All bloggers are liars

I thought the best bit of Jordan’s article was the story about his fourth-grade class on peanut butter sandwiches. Either his teacher had read Quine as a student, or they had some real philosophical talent.

I hope the Slaters let Jordan give a good talking to Tim Noah, who said (of David Brock), “‘Everything I say is a lie’ is a standby of late-night bull sessions in college dormitories, sometimes attributed to Episode 41 of Star Trek, titled ‘I, Mudd.’ In fact, the true provenance of such riddles is Kurt Gödel’s Incompleteness Theorem.”

“Everything I say is a lie” is just a version of the Epimenedes paradox, which is of ancient origin.

Well, read the article. I don’t know where I got my formulation above, maybe 20 years ago in Hofstadter. I will admit that I look at theoretical physics(string theory) and cosmology thru my lay psuedo-Godel sunglasses and shake my head in sadness.

“Unified Theorey of Everything? Someday Godel will break their hearts.”

I find a kind of comfort there.

Peter Novick’s That Noble Dream is good on how some early 20th century historians rushed to embrace the uncertainty principle but sadly had no idea what they were talking about.

OK, count me among the heretics on this one. I believe that Searle, Lucas and Roger Penrose are fundamentally correct when they argue (make arguments which imply) that one implication of Goedel’s Theorem is that no Turing machine is a syntatic system.

I also think that the Axiom of Choice is transparently wrong, but that’s for another time. Come to think of it, I do think that there are particular ethical conclusions that can be drawn from Special Relativity, but those are also for another post.

My “7:00pm” post was made after Geoff Robinson’s “10:26pm” post. Seems that WordPress and/or some sysadmin has reordered the timeline. (My comment makes a bit more sense after the thread drift of the then-preceding eight.) I’d make a physics analogy, but that would just be silly…

How else could the smallest possible change be interpreted to mean a paradigm-smashing event?

The transition from one quantum state to another has no intermediate states. Hence.

Jordan Ellenberg is right about two things: most mathematicians can get along just fine without ever thinking about Gödel, and most people who cite Gödel’s theorem outside of mathematics are cranks. It seems that he is not entirely immune to the second point.

The reductionist (or do I mean formalist?) program was not killed by Gödel. Far from it. Whether they know it of not, whether they’re are trying to or not, (almost) all mathematicians are doing work that can easily be formalized within set theory and all of their theorems can be proved in ZFC. So they are “limited” by Gödel’s theorem, even if (as Jordan points out) they are not likely to ask provably undecidable questions in their everyday work. But it’s not as if Jordan Ellenberg has insight into the natural numbers — however much they may exist despite our inability to axiomatize them — which would allow him to prove something which is not provable in ZFC. His colleagues would not accept such a proof as valid without some sort of philosophical upheaval. Not because they think about of care about ZFC, but because the formalist (or do I mean reductionist?) program was so successful.

What I’m saying is: the axiomatic fondations of mathematics are strong, Gödel didn’t weaken those foundations (he only illuminated their nature) and the conclusion of Jordan’s article is complete bunk.

(But he’s allowed his opinion. It’s not like I can prove any of what I just stated.)

I think Ellenberg overstates his case by a good bit. It’s true, I suppose, that Goedel’s incompleteness theorem is more of a confirmation of something that seemed likely from mathematical experience than a world-shattering, revolutionary surprise. The formalist claim of completeness was made partly to astonish, and while ultimately it turned out to be false, the trip it took mathematics on was worth making. But Ellenberg’s “at least so far” rings very hollow in a field where we still rely, day-to-day, on ideas thousands of years old; if incompleteness still makes no difference to number theory in a few thousand years, then we can let that go by.

And, by the way, Ted Chiang covered all this ground to enormous effect in his story “Division by Zero”. It’s collected in *Stories of Your Life and Others”, which I think would be of interest to all the *Crooked Timber* authors.

Historian K.N. Chaudhuri wrote an excellent work Trade and Civilisation in the Indian Ocean: An Economic History from the Rise of Islam to 1750. Enough for me to hunt down his later book Asia Before Europe from the library…

What do I find? He’s been to a conference full of lunatic French philosophers, and is now using everything from Gödel’s Theorem to the Axiom of Choice as arguments for his historical theses. Aaaarrgggh!

Anyway, I’d add the Axiom of Choice to the metatheorem: it has no (direct) application outside mathematics (how many uncountable collections are there in the real world?)

Ellenberg’s argument would be more compelling if he avoided a compositional fallacy:
But most pure mathematicians can easily go through life with only a vague acquaintance with Gödel’s work. So far, I’ve done it myself.

I humbly agree with Jordan Ellenberg, though I may be accused of not being the sort of person who cares too much for Godel’s incompleteness theorem. On the other hand, Godel’s completeness theorem (equivalent to the compactness theorem of first order logic) has very important applications in mathematics, via model theory. I also rely very heavily on the Axiom of Choice. So I’m looking forward anxiously to dsquared’s critique. I’m sure it will be enjoyable to read, as is the case with everything he writes.

> Next up, evidence that water is wet.

If you ask Kent Bach, he’ll tell you that water isn’t wet. It makes things wet.

Grrr. As anyone who has dealt with sensors with limited resolution can attest, it is in fact the case that 2+2=5, for sufficiently large values of 2.

So what is the opinion of John R Boyd around here?

Actually, I’m pretty sure that we can prove both 2+2=4 and 4≠5.

Clearly you’re an Ivory Tower Liberal Elitist who’s never dealt with real-life finance, media or politics.

The axiom of choice isn’t so bad. It’s unavoidable in the countable case– Russell’s example (constructing a set of ‘right’ socks from an infinite set of pairs of socks requires an axiom of choice) persuades me.

And it’s not so bad in the in the uncountable case either. Descriptions of the various notorious examples (Banach-Tarski, et. al.) tend to leave out the basic point that unmeasurable sets are not physically realizable. At least in the negative form (there are uncountable sets on which one cannot consistently define a measure) AC seems OK to me.

Quiggin’s metatheorem: Any interesting conclusion derived with reference to Godel’s theorem is unfounded.

If that is understood as “derived with reference to Godel’s theorem alone” then I agree. Indeed it would be merely an application of Benacerraf’s dictum “No philosophy in, no philosophy out” or you can’t derive a philosophical conclusion for a mathematical result without supplementary philosophical assumptions.

I love this kind of talk.

(Recently at Pharyngula P.Z. Meyers set his comments to run from newest comment to oldest. Reading down, it was startling to run across the contradiction of a point that hadn’t been made yet. But this G&oum;del mash-up is way better)

A question to brock sides. You say,

…if we can prove that the axioms are consistent, then they aren’t.

Does this mean the same thing as saying they aren’t all “axioms,” that some irreducible number of them can always be provably swapped out for another likely formulation, as with the non-Euclidean geometries?

I look forward to seeing you answer my question before I post it.

Penrose’s first book struck me as amazingly badly
reasoned for a man with so many genuine accomplishments.
A friend of mine who does work on quantum cryptography
said he had asked Penrose at a lecture:

“You say that consciousness is impossible for a machine
because it exploits certain quantum effects. I have built
machines that exploit quantum effects in interesting ways,
such as to send bits across a channel that cannot be intercepted.
Why couldn’t a machine be built to exploit the quantum
effects you speak of, and thus create consciousness?”

Penrose, he says, essentially didn’t answer…

I think I’ve nailed what I don’t like about Jordan Ellenberg’s dismissal of Gödel’s incompleteness theorem. Ellenberg writes:

One way to understand Gödel’s theorem (in combination with his 1929 “completeness theorem”) is that no system of logical axioms can produce all truths about numbers because no system of logical axioms can pin down exactly what numbers are. My fourth-grade teacher used to ask the class to define a peanut butter sandwich, with comic results. Whatever definition you propose (say, “two slices of bread with peanut butter in between”), there are still lots of non-peanut-butter-sandwiches that fall within its scope (say, two pieces of bread laid side by side with a stripe of peanut butter spread on the table between them). Mathematics, post-Gödel, is very similar: There are many different things we could mean by the word “number,” all of which will be perfectly compatible with our axioms. Now Gödel’s undecidable statement P doesn’t seem so paradoxical. Under some interpretations of the word “number,” it is true; under others, it is false.

The parable of the peanut-butter sandwiches seems to be referring to (or, at least, it reminds me of) Abraham Robinson’s nonstandard analysis. A mathematician can step out of strict number theory into a realm where P is provable and where P does not encode “P cannot be proven”, so the proof of P does not result in contradiction. That’s all well and good for the number theory we started with; but in the new nonstandard realm there exists a statement P’ which does encode “P’ cannot be proven”. You can continue this recursively, but at every stage there will always be a statement that cannot be proven without contradiction until the next iteration.

(I’m not actually a mathematician; I just know a few bits and pieces of mathematics.)

In the end, Ellenburg trivializes Gödel’s theorem by saying that statement P might not be provable, but it’s not an interesting statement in the first place. (Those grapes were sour anyway.) This reminds me of the proof that there are no uninteresting integers.

Does anyone agree with me that the question of whether God knows the future can be reduced to the halting problem, and thus answered “no”?

I thought that something significant could be derived from the observation that it is not apriori that it was Godel who was responsible for the discovery of Godel’s theorem. Is this the undoing of Quiggin’s metatheorem?

Oh, and what Brock said above about Quiggin’s conjecture!

That we can prove 2 and 2 isn’t 5 doesn’t show we can’t prove that it is. Which is interesting!

As for 2 + 2 = 4 and 4 not equal 5, I remember vaguely peano aritmetic. It wouldn’t matter at all that 2 + 2 = 4 and 4 = 5 if you simply defined S(3) = 4, 4 = 5, and S(5) = 6.

Not that it would be useful in any way, but it would introduce no (new) inconsistencies in simple arithmetic.

And these abstractions aren’t entirely useless because irl you have to be able to understand that 12 can be equal to 10 and equal to A depending on the context.

Something as simple as the principle of well-ordering for the natural numbers is not expressible in first order logic. How can one say that every subset of the natural numbers has a least element in a finite-length first order sentence? The reason why first-order logic is useful in mathematics is that it satisfies compactness and the Lowenheim-Skolem theorems. One can go to higher order logics and capture more aspects of mathematics (one gains expressive power), at the huge expense of losing compactness.

“Quiggin’s Metatheorem: Any interesting conclusion derived with reference to Godel’s theorem is unfounded.”

That’s almost certainly right, if by “interesting conclusion” you mean of practical interest. Godel’s theorem (trying to make an umlaut obviously causes browsers to spit out garbage, so I’m not going to try) just tells us that certain things are in principle impossible, but we don’t need Godel’s theorem to know that they are in practice impossible. I find, though, that in principle impossibility is interesting in itself—like perpetual motion, exceeding the speed of light, trisecting an angle, etc.

With all respect, dsquared, Penrose and Lucas are mistaken. Godel’s theorem doesn’t say anything at all about whether AI is possible. It says that computers can never do certain things, but there is absolutely no reason to believe humans can do them, either.

My question to brock sides was about “they aren’t” in the phrase “if we can prove that the axioms are consistent, then they aren’t.” Did it mean they aren’t consistent, or they aren’t “axioms” or a little of both?

Aren’t consistent.

Randolph Fritz, CT has already recommended Ted Chiang — that’s where I heard of him, IIRC.

Still, I think Divide By Zero is his weakest, and is not worth reading. I don’t know anyone who likes it. But I do recommend that everyone buy that collection and read the rest of the stories. Chiang is simply smarter than other authors.

Here’s an interesting reference to Goedel, Sraffa, and Einstein:

http://ssrn.com/abstract=608642

I find it highly questionable, to be nice.

But you might think, “It would be nice if the axioms could prove themselves consistent.” Godel’s answer: “No, it wouldn’t.”

Think about it: if the axioms proved themselves consistent, what would that tell you? After all, if they are inconsistent, they could prove anything at all, including their consistency! So why would their proof of their own consistency give you any more grounds to trust them?

It’s not the fact that a system can’t prove its own consistency that’s really interesting; it’s the fact that a system that seemingly formalises our intuitive finitistic reasoning about numbers can’t do it. That is what buried the formalit program, or at least has been widely considered to have done so.

Cosma and Brian noticed by attempt to subvert the dominant paradigm by cunningly typing “syntactic” instead of “semantic” …
My understanding is that Searle’s current position (which which I agree, and I don’t agree that the Systems Reply is a “devastating critique”, or that the Lucas/Penrose argument has been refuted by anything Putnam’s published) is that there is no such thing as a syntactic system which is not a semantic system, because syntactic systems are only syntactic in as much as we interpret their output as having content.
In other words, what a computer (or a Chinese room) does is to turn switches on and off, and this can only be called “syntactic manipulation” if there is somebody around to interpret it as such; a symbol isn’t a symbol unless it signifies. So I’d argue that Brian’s “connected up to the world in the right way” has to mean “connected up to some part of the world which has ‘real’, non-derived intensionality”. I do think that there is such a thing as “intrinsic semanticness”, because I’ve got it and so (I presume) has my wife. My son appears to be developing it.
But we Searleans, rather like we Welsh, are a small and unpopular nation, more keen on backbiting against our more powerful neighbours than taking them on in open combat.
On the Axiom of Choice, I just don’t like it because I think it’s at the root of a lot of the difficult paradoxes in probability; I’m not happy with the idea of being able to select a set of points on the basis of no particular criterion of selection. In particular, a lot of the “sadistic angel” problems previously posted on CT disappear if you assume the negation of the AoC, as it becomes clear that the angel is offering you a pretend-gamble based on pretend-probabiliies.
Meanwhile, the particlar ethical conclusions from Special Relativity relate to a weak version of consequentialism; I think that the role of c in special relativity, based on the requirement to preserve the order of causation in all frames of reference, rules out “backward-causal” moral principles like, for example, the view that murders of Kurds in 1989 can be part of the rationale for a war in 2003.

” On the Axiom of Choice, I just don’t like it because I think it’s at the root of a lot of the difficult paradoxes in probability; I’m not happy with the idea of being able to select a set of points on the basis of no particular criterion of selection. In particular, a lot of the “sadistic angel” problems previously posted on CT disappear if you assume the negation of the AoC, as it becomes clear that the angel is offering you a pretend-gamble based on pretend-probabiliies.”
Actually, no. The “sadistic angel” paradoxes, as I recall them, don’t have anything to do with the AoC. For instance, on the one where you are choosing one point out of the unit interval you could replace the unit interval with a countable dense subset and the single point with, well, any finite set, and the paradox would be the same, but now the AoC wouldn’t be necessary to choose your points.

Luc, this looks more like the position Dennett wrongly attributes to Searle than his actual work. Apologies if I am wrong, but experience has led me to generally suspect that when I see the words “fuzzy feelgood notion”, then a) I’m talking to someone whose main exposure to Searle’s argument has been through Dennett’s summaries and b) the argument is going to get really personal and unpleasant pretty soon, because lots of people who take on board Dennett’s arguments also appear to adopt his polemic style.
Just as a set of preliminaries, I would like to make it clear that the phrase “strong AI” is just as fuzzy as anything in Searle, and that whatever makes me feel good, I suspect that the possibility of intensional computers makes you feel just as good. So can we please not use phrases like “fuzzy feelgood notion”, which don’t advance anything and really only serve as a form of intellectual insult.
Searle’s actual point (and to be hoenst, he hasn’t helped matters by changing his mind on the importance of syntax/semantics, or by some of his examples like the one about simulated fires) is one that can be put into terms that don’t look particularly objectionable to anyone who believes that intensionality is important. Viz:
Brian talks about things being “connected up to the world in the right way”. But a Turing machine can’t be connected up to the world at all, in any way, because it’s an abstract entity[1]. An instantiation of a particular Turing machine can be connected up to the world (and thus, potentially, connected up to the world in the right way), but if it is, then it is connected up to the world as a physical object, not as a Turing machine, and a similarly constructed physical object which was not (which we did not choose to view as) a Turing machine would also be “connected up to the world in the right way”. Therefore, while some physical instantiations of Turing machines might be intensional, nothing is intensional by virtue of being a Turing machine.
[1] This point is arguable, but I would personally defend the view that a connection to the world via some predicate of the kind “is an instantiation of” is not really much of an advance from the mind/body problem.

For those interested in Goedel’s theorems, Prof. Peter Smith of Cambridge has a partial draft of his book An Introduction to Goedel’s Theorems online at http://www.godelbook.net.
The chapters posted take you all the way through both of Goedel’s incompleteness theorems. The book does require a background in symbolic logic.

Daniel,
You’re right about the fuzzy thing, though a. is not the case and b. is not my intention.
But a dispute where after years and years of debate the different positions just grow further apart, is bound towards somewhat more emotional responses, as reason is unable to prevail.
The easy way out of this dispute is to just stick with weak AI, call a chess program a database with a search algorithm, and move on. But that doesn’t give much satisfaction. Just as I as i can say I have money in my bank account, and I can use it to trade in objects in role playing games, i would call at some point automated systems intelligent. Artificial, not human, but intelligent none the less.

nothing is intensional by virtue of being a Turing machine

But this gets close to a statement like a TM can’t send any email. So what? What is the relevance of it? Do we need to create another model? Or do we regress to the other end of the spectrum with Searle’s opinion that it is linked with biology?
I referred in my previous message to an interview from 2001 so this opinion is his, and fairly recent.
Quote in full (as the website is somewhat unreliable)

G5: If a computer manipulating symbols via a specified algorithm is not understanding, how do you believe humans process information? Do you believe this to be non-computable?
This question can be answered briefly. The brain processes information by biologically very specific, though still largely ill-understood, neuronal processes. The brain is, after all, a machine that produces consciousness, intelligence, intentionality, etc., by machine processes such as neuron firings at synapses. I regard the notion of what is “computable” as, for the most part irrelevant to the operations of the brain. The brain is a physical system like the stomach. You can describe many, perhaps most, brain processes in a precise enough way that you can simulate them on a computer, just as you can describe the stomach in such a way that you can give a computational simulation of the processes in the stomach. But the computability of these processes at the formal level is not what is essential to the corresponding physical processes at the biological level. At the biological level there are actual causal mechanisms that produce consciousness, intentionality, and all the rest of it.
Basically I think the brain is important. We might be able to do what the brain does using some other medium, but we would have to duplicate the specific causal powers of the brain, and not just do a simulation or a modeling of brain processes, you actually have to duplicate the causal powers. An analogy will make this point clear:.You do not have to use feathers in order build a machine that can fly, but you do have to duplicate the bird’s causal power to overcome the force of gravity in the earth’s atmosphere, and a computer simulation of flight is not a flight. The notion of what is computable and what is not computable is a mathematical notion having to do with what sorts of problems you can solve with algorithms, and I do not see any difficulty posed for artificial intelligence by the limits of computability. The problem is not computability, the problem is psychological reality. I realize that some people have made a great deal out of the fact that there are non-computable problems that humans can solve, but this argument seems to me irrelevant to AI because humans may be using algorithms to solve those problems even though the algorithms are not, for example, theorem-proving algorithms.

But this gets close to a statement like a TM can’t send any email.
I’ll go a bit further; a Turing machine can’t do anything at all. Turing machines are abstract entities.
A physical machine can act in a way that can be interpreted as an instantiation of a Turing machine, but this is very different. In particular, my view is that the passive-voice “can be interpreted” is covering up a big problem here; any syntactic manipulation (still less semantic) carried out by such a physical machine is entirely parastic on the decision to interpret particular physical events as having a formal role, by someone whose intensionality is not “second-hand” in this way.