Planet Musings

It's going to be a summer of student projects for me! I spoke with Winston Harris (Middle Tennessee State) about our summer project to automate the detection of planets in RV data; he is installing software and data now. I spoke with Abby Shaum (NYU) about phase modulation methods for finding planets; we may have found a signal in a star thought to have no signal! I spoke with Abby Williams (NYU) who, with Kate Storey-Fisher (NYU), is looking at measuring not just large-scale structure but in fact spatial gradients in the large-scale structure, with Storey-Fisher's cool new tools. With Jonah Goldfine (NYU) I looked at his solutions to the exercises in Fitting a Model to Data, done in preparation for some attempts on NASA TESS light curves. And I spoke with Anu Raghunathan (NYU) about possibly speeding up her search for planet transits using box least squares. Our project is theoretical: What are the statistical properties of the BLS algorithm? But we still need things to be much much faster.

Applied Category Theory 2020 is coming up soon! After the Tutorial Day on Sunday July 6th, there will be talks from Monday July 7th to Friday July 10th. All talks will be live on Zoom and on YouTube. Recorded versions will appear on YouTube later.

Here is the program—click on it to download a more readable version:

Here are the talks! They come in three kinds: keynotes, regular presentations and short industry presentations. Within each I’ve listed them in alphabetical order by speaker: I believe the first author is the speaker.

This is gonna be fun.

Keynote presentations (35 minutes)

• Henry Adams, Johnathan Bush and Joshua Mirth, Operations on metric thickenings.

• Nicolas Blanco and Noam Zeilberger: Bifibrations of polycategories and classical linear logic.

• Bryce Clarke, Derek Elkins, Jeremy Gibbons, Fosco Loregian, Bartosz Milewski, Emily Pillmore and Mario Román: Profunctor optics, a categorical update.

• Tobias Fritz, Tomáš Gonda, Paolo Perrone and Eigil Rischel: Distribution functors, second-order stochastic dominance and the Blackwell–Sherman–Stein Theorem in categorical probability.

• Micah Halter, Evan Patterson, Andrew Baas and James Fairbanks: Compositional scientific computing with Catlab and SemanticModels.

• Joachim Kock: Whole-grain Petri nets and processes.

• Andre Kornell, Bert Lindenhovius and Michael Mislove: Quantum CPOs.

• Martha Lewis: Towards logical negation in compositional distributional semantics.

• Jade Master and John Baez: Open Petri nets.

• Lachlan McPheat, Mehrnoosh Sadrzadeh, Hadi Wazni and Gijs Wijnholds, Categorical vector space semantics for Lambek calculus with a relevant modality.

• David Jaz Myers: Double categories of open dynamical systems.

• Toby St Clere Smithe, Cyber Kittens, or first steps towards categorical cybernetics.

Regular presentations (20 minutes)

• Robert Atkey, Bruno Gavranović, Neil Ghani, Clemens Kupke, Jeremy Ledent and Fredrik Nordvall Forsberg: Compositional game theory, compositionally.

• John Baez and Kenny Courser: Coarse-graining open Markov processes.

• Georgios Bakirtzis, Christina Vasilakopoulou and Cody Fleming, Compositional cyber-physical systems modeling.

• Marco Benini, Marco Perin, Alexander Alexander Schenkel and Lukas Woike: Categorification of algebraic quantum field theories.

• Daniel Cicala: Rewriting structured cospans.

• Bryce Clarke: A diagrammatic approach to symmetric lenses.

• Bob Coecke, Giovanni de Felice, Konstantinos Meichanetzidis, Alexis Toumi, Stefano Gogioso and Nicolo Chiappori: Quantum natural language processing.

• Geoffrey Cruttwell, Jonathan Gallagher and Dorette Pronk: Categorical semantics of a simple differential programming language.

• Swaraj Dash and Sam Staton: A monad for probabilistic point processes.

• Giovanni de Felice, Elena Di Lavore, Mario Román and Alexis Toumi: Functorial language games for question answering.

• Giovanni de Felice, Alexis Toumi and Bob Coecke: DisCoPy: monoidal categories in Python.

• Brendan Fong, David Jaz Myers and David I. Spivak: Behavioral mereology: a modal logic for passing constraints.

• Rocco Gangle, Gianluca Caterina and Fernando Tohme, A generic figures reconstruction of Peirce’s existential graphs (alpha).

• Jules Hedges and Philipp Zahn: Open games in practice.

• Jules Hedges: Non-compositionality in categorical systems theory.

• Michael Johnson and Robert Rosebrugh, The more legs the merrier: A new composition for symmetric (multi-)lenses.

• Joe Moeller, John Baez and John Foley: Petri nets with catalysts.

• John Nolan and Spencer Breiner, Symmetric monoidal categories with attributes.

• Joseph Razavi and Andrea Schalk: Gandy machines made easy via category theory.

• Callum Reader: Measures and enriched categories.

• Mario Román: Open diagrams via coend calculus.

• Luigi Santocanale, Dualizing sup-preserving endomaps of a complete lattice.

• Dan Shiebler: Categorical stochastic processes and likelihood.

• Richard Statman, Products in a category with only one object.

• David I. Spivak: Poly: An abundant categorical setting for mode-dependent dynamics.

• Christine Tasson and Martin Hyland, The linear-non-linear substitution 2-monad.

• Tarmo Uustalu, Niccolò Veltri and Noam Zeilberger: Proof theory of partially normal skew monoidal categories.

• Dmitry Vagner, David I. Spivak and Evan Patterson: Wiring diagrams as normal forms for computing in symmetric monoidal categories.

• Matthew Wilson, James Hefford, Guillaume Boisseau and Vincent Wang: The safari of update structures: visiting the lens and quantum enclosures.

• Paul Wilson and Fabio Zanasi: Reverse derivative ascent: a categorical approach to learning Boolean circuits.

• Vladimir Zamdzhiev: Computational adequacy for substructural lambda calculi.

• Gioele Zardini, David I. Spivak, Andrea Censi and Emilio Frazzoli: A compositional sheaf-theoretic framework for event-based systems.

Industry presentations (8 minutes)

• Arquimedes Canedo (Siemens Corporate Technology).

• Brendan Fong (Topos Institute).

• Jelle Herold (Statebox): Industrial strength CT.

• Steve Huntsman (BAE): Inhabiting the value proposition for category theory.

• Ilyas Khan (Cambridge Quantum Computing).

• Alan Ransil (Protocol Labs): Compositional data structures for the decentralized web.

• Alberto Speranzon (Honeywell).

• Ryan Wisnesky (Conexus): Categorical informatics at scale.

It was a very low-research day! But I did get in an hour with Bedell (Flatiron), discussing our plan to automate some aspects of exoplanet detection and discovery. The idea is: If we can operationalize and automate discovery, we can use that to perform experimental design on surveys. Surveys that we want to optimize for exoplanet yield. We discussed frequentist vs Bayesian approaches.

Any group acts as automorphisms of itself, by conjugation. If we differentiate this idea, we get that any Lie algebra acts as derivations of itself. We can then enhance this in various ways: for example a Poisson algebra is both a Lie algebra and a commutative algebra, such that any element acts as derivations of both these structures.

Why do I care?

In my paper on Noether’s theorem I got excited by how physics uses structures where each element acts to generate a one-parameter group of automorphisms of that structure. I proved a super-general version of Noether’s theorem based on this idea. It’s Theorem 8, in case you’re curious.

But the purest expression of the idea of a “structure where each element acts as an automorphism of that structure” is the concept of “rack”.

Even simpler than a rack is a “shelf”.

Alissa Crans defined a shelf to be a set $S$ equipped with a binary operation $\triangleright \colon S \times S \to S$ such that each element $a \in S$ gives a map $a \triangleright - \colon S \to S$ that is a shelf endomorphism.

If you don’t like the circularity of this definition (which is the whole point), what I’m saying is that a shelf is a set $S$ equipped with a binary operation $\triangleright \colon S \times S \to S$ such that

$$a \triangleright (b \triangleright c) = (a \triangleright b) \triangleright (a \triangleright c)$$

This is called the self-distributive law, since it says the operation $\triangleright$ distributes over itshelf. (Sorry: “itself”.)

Actually this is a left shelf; there is also a thing called a “right shelf”. Any group is both a left shelf and a right shelf in a natural way, via conjugation.

We can similarly define a rack to be a set $S$ equipped with a binary operation $\triangleright \colon S \times S \to S$ such that each element $a \in S$ gives a map $a \triangleright - \colon S \to S$ that is a rack automorphism.

In other words, a rack is a shelf $S$ where each map $a \triangleright - \colon S \to S$ is invertible.

It’s common to write the inverse of the map $a \triangleright -$ as $- \triangleleft a$. Then we can give a completely equational definition of a rack! It’s a set $S$ with two binary operations $\triangleright, \triangleleft \colon S \times S \to S$ obeying these identities:

$$(c \triangleright b) \triangleright a = (c \triangleright a) \triangleright (b \triangleright a), \qquad a \triangleleft(b \triangleleft c) = (a \triangleleft b) \triangleleft(a \triangleleft c)$$

$$(a \triangleleft b) \triangleright a = b , \qquad a \triangleleft(b \triangleright a) = b$$

These axioms are slightly redundant, but nicely symmetrical.

If we have a shelf that’s a smooth manifold, and the operation $\triangleright$ is smooth, and an element $e \in S$ obeying $e \triangleright a = a$ for all $a \in S$, then I think we can differentiate the shelf operation at $e$ and get a “Leibniz algebra”.

A Leibniz algebra is a vector space $L$ with a bilinear operation $[-,-] \colon L \times L \to L$ such that each element $a \in L$ gives a map $[a,-] \colon L \to L$ that is a Leibniz algebra derivation. In other words, the Jacobi identity holds:

$$[a, [b,c]] = [[a,b], c] + [b, [a,c]]$$

Again, this is really a left Leibniz algebra, and we could also consider “right” Leibniz algebras. Any Lie algebra is naturally both a left and a right Leibniz algebra.

I’ve been trying to understand Jordan algebras, and here’s one thing that intrigues me: in a Jordan algebra $A$, any pair of elements $a,b \in A$ acts to give a derivation of the Jordan algebra, say $D_{a,b} \colon A \to A$, as follows:

$$D_{a,b} (c) = a \circ (b \circ c) - b \circ (a \circ c)$$

If $L_a$ means left multiplication by $a$, this says $D_{a,b} = L_a L_b - L_b L_a$.

This property of Jordan algebras seems nicer to me than the actual definition of Jordan algebra! So we could imagine turning it into a definition. Say a pre-Jordan algebra is a vector space $A$ equipped with a bilinear map $\circ : A \times A \to A$ such that for any $a,b \in A$, the map $D_{a,b} \colon A \to A$ defined above is a derivation, meaning

$$D_{a,b}(c \circ d) = D_{a,b} (c) \circ d + c \circ D_{a,b}(d)$$

for all $c, d \in A$.

Puzzle. Are there pre-Jordan algebras that aren’t Jordan algebras? Are any of them interesting?

Most of us have been staying holed up at home lately. I spent the last month holed up writing a paper that expands on my talk at a conference honoring the centennial of Noether’s 1918 paper on symmetries and conservation laws. This made my confinement a lot more bearable. It was good getting back to this sort of mathematical physics after a long time spent on applied category theory. It turns out I really missed it.

While everyone at the conference kept emphasizing that Noether’s 1918 paper had two big theorems in it, my paper is just about the easy one—the one physicists call Noether’s theorem:

• Getting to the bottom of Noether’s theorem.

People often summarize this theorem by saying “symmetries give conservation laws”. And that’s right, but it’s only true under some assumptions: for example, that the equations of motion come from a Lagrangian.

This leads to some interesting questions. For which types of physical theories do symmetries give conservation laws? What are we assuming about the world, if we assume it is described by a theories of this type? It’s hard to get to the bottom of these questions, but it’s worth trying.

We can prove versions of Noether’s theorem relating symmetries to conserved quantities in many frameworks. While a differential geometric framework is truer to Noether’s original vision, my paper studies the theorem algebraically, without mentioning Lagrangians.

Now, Atiyah said:

…algebra is to the geometer what you might call the Faustian offer. As you know, Faust in Goethe’s story was offered whatever he wanted (in his case the love of a beautiful woman), by the devil, in return for selling his soul. Algebra is the offer made by the devil to the mathematician. The devil says: I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.

While this is sometimes true, algebra is more than a computational tool: it allows us to express concepts in a very clear and distilled way. Furthermore, the geometrical framework developed for classical mechanics is not sufficient for quantum mechanics. An algebraic approach emphasizes the similarity between classical and quantum mechanics, clarifying their differences.

In talking about Noether’s theorem I keep using an interlocking trio of important concepts used to describe physical systems: ‘states’, ‘observables’ and `generators’. A physical system has a convex set of states, where convex linear combinations let us describe probabilistic mixtures of states. An observable is a real-valued quantity whose value depends—perhaps with some randomness—on the state. More precisely: an observable maps each state to a probability measure on the real line. A generator, on the other hand, is something that gives rise to a one-parameter group of transformations of the set of states—or dually, of the set of observables.

It’s easy to mix up observables and generators, but I want to distinguish them. When we say ‘the energy of the system is 7 joules’, we are treating energy as an observable: something you can measure. When we say ‘the Hamiltonian generates time translations’, we are treating the Hamiltonian as a generator.

In both classical mechanics and ordinary complex quantum mechanics we usually say the Hamiltonian is the energy, because we have a way to identify them. But observables and generators play distinct roles—and in some theories, such as real or quaternionic quantum mechanics, they are truly different. In all the theories I consider in my paper the set of observables is a Jordan algebra, while the set of generators is a Lie algebra. (Don’t worry, I explain what those are.)

When we can identify observables with generators, we can state Noether’s theorem as the following equivalence:

$$The \; generator \; a \; generates \; transformations \; that \; leave \; the observable \; b \; fixed\!.$$

$$\Updownarrow$$

$$The \; generator \; b \; generates \; transformations \; that \; leave \; the \; observable \; a \; fixed\!.$$

In this beautifully symmetrical statement, we switch from thinking of $a$ as the generator and $b$ as the observable in the first part to thinking of $b$ as the generator and $a$ as the observable in the second part. Of course, this statement is true only under some conditions, and the goal of my paper is to better understand these conditions. But the most fundamental condition, I claim, is the ability to identify observables with generators.

In classical mechanics we treat observables as being the same as generators, by treating them as elements of a Poisson algebra, which is both a Jordan algebra and a Lie algebra. In quantum mechanics observables are not quite the same as generators. They are both elements of something called a ∗-algebra. Observables are self-adjoint, obeying

$$a^* = a$$

while generators are skew-adjoint, obeying

$$a^* = -a$$

The self-adjoint elements form a Jordan algebra, while the skew-adjoint elements form a Lie algebra.

In ordinary complex quantum mechanics we use a complex ∗-algebra. This lets us turn any self-adjoint element into a skew-adjoint one by multiplying it by $\sqrt{-1}$. Thus, the complex numbers let us identify observables with generators! In real and quaternionic quantum mechanics this identification is impossible, so the appearance of complex numbers in quantum mechanics is closely connected to Noether’s theorem.

In short, classical mechanics and ordinary complex quantum mechanics fit together in this sort of picture:

To dig deeper, it’s good to examine generators on their own: that is, Lie algebras. Lie algebras arise very naturally from the concept of ‘symmetry’. Any Lie group gives rise to a Lie algebra, and any element of this Lie algebra then generates a one-parameter family of transformations of that very same Lie algebra. This lets us state a version of Noether’s theorem solely in terms of generators:

$$The \; generator \; a \; generates \; transformations \; that \; leave \; the \; generator \; b fixed\!.$$

$$\Updownarrow$$

$$The \; generator \; b \; generates \; transformations \; that \; leave \; the \; generator \; a \; fixed\!.$$

And when we translate these statements into equations, their equivalence follows directly from this elementary property of the Lie bracket:

$$[a,b] = 0$$

$$\Updownarrow$$

$$[b,a] = 0$$

Thus, Noether’s theorem is almost automatic if we forget about observables and work solely with generators. The only questions left are: why should symmetries be described by Lie groups, and what is the meaning of this property of the Lie bracket?

In my paper I tackle both these questions, and point out that the Lie algebra formulation of Noether’s theorem comes from a more primitive group formulation, which says that whenever you have two group elements $g$ and $h$,

$$g \; commutes \; with h\!.$$

$$\Updownarrow$$

$$h \; commutes \; with \; g\!.$$

That is: whenever you’ve got two ways of transforming a physical system, the first transformation is ‘conserved’ by second if and only if the second is conserved by the first!

However, observables are crucial in physics. Working solely with generators in order to make Noether’s theorem a tautology would be another sort of Faustian bargain. So, to really get to the bottom of Noether’s theorem, we need to understand the map from observables to generators. In ordinary quantum mechanics this comes from multiplication by $i$. But this just pushes the mystery back a notch: why should we be using the complex numbers in quantum mechanics?

For this it’s good to spend some time examining observables on their own: that is, Jordan algebras. Those of greatest importance in physics are the unital JB-algebras, which are unfortunately named not after me, but Jordan and Banach. These allow a unified approach to real, complex and quaternionic quantum mechanics, along with some more exotic theories. So, they let us study how the role of complex numbers in quantum mechanics is connected to Noether’s theorem.

Any unital JB-algebra $O$ has a partial ordering: that is, we can talk about one observable being greater than or equal to another. With the help of this we can define states on $O,$ and prove that any observable maps each state to a probability measure on the real line.

More surprisingly, any JB-algebra also gives rise to two Lie algebras. The smaller of these, say $L,$ has elements that generate transformations of $O$ that preserve all the structure of this unital JB-algebra. They also act on the set of states. Thus, elements of $L$ truly deserve to be considered ‘generators’.

In a unital JB-algebra there is not always a way to reinterpret observables as generators. However, Alfsen and Shultz have defined the notion of a ‘dynamical correspondence’ for such an algebra, which is a well-behaved map

$$\psi \colon O \to L$$

One of the two conditions they impose on this map implies a version of Noether’s theorem. They prove that any JB-algebra with a dynamical correspondence gives a complex ∗-algebra where the observables are self-adjoint elements, the generators are skew-adjoint, and we can convert observables into generators by multiplying them by $i$.

This result is important, because the definition of JB-algebra does not involve the complex numbers, nor does the concept of dynamical correspondence. Rather, the role of the complex numbers in quantum mechanics emerges from a map from observables to generators that obeys conditions including Noether’s theorem!

To be a bit more precise, Alfsen and Shultz’s first condition on the map $\psi \colon O \to L$ says that every observable $a \in O$ generates transformations that leave $a$ itself fixed. I call this the self-conservation principle. It implies Noether’s theorem.

However, in their definition of dynamical correspondence, Alfsen and Shultz also impose a second, more mysterious condition on the map $\psi$. I claim that that this condition is best understood in terms of the larger Lie algebra associated to a unital JB-algebra. As a vector space this is the direct sum

$$A = O \oplus L$$

but it’s equipped with a Lie bracket such that

$$[-,-] \colon L \times L \to L \qquad [-,-] \colon L \times O \to O$$

$$[-,-] \colon O \times L \to O \qquad [-,-] \colon O \times O \to L$$

As I mentioned, elements of $L$ generate transformations of $O$ that preserve all the structure on this unital JB-algebra. Elements of $O$ also generate transformations of $O,$ but these only preserve its vector space structure and partial ordering.

What’s the meaning of these other transformations? I claim they’re connected to statistical mechanics.

For example, consider ordinary quantum mechanics and let $O$ be the unital JB-algebra of all bounded self-adjoint operators on a complex Hilbert space. Then $L$ is the Lie algebra of all bounded skew-adjoint operators on this Hilbert space. There is a dynamical correpondence sending any observable $H \in O$ to the generator $\psi(H) = i H \in L,$ which then generates a one-parameter group of transformations of $O$ like this:

$$a \mapsto e^{i t H/\hbar} \, a \, e^{-i t H/\hbar} \qquad \forall t \in \mathbb{R}, a \in O$$

where $\hbar$ is Planck’s constant. If $H$ is the Hamiltonian of some system, this is the usual formula for time evolution of observables in the Heisenberg picture. But $H$ also generates a one-parameter group of transformations of $O$ as follows:

$$a \mapsto e^{-\beta H/2} \, a \, e^{-\beta H/2} \qquad \forall \beta \in \mathbb{R}, a \in O$$

Writing $\beta = 1/k T$ where $T$ is temperature and $k$ is Boltzmann’s constant, I claim that these are ‘thermal transformations’. Acting on a state in thermal equilibrium at some temperature, these transformations produce states in thermal equilibrium at other temperatures (up to normalization).

The analogy between $i t/\hbar$ and $1/k T$ is often summarized by saying “inverse temperature is imaginary time”. The second condition in Alfsen and Shultz’s definition of dynamical correspondence is a way of capturing this principle in a way that does not explicitly mention the complex numbers. Thus, we may very roughly say their result explains the role of complex numbers in quantum mechanics starting from three assumptions:

• observables form Jordan algebra of a nice sort (a unital JB-algebra)

• the self-conservation principle (and thus Noether’s theorem)

• the relation between time and inverse temperature.

I still want to understand all of this more deeply, but the way statistical mechanics entered the game was surprising to me, so I feel I made a little progress.

I hope the paper is half as fun to read as it was to write! There’s a lot more in it than described here.

I gave a talk on some work I did with Kenny Courser. You can see slides of the talk, and also a video and the papers it’s based on:

• Coarse-graining open Markov processes.

Abstract. We illustrate some new paradigms in applied category theory with the example of coarse-graining open Markov processes. Coarse-graining is a standard method of extracting a simpler Markov process from a more complicated one by identifying states. Here we extend coarse-graining to ‘open’ Markov processes: that is, those where probability can flow in or out of certain states called ‘inputs’ and ‘outputs’. One can build up an ordinary Markov process from smaller open pieces in two basic ways: composition, where we identify the outputs of one open Markov process with the inputs of another, and tensoring, where we set two open Markov processes side by side. These constructions make open Markov processes into the morphisms of a symmetric monoidal category. But we can go further and construct a symmetric monoidal double category where the 2-morphisms include ways of coarse-graining open Markov processes. We can describe the behavior of open Markov processes using double functors out of this double category.

For more, look at these:

• John Baez, Brendan Fong and Blake Pollard, A compositional framework for Markov processes. (Blog article here.)

• John Baez and Kenny Courser, Coarse-graining open Markov processes. (Blog article here.)

• Kenny Courser, Open Systems: A Double Categorical Perspective.

I'm writing a paper on uncertainty estimation—how you put an error bar on your measurement—and at the same time, Kate Storey-Fisher (NYU) and I are working on a new method for estimating the correlation function (of galaxies, say) that improves the precision (the error bar) on measurements.

In the latter project (large-scale structure), we are encountering some interesting conceptual things. For instance, if you make many independent measurements of something (a vector of quantities say) and you want to plot your mean measurement and an uncertainty, what do you plot? Standard practice is to use the square root of the diagonal entries of the covariance matrix. But you could instead plot the inverse square roots of the diagonal entries of the inverse covariance matrix. These two quantities are (in general) very different! Indeed we are taught that the former is conservative and the latter is not, but it really depends what you are trying to show, and what you are trying to learn. The standard practice tells you how well you know the correlation function in one bin, marginalizing out (or profiling) any inferences you have about other bins.

In the end, we don't care about what we measure at any particular radius in the correlation function, we care about the cosmological parameters we constrain! So Storey-Fisher and I discussed today how we might propagate our uncertainties to there, and compare methods there. I hope we find that our method does far better than the standard methods in that context!

Apologies for this rambling, inside-baseball post, but this is my research blog, and it isn't particularly intended to be fun, useful, or artistic!

If there were ever a time for liberals and progressives to put aside their internal squabbles, you’d think it was now. The President of the United States is a racist gangster, who might not leave if he loses the coming election—all the more reason to ensure he loses in a landslide. Due in part to that gangster’s breathtaking incompetence, 130,000 Americans are now dead, and the economy tanked, from a pandemic that the rest of the world has under much better control. The gangster’s latest “response” to the pandemic has been to disrupt the lives of thousands of foreign scientists—including several of my students—by threatening to cancel their visas. (American universities will, of course, do whatever they legally can to work around this act of pure spite.)

So how is the left responding to this historic moment?

This weekend, 536 people did so by … trying to cancel Steven Pinker, stripping him of “distinguished fellow” and “media expert” status (whatever those are) in the Linguistics Society of America for ideological reasons.

Yes, Steven Pinker: the celebrated linguist and cognitive scientist, author of The Language Instinct and How the Mind Works (which had a massive impact on me as a teenager) and many other books, and academic torch-bearer for the Enlightenment in our time. For years, I’d dreaded the day they’d finally come for Steve, even while friends assured me my fears must be inflated since, after all, they hadn’t come for him yet.

I concede that the cancelers’ logic is impeccable. If they can get Pinker, everyone will quickly realize that there’s no longer any limit to who they can get—including me, including any writer or scientist who crosses them. If you’ve ever taken, or aspire to take, any public stand riskier than “waffles are tasty,” then don’t delude yourself that you’ll be magically spared—certainly not by your own progressive credentials.

I don’t know if the “charges” against Pinker merit a considered response (Pinker writes that some people wondered if they were satire). For those who care, though, here’s a detailed and excellent takedown by the biologist and blogger Jerry Coyne, and here’s another by Barbara Partee.

So, it seems Pinker once used the term “urban crime,” which can be a racist dogwhistle—except that in this case, it literally meant “urban crime.” Pinker once referred to Bernie Goetz, whose 1984 shooting of four robbers in the NYC subway polarized the US at the time, as a “mild-mannered engineer,” in a sentence whose purpose was to contrast that description with the ferocity of Goetz’s act. Pinker “appropriated” the work of a Black scholar, Harvard Dean Lawrence Bobo, which apparently meant approvingly citing him in a tweet. Etc. Ironically, it occurred to me that the would-be Red Guards could’ve built a much stronger case against Pinker had they seriously engaged with his decades of writing—writing that really does take direct aim at their whole worldview, they aren’t wrong about that—rather than superficially collecting a few tweets.

What Coyne calls the “Purity Posse” sleazily gaslights its readers as follows:

We want to note here that we have no desire to judge Dr. Pinker’s actions in moral terms, or claim to know what his aims are. Nor do we seek to “cancel” Dr. Pinker, or to bar him from participating in the linguistics and LSA communities (though many of our signatories may well believe that doing so would be the right course of action).

In other words: many of us “may well believe” that Pinker’s scientific career should be ended entirely. But magnanimously, for now, we’ll settle for a display of our power that leaves the condemned heretic still kicking. So don’t accuse us of wanting to “cancel” anyone!

In that same generous spirit:

Though no doubt related, we set aside questions of Dr. Pinker’s tendency to move in the proximity of what The Guardian called a revival of “scientific racism”, his public support for David Brooks (who has been argued to be a proponent of “gender essentialism”), his expert testimonial in favor of Jeffrey Epstein (which Dr. Pinker now regrets), or his dubious past stances on rape and feminism.

See, even while we make these charges, we disclaim all moral responsibility for making them. (For the record, Alan Dershowitz asked Pinker for a linguist’s opinion of a statute, so Pinker provided it; Pinker didn’t know at the time that the request had anything to do with Epstein.)

Again and again, spineless institutions have responded to these sorts of ultimatums by capitulating to them. So I confess that the news about Pinker depressed me all weekend. The more time passed, though, the more it looked like the Purity Posse might have actually overplayed its hand this time. Steven Pinker is not weak prey.

Let’s start with what’s missing from the petition: Noam Chomsky pointedly refused to sign. How that must’ve stung his comrades! For that matter, virtually all of the world’s well-known linguists refused to sign. Ray Jackendoff and Michel DeGraff were originally on the petition, but their names turned out to have been forged (were others?).

But despite the flimsiness of the petition, suppose the Linguistics Society of America caved. OK, I mused, how many people have even heard of the Linguistics Society of America, compared to the number who’ve heard of Pinker or read his books? If the LSA expelled Pinker, wouldn’t they be forever known to the world only as the organization that had done that?

I’m tired of the believers in the Enlightenment being constantly on the defensive. “No, I’m not a racist or a misogynist … on the contrary, I’ve spent decades advocating for … yes, I did say that, but you completely misunderstood my meaning, which in context was … please, I’m begging you, can’t we sit and discuss this like human beings?”

It’s time for more of us to stand up and say: yes, I am a center-left extremist. Yes, I’m an Enlightenment fanatic, a radical for liberal moderation and reason. If liberalism is the vanilla of worldviews, then I aspire to be the most intense vanilla anyone has ever tasted. I’m not a closeted fascist. I’m not a watered-down leftist. I’m something else. I consider myself ferociously anti-racist and anti-sexist and anti-homophobic and pro-downtrodden, but I don’t cede to any ideological faction the right to dictate what those terms mean. The world is too complicated, too full of ironies and surprises, for me to outsource my conscience in that way.

Enlightenment liberalism at least has the virtue that it’s not some utopian dream: on the contrary, it’s already led to most of the peace and prosperity that this sorry world has ever known, wherever and whenever it’s been allowed to operate. And while “the death of the Enlightenment” gets proclaimed every other day, liberal ideals have by now endured for centuries. They’ve outlasted kings and dictators, the Holocaust and the gulag. They certainly have it within them to outlast some online sneerers.

Yes, sometimes martyrdom (or at least career martyrdom) is the only honorable course, and yes, the childhood bullies did gift me with a sizeable persecution complex—I’ll grant the sneerers that. But on reflection, no, I don’t want to be a martyr for Enlightenment values. I want Enlightenment values to win, and not by vanquishing their opponents but by persuading them. As Pinker writes:

A final comment: I feel sorry for the signatories. Moralistic dudgeon is a shallow and corrosive indulgence, & policing the norms of your peer group a stunting of the intellect. Learning new ideas & rethinking conventional wisdom are deeper pleasures … and ultimately better for the world. Our natural state is ignorance, fallibility, & self-deception. Progress comes only from broaching & evaluating ideas, including those that feel unfamiliar and uncomfortable.

Spend a lot of time on Twitter and Reddit and news sites, and it feels like the believers in the above sentiment are wildly outnumbered by the self-certain ideologues of all sides. But just like the vanilla in a cake can be hard to taste, so there are more Enlightenment liberals than it seems, even in academia—especially if we include all those who never explicitly identified that way, because they were too busy building or fixing or discovering or teaching, and because they mistakenly imagined that if they just left the Purity Posse alone then the Posse would do likewise. If that’s you, then please ask yourself now: what is my personal break-point for speaking up?

Gravity! This one’s sure to be a crowd-pleaser. We take advantage of some of our previous discussion of curvature and spacetime, but we also talk about Einstein’s physical motivations for inventing general relativity, and the origin of gravitational time dilation and the like.