The Mathenauts is set in a near-future where "Brill-Cohen flight" has been discovered: the ability to take a ship into the raw mathematical space underlying reality. Ships, which look like a radio minus the casing, are crewed by eccentric high-flyer mathematicians, but their internal reality is held together by a "psychic ecology" of students with more mundane mindsets.
The ship, the Albrecht Dold, was a twelve-googol scout that Ed Goldwasser and I'd picked up cheap from the NYU Courant Institute. She wasn't the Princeton IAS Von-Neumann, with googolplex coils and a chapter of the DAR, and she wasn't one of those new toys you've been seeing for a rich man and his grandmother. Her coils were DNA molecules, and the psychosomatics were straight from the Brill Institute at Harvard. A sweet ship. For psychic ecology we'd gotten a bunch of kids from the Bronx College of the New York City University, commonsense types - business majors, engineers, pre-meds.Jimmy, the mathematician narrator, tells how there is a horrific in-flight accident when the "isomorphomechanism" (that keeps the internal reality stable) fails, affecting another member of the crew.
Instrument racks and chairs and books shrank and ballooned and twisted, and floor and ceiling vibrated with my breath. It was horrible. Ted Anderson was hanging in front of the immy, the isomorphomechanism, but he was in no shape to do anything. In fact, he was in no shape at all. His body was pulsing and shaking, so his hands were too big or too small to manipulate the controls, or his eyes shrank or blossomed.Jimmy repairs the fault, but Anderson, occupying the same space as the "immy", is rejected by the "commonsense circuits" and disappears. After an unsuccessful search of a ship and mathematical discussion of where Ted might have gone, there's a further scare when the ship's psychic ecology - students in a facsimile of a New York streetcar - breaks down.
The walls were firm, the straw seats scratchy and uncomfortable. The projectors showed we were just entering the 72nd Street stop. How real, how comforting! I slid the door open to rejoin Johnny and Ed. The subway riders saw me slip into freefall, and glimpsed the emptiness of vector space. Hell broke loose! The far side of the car bulged inward, the glass smashing and the metal groaning. The CUNYs had no compensation training!Johnny Pearl, the ship's "psychist", restores the ecology by singing a college anthem, and the crew finish their search and begin the tests the voyage is intended for. At that point, a ghostly Ted Anderson reappears and reveals an uncomfortable truth: that the raw mathematical space is the real universe. He gives Jimmy a glimpse of the creatures that inhabit it.
— I saw a set bubbling and whirling, then take purpose and structure to itself and become a group, generate a second-unity element, mount itself and become a group, generate a second unity element, mount itself and become a field, ringed by rings. Near it, a mature field, shot through with ideals, threw off a splitting field in a passion of growth, and became complex.Reality as humans know it, Anderson explains, is the creation of a "Great Race" in this mathematical universe, who lost their powers but left mathematics as a "seed" by which humans might regain the ability to inhabit that reality. He then disappears permanently, leaving his notebook.
— I saw the life of the matrices; the young ones sporting, adding and multiplying by a constant, the mature ones mating by composition: male and female make male, female and male make female — sex through anticommutivity! I saw them grow old, meeting false identities and loosing rows and columns into nullity.
— I saw a race of vectors, losing their universe to a newer race of tensors that conquered and humbled them.
The surviving characters go on to live their lives, dealing with that revelation in different ways: one marries and has 15 children; one gets religion and writes a book about Ted's views; and Jimmy, the narrator, concentrates on the business side of marketing Ted's ideas, multidimensional products that make a paradise of Earth.
Me, I'll stick to the Earth. The "real" planet is a garden spot now, and the girls are very lovely.The story, at one level, is a pastiche. Kagan wrote it when he was a mathematics student, and it abounds in punning use of mathematical terms and its characters' hardboiled mathematical exclamations ("Great Gauss!", "Holy Halmos!" etc). For that reason, I guess, it's often classed as humorous SF (as in the 1982 anthology Laughing space: funny science fiction). Its extensive mathematical content, and its excellent portrayal of the ethos of mathematicians, also accounts for its presence in Rudy Rucker's 1987 niche anthology Mathenauts: tales of mathematical wonder.
Ted Anderson was recorded lost in topological space. He wasn't the first, and he was far from the last. Twiddles circuits have burned out, DaughtAmsRevs have gone mad, and no doubt there have been some believers who have sought out the Great Race.
Nevertheless, I think it's an altogether better story than that, as an examples of "conceptual breakthrough" story (see PK Dick, Ubik and conceptual breakthrough and Breaking out of the game) that, despite the pastiche and generally positive outcome, leaves the reader with a continuing sense of unease that the assumptions of reality have been revealed hollow. It tackles the still-topical questions of Platonism and Neoplatonism: the idea that reality may have a mathematical structure, since mathematics describes it so well. An example of such theories is Antony Garrett Lisi's Exceptionally Simple Theory of Everything, which proposed that elementary particles correspond to the symmetries of a vast mathematical group called E8. All that aside, though, the strange and bold vision in The Mathenauts still makes it as fresh reading as when I first encountered it. It's a pity it's not online anywhere, but secondhand anthologies containing it aren't too hard to find.
Norman Kagan wrote several other mathematical stories, notably the 1964 Four Brands of Impossible, but didn't go on to an SF writing career. According to the blurb in Rudy Rucker's 1987 Mathenauts anthology:
What ever became of Norman Kagan anyway? He left math for film and still lives in Manhattan. He's written a number of books on cinema, and is currently involved in putting together a TV science news magazine to be called "Spacetime Continuum News." When I pressed having once written a mathematical mystery story called The Venn Data Vendetta — but he lost the only copy.I don't know what he's doing now (or even if he's still alive), but I assume these books - on the cinema of Stanley Kubrick, Oliver Stone, Robert Zemeckis and Roibert Altman- are by the same Norman Kagan.
Addendum: a personal note, via discussion with Felix Grant. The story is particularly memorable for me as one of the first SF stories I read. I got into the genre via my great-uncle Dennis, a nice autodidact polymath among my Wiltshire relatives who I regret not fully appreciating at the time - but I was only 11, so I guess it's excusable - and who gave me heaps of secondhand SF Book Club editions (including the brilliant The Hole in the Zero) and copies of Analog magazine. On those grounds, this post is dedicated to my late and excellent Uncle Den.