Skip to comments.Truth, Incompleteness and the Gödelian Way
Posted on 05/21/2005 2:42:12 AM PDT by infocats
Is there a more powerful modern Trinity? These reigning deities proclaim humanity's inability to thoroughly explain the world. They have been the touchstones of modernity, their presence an unwelcome burden at first, and later, in the name of postmodernism, welcome company.
Their rule has also been affirmed by their once-sworn enemy: science. Three major discoveries in the 20th century even took on their names. Albert Einstein's famous Theory (Relativity), Kurt Gödel's famous Theorem (Incompleteness) and Werner Heisenberg's famous Principle (Uncertainty) declared that, henceforth, even science would be postmodern.
(Excerpt) Read more at r-s-r.org ...
It discusses, tangentially, the impact of Godel's theorem on mathematics. I had been aware of him since back in the '70's or '80's when Godel, Escher, Bach came out but didn't appreciate how profoundly he changed everything
It discusses, tangentially, the impact of Godel's theorem on mathematics. I had been aware of him since back in the '70's or '80's when Godel, Escher, Bach came out but didn't appreciate how profoundly he changed everything.
Profound indeed! What Goedel demonstrated was that in any linear logical system (mathematics, law etc.), by using only the initial assumptions, laws, and understandings of that system, a point would be reached where the system would contrdict itself.
Only by resorting to a new law or understanding taken from outside the orignal rule set, could this contradiction (at least temporarily) be overcome.
orignal -> original
This is totally off topic, but this seems to be a thread where someone might know the answer. If F=ma then does this not state that there would be zero force if an object is not accelerating? But if a car is moving at a constant 60 mph (no acceleration) wouldn't it have a force were it to slam into me?
|if a car is moving at a constant 60 mph (no acceleration) wouldn't it have a force were it to slam into me?
You are confusing force with energy.
But E=mc**2 where
c is squared because you would c stars 2 if you got hit by a car.
I hope this clears things up.
What your forgetting is that when this hypothetical car slams into that hypothetical you, it will accelerate (negatively) and all that energy that went into accelerating it up to 60 MPH will be transmitted into YOU, as you slow the vehicle from 60 to zero, in a VERY short time (high acceleration rate).
The force that the car will impose upon your body will be sufficient to cause the rigid structures of your body to fracture and the soft structures to burst.
The force of transmitted to you by the car will cause your body to accelerate. Of course the part that is in contact with the car will accelerate faster than the parts that are not. As a result of this your arms might be torn from their sockets, shoes ripped from your feet, and watched thrown from your wrists.
What I am describing is entirely the result of the car's deceleration and transmittal of force (and energy to you). I have to tell you that only a fraction of the car's speed need be lost to cause all sorts of havoc and mayhem to your hypothetical body, due to the huge difference in mass between you and a car, as well as the energy associate with speed.
Good luck with your experimentation!
Ah, thanks. Potential energy is just that - potential until an interation occurs. It's the interaction which creates the transfer of energy and force.
Nick, you're in danger of many things, but clarification is not one of them. ;0)
no way. conservation of momentum. you will be flung at a sizable fraction of that 60 mph for a hundred feet or so (until you hit the ground and the law of friction takes over) but the car will scarcely lose velocity.
Actually, it kind of leaves me lying in the road, bleeding :-)
Not exactly. You can still have a consistent (non contradictory) system. What Goedel showed was that any such system would contains "undecidable" statements. In OTW, there will always be some statement S, that cannot be proven neither True nor False.
That was obviously one of the senarios I had in mind, when I referred to a very small reduction in the velocity of the car.
The other scenario, where the cars velocity was reduced to zero in a very short time presupposed a solid brick wall immediately behind the experimenter. Now it is true that the brick wall would ultimately be responsible for the deceleration of the car (if it is possible to speak of a brick wall as "responsible") but the action-reaction of the car-wall combination wwould be mediated through and by the body of the experimenter.
Result: severe compression.
You will be flung at far greater than 60 mph, actually, but the poster is correct about conservation of momentum.
Don't forget: "Keep it up constantly" and that it's both a necessary and sufficient condition for eternal life!
(Which is why Protestants and evangelicals rightly view it as a spiritual activity that doesn't depend on a ceremony to make it happen)
Not exactly. You can still have a consistent (non contradictory) system. What Goedel showed was that any such system would contains "undecidable" statements. In OTW, there will always be some statement S, that cannot be proven neither True nor False.
Hmmm. I'll have to research your assertion. Unfortunately, I already packed both my copy of Goedel's Proof and The World of Mathematics in preparation for a move to Albany.
Postmodernism in that context then means, where people arrive after they come to doubt the knowledge claims of modernism, as well. Typically because they see no rigorous distinction between metaphyiscal and scientific knowledge, or they reduce both to opinion, or deny knowledge properly so called applies to either. Skepticism alone does not have to arrive here, but subjectivist idealisms often do, including skeptical idealisms. Relativism arrives here, historicism likewise. The basic diagnosis is the reason's attempts to reliably and exhaustively grasp a real external world fail. For logical reasons, or because of limits on knowledge or certainty, or because of subjective distortions, or because there isn't an unambiguous external reality to grasp in the first place, or because real relations don't have the necessary stability or determinism to be grasped unless we artificially impose it on nature, etc. Arguments of that sort.
In the continental philosophy tradition, modernism would end around Hegel and Marx (and later neo-Kantians) and post modernism would arrive with Nietzsche and Heidegger, and would include present day French versions of their thought like Foucault and Derrida. In the English philosophy tradition, modernism would include Russel and Whitehead and the early Wittgenstein, post modernism would come in with the later Wittgenstein, Quine, and Kuhn. In mathematics, Hilbert is the paradigmatic modernist and Godel passes for "post". In reality, Godel was a mathematical platonist who went back to pre-modern philosophy for his philosophic opinions, but he certainly rejected Hilbert's "constructivist" program to ground mathematics in logic. He thought it is essentially bigger. (In pure formal computation terms, he was right - basic logic is not Turing complete and Peano arithmetic is).
Perhaps you also could tell me
what comes after postmodernism?
I think the fight between positivists and post modernists is a case of both of them being wrong, while having sufficient arguments to show the other side is overstating its own case. A tradition that avoided both mistakes, in my opinion, broke of from both a century ago and is still alive and well - the pragmatic or "fallibist" tradition. Which means people like Peirce, Santayana, Popper, and Hintikka. Basically they drop the pretense of certainty but allow for the existence of something that deserves to be called knowledge without being certainty. They rehabilitate opinion, refusing to consider it a synonym for "error". We guess, our guesses are not blind but educated, then we iterate on our guesses - that is reason. All knowledge rests on faith, they aren't opposites. But some faiths are more reasonable than others, have the characteristics of knowledge rather than of wishing.
This is not a widely enough held opinion to constitute a period. It is more like a selective set of agreements and brickbats for the various contending parties the periodization refers to.
Yes the post moderns are right that positivism advanced indefensible claims about certainty, reduction of everything to supposedly unambiguous logic, the violence of its attacks on metaphysics (which recoil on itself when examined closely), etc. Enough logic isn't unambiguous, and all theories make use of things besides sense experience or logical reasoning, things that are not remotely certain, guesses. Yes there is fundamental ambiguity in the external reference of any formal structure, different ways any given picture can map onto the real world. Yes there are limits to logic (problems too large to be exhausted e.g.), to determinism, etc. But no, none of this means there is no knowledge only subjective opinions. "Subjective opinion" is not an adequate description of how space probes get to Mars.
Yes the modernists are right that there are real methods for arriving at reliable knowledge of real external world. But no, those methods are not infallible, they are not deductive-logical, they are based on more than sense data, they require philosophical constructs and leaps that are at bottom metaphysical opinions, if ones we are "programmed" for or that nature itself prompts in us. And nowhere is it written that they must exhaust the world. As a set, the knowable is not empty, but it need not coincide with the existent, either.
I hope this helps.
Boiled down to one sentence: Postmoderns suffer from cognitive dissonance (the mental confusion that results from actually holding polar opposite attitudes and beliefs simultaneously).
They don't believe that there is such a thing as "absolute truth", EXCEPT the absolute truth that there us no such thing as absolute truth.
Such confused mentalities can be found on forums like Free Republic actually claiming to want to uphold and defend the Constitution --- which, of course, is a meaningless document unless it is actively guarding the absolute moral (self-evident) truth that man has inalienable (because they're God-given) rights.
Any who are foolish enough to think it is possible to use reason and logic in a debate with them are also suffering from cognitive dissonance.
Can someone tell me what postmodernism is??
Funny I was asking the same question this week and my good friend directed me to this outstanding essay.
Me thinks you protest too much. LOL
Would it be too simple-minded to say
that Post-Modernism is a Western form of Buddhism?
(Somewhat watered down, perhaps)
Maybe if you see the similarity of what sounds like nonsense from postmodernism and what sounds like nonsense in a koan, then you could expand that to "both see the world as nonsense."
There are many forms of buddhism, but I think it would be accepted that all stress knowing beyond concepts - that enlightenment consists in seeing the universe without the prejudices and constrants of conceptual naming.
However, IMHO again, buddhists wouldn't say this enlightenment, or buddha mind, is subjective-only truth, varying with the observer. I believe they have spent a great deal of time in the certainty that it is the same, but not describable in words/concepts. They invite each to look for himself and see.
And I think a postmodernist would be lost without his plethora of high-sounding concepts.
BTW, the Western form of Buddhism is buddhism.
Well I would think an honest post-modernist would have to admit
that his plethora of concepts is merely the ship
he uses to cross the river.
If I may be so bold
I should like to give a definition of post-modernism here:
"Post-Modernism is Buddhism without Nirvana"
Sure it has. Gödel's Incompleteness Theorem applies to any axiom system at least as strong as Peano arithmetic, as long as that theory is consistent and is recursively axiomatizable (which means that there is a computer program that would list precisely the axioms of the system, if allowed to run forever -- in other words, it means that you actually can tell what the axioms are). In fact, there are weak subsystems of Peano arithmetic that will do.
Anyway, this applies to Zermelo-Fraenkel set theory, for example, as well as any common variant, extension, etc. If ZF is consistent, then the sentence that asserts "ZF is consistent" is neither provable nor disprovable in ZF.
That's what I thought too.
I've never understood why G's theorem is thought to be profound.
To paraphrase it slightly
it simply says that certain chess positions can not be reached
by a series of legal moves.
Ah. "The Way To Where There's No There."
Sort of like Christianity without the salvation part..
I agree with the post that describes post-modernism as two roads taken to excess. There's some truth in post-modernism, certainly something had to give after positivism collapsed, but I'm a believer in transcendent objective truths and values - something I don't see a lot of in post-modernism.
In its larger interpretation, if you substitute "knowledge of reality" for "chess positions" and "means of knowing" for "series of legal moves" is the conclusion that any and all axiomatic systems are forever limited in their scope of knowledge.
It's value is in its concreteness, IMHO, in demolishing scientism and rationalism as holy grails for All Knowing.
This doesn't happen with arithmetic because you can enumerate the theorems of arithmetic. If you try to enumerate the theorems of set theory in the same way, you hit a snag once you have sets of cardinality greater than the integers and start having theorems about each element of them. Sure you can set up some enumeration scheme, based on applying whatever operations are allowed in some definite order. But this enumeration won't be "onto" for the possible theorems of set theory.
If it were about chess, it would be of no great significance :-).
At the time Gödel proved it, the Incompleteness Theorem put to rest Hilbert's program of proving the consistency of formal number theory by finitary means.
If one is a Platonist, who believes in the existence in some sense of the mathematical universe (as Gödel himself did, notwithstanding all the discussion in this thread of post-modernism), then the Incompleteness Theorem says that solving some problems will require more than just proving things from a well-established set of axioms. We must use our mathematical intuitions to develop new insights into the mathematical universe; the idea is that these new insights will lead to new axioms that can be accepted as self-evidently true.
This approach led directly to the current situation where people have added various "large cardinal axioms" to Zermelo-Fraenkel set theory. These large cardinal axioms tend to be statements that extremely large infinite sets exist, but, surprisingly, they have interesting consequences for arithmetic and analysis.
By the way, Gödel's work directly answered Hilbert's 2nd problem (in which Hilbert called for a finitary consistency proof of arithmetic) and partially answered Hilbert's 1st problem (the continuum hypothesis), but neither one in the way Hilbert intended. Gödel solved the 2nd problem by showing that there was no such finitary consistency proof. He half-solved the 1st problem by showing that the continuum hypothesis was consistent with the rest of set theory (assuming that set theory itself is consistent); Cohen later showed that the negation of the continuum hypothesis is also consistent. Of course, this leaves the continuum hypothesis itself unsolved; it's independent of the axioms of set theory, and we're waiting for further insight into whether it's true or false. (Incidentally, Gödel believed that the continuum hypothesis was false.)
It would be very surprising if ZF were not consistent. However, the point is that Gödel's Incompleteness Theorem applies to it in exactly the same way as to first-order Peano arithmetic: if the theory in question is consistent, then it cannot prove its own consistency.
If you believe that Peano arithmetic is consistent, then your reason for believing so must transcend Peano arithmetic.
If one believes that ZF set theory is consistent, then one's reason for believing so must transcend ZF.
This doesn't happen with arithmetic because you can enumerate the theorems of arithmetic. If you try to enumerate the theorems of set theory in the same way, you hit a snag once you have sets of cardinality greater than the integers and start having theorems about each element of them.
This isn't true. Your confusing the theorems with the subject that they're intended to be about. Theorems themselves are just finite strings of symbols from a finite (or countable) alphabet. The theorems of ZF can be enumerated in exactly the same way as those of Peano arithmetic. In each case, you start with the axioms (which can be specified very simply) and then systematically write down every possible proof; the last line of each proof is a theorem. This gives you a way to enumerate all theorems of the system. (You can write a computer program to do this.)
The catch is that there's no way to enumerate (via a computer program or algorithm) all the non-theorems.
I don't quite understand this.
If it's independent of the axioms of set theory
then we are free to accept it as either true or false.
Sort of like
if I don't like playing chess tonight
I can always look for a bridge party.
Your -> You're
Sorry for the typo.
By the way, the reason your argument doesn't work is that you don't "hit a snag once you have sets of cardinality greater than the integers and start having theorems about each element of them." In fact, most of these elements cannot be individually defined via a formula of set theory. If a real number, say, is undefinable in set theory, there is no way to write a sentence of set theory that talks about it specifically. You can write a sentence of set theory that applies to it (because it says, for example, that "every real number has some property"), but you can't write a sentence that distinguishes it from all other real numbers.
The full picture is trickier than this, since you can't define "definability" in set theory, but that's the idea.
Why do you dismiss the game of chess so lightly?
OK, maybe chess is not sufficiently complex
but I think it is a reasonable hunch to say
every proof in any finite axiomatic system can be reduced
to finding the correct move in some position in the game of GO.
If this seems outlandish
don't forget that Conway has shown
that his game of LIFE is sufficiently complex
that it can produce a Turing machine.
I take it you're talking about socialism.
IMO, it would be closer to salvation without God.
I think I'd rather define socialism as salvation by man - or the theory that man can be perfected on earth (by the right institutions and systems).
Socialism and utopianism share alot as I see them.
Farking engineers - we're all the same.
Play some Ray Charles music, and watch the movie 'Ray'. It's pretty good and deserved some Oscars. I'm sorta on Rays' side, 'cause I've been enjoying his music since 1952.
R.I.P., Ray Charles Robinson, and thank you for the music..........Barry/gonzo
Hey Dave - we're drowning in a sea of FReepers now!!!!!!!!!!!!!!!!!!!
I don't think its a matter of complexity, but one of "proof" and knowing all the statements within the system using the system itself, the validity of an axiomatic system to be "complete" in that sense. Something beyond the system is needed, and then the incompleteness is encountered again.
I can't really illustrate using your chess analogy effectively off the top of my head, maybe someone else will.
... it's independent of the axioms of set theory and we're waiting for further insight into whether it's true or false.
I don't quite understand this. If it's independent of the axioms of set theory then we are free to accept it as either true or false.
Sort of like if I don't like playing chess tonight I can always look for a bridge party.
This is an interesting question. There's clearly a sense in which one can do exactly as you say. If a sentence is independent of set theory, both the sentence and its negation are consistent with set theory, so you can develop two different versions of mathematics without hitting a contradiction in either one. (This is much like Euclidean and non-Euclidean geometries.)
But people generally want their mathematics to mean something, not just to be a formal game with symbols. (Consistency is just a formal, syntactic property.) The axioms of set theory are intended to be true of the mathematical universe as a whole, whatever that might consist of.
Plenty of axiomatizations are consistent but are of no particular interest. For example, by Gödel, assuming that ZF is consistent, the following is also a consistent theory: ZF together with the sentence "ZF is inconsistent". But this isn't really a very interesting theory, and no one would propose adopting it as a base for mathematics. Why? Because it's not satisfied by the actual mathematical universe (since ZF really is consistent).
Why do people take Zermelo-Fraenkel set theory (possibly with the Axiom of Choice, depending on your taste) as being a description of the general accepted principles of mathematics? ZF isn't set in stone, after all. It's just that people know what mathematical principles they accept, and ZF seems to embody them.
In fact, when Zermelo first compiled his axiomatization of set theory, he didn't include what is now known as the Replacement Axiom Schema. Fraenkel pointed this out, and it's reported that Zermelo agreed that it should be there and said that he just forgot to put it in.
The thing is that mathematicians have a good idea as to what the mathematical universe looks like, and they want to use an axiom system that correctly describes that universe and that lets them prove as much as possible. By Gödel's Incompleteness Theorem, no consistent axiomatization of mathematics can be complete, so there's always the possibility of realizing that there are mathematical principles that you would accept but that haven't been included in the theory, and then adding them in.
From a formal perspective, one can add a new axiom "ZF is consistent". (The consistency of ZF is widely accepted, although it's not provable within ZF.) One can then add the consistency of the new theory. And one can then add the consistency of that new theory, and so on. Each one of these axioms can be seen informally to be true, but cannot be proven from the axioms admitted before it.
But it's more interesting to look at one's intuition about the mathematical universe and use that to find new axioms that one can argue are true in the mathematical universe but that ZF doesn't prove. This would be very much like the situation with the Replacement Axiom Schema -- you may think that you have a good axiom system, but a little reflection indicates that the mathematical universe is richer in structure than the axiom system requires, so you decide to add a new axiom.
The basic intuition of set theory is that the universe of sets ought to be as large as possible and as rich as possible, since it's supposed to contain all conceivable mathematical objects. Several of the axioms of ZF can be thought of as expressing (some small part of) this idea.
Many axioms that go beyond ZF are known, and people have varying amounts of confidence in whether these axioms are true in the universe of sets. Most of these axioms are so-called large cardinal axioms; a large cardinal axiom requires the existence of sets larger than can be proven to exist without that axiom.
A simple example of a large cardinal axiom is the axiom of inaccessibility, which states that there exists a "strongly inaccessible" cardinal. [This is an uncountable cardinal kappa so large that: (1) the union of fewer than kappa-many sets of size less than kappa must have size less than kappa; and (2) the power set of any set of size less than kappa must have size less than kappa.] One gets the feeling that if the universe is to be as large as possible, then strongly inaccessible cardinals ought to exist. If they don't exist, it means you just didn't include everything you could have. It turns out that the existence of a strongly inaccessible cardinal proves the consistency of ZF. It follows that one can't prove, assuming just ZF, that strongly inaccessible cardinals exist. But one's intuition says that they should exist; ZF is just too weak to prove it.
And so on to larger and larger cardinals.... Mahlo cardinals, weakly compact cardinals, Ramsey cardinals, measurable cardinals, strongly compact cardinals, supercompact cardinals, huge cardinals, ..., the existence of these being progressively more powerful assumptions. Strongly inaccessible cardinals are, in fact, very small in the hierarchy of large cardinals.
It's certainly possible for a game to be complex enough to embed mathematics in. If so, however, the interest in the game would be due primarily to the fact that you can embed mathematics in it, not for the game itself. (Well, the game might be interesting or entertaining as a technical problem anyway, but you had asked why Gödel's incompleteness theorem is considered profound. I don't think any analysis of chess or Go or whatever could be considered profound -- again, unless you could embed something of independent interest in it, in which case that would be the reason for considering the result to be profound.)
Incidentally, I don't think that Go is complex enough for what you're suggesting, since it's played on a finite board, unlike Conway's game of life. (I would guess that Go might be PSPACE-complete, which is pretty complicated, but nowhere near complicated enough to embed a universal Turing machine in.)
True. The game would have to be modified in some way
either by allowing for an arbitrarily large board
or else by allowing for the use
of an arbitrarily large number of stones
(The latter would be preferable if one actually wanted to play it)
as it stands
GO is a fiendishly complicated game
I've only played it a few times
and I don't see how anyone could master the strategy.