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I think it's time to elevate the function into the same status as the number. A number is ultimately founded on sets. A number is a most basic way to structure
by bsedlm 4y ago
I think it's time to elevate the function into the same status as the number.
A number is ultimately founded on sets. A number is a most basic way to structure sets. Every number can be made out of pure sets (and later on, from tuples of sets-as-numbers)
At the moment, functions are also founded in terms of sets. But I think that mathematics can be constructed based on functions instead of sets. As far as I understand, this can be done using simple type theory; but I still have some studying to do about STT.
In any case, my point is that most mathematics since around the 18th century are not really working with numbers, but with functions as if they were numbers.
- carnitine 4y agoThis is what category theory attempts to do, except in an even more abstract sense, based on morphisms which generalise functions.
- layer8 4y agoWouldn’t relations be the more fundamental notion?
- bsedlm 4y agorelations are also defined in terms of sets: As a sets of pairs, e.g. an element of a set like {x ,{x,y}} is just the pair (x,y), a set of such pairs IS a relation. moreover, in terms of sets functions are a relation with some restrictions: - That they are binary relations - The "functional-requirement" (as I call it) refers to the difference between a relation and a function. for every x, f(x) = (only one thing); e.g. a relation can have pairs (x,a) and (x,b) but no function can do this. As far as I've understood, this gets really "fun" when characteristic-functions enter the scene... With these boolean-valued functions it's possible to define a set.
- layer8 4y agoI’m aware of all that, but it doesn’t really answer my question. If functions can be used as a fundamental concept not based on sets (maybe similar to arrows in category theory), what prevents generalizing that approach to relations? While relations lose the guarantee of single-valuedness, they gain the guarantee of being invertible (or in the case of n-ary relations, arguably a further generalization, permutable).
- housecarpenter 4y agoSince all three notions are closely interrelated, it's fairly trivial to take any one you choose as "fundamental" and define the others in terms of it: - Sets as fundamental: define ordered pairs via Kuratowski's definition, define relations as sets of ordered pairs, define functions as relations not containing any two ordered pairs with the same first projection but different second projections - Relations as fundamental: define sets as unary relations, define functions as relations the same way as before - Functions as fundamental: define sets as functions where the codomain is some fixed singleton (characteristic functions are more appropriate for subsets of a set, rather than sets on their own), define relations as functions whose codomains are power sets Because of this sort of thing, I don't think any notion of "fundamental" that's purely based on interdefinability really makes much sense. I personally have a sense that some definitions are "constructive" and others are "destructive" (my use of "constructive" here is in no way related to "constructive" as in "constructive logic", it's just based on the English meaning of "construct"). Of the definitions I gave above, I perceive the definitions of functions and relations from sets, of functions from relations, and relations from functions as "constructive", while the definitions of sets from functions and from relations are "destructive". Since I can't think of any "constructive" definition of sets from functions or relations, while I can think of "constructive" definitions in the opposite direction, I would say sets are more fundamental than functions and relations. Whereas since both relations and functions can be "constructively" inter-defined, I'd say they're both on the same level. Admittedly, I have no idea how to define "constructive".
- xigoi 4y agoIsn't that just lambda calculus?
- yakubin 4y agoLambda calculus models only computable functions.