June 29, 2005 2:59 AM Subscribe

Godel's theorems have been used to extrapolate a great many "truths" about the world. Torkel Franzen sets the record straight in his new book Godel's Theorem: An Incomplete Guide to Its Use and Abuse. Read the introduction (PDF). If you want, check out his explanation of the theorems.

posted by Gyan (65 comments total) 1 user marked this as a favorite

posted by Gyan (65 comments total) 1 user marked this as a favorite

I just recently read Godel, Escher, Bach: An Eternal Golden Braid. It's all about relating the self-referentialness of all three of their lives' works to how the human conciousness runs. It's terribly interesting, but a real hard read, I did a 20 page paper on it last semester. I got a B- (even though she called my paper well written and intelligent).

posted by Mach5 at 4:58 AM on June 29, 2005

posted by Mach5 at 4:58 AM on June 29, 2005

Achilles: What, they haven't mentioned Gödel, Escher, Bach: An Eternal Golden Braid?.

Tortoise: Well, they did say the new black. Hofstadter is rather old school.

Achilles: Do tell. Old school or not, it remains an excellent treatise on the subject, and it keeps up in work.

Tortoise: Heavens, yes, and there's no arguing that. Still, though, it's to be expected. "The world will little note, nor long remember, what we say here," after all.

Achilles: Drat. You're right again, Mr. T.

Tortoise: ?. On preview, it looks like we've be trumped.

posted by eriko at 5:01 AM on June 29, 2005

Tortoise: Well, they did say the new black. Hofstadter is rather old school.

Achilles: Do tell. Old school or not, it remains an excellent treatise on the subject, and it keeps up in work.

Tortoise: Heavens, yes, and there's no arguing that. Still, though, it's to be expected. "The world will little note, nor long remember, what we say here," after all.

Achilles: Drat. You're right again, Mr. T.

Tortoise: ?. On preview, it looks like we've be trumped.

posted by eriko at 5:01 AM on June 29, 2005

Well ... Mach 5 has to be faster than either a Tortoise or Achilles.

posted by TimothyMason at 5:15 AM on June 29, 2005

posted by TimothyMason at 5:15 AM on June 29, 2005

Hofstadter often gets the blame for the wild misrepresentations of Godel's incompleteness theorm, but unfairly so in my opinion.

Godel's theorem says no more and no less than what it says. What it says is very counter-intuitive. That it is very counter-intuitive may be important...or it may not be important. You can be rigorous about it. If so, then you're doing math and if it's important, it's important in the*realm of mathematics*. Alternatively, you can be philosophical about it in the hand-waving sense. If so, then you're not doing math. That *doesn't* mean that it's necessarily not in some way important outside of math. It well may be. But the math alone doesn't get you that far. It's a mistake to claim that it does.

By nature, I'm the type to think it's important because I'm the type to be immensely fascinated by the very notion of "intuitive" and "counter-intuitive",*especially* in the context of math. No doubt because I'm very intuitive. And my intuition tells me that the ways in which our intuition leads us to deeply counter-intuitive results is meaningful. But then, I would think that, wouldn't I?

Maybe someone will come along who will rigourously connect Godel's proof to something of wider philosophical import. Lots of people have tried, are trying, and will try because, by gosh, their intuition says that there's something there. But Godel's incompleteness theorem is very, very specific. It very well may have no relevance to anything beyond the branch of mathematics within which it exists.

And this may be perverse, but over my lifetime I've come to strongly believe that you can't productively search "intuition space" without having a strong grasp, and constantly keeping in mind, what is known and not known rigourously. Because I'm by nature a hand-waver, this was a lesson hard won.

Douglas Hofstadter is also deeply intuitive, ruminating, "philosophical". GEB spoke extremely deeply to me for this reason. And he had to write that book because it was a labor of love and it was a core expression of his being. But I can't help but think that had someone else written it, he, too, would be annoyed at the popularization of Godel incompleteness brought by that book and the misrepresentations and misunderstandings it spawned.

posted by Ethereal Bligh at 5:19 AM on June 29, 2005

Godel's theorem says no more and no less than what it says. What it says is very counter-intuitive. That it is very counter-intuitive may be important...or it may not be important. You can be rigorous about it. If so, then you're doing math and if it's important, it's important in the

By nature, I'm the type to think it's important because I'm the type to be immensely fascinated by the very notion of "intuitive" and "counter-intuitive",

Maybe someone will come along who will rigourously connect Godel's proof to something of wider philosophical import. Lots of people have tried, are trying, and will try because, by gosh, their intuition says that there's something there. But Godel's incompleteness theorem is very, very specific. It very well may have no relevance to anything beyond the branch of mathematics within which it exists.

And this may be perverse, but over my lifetime I've come to strongly believe that you can't productively search "intuition space" without having a strong grasp, and constantly keeping in mind, what is known and not known rigourously. Because I'm by nature a hand-waver, this was a lesson hard won.

Douglas Hofstadter is also deeply intuitive, ruminating, "philosophical". GEB spoke extremely deeply to me for this reason. And he had to write that book because it was a labor of love and it was a core expression of his being. But I can't help but think that had someone else written it, he, too, would be annoyed at the popularization of Godel incompleteness brought by that book and the misrepresentations and misunderstandings it spawned.

posted by Ethereal Bligh at 5:19 AM on June 29, 2005

For anyone else who was confused, it took a wikipedia entry to help me decipher what this Godel thing is that everyone keeps talking about.

(Please don't let me be the only one who was confused, I'm really quite smart, really ...)

posted by forforf at 5:29 AM on June 29, 2005 [1 favorite]

(Please don't let me be the only one who was confused, I'm really quite smart, really ...)

posted by forforf at 5:29 AM on June 29, 2005 [1 favorite]

forforf, that's not the relevant theorem(s). This one, is.

posted by Gyan at 5:40 AM on June 29, 2005

posted by Gyan at 5:40 AM on June 29, 2005

It's *tortois and the hare*. I have much respect for the **Eternal Golden Braid**, but still, it's *tortois and the hare*. Isn't it? I was there for my childhood. You can't fool me twice. Or something. And who keeps putting an s on math, for that matter?

posted by nervousfritz at 5:50 AM on June 29, 2005

posted by nervousfritz at 5:50 AM on June 29, 2005

Man, I wish forforf hadn't linked to that.

posted by Ethereal Bligh at 5:57 AM on June 29, 2005

posted by Ethereal Bligh at 5:57 AM on June 29, 2005

Good lord. I thought the ontological proof was stupid back when I was fourteen and I haven't had any reason to change my mind since. Gödel took it seriously?! What thin partitions sense from crackpottery divide.

posted by languagehat at 6:08 AM on June 29, 2005

posted by languagehat at 6:08 AM on June 29, 2005

Thankyouthankyouthankyou Torkel Franzen (and Gyan for the great links!). Intellectual overenthusiasm mixed with hubris and laziness of thought frustrates me more than I can say, and one of the most egregious and common examples is the willy-nilly invocation of Gödel's theorem where it clearly doesn't belong. I've heard it mentioned more than once in my *law* classes. Kills me.

nervousfritz: Zeno's paradox, so far as I know, involved Achilles. Lewis Carroll's version, though, was all about "A Kill-Ease".

This is good.

On preview: eriko is clearly my Achilles.

posted by dilettanti at 6:14 AM on June 29, 2005

nervousfritz: Zeno's paradox, so far as I know, involved Achilles. Lewis Carroll's version, though, was all about "A Kill-Ease".

This is good.

On preview: eriko is clearly my Achilles.

posted by dilettanti at 6:14 AM on June 29, 2005

I like Goedel because he was a Platonist. Not enough of those.

posted by mokujin at 6:17 AM on June 29, 2005

posted by mokujin at 6:17 AM on June 29, 2005

From Wikipedia, "It is often abbreviated maths in Commonwealth English and math in American English." There is a world outside your borders, you know.

posted by salmacis at 6:17 AM on June 29, 2005

mokujin : *"I like Goedel because he was a Platonist. Not enough of those."*

Stewart Shapiro, in his book, Philosophy of Mathematics, says that most mathematicians are closet platonists.

posted by Gyan at 6:21 AM on June 29, 2005

Stewart Shapiro, in his book, Philosophy of Mathematics, says that most mathematicians are closet platonists.

posted by Gyan at 6:21 AM on June 29, 2005

Someone else that I've read, but can't recall their name, said the same thing. Rubens? Of course, "platonists" in the idealism with regard to mathematics sense. Not real platonists.

Anyway, contemporary mathematicians have to ride the razor's edge I alluded to in my comment. They probably wouldn't be mathematicians if they didn't

This has a bearing on what's happening when we talk about what Godel's theorem "means".

posted by Ethereal Bligh at 6:59 AM on June 29, 2005

This sentence confuses "means" with "use".

(Sorry, EB, but you've triggered the self referential part of my brain.)

Why is it that this noun phrase doesn't mean what this noun phrase does?

Does this sentence remind you of quonsar?

This sentence, although not a question, neverless ends in a question mark?

This sentence no verb, or relevance.

Sorry.

Sorry.

posted by eriko at 7:20 AM on June 29, 2005

(Sorry, EB, but you've triggered the self referential part of my brain.)

Why is it that this noun phrase doesn't mean what this noun phrase does?

Does this sentence remind you of quonsar?

This sentence, although not a question, neverless ends in a question mark?

This sentence no verb, or relevance.

Sorry.

Sorry.

posted by eriko at 7:20 AM on June 29, 2005

Mr Bluesky's take: Tea + G5 + 7:30 - Full 8 Hrs + Godel's Theorem =

Headache!

Will return after tea has taken hold of my senses.

posted by Mr Bluesky at 7:28 AM on June 29, 2005

Headache!

Will return after tea has taken hold of my senses.

posted by Mr Bluesky at 7:28 AM on June 29, 2005

Oh, and by the way languagehat, I'm glad you said it (about Anselm's proof). I wanted to, but didn't, because I'm probably being enough of a smartypants already. But Anselm's proof annoys the hell out of me. I've known smart people who take it seriously. Okay, worse, I've known smart people who are normally very careful thinkers who take it seriously. Or at least more seriously than it deserves. It's ugly and sloppy and even Leibniz and Godel can't clean it up. Why would they try? That's a good question.

And really, really smart people who are careful thinkers take Newcomb's paradox seriously, too. It is this sort of problem that leaves me feeling as if I'm way smarter than some really smart people...or way dumber. Some kinds of things which appear to be deep mysteries or paradoxes to others almost always look to me like an obvious category error of some sort. I see*positive property* in Godel's ontological "proof" and I see a placeholder for meaning, but no meaning. I see *choice* in Newcomb's paradox and I ask "what the hell is *choice*?"

On Preview:*"Sorry, EB, but you've triggered the self referential part of my brain"*

Exactly which part would that be?

posted by Ethereal Bligh at 7:28 AM on June 29, 2005

And really, really smart people who are careful thinkers take Newcomb's paradox seriously, too. It is this sort of problem that leaves me feeling as if I'm way smarter than some really smart people...or way dumber. Some kinds of things which appear to be deep mysteries or paradoxes to others almost always look to me like an obvious category error of some sort. I see

On Preview:

Exactly which part would that be?

posted by Ethereal Bligh at 7:28 AM on June 29, 2005

It appears Godel's theorems are abused so often because they are so useful and novel as ideas.

And it does have implications for human consciousness, since humans do work with math(s).

I envy anyone reading GEB for the first time. You can only do that once....er.

posted by Smedleyman at 7:35 AM on June 29, 2005

And it does have implications for human consciousness, since humans do work with math(s).

I envy anyone reading GEB for the first time. You can only do that once....er.

posted by Smedleyman at 7:35 AM on June 29, 2005

But Goedel was a vocal Platonist who believed that concepts and ideas have some kind of objective reality independent of the human mind. And that is probably why he liked the ontological proof.

posted by mokujin at 7:47 AM on June 29, 2005

posted by mokujin at 7:47 AM on June 29, 2005

This is where I lose a lot of interest. Maybe it's just because I am an applied mathematician. It's not clear to me that such a correspondence can actually be established outside of theoretical hand-waving, so it's never been entirely clear to me that I should really care about the incompleteness theorem.

Trying to formalize theorem provers is very non-trivial. Most non-trivial proofs seem to require a lot of language that it is very difficult to formalize. So I don't know. But I'll have to read up on him one day, as it's probably just my own ignorance.

posted by teece at 7:50 AM on June 29, 2005

Eriko wins!

Great tortoise/hare conversation.

Hofstadter would be proud.

Torkel Franzen wins also with the book title employing the word "incomplete."

Self reference can be such fun.

posted by nofundy at 7:54 AM on June 29, 2005

Great tortoise/hare conversation.

Hofstadter would be proud.

Torkel Franzen wins also with the book title employing the word "incomplete."

Self reference can be such fun.

posted by nofundy at 7:54 AM on June 29, 2005

Maybe.

"...since humans do work with math(s)."

Well, no. That doesn't follow.

On Preview:

On Second Preview:

posted by Ethereal Bligh at 7:58 AM on June 29, 2005

Interesting, I had no idea Torkel had a book coming out. He's working at the same department I did before I left the university 5 weeks ago. In fact, he recently moved into my old office...

I think I have to get the book - I have tried to discuss Gödels theorems and different misconceptions with people several times.

posted by rpn at 8:14 AM on June 29, 2005

I think I have to get the book - I have tried to discuss Gödels theorems and different misconceptions with people several times.

posted by rpn at 8:14 AM on June 29, 2005

Ah then, I will definitely have to head to the library and wade into the rarified heights of such abstraction, then. Thanks.

posted by teece at 8:16 AM on June 29, 2005

When studying it in college, I remember the theorem only referring to "completeness" and not "consitency". This Franzen fellow refers to all of the theorems as determining "consistency" in a system. I've never heard that term in mathematics before. Is that the same as completeness?

posted by destro at 8:16 AM on June 29, 2005

posted by destro at 8:16 AM on June 29, 2005

Indeed. And one of the worst abuses (because it comes from someone I admire who should really know better) is this.

P.S.: Love the title.

I'd like to make my last sentence self-referential, but couldn't - oh wait.

posted by spazzm at 8:37 AM on June 29, 2005

Excellent post. I've seen Godel's theorem(s) invoked rather haphazardly in the blue to support any number of claims and it's gotten under my skin. Glad to hear I'm not the only one bugged by this. I even vaguely recall its use to "attack" any meaningful distinction between, e.g., noetic and empirical approaches to knowledge. I'd provide the set of all sets containing the aforementioned posts, but it's lunchtime. Also, if you take the ontological proof seriously, I have a perfect island I'd like to sell you. Guanillo Charter Tours, all of that.

posted by joe lisboa at 9:19 AM on June 29, 2005

posted by joe lisboa at 9:19 AM on June 29, 2005

the incompleteness result can be expressed as "any formal system of sufficient power as to describe arithmetic is either incomplete or inconsistent".

inconsistent means that the system proves some statements to be both true and false. incomplete means that there are some statements which cannot be proven to be either true or false.

the original (and only?) proof was formulated entirely in terms of rock solid number theory and first order logic. no flakey hand waving at all.

posted by paradroid at 9:19 AM on June 29, 2005

Paraconsistent logics are the future, people. Well, they're fun anyway.

posted by sonofsamiam at 9:32 AM on June 29, 2005

posted by sonofsamiam at 9:32 AM on June 29, 2005

Hear hear. Along with quantum mechanics and the second law of thermodynamics. Maybe someone will take on those in the same vein.

posted by DevilsAdvocate at 10:13 AM on June 29, 2005

Assertion: Godel's Incompleteness theorem has been superseeded by Turing's Halting problem, which is much easier to understand.

Discuss.

posted by delmoi at 1:17 PM on June 29, 2005

Discuss.

posted by delmoi at 1:17 PM on June 29, 2005

Good summation of Godel, platonism and some other stuff in Edge's interview with Rebecca Goldstein.

Delmoi, inasmuch as the halting problem represents incompleteness, it's just a symbolic representation of the concept. Boiling water getting colder at room temperature is easy to envisage but that doesn't make thermodynamics any less relevent.

posted by Sparx at 2:37 PM on June 29, 2005

Delmoi, inasmuch as the halting problem represents incompleteness, it's just a symbolic representation of the concept. Boiling water getting colder at room temperature is easy to envisage but that doesn't make thermodynamics any less relevent.

posted by Sparx at 2:37 PM on June 29, 2005

I should have guessed as much. Those English bastards are always changing the spelling of everything. Next it will be "mauths" They love putting extra 'u's in the way, almost as much as they love their damn tea.

posted by nervousfritz at 3:06 PM on June 29, 2005

Godel's theorems are pure mathematical masturbation.

I dont believe in any natural numbers bigger than A^A^A^A^A or less than -A^A^A^A^A where A is the number of sub-atomic particles in the known universe. I define my arithmetic system accordingly. If for some reason, I need to believe in a natural number of greater magnitude than these, let me know and I will redefine my system to accommodate.

Bingo. My belief system is totally consistent and complete and Godel and all his disciples and pretenders disappear in a puff of logic.

posted by DirtyCreature at 4:36 PM on June 29, 2005

I dont believe in any natural numbers bigger than A^A^A^A^A or less than -A^A^A^A^A where A is the number of sub-atomic particles in the known universe. I define my arithmetic system accordingly. If for some reason, I need to believe in a natural number of greater magnitude than these, let me know and I will redefine my system to accommodate.

Bingo. My belief system is totally consistent and complete and Godel and all his disciples and pretenders disappear in a puff of logic.

posted by DirtyCreature at 4:36 PM on June 29, 2005

Possibly, and it's kind of funny to watch these pedantic math types correct the supposedly egregious popular misunderstandings of Godel when I suspect that many of them are not familiar with the halting problem, which does say that there are true things that can't be proved. Well, computed.

posted by transona5 at 6:44 PM on June 29, 2005

What if one could conceivably need an indefinite -- perhaps ongoing -- number of such extensions?

posted by weston at 7:25 PM on June 29, 2005

LOL. I knew I shouldn't have put that caveat in. Ok ok - I'll let you extend my system A^A^A^A^A times but that's IT.

But first I'd like to know just ONE possible reason why you would ever need to extend my system.

posted by DirtyCreature at 10:26 PM on June 29, 2005

To calculate the (A^A^A^A^A + 1)-th digit of pi, perhaps?

Because, simply, your number system makes no sense? I mean, I'm pretty sure you're joking, but still. While every practical use of numbers will fall well shy of that (including, if I'm correct, calculating the entropy of the whole universe, which should be of the order of A! - an indescribably huge number, which is still smaller than A^{A^(A^(A^A))}), that doesn't make it right. For example, in your universe, you can't add any two number or multiply any two numbers arbitrarily - which for any reasonable requirement of a number system seems to be a bit of a failiure.

posted by vernondalhart at 11:03 PM on June 29, 2005

Because, simply, your number system makes no sense? I mean, I'm pretty sure you're joking, but still. While every practical use of numbers will fall well shy of that (including, if I'm correct, calculating the entropy of the whole universe, which should be of the order of A! - an indescribably huge number, which is still smaller than A

posted by vernondalhart at 11:03 PM on June 29, 2005

spazzm writes *"And one of the worst abuses (because it comes from someone I admire who should really know better) is this."*

Agreed. Great tiles, but his explorations of consciousness are so much hand waving. "Quantum effects in microtubules" just takes a*mystery* that's become a *problem* and turns it back into a *mystery*. Feh.

posted by orthogonality at 11:21 PM on June 29, 2005

Agreed. Great tiles, but his explorations of consciousness are so much hand waving. "Quantum effects in microtubules" just takes a

posted by orthogonality at 11:21 PM on June 29, 2005

Of course we're familiar with the halting problem. I knew of the halting problem long before Godel's theorem. Anyway, it too only proves what it proves and nothing more or less. It's no more explicitly philosophically far-reaching than Godel's theorem.

Yeah. Ugh. Penrose.

posted by Ethereal Bligh at 12:17 AM on June 30, 2005

Yeah. Ugh. Penrose.

posted by Ethereal Bligh at 12:17 AM on June 30, 2005

For what purpose?? For mathematical masturbation? Sure - go nuts. But they got to the moon with 15 decimal places of pi. A little less than A^A^A^A^A no?

It makes complete sense. Tell me the two biggest numbers you will ever multiply by each other and I'll build a number system which satisfies your needs way more than adequately. Ok certain incredibly large unfathomable numbers in the system can't be multiplied or added together - so? Its just another axiomatic system.

Not pretty? Not elegant? Trust me, for all intents and purposes my system will function as sexily as normal number theory for any practical problem that can ever be considered. Sure the concept of the countably infinite is elegant but as we've seen, it gets you in a lot of strife and is completely unnecessary.

My point is it's incredibly easy to prove consistency and be complete at the same time as long as your belief system stays finite - no matter how unbelievably incredibly unthinkably large you choose your finite universe to be. Anything more is masturbatory and problematic.

posted by DirtyCreature at 3:43 AM on June 30, 2005

DirtyCreature:

Actually, your system will be incredibly complicated:

'Normal' number systems allow any two numbers to be multiplied and added. Any number can be divided by any other number.

Exception: You can't divide by zero.

Your system would be pretty similar, but your list of exceptions would be very, very long (infinite if you include real number, as a matter of fact) since you have to make an exception for any addition or multiplication that exceeds the bounds of your system.

Why can't you simply disallow any operation that produces a result larger than the bound? Because then you must define "larger than the bound", which is a number outside the bound. Catch-22, amigo.

But I assume you're joking, of course.

posted by spazzm at 4:07 AM on June 30, 2005

Actually, your system will be incredibly complicated:

'Normal' number systems allow any two numbers to be multiplied and added. Any number can be divided by any other number.

Exception: You can't divide by zero.

Your system would be pretty similar, but your list of exceptions would be very, very long (infinite if you include real number, as a matter of fact) since you have to make an exception for any addition or multiplication that exceeds the bounds of your system.

Why can't you simply disallow any operation that produces a result larger than the bound? Because then you must define "larger than the bound", which is a number outside the bound. Catch-22, amigo.

But I assume you're joking, of course.

posted by spazzm at 4:07 AM on June 30, 2005

The reals are defined separately and aren't a problem for Godel. To know where I'm coming from requires an understanding of the machinery of the Godel proof. It's the natural numbers (1,2,3,4,.....) that are the problem.

No thats not how you do it. Define two sets - the touchables and the untouchables. Any two numbers in the touchables can be added or multiplied, the results of which may be an element of either set. There is no multiplication or addition defined on elements of the untouchables. The "bound" as you describe it is just the sum of the sizes of the two sets.

posted by DirtyCreature at 4:26 AM on June 30, 2005

Ahh, so you're inconsistent, thus, you are immune to Gödel's discovery.

Why your system is inconsistent: Operations give different answers at different scales. If we let "N" equal the largest possible positive integer, we find that N!=N and (N-1)!=N, and (N-2)!=N, but 3!=6, and (3-1)!=2, and (3-1)!=1.

By the way, does the set of all non-self-inclusive sets include itself in your system?

posted by eriko at 5:20 AM on June 30, 2005

DirtyCreature:

I bow before your superiour intellect. Clearly, your new system of natural positive numbers smaller than or equal to X has no flaws. And pay no heed to eriko (that snarky cad!), he's just jealous because he didn't think of it first.

Incidentially, here's a related research project. Se also here.

posted by spazzm at 6:55 AM on June 30, 2005

I bow before your superiour intellect. Clearly, your new system of natural positive numbers smaller than or equal to X has no flaws. And pay no heed to eriko (that snarky cad!), he's just jealous because he didn't think of it first.

Incidentially, here's a related research project. Se also here.

posted by spazzm at 6:55 AM on June 30, 2005

" "...since humans do work with math(s)."

Well, no. That doesn't follow. "

Ok, humans don't work with math. Fine.

This isn't a post.

posted by Smedleyman at 1:14 PM on June 30, 2005

Well, no. That doesn't follow. "

Ok, humans don't work with math. Fine.

This isn't a post.

posted by Smedleyman at 1:14 PM on June 30, 2005

I said the argument was flawed--that the conclusion was not warranted from the premises. Furthermore, the conclusion of the argument

And you're right: that isn't a post. It's a comment.

posted by Ethereal Bligh at 1:51 PM on June 30, 2005

spazzm

I bow to your perceptiveness at being able to recognize someone with a little more training in this area than some from just a few sentences.

eriko

Factorial isn't a fundamental operator of arithmetic but ok if you want factorials, let them be defined only on the touchables and we will extend the untouchacles to include the factorials of all numbers in the touchables.

posted by DirtyCreature at 2:33 PM on June 30, 2005

I bow to your perceptiveness at being able to recognize someone with a little more training in this area than some from just a few sentences.

eriko

Factorial isn't a fundamental operator of arithmetic but ok if you want factorials, let them be defined only on the touchables and we will extend the untouchacles to include the factorials of all numbers in the touchables.

posted by DirtyCreature at 2:33 PM on June 30, 2005

Constructivism is a (somewhat) legitimate school of thought in mathematics.

posted by Ethereal Bligh at 2:47 PM on June 30, 2005

posted by Ethereal Bligh at 2:47 PM on June 30, 2005

What the heck is the use of a number system that does not define operations on 1 (factorial(1)), 2 (factorial(2)), 6 (factorial(3)) or 24 (factorial(4))?

What you're proposing regarding disallowing numbers larger than a certain limit is what digital computers do - if you keep adding one to an integer you'll end up with a negative number after a while, because addition is undefined beyond a certain limit.

While this is useful, it doesn't disprove Godel.

To put it bluntly: Your system may be consistent, but it is incomplete.

It's incomplete because there are true statements that it cannot prove.

For example:

If the limit (the largest number defined in the "touchables" or "untouchables") of your system is X, you cannot prove or disprove (for example) "Y is prime", where Y > X.

You may hold the opinion that no-one would ever "need" a number as large as Y, but that's not the point. Completeness, as defined by Godel, is not subject to "need".

posted by spazzm at 3:45 PM on June 30, 2005

I'm not trying to disprove Godel's theorems. I'm trying to show they are masturbatory and aren't really very useful or applicable.

This is not the definition of completeness. Completeness is determined with respect to expressible statements within the language of the system.

"Y is prime" is not expressible in my system because Y is not even defined. I don't believe in Y. It's completely useless to anyone except masturbators.

Trust me. My system is complete and consistent. For those who want to learn more, I suggest looking up Wikipedia as a starting point.

posted by DirtyCreature at 5:06 PM on June 30, 2005

Nice try. Godel defined completeness only for systems

You seem to try to get around this by redefining the meaning of "natural number" by claiming that there is a largest possible natural number. Unfortunately (?), there is no largest natural number.

There are systems that are both complete and consistent (Peano arithmetic comes to mind), but they cannot define the natural numbers.

There's some pretty compelling evidence that there are infinitely many primes. This means that no matter how large X (the largest number in your system) is, there will always be a prime Y such that Y > X. Whether you believe in it or not.

I'll trust you once you explain how a number system that does not allow operations including the numbers 1, 2, 6 or 24 is 'complete' in the sense Godel uses. For some reason you seem to be avoiding the question.

posted by spazzm at 5:51 PM on June 30, 2005

Um - that should be "cannot define multiplication and addition on all natural numbers".

posted by spazzm at 6:11 PM on June 30, 2005

In all honesty, DirtyCreature, your system would work for everyday life just fine. I can't think of a single number that one could ever need for any use in every day life, or in any science other than mathematics that would be larger than you upper bound. I'll grant you that.

However, it doesn't work for mathematics. While you might be able to produce some version of the Reals out of it, your upper bound on the integers would also provide an upper bound on the reals, and hence it would also provide a lower bound in size of integers too - which would yield major problems with notions of convergence and continuity. Although in that sense, you couldn't really do calculus in such a system, and hence basically any science that would ever use it would be out.

You could try get around this by just using the techniques of calculus and call the rest 'mathematical masturbation' but you'd be grossly deluding yourself. The reason that these things work implicitly assumes a multitude of facts about the number system that they are based on - and there's no way around that. A major part of the 19th and 20th centuries work in mathematics was coming to grips with all this. The world you live in is built on such masturbation, I'm afraid.

posted by vernondalhart at 8:06 PM on June 30, 2005

However, it doesn't work for mathematics. While you might be able to produce some version of the Reals out of it, your upper bound on the integers would also provide an upper bound on the reals, and hence it would also provide a lower bound in size of integers too - which would yield major problems with notions of convergence and continuity. Although in that sense, you couldn't really do calculus in such a system, and hence basically any science that would ever use it would be out.

You could try get around this by just using the techniques of calculus and call the rest 'mathematical masturbation' but you'd be grossly deluding yourself. The reason that these things work implicitly assumes a multitude of facts about the number system that they are based on - and there's no way around that. A major part of the 19th and 20th centuries work in mathematics was coming to grips with all this. The world you live in is built on such masturbation, I'm afraid.

posted by vernondalhart at 8:06 PM on June 30, 2005

Sorry, that should be

... and hence it would also provide a lower bound in size of**real numbers** too ...

posted by vernondalhart at 8:07 PM on June 30, 2005

... and hence it would also provide a lower bound in size of

posted by vernondalhart at 8:07 PM on June 30, 2005

Again, some mathematicians take mathematical constructivism seriously. Google it. I'm not paying much attention to DirtyCreature's comments, or this argument, and skimming leads me to view him unfavorably--but, even so, it is simply wrong to dismiss out-of-hand (what appears to me) to be his underlying mathematical philosophy.

posted by Ethereal Bligh at 8:14 PM on June 30, 2005

posted by Ethereal Bligh at 8:14 PM on June 30, 2005

Thank you.

As defined by masturbators, agreed. (Don't take offence by the term "masturbators". The analogy is that masturbating might be fun but it doesn't produce anything much useful and can lead to significant embarassment.)

Sorry no, this is wrong. I know this seems a strange result but an axiomatic system defining all the real numbers as we know and love them can be consistent and complete. This is confusing given that the reals contain the natural numbers but nevertheless true. It's just the natural numbers that are the problem. (Should be somewhere in Wikipedia)

Ok ok I'm a complete, consistent asswipe. I can live with that ;)

posted by DirtyCreature at 9:19 PM on June 30, 2005

Taski's theorem on real closed fields (See Model Theory) which proves an axiomatization of the reals exists which is complete.

I'm sure this level of detail is going way too far for many here but I'm a sucker for punishment.

posted by DirtyCreature at 10:39 PM on June 30, 2005

I'm sure this level of detail is going way too far for many here but I'm a sucker for punishment.

posted by DirtyCreature at 10:39 PM on June 30, 2005

Sorry - "Tarski's theorem"

posted by DirtyCreature at 10:40 PM on June 30, 2005

posted by DirtyCreature at 10:40 PM on June 30, 2005

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posted by foxy_hedgehog at 3:35 AM on June 29, 2005