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"As a matter of fact, the feeling of witchcraft that has hovered over umbral calculus is probably what kept it from dying altogether. ... At long last, it was

"As a matter of fact, the feeling of witchcraft that has hovered over umbral calculus is probably what kept it from dying altogether. ... At long last, it was realized that umbral calculus could be made entirely rigorous by using the language of Hopf algebras, and this was done in a lengthy treatment. However, although the notation of Hopf algebra satisfied the most ardent advocate of spic-and-span rigor, the translation of 'classical' umbral calculus into the newly found rigorous language made the method altogether unwieldy and unmanageable. Not only was the eerie feeling of witchcraft lost in the translation, but, after such a translation, the use of calculus to simplify computation and sharpen our intuition was lost by the wayside." (Gian-Carlo Rota, 1994)

foxes 4y ago
Why does something have to be portrayed as spooky and magical. If it works it works, then it’s our job to explain clearly why. That quote seems strange. I have not seen the explanation in terms of Hopf algebras but if the translation makes it unwieldy that seems like it’s not a very good language to use to describe it, Most of these umbral calculations seem pretty clean.
- goldenkey 4y ago
  It was spooky and magical because it worked so well and it was based on symbolic transformations -- not mathematical ones. It'd be liking having calculus' basic derivative rules without their proofs. Without the rigorous backing, Umbral calculus was used for research and exploration but then additionally work had to be done to prove the results. Now, with the rigorous backing, the results stand alone - as the umbral transformations have been proven to be rigorously valid. I am not too knowledgeable on the proofs of validity of Umbral calculus but my guess would be that only part of the vast library of umbral tricks, hacks, and symbolic bashing have been formalized - so the rigorous subset of provably correct transformations is likely more limited and thus cumbersome.

foxes 4y ago

Why does something have to be portrayed as spooky and magical. If it works it works, then it’s our job to explain clearly why. That quote seems strange. I have not seen the explanation in terms of Hopf algebras but if the translation makes it unwieldy that seems like it’s not a very good language to use to describe it, Most of these umbral calculations seem pretty clean.

goldenkey 4y ago
It was spooky and magical because it worked so well and it was based on symbolic transformations -- not mathematical ones. It'd be liking having calculus' basic derivative rules without their proofs. Without the rigorous backing, Umbral calculus was used for research and exploration but then additionally work had to be done to prove the results. Now, with the rigorous backing, the results stand alone - as the umbral transformations have been proven to be rigorously valid. I am not too knowledgeable on the proofs of validity of Umbral calculus but my guess would be that only part of the vast library of umbral tricks, hacks, and symbolic bashing have been formalized - so the rigorous subset of provably correct transformations is likely more limited and thus cumbersome.