> On Oct 21, 2015, at 2:11 PM, Benjamin Udell <bud...@nyc.rr.com> wrote: > > I think that the relevance of the classification of research is in the light > shed on the logical supports among fields in the build-up of knowledge. > Physics doesn't decide which math is mathematically right, which combined > mathematical postulates are consistent and nontrivial, and so on, instead it > decides which maths are applicable to, and illuminating in, physics. How far > can one trace such structures of logical dependence and independence?
This seems to rest on a problematic division of who decides what’s right. It’s precisely because of cases like this that I find Peirce’s focus on taxonomy (especially of the sciences) problematic. I completely agree with your later comments about how taxonomies help keep logical divisions clear. Where that can be done we must do it. Think for instance of the analysis of interpretations of quantum mechanics where we must keep epistemological and ontological categories clear. My problem with the above is that what determines something is right is largely a community response. An informed community focusing in on logic. But a social community nonetheless. Most problematically I’m not sure the community is neatly split among physicists and mathematicians. Clearly mathematicians do physics. (Think Peter Woit for instance who has been a constant skeptic of string theory keeping physicists honest about the empirics of their work) I think the opposite happens as well. Now we might be tempted to say that we must distinguish between a physicist doing mathematics and a mathematical doing physics. But that then leads us to a situation where I sense a vicious regress. A physicist is a mathematician when judging what is mathematics when… I think we can separate the fields but of course what decides what is right is separate from its nature. Physics as a social field blends with mathematics so that areas get pushed by physics. Think certain subsets of abstract algebra of use in dealing with symmetries. It’s also the case that the very nature of mathematics changes with this influence. I think this was noted in the 70’s by Putnam with his justly famous paper on semi-empirical methods in mathematics. The “taking as true” mathematical theorems for which there is no formal proof but which most regard as true became much more common. Then there were the quasi-proofs starting with things like the Four Color Theorem where computers were so utilized such that no individual really could trace the steps in the proof. While not exactly what Putnam was getting at it became a good illustration of semi-empirical methods. Since then such proofs have become more common. Likewise unproven mathematical theorem can be treated as true due to influence from physics. I believe the Corbordism Hypothesis is an example of this - one can approach it via algebraic topology or from quantum field theory. This is all quite loose though. Clearly physicists typically approach things quite different from how mathematicians typically do. The above to me is more an example of Peirce’s principle of continuity. I bring all this up simply because it seems like the sciences are so messy like this that even two fields between which it seems easy to draw boundaries shows that in practice the boundaries break down. Famously this is a problem dividing the scientific from the non-scientific as well. I suspect it will be true of any category of this sort which rests in part upon a social aspect. Any logical analysis which depends upon the maintenance of strong boundaries would thus be an incorrect analysis. (Although perhaps useful as a first or second order approximation for thinking through issues) Even clear divisions such as between ontology and epistemology sometimes break down in similar ways. (What if the epistemological limits are just ontological for instance? Certainly possible if one is an idealist of certain stripes.)
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