Castles in the Air
Posted by Mike Shulman
The most recent issue of the Notices includes a review by Slava Gerovitch of a book by Amir Alexander called Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. As the reviewer presents it, one of the main points of the book is that science was advanced the most by the people who studied and worked with infinitesimals despite their apparent formal inconsistency. The following quote is from the end of the review:
If… maintaining the appearance of infallibility becomes more important than exploration of new ideas, mathematics loses its creative spirit and turns into a storage of theorems. Innovation often grows out of outlandish ideas, but to make them acceptable one needs a different cultural image of mathematics — not a perfectly polished pyramid of knowledge, but a freely growing tree with tangled branches.
The reviewer makes parallels to more recent situations such as quantum field theory and string theory, where the formal mathematical justification may be lacking but the physical theory is meaningful, fruitful, and made correct predictions, even for pure mathematics. However, I couldn’t help thinking of recent examples entirely within pure mathematics as well, and particularly in some fields of interest around here.
Here are a few; feel free to suggest others in the comments (or to take issue with mine).
Informal arguments in higher category theory. For example, Lurie’s original paper On infinity topoi lacked a rigorous formal foundation, but contained many important insights. Because quasicategories had already been invented, he was able to make the ideas rigorous in reasonably short order; but I think it’s fair to say the price is a minefield of technical lemmas. Nowadays one finds people wanting to say “we work with -categories model-independently” to avoid all the technicalities, but it’s unclear whether this quite makes sense. (Although I have some hope now that a formal language closer to the informal one may come out of the Riehl-Verity theory of -cosmoi.)
String diagrams for monoidal categories. Joyal and Street’s original paper “The geometry of tensor calculus” carefully defined string diagrams as topological graphs and proved that any labeled string diagram could be interpreted in a monoidal category. But since then, string diagrams have proven so useful that many people have invented variants of them that apply to many different kinds of monoidal categories, and in many (perhaps most) cases they proceed to use them without a similar justifying theorem. Kate and I proved the justifying theorem for our string diagrams for bicategories with shadows, but we didn’t even try it with our string diagrams for monoidal fibrations.
Combining higher category with string diagrams, we have the recent “graphical proof assistant” Globular, which formally works with a certain kind of semistrict -category for . It’s known that semistrict 3-categories (Gray-categories) suffice to model all weak 3-categories, but no such theorem is yet known for 4-categories. So officially, doing a proof about 4-categories in Globular tells you nothing more than that it’s true about semistrict 4-categories, and I suspect that few naturally-ocurring 4-categories are naturally semistrict. However, such an argument clearly has meaning and applicability much more generally.
And, of course, there is homotopy type theory. Plenty of it is completely rigorous, of course (and even formally verified in a computer), but I’m thinking particularly of its conjectural higher-categorical semantics. Pretty much everyone agrees that HoTT should be an internal language for -topoi, but with present technology this depends on an initiality theorem for models of type theories in general that is universally believed to be true but is very fiddly to prove correctly and has only been written down carefully in one special case. Moreover, even granting the initiality theorem there are various slight mismatches between the formal theories in current use and what we can construct in higher toposes to model them, e.g. the universes are not strict enough and the HITs are too big. Nevertheless, this relationship has been very fruitful to both sides of the subject already (the type theory and the category theory).
The title of this post is a reference to a classic remark by Thoreau:
“If you have built castles in the air, your work need not be lost; that is where they should be. Now put the foundations under them.”
Re: Castles in the Air
I know you were only using the NAMS article as a springboard for the body of your post, but all those parts – which I assume to be the reviewer’s paraphrase of things in the book – which had a neat opposition between the purported Jesuit conformism and the Royal Society’s supposed pluralism really caused me to make Marge Simpson noises… Especially passages such as
… pitted a champion of social order against an advocate of intellectual freedom…
I think my anachronism klaxon just went off so loudly that I might be temporarily deafened :(
Anyway, I don’t want the thread to get derailed before it even starts, but just had to get that one off my chest. I see from Wikipedia that Alexander is a professional historian with training to match, so hopefully the book takes a more cautious/nuanced line (see here for a Times HE review).