Aah, I love the smell of conjecture-and-refutation in the morning

I took an exam today, so here are two final cool-down tirades:

1. I could add to my critique of Kuhn that even a non-“revolutionary” has a natural incentive to refine or re-shape the paradigm governing his activity—fame and glory, certainly, if it’s a big breakthrough, but also just solving his problems, be they great or small. So in one sense you could say that the work of the individual scientist resembles more Popper’s conception of science as a whole, i.e. conjecture followed by refutation, as long as that is qualified by noting that refutation is actually where problem-solving begins, not where it ends. I think that Popper was wrong to think in terms of universal, objective, direct-experiment-produced refutations, and I think Kuhn was right to criticize it, but if it is not right to imagine an objective standard governing refutation of ideas, it still seems clear that on an individual basis scientists (and people at large, for that matter) have personal criteria for continuing to accept or discarding a view. But obviously if an idea which has been accepted is contradicted by evidence or another idea that contradicts it, in a subjective sense refutes it, the scientist won’t just abandon the whole issue, unless he just doesn’t feel up to it. He will try to solve the problem, either by reconciling his views, or replacing his old views with new ones, etc. That’s why I think it’s also wrong to talk about paradigms or models as “research programmes,” as Lakatos did. I don’t doubt that research programs exist, but they are chosen by the scientist who has accepted the paradigm but run into problems integrating it with his other views, not by the formulater of the paradigm. Because problem-solving is an effect of, well, problems, a list of problems to solve implies areas where the solution has failed. Obviously when someone offers a new paradigm they are offering a solution, not an unresolved problem. They may add mention of unresolved ambiguities or problems as a caveat, but that’s not the essence of their contribution, and has no imperative effect on what problems the future researcher takes up. In fact many of the problems that the solution eventually raises aren’t or can’t be anticipated by the creator initially. So all of these issues that philosophers of science have addressed, revolutions, conjectures and refutations, research programs, have validity, but those that formulate them have a tendency, like so many others, to over-emphasize the collective over the individual, and to see broad-scale processes rather than individual human activity.

2. Last point: I’m getting sick of people saying that mathematics is abstract and essentially divorced from reality. I know, I’ve done it myself repeatedly, but when my philosophy of science professor repeated this old cliché again, I realized that in some ways it seems awfully ludicrous. I realize that mathematics is often not intentionally an attempt to model or describe processes observed in reality, but I think it would be difficult to conceive of an element of reality more fundamental than quantity. In fact to me quantity (as opposed to quality) seems virtually synomymous with external reality and objects. And so the basic properties of math are fundamentally real, even if the possibilities spun out from these properties haven’t yet been observed. I get the sense that the great advantages of math as a real property by some perversity are actually used against it. For example, the fact that everyone knows exactly what some number, say 6, is but can’t explain why is used as evidence for the view that it is self-defining and therefore fundamentally insular, that its identity is not rooted in the real. In fact, you could reverse that and say that it is simply a defect of language. Again, it’s often said that math is a language, but if we really take that seriously we must think of it as a language different than other human languages. Defining a quantity is like trying to render one language into another. There’s no such thing as a true translation, and between math and language the gap is far more profound than between English and French. I would say that language basically describes qualities and math quantities—math is so fundamentally objective that it simply can’t be described in terms of the subjective qualities that language describes, and vice versa. Again, my professor used the fact that 3+3=6 no matter whether such a combination actually exists anywhere in the world and will never change no matter how the world changes as evidence that math is fundamentally cut off from the world. But that’s like saying the expression “it will snow tomorrow” has no meaning or connection to reality if it doesn’t actually snow tomorrow. The only explanation I can think of for this view is that the correspondence theory has yet again been pushed way too far—you know, the view that the word “rock” must basically correspond to an actual rock. In this case, the fact that the word, or the number, is not actually the object itself is being used as evidence that it has no relationship to reality. But I think we can at least say this much: if external reality is held to be essentially objective, i.e. pertaining to objects, than math is more connected to reality than language. Unless anyone has any alternatives to those two, I think the pertinence of math to reality has been demonstrated. I assume most scientists concur, which is why the scientists who my professor praises as being more addressed to reality than mathematicians basically conduct their work in math. Granted math is not a good language for confessions, but fortunately for those, like me, more interested generally in working out what’s in their heads than in what’s going on around them, we still have language to deal with that.