Is Everything in the Universe Truly a Mathematical Structure?

[MathematicalPhysicist]

So in other words, Goedel used rules that are not of this universe. I must say that was quite an accomplishment!

the term rules of universe is used too loosly here.
in the same manner i could say that the rules of the state are rules of the universe.

i thought that when you said rules of the universe you meant physical rules, and i wonder how quantum mechanics has got to do with godel's theorems or to any other scientifical theory which ascribe to the universe its rules.

 

[Demystifier]

I don't say that's hard to grasp, but if you knew the details of the proofs and the lemmas and postualtes being used then you could have told me what rules of the universe are being used there if there are such rules.

One of the main assumption is computability, i.e., that every result can be obtained by applying an algorithm and performing a FINITE number of steps with it. By this criterion, even the circumference of a unit circle cannot be computed, because pi=3.14159265...
cannot be calculated by a finite number of steps. In my opinion, this implies that Nature is not such an algorithm (a Turing machine), so the Godel theorem is not applicable to the behavior of Nature. (Another possibility is that Nature does not really work with pi, but with a rational number that only approximates pi. Such a Universe could be computable, but not elegant.)

 

[fleem]

the term rules of universe is used too loosly here.
in the same manner i could say that the rules of the state are rules of the universe.

i thought that when you said rules of the universe you meant physical rules, and i wonder how quantum mechanics has got to do with godel's theorems or to any other scientifical theory which ascribe to the universe its rules.

I ask the readers to take a closer look at those four points I made a couple posts ago. Here I'll partly repeat myself and maybe expand on it a bit:

All rules are physical. There's no such thing as unreal rules. There is also only one set of rules. We cannot claim there is a set of rules outside the universe, for to do so is saying those rules cannot be described (cannot interact) with our rules--and that's just silly. Its like saying an object continues to exist while it does not interact with anything else in the universe--thus a proof of Mach's principle. Even our supposition that "the behavior of the universe" is somehow profoundly different from "abstract mathematics" is wrong. Abstract mathematics is just as physical and really occurring in the (very physical!) neurons of a genius' brain, as are any other behaviors of the universe. Note also that I said the universe certainly allows us to make false statements (which are merely statements inconsistent with the postulates we have pulled out of a hat), whether we can prove the statement false, or not, based on those postulates.

I've always kind of wondered why people consider Goedel's incompleteness theorems so enlightening. I admit I may not understand the path he took very well, but what he says is still obvious: Unprovability applies to everything no matter what set of rules we use to examine a statement. This is because all the "rules" we play with are based on unproven presumptions (axioms, postulates) that we pulled out of a hat. No logic can be circular, so it must have a beginning based on unproven assumptions. Yet my point is that even this logic is based on the rules of logic I learned from the universe. So even this paragraph is unprovable.

So what is "consistent" really becomes, like all decisions we make throughout our lives, a problem in statistics based on guesses--we make decisions, and resolve the probability of something being true, by guesstimating and combining probability distributions. And all ideally are based on some geusstimate that the postulates we use are "good for what we use them for".

So its important for scientists to always have a nagging voice in the back of their heads reminding them that, "All your work is based on postulates that might need changing someday".

 

[Mike2]

...
So its important for scientists to always have a nagging voice in the back of their heads reminding them that, "All your work is based on postulates that might need changing someday".

Godel also proved that deductive logic (propositional calculus) IS complete and consistent. The postulates and axiom of deductive logic are NOT going to change. For you will always be backed into a corner as to whether some theory is true or false, thus making the algebra of true and false, namely deductive logic to be the deciding factor in all decisions.

If the laws of physics can be derived from deductive logic or at least given as a mathematical representation of deduction, then there is no arguing about it anymore.

See my home page by clicking on Mike2 in this post and choosing "View Public Profile".

 

[MathematicalPhysicist]

Godel also proved that deductive logic (propositional calculus) IS complete and consistent. The postulates and axiom of deductive logic are NOT going to change. For you will always be backed into a corner as to whether some theory is true or false, thus making the algebra of true and false, namely deductive logic to be the deciding factor in all decisions.

If the laws of physics can be derived from deductive logic or at least given as a mathematical representation of deduction, then there is no arguing about it anymore.

See my home page by clicking on Mike2 in this post and choosing "View Public Profile".

well, if you mean godel's completeness theorem then we assume already that theory is consistent, the theorem says on hilbert's system, that every statetment is provable in a theory T iff for every model of T this model satisfies it, if a theory is not consistent then you can prove that every statement is provable from T as well.
but the theorem is for first order predicate logic, obviously it also works for propositional calculus, but you know already what is stronger.

as i see it you argue something else than the other poster, he argues that the laws of logic are derived from the laws of the universe and not vice versa.

 

[MathematicalPhysicist]

One of the main assumption is computability, i.e., that every result can be obtained by applying an algorithm and performing a FINITE number of steps with it. By this criterion, even the circumference of a unit circle cannot be computed, because pi=3.14159265...
cannot be calculated by a finite number of steps. In my opinion, this implies that Nature is not such an algorithm (a Turing machine), so the Godel theorem is not applicable to the behavior of Nature. (Another possibility is that Nature does not really work with pi, but with a rational number that only approximates pi. Such a Universe could be computable, but not elegant.)

should i recall you calculus where pi is an irrational number which is the limit of a sequence of rational numbers, i think you and others mistakingly mix between the use of maths in physics and maths by its own merits.

 

[Demystifier]

should i recall you calculus where pi is an irrational number which is the limit of a sequence of rational numbers, i think you and others mistakingly mix between the use of maths in physics and maths by its own merits.

I do not see how it is related to what I said.

 

[Demystifier]

http://arxiv.org/abs/0704.0646

I predict that this paper will become famous and frequently cited, not only by those who will like it, but also by those who will not.
Does anyone wants to take a bet?

By the way, I am not one of those who will particularly like it.

Now more then 2 years later let us see if I was right.

At the moment it has 15 SPIRES citations. It is not very impressive, so I was not right that it will become frequently cited.

Still, I think I was right that it will become famous. For example, it has 19 blog links, which is quite impressive.

 

[Demystifier]

Now I've noticed that this thread is moved on Math and Science Software forum from Beyond the Standard Model forum. This thread certainly does not belong to Math and Science forum. Could someone in charge move it on a more appropriate place?

 

[pellman]

I just had a brief look at the paper for the first time today.

The most weird thing with the mathematical universe is that, according to Tegmark, EVERY mathematical structure exists, and no mathematical structure is more real then other. Thus, not only quantum mechanics is real, but also classical mechanics is real, Ptolomei mechanics is real, a model in which the universe is a dodecahedron is real, anything. It seems that almost every paper on physics is correct, provided that a purely mathematical mistake has not been made. In fact, such papers with mathematical mistakes are also correct, because these papers themselves are also mathematical structures (because everything is a mathematical structure).

this is essentially the same as my thoughts on it. If we ask, "why is our universe the way it is, and not some other way?" or "why is there a universe and not merely nothing?" the answer has to be contingent, "baggage" in Tegmark's view, and not in the math itself. The alternative is that all possible universes exist (and maybe some impossible ones). everything happens. There is no answer.Another problem is multiplicity itself. On page 4 Tegmark notes that "if two structures are isomorphic, then there is no meaningful sense in which they are not one and the same". so how do we have a universe containing such a vast number of phenomena described by the same mathematics if there is nothing underneath but the mathematics itself? How are they distinguished? I admit, though, it is likely that I pose this question because I don't really understand Tegmark well enough.

So, if Tegmark is right, what is the point in doing science?

It's fun.