GUEST BLOGGER BILL GASARCH TOPIC: How much is what we do influenced by non-scientific criteria? (BEFORE I START TODAYS BLOG- A REQUEST. EMAIL ME OTHER LUDDITE QUESTIONS- I WILL POST THE BEST ONES ON FRIDAY) I) AN INCOMPLETE SUMMARY OF Thomas Kuhn's book The Structure of Scientific Revolution: For long periods of time a field of science will agree on the basic terms and problems of the field and will all work with that worldview (also called a paradigm). This is called Normal Science. This is GOOD since if people were working with different paradigms progress would be hard. BUT there comes a time when some problems just cannot be solved using the usual techniques. There will be an effort to jam this problem and some approaches to it into the current paradigm, but eventually, the old paradigm will fall and a new one will take its place. The new one will help to answer some old questions, and pose new ones that could not have even been asked in the old one. Newtonian Phy vs Einstein is the usual example, though there are others on a much less cosmic scale. II) People after him have misconstrued his work to saying that science has NO objective truth, that it ALL depends on the Paradigm. This is, of course, hogwash. More so when they claim that its a tool by the elite to dominate the masses, or some such (look up SOKAL HOAX on google for one view of this view). III) But a fair question CAN be raised along these lines: How MUCH of what scientists do depends on political or personality or other factors VERSUS how much is driven by objective scientific principles? A few examples a) What if in response to Russell's paradox the math world essentially axiomized what set theorist now call V=L (every object is constructable). Then we would know LOTs more about L, we would KNOW that the Axiom of Choice is true, and we would know that Cont Hyp is true. We might know that there were these weird other models that are unnatural where CH is false, but we wouldn't care. (Some Set Theorists tell me this could never happen- that people would be interested in other models. They are wrong.) b) What if in response to the Banach Tarski paradox mathematicians rejected some version of the axiom of choice? This would have been quite possible before AC began being used in so many places. c) The people who believe in constructive methods only (e.g, Brouwer) are portrayed as cranky old men holding onto an old paradigm that no longer worked. But if they had won then people like Hilbert would be viewed as crazy rebels who fortunately were never taken seriously. (This one I am less sure of- nonconstructive techniques are SO powerful that I think they may be inevitable.) d) If Computing Devices were invented either earlier or later then they were would have a drastic effect on Theory. While we think that P vs NP is a natural problem, it only came out once the technology was in place. Was it inevitable that it arise? Probably Was it inevitable that it be considered important? Hard to say. e) There is ALOT of work in Quantum Computing because (i) Peter Shor proved FACTORING in Quantum P hence giving the problem new interest, or (ii) There is (or actually was) lots of Grant money in it. (of course these two are linked) f) Do schools like MIT have too big an influence on what gets studied? (They have less influence now than the used to.) MORE GENERALLY, if I had the time and the energy I would do research on history/phil of math asking the question HOW MUCH DO EXTERNAL FORCES EFFECT WHAT IS STUDIED ? and I would do it WITHOUT an ax to grind.

I am surprised you didn't list under non-scientific criteria: Where the money is

ReplyDeleteYou claim to be describing the Kuhnian philosophy, but what you have described seems quite Lakatosian to me. The central question is:

ReplyDeleteWhen a scientist chooses to switch from one paradigm to another, can s/he be said to be doing so for RATIONAL criteria? That is, although people are affected by a variety of outside influences, the "correct" choice will be made by the majority in the long run.

By correct I mean, can we say, at least in hindsight, that the new paradigm is OBJECTIVELY more correct, by some specified criteria, than the old one? An example of such a criterion might be "a closer approximation to reality" or "more empirically adequate" (at least for scientific theories - the situation in Math is a bit different). If you think so, then you are a Lakatosian and if not then you are a Kuhnian.

This is related to whether one accepts Kuhn's notion of incomensurability or not, e.g. is the meaning of "mass" in Newtonian and Relativistic mechanics so different that we can't compare them objectively, or is the Newtonain concept really subsumed by the Relativistic one, as most physicists believe.

The Kuhnian view does seem to me to lead to relativism about science and the whole postmodern attack. Kuhn might have rejected this, but I don't think he managed to argue convincingly against it in his book.

This seems to be a confusion of reality with the models we devise to explain reality. In philosophy this is called the error of reification.

ReplyDeleteWhen we move from one model to another model of reality we are not moving from one reality to another. The reality we attempt to model is the same reality. Only the model has changed. And we only change models if there are inadequacies in the older model and useful things we can do with the new model we could not do with the old.

We should also be aware that newer models of reality are not necessarily better models. The new model of Eugenics in the early part of the 20th century was not a better model scientifically as it allowed some political forces to get away with murder literally. It was also false and a poor scientific model that was eventually replaced by a genetic model.

I could level similar criticism toward the string "theory" of physics as it has yet to come up with a testable hypothesis and thus is certainly NOT a theory. People, who otherwise have good sense, want the string "theory" to be correct so badly they are opening themselves to ridicule.

Somehow science can now be done with just mathematical a priori reasoning and no laboratory or testing is needed? Some people are making silly assumptions string theory is true and thus the need for verification has suddenly been eliminated in science. This is no different than the fundamentalists' claim that belief is all we need to make truth. And THAT is definitely a very poor model of reality.

Completeness and consistency of any mathematical system by themselves tell us nothing about the world. Only testing can verify how close our models are to reality.

Re: P vs NP a natural problem.

ReplyDeleteThe P vs NP question (or things closely related to it) arose several times, in several different contexts. They came out of diverse technologies and cultures, not as a result of a single scientific subculture--that is one of the reasons why it is an important problem.

(A partial list:

1. Goedel's letter to von Neuman, suggesting that it would be important to know whether tautologies have short proofs.

This is NP=?co-NP, but it is closely related.

2. Many reductions among difficult combinatorial problems were known, and actively pursued in the context of 01 integer programming before the formal definition of the P vs NP problem.

3. The definitions of P and NP given by Edmonds arose in the context of trying to explain why algorithms that were too slow on the then existing computers were still valuable.

4. In the late 60s, early seventies there was a research agenda in the OR community to prove that NP intersect coNP = P, as the natural generalization of duality in Linear Programming and graph minimax theorems, or prove that the conjecture is false.)

I believe that the characteristic of natural and important problems is that they arise in several independent ways.

For example, if Newton was never born, we KNOW that we would still have Calculus, courtesy of Leibnitz.

There is no question in my mind that the underlying big question "How do we prove that a computation on a

generalmodel is hard?" is a fundamental one. P vs NP is important because it is a manifestation of it. I bet that a useful technique will change our ideas of what computation is, in the Kuhnian sense. That's why the question is important.