I've just become aware of the concept of "Steampunk", which is an alternate universe in which Babbage-style steam computers actually worked in the 19th Century & thereby changed the world. (Yes, I know; I'm Rip Van Winkling myself; no snarky remarks!) Steampunk today means an exceedingly retro anachronistic concept -- e.g., a spring-wound software-defined digital radio -- so it doesn't actually require steam, per se. https://en.wikipedia.org/wiki/Steampunk Prior to the discovery of the Antikythera Mechanism, any mention of Archimedes in conjunction with geared wheels might have initially seemed "steampunk", but modern X-ray & gamma-ray images and recreations have proved it to be a remarkably sophisticated analog computer. https://en.wikipedia.org/wiki/Antikythera_mechanism Probably one of the coolest "steampunk" exercises of the 20th Century was economist William Phillips's 1949 "MONIC" water-based analog computer which represented the macro economy of the UK (called by one wag "hydraulic Keynesianism"). (Phillips could easily have been the inspiration for the TV series MacGyver -- parodied by SNL's "Gruber" -- as Phillips built all sorts of stuff while a Japanese POW -- including a radio in his shoe.) https://en.wikipedia.org/wiki/William_Phillips_%28economist%29 https://en.wikipedia.org/wiki/MONIAC I would propose that "steampunk mathematics" would be math that Cayley or Hamilton would not only understand, but might have written themselves. No Bourbaki, no axioms of choice, no Goedel (or at least Goedel only in an analog computer world). No Hilbert (especially no Hilbert's Tenth Problem), no quantum theory, very little topology (except early knot theory, thanks to Cayley, etc.). Unfortunately, my emphasis on 19th Century technology leaves out things like relay computers & Strowger switches, which I think would have been understood by, and really appealed to, the Victorians, so long as they don't include vacuum tubes or germanium diodes. I think that a lot of computer graphics technology might have appealed to Cayley & Hamilton -- particularly the use of Hamilton's quaternions for "in-betweening". It is conceivable that _fractal_ graphics, such as some of Gosper's cool pix, would have been understood by Cayley, at least. Cayley was perhaps the first to have notice the chaotic behavior of Newton's method for finding polynomial roots in the complex plane. Computer algebra systems would have been very appealing to Cayley & Hamilton, as the 19th Century mathematicians all seem to have waded through prodigious amounts of symbolic calculations. I'd be interested in any other proposals for "steampunk" mathematics.