Thus spake Russ Abbott circa 12/01/09 04:18 PM: > No doubt both are important. The symbols (and formalisms) keep us honest. > The concepts, though, are what the symbols are about. By themselves they are > not about anything.
I disagree. I think the emphasis on concepts is a peculiar form of anthropocentrism (or, at worst, narcissism ;-). An explicit and eminent objective in both math and science is to make processes explicit so that those processes can be argued about, falsified, justified, repeated, and taught. So, while any particular symbol may be ancillary, the methods of extracting and symbolizing the peculiar and particular "concepts" in any given occult process is critical. Symbolization is not just a necessary evil. It is what scientists and mathematicians do for a living. It's their job. For science, this is especially true because the goal is to reify processes out there in the world, to permanently and consistently remove the "concepts" peculiar and particular to any one (small collection of) person(s) and make them available for everyone to play with. >From that perspective, the symbols (as a tool for externalizing the internal) are way more important than the concepts. Looked at from another perspective, the syntax any given person chooses to represent their concepts has a huge and lasting impact on the external (somewhat objective) artifacts she creates, e.g. the battle between Leibniz and Newton. A more modern example is the Baez and Stay paper cited by Roger back in August: http://www.mail-archive.com/friam@redfish.com/msg04285.html I don't pretend to understand it; but it seems to me that much of the unification attempted by that paper revolves around the symbols they're using to represent the concepts peculiar to each domain. Further, it's entirely possible that, although we _think_ the concepts are actually the _same_ and unifying the domains around monoidal categories may be a more general "notation" for the same concepts, it's possible that the instantiation of the common "notation" for all these different concepts is really just building _consensus_ between the various (peculiar and particular) minds involved. I.e. the notation _causes_ the concepts, not vice versa. I know that's reaching; but, it's at least reasonable that the language in which we choose to represent something modifies/guides the concepts in our heads. And it's a primary goal of math and science to remove the peculiarities and particularities associated with any single (small collection of) human(s). Any attempt to make symbols more or less important than concepts in math will fail because they are both part of the same thing. The concepts are Yin and the symbols are Yang. -- glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org
