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 Jul 31 comment If an inequality is true for all natural numbers, is it necessarily true for all real numbers inbetween? ...except for 0. Which is excluded from some definitions of the natural numbers for murky reasons, but the question seems to assume it's included. (i.e., equating the positive reals with the ones "between the naturals") May 17 comment Is “cofunctor” an accepted term for contravariant functors? I think "covariant" is the outlier here, and is using the co- prefix differently. "Contrafunctor" appears to be a contraction of "contravariant functor" and isn't used by mathematicians that I'm aware of. At any rate, there isn't necessarily a tidy duality here, because to make sense of arrows being "flipped" requires a context establishing a preferred direction for arrows, which a functor in full generality doesn't have. May 7 comment A natural example in category theory Does the requirement given include the case where $Y$ is $X$ and the morphism $X \to Y$ is the identity, or must they be distinct? Apr 30 comment distribution of categorical product (conjunction) over coproduct (disjunction) The canonical morphism is so named because it follows directly from the definitions of the product and coproduct, so it doesn't really tell you anything about its inverse (which may not exist). Apr 30 comment distribution of categorical product (conjunction) over coproduct (disjunction) Dually, $(A \times B) + C \to (A + C \times B + C)$ always exists as well. Going in the other direction is where things get trickier. Apr 30 comment distribution of categorical product (conjunction) over coproduct (disjunction) @JoshuaTaylor: If you're working in an intuitionistic setting, quite a few tools along these lines already exist in the context of typed lambda calculi--e.g., the proof in your question can be read off directly as a lambda term whose type is the formula being proved, and a program that can convert it to use only function composition and application exists (and is used largely to entertain other programmers). Apr 30 comment distribution of categorical product (conjunction) over coproduct (disjunction) @JoshuaTaylor: The definition of a monoidal closed category includes that currying is a bijection, so it isn't actually necessary to derive that. I believe it's also possible to derive both directions using only the eval arrow and without relying on the product projections. This is all pretty easy to reconstruct in a cartesian closed category, though. Apr 25 comment Given a functor between categories, how to denote a morphism between particular objects of that category @smartcaveman: Morphisms can be just about anything depending on the category. There's no reason in general for a mapping from one object to another to have anything to do with the morphisms of the category they live in. Apr 25 comment Given a functor between categories, how to denote a morphism between particular objects of that category @smartcaveman: So in general, you're talking about relating a functor between two categories with morphisms in a third category that contains all the objects of the other two? Apr 25 comment Given a functor between categories, how to denote a morphism between particular objects of that category @smartcaveman: So in that case, $C_1$ and $C_2$ are distinct categories whose objects are also categories, right? Apr 1 comment Why is propositional logic not Turing complete? @Hypercube: I mean circuits where an input to a gate can logically depend on its own output. For example, flip-flops implement simple memory cells using feedback loops. Jan 7 comment Natural numbers as types. A 0-tuple or record with no fields, i.e. an empty product type, are also common ways to describe the unit type. And of course an uninhabited type is 0. Dec 28 comment What is the practical difference between a differential and a derivative? In the version of that joke I originally heard, the second "problem" had the faucet already running and making a mess, which the mathematician reduces to a solved problem by (of course) setting the trash on fire. Dec 28 comment A new imaginary number? $x^c = -x$ Hm. Seems to me you only lose that rule about exponents half the time--if you preserve parity of the number of c exponents it should work, I think? Note that introducing an extra factor of 1^c would always suffice. You still end up with x^2c == x^2, of course, which seems peculiar. It's sort of a hypernegative unit, that constructs the inverse of two hyperoperations down, like -1 does one hyperoperation down. Unfortunately, multiplying by c doesn't give you the predecessor operation... Dec 5 comment How can I introduce complex numbers to precalculus students? @AlistairBuxton: Consider which of the following i represents: Half of a reflection along the real line, an axis in 2D space, or a direction of rotation in 2D space? Now, is there any "trick" that will similarly represent half of a complex conjugation, an axis in 3D space, and a direction of rotation in 3D space (recall that you need a scalar component here to get a magnitude of 1!)? Nov 30 comment Notation for repeated application of function Oh, nice! Unconventional, but unambiguous and nearly self-explanatory to anyone familiar with ∘ and the standard superscript notation. Nov 30 comment how to prove $437\,$ divides $18!+1$? (NBHM 2012) @TimPietzcker: Also, the significance in this context is that if a and b are relatively prime, and both divide c, then their product also divides c for hopefully obvious reasons. Nov 16 comment Why do some people state that 'Zero is not a number'? Yeah, that's part of the distinction I'm trying to make--the expression 0/0 has no defined value, i.e., the division operation is not defined on those arguments in the same way that division by "cucumber" is not defined. There is nothing denoted by 0/0 to be a number or not. Nov 16 comment Why do some people state that 'Zero is not a number'? Strictly speaking "NaN" is distinct from "is not defined". NaN comes from the specification for floating point values, which computers generally use as an approximation of the real numbers. NaN is a well defined value (actually several, as there are distinct flavors of NaN) with specific properties (such as being unequal to all floating point values, including itself). Nov 15 comment Isn't every Endofunctor an identity Functor? @drozzy: I was talking about the constant mapping example, but the same applies to both. Perhaps a more straightforward example would be an ordering on the natural numbers, and a functor that maps each number to its successor?