david petry wrote: [...] > > The Marxists produce beautiful theories. They produce complex, clever, > precise, and apparently logically consistent arguments, which must pass > a rigorous peer review process. The Marxists believe themselves to be > open minded, unbiased, compassionate, independent thinkers. But, of > course, the skeptics don't see it that way at all. According to the > skeptics, market forces are simply part of reality, and responding to > those forces is a natural and compassionate thing for humans to do, and > hence the implementation of Marxism (which tries to replace market > forces with governmental planning) requires a brutally oppressive and > intrusive government willing to criminalize human nature, so that no > matter how beautiful the Marxist theories may be, there is something > fundamentally very very wrong with the Marxist world view. Many > skeptics in academia believe that they are severely discriminated > against by the Marxists. >
This is not really true, of course. It's just market forces at work. There is more demand for Marxists in humanist circles than for anti-Marxists.
[...] > For the sake of this argument, let's consider an idealized skeptic. Our > skeptic will be intelligent and honest to a fault. She will have a > technical background, and will be fully aware of the power of > mathematics in technology. She will have no axe to grind, and she will > have no philosophical, religious or political biases, other than a > propensity for skepticism. That is, she looks for concrete evidence, > observable implications, and testable consequences. She is skeptical of > mere clever argumentation. And she refuses to be intimidated by appeals > to authority or ad hominem attack. > > So let's say that our skeptic has been reading popular books about > mathematics. For example, books by Hofstadter, Penrose, Rucker, > Smullyan, Kline, Singh, Aczel, and maybe others. She is troubled that a > lot of the modern mathematics she has been reading about seems to be > nothing more than clever argumentation with no concrete content and no > testable consequences. She wants to know how the ideas she has been > reading about can help us to understand the world in which we live. > > Let's look at specific examples of the mathematical ideas she is > skeptical about. > > 1) Set theory and Cantor's Theorem. It seems obvious to our skeptic > that the mathematical constructs we actually deal with must be > identifiable, and that we can only identify a countable number of such > constructs (since our language is countable). So Cantor's Theorem > asserting the existence of uncountable sets (and hence the existence of > objects which cannot be identified) cannot have any concrete content. > The idea that there must exist more unidentifiable objects than > identifiable objects appears to be silly word play.
Sounds like our skeptic is a staunch positivist (and not even a logical positivist, since she apparently doesn't accept the validity of logic), if she thinks "The idea that there must exist more unidentifiable objects than identifiable objects appears to be silly word play." If she really had no "axe to grind", she would be perfectly willing to accept this claim, given sufficient evidence. Now I will grant that she might feel that Cantor's argument is insufficient evidence for the claim, but that of course is a different matter than rejecting the claim out of hand, as you describe her doing.
> Clearly, to our skeptic, set theory includes an element of make-believe. So > she concludes that much of what she has been reading is nothing more than > clever argumentation with no concrete content; what possible testable > consequences are there to the assertion that unidentifiable objects > exist? And why don't the books she has been reading address the obvious > skeptical objections to such ideas?
It is usually estimated that there are roughly 10^80 atoms in the universe (give or take a few orders of magnitude). Obviously we cannot hope to "identify" all of them. So evidently, our "skeptic" must either reject atomic theory, or accept that the existence of unidentifiable objects has testable consequences.
> 2) Godel's Theorem (loosely, no consistent formalism can prove its own > consistency) Informally, our skeptic claims, a proof is a compelling > argument. It seems clear to our skeptic that if we are to believe that > the formal theorems in our formalism should be accepted as compelling > arguments, then at the very least it must be the case that we already > believe that our formalism is consistent, and hence, no possible formal > proof within that formalism could be considered to be the evidence that > compels us to believe that the formalism is consistent. And our skeptic > asks, is that not already the essential content of Godel's theorem?
Godel's theorem shows that there cannot be a formal proof that the formalism is consistent. Our non-logical positivist's reasoning, while valid, only shows that such a formal proof would be useless, not that it cannot exist.
[...delete argument from false premises...] > > 4) Fermat's Last Theorem. Our skeptic notes that while FLT itself has > clear meaning and a concrete content, there's nevertheless something > fishy about the idea that it has been proved. There is something that > is immediately clear to anyone who has dared to search for a > counterexample to FLT: just due to chance alone, it seems unlikely that > there is a counterexample. That is, for an exponent 'p' of modest size > or larger, the set of integers which are p'th powers is a very very > sparse set of integers, and for an arbitrary set of integers that is > that sparse, straightforward probabilistic reasoning tells us that it > is very unlikely that the sum of two of its elements will turn out to > be another element of the set. In fact, for example, a > back-of-the-envelope calculation suggests that for a set as sparse as > the set of 50'th powers, the probability that two of its elements will > sum to a third element of the set is about 1/10^200, and this can be > loosely interpreted as giving a probabilistic proof that FLT is almost > certainly true for exponent 50.
Well, this is certainly a consistent argument for our non-logical positivist. Or maybe not. She does not care about "proof", but only about what can be identified, so it is only natural that a proof of FLT would hold no interest to her. On the other hand, in her eyes, FLT is proven by the fact that noone has ever found a counterexample, quite independently of any probability arguments. So even your probability argument is of no value to her. In any event, she likely rejects the existence of probabilty because it cannot be tested without circularity. But perhaps she is a "heuristic positivist".
[...] > Our skeptic notes that "real" mathematics (i.e. the mathematics which > has the potential to help us understand the observable world in which > we live; the mathematics used by physicists, computer scientists, > statisticians, economists, and applied mathematicians) has testable > consequences (see appendix).
i.e. mathematics with testable consequences has testable consequences. I have to agree with this...
> But how can we test statements about a world of the infinite lying > beyond what we can observe?
How can we test statements about atoms that are too small to see? I say that the validity of calculus demonstrates the "existence" of the infinite in the same sense that the validity of modern chemistry and thermodynamics proves the existence of atoms.
> What test could we perform to compel us to believe that Godel's proof tells us more > than what a simple and immediate informal argument compels us to > believe?
A test can't "compel" us to beleive anything, it can only support or refute a claim. But one is always free to accept or reject the claim anyway, by adding auxiliary claims if necessary.
> How can we test the assertion that paradox is something other > than pure nonsense?
If consideration of the paradox leads to useful or interesting results, such as Godel's theorem, then it must not be _pure_ nonsense.
> How can we test the proof of a theorem which tells > us nothing more than what we should expect from simple probabilistic > reasoning?
Look at individual steps of the proof, to see if they are valid.
> Should we not be skeptical of modern mathematics?
You have not given any reason to be any more skeptical of "modern mathematics" than of, say, the mathematics of Euclid. Euclid also included lengthy and complicated proofs of apparently self-evident results.
> Is it possible that modern mathematics is built on a flawed model of > reality?
Conventional wisdom among mathematicians is that it is not based on a model of reality at all. So I would have to say no. If it were based on some world model, they would surely know that, no?
[...] > So now, let's pretend that our skeptic ventures into sci.math and > sci.logic to explain her reasons for being skeptical of modern > mathematics. What will happen? It's not a pretty picture; the > mathematicians will go on the offensive. They'll call her a crackpot.
Quite rightly, too, seeing as she rejects the validity of logic, and possibly atomic theory as well.
> They'll claim that she is unqualified to even have skeptical thoughts > about mathematics, and that she is unreasonably demanding that the > experts come down to her level of understanding. They'll claim that she > is trying to impose her religion on others, and that she is trying to > take away the mathematicians' freedom.
Quite rightly; she is indeed trying to impose her heuristic positivism on other people, and doing so under the false pretense that it is identical with "skepticism".
[...] > We can think of mathematics as a science which studies the phenomena > observed in the world of computation. All of the mathematics that has > the potential to be applied can be thought of that way.
True. In fact, all known mathematics can be thought of that way, although in some cases, the world of computation must be generalized. (For example, recursion theory routinely deals with the concept of relative computability, where one considers computability given a single non-computable primitive.)
> As a conceptual aid, we can think of the (abstract) computer as both a > microscope and a test tube: it helps us peer deeply into the world of > computation, and it gives us a way to perform experiments within the world of > computation.
No, you need a real computer to do experiments. You can't do experiments on something that doesn't acutally exist.