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May 20, 2016

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 (,1)(\infty,1)-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 \infty-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 nn-category for n4n\le 4. 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 (,1)(\infty,1)-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.”

Posted at May 20, 2016 3:39 AM UTC

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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).

Posted by: Yemon Choi on May 21, 2016 2:05 AM | Permalink | Reply to this

Re: Castles in the Air

I quite enjoyed reading this book, and yes, the reviewer is far more hysterical than the book is. There is even one serious typo in the review “The Jesuits even went so far as to engineer the dissolution of a small monastic order, the Jesuits, which had sheltered Cavalieri and Stefano degli Angeli”. This should be “a small monastic order, the Jesuats”. Slightly different name, totally different organisation.

I read somewhere that Newton was forced to write his Principia in a geometric style, without using calculus, in order to appease the mathematicians of the day. This makes the Principia quite a big complicated work. Reading Alexander’s book helps to illuminate the environment Newton was working in, and also that the ideas behind calculus were already growing before Newton & Leibniz came along.

Posted by: Simon Burton on May 21, 2016 1:23 PM | Permalink | Reply to this

Re: Castles in the Air

Thanks for explaining that typo. I had been scratching my head over the idea of the Jesuits going so far as to engineer their own dissolution.

Posted by: John Baez on May 22, 2016 6:19 PM | Permalink | Reply to this

Re: Castles in the Air

The typo is indeed ridiculous. Needless to say, in the original review sent to the journal, as well as in the page proofs returned to me, the word “Jesuats” was spelled correctly.

Posted by: Slava Gerovitch on May 23, 2016 1:16 AM | Permalink | Reply to this

Re: Castles in the Air

the word “Jesuats” was spelled correctly.

Google searches tend to give “Jesuates” or “Jesuati” as the correct spelling, but that is no excuse for changing “Jesuats” to “Jesuits” at the last moment even if “Jesuats” wasn’t in a list of known words.

Posted by: RodMcGuire on May 23, 2016 3:07 AM | Permalink | Reply to this

Re: Castles in the Air

I find plenty of references to Jesuats in a Google search.

On another topic: the example of quantum field theory or more specifically of Feynman integrals also came up in an interview of Manin, who described it with the arresting image “Eiffel tower hanging in the air with no foundation”; as recounted in this Café post.

Posted by: Todd Trimble on May 23, 2016 5:28 AM | Permalink | Reply to this

Re: Castles in the Air

what wikipedia tells of them suggests the Jesuates as-such cannot have been involved in the suppresion of the Society of Jesus, as the former order had itself been suppresed more than fifty years earlier. There can’t even have been many devoted former members around… Is it just a funny thing that neither order seems to have been monastic? (certainly not the SJ, and not evidently the Ap.Cl.S.Jerome)

Or maybe there’s a third group that might be involved?

Posted by: Jesse C. McKeown on May 25, 2016 1:48 AM | Permalink | Reply to this

Re: Castles in the Air


  1. I was sure I was responding to an earlier comment.
  2. I also seem to have got the tenses, subjects and objects mixed up, so that the chronology actually works very well; apologies to the historians.
Posted by: Jesse C. McKeown on May 25, 2016 1:51 PM | Permalink | Reply to this

Re: Castles in the Air

Nor could there ever be any excuse for making a change like that after the author has approved the page proofs.

Posted by: Mike Shulman on May 23, 2016 7:51 AM | Permalink | Reply to this

Re: Castles in the Air

That’s interesting, and it makes me want to read the book - I’ve thought it a little strange that in the standard potted histories its usually claimed that calculus was invented roughly around the same time by Newton & Liebniz independently; the coincidence in timing suggested, at least to me, that there was some prior art or circle of ideas that they were both dependent on. Its nice to see that my hunch wasn’t misplaced.

Can a similar story told about gravity? If memory serves, the story that I know is of Halley (or was it Hooke) visiting Newton asking what locus would an object travel in if it were attracted to the sun by an inverse square law, which suggests too that the ideas and notions about gravity were in the air, so to speak.

Posted by: Mozibur Rahman Ullah on July 4, 2016 3:12 PM | Permalink | Reply to this

Re: Castles in the Air

Yes, well. As you said, let’s not get sidetracked.

Posted by: Mike Shulman on May 21, 2016 2:07 AM | Permalink | Reply to this

Re: Castles in the Air

There’s an interesting connection between two castles in the air:

1) the very fruitful but so far mostly nonrigorous path integrals in quantum field theory, and

2) the cobordism hypothesis, which was formulated before the necessary concept of “nn-categories with duals” had been precisely defined.

Namely: Atiyah’s axioms for a TQFT are nicely motivated by calculations that should be true if path integrals make sense. These axioms wind up revealing, and exploiting, a connection between cobordism categories and categories of finite-dimensional Hilbert spaces: namely, they’re both symmetric monoidal categories with duals.

To formalize more sophisticated path integral calculations one wants to chop manifolds up into pieces with corners. This leads to a concept of ‘extended TQFT’, which would be algebraically beautiful if we had a purely algebraic description of the nn-category of points, cobordisms between points, cobordisms between cobordisms between points, and so on. That description is the cobordism hypothesis.

The cobordism hypothesis is by now well on the way to being proved, though not quite there yet. Jacob Lurie and others needed to develop an impressively large amount of higher category theory to get this far.

But the most interesting thing, to my mind, is how this simultaneously provides a foundation for both 1) some ideas about path integrals, and 2) some ideas about a purely algebraic approach to differential topology.

It would be very nice if further developments along these lines could help mathematicians understand the path integrals that physicists find most practical, namely those that arise in nontopological quantum field theories like the Standard Model.

Posted by: John Baez on May 21, 2016 6:46 AM | Permalink | Reply to this

Re: Castles in the Air

Not my field of course, but I’ve gleaned from Urs something of the idea that boundaries and defects can generate geometric, i.e., non-topological, forms of QFT, such as this from ‘Differential cohomology in a cohesive \infty-topos’:

This quantization process is particularly interesting for the boundary prequantum field theories discussed in, where it yields quantization of geometric (non-topological) field theories as the “holographic” boundaries of topological field theories in one dimension higher.

But perhaps there’s still a way to go.

Posted by: David Corfield on May 21, 2016 12:07 PM | Permalink | Reply to this

Re: Castles in the Air

There’s a long way to go, since none of the quantum field theories physicists like best (for example, the Standard Model) have been given a rigorous formulation. In the current approach — that is, the one people have been working on since the 1940s — this seems to require significant advances in analysis. However, it’s possible that better ways of looking at the problem could lubricate the solution. There’s a million dollars waiting for someone who figures this out.

Posted by: John Baez on May 22, 2016 6:22 PM | Permalink | Reply to this

Re: Castles in the Air

If you seek examples of mathematicians who cherish creativity even at the expense of rigor, it’s good to look at the more rebellious responses to Jaffe and Quinn’s controversial paper on “theoretical mathematics” in the Bulletin of the AMS. This paper warned of the dangers of mathematics becoming less rigorous under the influence of physics, and tried to set up a system for preserving the purity of mathematics. They reviewed the history of mathematics in a collection of “success stories” and “cautionary tales”.

Surprisingly, Poincaré’s work on algebraic topology falls into the latter category. Apparently initiating the subject, inventing the fundamental group and finding the first homology sphere was not enough:

In spite of its obvious importance it took fifteen or twenty years for real development to begin. Dieudonné expresses surprise at this slow start [D, p. 36], but it seems an almost inevitable corollary of how it began: Poincaré claimed too much, proved too little, and his “reckless” methods could not be imitated. The result was a dead area which had to be sorted out before it could take off.

They wag a similar finger of reproof at Dennis Sullivan, René Thom, William Thurston, Edward Witten and Andrej Kolmogorov. These gentlemen were insufficiently careful, and thus held back progress.

Naturally, this assessment irritated some mathematicians. Here are a few amusing quotes from their responses.

Atiyah wrote:

I find myself agreeing with much of the detail of the Jaffe-Quinn argument, especially the importance of distinguishing between results based on rigorous proofs and those which have a heuristic basis. Overall, however, I rebel against their general tone and attitude which appears too authoritarian.

My fundamental objection is that Jaffe and Quinn present a sanitized view of mathematics which condemns the subject to an arthritic old age. They see an inexorable increase in standards of rigour and are embarrassed by earlier periods of sloppy reasoning. But if mathematics is to rejuvenate itself and break exciting new ground it will have to allow for the exploration of new ideas and techniques which, in their creative phase, are likely to be as dubious as in some of the great eras of the past. Perhaps we now have high standards of proof to aim at but, in the early stages of new developments, we must be prepared to act in more buccaneering style.

(I remember a conference at which Saunders Mac Lane complained about this paragraph, and sang a song about pirates, maybe from Gilbert and Sullivan.)

Karen Uhlenbeck wrote:

I feel that the article makes exactly the wrong point about influence on young mathematicians. I well remember that as an undergraduate I was initiated into the mysteries of distributions by being told by a graduate student that physicists had used them, but understood nothing important about them. Only an innovative and brilliant mathematician like the idolized Laurent Schwartz could make sense of the physicists’ nonsense. Unfortunately, this attitude was reinforced during my formative years by both mathematicians and physicists. Mathematicians seemed to think that physicists did not do physics “right”, while physicists thought of mathematicians as worthless insects. Only after taking part in the mathematical development of gauge theory could I comprehend the essential importance of outside ideas in mathematics and the contrary possibility of mathematical language being of real use outside the discipline itself.

I find it difficult to convince students—who are often attracted into mathematics for the same abstract beauty and certainty that brought me here—of the value of the messy, concrete, and specific point of view of possibility and example. In my opinion, more mathematicians stifle for lack of breadth than are mortally stabbed by the opposing sword of rigor.

Benoit Mandelbrot quite tartly wrote:

Unfortunately, hard times sharpen hard feelings; witness the discussion (to be referred to as JQ) that Arthur Jaffe and Frank Quinn have devoted to diverse tribal and territorial issues that readers of this Bulletin usually leave to private gatherings. Those readers — you! — like to be called simply “mathematicians”. But this term will not do here, because my comment is ultimately founded on the following conviction:

For its own good and that of the sciences, it is critical that mathematics should belong to no self-appointed group; no one has, or should pretend to, the authority of ruling its use.

Therefore, my comment needs a focussed term to denote the typical members of the AMS. Since it is headquartered on Charles Street, I propose (in this comment, and never again) to use “Charles mathematicians”.


Many Charles mathematicians are offended that a few major players have been well rewarded for “passing” statements as if they were proven theorems. The fear is expressed, that the person who will actually prove these assertions will be deprived of credit. I think this is an empty fear because the AMS already has countless ways of rewarding or shunning whomever it chooses.

Even more acidly, René Thom wrote:

[…] I think the proposal of the authors, to establish a “label” for mathematical papers with regard to their rigor and completeness, is an excellent idea.

Rigor is a Latin word. We think of rigor mortis, the rigidity of a corpse. I would classify the (would-be) mathematical papers under three labels:

1) a crib, or baby’s cradle, denoting “live mathematics”, allowing change, clarification, completing of proofs, objection, refutation.

2) the tombstone cross. Authors pretending to full rigor, claiming eternal validity, may use this symbol as freely as they wish. This kind of work would constitute “graveyard mathematics”.

3) the temple. This would be a label delivered by an external authority, the “body of high priests”. This body could initially be made up of the editors in chief of the “core” papers as suggested by Jaffe-Quinn.

Posted by: John Baez on May 21, 2016 7:10 AM | Permalink | Reply to this

Re: Castles in the Air

John wrote:

(I remember a conference at which Saunders Mac Lane complained about this paragraph, and sang a song about pirates, maybe from Gilbert and Sullivan.)

The song was undoubtedly “The Ballad of Captain Kidd”, a cautionary tale of the 17th-century privateer William Kidd who was executed for piracy (and murder), which Mac Lane included in his defence of proof “Despite physicists, proof is essential in mathematics”. Nevertheless, one may be tempted to interpret the following sentiment of Gilbert and Sullivan’s Pirate King in the context of this debate:

Although our dark career sometimes involves the crime of stealing,
We rather think that we’re not altogether void of feeling.
Although we live by strife, we’re always sorry to begin it:
For what, we ask, is life without a touch of Poetry in it?

Posted by: Alexander Campbell on May 21, 2016 2:04 PM | Permalink | Reply to this

Re: Castles in the Air

Thanks! Gilbert and Sullivan’s Pirate King makes a strong argument for his side of the debate, but Mac Lane was unforgiving of mathematical pirates.

A free version of Mac Lane’s paper is currently available here, and in a buccaneering style I’ll quote some:

But the leading response to Jaffe–Quinn came from the person who is undoubtedly leader of the mathematician’s current fruitful interaction with physics, Sir Michael Atiyah, Fields medalist and now Master of Trinity College, Cambridge, who thunders (1994): “Perhaps we now have high standards of proof to aim at, but, in the early stages of new developments, we must be prepared to act in a more buccaneering style”. (I have never observed Sir Michael in such a style.)

However, a buccaneer is a pirate, and a pirate is often engaged in stealing. There may be such mathematicians now. Moreover, America can clearly recall the days when such British buccaneers operated off our coasts. One such was the notorious Captain William Kidd; before the bibliography we append a poetic summary of the style of his doing. We do not need such styles in mathematics.

He appends the cautionary tale, The Ballad of Captain of Kidd:

My name was William Kidd, when I sailed, when I sailed,

My name was William Kidd, when I sailed,

My name was William Kidd; God’s laws I did forbid,

And so wickedly I did, when I sailed.

My parents taught me well, when I sailed, when I sailed,

My parents taught me well, when I sailed,

My parents taught me well, to shun the gates of hell,

But against them I rebelled, when I sailed.

I murdered William Moore, as I sailed, as I sailed,

I murdered William Moore, as I sailed,

I murdered William Moore, and laid him in his gore,

Not many leagues from shore, as I sailed.

I spied three ships from Spain, as I sailed, as I sailed,

I spied three ships from Spain, as I sailed,

I spied three ships from Spain, I looted them for gain,

Til most of them were slain, as I sailed.

I’d ninety bars of gold, as I sailed, as I sailed,

I’d ninety bars of gold, as I sailed,

I’d ninety bars of gold and dollars manifold,

With riches uncontrolled, as I sailed.

Farewell, the raging main, I must die, I must die,

Farewell, the raging main, I must die,

Farewell, the raging main, to Turkey, France and Spain,

I shall never see you again, for I must die.

To the Execution Dock, I must go, I must go,

To the Execution Dock, I must go,

To the Execution Dock, while many thousands flock,

But I must bear the shock, and must die.

Take a warning now by me, for I must die, I must die,

Take a warning now by me, for I must die,

Take a warning now by me, and shun bad company,

Lest you come to hell with me, for I must die.

Posted by: John Baez on May 22, 2016 6:37 PM | Permalink | Reply to this

Re: Castles in the Air

If we look past the hyperbole and rhetoric about pirates, I wonder how much actual disagreement there is. Atiyah admits “we now have high standards of proof to aim at” while Mac Lane “does not deny the many preliminary stages of insight, experiment, speculation or conjecture which can lead to mathematics.” Are there concrete modern examples that the two sides of this debate disagree about?

Posted by: Mike Shulman on May 22, 2016 10:32 PM | Permalink | Reply to this

Re: Castles in the Air

Mike’s question is a good one.

I wonder how much of this debate was driven by specific personal grievances, or perhaps by some people simply finding the operating style of others to be distasteful or annoying. Maybe I’m cynical, but it wouldn’t be the first time that personal matters have been dressed up as high-flown academic discourse.

In any case, I’m now reading that AMS debate to see what actual substance I can extract.

Posted by: Tom Leinster on May 23, 2016 9:22 PM | Permalink | Reply to this

Re: Castles in the Air

Here’s what Jaffe and Quinn, in their original article that set it all off, list as “the problems”. They use the word theoretical to mean “speculative and intuitive work”:

(1) Theoretical work, if taken too far [my emphasis], goes astray because it lacks the feedback and corrections provided by rigorous proof.

(2) Further work is discouraged and confused by uncertainty about which parts are reliable.

(3) A dead area is often created when full credit is claimed by vigorous theorizers: there is little incentive for cleaning up the debris that blocks further progress.

(4) Students and young researchers are misled.

I think these are interesting points. They seem entirely uncontroversial if interpreted with a cool head — but, of course, everything depends on what constitutes “too far”.

Posted by: Tom Leinster on May 23, 2016 9:52 PM | Permalink | Reply to this

Re: Castles in the Air

I wonder whether the responses would have been so acrimonious if Jaffe and Quinn hadn’t pointed fingers of blame at specific mathematicians. It’s not surprising that the people whose names were named, like Thom, would be emotionally inclined to “fight back”.

I don’t know enough about any of the particular instances that J+Q pointed at to have an opinion about them, but I can say from personal experience that these problems have not been entirely absent from the story of HoTT. Part of the problem there was, I think, inherited from a tendency in some type-theoretic circles to regard the “initiality theorem” as known in general, since it was “clearly” a straightforward extension of the cases that had been done carefully. Perhaps this in turn was partially due to influence from computer science; I can’t really say for sure. I give Voevodsky a lot of credit for emphasizing the unpopular point that this theorem actually still needs to be proven; but at the same time I don’t think we need to halt all progress until it is, as long as we are clear about when we depend on it. I also take some blame on myself for claiming that Peter Lumsdaine and I had a general semantic construction of HITs for a long time but not actually getting around to publishing it — but fortunately, that doesn’t seem to be deterring other people from pursuing their own approaches to HIT semantics.

I do think Mandelbrot is wrong and there is a real danger of credit accruing to the one who makes a bold conjecture at the expense of the one who does the work to prove it. After something has been known as “Alice’s conjecture” for many years, it’s unlikely to become widely knows as “Eve’s theorem” even after Eve proves it. It would be nice if there were a more concise way to say “The statement conjectured by Alice and proven by Eve.”

Posted by: Mike Shulman on May 23, 2016 10:36 PM | Permalink | Reply to this

Re: Castles in the Air

How about ‘the Alice-Eve theorem’!

Posted by: Jamie Vicary on May 25, 2016 1:49 PM | Permalink | Reply to this

Re: Castles in the Air

To me “the Alice-Eve theorem” implies that Alice and Eve proved the theorem working together, or perhaps that they proved it independently — not that one of them conjectured it and the other proved it.

Posted by: Mike Shulman on May 25, 2016 2:26 PM | Permalink | Reply to this

Re: Castles in the Air

“the EveAlice Theorem”?

Posted by: Jesse C. McKeown on May 25, 2016 3:31 PM | Permalink | Reply to this

Re: Castles in the Air

What’s a proof for one person may be only a plausibility argument for another. And there is no nice or graceful way to deal with such a situation.

Posted by: Eugene Lerman on May 25, 2016 8:04 PM | Permalink | Reply to this

Re: Castles in the Air

Always worth a call out for Pierre Cartier’s Mathemagics* (A Tribute to L. Euler and R. Feynman) when this topic is raised.

Posted by: David Corfield on May 21, 2016 12:14 PM | Permalink | Reply to this

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