Nature of Mathematical Truth

July 29, 2005 10:24 AM

Gödel and the Nature of Mathematical Truth : A Talk with Verena Huber-Dyson
posted by Gyan (77 comments total)

Awesome post. Thanks.
posted by Rothko at 11:31 AM on July 29, 2005

Wow, this looks great! I'm only through the first section, though, and it's increasingly obvious that I'm going to have to take some time and sit down and read this slowly and check references. As someone who's highly interested in the potential implications of Gödel's proof while simultaneously not really understanding it I'm really looking forward to it.
posted by nTeleKy at 11:50 AM on July 29, 2005

"the facts of platonic reality."

Ah, yes. The "facts" of platonic "reality". What facts might those be? And how does one so practiced in intellectual rigor and clarity apply the word "reality" to the formally ideal? I have never understood this. Backwards, it seems to me. Sometimes I think Plato was the worst thing that ever happened in the course of our collective intellectual development—the Apology and Phaedo naturally excepted.

...

Was it Quine who asked how many non-existent men there were in the doorway?

</derail>
posted by dilettanti at 12:28 PM on July 29, 2005

I agree with nTeleKy, I'm going to need some time to digest this, but what I've read so far has me excited to do just that. Thanks for the post.
posted by OmieWise at 12:28 PM on July 29, 2005

I have been trying to get myself to get off my ass and learn more about logic for various reasons; curiosity, as a road into learning about systems like Coq and Metaprl, and curiosity again. I haven't done any of the work required yet and I have zero actual understanding of this thread's topic but I can point to something related to this thread.

A book has recently been published that is about the abuse of Godel's Theorem.
posted by rdr at 12:36 PM on July 29, 2005

Sometimes I think Plato was the worst thing that ever happened in the course of our collective intellectual development

I used to feel the same way, but have since come around to the feeling that the ideal is prior to the real. I of course cannot support this with a purely formal argument, but it seems that a blind faith in the absolute validity of formalism and consistency is an arbitrary one, formed more by the contingencies of history than any a priori desiderata.

The work of Godel, Madhu Sudan, Chaitin, Buckminster Fuller and the new movement in paraconsistent logics have made me doubt my previous, more Aristotelian stance.

Was it Quine who asked how many non-existent men there were in the doorway?

All of them, obviously.
posted by sonofsamiam at 12:46 PM on July 29, 2005

I used to feel the same way, but have since come around to the feeling that the ideal is prior to the real.

sonofsamiam: Same here.

Gyan: Great post!
posted by all-seeing eye dog at 12:50 PM on July 29, 2005

rdr : "A book has recently been published that is about the abuse of Godel's Theorem."

See here.

sonofsamiam: the new movement in paraconsistent logics

Does 'in' signify within or of?

An unconvincing and sleight-of-hand defense of 'real' worlds is provided by the Churchlands in the paper,

Neural worlds and real worlds (PDF):

States of the brain represent states of the world. But at least some of the mind–brain’s internal representations, such as a sensation of heat or a sensation of red, do not resemble the external realities that they represent: mean kinetic energy (temperature) or electromagnetic reflectance (colour). The historical response has been to distinguish between objectively real properties, such as shape and motion, and subjective properties, such as heat and colour. However, this approach leads to trouble. A challenge for cognitive neurobiology is to characterize, in general terms, the relationship between brain models and the world. We propose that brains develop high-dimensional maps, the internal distance relationships of which correspond to the similarity relationships that constitute the categorical structure of the world.

posted by Gyan at 1:05 PM on July 29, 2005

I used to feel the same way...

I used to feel the opposite way, but then I read Wittgenstein.





Seriously, thanks for this post.
posted by voltairemodern at 1:31 PM on July 29, 2005

voltairemodern: "I used to feel the opposite way, but then I read Wittgenstein."

Can you elaborate?
posted by Gyan at 1:32 PM on July 29, 2005

i took a graduate-level course on wittgenstein as an undergrad. i liked the philosophical investigations more than the tractatus. seems to me his notion of the 'unsayable' (or however you prefer to translate it) had a lot in common with the ideal realm of plato. i've seen it suggested elsewhere that wittgenstein may have been a horribly misunderstood platonist (somewhere in here i think; and this may actually be the article Huber-Dyson alludes to at one point in her piece).
posted by all-seeing eye dog at 2:26 PM on July 29, 2005

An unconvincing and sleight-of-hand defense of 'real' worlds is provided by the Churchlands in the paper,

I'm pretty sure Godel's point was that 'ideal worlds' are real, not that 'real worlds' aren't. that's actually a really important distinction. One view leads to the non-sensical scourge of 'absolute relativism'; the other doesn't.
posted by all-seeing eye dog at 2:30 PM on July 29, 2005

Amazing how many brainy, interrelated Dyson's there are .
Esther Dyson, Electronic Freedom Foundation, George Dyson, technology historian and former eccentric (read 'Starship and the Canoe'), Freeman Dyson, nobel-prize winning physicist, the father of both and former husband of Verena. 'Mazin. . .
posted by mk1gti at 2:37 PM on July 29, 2005

all-seeing eye dog : "I'm pretty sure Godel's point was that 'ideal worlds' are real, not that 'real worlds' aren't."

What would an 'ideal world' constitute of?
posted by Gyan at 2:40 PM on July 29, 2005

Gyan: well, any mathematically or otherwise formally describable worlds, really, as i understand it... but i'm open to alternate interpretations.
posted by all-seeing eye dog at 2:45 PM on July 29, 2005

A pretty informative piece about which, as I've written elsewhere, I have one major gripe:

Just one gripe... Goldstein, referring to the friendship between Godel and Einstein says:

"... Both of them saw their work in a certain philosophical context. They were both strong realists: —Einstein in physics, and obviously Godel in mathematics. That philosophical perspective put them at odds with many of their scientific peers..."

Well, mathematical realism (aka Platonism) and scientific realism are not quite the same thing. They are not related. Mathematical realism is a species of idealism, whereas scientific realism is strictly a (mutated) nephew of logical positivism, the antithesis of Platonism as explicitly mentioned by Goldstein herself in the linked article when she discusses Godel's differences with the Vienna circle. It's probably true furthermore, that calling Einstein a "realist" oversimplifies his somewhat eclectic and idiosyncratic philosophy of science. So the parallelism fails completely.

posted by talos at 2:45 PM on July 29, 2005

all-seeing eye dog : "well, any mathematically or otherwise formally describable worlds,"

Real, in what sense?
posted by Gyan at 3:04 PM on July 29, 2005

Gyan: can't respond now. will try to make it back soon.
posted by all-seeing eye dog at 3:12 PM on July 29, 2005

Great post. I'll have to come back and give this the time it deserves. Thanks.
posted by elwoodwiles at 3:42 PM on July 29, 2005

I keep trying to tell you people, that there is no real deep mystical philosophical implication of Godel's theorems to mankind, the human mind, life, the universe and everything. They say nothing more than "you cannot compute the infinite". Surprise surprise. You can build your universe to as insanely large as you like it, just don't mess with infinity or you won't be able to prove everything that comes out of your mouth.

As pointed out in the article, Hilbert made a lot of spurious assumptions when trying to formalize mathematics through essentially finite means. Godel simply came up with a construction that formalized people's intuition and destroyed Hilbert's grand project. Godel is not the Messiah. He is just a very naughty boy.

Get over it. Eat as many donuts as you like as long as you know when to stop.
posted by DirtyCreature at 4:59 PM on July 29, 2005

Thanks Gyan, again.
posted by semmi at 7:19 PM on July 29, 2005

Hmm... Read like a bunch of gobltygook to me.
posted by delmoi at 8:28 PM on July 29, 2005

I keep trying to tell you people, that there is no real deep mystical philosophical implication of Godel's theorems to mankind, the human mind, life, the universe and everything. They say nothing more than "you cannot compute the infinite".

You're an idiot.

Godel's incompleteness theorem has nothing to do with infinity. "you cannot compute the infinite" is obviously true with a simple proof, namely that any computing device has a finite amount of storage and time.

Godel's proof certainly had real world consequences for a couple of people.
posted by delmoi at 8:32 PM on July 29, 2005

Oh, also no computer which is not stuck in a loop can ever take more then 2^M steps where M is the number of bits of information it takes to encode the state of said computer. So space means finite time, or detection of an infinite loop.
posted by delmoi at 8:35 PM on July 29, 2005

Delmoi:

Why the rudeness? As I understand it Godel's Theorem doesn't hold in finite models so in what sense isn't Godel's Theorem pretty intimately tied to reasoning about infinite systems?
posted by rdr at 9:16 PM on July 29, 2005

rdr: because he claimed that godel's incompleteness theorem said "nothing more" then that simple statement, which is clearly wrong.
posted by delmoi at 10:20 PM on July 29, 2005

It's odd. As I find myself doing more and more math at higher and higher levels my tolerance for this sort of thing has dropped.

Godel's theorem is a neat way of encoding the statement "This statement is false" in mathematical notation. That's it folks, nothing to see here, move along. Yes, there are true things in math that cannot be proved, but that doesn't mean a thing for all the things we have proved or are going to prove. It just helps along things like the independence of the Continuity Hypothesis and the Axiom of Choice from the rest of math. (I tend to be rather attached to the Axiom of Choice as it allows the Banarch-Tarski paradox which I'm a big fan of.)

On preview, what DirtyCreature said. Except that he (she?) stated it wrong. I suspect that she (he?) saw the Turing machine proof of the theorem which relies on a diagonalization arguement similar to that used to prove the various sizes of infinity. But more than anyone else here, DirtyCreature sees the signifigance of the theorem: not much, but isn't a neat thing to look at.

Why doesn't anyone ever get this excited about Cantor? His ideas were just as revolutionary and actually did change a lot in math.

To summarize: Godel means a whole lot to philosophers and not a lot to most mathematicians. (My Mathematical Logic class is the only class that I have had no application for outside of that class itself. I have had to use Abstract Algebra, both Real and Complex Analysis, et. al. in many other math classes I have taken and in my senior research project) If you are working on the very fundamentals of math this matters. If you're doing anything else, this is just an odd note, a minor pit fall. This does not prove the existance of god, say that there are problems in the fundamentals of physics, etc.

I'll stop ranting now.
posted by Hactar at 10:23 PM on July 29, 2005

Gyan: an ideal world would consist of unchanging things -- as I understand a very general form of Platonism.

And I think that any reading that makes Wittgenstein a Platonist would alter history in a really bad way. At least later Wittgenstein.
posted by ontic at 10:24 PM on July 29, 2005

Also, could someone please define all those German words she's using? (Especially "Königsweg" and "inhaltlich" and "intuitionistisch." I think I'd be able to completely understand the paper if the author had not resorted to unstranslated foreign words either for precision (benefit of the doubt) or to sound better (cynical view).
posted by Hactar at 10:30 PM on July 29, 2005

"Real, in what sense?"

Gyan: let's start with what "ideal" means, and i think i'll probably answer your question along the way.

one of the classical illustrations of the concept of "ideals" (in the philosophical sense) goes something like this: think about a triangle, in the abstract. as anyone who's been forced to take some basic geometry knows, there are some mathematically-describable features common to all triangles (the interior angles always sum to 180 degrees, for example). without having to measure a specific triangle, we always know that the interior angles (regardless of the specific measure of each individual angle) will add up to 180 degrees. this is true of every triangle we can find, which suggests that all triangles--as much as they may vary in other respects--exhibit a kind of underlying unity. for plato, this unity originated in an "ideal form." basically, a form is a preexisting idea of a thing ("idea" here only meaning, basically, not made of stuff you can actually touch or see) that exists somewhere beyond the physical world. in this view, every specific triangle is just a variation on (really, in plato's view, a failed attempt to copy) the ideal form of the triangle. i'm not quite as judgmental as plato, so i don't think of all real-world triangles as "failed versions" of their ideal forms, but otherwise, i think there's something to the idea. in fact, goedel originally set out to prove the existence of ideal mathematical realities that exist independent of the formal systems used to express them in his famous proof, and arguably, that's exactly what he did. (although many have (mis?)interpreted his results to be concerned with pointing out the inherent limitations of formal systems.)

whew. that was a mouthful, eh?
posted by all-seeing eye dog at 11:10 PM on July 29, 2005

"konigsweg" = literally: "King's Way"

"inhaltlich" = eh... eludes me at the moment... something like, take-in-able? maybe conceivable?

"intuitionistisch" = "intuitive"

so, yeah. she could have just used english to say those things.
posted by all-seeing eye dog at 11:14 PM on July 29, 2005

according to systran:

inhaltlich = "contentwise"?

so that may actually have been a legitimate use...

and this one:

intuitionistisch = "intuitionistic"

so, i guess i should retract what i said before; these do legitimately seem to be terms that don't translate well. not sure what konigsweg means beyond its literal translation.
posted by all-seeing eye dog at 11:47 PM on July 29, 2005

A "Königsweg" is a short cut, an easy way. There's an old quote saying "there's no Königsweg into philosophy"; if a king wants to be wise he has to study just as hard as anybody else. (Quick Google search: if the page I found is correct it's a Marx quote.)
posted by Termite at 1:42 AM on July 30, 2005

all-seeing eye dog : "basically, a form is a preexisting idea of a thing ('idea' here only meaning, basically, not made of stuff you can actually touch or see) that exists somewhere beyond the physical world."

Exists, in what sense?
posted by Gyan at 5:01 AM on July 30, 2005

'Exists' in more or less the same sense as the question you just asked me does. Now you tell me, does your question 'exist'? If not, I'm afraid I can't answer it for you.
posted by all-seeing eye dog at 8:43 AM on July 30, 2005

all-seeing eye dog : "'Exists' in more or less the same sense as the question you just asked me does. Now you tell me, does your question 'exist'?"

A question is just a disposition, i.e. a trigger to elicit communication that is semantically connected to it. In any case, you've provided an analogy rather than a description.
posted by Gyan at 10:12 AM on July 30, 2005

Gyan: to me, that's all language (which is just a formal representation of its semantic object(s)) can really do. what would you suggest a question is a 'trigger' to 'communicate' if this isn't the case?
posted by all-seeing eye dog at 10:28 AM on July 30, 2005

to clarify a little: "that's all language... can do" in my previous comment was meant to say that all language can do is provide good analogies to its semantic objects. put crudely, that's what godel set out to prove: that no formal system is powerful enough to describe certain logically necessary features of its objects, which implies that those objects actually do exist independently of the systems used to express them.
posted by all-seeing eye dog at 10:53 AM on July 30, 2005

A bit late, but: Hactar's got it dead right (rock on Reedies!). Working mathematicians just don't ever worry about Gödel. It was, and remains, an astounding piece of mathematics (much more impressive than DirtyCreature implies), but largely irrelevant to the practice of mathematics.
posted by gleuschk at 11:29 AM on July 30, 2005

Hactar : "Godel's theorem is a neat way of encoding the statement 'This statement is false' in mathematical notation."

Isn't that a contradiction? Godel seems more like "This statement has no proof".

all-seeing eye dog : "that no formal system is powerful enough to describe certain logically necessary features of its objects, which implies that those objects actually do exist independently of the systems used to express them"

How do you know that they are 'logically necessary'? They are provable or evident in some other way.
posted by Gyan at 11:35 AM on July 30, 2005

"How do you know that they are 'logically necessary'? They are provable or evident in some other way."

as i understand it, they are only provable or evident if you invent a new formal system that isn't logically consistent with the original system to prove them. but that's just my intuition, and i may be oversimplifying the finer points (as i said before, it's been a long time since i've invested sustained energy thinking about this stuff).
posted by all-seeing eye dog at 12:08 PM on July 30, 2005

a couple of related thoughts: think of a formal system (mathematical, logical, linguistic, etc.) as a map to a particular location. obviously, reading a map is no substitute for actually making the trip to that location, although the map can be a really useful tool along the way.

And I think that any reading that makes Wittgenstein a Platonist would alter history in a really bad way. At least later Wittgenstein.

ontic: can you explain this comment a bit more?
posted by all-seeing eye dog at 12:32 PM on July 30, 2005

...but then I read Wittgenstein.

Just an aside: Wittgenstein had the misfortune of introducing his Tracticus at the same conference that Godel introduced his proof, thereby losing the thunder.

That must have sucked, kind of like having your movie open on 9/11.
posted by StickyCarpet at 12:50 PM on July 30, 2005

Isn't that a contradiction? Godel seems more like "This statement has no proof".

There's the interesting part, and the reason Godel's proof is interesting. Prior to Godel, truth was identified with provability. He tells us that "truth" as is intuitively understood can never be captured in any formal system. That may imply a couple of things, both of which I tentatively buy: a) human thought is necessarily paraconsistent, and b) the phenomenal universe will not be captured by any formal system. The quantitative universe described by science (at any given resolution) cannot help but be described by some formal system.

Incidently, I'd hardly say that proof theory and foundational stuff is irrelevant to working mathematicians; every computer program defines an axiomatic system and an execution of that program is isomorphic to a proof of the output in that system. Considerations of consistency and provability are extremely relevant, imo. Since most people "doing math" for a living are doing statistics and/or linear algebra, it doesn't matter to them. Incompleteness is very relevant to computer science.

To see what the future is going to look like, check out Madhu Sudan's work, and think about it for a while. They don't hand out Fields medals for nothing. The implications for society are staggering, but almost nobody's hip to it yet, except a few cryptoanarchists. I also do seriously recommend Chaitin, if you can stomach his idiosyncratic writing style; the ideas in there are worth it.

----

I think Plato's thought was a little subtler than that the forms are ephanized only imperfectly; they are instantiated according to their respective situation and environment, which is necessarily unique. "Imperfection" didn't hold the same stigma to him.
posted by sonofsamiam at 2:42 PM on July 30, 2005

Metafilter: Not a formal system, and therefore not necessarily incomplete or inconsistent.
posted by cleardawn at 3:59 PM on July 30, 2005

He tells us that "truth" as is intuitively understood can never be captured in any formal system.

Completely and utterly false.

That may imply a couple of things, both of which I tentatively buy: a) human thought is necessarily paraconsistent, and b) the phenomenal universe will not be captured by any formal system. The quantitative universe described by science (at any given resolution) cannot help but be described by some formal system.

This is what I mean by these people who misinterpret Godel's theorems and try to make it into mystically and all-encompassing - it is not. However I don't see why it is useful at all to me to educate random people on some anonymous internet forum whose background is unknown to me, who I am never likely to meet and who want to claim they understand something they don't. It is too easy for someone when completely exposed as false, to pay another $5 and reinvent themselves.

But I will leave you with one hint and one hint only : you forgot the proviso "any formal system as least as powerful as standard arithmetic".

If anyone wants to keep arguing and contradicting, go for it. I'm not going to help anyone that doesn't admit their ignorance about a subject they have never studied or been examined on by an accredited instiution. That's the problem with these forums - you never know who you are dealing with or why and everyone can pretend to be a legend in his own lunchtime.
posted by DirtyCreature at 5:43 PM on July 30, 2005

DirtyCreature, do you really not see "... why it is useful to me ... to educate random people ... whose background is unknown to me, who I am never likely to meet..." ???

Judging from the lamentable inaccuracy and ill-humor of some of your comments here, your own education is as dependent as anyone else's on the random kindnesses of strangers.

I don't know where your camel is, but if I find it, I promise I'll let you know. :-))
posted by cleardawn at 6:39 PM on July 30, 2005

Judging from the lamentable inaccuracy and ill-humor of some of your comments here, your own education is as dependent as anyone else's on the random kindnesses of strangers.

For those accustomed to the ambiguity and uncertainty of subjective political discussion, mathematical logic is not a safari they should be exposing their soft fleshy thighs on without suitable ammunition. As Hilbert himself discovered, there is no Amnesty International equivalent to come to your aid when your life's entire body of work has been dismembered as a result of a mauling by the unsympathetic jaws of a superior logical argument.
posted by DirtyCreature at 7:25 PM on July 30, 2005

sonofsamiam: the new movement in paraconsistent logics...

About 90% of the logic I took at a tertiary level was with Graham Priest. It's well worth the effort if you ever get a chance to sit in on one of his lectures.
posted by snarfodox at 1:07 AM on July 31, 2005

I'd hardly say that proof theory and foundational stuff is irrelevant to working mathematicians;

And yet it's so. Hike yourself down to the local university, go down the hall of the math building, and ask. If it happens to be my university, I'll buy you a cup of coffee and we can talk about it. Otherwise, you'll mostly get a bunch of people who would like you to get out of there with your Godel-nonsense and let them get back to work.
posted by gleuschk at 5:29 AM on July 31, 2005

um, excuse me, gleuschk, dirtycreature, et al., but why bother coming to a thread about godel just to godel bash, or to go on and on about how insignificant his work is? if you really think that's true, why bother? just to enlighten us all with your keen lack of insight? why do so many people around here like to join discussions they claim not to be interested in only to start bitching about how the people that are interested in them are wrong, and btw, shouldn't be interested in them either? get a life. that's just rude and pointless. just because you're interacting with seemingly anonymous people on the internet doesn't diminish that fact.
posted by all-seeing eye dog at 6:22 AM on July 31, 2005

Shouldn't feed the trolls like this, but...

"This is what I mean by these people who misinterpret Godel's theorems and try to make it into mystically and all-encompassing - it is not."

and another thing: what are you talking about in the first place? we've been talking about what Godel himself intended his proofs to mean jack ass, as discussed in a related link here.

are you telling me he's not a reliable authority on what his own proofs meant? oh never mind. of course you will.
posted by all-seeing eye dog at 6:31 AM on July 31, 2005

DirtyCreature: yes, yes, sufficiently powerful, consistent. Thanks for reminding us. You're very clever.

gleuschk: what did I just say? The work that followed from considerations of Godel's ideas will be popular in the future. I guess I wasn't very clear what I was getting at.

Now, if that's settled, I would like to float the opinion that there are no "rough corners" or "useless tricks" in math. Those tricks are implicit in the axioms we choose. Anomalous results are the boundaries of what we can usefully predict. Revolutions in math and physics happen when someone sees a hidden assumption they've been making, and edge cases help reveal those assumptions.
posted by sonofsamiam at 7:38 AM on July 31, 2005

I think Plato's thought was a little subtler than that the forms are ephanized only imperfectly; they are instantiated according to their respective situation and environment, which is necessarily unique. "Imperfection" didn't hold the same stigma to him.

sonofsamiam: that's a valid point; i guess i'm actually thinking of some of the later christian idealists who framed idealism in heirarchical terms with god at the top of the pyramid representing "perfection" in the more common, value-laden sense.

see that, dirtycreature? how i made a point, and sonofsamiam offered a valid counterpoint, so i modified my original position? that's the dialectic method. it's a really good approach for getting at the truth, when that's the point of what you're doing (as opposed to the point being to pound your chest, clack your jaws and look ferocious).
posted by all-seeing eye dog at 8:16 AM on July 31, 2005

The work that followed from considerations of Godel's ideas will be popular in the future.

I understood your point, but perhaps I didn't make my objection clear. One of my unspoken assumptions (and perennial stumbling blocks in this sort of discussion) is that the word "mathematics" means "pure mathematics". Applied mathematics, crypto, computer science, etc., are all very nice, but they're not what I was talking about.

That said, here's a restatement of my post above: Working pure mathematicians do not give a hoot about the "metamathematical implications" of Gödel's theorems. They do not act as if completeness or consistency is at issue in their work.

all-seeing eye dog, you're being a little bit rude. It'd be nice if you didn't do that. I don't think that anyone here has come into this thread to Gödel-bash. Gödel is a notoriously difficult topic to discuss (a) with non-mathematicians, or (b) with non-philosophers, or (c) with a mixed crowd of mathematicians and philosophers. Part of the reason for this is popular articles on Gödel, which stretch to find "real-world applications" of highly abstract ideas. This drives me nuts, and I think it drives a lot of mathematicians nuts. (For a more prosaic example of this sort of thing, look at any popular article on the Riemann Hypothesis and watch for the phrase, "destroy internet commerce as we know it". This makes my teeth hurt.) (Also, I'm not saying the linked article did this, because it totally did not. The price it pays for this is that it's quite difficult to read, and so won't reduce the number of foolish popular articles.)

Having seen this so many times, I make a point of trying to point it out when I see it. To be clear (and I said this in my first post, but it's worth trying to say better): Gödel's work is great mathematics. But to a working pure mathematician in another field, it is no more interesting than any other piece of great mathematics. For the vast majority of researchers, it is irrelevant to their own inquiries.
posted by gleuschk at 8:52 AM on July 31, 2005

or (c) with a mixed crowd of mathematicians and philosophers.

gleuschk: i apologize if i overreacted to your comments, or came across as rude. there have been some over-hasty generalizations and ad hominem attacks throughout this thread, and now that you've clarified your remarks, i'll grant many of your points. obviously, i'm coming at godel from a philosophical background, but godel's work was unambiguosly intended to have philosophical implications, so i tend to bristle when it's suggested that people shouldn't be doing philosophy concerned with godel's results (and that's not a straw man--it happens). on review, i see you don't seem to be making that assertion. But other participants in this discussion (like dirtycreature) seem to be dismissively conflating well-established, legitimate philosophical arguments with "mystical" mumbo-jumbo, perhaps out of ignorance. you're right: it is hard for philosophers and mathematicians to see eye-to-eye when it comes to godel. because both camps try to assert exclusive ownership of the authority to interpret his results. since godel himself considered his work more relevant for its philosophical implications, i don't quite understand why mathematicians sometimes seem so intent on claiming his work as their exclusive domain (particularly since many of those same mathematicians often in the same breath tout the fact that his results are useless from the perspective of a working mathematician).
posted by all-seeing eye dog at 12:51 PM on July 31, 2005

For example, gleuschk, dirtycreature wrote:

If anyone wants to keep arguing and contradicting, go for it. I'm not going to help anyone that doesn't admit their ignorance about a subject they have never studied or been examined on by an accredited instiution.

not only does this statement commit the fallacy of argument from authority, it makes a lot of unfounded (and, as it turns out, at least partly innacurate) assumptions and generalizations. for example, i have studied and "been examined" on philosophy at an "accredited instiution [sic]" (although i am not a "working philosopher," in the academic sense) and i suspect at least a couple of others involved in this discussion have as well (and even if that's not the case, some could be self-taught).
posted by all-seeing eye dog at 1:16 PM on July 31, 2005

one last point on this (sorry for not taking this to metatalk, but it needs to be seen in context): dirtycreature's "Amnesty International" remark was inappropriate and rude, too. (i'm not sure who or what this remark was directed at in the first place, but having humanitarian interests certainly shouldn't preclude someone from bringing ideas to the table; Einstein spent much of his life championing humanitarian causes, you might recall. i'm curious to know why dirtycreature has to bring the spectre of personal political committments into this discussion at all. it makes it difficult not to conclude that his remarks are rooted more in certain unexamined personal biases than in any principled theoretical positions he might hold.
posted by all-seeing eye dog at 1:32 PM on July 31, 2005

all-seeing eye dog : "dirtycreature's 'Amnesty International' remark was inappropriate and rude, too. (i'm not sure who or what this remark was directed at in the first place, but having humanitarian interests certainly shouldn't preclude someone from bringing ideas to the table"

As I understand it, DirtyCreature was making the point that politics is subjective, whereas logic is rigid and demanding. If you agree to the rules (of logic) then a logical defeat remains a defeat with no wiggle-room for escape.
posted by Gyan at 2:35 PM on July 31, 2005

okay, i can see how you arrived at that interpretation. but dirtycreature never actually offers a specific logical argument. nor does he demonstrate that anyone else has made a subjective claim. and what he has posited isn't quite accurate, in my view. even the rigor of logic has its limits. for example, all logical arguments are predicated on assumptions (premises). the premises of an argument can be qualified with additional supporting arguments, but to construct a supporting argument, you first have to stipulate additional premises. in order to fully qualify a series of logical arguments to the degree of rigor dirtycreature implies, you'd have to be able to build an infinite number of supporting arguments to fully-qualify all of your premises. that just isn't possible. so no logical argument can ever achieve absolute rigor. that's where logical intuition necessarily comes in to the picture.
posted by all-seeing eye dog at 2:54 PM on July 31, 2005

...unless of course you're talking about purely formal logic, which doesn't make statements about any particular real-world domain outside of the formal logical system being described...
posted by all-seeing eye dog at 2:59 PM on July 31, 2005

all-seeing eye dog : "but dirtycreature never actually offers a specific logical argument."

I didn't claim he did.

all-seeing eye dog : "for example, all logical arguments are predicated on assumptions (premises)."

Which is why I said, "If you agree to the rules (of logic) then a logical defeat remains a defeat with no wiggle-room for escape.".
posted by Gyan at 3:01 PM on July 31, 2005

huh? i'm having trouble following you. i just explained why there's no such thing as a 'logical defeat' with 'no wiggle-room' for escape in my view. all logical arguments come with wiggle room, and evaluating whether one particular logical argument is superior to another requires the use of intuition, unless you're only evaluating the purely formal features of the arguments (which tells you nothing about the soundness of the premises). please clarify what you're getting at...
posted by all-seeing eye dog at 3:14 PM on July 31, 2005

If you agree to the axioms and inference rules, then there's no wiggle-room.
posted by Gyan at 3:16 PM on July 31, 2005

ah--got you. but establishing axioms and inference rules also requires the use of intuition, if the formal system is supposed to be meaningful.
posted by all-seeing eye dog at 3:23 PM on July 31, 2005

But not if it's just supposed to be interesting.
posted by gleuschk at 7:02 PM on July 31, 2005

fair enough.
posted by all-seeing eye dog at 7:58 PM on July 31, 2005

eye dog,

FACT : There is no complete, consistent formal system that defines the natural numbers as we know them.
FACT : There is a complete, consistent formal system that defines the real numbers as we know them.

Yet the natural numbers are a "subset" of the real numbers and the real numbers are infinitely more numerous than the natural numbers. How can this be?

This little example in itself should show why trying to "interpret" Godel's theorems into some mystical philosophical framework, is fraught with difficulties for those who do not understand the core mathematical "message" of the theorems.
posted by DirtyCreature at 4:37 AM on August 1, 2005

mystical philosophical framework

please explain what "mystical" framework you're referring to? mathematics originated in philosophy. if you're equating philosophy with mysticism you misunderstand philosophy in a fundamental way.

Godel's theorems into some mystical philosophical framework, is fraught with difficulties for those who do not understand the core mathematical "message" of the theorems.

you still haven't addressed my earlier point: the man who constructed the proof probably has a better grasp of its "core mathematical [sic] message" than you do, and by all accounts, he was attempting to make a philosophical argument using mathematics. why won't you respond to that point?
posted by all-seeing eye dog at 7:11 AM on August 1, 2005

Yet the natural numbers are a "subset" of the real numbers and the real numbers are infinitely more numerous than the natural numbers.

Here is the error, a "type error". My math teachers would never let us use natural numbers as reals, since they have (potentially) completely different axiomatizations.

The natural numbers are recursively enumerable and thus subject to incompleteness stuff like Rice's theorem, Chaitin/Kolmogorov complexity, and Godel. The reals are not recursively enumerable under any formalization.
posted by sonofsamiam at 7:14 AM on August 1, 2005

and by all accounts, he was attempting to make a philosophical argument using mathematics. why won't you respond to that point?

Because a) from what I have read it is probably not true and b) I am not up with the details of Godel's life. I vaguely recall reading that he didn't utter a word about any philosophical implications of the theorems until 1942. I do remember reading he had held platonist convictions at University. However my guess is he was just motivated by the works of Alfred Tarski (a logician at least as accomplished as Godel - and quite possibly more accomplished) who he met in Vienna, as well as the work of Hilbert that was big news at the time. Whatever convictions he had were largely shaped by the results of his work rather than the other way around (Disclamer : guessing)

Even so, suppose he was trying to make some philosophical point, the philosophical point must not overextend itself past the core mathematical truth. Its quite clear that even philosophers like Wittgenstein embarassed themselves greatly by their misunderstanding of the implications.

I have a book about this which includes a series of essays discussing the implications - philosophical and otherwise of the work (which I haven't really bothered to read very thoroughly but which I am sure would answer your question). If you are interested in this topic for more than just the purposes of this discussion, I will track it down for you and give you the details. Let me know.
posted by DirtyCreature at 1:44 PM on August 1, 2005

If you are interested in this topic for more than just the purposes of this discussion, I will track it down for you and give you the details. Let me know.

DirtyCreature: That'd be great. Actually I would be interested. Thanks!
posted by all-seeing eye dog at 3:58 PM on August 1, 2005

Ebook

Hardcopy

Amazon has a table of contents and an excerpt. The book includes the original Godel paper (translated) as well as interpretations, Godel's own thoughts and history, and significant philosophical discussion.
posted by DirtyCreature at 4:43 PM on August 1, 2005

Thank you DirtyCreature!

I love it when that happens. :-))

My own background is in computer science, occasionally using formal logic for proofs of algorithmic computability, and more often bog-standard Boolean algebra and set manipulation, so obviously, Gödel's incompleteness theorem isn't exactly part of my daily work, either. :-))

But to suggest it has NO mystical importance is incorrect at several levels.

To claim that some things are somehow "more mystical" than others, which are somehow "less mystical", is to ignore the fundamental interconnectedness (the underlying unity) of all things, surely a grave error when discussing mysticism.

It follows that mystical importance resides in one of two places; either in "Nothing", or in "Everything".

For reasons that are purely subjective, I would prefer the latter, as would you, dear reader, I expect.

So, let us consider that every leaf that rustles in the wind is pregnant with mystical implications; and that surely applies to every statement we make, as well, including both true and false mathematical statements, as well as their truth or falsehood.

Gödel's work is intuitively interesting to so many people, I think, because it's a simple, elegant, logical proof of a very small, partial subset of the first line of the Tao te-Ching: "The Tao that can be expressed in words is not the eternal Tao".

(Or, "Words cannot express the Eternal.")

That doesn't mean words are useless, any more than Gödel means that formal logical systems involving natural numbers are useless. But it is surely a beautiful and mystical concept, worthy of the pleasure we take in it.

To interpret Gödel's work as "proving" statements such as "All formal systems are inherently unreliable", or "All rules must sometimes be broken", or "All the structures produced by our conscious minds are incomplete and inconsistent, and fail to grasp the full beauty of Reality" is, admittedly, not remotely justified by the mathematics.

But Gödel is perhaps the only mathematician (as distinct from philosopher, poet, saint, or social scientist) to provide some evidence to support (albeit in a limited way) the intuitive claim to veracity of such sweeping calls to conscious humility.

I like Gödel because he used logic to remind us of the fallibility of mathematics. Consciousness of our own fallibility is usually a healthy thing, I think. Especially for intellectuals and scientists.

Though of course, I could be wrong. :-))
posted by cleardawn at 5:57 AM on August 2, 2005

Here's an English translation of the Tao te Ching.
posted by cleardawn at 6:23 AM on August 2, 2005

*Tiptoes around trying not to tread too heavily on all the pretty little flowers cleardawn has planted all around.*

As a corollary to your mystic interpretation - I don't believe in infinities, my world is finite even though I'm willing to extend it well beyond anything anyone will likely ever conceive of in the next trillion years, and because I don't run foul of Godel's theorems, I have no cause to doubt that I am infallible or to be humble in any way and will carry on believing such. Ok umm thanks.

You have yourself a highly mystical day, sunshine.
posted by DirtyCreature at 12:53 PM on August 2, 2005

DirtyCreature, if you choose to keep your eyes closed tight, no-one will ever be able to prove to you the beauty of the clear, bright dawn.

But I'm glad to see you're thinking about flowers and sunshine. It's a start. :-))
posted by cleardawn at 4:27 PM on August 20, 2005