle O. That gives a relativise theory of computability.

As O is not computable, that is akin to a definition of computable with the help of some “god”. That one has a non computable ability, like knowing H, or TOT. All limitations results remains valid. It was thought that such gods/oracles could help to overcome incompleteness, but that is not the case… Turing is the one introducing such “God" in computability theory, and he called them Oracle.

Bruno






Telmo

Assuming some non-mechanism, all positions remains open.

Bruno







Telmo

Brent
As far as the laws of mathematics refer to reality, they are not
certain, and as far as they are certain, they do not refer to
reality.
        -- Albert Einstein


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