Nothing at all hard about the conjectures or the theorems. Be good if you payed attention to the words. It would appear you're getting your "completeness" from problems that are considered NP-complete. There are many Youtube videos on NP-completness, for almost every level of student. Give some a try . — tim wood

Yet, try and specify this syntactically/quantifiably in a formal system, if you can even begin to. Non-trivial stuff...

Interesting ideas, but to my mind it's putting the cart before the horse. I don't see how this sort of knowledge would help prove a theorem, which is the essence of mathematics — jgill

There's something in my mind that I haven't disclosed, and this is more of a sentiment if you don't care or whatnot; but, without knowing that a theorem's proof is more complex or less, that this contributes to mathematics seeming a lot more, how to put this lightly, disorganized or uncategorized? I'm not sure if you follow me on this sentiment. Feel free to comment if not.

What do you mean by the complexity of a proof? I've already suggested that it could be defined as the length of a proof in some formal proof system. What do you mean by it? — fishfry

I keep on repeating myself here; but, I discussed earlier that there's some lack of language with a syntax that could ascertain this. The only example I keep on mentioning is related to Kolmogorov complexity, which is non-computable.

What is a QED? Do you mean a proof? — fishfry

Yes.

I've already suggested a way. I don't know that it's regarded as particularly important but I could be wrong about that. For sure it's important in computer science, and I pointed you to complexity theory. — fishfry

Well, this much I understand; but, the point of this thread is in regards to whether there is any relation to an axiomatic system that is either or consistent and complete and the determination of "truth" as to whether the axiomatic system is subject to degrees of "complexity"? Which, seemingly can only be entertained when providing proofs, and those proofs being subject to a formal measure of complexity...

I'm thinking again of having proven a theorem, putting into a computer algorithm, then counting symbols to ascertain "complexity." I guess that would be some sort of definition, but why one would wish to do that is not clear to me. I suppose one could say, "Oh, Gill's proof of Theorem X has complexity 32, whereas Kojita's proof has complexity 56." Then what? — jgill

Yes, and as to this, I think that it is important in terms of making mathematics more formal in saying that a less complex formal proof for a theorem is important, no?

And by 'complexity' do you mean 'the sum of the lengths of the formulas in the proof'? — TonesInDeepFreeze

Here's the inherent problem as I see it, in that there's no syntactically quantifiable method, not least a semantical qualitative method to make "complexity" determinate. What do you think?

Mathematicians often talk about a theorem or conjecture being qualitatively "hard", yet for "complexity", there's no syntax, or is there?

If all one wants to do is count primitives or the length of the proof, then there's still the burden of determining how to estimate its Turing complexity, yes?

And, based on past reasoning, whether a theorems proof is determinate in regards to complexity in a formal axiomatic system, then the method at determining this is dependent on XYZ, with XYZ helping in determining complexity of a theorem proof's least exhaustive Q.E.D. to be performed by a Turing machine or simply computable.

Let me know if that's a word salad again. :sweat:

"XYZ"-In my mind is either or related to completeness or consistency, which is where my confusion seems to arise.

As far as I can guess, you are asking what I already mentioned:

Given P, is "What is the length of the shortest proof of P?" computable? — TonesInDeepFreeze

Well, yes. I'm concerned with complexity of a theorems proof or what I think is how you can gauge complexity in mathematics, through determining how simple the proof is for any theorem.

Does that sound about right?

Are you asking whether L(P) is computable? — TonesInDeepFreeze

I'm asking if n or m<n is determine in shortest Turing Computable length given P, and L(P) being P's proof.

How long is this stick? Oh, by the way, no means or methods of measuring allowed. — tim wood

Yeah, and that's not even counterintuitive, until I bring into the discussion completeness and consistency of a theorem, which seemingly, as I view it, are needed to determine the shortest length of a proof to determine the complexity of it. And, brute forcing the result of the shortest Q.E.D. is not really that interesting to talk about.

It's interesting @tim wood and @fishfry, that Kolmogorov complexity is not computable, so not sure if this is moot.

May I try? Is it a fair translation to say that you're interested in the length of proofs and determining when a given proof is the shortest? — tim wood

Yes.

As to determining complexity itself, that measure is pre-given, yes? — tim wood

No, according to how I view complexity, it's ascertained or determined by the shortest Q.E.D.

E.g., as the count of the symbols in the proof? In any case you cannot measure anything without first having established a method of measuring and quantifying it, yes? — tim wood

Well, yes. Meaning that the amount of primitives and recursions leads to the least exhaustive outcome for a Turing machine.

A rising tide lifts all 'oats.

Brotherly love:

This sentence repeats the same misunderstanding. Individual theorems do not have the consistency or completeness attributes, in the same sense that automobile tires don't have the horsepower attribute. — fishfry

So, when I say that a proof of a theorem is subject to not being able to determine its complexity does that mean anything?

But completeness and consistency of axiomatic systems seem to have little or nothing to do with your question. If a theorem has a proof and you want to find the simplest one, that has nothing to do with the axiomatic system being incomplete. The axiom of choice has no proof in ZF so it's meaningless to ask about the simplest proof. And if a system is inconsistent, then everything has a one-line proof following directly from the inconsistency. So completeness and consistency are irrelevant to your question. — fishfry

What I'm trying to determine is whether there is any possibility to determine the complexity of proofs by reasoning that a Q.E.D. would occur at the least exhaustive method of determining it. Does that make sense? Following with this logic, if you don't have a method of doing this, then how can you determine complexity in mathematical proofs?

A theorem can neither be consistent nor complete not by virtue of Gödel, but rather by virtue of the fact that the terms consistency and completeness apply to axiomatic systems and NOT to theorems. You seem unclear on this point. — fishfry

Please forgive my lack of knowledge on the matter; but, what I wanted to say that a theorems proof for an axiomatic system in math is unable to be ascertained in complexity with regards to being neither complete and consistent.

With this fact being true, then how does one ever hope to gauge a mathematical theorem's proof as measurably complex or less complex.

Is it possible for you to focus your subject? Can you perhaps give a specific example of what you're getting at? — fishfry

The subject or point in question is whether one can determine a theorem's complexity from the fact that a theorem cannot both be consistent and complete, according to Gödel's Incompleteness Theorems?

Actually, in even more simpler terms to determine the complexity of theorems, supposedly, of greater importance would be a focus on completeness rather than consistency of a theorem, as the proof of a theorem is more concerned with maintaining consistency by all measures.

What do others think? Does this make things sounds somewhat disorganized in how proofs are written?

You are not talking about deriving the proof of a math theorem, you are asking that if I were to conjecture a theorem, then prove it, and then put my proof into some sort of computer algorithm - which is just another way of writing it out - is there a way to determine the "complexity" of my proof by examining the algorithm? Is this what you mean? — jgill

Yes, jgill, that's what I mean. In simpler terms, I don't see how logically you can begin to ascertain "complexity" or Turing complexity of a theorem's proof by working at it in any way without knowing it being complete and/or consistent, which Gödel proved it being impossible to satisfy it being both complete and consistent.

Are you saying a theorem is complete if it has been proven? — jgill

No, I'm saying that to work on a theorem in terms of "complexity", which doesn't even have unitary measure or quantifiable methodology to start with, then how does one ascertain this with the proof being moot in terms of both completeness and consistency.

Hope that makes some sense.

Then provide your reasoning. I don't see what your getting at. Everything written in the OP is just what it says.

What do you mean by a theorem being complete? — TonesInDeepFreeze

There aren't many to talk if any, and Godel's Incompleteness theorems apply retroactively, to all that have denumerable and countable alphabets. However, some uncountable alphabets exist, which leaves open the possibility that a theorem may be complete, yet possibly incomprehensible.

What do you mean by "determine a QED result for a Turing machine"? — TonesInDeepFreeze

That the Turing complexity is determinate or definite.

What is a theorem's complexity? — tim wood

I think I can help with this question. It simply means the number of steps to determine a Q.E.D. result for a Turing machine. Of course there's no units to measure this by, which reverts us back to the original question.

Complete? At infinity? That means it never is, yes? — tim wood

Well, hypothetically, and I firmly believe that completeness isn't impossible to entertain, that by expanding the Gödel alphabet up to a certain integer, that we might arrive at a proof that is least complex for a theorem.

He did write an article titled, "On the Length of Proofs," the gist of which is that by adding axioms 1) unprovable theorems may become provable, and 2) many long proofs may be shortened. — tim wood

Yes, you did mention this previously, which is of interest to me. Yet, as mentioned that given the Second Incompleteness Theorem, you cannot have consistency, which aligns with no way of determining complexity, even without the previous issue with no unitary measure of determining it at all!

And, another approach is to determine the amount of information a proof "contains" informationally and then be able to ascertain this with a metric of a quantum computer, I think?

Or in other words how are mathematical theorems and proofs scrutinized in terms of complexity if completeness cannot be determined first.

Another way of stating the OP in terms of a question seemingly would be:

At which point when the entropic state of a localized system cause decoherence between entangled states of two systems?

It's actually not that trivial.

See, the sparse google results for an inquiry on the issue.

https://www.google.com/search?q=quantum+mechanics+and+entropy

Yeah, I suppose my question is simply the relation between quantum mechanics and entropy in general for localized systems, and seemingly nobody can provide any information on that relation...

Thanks anyway?

I'm inclined to suppose that time evolution of stochastic systems describe the tendency of fit for stochastic measurements to produce decoherence as T grows minutely, which might be what you are mentioning about the Yang-Mills gap, yes?

Well, I know of only von Neumann entropy in stochastic Quantum Mechanics. Yet, there's not really any literature out there pointing towards the entropic states of systems subject to Quantum Mechanics, which I find puzzling.

Do you know of any mention of any co-linearization between von Neumann stochastic Quantum Mechanics and entropy in general?

Thanks.

@Wayfarer, what do you think?

You mean observing people in your dream as if you are separate or outside the dream itself - or the people in the dream as having separate minds from you, or what ? — Amity

The people in the dream. Those mysterious homunculi that populate your dreams. Who the fuck are those people? =]

Dreaming entails producing sensations or what I call 'qualia' from past experiences in an amalgamate manner. Manny of the experiences are qualitative in how they can reproduce sounds in a coherent manner from music you may have heard or to perceiving the nature of light in a dream.

It's fascinating to me to witness other people as if with a separate mind in a dream. What do you think?

That's interesting about identity. But, I surmise that due to regulations and oversight much of this will be limited to the ultra-rich initially and then slowly enter the market.

How fast or how long do you think humanity will switch over to transhumanism?

I wouldn't consider that an experience on top of another experience; it's just part of the dream experience. There is something it is like to have a dream, just as there is something it is like to listen to Mozart in reality. This is the distinction I would draw between the two qualia or qualitative experiences. — Luke

But, it's still a qualia to experience whilst dreaming, no?

or are you dreaming of hearing Mozart? — Luke

Yes

I don't understand what having one experience on top of another means. — Luke

Let me provide an example:

Listening to Mozart whilst in a dream is what can be called a qualia?

How does one experience something on top of what one experiences? — Luke

In a dream this happens all the time. Everything in a dream is qualitative in manner to be more clear, with the observer or subject further experiences these qualia...

When you say "these amalgamates of past experiences", are you referring to dreams? If so, then you have already told us that these "are not entirely qualia". — Luke

I am not entirely certain. It seems to me that when one experiences something on top of what is or has been experienced, then the denotation of 'qualia' applies to the phenomenon of experiencing, doesn't it?

And something else that's pretty neat from the Tractarian:

5.64, Wittgenstein asserts that “Here it can be seen that solipsism, when its implications are followed out strictly, coincides with pure realism. The self of solipsism shrinks to a point without extension, and there remains the reality co-ordinated with it.”

"Epistemic solipsism?" I refer you again to Witty's "Private Language Argument" which makes the case that any discourse which is not public – not accessible by others – is nonsensical (e.g. babytalk), which includes "epistemic solipsism". The eye is not in its own field of vision, the hand can grasp anything except itself; other eyes and hands are entailed respectively in order to see eyes and to grasp hands. Touching involuntarily touches back – "solipsism", as Samuel Johnson quipped, it's refuted thus. — 180 Proof

Thanks for the comment on the PLA. In like here's one:

P.M.S Hacker provides the following:

What the solipsist means, and is correct in thinking, is that the world and life are one, that man is the microcosm, that I am my world. These equations... express a doctrine which I shall call Transcendental Solipsism. They involve a belief in the transcendental ideality of time. ... Wittgenstein thought that his transcendental idealist doctrines, though profoundly important, are literally inexpressible.

— Hacker, Insight and Illusion, op cit., n. 3, pp. 99-100.

Existing forever. What about the loved ones forever gone? Would we not eventually be bored beyond reason? To be fair, I think that those that always say that it is because we are finite that we are able to give meaning to life is not that clear to me. — Manuel

Yeah, there's a lot of philosophical questions here to go about. I mean, life with respect to death or boredom, which are prominent topics in philosophy. I suppose my point would be to look at life as a limited thing, as it always is perceived and then go about making a decision as to whether one would want to prolong it or try as to.

"The solipsist achieves no practical advantage from advancing his views". — Banno

Very true. But, technically, we have some idea of what solipsism may mean by the mere fact that we experience a world like a solipsist would every night when we dream. Ya, no?

Except solipsism is a commitment to a 'metaphysical fantasy' and not itself a mental state. Perhaps REM sleep / lucid dreaming "seems" solipsistic but they need not "seem" so in every case. — 180 Proof

Enabled by REM sleep, perhaps?

Epistemic solipsism seems to me to be a viable position to hold in regards to the indubitability of the solipsist's world, where he or she alone is the observer of the observed.