the absoluteness of mathematics (and laws)

[angelofdeath9308]

So I was talking to a friend about various topics, and physics came up. We talked about big bangs and other things. In the discussion he mentioned something that I read about a while back, and a question popped into my mind. If the laws of physics break down in a singularity, does that mean that laws of logic (i.e. Something cannot be both "A" and "not A") and mathematics (1+1=2) break down too?

 

[Cythraul]

That's an interesting question. I guess one of the questions that needs to be answered is what exactly the laws of logic govern. Are they just a condition for intelligibility; are they required simply for a propostion or argument to make sense, or does physical reality abhor contradictions? I think it was Kant who first posited the existence of a noumenal reality inaccessible to our comprehension which is not comformable to the laws of logic. I think it's a mistake to regard the laws of logic as anything more than the rules not just governing arguments and propositions but presupposed in arguments and propositions. Take the law of non-contradiction as an example: It is not the case that it is (both) the case that p and not the case that p. Whatever proposition you substitute for p, you always end up with a true statement. So substitute "it is raining" for the variable p and you get "It is not the case that it is both the case that it is raining and not the case that it is raining." The resulting statement tells you nothing about whether it is in fact raining or not so in that sense I think the laws of logic tell you nothing about the world; they place no constraints on reality. But then I think that this is not really answering your question but I really haven't delved into this question a whole lot so...Oh and the math thing. Well, basic arithmetic is just tautological. 2 + 2 = 4 is true just by virtue of the definitions of "2" "+" "=" and "4". I'm not sure if any of this goes any way towards answering your question but I'll think some more about it and post again.

edit: Honestly, I'm not really sure how to go about answering your question. I don't really know how to picture a world where physical laws completely broke down. I think it's just always incoherent to assert a contradiction regardless of the state of the world. There is a strand in modern logic called "tri-value logic" where the conjunction of p and ~p is regarded as a genuine possibility. However, I think that 1) tri-value logic was created to deal with certain paradoxes and as far as I'm concerned paradoxes are usually peculiar turns of language, not genuine threats to the rules of rationality and 2) tri-value logic just seems straightforwardly self-refuting. If you accept the notion of truth and falsity then the conjunction of p and ~p is never an option.

 

[angelofdeath9308]

Its late at night, and I'm trying to figure something (anything)about this out. I'm no physics major, so I'm going off pure reasoning and any physics I already know. (question directed to anyone) But would agree that it is safe to say that physics is based on mathematics. I'm guessing some of it is, since I dont know everything about it, but take a look at these laws (I guess you dont have to read all of it if you already know the jist of it):

thermodynamic laws
First law of thermodynamics
The change in internal energy of a system is the sum of the heat transferred to or from the system and the work done on or by the system.
Second law of thermodynamics
The entropy -- a measure of the unavailability of a system's energy to do useful work -- of a closed system tends to increase with time.
Third law of thermodynamics
For changes involving only perfect crystalline solids at absolute zero, the change of the total entropy is zero.
Zeroth law of thermodynamics
If two bodies are each in thermal equilibrium with a third body, then all three bodies are in thermal equilibrium with each other.

“Archimedes' Principle”
Glossary
Definition: When a body is wholly or partly immersed in a fluid, it experiences an upthrust or buoyant force equal to the weight of fluid it displaces.

I'm assuming that mathematics was involved in the formation of these laws. If it is so that mathematics is the basis of physics, and if it is true that the laws of physics break down in a singularity (which I'm still having trouble comprehending), then mathematics too would break down. To bad I'm not a physics major or anything, maybe I'd understand this stuff a little better. Anyways, I'd still like to hear other's thoughts on this, maybe they'd know more and could shed a little more light.

 

[angelofdeath9308]

edit: Honestly, I'm not really sure how to go about answering your question. I don't really know how to picture a world where physical laws completely broke down. I think it's just always incoherent to assert a contradiction regardless of the state of the world. There is a strand in modern logic called "tri-value logic" where the conjunction of p and ~p is regarded as a genuine possibility. However, I think that 1) tri-value logic was created to deal with certain paradoxes and as far as I'm concerned paradoxes are usually peculiar turns of language, not genuine threats to the rules of rationality and 2) tri-value logic just seems straightforwardly self-refuting. If you accept the notion of truth and falsity then the conjunction of p and ~p is never an option.

yeah, I agree with you man, I can't really picture a universe where laws break down either. question; What is p and ~p? does it mean "p" and "not p," if so, I cant say I've ever heard of a system that says these two can coexist. But I'm too tired to research for now. I'll be back later.

 

[Cythraul]

yeah it means p and not p.

 

[Alwin]

a small example from maths: suppose we have a variable x and a constant a who satisfy

ax = 3, then x = 3/a if a is not zero

consider x = 3/a as the world outside the singularity

then a = 0 is the singularity; in this case x can be any number and the "law" x = 3/a breaks down

 

[Majesty]

Since we created maths of course it can be exact.

 

[veil the sky]

So I was talking to a friend about various topics, and physics came up. We talked about big bangs and other things. In the discussion he mentioned something that I read about a while back, and a question popped into my mind. If the laws of physics break down in a singularity, does that mean that laws of logic (i.e. Something cannot be both "A" and "not A") and mathematics (1+1=2) break down too?

no they don't.

simply because physics exists by virtue of the physical world as a factual refernent. a singularity is ontologically different to the physical world in which the laws of physics have been recognised.

mathematical principles refer to intellect before they refer to the physical world.

as cythraul suggests this is an essentially kantian notion, but i think a correct one.

fundamental mathematical principles are primary notions through which we subsume our understanding of the world, rather than things we learn from it. they may not be meaningful in a hypothetical 'singularity' but only because of the lack of intelligent observer, which is presupposed in talking about mathematics in the first place! so it stands and falls as it is, not contingent upon conditions of the cosmos.

 

[Birkenau]

I'm rather dumb when it comes to physics, but I was under the impression that under Einstein's law of special relativity, only length and time become variables, and light remains a constant. Because numbers are simply a split of individuals or a count of something, they wouldn't be a changed in a singularity, however if we accounted for something here on earth, like say I had 2 apples, in a singularity there would still be two apples, but of totally different size and nature, and these could be split into 4 due to their comparitive size on earth. Fuck I don't think any of that makes any sense or has any scientific backing, but I'll post this anyway.

 

[Ganymede]

I'm rather dumb when it comes to physics, but I was under the impression that under Einstein's law of special relativity, only length and time become variables, and light remains a constant. Because numbers are simply a split of individuals or a count of something, they wouldn't be a changed in a singularity, however if we accounted for something here on earth, like say I had 2 apples, in a singularity there would still be two apples, but of totally different size and nature, and these could be split into 4 due to their comparitive size on earth. Fuck I don't think any of that makes any sense or has any scientific backing, but I'll post this anyway.

Of course, because of quantum mechanics, Einstien's theory os spatial relativity is completely obsolete. Quantum mechanics of course does create many paradoxes that essentially built a wall in the path to unified theory. All variables can exist simultaneously (i.e. x=a and x=b, b being diffent from a). Breaking physics down to a state of singularity, however, is something I don't exactly understand. Of course, when it comes to physics, I now about as much of that as speaking Chinese.

 

[Demiurge]

yeah, I agree with you man, I can't really picture a universe where laws break down either. question; What is p and ~p? does it mean "p" and "not p," if so, I cant say I've ever heard of a system that says these two can coexist. But I'm too tired to research for now. I'll be back later.

There is such a thing. This is a good article on dialetheism: http://plato.stanford.edu/entries/dialetheism/

 

[derek]

I've always liked the idea that the rules of logic and mathematics could just, one day, flip and reverse themselves...makes for some interesting situations

However, that was just filler. I like this thread, but I'm far too stoopid to contribute. ^Nice article tho, cheers.

 

[Ravenous Enemy]

e^(i&#960 = -1 ... explain THAT!

(π = pi)

 

[Lord Foul]

If the laws of physics break down in a singularity, does that mean that laws of logic (i.e. Something cannot be both "A" and "not A") and mathematics (1+1=2) break down too?

I contest the idea that the laws of Physics break down in a singularity. A singularity is a methematical construct that fits certain hypotheses well, not an observed phenomena. It is actually the mathematical description of natural laws that breaks down in a singularity, as our system of math (as well as our intellect) is ill-equipped to handle the idea of infinities.

If there are indeed singularities, we presently lack the means to describe what happens "inside" them (if they can be said to have an inside). That is not the same as saying that there are no laws governing singularities.

As for logic and mathematics, they are not exaclty the same thing, but in essence one is an extension of the other. Logic (and by extension, math) doesn't need any special cosmological event in order to break down, you can do it in your own home. Paradoxes abound: http://en.wikipedia.org/wiki/Autological

Edit: I would be doing a great disservice to a smart man if I did not also mention http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorem

 

[Alwin]

A singularity is a mathematical construct that fits certain hypotheses well, not an observed phenomena. It is actually the mathematical description of natural laws that breaks down in a singularity, as our system of math (as well as our intellect) is ill-equipped to handle the idea of infinities.

Indeed, issue solved. Next, please!

 

[Alwin]

e^(i&#960 = -1 ... explain THAT!

(π = pi)

go see your tutor

 

[Ravenous Enemy]

Thanks but no thanks.

Not really anybody can explain this - it just so happens that when the exponential funtion is raised to the power of the product of the square root of -1, i, and pi, we get -1... why should all these things come to this? Pretty much inexplicable, that's all.

 

[Mormagil]

It's not inexplicable if you use infinite series.