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The Axiom of Choice Is Wrong (2007)

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Re: The Axiom of Choice Is Wrong (2007)

#141

Post by Tloewald »

cygx wrote: post #93
Tloewald wrote: post #59

Earlier quoted context omitted.

No incompleteness proves that there will be statements that are true or false that cannot be proven to be true or false. Not being able to write all the rules is a completely different and unrelated thing.

No incompleteness proves that there will be statements that are true or false that cannot be proven to be true or false. Careful there: Gödels first incompleteness theorem proves that in a consistent effectively axiomatized formal system, there will be statements that are true or false that cannot be proven to be true or false. So let's assume there were such a thing as objective mathematical reality and that it coul…

I'm being quite careful: 'consistent effectively axiomatized formal system' is implied by my use of the word 'proves'.

Re: The Axiom of Choice Is Wrong (2007)

#142

Post by Jweb_Guru »

naasking wrote: post #138
cygx wrote: post #135

Earlier quoted context omitted.

The point is that according to ordinary QM, any of the states you can construct via superposition are physical. Eg for the qubit, the state space is the Bloch sphere, which is continuous.

Which is a property of our formalism, not reality. Any state we'd care to actually observe necessarily has finite precision, so the internal mechanics of our formalism to model this process utilizing infinities is an artifact only of our formalism, not of reality. Furthermore, quantum computing and classical computing are known to have the same computational power, just different computational complexity, ie. any qua…

> Which is a property of our formalism, not reality. Any state we'd care to actually observe necessarily has finite precision, so the internal mechanics of our formalism to model this process utilizing infinities is an artifact only of our formalism, not of reality.

But this is exactly my point. The fact that our observations have finite precision doesn't say anything about the underlying reality. That's the part that you're asserting. If it also turns out that assuming the underlying system is continuous results in much more compact theories than assuming otherwise, why is the obvious conclusion that spacetime is in fact discrete?

Re: The Axiom of Choice Is Wrong (2007)

#143

Post by monochromatic »

vostok wrote: post #47
llamaz wrote: post #27

Earlier quoted context omitted.

Mathematics is formalised is to avoid this sort of philosophizing. I used to think, for example, that the dirac delta function was mathematical fiction - a mathematical "hack". But then in an engineering control systems class, we did an experiment where we used a step function to approximate a dirac delta function. I could see the results both on the computer screen and in physical reality through a mass-spring-dampe…

> I used to think, for example, that the dirac delta function was mathematical fiction - a mathematical "hack". Can you expand on this? The Dirac delta function is not a function from R to R for example.

Treating it as a function is the fiction (and the reason is probably that the average college freshman doesn't know what a distribution is). But it's "enough" like a function that unless you look too closely, no real problems arise from glossing over the function/distribution distinction.

Re: The Axiom of Choice Is Wrong (2007)

#144

Post by monochromatic »

thethirdone wrote: post #106
IshKebab wrote: post #104

How can an axiom be 'wrong'?

Suppose, I choose "This sentence is false" as an axiom. having it as an axiom allows me to reason from its content that it is false and thus I have a contradiction. In this way at least, that axiom is "wrong".

But the Axiom of Choice doesn't lead to contradictions (unless Zermelo Fraenkel set theory itself contains contradictions). The Axiom of Choice just leads to some non-intuitive results.

Re: The Axiom of Choice Is Wrong (2007)

#145

Post by IshKebab »

dvt wrote: post #110
IshKebab wrote: post #104

How can an axiom be 'wrong'?

An axiom is simply a statement that is defined as being true. Often times on grounds of being self-evident. For example, one of Euler's axioms is "Things that are equal to the same thing are also equal to one another". Whereas Euler's may seem pretty tame, the Axiom of Choice is not. That's why some people think it might be wrong.

Exactly, so unless it leads to contradictions it may be weird and unintuitive but it's not wrong.

Re: The Axiom of Choice Is Wrong (2007)

#146

Post by DigitalPhysics »

If you're interested in the Axiom of Choice, Godel, finitism, pseudo-randomness, complexity, information, and other foundational topics, check out the indie film "Digital Physics" on iTunes, Amazon, or Vimeo. Free packs of trading cards (with gum!) are available too! Check the website.

Re: The Axiom of Choice Is Wrong (2007)

#147

Post by monochromatic »

skissane wrote: post #114
monochromatic wrote: post #112

Earlier quoted context omitted.

Having only read the wiki page about dialetheism, it seems a lot more like philosophy than math or logic. Meh.

The whole topic has both its more philosophical aspects and its more mathematical aspects. If you want to explore the more mathematical aspects, you probably want to start here https://en.wikipedia.org/wiki/Paraconsistent_mathematics and then end up reading something like https://www.amazon.com/Inconsistent-Mathematics-Its-Applicat... EDIT: see also https://ir.canterbury.ac.nz/bitstream/handle/10092/5626/1263...

Thanks for that last link. I'm working my way through it, and there's interesting material here.

Re: The Axiom of Choice Is Wrong (2007)

#148

Post by naasking »

Jweb_Guru wrote: post #142
naasking wrote: post #138

Earlier quoted context omitted.

Which is a property of our formalism, not reality. Any state we'd care to actually observe necessarily has finite precision, so the internal mechanics of our formalism to model this process utilizing infinities is an artifact only of our formalism, not of reality. Furthermore, quantum computing and classical computing are known to have the same computational power, just different computational complexity, ie. any qua…

> Which is a property of our formalism, not reality. Any state we'd care to actually observe necessarily has finite precision, so the internal mechanics of our formalism to model this process utilizing infinities is an artifact only of our formalism, not of reality. But this is exactly my point. The fact that our observations have finite precision doesn't say anything about the underlying reality. That's the part tha…

> But this is exactly my point. The fact that our observations have finite precision doesn't say anything about the underlying reality. That's the part that you're asserting.

I'm not sure what you think I've asserted, but my only actual assertions are that all of modern physics requires only computable abstractions, and thus, continuous abstractions aren't strictly necessary.

I can certainly make the case that a continuous ontology requires more ontological commitments than a discrete ontology, but I haven't said anything along those lines thus far.

> If it also turns out that assuming the underlying system is continuous results in much more compact theories than assuming otherwise, why is the obvious conclusion that spacetime is in fact discrete?

What is the meaning of "compactness"? A proper comparison of theoretical parsimony requires comparing the Kolmogorov complexity. Compressing ontologies positing reals and/or rationals are definitely less parsimonious than ontologies positing only naturals.

Re: The Axiom of Choice Is Wrong (2007)

#149

Post by Jweb_Guru »

naasking wrote: post #148
Jweb_Guru wrote: post #142

Earlier quoted context omitted.

> Which is a property of our formalism, not reality. Any state we'd care to actually observe necessarily has finite precision, so the internal mechanics of our formalism to model this process utilizing infinities is an artifact only of our formalism, not of reality. But this is exactly my point. The fact that our observations have finite precision doesn't say anything about the underlying reality. That's the part tha…

> But this is exactly my point. The fact that our observations have finite precision doesn't say anything about the underlying reality. That's the part that you're asserting. I'm not sure what you think I've asserted, but my only actual assertions are that all of modern physics requires only computable abstractions, and thus, continuous abstractions aren't strictly necessary. I can certainly make the case that a cont…

> What is the meaning of "compactness"? A proper comparison of theoretical parsimony requires comparing the Kolmogorov complexity. Compressing ontologies positing reals and/or rationals are definitely less parsimonious than ontologies positing only naturals.

As far as I can tell, by this argument there is literally nothing that would convince you that anything but a discrete theory was correct. So I don't think it's a very interesting argument, sorry. But thank you for explaining your viewpoint.

Re: The Axiom of Choice Is Wrong (2007)

#150

Post by vostok »

monochromatic wrote: post #143
vostok wrote: post #47

Earlier quoted context omitted.

> I used to think, for example, that the dirac delta function was mathematical fiction - a mathematical "hack". Can you expand on this? The Dirac delta function is not a function from R to R for example.

Treating it as a function is the fiction (and the reason is probably that the average college freshman doesn't know what a distribution is). But it's "enough" like a function that unless you look too closely, no real problems arise from glossing over the function/distribution distinction.

Right. That was kind of my point. It is a "mathematical trick" in the sense that it's not a function, but a distribution.
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