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The Generalist – On Alexander Grothendieck
- auntienomen 12y agoLovely writing. This is better than most of what popular science writers have produced on the subject.
- elliptic 12y ago"What else falls when you drop it?" A duck!
- tel 12y agoCan anyone who understands sheaves clarify the whole "point is a place where all functions are constant" analogy?
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- vzcx 12y agoYou actually don't need all the machinery behind sheaves to understand this! The general concept is that of a terminal object in a category. In the category of sets, the terminal objects are the singletons: given a set S, there is only one function f:S -> {a}... send everything in S to a! In the category of topological spaces, the terminal objects are the single-point spaces.
- tel 12y agoThat was my first thought as well, but the author directly linked this notion to sheaves and also was talking about mapping from points.
- qcoh 12y agoI guess this is because the structure sheaf of the spectrum of a field is a constant sheaf.
- SteveLandsburg 12y agoYes, this is exactly what I intended.
- westoncb 12y agoI've been devoting myself for a while now to an approach to thinking that seems along the lines of Grothendieck's project (as described in the blog), where difficulty should only come in defining the framework to work within, while work within the framework is easy. In defining a framework, fundamentally you are coming up with a set of types and describing how things of these types are able to interact with one another; if it's a framework that's easy to work with, the number of types, relations, and operations is small (relative to other formulations). If this framework is concise, but the universe of things described by it is large, then the types must be very general. Therefore, using an approach of finding an elegant framework, rather than mastering the complexity of an inelegant framework, relies heavily on one's ability to generalize. My issue with spending a lot of time thinking this way is that it gives me a sort of uneasy feeling... What are these super abstract generalizations? From what I can tell, this mostly comes from wondering whether with all this 'zooming out' we are getting closer to truths about the universe, or our own minds—and the latter possibility feels a little suffocating. Anyone know any books, essays, whatever taking up that issue?
- nabla9 12y agohttps://en.wikipedia.org/wiki/Category_theory https://en.wikipedia.org/wiki/Category_theory
- westoncb 12y agoCategory theory is probably the epitome of what I'm talking about. Isn't it somewhat disturbing that what we can ground so much mathematics in something that so closely resembles a structure in the human mind? Thinking about it directly again has given me a slightly different perspective, though. Maybe this resemblance of extremely general structures to human mental structures just relates to facility of expression, and isn't necessarily moving us further from objectivity—it's just easier for us to express our concepts as parameterizations of a structure that resembles the abstract structure of human concepts. The different parameterizations we can supply constitute all possible human concepts—but the important thing is that some of those concepts will be valid descriptions of objective phenomena, and those are the ones we attempt to supply.