Was the convolution product invented or discovered?

Question

In analysis textbooks and classes I sometimes see the convolution product introduced as a sort of artificial tool - just a clever method for constructing functions that somebody smart came up with at some point.

My question is, is this an accurate description of how it first came to light historically? That is, was the convolution product:

1. Discovered, in the sense that it arose naturally in a mathematics problem, given to the mathematician by nature, so to speak.

2. Invented, somebody went out of their way to develop a construction that would have certain properties.

To illustrate the difference between (1) and (2), consider the complex numbers (discovered accidentally as solutions to equations) and the quaternions (invented after years of painstaking labour specifically in order to suit some purpose).

Answer 1

Neither was discovered nor invented. A particular form of the convolution operation arised "naturally" in the solution of diferential equations by certain definite integrals, as described on Section 1049 of Euler's book "Institutionum Calculi Integralis (Vol. 2)."

A more complete history of convolution product may be read on the paper "A History of the Convolution Operation" (http://pulse.embs.org/january-2015/history-convolution-operation/).

Answer 2

This is not a very historically oriented answer, but it grew much too long for a comment and I hope it sheds some light on the issue.

I'd find it highly surprising if the convolution was not what you would call 'discovered', since it arises extremely naturally in probability theory. In fact, I 'discovered' it in this context a few years ago, even though I was already aware of what a convolution is in other contexts (notably analysis).

Consider a variable $X$ and a variable $Y$, with continuous probability distributions $\rho_X$ and $\rho_Y$ respectively. Then, the probability distribution $\rho_{X+Y}$ for $X+Y$ is given by $\rho_{X+Y}=\rho_X\star \rho_Y$. This is quite obvious from the fact that, if $X+Y=Z$ and $X=A$, then $Y=Z-A$. Thus, $$ \rho_{X+Y}(Z)=\int_{-\infty}^\infty \rho_X(A)\rho_Y(Z-A)\ \mathrm{d}A$$ and we find the convolution to be one of the most basic tools one needs in the context of probability theory.

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Historical edit:

Through wikipedia, one finds this source, which suggests the convolution actually was used first in the context of analysis, specifically in a work by D'Alembert, while deriving Taylor's theorem. In this case, D'Alembert actually only used the particular case where one of the two functions is just $1$, and he certainly doesn't do anything to develop the general theory that has, by now, been established.

Answer 3

This question does not belong to history of sciences and math in the strict sense. This is a question about philosophy of mathematics and it has nothing to do with convolution. Convolution is just an example of a mathematical object.

You can ask about any mathematical object whether it was invented or discovered. And the answer depends on the philosophical views of the person who answers.

Therefore, there cannot be an objective answer to such a question.

Personally, like most mathematicians, I am a Platonist, and I believe that mathematics exists independently of us, and we discover it.

Some others ridicule this point of view. But there is no way to reach a consensus, like on any philosophical question. So this question has no answer.

Convolution is as "natural" as addition or multiplication. So if addition and multiplication (say, of integers) "exists" in the real world, and we discover it, then the same can be said about convolution.

But of course there are people who will say that addition and multiplication were invented. (Did not exist before humans started to add and multiply). And history shows that there is no reconciliation between these two sets of people.

EDIT. The answer of Danu which was simultaneous with my answer gives an example how convolution arises in probability theory. But convolution arises everywhere. If you think for example of multiplication of decimal numbers, this is nothing but a special case of convolution.

As I said, from my point of view all this "exists" and we discover it. But some people (mostly philosophers) will say that we "invented" probability and multiplication, and they "did not exist" before we invented them:-)