As Promised, Proof Reality is more Consistent, Complete than Math

In another forum ("Physics and Math") someone asked me to make good on my claim to have proven that reality and the universe is both more consistent and complete than math. This is a counterpoint to recent popular books, such as Max Tegmark's "Our Mathematical Universe".

My math background is diverse, and one of the largest influences on it was one R.W. Hamming, whose essay on the subject of the "Unreasonable Effectiveness of Mathematics" may be found here:

https://www.dartmouth.edu/~matc/MathDrama/reading/Hamming.html

The inspiration for Hamming's essay was an earlier one of the same title by Wigner.

My group at the former Comsat Laboratories implemented the first Hamming error correction code for use with the infrastructure of the Intelsat satellites. It was in fact my first assignment, together with a team that included coding theory giant Lin-nan Lee, who came to Comsat from Linkabit, and worked with Mr. Viterbi himself. Comsat Corporation was originally the United States signatory to the Intelsat consortium by the Telecommunications act of 1963, signed into law just a month before John Kennedy was assassinated.

In his essay, Mr. Hamming points out that we often adapt our math to fit whatever it is we find works. In particular, for much of the error correcting codes that were implemented after the algorithm that bears his name, 1+1=0. Show me anywhere else in nature this happens. I'm sure it probably does, but it is both rare and counterintuitive. I can't think of anything that is natural about other coding theory applications like Galois fields and cyclotomic cosets either, but I don't wish to belabor the point Mr. Hamming has already aptly made in his essay.

In the early 20th century, there evidently was a "no holds barred" feud between the likes of Albert Einstein and his friend Kurt Gödel and Einstein's former calculus professor Minkowski and David Hilbert, who Minkowski considered to be his best student. Minkowski died suddenly from a ruptured appendix in 1909, but not before implementing a torturously complex explanation for Lorentz covariance, Minkowski rotation, and the warpage of spacetime in an effort for his ideas to prevail in physics over his duller student Einstein, who only with difficulty and a lot of heckling from Hilbert managed to come up with the corrected field equations we use today to make the clock synchronization on GPS work. Whether or not Hilbert had any influence in deriving these is unclear, but apparently the experience took a toll on Einstein because Hilbert either wanted to take it in a different direction, or take some of the credit for developing General Relativity.

Hilbert went on to develop the idea of the "axiomization" of mathematics, after the earlier work done by Russell and Whitehead's "Principia Mathematica". The main idea was to completely generalize and codify as many of the rules of mathematics as possible, so that mathematical proof could be accomplished in a more mechanistic fashion that would ELIMINATE AS MUCH HUMAN ERROR IN THE PROCESS as possible. There was hope that eventually the entire field of mathematics would be accomplished by means of mechanized or electronic means, or by some combination of both.

Eventually Kurt Gödel managed to immigrate to the United States to join his friend Einstein at Princeton and in the process of joining Hilbert's axiomization efforts, developed both his first and stronger second Incompleteness theorems which rendered Hilbert's entire process less than worthless.

You see, an inconsistent system of axiomatic math would be one which, like division by zero, would allow Hilbert's program to validate all sorts of perfectly logical proofs which were sheer utter nonsense. On the other hand, an incomplete system of axiomatic math would be unable to prove a great number of such proofs, but for the self-referential ones, there would ALWAYS be at least ONE true proposition which could not be proved or demonstrated by means of the original set of axioms.

To try and save face, Hilbert added to Gödel's incompleteness another condition, that of "decideablilty", possibly because Hilbert knew that Gödel had a disdain for quantum theory in general, and the Principle of Uncertainty in particular, possibly put there by his friend Einstein. Decidability and Uncertainty of course go hand in hand. At any rate, Hilbert's "decidability" is moot, and of no effect either way on the validity of the process of axiomization, which thanks to Gödel, was now dead.

So the question at hand is, is there any reason we should expect that the universe itself should follow the pattern of mathematics or was it created by means of the manifestly restricted logic that a race with finite minds would construct a language?

Look at something from Hamming's essay one more time:

"From the above we see that one of the main strands of mathematics is the extension, the generalization, the abstraction... But note that in the very process the definitions themselves are subtly altered. Therefore, what is not so widely recognized, old proofs of theorems may become false proofs. The old proofs no longer cover the newly defined things. The miracle is that almost always the theorems are still true; it is merely a matter of fixing up the proofs. The classic example of this fixing up is Euclid's The Elements [4]. We have found it necessary to add quite a few new postulates (or axioms, if you wish, since we no longer care to distinguish between them) in order to meet current standards of proof. Yet how does it happen that no theorem in all the thirteen books is now false? Not one theorem has been found to be false, though often the proofs given by Euclid seem now to be false. And this phenomenon is not confined to the past. It is claimed that an ex-editor of Mathematical Reviews once said that over half of the new theorems published these days are essentially true though the published proofs are false. How can this be if mathematics is the rigorous deduction of theorems from assumed postulates and earlier results? Well, it is obvious to anyone who is not blinded by authority that mathematics is not what the elementary teachers said it was. It is clearly something else."

Indeed, it is NOT THE PATTERN OF THE UNIVERSE THAT HAS CHANGED in these older proofs. It is the math. Therefore, I assert with authority that the universe is both more consistent and more complete than the language of mathematics that ANY finite mind can devise to describe it. Q.E.D.

Anyone who claims it is otherwise is either a liar, or has miscalculated, or perhaps their communications channel has excessive noise without an effective means of error correction.

 

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First, Minkowski likely derived his work on spacetime from his work in number theory. Second, to describe it as "torturously complex" is a bastardization of English, a gross mistake compared to other approaches to do the relevant physics, and a horrid attempt to detract from the points of legitimate scholarship with a non sequitor attack on the character of the work.

Third, the work Einstein had was difficult in part because he had to invent a new way of doing mathematics.

I think we learn less about mathematics and physics than we do about the character and capabilities of the author of the post.

The above seems like an incredibly anachronistic description of events and motivations.

Sadly, no.

This is not what Godel showed. Godel's results are entirely constructive, so "true" is not something that can necessarily be said of the proposition in question.

Tell this to the many people in computer science working on decision problems!

[quote\Look at something from Hamming's essay one more time:

"From the above we see that one of the main strands of mathematics is the extension, the generalization, the abstraction... But note that in the very process the definitions themselves are subtly altered. Therefore, what is not so widely recognized, old proofs of theorems may become false proofs. The old proofs no longer cover the newly defined things. The miracle is that almost always the theorems are still true; it is merely a matter of fixing up the proofs. The classic example of this fixing up is Euclid's The Elements [4]. We have found it necessary to add quite a few new postulates (or axioms, if you wish, since we no longer care to distinguish between them) in order to meet current standards of proof. Yet how does it happen that no theorem in all the thirteen books is now false? Not one theorem has been found to be false, though often the proofs given by Euclid seem now to be false.[/quote]
I'm not sure what this is discussing other than the possibility of considering additional contexts of inference. Yes, we can keep adding on additional requirements for logical reasoning ad infinitum, but this doesn't really get us anywhere. The is no sense in this that the proofs that we saw were "false", just inadequate to a different standard of justification.

If there are no proofs, then how can we know the truth of these theorems?

Ah, yes, the crank call of the conspiracy of authority and the call to "open mindedness".

Facts not in evidence: the math has changed.

QED? WTF?

How can someone who is calling for an inanely higher standard for proof offer the above mess of innuendo, outright attack on character, vague hand-waving at actual results, and bastardization of the history of mathematics in the Twentieth Century and then have the temerity to write, "QED"?

 

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I'm sorry that you don't understand, mathematics is nothing more than a narrative about a universe that is not as prone to making errors as we are.

It is also much more relentless and merciless than the most rigorous form of proof, which is the only reason that finite mind of yours endured for long enough to be able to do any math.

There are many more mistakes in you analysis than in mine. I certainly would hope your math is better, but somehow I doubt it.

 

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Still not taking the anti-arrogance pills Dan? Still, it is reassuring to see that nothing has changed in the year (?) since I last had the misfortune to run across one of your posts.

I do have one question for you: why have you chosen to offer a proof that is free of logic?

 

OK, if that was your point, why didn't you argue for that point in your screed? Instead, you laid out a series of mistakes of the history of mathematics and then forsook any attempt to tie it to your supposed thesis.

OK, find one.

Look, I'm very sympathetic to the idea that mathematics cannot fully capture all of physical reality. I would like to see a decent argument for that position, however.

 

You said: "OK, find one".

I SAID:

"Eventually Kurt Gödel managed to immigrate to the United States to join his friend Einstein at Princeton and in the process of joining Hilbert's axiomization efforts, developed both his first and stronger second Incompleteness theorems which rendered Hilbert's entire process less than worthless."

TO WHICH YOU SAID:
"Sadly, no."

http://www.boston.com/news/globe/ideas/articles/2005/04/03/the_mathematicians_lament/?page=full

"Gödel immigrated to the United States in 1940 and took up residence at the Institute for Advanced Studies in Princeton, where he found a conversation partner and confidant in Albert Einstein"

http://plato.stanford.edu/entries/hilbert-program/

(the reference for most of my OP, by the way. After all, this IS the "Philosophy" forum

"Gödel's work is generally taken to show that Hilbert's Program cannot be carried out."

So, "sadly no" is a response to what part of my text?

Notice that I'm not taking the bait of trying to do a formal mathematical proof because frankly, it is far too limited a language in which to prove much of anything. Hamming's essay already did most of the work of what proof can be done for me. I just connected the dots. If you don't see it, I don't fault you. The universe is perfectly complete and consistent. It is mathematics that has been in a state of continual change since ancient Greece to try and capture the most basic elements of it, nor can it be said that it has entirely succeeded.

Mathematics is a narrative written in a symbolic language that is as flawed as the mathematicians who use and abuse it are. If mathematicians were called upon to implement a program that would result in the creation of their own minds using the limited tools they themselves had crafted, they would fail. They have already done so. Artificial Intelligence modeled with symbolic logic is beset by halting problems. Gödel had many correspondences with Von Neumann about this issue.

Meanwhile, there is both inconsistency and incomplete parts of everything and anything they believe they have figured out, and that includes everything and anything in physics that involves a constant that was not derived from first principles, free parameters, or hidden variables. I can't make it any simpler than that.

Tossing in models of multiverses, block universes, and models of higher dimensions and Hilbert spaces really doesn't impress anyone who understands, they do not have an adequate working model of the entirety of a SINGLE universe yet; the one we actually live in.

 

If someone has taught you that the history of mathematics associated with physics is different, my advice is to seek instruction elsewhere.

Minkowski was a man obsessed with the mathematical beauty of quadratics in particular (and I have read his own words to this effect). So, naturally, when his C student Einstein developed relativity, he wanted some credit for the accomplishment. Where Einstein saw distances as another way of expressing light travel TIME (light years, the current NIST standard defined in terms of wavelengths at rest for 1 meter), Minkowski wanted to see if he could throw in a quadratic exression to relate space and time. By commingling the strong invariant speed of light, the foundation of relativity to make an invariant 4D quantity he called the interval, he saw a way to combine both his beloved quadratics AND the work of the student he most respected, Hilbert.

Nonsense like this had happened in mathematical physics before. Hooke's insistence to Newton that celestial mechanics was based on DIRECT PROPORTIONAL RELATIONSHIPS HE FOUND EMBODIED IN THE MOTION OF MECHANICAL SPRINGS is another example of how mathematics sometimes gets carried away with new ideas. Minkowski's warpage of spacetime is no better. The springs have been replaced by illustrations of a 2D fabric that is basically an expression of the same idea. Inertialess springs, like inertialess rubber sheets, are the mathematical equivalent of fantasy.

Before I started complaining about them, the internet was full of graphic images of Minkowski relativistic rotation / distortion of spheres and cubes. Now I can find none anywhere. Even Wikipedia has replaced those images with stick figures and scaled back graphics.

Lengths do not "rotate" into the dimension of time at relativistic speeds as Minkowski suggested. The only thing that rotates in this universe is the propagation of energy, and it does so in a manner that is consistent with propagation of energy in three dimensions, ALL of which can be described as the dimension of light travel time. The Lorentz contraction of lengths of the spaces between atoms and between electron clouds and nuclei that is due to relative motion is the same phenomenon as the contraction of wavelengths of energy we call Doppler shifts. That idea, combined with the speed of light which limits energy propagation in this universe, are the only ideas you will ever need to deal with in order to understand, space IS time. Inertialess spacetime does not "warp" around gravitating objects. But time does dilate.

Simultaneity in a universe of energy transfer events does not even occur without the involvement of quantum entanglement, and that effect requires no bulk propagation of energy, nor is it limited by the speed of light. Entanglement's instant of "now" is the basis of the origin of time itself. But Minkowski's belabored and extravagant ideas about quadratics, the Pythagorean theorem, and complex numbers will never help you understand this, even in small measure.

In this universe, Einstein and Gödel rule. Minkowski and his star student Hilbert aren't even worthy of footnotes in mathematical physics, or in mathematics. File them both under multiplication using Roman numerals.

 

Yes, I said, "Sadly, no," because you said that, "[Gödel] developed both his first and stronger second Incompleteness theorems which rendered Hilbert's entire process less than worthless."

It would be stupid to think that I was objecting to the immigration when the point of this essay was about the nature of mathematics. Heck, it would be stupid to include the information about the immigration in the first place, since it clearly has no bearing on the topic.

So if the only problem you can find is this imagined one that has no relation to your argument, then you and your argument are in serious trouble.

We do not want a mathematical proof about this subject. We want coherent reasoning in English on the topic. Immigration has nothing to do with the topic.

So, essentially, your argument is to loudly state your conclusion and to claim that somebody else also loudly stated your conclusion. Good to know.

But do you understand what these results mean? And do you understand the limitation of many of these results to first-order logic for certain ranges of statements? Why can't we use second-order logic and mathematical practice to produce a system of statements as consistent and complete as reality? What does it mean to say that reality is consistent and complete?

 

Just a note, that a science like math cannot be both "complete" and "free of contradictions"

https://en.wikipedia.org/wiki/Gödel's_incompleteness_theorems

So your claim "the universe is both more consistent and complete than math" is not new and there is little to do about it. Any science that uses logic, algebra or the like will either be incomplete or contradictionary.

And we know that since about 80 years.

 

Perhaps you should seek instruction elsewhere? You seem to have learned much from just random pop claims on the internet.

When was Einstein ever a C student? Do you have any source for the claim that Minkowski attempted to get credit?

How is this nonsense? The Minkowski metric is excellent for doing physics. Just because you find the history of the development aesthetically displeasing is not evidence that the mathematical physics is bad.

Yeah, Hooke was a dick. Though Newton went to great pains to show what would happen in orbits if Hooke was right and that it did not fit the evidence.

No, Minkowski's spacetime is obviously better as it captures, elegantly, everything in Special Relativity in theory as well as practice while Hooke's spring theory produces different physical behavior than that observed in the solar system.

Holy delusions of grandeur, Batman!

That isn't what he suggested. That is not what anyone suggests. What they suggest is that one can translate from one set of coordinates to another with a rotation. This uses a specific mathematical definition of the word "rotation".

 

Euclidean geometry only works for solids at rest, since the days of Ancient Greece and Pythagorus, who also believed that numbers have souls. Such geometry likewise only works if your coordinate system origin is an anchor stuck in a solid. Nothing in the geometrical constructs of Ancient Greece can ever move, and their constructs are for people whose minds have hardened to the consistency of a block of cement or a marble statue of Zeus.

Well, have a happy holiday, at any rate. Please don't gift me any math books.

 

Tell that to the thousands of engineers that get by with Newtonian mechanics in their millions of applications.

Why stick with attacking the character of mathematicians rather than address the mathematics as used?

 

Um.. Those would not be GPS satellite engineers. I am a retired satellite engineer whose specialty was in applications of coding theory.

I have attacked "the mathematics used" including the math used to create LIGO in another thread. They will never detect a gravity wave with an interferometer that has a baseline shorter than 1/3 of the wavelength of the gravity wave you are trying to detect. For the Sun-Jupiter system, this would be 4 light years. This is an alternative calculation to supplement Special Relativity's definition of "local". And only a telecom engineer would understand this.

Whatever math is being used to make more and more sensitive interferometers to detect gravity waves is not going to work. It is still impossible by means of local measurement to determine motion through absolute space or absolute time, or equivalently, to detect a gravity wave locally. You would need to compare time dilations at points very far removed from the gravity wave, and I have just calculated how distant that would need to be.

 

Um... I may have missed it in the 13 or so posts so far but wasn't there supposed to be a proof put forward at some point? I see a claim. I see the phrase "Q.E.D." and some history stuff, but no actual proof.

I might as well say that this thread has no such proof, and end it with Q.E.D. as that seems to suffice, right?
Or am I missing something?

 

You are under the impression that any axiomatic proof that the universe is more consistent and/or complete than mathematics is something that can be proven in mathematics? Then you aren't paying attention.

I'm saying it isn't possible because mathematics is by its very nature either incomplete or inconsistent. So, take Gödel's incompleteness theorem as an AXIOM.

That being the case, you can't possibly expect to prove that something like the universe, which IS perfectly complete AND consistent, meet those criterion starting from an axiomatic system of logic (mathematics) that does not possess such properties itself. To do so would require better math than is available.

But if you really insist on doing a partial proof, that may is possible:

(1-1) The universe contains no instance of DIVISION BY ZERO.

(1-2) The universe contains no instance of ANY TRUTH THAT IS AN ABSOLUTE. Hence, no liar's paradox is possible.

(1-3) The universe contains no instances of SELF-REFERENTIAL SYMBOLIC REFERENCES.

(1-4) The universe contains no instances of CONSTANTS, FREE PARAMETERS, OR HIDDEN VARIABLES which cause inconsistencies in the way anything and everything in it manifestly works.

There may be other examples of how the universe is more complete and consistent than any math that describes it. However, there is no guarantee that all can be enumerated, and that is another basic completeness problem that current mathematics isn't even marginally able to address.

I'm sorry if this doesn't meet your definition of a proof, but it is literally the best that can be done. This topic was posted in the Philosphy forum, not Math and Physics. As far as I am aware, philosophy has never required anything like a proof. And that's too bad, because philosophy really could use such a check on its veracity occasionally.

 

What is it with engineering that creates a certain kind of individual focused on their own narrow specialty with contempt for everything else anyone does?

Please stop treating all other engineers with contempt. Have some respect for the Newtonian mechanics that they use that is an amazing approximation to the behavior of the physical systems that they use.

This is irrelevant to your screed in the original post. Please provide an argument for your point instead of meaningless attacks on character.

 

No, you just attacked a straw man.

You are using the language of formal proof, i.e., "Q.E.D.", yet you have no real argument, let alone a proof.

This is the problem that every reader will have with your screed.

An existing being cannot be complete in the sense given by Gödel; only deductive systems can be complete. What you need to show is that there is no statement about the universe that we can deduce and that we cannot deduce in mathematics.

It is not a proof because it has no conclusion, it is merely a collection of premises. Nor is there any attempt to link these premises to each other. It fails to match the form of a proof or even an argument.

Philosophy requires argument. Too often, people who do physics think that they can take a vacation from reason when they do philosophy. Philosophy is not about merely accepting the angry rants of someone who can calculate in some other venue.

 

I would grant you that e^(i*pi) = -1, or that e^(i*theta) = cos(theta+i*sin(theta),
Is something that the universe will not teach you, but whatever are the consequences of that identity are surely manifest. At least, mathematics is complete enough to show this is true, and that is saying a lot.

To be complete, a deductive system need only demonstrate that there is no statement which, while true, cannot be proven using only the axioms of that system. So in a sense, the universe itself is not complete without mathematicians.

So what we have left to evaluate is the proposition of whether or not the minds of mathematicians can be created in this universe. I believe I have already covered this, and it has a value designated as true, by means of the anthropic principle.

If I need to list all of the axioms, I suppose what we have is the equivalent of a divide by zero, until or unless mathematicians somehow manage to work it ALL out, with a finite number of axioms. I think, in that case, that would be a SECOND divide by zero on the same proposition, which actually does make sense, particularly if G-d him or herself is a mathematician. No wonder he or she is inscrutable, and moves in mysterious ways. An infinite mind wanting to be finite. Hmmm... Infinity having affinity for finity.

If you can think or do math that is further outside "the box" than that, you have earned my respect, support and encouragement.

 

What you have written makes absolutely no sense. It is not "outside the box" reasoning, it is not reasoning.

 

Dwell on danshawen's above post a little longer.