This function D on the integers might be interesting, but why call it a derivative? I can’t see much conceptual similarity with calculus derivatives.
Broadly, there is a class of functions refered to as "derivations" that can be viewed as a generalization of the derivative. In particular, a derivation satisfies 2 properties:
1) It is linear
2) It satisfies the product rule.
Any function that satisfies these rules is often called a "derivative".
Notably, the function discussed in the article fails the linearity test, which is a pretty big problem for calling it a derivative.
The definition presented here is a loose analogy to derivatives rather than an actual generalization, which doesn't fully justify using the name IMO.
It does, but this one's on a rig (= ring - negatives), not a vector space over any field.
EDIT: Actually, according to wikipedia, that is exactly what is done in differential algebra
Every ring is a module over itself. But you wouldn't want the definition you propose; instead, you'd want `D(ab) = aD(b) + D(a)b`. If you really like some sort of linearity to be present, you could observe that this property forces every derivation to be linear as a transformation of `R_0`-modules, where `R_0` is the subring `ker(D)` of "constants".
It's been many years since I had calculus at uni, and never any abstract math. Why is the product rule picked as the "interesting" attribute of derivatives, ie to serve as the basis for the generalization?
Is there some deeper connection of the product rule in ordinary derivatives that singles out the product rule over the other properties a derivative has?
For me, a key aspect of derivatives is that it allows for something like Taylor expansion or integrals to exist. Are there any equivalent things to these product-rule-generalized derivatives?
Having said that, there are many usages of other derivations where the calculus inspiration is clear, even if the geometric meaning that motivated the calculus is lost.
For instance, we often talk about polynomials over arbitrary fields. In general, there is no way to graph such polynomials. There is no notion of tangent lines, slope, "continuous", or even "less than". There is, however, still the notion of roots and the multiplicity of roots. These notions turn out to be quite important.
When working with any polynomial, you can define the "formal derivative" as a derivation that also satisfies D(x) = 1, D(a) = 0 (where a is an element of the underlying field). This operator behaves as you would naively expect a derivative to behave over polynomials. In Galois theory, it is important to distinguish between polynomials that have repeated roots, and those that do not. If you have a polynomial f(x), you can determine this by taking its formal derivative f'(x). Then, you can easily compute their greatest common divisor [0]. If this is a constant, then you know f has no repeated roots.
[0] Using Euclid's algorithm, this is a purely mechanical process that can be done without needing to factor either f or f'. A similar trick has actually been used to attack real word cryptography. If there are secret primes p and q, and a public number pq, many cryptosystems assume that it is infeasible to determine what p and q are. However, if there is a bad random number generator, you might get 2 different keys that share a prime, so have the public numbers pq and ps, then you can easily determine that p is a common factor, from which you can easily recover q and s. This means that you can look for a large collection of public keys and try this attack on possible pair of them.
Other basic rules would be addition:
(f(x) + g(x))' = f'(x) + g'(x)
This is just linearity (together with constant multiplication)
And function composition (the chain rule):
(f ∘ g)' = (f' ∘ g)⋅g'
We would need to somehow figure out what should correspond to function composition.
So if we want something that captures some important algebraic properties of derivatives the product rule would be a good place to look.
This ”derivative” is not linear though, and that was sort of what motivated my question.
I was just skimming the wikipedia article but this seems like a good argument.
> number derivative is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis.
No, that’s the derivative of the function that return n whatever its argument, which also can be written as λx.n or in a zillion different ways.
That can be written as n, but is different from the number n.
Also, one man’s “pretty confusingly is another man’s “similar things should have similar names”. There’s a rich history in mathematics of overloading the meaning of terms and symbols as long as there’s some similarity between them, for example when using × for both the multiplication of numbers and of matrices (where the former is commutative, but the latter isn’t, barring some exceptions such as 1 × 1 matrices).
(See also the comment elsewhere in this thread which says “Mathematicans like to call two things with the same "structure" by the same name, even if it's not obvious how they're otherwise related” (https://news.ycombinator.com/item?id=40327885)