7 ms·
I find the following remark by Rota interesting: Insofar as the Laplace transform goes, two radically different uses of the word “function” are dangerously con
by ogogmad 4y ago
I find the following remark by Rota interesting:
Insofar as the Laplace transform goes, two radically different uses of the word “function”
are dangerously confused with each other. The first is the ordinary notion of function as a
something that has a graph. The second is the radically different notion of function as density, whether mass density or probability density. For the sake of the argument, let us agree
to call this second kind of function “density function.” Professional mathematicians have
avoided facing up to density functions by a variety of escapes, such as Stieltjes integrals,
measures, etc. But the fact is that the current notation for density functions in physics and
engineering is provably superior, and we had better face up to it squarely
I've suggested to mathematicians that the notation
⌠
| f(x) dμ(x)
⌡
in measure theory where μ is a measure should be replaced with
⌠
| f(x)⋅μ(x) dx
⌡
where the measure μ is treated as a density function. What I've found is a mixture of either not understanding the logic, or the response "What's the point?" So it's interesting seeing Gian-Carlo Rota suggest the same idea AFAICT.
As a bonus, the Radon-Nikodym "derivative" would be written as μ/λ instead of dμ/dλ. The RN derivative then doesn't appear like a derivative - it looks simply like division.
- iamcreasy 4y agoWhat is wrong with the two difference usages of a function? If you treat function as a system that produces an output given an input - does it matter if it is a graph or a probability distribution?
- deleted 4y ago[deleted]
- ogogmad 4y ago> If you treat function as a system that produces an output given an input - does it matter if it is a graph or a probability distribution? Because that's wrong. For example, take the Dirac delta function (point mass). You can only integrate it, not evaluate it. GCR was saying that it's OK to use the same notation for both kinds of function. Mathematicians use separate notations, which is ugly and awkward. Engineers and physicists conflate these objects in their terminology, which is misleading. The right answer is to use the same notation but explain there's a difference.
- thrown_22 4y agoNow I'm thinking what the type of the Dirac delta function is. I'm leaning to something like: (X (R -> R) ((R -> R) -> (R -> R))) -> R where we are given a function (R -> R) on which it acts and a linear operation (R -> R) -> (R -> R), like integration, which acts on it and gives us a single real number as the answer.
- johnbcoughlin 4y agoThis is basically correct, the theory of distributions can be presented in terms of linear functionals. A functional is a map that takes a function and returns a scalar. I'm not sure how to encode linearity in your type system though...
- thrown_22 4y ago>I'm not sure how to encode linearity in your type system though... You don't really, the same way that a regular function R->R has no specific type information that tells you ti's linear or not, e.g. both x and sin(x^2) have the same type.
- patrick451 4y agoWhat do you mean you can't evaluate it? It evaluates to zero everywhere but the origin.
- WoahNoun 4y agoThe rigorous definition of dirac delta is a function that doesn't take points as inputs. It's a function that takes in sets and then integrates over that set to product a value.
- deleted 4y ago[deleted]
- BeetleB 4y agoThis is what is often taught in engineering/physics, but is not correct. One physics professor tried to explain it by pointing out that it only has meaning under an integral sign (not sure it's correct). The other obvious problem is its value at the origin. Infinity as a value is not something mathematicians like - at least not the way it is used here. And integrating it to get 1 makes no sense mathematically merely by setting its value to infinity.
- asdf_snar 4y agoI think I am one of these mathematicians that doesn't understand the logic. How can I write μ(x)dx instead of μ(dx) without risking the confusion that dx is Lebesgue measure? You may have explained this in your other reply, but I don't quite follow. P.S. Beautiful integral signs.
- politician 4y agoPlease excuse my imposition, as I am a humble programmer who spends his days adding and subtracting 1, and not a mathematician. Yet, this discussion of the confusion and potential confusion of misinterpreting notation strikes me as something that has long (well, in the sense of programming) been solved in my area with type systems and syntax highlighting. Do mathematicians not have these tools?
- Sharlin 4y agoThere is no syntax highlighting on the blackboard. Math notation is not designed. It has haphazardly evolved over centuries. It is not rigorous even though math itself is (attempts to be) rigorous, it is a language as imperfect as its users. But it does its job well enough.
- ABeeSea 4y agoThey do! It’s called abstract algebra and it’s very similar to type theory in a lot of ways. But to get to the rigorous mathematician definition of manipulating dx and dy, it requires a large amount of the machinery from abstract algebra that’s hard to quickly absorb or explain.
- ogogmad 4y agoI'm suggesting maybe writing the Lebesgue measure as 1 so the Lebesgue integral of a function f becomes ⌠ | f(x)⋅1 dx ⌡ The logic is that the Lebesgue measure is a density which is everywhere equal to 1. Given a measurable space over \mathbb R^n, I think there is only one such measure. Another example is that δ(x) in ⌠ | f(x)⋅δ(x) dx ⌡ represents the Dirac measure. For producing this ASCII art, I use Sympy. I write for instance pprint(Integral(f(x) * delta(x)))