For a more current example of a mathematical technique that preceded formalization by a considerable amount, consider renormalization. Particularly renormalization over a calculation that takes place over Feynman diagrams.<p>For decades physicists were happily using this to predict experiment, while mathematicians were tearing their hair out trying to make some formal sense of this, even if only in a limited context. I'd have to do some poking around to find out whether mathematicians are happy about it yet, even though the idea is older than I am.
Related:<p><i>Heaviside’s Operator Calculus</i> - <a href="https://news.ycombinator.com/item?id=569934">https://news.ycombinator.com/item?id=569934</a> - April 2009 (6 comments)<p>This is also interesting: <a href="https://www.johndcook.com/blog/2022/10/12/operational-calculus/" rel="nofollow">https://www.johndcook.com/blog/2022/10/12/operational-calcul...</a> (via <a href="https://news.ycombinator.com/item?id=33179121">https://news.ycombinator.com/item?id=33179121</a>, but no comments there)
There are several similar variants of different kinds of math that make just as much sense as mainstream methods to me. It all feels very arbitrary.<p>I think that's what got me into software. If we're just making shit up either way, then useful artifacts is a nice bonus.
I always loved that the derivative of the heavyside operator is equivalent to the dirac delta operator. The idea of impulse and how to apply that to a system is such a unique and useful unlock in E&M and has such a nice analog of connecting the circuit.<p>One of those things that made it click for me that math truly is defined rules of operations over definitions and could be constructed as to be useful for us, and not just a handed down pure concept. We need to model this very specific thing, here's an operator for it.
Gosh this takes me back to my EEE degree. Very difficult to understand at first, way to abstract if you have not seen the electromagnetic phenomena play out in real life and you are not well versed in engineering mathematics.
Beautiful, I love the irreverence. Reminds me a lot of the "umbral calculus" for computing combinatorial identities. It proceeds in much the same way - deliberately make an (unjustified) abuse of notation, work with it at face value regardless, and reap the rewards...<p>...or spend hours debugging the mess you've made if it doesn't work =P