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Medieval Theories of Haecceity
- zem 12y agoone of my favourite words!
- Animats 12y agoAristotle created some messes in logic that weren't untangled until the 19th century. This is one of them. This concept has some practical applications in computing. When are two objects equal? If they are immutable, and there is no way to distinguish them via their public interfaces, are they equal? Should they be considered the same object? If so, copies can be shared, or single objects can be duplicated without harm. In logic, this is a statement in the theory of uninterpreted functions: (forall f, x, y f(x) = f(y)) => x = y Here, if any function f applied to x and y produces the same answer, x=y. x and y are indistinguishable, and in mathematics, that implies they are identical. This has applications in constructive mathematics, where all the functions already in existence can be enumerated and tried. Any new function must be a composition of existing functions, so if that property holds for the existing functions, it must hold for all new functions. (I figured this out ten years after I had a paper rejected by JACM, where I did bulky machine proofs of the theory of arrays without using set theory. There's a better way that's fully constructive using the argument above.) The medievalists apparently didn't pick up on the immutable angle. That led them into circular reasoning: "Maintaining that an object is singular or individual because it results from or is endowed with a proper principle of individuation which is itself singular, amounts to nothing more than duplicating what should be explained."
- jsprogrammer 12y ago>Here, if any function f applied to x and y produces the same answer, x=y. x and y are indistinguishable, and in mathematics, that implies they are identical. That is an error. They are only indistinguishable with respect to f, but it does not preclude the existence of a `g` where (exists f, g, x, y f(x) = f(y), g(x) != g(y)). >Any new function must be a composition of existing functions Based on what? This might apply for limited sets like natural numbers, but I imagine an argument similar to Cantor's diagonal can be made for others.
- twic 12y agoThe 'forall f' precludes the existence of such a g. If there is no reasonable way to enumerate all f, then the original statement doesn't apply, and indeed, your g might exist.
- pfortuny 12y agoI am no formal logician but I guess your argument implies that one assumes Zermelo-Frankel's set theory (otherwise one can devise a finite collection of "sets" and "functions" for which your "axiom" does not hold). Now, if you are in ZF, then you have infinite sets and as far as I know, your argument is true only if you accept the axiom of choice (which you may or may not but is not a "trivial" thing). Also notice that the mathematical world is not exactly the same as the "real" world... Let us imagine there are parts of the Universe which are farther than any possible interaction with them. Are they reduced to a single point? Why not? Would really like to receive an answer.
- dalke 12y agoI'll make some idle musings tangential from the main point. The instantiation of abstract concepts in computer hardware leads to complications. The access time for x and y can vary, as can the power use, or who gets charged for measuring it. I can think of an augmented f(x) which includes some those measurables as part of the public interface, but then the result would be probabilistic, and would need to exclude functions which can measure exactly where each object is located. It might possible to encode multiple values at the same location, eg, via quantum spin states, so treat "location" as more than a position in space-time. At the end though, this says that an object's composition and location in space-time is the only "public interface". All else is artificial. Copyright law is one of these artificial distinctions. If you have a movie on your computer, and I have one on mine, my copy might be in violation of copyright law. A duplicated movie can lead to harm, even though the public interfaces for both give identical results.
- mjburgess 12y agoThe problem is we're dealing with real objects, conflating real objects and mathematical objects is begging the question against the person saying we need haecceity (also called "thisness" - the property of being "this" thing and not "that" thing). If real objects are just like mathematical objects then all they are is bundles of properties: object A is no more than the set of properties that can be ascribed to A. There are a few problems with that, but here's an interesting one. Consider a world in which there are only two identical iron spheres, say 1 mile wide, 1 mile apart from each other. Now each sphere has exactly the same properties - are they the same spheres? Most people would say no. So what individuates them? Well, some property that makes it the case that one sphere is that sphere and not any other. To put it in your terminology, for every boolean function f(x) which is true iff x has property f and false if it doesn't have f (eg. IsIron(x) = True), then both spheres have exactly the same set of true/false values for all properties (forall f.). Unless we introduce a haeecity functions: a(x = A) = True for A, b(x = B) = True for B - ie. properties which are only true when their arguments pick out the objects which they are about. Then we can distinguish between objects, because they have different values for haecities (property functions) a, b.
- anentropic 12y agoFirst time I have read anything about haecceity. The first thing that pops into my head regarding the two iron spheres is that they must differ in their position in the universe, so they differ in something other than their haecceity. If there is no way for two identical real objects to occupy the same space then maybe this concept never applies to real objects...?