Daniel Dare is one of those web personalities you come across who you might initially think is a crank, but after integrating over time you see that they have a subtle depth. He’s largely hostile to belief and subscribes to the Dawkins ultra-rationalist condescension of spiritualism. I haven’t asked him directly but I suspect he would strenously disagree with the inconvenient truth that rationality itself is flawed. At any rate, he’s got a blog and it’s worth subscribing to on RSS.

I think he’d be one of the most interesting people in Twitter if he were to get the Zen. In the meantime, I’ll settle for his Tao.

Hi Aziz, I’m sorry I don’t know your email.

Thanks for the link. I am well aware of Godel’s theorem. But it is not a flaw, so much as a constraint on rational systems.

Personally, I think it actually gives us a some fascinating clues about what the nature of an ultimate theory must be.

For example: Godel’s theorem only applies to systems complicated enough to include arithmetic. So theories simpler than arithmetic are not affected by it. For them, Godel’s proof breaks down.

That means, provided your ultimate theory of everything is simpler than arithmetic, then you don’t have to worry about Godel.

That’s not quite a correct description of it.

Godel’s theorem was that in any system which is sufficiently rich to permit the question “is this system complete?” to be formulated

within the systemthen the answer will always be “no”.But a system which is less rich than that cannot be the “ultimate theory of everything”.

It’s like Turing Machines and more complex computers. If you can show that a Turing Machine can simulate the more complex computer, then anything the more complex computer can do, the Turing Machine can also do. Likewise, if hypothetical system “X” is able to contain the physical universe, then it has to contain all the mathematical systems which do fall under Godel’s theorem, and that means that system “X” itself must fall under Godel’s theorem, since it can formulate the completeness question.

If it cannot formulate the completeness question, then it will not be sufficiently rich to describe the physical universe.

Steven I think you are oversimplifying this. It is quite possible that the ultimate ToE is very, very simple.

Then more-complex laws, like arithmetic (and self-reference) could be emergent properties. Like thermodynamics.

Maybe at the deepest level, even arithmetic is only approximately true, like Euclidean geometry in GR. If it is that simple then Godel would not apply.

The higher-order theories, OTOH can be Godel-incomplete. It just means no single, higher-order theory will explain the whole world. You will always need multiple higher-order theories to cover all of (known) reality.

I am really saying that Godel is implying that there are two distinct classes of theories: Base-level ToE which is v.simple but G-complete.

And emergent, higher-level theories, which are rich and complex, but G-incomplete, and therefore each only covering a limited domain.

Dan, I don’t buy it. You are arguing on faith! Why would the ToE be intrinsically simple to the point where Godel would not apply? I think you are trying to evade the implications of Godel on a strictly rationaliast worldview by simply postulating a striped down ToE. You cant wave your hands about emerent properties; emergent properties are a property of complex (ie, rule-based) systems, not simple ones.

A great site on Godel is here: http://www.miskatonic.org/godel.html

which really gives one a better intuition about Godel.

IT is not me who implies there are two different kinds of theory. It is Godel’s theorem. It is the theorem that establishes that there are Godel-complete systems and Godel-incomplete ones. The are distinguished by complexity – Whether they support self-referential statements.

The only “faith” would be if I asserted that Godel-complete theories are also realized in nature. And you are right that I have no proof of that. So all I can assert is the possibility.

I suppose it’s possible that there are no Godel-complete theories in Nature. But if that’s true then all that leads you to is the conclusion that a genuine Theory of EVERYTHING is impossible.

If that’s true, then physics will always consist of multiple but individually-incomplete theories.

There is nothing “hand-waving” about emergent properties. Phenomenological Thermodynamics is an emergent theory with respect to Statistical Mechanics. I take that as the archetype. It usually happens because of some kind of coarse-graining approximation.

Population Ecology is emergent wrt Biology which is emergent wrt Particle Physics.

BTW you may be interested in this paper which was published in 2006 that purports to prove that quantum computational systems aren’t bothered by Godel incompleteness, because the undecidable parts cannot be consistently defined, and so don’t exist.

http://arxiv.org/abs/quant-ph/0602009

If the universe is some kind of natural quantum computer, then we will never encounter a genuine undecidable proposition that is realized in nature. (Assuming the proof, or something like it is valid).

fledgling otaku,

I’ve been thinking about this comment too:

“I think you are trying to evade the implications of Godel on a strictly rationaliast worldview by simply postulating a striped down ToE.”

I think one of the things you have to understand about the rationalist worldview, is that the rationalist worldview is rigorously empirical. No matter how persuasive a theory may seem mathematically, it ain’t worth a can of beans compared to a single, well-constructed experiment.

In fact, I would argue the opposite: The more radical the mathematical conclusion, the more carefully every step in the analysis should be validated empirically – at least if you are planning to apply it to physics. This is necessary to ensure that we aren’t building a skyscraper of speculation on foundations that can’t hold the weight we are trying to put on them.

For instance: I’m perfectly happy to accept Godel’s Incompleteness Theorem, as a theorem in number theory, but I’m far less certain about the unlimited validity of number theory in physics.

For one thing I would ask:”Which number theory?”

I’ll bet you wouldn’t have to change the axioms of number theory all that much, to yield radically different theoretical conclusions. And only extremely cautious empirical validation could show us, which number theory applies in nature, at the depth of ToE.

“the rationalist worldview is rigorously empirical.”

understand that perfectly, and that is the flaw. Godel is the key to understanding that the empirical realm is limited. You err in postulating that all things that are True are either empirically verifiable or be swept under the rug of irrational. But there are also things that are True which can not be empirically verified. Thats the key of Godel. I really urge you t read the essays at the link on Godel I sent you so you really appreciate the power and nuance of the implications of Godel’s theorem on reality itself. It’s not an argument that applies solely to number theory because the proof is universal. I will post a generic proof of Godel tomorrow morning, so lets table the discussion until then.

One more comment, though, Dan – I read the abstract at arXiv but am not really impressed. If the paper really implies what it claims, it would be tantamount to a corrolary on Godel. And thats big league stuff, worthy of publication in a leading journal. We may be on teh cusp of a tremendous change in our understanding of mathematics, but i call foul on your invoking it as having already occurred.

For teh time being, I won’t accept any argument that the principles of basic arithemetic can be formulated in a fashion that exempts them from Godel. When I post the proof tomorrow you will see why.

“But there are also things that are True which can not be empirically verified.”

Sure mathematics is true but not empirically verified. No question about it. The problem is which theory of mathematics applies to physics? That’s the part that needs to be empirically verified.

“I wonâ€™t accept any argument that the principles of basic arithemetic can be formulated in a fashion that exempts them from Godel.”

Agreed, but what if the appropriate number theory at the level of ToE is not basic arithmetic? Maybe it’s one of the non-standard forms of arithmetic? (http://en.wikipedia.org/wiki/Non-standard_arithmetic), Maybe it’s the hypercomplex numbers.

“but i call foul on your invoking it as having already occurred”

Otaku, did you read where I said, “purports to prove” and “Assuming the proof, or something like it is valid”? – That was intended as a caution. Like I said I thought the paper was interesting – it raises interesting possibilities.

“But there are also things that are True which can not be empirically verified.”

Sure Mathematics is true but cannot be empirically verified, no question about it. But the real question is: Which part of mathematics applies to physics? That’s the part that needs to be empirically verified.

“I wonâ€™t accept any argument that the principles of basic arithemetic can be formulated in a fashion that exempts them from Godel.”

Agreed. But what if the number theory that applies at the level of ToE is not basic arithmetic?

“but i call foul on your invoking it as having already occurred”

I seem to recall having said, “purports to prove” and “Assuming the proof, or something like it is valid”. They were intended to be cautionary. I still think the paper is “interesting”. So where’s the foul?

meh.

Dare, Sir Roger is a theist.

get with the program.

do u know wat i think about a lot lately?

the reduced curvature of spacetime, like in Sir Rogers book.

i think…mebbe…our slice of spacetime is a gynormous ring….and the curve is barely perceptable becuz our crude viewfinders.

😉

and it’s almost May!!!!!

CERN FTW!!!!!!!!!!

“Which part of mathematics applies to physics?”

Dan. Are you playing devil’s advocate here? Surely you realize that physics and math are not fundamentally separable. The biggest questions in physics are mathematical ones.

As far as the arXiv abstract goes, i call it a foul because the existence of the abstract does not justify any skepticism towards the applicability of Godel to formal systems. Its not sufficient.

But the paper (and it may well be wrong) doesn’t argue that Godel doesn’t apply to formal systems. It argues that undecidable propositions aren’t actually realizable in quantum computation, because they can’t be defined consistently. It is saying that q computation skirts around the troublesome parts without encroaching on them. Like I say the paper may be wrong but it’s interesting.

“The biggest questions in physics are mathematical ones.”

They are like twins. About this we agree. But the overlap is not to say they are the same. Many things are true in mathematics that have never been observed in nature, and the history of physics is littered with the bodies of mathematically-flawless theories that nature, sadly, didn’t choose to obey.

“Itâ€™s not an argument that applies solely to number theory because the proof is universal.”

Well, I just think it’s polite to ask the universe, if IT agrees with YOU about what is universal. That’s why physicists have to do the experiments.

Daniel, are you aware that Aziz has a Ph.D in physics? That kind of double-talk won’t fly against someone like him.

…and his specialty is quantum mechanics…

I don’t know what you are calling double-talk Steven. Can you be more explicit?

I have spoken to Aziz many times. It was Aziz that told me Haibane was linking to my weblog.

I very much doubt that a physicist wouldn’t understand what I’m talking about. The theories that you are most certain about, are the ones that most desperately need empirical checking. Because the consequences if the experiments showed a discrepancy would be so serious.

The notion that physics can be free of empirical validation is the most profound arrogance. Nature decides what is true. Man struggles to understand it. Many times in the past, our best efforts have crumbled in our hands. Theories we had great confidence in have fallen apart when confronted by the test of empirical reality. Physicists have learned to be humble before nature.

pfft.

im with tegmark

you eggsheads

😉

“Godel is the key to understanding that the empirical realm is limited”

NO.

Godel proves that THEORIES about the empirical realm are limited.

It is propositions that are unprovable in Godel’s theorem.

The objective world can still be explored by experiment.

It’s just that we can’t be sure that any given theory will be able to explain what we see, because formal system contains propositions that are unprovable from within.

At worst, this means that if nature instantiates certain tricky propositions then your theory might fail, and you will need a different theory.

It’s a challenge to the idea of a single unified theory of nature. But it is not a challenge to rationality itself, because we can invent as many theories as we need.

And even a unified theory of nature might be valid, if you could find one that doesn’t have any unprovable parts, in all the cases that nature instantiates.

Or as I argued above: you may be able to use the “emergence hack” to generate multiple, related theories that collectively describe all of objective reality on all scales. Each sub-theory being specialized to work well in its own domain.

By “emergence hack” I mean approximation schemes or “correspondence principles” etc, etc.

It’s basically a plausible creation-myth for generating one theory from another. – But a lot of physics works this way.

And that is why you need to test this thing empirically, because it is far from obvious that any current theory contains unprovable propositions in any part that makes a difference to the success of the theory.

Physics just seems to steam ahead, worrying far more about funding than Godel’s Theorem.

Dan, you say that the abstract

however, the abstract says explicitly:

(Note that “qm-arithmetic” is something new that they define in the paper)

And here they are suggesting that the entire edifice of arithmetic can be rebuilt using their invention. Or maybe not the entire edifice; tey say “problems of the foundation” which is vague. Some problems? All? its hard to tell.

The abstract is written in such vague terms that its clear that they want the reader to think they have done an end run around Godel to get to the foundations of math. In other words, they are arguing quite directly that they have provided a means to define a formal system – specifically, arithmetic!! – so that Godel doesn’t apply. “skirts around the troublesome parts” ?? Godel is not a speed bump, its a fundamental property of formal systems. You can’t wave your hands about qm and say you “evade” it. Or, if you can, then you submit the proof to peer review and join the pantheon of Math Gods. Not release it on arXiv.

let me address some of the other tangential points you also raised:

I think you are confusing things by anthropomorphising the Universe in this way. The reason physicists do experiments is because they test theories. Theories are fundamentally nothing more than “statements” in a formal system. Godel says that there exist in ANY formal system certain statements whose truth cannot be determined within the system. That means that there are NO experiments you can do to assess the truth of certain theories. Thats not a cosmic conspiracy! its an intrinsic property of the system itself.

It’s not clear to me what you are referring to here. A theory we are “certain” about is one that has already been empirically tested. Godel says that there are some theories that cannot be empirically verified. Obviously we arent certain about those.

Its not clear to me where I said “physics” can be free of empirical validation. Godel says that certain statements lie outside the realm of empirical validation. Whether those statements are true or not is not the issue; only that they can be

verified. I think it far more arrogant to presume that Truth is subject to our verification. The Truth of a given statement – in math or physics – is an absolute, decided as you put it by Nature (or God, but lets not go there). Validation is just what we simple creatures do within the Labyrinth of Reason in which we are imprisoned.matoko, awesome avatar 🙂 Not too clear on the context of your Tegmark quote.

Steven you give me too much credit. My PhD is in medical physics, specifically MRI imaging, not quantum 🙂 I do have to know a it about QM in my line of work but Im not qualified to write, or review, the kind of paper to which Dan linked. my skepticism is more meta.

Dan, look sliek you posted a few more comments while I was drafting my longer reply. I am tabling the discussion for now. Lets pick this up tomorrow when i publish the proof of Godels theorem and start fresh.

Sure deal with it when you like

You said, ‘Godel is not a speed bump, its a fundamental property of formal systems. You canâ€™t wave your hands about qm and say you â€œevadeâ€ it.”

Yes, it is a property of formal systems that they contain unprovable propositions, but that doesn’t mean that every proposition is unprovable.

Theories work over very large domains. As long as the parts that are relevant to the theory don’t contain any badly-behaved parts, then you have a successful theory.

There may be badly-behaved parts of the theory, but as long as they don’t impinge on the physically-relevant solutions, say as defined by the boundary conditions. then what happens outside that is of no consequence.

It occurs to me that this may be generally true. That when you impose boundary conditions to eliminate the unphysical solutions, you may also be eliminating the unprovable propositions or at least confining their effect to the unphysical parts of the solution state-space and so ensuring that they are not a problem to the theory.

Otherwise how can you explain that Godel-incomplete theories (and every physics theory is Godel-incomplete), work as well as they do?

I can set up a physical model, like a simple harmonic oscillator, and not a single point in the physically-relevant phase-space is badly-behaved. So where are the unprovable propositions that are preventing the theory from being complete?

that doesnâ€™t mean that every proposition is unprovable.nowhere in this discussion, am I aware that anyone has suggested this.

There is a certain vibe that you get from some pseudo-science fanatics, a kind of manner and style that’s pretty common. Aziz, remember those guys who claimed they’d proved the existence of God using Set Theory?

I’m getting that kind of vibe here. Entirely too much enthusiasm, entirely too little intellectual caution or rigor. Too much tossing around of obscure jargon in ways that sound like doubletalk. Too much “baffle them with bullshit”. Sorry, but I’m not buying any of this.

For one thing: arithmetic isn’t real. It doesn’t exist. Physics uses arithmetic, but arithmetic is a pure mental construct, that if it can be argued to exist at all (that depends on how you define existence) then its existence is in a pure mental space which is unrelated to the real world. That’s true of all mathematics.

These are consistent mental constructs, built from axioms and constructed using deductive reasoning involving applying rules which are part of the axiomatic foundation. It is possible to build parallel mathematical systems which have different axioms which will yield different answers to the same question.

What is the ratio of the circumference of a circle to its diameter? In Euclidean geometry it is pi. In elliptical geometry it is less than pi but there’s no constant answer. In hyperbolic geometry it is greater than pi, and again there is no constant answer.

Those answers contradict, but each answer is true within its own system. The point is that mathematical deductions are based on the axioms of that system, and if you change the axioms, you change the conclusions.

But they’re not based on physical reality. None of them are. We

usethem in some cases in physical reality when they are isomorphic to physical reality, because they make it easier to derive certain answers we need. But if they’re not isomorphic, they aren’t useful. try to navigate the oceans using Euclidean geometry and you’ll get lost, because the surface of the Earth isn’t a plane. You need elliptical geometry.A claim to find a quantum mechanical basis for arithmetic sounds like doubletalk to me, because no mathematics is based on physical phenomenon. I don’t buy that.

“That when you impose boundary conditions to eliminate the unphysical solutions, you may also be eliminating the unprovable propositions or at least confining their effect to the unphysical parts of the solution state-space and so ensuring that they are not a problem to the theory.”

That sentence is grammatically correct, but it doesn’t mean anything. (I think the sardonic term was “a meaningless noise”.) Or rather, it does mean something, but it isn’t profound.

See, if you’re permitted to impose boundaries on arithmetic, it’s easy to reduce it to the point where it is Godel-complete. There’s no particular surprise about this, nor does that contradict the theory. In fact, the theory points the way to how to do this. All you need to do is eliminate enough of arithmetic so that it is no longer possible to encode the completeness question within it.

But if you do that, you’ve also crippled it enough so that it is no longer isomorphic to enough of the real world situations where we use it to be worth studying any longer. That’s the problem. It’s not just a matter of confining it to “physical solutions”; you have to constrain it a lot more than that.

Hello Haibane-ists. Dan alerted me to this discussion, I think hoping I’d back him up on something. In fact, I think I have major disagreements with everyone here. The ontology of natural science is radically incomplete, but the problem is certainly not “rationality” per se, and the solution isn’t faith in anything. Basically, we have developed only a few of the “eidetic sciences”, and we think that the ones we

havedeveloped are the whole story, and that the ontological aspects which they reveal are all that exists, or all that can be investigated “rationally”. An eidetic science is something like an apriori discipline, except that it also includes primary data deriving from “intuition”, which in Husserl’s phenomenology is a technical term denoting a mode of direct awareness of something. Because we are so bad, philosophically, at understanding consciousness, we block out the intuitive roots of even the eidetic disciplines whicharerecognized (logic, mathematics, theoretical computer science), and instead treat their founding insights as arbitrary “axioms”, and their native methods of deduction as transformations of merely formal (syntactic) validity; and together with a mathematized and completely objectified notion of physical ontology, it all adds up to what looks like the culminating intellectual synthesis to its proponents (e.g. David Deutsch,The Fabric of Reality), but is actually just another stage in the development of knowledge, adulterated with arguments as to why thereareno further stages. And among those who can grasp this pseudo-synthesis, you either have people who are happy with it and see no need for anything else, or you have people who sense that there is something more to be explained but resort to quasiplatonic mathematical mysticism, since this is all that the consensus ontology leaves them to work with. Whereas to really progress, we need to step back to that Cartesian state of consciousness in which nothing is certain, take it in the Husserlian direction of attending to consciousness itself for long enough to discern its nature, constituents and structure, and then to reconstruct the natural sciences, which were generated through the process of hypothetico-deductive empiricism, in a way which does not throw out any of the ontology revealed in the phenomenological phase of investigations. That may still not be enough to get us within range of answering questions like “why does anything exist”, but it has to be better than where we are now.Actually, if I can venture to categorize the views expressed in this thread to date, I take Steven to be a pragmatic materialist, for whom numbers are convenient fictions, physics and computation explain everything, and there is (for example) no mystery about consciousness, or how properties inhere in things; Dan is very close to that position, but still hopes that some permutation of our existing concepts might magically explain, say, the existence of the universe itself; Matoko is still further out on the spectrum of mathematical mysticism; and Aziz may be willing to say that some things are entirely beyond the reach of reason in any form. (But I know his views the least.) Whereas I say we need a revival of

rationalist metaphysics, with a rigorous phenomenological epistemology at its base.I ought next to express some views on the specific topics at hand, such as what it is that Godel’s theorem says, whether it indicates a limit to rationality, and whether there’s anything to that paper which Dan found in the physics arxiv. But I might take a little breather first.

“Dan alerted me to this discussion, I think hoping Iâ€™d back him up on something.”

No. LOL

I know you well enough to know you’d bring your own unique take.

Thanks for coming.

Mitchell, welcome. Your link to eidetic sciences was broken so i have edited it to remove the hyperlink, fyi.

I appreciate the massive effort you took in writing your comment but I have only given it a cursory read so far, and replying in detail would be counterproductive. To be honest i would rather refocus on the specific question of the direct implications of Godels Theorem are rather than get pulled further into the metaphysical wilderness. Please do chime in on the latest post (the proof).

FYI if you want a sample of my thinking I will point you to a post on my deep skepticism towards the concept of Singularity, which in fact was partly a response to something at Overcoming Bias if I recall correctly.

BTW On consciousness I am currently into Thomas Metzinger. But I love the zombie argument.

ok, the new post is up…

On that “qm-arithmetic” paper from the arxiv:

In effect, what they do is take a vector space (a Hilbert space), identify a particular infinite orthogonal basis with the integers, and identify linear mappings between those basis vectors with the analogous mappings between integers. E.g. the infinite-dimensional linear transformation which maps vector |n> onto vector |n+1> for all n, corresponds to the mapping n -> n+1, i.e. to the operation of adding one. For many-to-one or many-to-many mappings, they use a Cartesian product of copies of the basic vector space.

The argument that their qm-arithmetic is decidable appears to be rather sloppy. They say: undecidability requires the possibility of diagonalization; but you can’t diagonalize our functions in a particular way. But they give no argument that it can’t be diagonalized another way.

They are also very casual in arguing that their functions can represent propositions

aboutqm-arithmetic. They merely say (in effect) that logical operators can be represented by truth-table functions, and those functions exist in their construction, QED. Well, that is not enough; sometimes you would want to employ those functions to refer to the logical operations, and sometimes to thearithmeticaloperations, and they say nothing about how the intended meaning is to be indicated.To critique the paper thoroughly, I think one would have to do the authors’ work for them – i.e. properly carry out the Godel argument for the system they have constructed – and then see what they skipped over or fudged.

Thanks Mitchell.