My music projects

The disagreement with Max Tegmark

Max Tegmark is my favorite living scientist.  I think that's a sign that you are a scientist or a musician or artist or poet: when you have a favorite one of that discipline.

(Not only if you get published, sell a record, or discover something new.)

He and I are at odds in one critical intersection of musings.  He thinks reality must be "computable" and that only Godelian-complete "space" is allowed to be part of reality.  He said straight up: Brian but formal languages are not mathematical objects.

Technically he said that Formal Systems aren't mathematical objects. That must be why he has it circumscribed by a red annulus.

Two things. Mathematical democracy and simpler theory compared to one in which only parallels which exist who meet certain strict criteria.

Formal systems, special cases of formal language, have the same ontological primacy as sets, strings, Neptune, and the HyperWebster which may be the context of any TOE. That means the hyperwebster is self similar and recursive (fractal).

The theory of formal languages indicates that formal language theory provides a type of model for classical set theory (i.e., ZF set theory).  Thus sets are contingent upon them (and vice versa) for their existence.

So if Tegmark believes in a level 4 universe where mathematical structures are parallels equipped by the rules of the structure (in essence), that mathematical existence equals physical existence, then there are more general structures than the ones in set theory which represent parallels larger than he considers to exist.

Yet parallel UNIVERSES should be as all-inclusive as possible, and that's not to mention mathematical democracy, that all structures exist.  Show the rules of formal languages to most mathematicians and I bet they would say it's no more or less a structure than groups... And if that caliber of peoples and organizations thinks formal systems (such as investigated by well known logician Raymond Smullyan) are mathematical objects.

For what it's worth, I think formal languages (hence, formal systems; the basis for his map of structure) are mathematical objects.  This is not to mention that this way it obeys Occam's razor: the two theories have intersecting conclusions (reality is a mathematical structure), but mine doesn't arbitrarily discriminate as to whether they are "computable" which is rather anthrocentric.

So we have that (1) Formal languages are mathematical objects and (2) formal languages are more general than "computable structures."  Computable structures would be a type of parallel but not a parallel universe. Give it a fancy name like quasi-parallel universe or quasi-parallel.  Since formal languages encompass computable structures, they are more towards the Universe.

Compression, Randomness, a theory of everything, and a 200TB proof

If the universe is describable (and I think it is if we allow neologisms), will its description fit in 1,000 pages? There is a computer that in two days apparently churned out a 200 TB proof yet my assumption is that the code is probably less than one GB. 200000GB to 1 GB is a mighty high compression ratio. Lossless compression at that. That 200 TB proof contains enough structure to be a proof. One could call the code that produced it a description of the proof.

In fact, 200,000:1 is exceptional levels of structure.

I tried using winRAR to compress various types of things. I attempted to compress white noise, speech, and a perfect 440Hz. I also compressed the output of a random number generator, plainext, and ciphertext. Finally images.

None of them is as high a compression ratio 200000:1 but that pales in comparison to infinite compression:

A sequence of all ones.

That description is a lossless compression of an infinite string to a finite string. So infinite to finite compression is attainable.

The question is will we ever find it?

Imagine the totality of all descriptions. This is included in the so called hyperwebster. Since the universe is describable, the hyperwebster contains that description.

Is that close?

Free Will, Omniscience, whether God plays dice, and a Theory of Everything

I'm having trouble accepting that random phenomena occur in the material universe but even less willing to accept that I don't have free will.  If truly random phenomena occur, that would imply that determinism is false.  Laplace had an idea called Laplace's demon which if feed the demon all the info about all the matter and energy in the material universe it will be able to use physics (or something) to predict what the state of the universe will be.  It seems that if there can be Laplace's demon (which remains to be seen), that would imply determinism, no free-will, and the non-existence of truly random phenomena.  If we assume that either determinism is false, or we do have free will, or truly random phenomena exist, then Laplace's demon does not exist.  

Laplace's demon would seem to be equivalent to an omniscient agent of some kind.  Basically the formulas in the TOE can predict the future and recall the past if fed absolutely every iota of information about the initial state of material such as every quark's momentum and position.  In a sense, the TOE itself is the omniscient agent.  It's just that the omniscience is not revealed until that TOE is programmed into a computer.

So, omniscience exists if and only if Laplace's demon (or something like it) exists.  Also, if we take the stance that we have free will then that strikes down the possibility of Laplace's demon, meaning that the existence of an omniscent being would mean we have no free will.  

And if we do have free will, that would suggest that there are no omniscient beings.

But maybe the universe is closer to a poem than a physics textbook (like a textbook from the future containing a TOE).  Maybe this dance we play with symbols and truth tables with a T and an F, depending on the truth values of the two propositional variable, has no inherent bearing on truth (what is actually true).  Maybe there is no such thing as simply black and white, false and true, maybe there is just relative grammatical correctness.

If the material universe is made of vibrating strings which exist not in the usual 3+1 dimensional space-time, then the set of states of the material universe is like a symphony.  But a symphony has sheet music and whether or not that sheet music has been written down yet, it might still exist.  This sheet music would be a TOE.

So we have a couple of options:
A. Laplace's Demon exists (or can exist)
B. There exists an omniscient agent
C. Determinism
D. Randomness does not exist
E. There is a TOE

OR

1. Laplace's Demon cannot exist (at least in the material universe)
2. Omniscience is impossible (though there may be a "maximally-scient" agent)
3. Non-determinism
3. God does play dice
4. There is no TOE

A scientist at NASA named David Wolpert seems to have proved that Laplace's Demon cannot exist.  He used purely mathematical arguments; there was no reference or appeals made to quantum mechanics.  I forget the details but something about having two omniscient agents try to emulate each other (i.e., copy what the other knows) and that leading to some kind of contradiction.

But argument by contradiction is not tautological in many-valued logics.

Incompleteness

Some curious results surrounding consistency and inconsistency.

A system is called inconsistent if it proves a statement is true while also proving the negation of that statement is also true.

A system is called consistent if it is not inconsistent.

result #1: in an inconsistent system, every statement formulated in that system is both provable (thus considered true) while its negation is also provable (thus considered false).

In an inconsistent system, every statement is both true and false, and provably so. Thus inconsistent systems are viewed as kind of useless.

result #2: Russell proved in 1901 that what was current set theory in his time, the basis for most math, was inconsistent. Thus set theory in Russell's time was rendered useless because the statement like 2+2=Pi is both provable and 2+2 doesn't equal Pi is also provable.

So set theory was revamped in such a way that Russell's result and all other known problems with naive set theory (as it is now called) were avoided. The hope was that set theory would be consistent and that its consistency would be provable. Also, the hope was that set theory would be complete: every true statement was provable.

Gödel came along and proved something that was considered shocking at the time:

result #3: if a system can both express arithmetic and is consistent, then it is incomplete. Incomplete means there is a statement in that system which is both true but not provable.

That's kinda crazy sounding: a system like set theory should be able to express arithmetic. And it can. Now if a system is also assumed consistent (and remember inconsistent systems are "useless"), then there is a statement in that system which is both true and not provable.

Said differently, any system in which all statements in that system which are true are also provable and which can express arithmetic are also inconsistent. This was considered to be a shocking result.

Gödel did so in 1931 by concocting/discovering a way to formalize the statement "This statement is not provable."

For ease/indolence, let S be the statement that says "S is not provable."

Again in bivalent logic, there are two possibilities:

S is true or

the negation of S is true.

Option 1: S is true. Then S is not provable.

Option 2: S is false. Then S is provable. We define every statement which is provable to be true (in that system).

Option 2 is ruled out because we assume the system is consistent in Gödel's theorem. Since the system is assumed to be consistent, it is never the case that a statement and its negation are both true. In option 2, we proved that S is false and true which would make the system inconsistent.

Therefore, option 1 must be the case. IOW, there is a statement (namely S) which is both true and not provable.

Loosely speaking, a system is called incomplete if there is a statement in that system that is both true and not provable. This concludes the sketch of the proof that if a system can express arithmetic and is consistent, then it is incomplete.

If you're working on a problem involving trying to prove something, you might be faced with the possibility that that something is true (which is good in a sense) yet cannot be proved (no matter how much time you have or how clever you are).

This is bad. You might have picked a statement which is true but you will never be able to prove it!

Now the statement S which says "S is not provable" might seem like a trick to push through this argument but several examples of more interesting statements like the "halting problem" are neither provable nor disprovable. Such statements are called formally undecidable.

Incidentally, the existence of a statement S = "S is not provable" is called a self-referencing statement. This is basically where arithmetic is used and needed in the hypothesis of Gödel's incompleteness result: one needs arithmetic in what's now called Gödel numbering in order for S to exist and be a statement.

Also, there are different versions of incompleteness in many-valued logics (ones that "extend" classical logic to more than two truth values) but I am not super familiar with the interaction of incompleteness and many-valued logic. It has been extensively studied; I can say that much.