The Mathematical Universe

Mike2

Godel seemed to prove that mathematics is not complete - we can always find an equation which is true but not provable within mathematics. But then again this is true of any axiom of mathematics; we always just accept the axioms of mathematics without proof.

However, finding a TOE is not the same effort as finding every equation of math that is true. We don't expect that it will take ALL of math to describe all of physics. So the incompleteness of math tells us nothing about the completeness of physics.

ccdantas

Apparently, his philosophy is to equate the physical world with mathematics (yes, equate, not a sort of mapping between the two). He argues that this direct equality solves many problems. Actually, he seems to argue that such an equality solves the philosophical problem of whether there is an ultimate reality. Yes, there is one and it is pure mathematics. And everything is revealed! Isn't it obvious?

I may have misunderstood it all (I've only read his shorter paper - Shut up and calculate). In any case, I didn't find any of his arguments brilliant nor convincing. Looks like a very bad philosophy to me.

The fact that we can describe physical phenomena through mathematical reasoning is something much deeper to me and equating both is no solution (again, to me). It's like turning a difficult question into a trivial one as the best way to actually avoid it.

garrett

Mike2: Yep.

lqg: "what nature is is a philosophical question"
No, nature is certainly not a philosophical question. :D

And physics is not the answer, physics is the question... "yes" is the answer. (stolen from W.A.)

Chronos

That is both a hilarious and unsettling thought by Bee, marcus!

Tegmark is not shy by nature. His is a bold paper that could be very important, or a house of cards. It's sufficiently well constructed to be forgiveable, even if it turns out to be a dead end. It is. IMO, the right way to propose new ideas these days. Taking an occasional hit for going out on a limb is not necessarily fatal. 50 or more years ago a flawed paper was a problem, mainly because it took years to get one widely circulated. This is no longer an impediment. In the current Arxiv age, you can publish to your heart's content without fear of it taking a decade to refine, or cast aside your ideas. Intellectual integrity is still a necessary element, but, impeccability is a luxury most researchers cannot afford. The trade off is papers tend to be rushed to press nowadays. I can live with that.

Demystifier

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).

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

rewebster

Old Tegmark wants to be a POPULAR guy --and make everyone happy and 'like' him for what his says.

fleem

Just some general responses to some of the comments I see in this thread:

1. Goedel used the rules of the universe to prove his incompleteness theorem. Therefore the truth in his theorem depends on the consistency of the universe. Therefore his theorem does not apply to the universe.

2. To say there are two separate sets of rules ("the universe" and "logic") is to say each must be describable using the other (otherwise we can claim there is a set of rules that never interacts with the universe--which is just plain silly), which is to say they are the same set of rules. Therefore there is only one set of rules and the "universelogic" is it. The universe is logic, incarnate. For this reason, that will be what we see whether there is a higher machinery or not. So noting it doesn't imply there is a higher machinery.

fleem

More general comments:

3. The universe allows us to make a false statement. So we should be careful to discriminate between false statements and statements that appear consistent with the universe (with apologies to Goedel).

4. Everything that happens in our brains, or in a computer, follows the rules of the universe because those things are part of the universe. For example, if I claim to be using "the square root of minus-one" in some calculation, I am lying (or mistaken). A close examination of the neural signals in my brain would prove the square root of minus-one never appears, nor anything implying that it has some reality to it.

MathematicalPhysicist

I really wonder how many of you actually read the various proofs of godel's incompletness theorems, and know what axioms are being applied there.

what are the rules of the universe? and which of them godel actually used in his proof?

Demystifier

I have been reading the simplified versions given by Penrose in his semi-popular books, which seemed relatively easy to grasp and understand even intuitively. It seems to me that Godel's theorems do not have true practical implications, although I am not completely sure about that.

Demystifier

Well, he certainly did not make ME happy. And according to the Bee's faces, it seems that it made her unhappy too.

fleem

So in other words, Goedel used rules that are not of this universe. I must say that was quite an accomplishment, considering that the neurons in his brain most likely were, in fact, part of this universe.

MathematicalPhysicist

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.

look at smullyans' book on godel's incompletness theorems, gives his twist to the proofs.

MathematicalPhysicist

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

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

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

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.

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