I got in a debate with a math student regarding Austrian methodology, and he argued that propostitions of a system cannot be proved within that system.
Now, my understanding of Godel is limited, but I know that his theorem states roughly the same thing. It's certainly an interesting argument, but seems to be completely irrelevant. For example, the implications of his argument (taken to the extreme) would be that we cannot prove that 1+1=2. I gave this example, and suggested that we should chuck the entire fields of basic arithmetic and geometry because they can not be proved, but was then told that I didn't understand Godel.
Do any of you have a better understanding then I? How do we refute this claim?
he argued that propostitions of a system cannot be proved within that system
This is incorrect. Godel proved that for any (sufficiently powerful) formal system F, there exist valid statements in F which cannot be proved true or false (undecidable).
Sufficiently powerful --> F must be able to express the basic operations of arithmetic. I'm not sure if anyone has identified what the "minimal power" of F is for Godel's theorems to apply to F but it is important to note that Godel's incompleteness theorems do not apply to first-order logic (aka "logic").
There exist --> Any statement which can proved true or false in F has been proven in F. There is no "larger sense" in which it remains unproven. So, once you prove 1+1=2 from the Peano axioms, for example, you have indeed proven 1+1=2 from the Peano axioms. The incompleteness theorems have nothing to do with logical nihilism.
What Godel proved is that not all statements in F can be proven either true or false in F. This doesn't mean they can't be proven true or false in F', which is precisely what Godel's theorems do. He constructs a statement (nowadays called "the Godel sentence") which is neither true nor false in F but which is true in F' ... for every F, thereby proving there exist statements which are true but unprovable within F for every F (sufficiently powerful).
In particular, the statement "F is consistent" is unprovable in F if F is consistent. What this means is that you can't prove your formal system is consistent. It is always possible that your formal system contains an undiscovered inconsitency which is a death-blow to the formalists who imagined they could reduce the entire world to a math equation and then centrally plan it all.
This explains why Godel is so neglected and misunderstood by modern mathematicians who don't specialize in computability and the fundamentals of mathematical logic. The consequences of his theorems are so profound that there is no subject of mathematics that is not affected by them. Yet few mathematicians are more than cursorily aware of Godel's work. Why? Well, simple: the Marxist bias in academia leads them to ignore anything that debunks the chief modernist value: "we'll centrally plan everyone's lives with a technocratic bureaucracy whose policies are established by academic philosopher Kings".
Godel -> No theories of everything in maths -> Turing/Chaitin -> No theories of everything in physics -> No theories of everything in economics (Oh noes!) -> No central planning -> What Mises said all along
Clayton -
But isn't Austrian-school economics purported to be a theory of everything? Or am I not understanding what you mean by that term?
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A theory of everything would be some kind of closed-form equation - or perhaps even a computer algorithm or numerical method - which reliably computes the future state of the object of the theory. You'd think that the fact we can only predict the weather at most a week or so ahead (sometimes not even that much, depending on conditions) would dampen such optimism but the Venus Project mentality is not obstructed by facts.
Thanks, Clayton. I stand corrected.
Thanks Clayton. Your response was very helpful in understanding Godel.
I was looking through HA because I figured Mises would address this, but he did not. Hoppe has written about Godel, but it's very brief. From "On Praxeology and the Praxeological Foundations of Epistemology" footnote 59:
on the irrelevance of the famous Gödel-theorem for a constructively founded arithmetic see P. Lorenzen, Metamathematik (Mannheim: Bibliographisches Institut, 1962); also Ch. Thiel, "Das Begründungsproblem der Mathematik und die Philosophie," in F. Kambartel and J. Mittelstrass, eds., Zum normativen Fundament der Wissenschaft, esp. pp. 99-101. K. Gödel's proof—which, as a proof, incidentally supports rather than undermines the rationalist claim of the possibility of a priori knowledge—only demonstrates that the early formalist Hilbert program cannot be successfully carried through, because in order to demonstrate the consistency of certain axiomatic theories one must have a metatheory with even stronger means than those formalized in the object-theory itself. Interestingly enough, the difficulties of the formalist program had led the old Hilbert already several years before Gödel's proof of 1931 to recognize the necessity of reintroducing a substantive interpretation of mathematics à la Kant, which would give its axioms a foundation and justification that was entirely independent of any formal consistency proofs. See Kambartel, Erfahrung und Struktur, pp. 185-87.
I've always found that, to understand Hoppe, you have to throroughly read all of his sources because he does not explain them to you. He expects you to understand them already. The problem is that all of these sources are in German and I cannot find an English translation for them.
Clayton, do you know what Hoppe is saying here? How is Godel's theorem irrelevant and how does praxeology evade its critisicm?
Yet few mathematicians are more than cursorily aware of Godel's work. Why? Well, simple: the Marxist bias in academia leads them to ignore anything that debunks the chief modernist value: "we'll centrally plan everyone's lives with a technocratic bureaucracy whose policies are established by academic philosopher Kings".
Yes. Mathematics. That sole remaining stronghold of Marxism. Errr.... ok.
Clayton, there seems to be an important caveat with Goedel's Incompleteness Theorems - they concern formal systems that consist of an infinite number of axioms (of which Peano arithmetic is one).
How is Godel's theorem irrelevant
I'm not sure either what Hoppe means by "constructively founded arithmetic" - but I'm quite sure that any arithmetic, however founded, that can express the basic operations that we are familiar with (addition, subtraction, multiplication and so on) is subject to Godel's incompleteness theorems because the theorem uses the basic operations of arithmetic to construct the problematical self-referential statements.
Chaitin has gone on to show that incompleteness is actually pervasive throughout mathematics and has got to the root of why there is incompleteness in mathematics: randomness. That this is surprising to us (human beings) is a result of the fact that the human brain automatically selects to study "interesting" mathematical truths, truths that are true for some reason, that is, which have some kind of logical depth to them.
For example, it is true that 277546033 + 1847391631 = 2124937664 but this is a rather uninteresting mathematical fact because it doesn't appear to have any impact on other mathematical facts. It has no depth, it's just an isolated, true fact. Chaitin proved that if you don't allow the human brain to filter mathematical facts and you enumerate them all in some kind of mechanical fashion, almost all of them are like the sum at the beginning of this paragraph - random, isolated, uninteresting (but true) facts.
More surprisingly, most such mathematical facts are unprovable. They're simply true for no reason simpler than themselves. Explaining what this means and how it is the case gets into very deep waters but suffice it to say that just because something is unprovable doesn't mean it's not true. And that's what Godel told us: there are true things which can't be proved. Chaitin extends this and shows that it is the case for almost all mathematical facts. We just have brains that happen to filter out those uninteresting facts and focus on the mathematical facts that are beautiful and interesting because they're true for a reason.
and how does praxeology evade its critisicm?
Well, I think that praxeology and incompleteness tell us the same story: the human brain isn't omniscient, that is, it can't deduce all true mathematical facts and it can't foresee the indefinitely long-run consequences of its choices - in short, uncertainty is provably an ineradicable feature of the human condition. The only case in which the human brain could be omniscient is if it actually contained an infinite amount of (true) information which, clearly, it does not.
demonstrates that the early formalist Hilbert program cannot be successfully carried through, because in order to demonstrate the consistency of certain axiomatic theories one must have a metatheory with even stronger means than those formalized in the object-theory itself.
And this is the crux of the issue - the formalists thought we were going to be able to reduce mathematics itself (the discovery and proof of new, heretofore unknown math theorems) to a mechanical process. From there, it follows that this pattern can be applied to the rest of politics, industry and the economy and voila! you have a centrally-planned Utopia. As Mises says in Anti-Capitalist Mentality, the socialists have been rebutted on every point, even their ventures into trying to centrally-plan mathematical abstraction have failed.
That sole remaining stronghold of Marxism. Errr.... ok.
Read what I wrote again for comprehension. Are you going to dispute that academics, as a group, are more Marxist or Marxism-sympathetic than the general population? I don't have any studies to prove otherwise but I think it's a fairly well known anecdotal fact that academics tend to be more left-leaning than the general population.
The implications of incompleteness are devastating to any project of central-planning-through-formal-systems - even if you had the world's most powerful super-computer programmed by the most highly educated and highest IQ mathematicians you wouldn't be able to mechanize the mathematician. The mathematician's objects are pure abstractions - not nearly as complicated as real things - so how much more is this the case of the real economy?
But no, you're right, it's all a delusion in my fevered, right-wing brain.
@Autolykos: No, that's a misunderstanding based on different schools of logic; certain schools of logic do not permit the use of "substitution symbols", i.e. the 'x' in algebra:
2x2+1 = 2
... so instead, they create a new "axiom" for each symbol and relation. Since you can have an infinite number of such symbols as placeholders, they have to have an infinite number of axioms. This way of thinking is confusing and needlessly obtuse, in my opinion - substitution is perfectly natural and holds no metaphysical surprises so there's no good reason to banish it.
If you use a saner system that permits substitution, you will see that Godel's incompleteness theorems actually say this: no formal system with a finite number of axioms can contain all true mathematical statements. And this isn't "axioms" as above (placeholders to avoid substitutions) but, rather, real axioms that introduce novel mathematical facts. Godel says that for any finite set of axioms F you choose, there will always be a true but not provable statement expressible in F. Since we choose axioms such that they are not reducible to one another (cannot be proven from one another), what the incompleteness theorems are really saying is that "all mathematical truth" is an infinite number of irreducible truths. This is why your brain would have to be able to hold an infinite amount of (incompressible and true) information in order to know all mathematical truths.
Full disclosure: This is where I've learned almost everything I know about Godel. If you're mathematically/computationally inclined, I highly recommend reading as much Chaitin as you can, it's an enlightening experience.
I'm not sure either what Hoppe means by "constructively founded arithmetic"
From my understanding he follows in the tradition of Paul Lorenzen, who formulated the whole concept of constructive mathematics. The only English work of his that I've found is here.
Basically, constructive philosphy and construtively founded arithmetic is our norms for establishing such signs. We say 1 means one; we say that 2 means two; 1+2=3, etc. (This is also relevant for definitions, colors, etc). Of course, we can also say that 1 means five or that 2 means seven, or that 1+2=12, but in constructing the units themselves, the truth cannot be undone.
I look at it this way: in order for us to accept Godel's theorem as correct, we have to agree to certain norms and constructive foundations (definitions, math, etc; though Lorenzen argues that not all mathematics can be constructively founded). If we do not do this, then communication breaks down, and there is no way for us to really do anything.
I've been slowly working through Godel's paper (On Formally Undecidable Propositions of Principia Mathematica and Related Systems), and it will take a while for the pieces to fall into place, but from my understanding, Hoppe is saying that Godel is irrelevent (though correct) for reasons that Lorenzen's constructive philosphy and aritmetic explains. This is similar to saying that Mises' action axiom is correct because we agree on action meaning purposeful behavior. We don't think that action means "flying monkey." Of course, we can say that action means "flying monkey" but that wouldn't disprove the action axiom.
Also, thanks for that Chaitin link. I will definitely be exploring it when I have the time.
There really is no "way out" of Godel's incompleteness theorems - it's not based on some kind of unspoken "agreement" except that, of course, we must agree to the definitions of logic and mathematics being used. But to say that the incompleteness theorem is irrelevant to praxeology is simply incorrect - praxeological truths are sufficient to perform basic arithmetic (the praxeologist must hold that one thing and another thing makes two things, etc.) and, thus, the conclusions of praxeology are subject to the same caveat: it is at all times possible that there are undiscovered inconsistencies in praxeological theory.
But this caveat is much less menacing to praxeology than it is to central-planning because the praxeologist does not make a pretense of knowledge - it is the central-planner whose central-plans need to be purged of all undiscovered inconsistencies. And Godel says, "Nope, sorry, you can never be sure you have a perfect theory."
Personally, I don't like Godel's method of proof - he didn't have the benefit of hindsight on his own theorem. I recommend this treatment, it is much more approachable. Also, check out this lecture which is substantially the same as the link above but Chaitin's speaking and mannerisms are highly entertaining in themselves!