Stephen Hawking, Theory of Everything, and Goedel’s Incompleteness theorem

This post has the following readability test scores:

Flesch Reading Ease: 63.63
Flesch-Kincaid Grade Level: 8.00
Automated Readability Index: 9.00

For more info about readability tests, check out my post about those here.

Whew, it’s been a while since my last post. Work and a project (during and continuing till after the end of a semester) kept me really busy.

Anyway, this post is about the webpage containing prof. Stephen Hawking’s talk about his latest (AFAIK) stand on the search for the theory of everything, how Kurt Goedel’s incompleteness theorem/s influenced his opinion/view.

The speech in text format (1st reference link below), as you may immediately notice, is quite messy and seems to be unedited. I think that the speech laid out in the web page was just a speech-to-text conversion of prof. Hawking’s speech, which by the way you can download, so you can listen to his very iconic “electronic” voice. This speech-to-text conversion is quite evident I think since the starting letter of each sentence is not capitalized, and difficult words (at least for an American English speech-to-text program) such as Laplace (for the French mathematician) turn out to be spelled as Le-plass, which is what you would most probably get if you spoke into a speech-to-text synthesizer just like prof. Hawking’s.

As you may well know, a theory of everything (ToE) in physics aims to unify the four basic forces in the known universe, and which can, in theory, be used to predict anything that ever happened or will happen in the universe. My previous weblog post further clarifies this theory, or the search for it. I found the 1st reference link below when I Googled for “Stephen Hawking Theory of Everything”, hoping that I would be further enlightened about how the man labelled as “the brightest theoretical physicist since Albert Einstein” is going about searching for the ToE. I never expected that he would give up, in a sense, which for a while made me feel uncomfortable, though the feeling eventually disappeared. What made me feel uncomfortable was not because the universe is full of uncertainty and randomness we can’t explain (or at least not yet for some). What made me feel uncomfortable was that the man who so passionately sought for the ToE from his books and studies in the 70s and 80s (an example is the 2nd reference link below) has given up the search for the ToE.

But as I re-read the speech, Goedel’s incompleteness theorem hits the point in between the eyes. The great professor starts out (as he usually does) by discussing briefly the ideas that will prove to be necessary for his lecture’s whole point, including Quantum Chromodynamics (QCD), scientific determinism, Newton’s gravitational law, wave functions, etc. This is a style which I think he is famous for, and which is why his lectures are very popular and well sought after. His funny and comical inserts pop out once in a while to stir things a little bit.

Essentially, the gist of his lecture (as it would seem to me) is that there cannot be a single ToE, one which, as I’ve mentioned previously, will let us know what exactly happened at a specific point in our universe’s history, or what will happen to its future. The ToE should be able to tell you what you were thinking last night, or what you will think tomorrow night. Yes, the ToE is the theory of all theories. The problem however is that the human brain, as prof. Hawking puts it, is composed of so many parts made up of so many particles that we couldn’t possible compute what you’ll be thinking the next minute, even with the most powerful computers in the world today.

Prof. Hawking admits that until he realized the implication of Goedel’s incompleteness theorem, he implicitly assumed that a ToE will be found, probably relying on what can be referred to as “scientific intuition”. According to prof. Hawking, the positivist philosophy of science is that every good physical theory is a mathematical model, which he probably gets from Karl Popper and other positivist thinkers. And since, according to Goedel’s incompleteness theorems, there are mathematical results that cannot be proven, then so must there be physical theories that cannot be proven as well, including the ToE. Goedel’s incompleteness theorems essentially say that, as mentioned by prof. Hawkings himself, math is either inconsistent or incomplete. The professor bets his money (as he did in a previous debate with Kip Thorne decades ago) on the incomplete part of mathematics. Goedel’s theorems are proved using self-referring statements such as

“This sentence is false”

If the statement is true, then (the idea it implies) it is thus false. If the statement is false, then it is the opposite, meaning the idea the statement implies is true. Another would be to apply Bertrand Russell’s paradox on the barber paradox:

In a town which has a rule that the (only) barber shaves only all men that do not shave themselves, and doesn’t shave those who shave themselves. One can then ask, does the barber shave himself? If he does, according to the rule, he shouldn’t. If he doesn’t shave, according to the rule, he must shave himself.

Lastly, prof. Hawkings apologizes if anybody got/gets disappointed on his current view about the ToE. As I’ve said, I was initially disappointed, but considering his rationale about the search for the ToE, I turn out to be fine, though I still think he still leaves room in himself in case he makes a wrong prediction (just as he did in the 2nd reference link) that we might find the ToE in this decade. In his lecture in the 1980s regarding the discovery of the ToE, he said that we may find the ToE, and he’ll give another lecture about the ToE and where we stand 20 years later (which is now). Well, he did tell us where we stand, or at least what his view of where we stand in the search for the ToE. He also said then that once the ToE is found, physicists will lose their jobs, and physics will end. But according to his latest lecture, thanks to Goedel, mathematicians he said will always have a job, and most probably physicists too.

References:

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14 Responses to “Stephen Hawking, Theory of Everything, and Goedel’s Incompleteness theorem”

  1. doc holiday Says:

    Random ponderings and rant:

    I do not think Goedel’s incompleteness theorem takes into full account the nature of elastic space/time. Schwarzschild coordinates, in terms of black holes seem to represent just one side of a geometric coordinate system, i.e, the surface area takes into account a singularity from dimensionality which is “outside” the singularity, versus the possibility of the singularity “inside”. Schwarzschild simplified the curvature of spacetime outside a spherical, nonspinning star, then he wondered what it would be like to be inside such a star. This radius spacetime center was used to breakdown calculations resulting in a centralized core where spacetime would infinitely curve in on itself.

    Thus, is the infinite curvature of spacetime which breaks down into a coordinate of zero seem incomplete? I think Hawking needs to

    In Schwarzschild coordinates, this singularity lies on the sphere of points at a particular radius, called the Schwarzschild radius

    Einstein used rectangular coordinates to approximate the gravitational field around a spherically symmetric, non-rotating, non-charged mass. Schwarzschild, in contrast, chose a more elegant “polar-like” coordinate system and was able to produce an exact solution.

    A singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.

    Re: Gödel’s first incompleteness theorem, states that:
    For any consistent formal, recursively enumerable theory that proves basic arithmetical truths, an arithmetical statement that is true, but not provable in the theory, can be constructed. That is, any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete.

    “One of the things that Goedel proved was that time does not exist, at least not in the physical world. It is an imaginary dimension, both in the ordinary sense, and in the mathematical sense. In math, imaginary numbers have the square root of minus one as a factor, which segregates them from real number. Only real numbers correspond to real dimensions, which allow movement.”

    http://en.wikipedia.org/wiki/Wormhole#Schwarzschild_wormholes

    Wormholes connect two points in spacetime, which means that they would in principle allow travel in time as well as in space. In a 1988 paper, Morris, Thorne and Yurtsever[3] worked out explicitly how to convert a wormhole traversing space into one traversing time.

    Sakharov:

    The energy that it takes to curve space is nothing but perturbation in the vacuum energy of fields plus particles brought about by that curvature according to Sakharov: Sakharov’s idea that gravitation is a manifestation of the alteration of the zero-point fluctuations of space.

    Also see: Einstein’s geometrodynamics

    Too late to re-write….

  2. doc holiday Says:

    Re: “Thus, is the infinite curvature of spacetime which breaks down into a coordinate of zero seem incomplete? I think Hawking needs to”

    >> I thought I should add this following passage from Wik to my previous incomplete (and poorly written) post — as a sort of additional observation related to Mr. Hawking’s crash into the barrier of Incompleteness):

    *** “Daniel Dennett defends scientific reductionism, which he says is really little more than materialism, by making a distinction between this and what he calls “Greedy reductionism”: the idea that every explanation in every field of science should be reduced all the way down to particle physics or string theory. Greedy reductionism, he says, deserves some of the criticism that has been heaped on reductionism in general because the lowest-level explanation of a phenomenon, even if it exists, is not always the best way to understand or explain it.
    Some strong reductionists believe that the behavioral sciences should become “genuine” scientific disciplines by being based on genetic biology, and on the systematic study of culture (cf. Dawkins’s concept of memes). In his book The Blind Watchmaker, Richard Dawkins introduced the term “hierarchical reductionism”[2] to describe the view that complex systems can be described with a hierarchy of organizations, each of which can only be described in terms of objects one level down in the hierarchy. He provides the example of a computer, which under hierarchical reductionism can be explained well in terms of the operation of hard drives, processors, and memory, but not on the level of AND or NOR gates, or on the even lower level of electrons in a semiconductor medium.
    Both Dennett and Steven Pinker argue that too many people who are opposed to science use the words “reductionism” and “reductionist” less to make coherent claims about science than to convey a general distaste for the endeavor. Furthermore, these opponents often use the words in a rather slippery way, to refer to whatever they dislike most about science. Dennett suggests that critics of reductionism may be searching for a way of salvaging some sense of a higher purpose to life, in the form of some kind of non-material / supernatural intervention. Dennett terms such aspirations “skyhooks,” in contrast to the “cranes” that reductionism uses to build its understanding of the universe from solid ground.”

    >>>> And as we go further out on the branches of entropy:

    On Artificial Intelligence: Dennett v. Gˆdel and Penrose

    Ghazal Zekavat

    http://serendip.brynmawr.edu/sci_cult/evolit/s05/web2/gzekavat.html

    “To Dennett, there exists a set of algorithms that yield a mathematical insight “even though that was not just what [they were] ‘for.'” (1) (p 441) Dennett poses the question, “how could Penrose have overlooked this retrospectively obvious possibility?” (1) (p 441) To “prove” the fallacy of Penrose’ argument, Dennett outlines an algorithm for playing perfect chess. (1) (p 439) The result of this algorithm would be an artificial sense of “insight” in the machine that models the algorithm. Dennett claims that since chess is a finite game, there are a finite number of possibilities. Were a computer equipped with an algorithm to account for each outcome, it would essentially pass the Turing test. Suppose that the ever common “fatal error” occurred in the computer, or the chip containing the algorithm was destroyed, would the computer’s insight be lost as well? Yes, however, this question is fundamentally incorrect in that something that never had insight to begin with cannot ever really “lose” insight. Were Bobby Fischer to break his arm, would he lose his insight? The answer is a resounding “No.” Were Bobby Fischer to suffer irreparable brain damage, would he lose his insight? Before this question can be answered, an even more important one must be asked: would Bobby Fischer still be Bobby Fischer if he suffered irreparable brain damage? It is not necessary to have an answer to this question before arriving at the conclusion that Bobby Fischer’s character traits, personality, and least of all, insights are a product of Bobby Fischer. An algorithm may not be written and consequently implanted into Bobby Fischer’s brain to bring Bobby Fischer back. An analogous example is the story of Phineas Gage, the foreman of a railway construction gang, who suffered brain damage when a three feet seven inch long tamping iron blew through the front side of his brain. Although Gage miraculously survived, his personality was drastically altered after the accident. Prior to the accident, Gage was described as an efficient, congenial and well-balanced man. Afterward, however, he became irreverent, profane and obstinate and was described as, “No longer Gage” by his coworkers. (3)

    Once again, would Bobby Fischer still be Bobby Fischer if he lost the qualities that made him Bobby Fischer in the first place? Yes, he would still be Bobby Fischer, but no, he would not be the same Bobby Fischer that he used to be. This case, although analogous to the Phineas Gage case, is not analogous to the damaged computer chip case. Will the computer chip still be a computer chip? Yes. Can its artificial insight be restored? Yes.”

  3. doc holiday Says:

    I’m not sure that you can beat the incompleteness theory to death, but this helps me sleep better:

    COMPUTER SCIENCE IN A THEORY OF SCIENCE DISCOURSE
    Gordana Dodig – Crnkovic

    http://www.idt.mdh.se/personal/gdc/work/cs_vetenskap.pdf

    “There is an important characteristic of a scientific theory or hypothesis that differentiates it from, for example, a religious belief: a scientific theory must be “falsifiable” (Popper). This means that there must be some observation/experiment or another well-approved theory that could disprove the theory in question. For example, Einstein’s theory of relativity made predictions about the results of experiments. These experiments could have produced results that contradicted Einstein, so the theory was (and still is) falsifiable.

    Re: …”math is either inconsistent or incomplete.”

    Also: “Prof. Hawking admits that until he realized the implication of Goedel’s incompleteness theorem, he implicitly assumed that a ToE will be found, probably relying on what can be referred to as “scientific intuition”.”

    See Also: Lazy Thinking Algorithm Synthesis
    in Gr¨obner Bases Theory
    Doctoral Thesis
    Vasile Adrian Cr˘aciun, M.Sc.

    http://www.risc.uni-linz.ac.at/publications/download/risc_3425/thesisAC.pdf

    Re: Bringing it All Together: Mathematical Knowledge Management
    While computer mathematics has seen tremendous developments in the last few decades, there is a lack of integration of the various tools available and lack of interest from “traditional mathematicians”.
    Mathematical Knowledge Management (MKM) is a new research field, established by the First International Workshop on Mathematical Knowledge Management, in 2001,
    see [Buchberger and Caprotti, 2001], which tries to bring together and focus all the efforts to reach this integration, i.e. to provide tools that allow efficient management of mathematical knowledge.

    FYI (OT):

    Physics and the law
    Stranger than truth
    Apr 3rd 2008

    http://www.economist.com/science/displaystory.cfm?story_id=10952790

    Re: The bugaboo this time is black holes. A black hole is an object so dense (and thus with such a strong gravitational field) that nothing—not even light—can escape it. Not surprisingly, no such object has ever been observed directly.

    The LHC is the proud creation of CERN, Europe’s main particle-physics laboratory, which is located near Geneva. It will create a zoo of new particles for those who study the fabric of reality to get to grips with. Among those objects may be some tiny black holes. The LHC’s physicists are particularly excited by these because they will allow for the experimental examination of gravity. They may also allow Stephen Hawking, a well-known British physicist, to receive a much-deserved Nobel prize. That would almost certainly happen if he turns out to have been right in his prediction that tiny black holes will evaporate in a spectacular burst of energy that has come to be known as Hawking radiation.

    Luis Sancho and Walter Wagner, however, are excited for a different reason. They fear that, far from evaporating in this way, any black holes created in the LHC will start sucking matter in—and will eventually swallow the Earth.

    Re: At this intensity of current the magnets are capable of guiding a 6 TeV proton beam.

    http://cdsweb.cern.ch/journal/article?name=CERNBulletin&issue=12/2008&number=1&category=News%20Articles&ln=en

    This one is for SH: “Tralfamadorians have the ability to experience reality in four dimensions; meaning, roughly, that they have total access to past, present, and future; they are able to perceive any point in time at will. Able to see along the timeline of the universe, they know the exact time and place of its accidental annihilation as the result of a Tralfamadorian experiment, but are powerless to prevent it. Because they believe that when a being dies, it continues to live in other times and places, their response to death is, “So it goes.” They are placid in their fatalism, and patiently explain their philosophy to Pilgrim during the interval he spends caged in a Tralfamadorian zoo. Eventually Pilgrim adopts their attitude, is returned to Earth, and tries to spread their philosophy.” (lol)

    See yah!

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