5 ms·

The skull splitting is a natural reaction because this has nothing to do with differentiation. Throw in Leibniz’s rule and the reader is reduced to reading the

The skull splitting is a natural reaction because this has nothing to do with differentiation. Throw in Leibniz’s rule and the reader is reduced to reading the fine print to understand. The articles (Wikipedia included) are as guilty of this as whoever chose this name.

eigenket 2y ago
This definition does actually have a surprising amount to do with differentiation. The definition works for any unique factorization domain and in particular for polynomials. It turns out that the definition here exactly matches the usual derivative for polynomials.
- xpe 2y ago
  Are you saying that for arithmetic derivatives, the definition (part "a" above) "1 for any prime; everything else calculated via the product rule" has a surprising amount to do with differentiation? If so, can you connect the dots? Or did you mean the properties (part "b" above)?
  - eigenket 2y ago
    Yes, > 1 for any prime; everything else calculated via the product rule does indeed have a surprising amount to do with differentiation! If you take the usual polynomial functions in one variable (lets say x is the variable and all our things are complex numbers) then these can be factored: e.g. x^2 + 3x + 2 = (x+1)(x+2). They form a (so called) unique factorization domain, which essentially means that factorization into "primes" works exactly the same as it does for integers. In the example above (x+1) and (x+2) are examples of prime factors which can't be factored any further. If you take the definition "1 for any prime; everything else calculated via the product rule" and apply it to this system where our "numbers" are polynomials and our "primes" are the polynomials we can't factor any further you get a definition of an "arithmetic derivative" for polynomials. The fun fact then is that this arithmetic derivative we just defined is exactly the same as the usual definition of the derivative from calculus: D[(x+1)(x+2)] = (x+1)D[(x+2)] + (x+2)D[(x+1)] = (x+1) + (x+2) = 2x+3 whereas d/dx (x^2 + 3x + 2) = 2x + 3
  - gjm11 2y ago
    More the first than the second. There are things other than the integers for which it makes sense to talk about "the primes". One example is: polynomials (with coefficients in, let's say, the complex numbers). In this case it turns out that the "primes" are exactly the linear polynomials (ax+b) where a is nonzero. There's a bit of ambiguity there, just as there is in the integers; 7 and -7 are "the same prime number", and x+3 and 5x+15 are "the same prime polynomial"; if we're going to say D(p)=1 then we need to pick which "version" of p has this property, and the obvious choice is the one of the form (x+a). So, now, if we apply the same definition as for integers to polynomials with these conventions, it says: (1) D(x+a) = 1 and (2) D(fg) = fD(g) + D(f)g when f,g are polynomials. And that turns out to give the exact same result as the "ordinary" derivative for polynomials. Whether "exactly identical to" implies "a surprising amount to do with" depends on how easily surprised you are, I guess. ... I glossed over the sense in which the "primes" are precisely the linear polynomials, so here are a few words about that for anyone who's curious. If we look at polynomials with complex-number coefficients, a beautiful theorem says that they can all be written as A (x-r1) (x-r2) ... (x-rk), and then one polynomial divides another if and only if its set of rj is a subset of the other's (handling repeated roots in the "obvious" way). It's pretty easy to get from this that the linear polynomials are (1) the irreducible ones, i.e., the ones that can't be factored into lower-degree polynomials, and (2) the prime ones, i.e., the ones with the property that if p divides ab then p divides either a or b. (These properties are equivalent for the integers, as well as for polynomials with complex coefficients, but there are other settings in which they come out different, and both of them are useful, so they have different names.) (What happens if we use real rather than complex coefficients? The Wikipedia "Arithmetic derivative" page claims that we still get the usual derivative, but that looks wrong to me, because if we work over the real numbers then x^2+1 is both prime and irreducible, but its derivative isn't 1. Maybe I'm missing something.)
    - eigenket 2y ago
      As far as your point in parentheses goes, I think wikipedia is either wrong or confusingly written (allowing complex factorisations of real polynomials makes what they're written consistent, but is a bit silly). See theorem (20) on page 18 of this pdf for a theorem along these lines https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.pdf https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarov...
xpe 2y ago
This comment demonstrates my point, does it not? It looks like more of the same: a battle of definitions. If one's goal is to win a definitional battle, what do you accomplish if you succeed? [1] But one will not consistently win such a battle. Many people will resist for various reasons, whether it be "stubbornness" or simply feeling like the other person shows no signs of trying to understand what they mean. I propose that better goals include: (i) understanding what people are saying; (ii) applying the concepts to some productive end. By "productive" I mean some forward progress in an empirical or mathematical sense, whether it be prediction or proof. So give up the battle. Why? Not because you are wrong. [2] Because "being right" about a definition is rather silly. We're talking about concepts being communicated by language and symbols. The goal is shared understanding of the concepts (which happens inside a brain), not merely enforcing a mapping of brain states to ink on a page (words) or vibrations in a physical medium (sound). [1]: Whether you win or lose, the distinction between (a: definition) and (b: properties) still exists. [2]: And not because you are "right" either. You can, at best, be consistent in your definitions and use them in useful ways.
- nobodyandproud 2y ago
  I’m saying that if confusion and annoyance has been the norm, then a quick translation for the general audience is in order. Even better would be a differentiating name, but I realize that’s unlikely.
xpe 2y ago
> The skull splitting is a natural reaction because this has nothing to do with differentiation. I recommend rephrasing that as "the definition (part "a" above) of arithmetic derivative is different than calculus definition of derivative." Do you see? Stating it this way reduces the war of words. Your point is made clear. [1] Then other people can say "Ok, sure, but don't you see how the properties (part "b" above) are the same? And isn't that interesting?" Think of this another way. Imagine an alternate history where the arithmetic derivative was discovered, named, and socialized first. Then imagine calculus came along later. If so, would calculus be wrong to use the same word, "derivative"? ... I won't answer that question because it is invalid. Better to dissolve the question [2]. My point? Let's try to shift away from historical battles over turf and terminology. Let's find ways to share insight. [1] Unless your intended point was: "how dare you use the word differently?" [2] https://www.lesswrong.com/posts/Mc6QcrsbH5NRXbCRX/dissolving-the-question https://www.lesswrong.com/posts/Mc6QcrsbH5NRXbCRX/dissolving...
- nobodyandproud 2y ago
  I have zero stakes in this other than stating perhaps it’s not the audience that’s at fault, when an in-circle term isn’t translated for a more general audience; and the audiences are consistently confused and annoyed. I woke up, noticed the title & blog (and references), went through the same tortured route and confusion, before stumbling on the fine print, and coming to same “oh, for crying out loud” reaction.
  - xpe 2y ago
    Is it fair to say the definitional confusion bothered you more than the interesting aspects (I'm presuming that part "b" above is more interesting) pleased you? If I were to guess... I'd say you (and many people, including myself, often) are weary of people redefining words in a way that seems wasteful, distortive (such as 'stealing' words that formerly had clear technical meanings), purely commercial, or self-promotional. For me, at least, the intent of the redefinition matters. But I detect no self-interest or obvious neglect in the case of the arithmetic primes.

eigenket 2y ago

This definition does actually have a surprising amount to do with differentiation. The definition works for any unique factorization domain and in particular for polynomials. It turns out that the definition here exactly matches the usual derivative for polynomials.

xpe 2y ago
Are you saying that for arithmetic derivatives, the definition (part "a" above) "1 for any prime; everything else calculated via the product rule" has a surprising amount to do with differentiation? If so, can you connect the dots? Or did you mean the properties (part "b" above)?
- eigenket 2y ago
  Yes, > 1 for any prime; everything else calculated via the product rule does indeed have a surprising amount to do with differentiation! If you take the usual polynomial functions in one variable (lets say x is the variable and all our things are complex numbers) then these can be factored: e.g. x^2 + 3x + 2 = (x+1)(x+2). They form a (so called) unique factorization domain, which essentially means that factorization into "primes" works exactly the same as it does for integers. In the example above (x+1) and (x+2) are examples of prime factors which can't be factored any further. If you take the definition "1 for any prime; everything else calculated via the product rule" and apply it to this system where our "numbers" are polynomials and our "primes" are the polynomials we can't factor any further you get a definition of an "arithmetic derivative" for polynomials. The fun fact then is that this arithmetic derivative we just defined is exactly the same as the usual definition of the derivative from calculus: D[(x+1)(x+2)] = (x+1)D[(x+2)] + (x+2)D[(x+1)] = (x+1) + (x+2) = 2x+3 whereas d/dx (x^2 + 3x + 2) = 2x + 3
- gjm11 2y ago
  More the first than the second. There are things other than the integers for which it makes sense to talk about "the primes". One example is: polynomials (with coefficients in, let's say, the complex numbers). In this case it turns out that the "primes" are exactly the linear polynomials (ax+b) where a is nonzero. There's a bit of ambiguity there, just as there is in the integers; 7 and -7 are "the same prime number", and x+3 and 5x+15 are "the same prime polynomial"; if we're going to say D(p)=1 then we need to pick which "version" of p has this property, and the obvious choice is the one of the form (x+a). So, now, if we apply the same definition as for integers to polynomials with these conventions, it says: (1) D(x+a) = 1 and (2) D(fg) = fD(g) + D(f)g when f,g are polynomials. And that turns out to give the exact same result as the "ordinary" derivative for polynomials. Whether "exactly identical to" implies "a surprising amount to do with" depends on how easily surprised you are, I guess. ... I glossed over the sense in which the "primes" are precisely the linear polynomials, so here are a few words about that for anyone who's curious. If we look at polynomials with complex-number coefficients, a beautiful theorem says that they can all be written as A (x-r1) (x-r2) ... (x-rk), and then one polynomial divides another if and only if its set of rj is a subset of the other's (handling repeated roots in the "obvious" way). It's pretty easy to get from this that the linear polynomials are (1) the irreducible ones, i.e., the ones that can't be factored into lower-degree polynomials, and (2) the prime ones, i.e., the ones with the property that if p divides ab then p divides either a or b. (These properties are equivalent for the integers, as well as for polynomials with complex coefficients, but there are other settings in which they come out different, and both of them are useful, so they have different names.) (What happens if we use real rather than complex coefficients? The Wikipedia "Arithmetic derivative" page claims that we still get the usual derivative, but that looks wrong to me, because if we work over the real numbers then x^2+1 is both prime and irreducible, but its derivative isn't 1. Maybe I'm missing something.)
  - eigenket 2y ago
    As far as your point in parentheses goes, I think wikipedia is either wrong or confusingly written (allowing complex factorisations of real polynomials makes what they're written consistent, but is a bit silly). See theorem (20) on page 18 of this pdf for a theorem along these lines https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarovski.pdf https://cs.uwaterloo.ca/journals/JIS/VOL6/Ufnarovski/ufnarov...

xpe 2y ago

This comment demonstrates my point, does it not? It looks like more of the same: a battle of definitions. If one's goal is to win a definitional battle, what do you accomplish if you succeed? [1] But one will not consistently win such a battle. Many people will resist for various reasons, whether it be "stubbornness" or simply feeling like the other person shows no signs of trying to understand what they mean. I propose that better goals include: (i) understanding what people are saying; (ii) applying the concepts to some productive end. By "productive" I mean some forward progress in an empirical or mathematical sense, whether it be prediction or proof. So give up the battle. Why? Not because you are wrong. [2] Because "being right" about a definition is rather silly. We're talking about concepts being communicated by language and symbols. The goal is shared understanding of the concepts (which happens inside a brain), not merely enforcing a mapping of brain states to ink on a page (words) or vibrations in a physical medium (sound). [1]: Whether you win or lose, the distinction between (a: definition) and (b: properties) still exists. [2]: And not because you are "right" either. You can, at best, be consistent in your definitions and use them in useful ways.

nobodyandproud 2y ago
I’m saying that if confusion and annoyance has been the norm, then a quick translation for the general audience is in order. Even better would be a differentiating name, but I realize that’s unlikely.

xpe 2y ago

> The skull splitting is a natural reaction because this has nothing to do with differentiation. I recommend rephrasing that as "the definition (part "a" above) of arithmetic derivative is different than calculus definition of derivative." Do you see? Stating it this way reduces the war of words. Your point is made clear. [1] Then other people can say "Ok, sure, but don't you see how the properties (part "b" above) are the same? And isn't that interesting?" Think of this another way. Imagine an alternate history where the arithmetic derivative was discovered, named, and socialized first. Then imagine calculus came along later. If so, would calculus be wrong to use the same word, "derivative"? ... I won't answer that question because it is invalid. Better to dissolve the question [2]. My point? Let's try to shift away from historical battles over turf and terminology. Let's find ways to share insight. [1] Unless your intended point was: "how dare you use the word differently?" [2] https://www.lesswrong.com/posts/Mc6QcrsbH5NRXbCRX/dissolving-the-question https://www.lesswrong.com/posts/Mc6QcrsbH5NRXbCRX/dissolving...

nobodyandproud 2y ago
I have zero stakes in this other than stating perhaps it’s not the audience that’s at fault, when an in-circle term isn’t translated for a more general audience; and the audiences are consistently confused and annoyed. I woke up, noticed the title & blog (and references), went through the same tortured route and confusion, before stumbling on the fine print, and coming to same “oh, for crying out loud” reaction.
- xpe 2y ago
  Is it fair to say the definitional confusion bothered you more than the interesting aspects (I'm presuming that part "b" above is more interesting) pleased you? If I were to guess... I'd say you (and many people, including myself, often) are weary of people redefining words in a way that seems wasteful, distortive (such as 'stealing' words that formerly had clear technical meanings), purely commercial, or self-promotional. For me, at least, the intent of the redefinition matters. But I detect no self-interest or obvious neglect in the case of the arithmetic primes.