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Stochastic Digital Circuits for Probabilistic Inference by Vikash Mansinghka, Eric Jonas, Josh Tenenbaum


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Robots that can adapt like animals

Nature 521, 7553 (2015). doi:10.1038/nature14422

Authors: Antoine Cully, Jeff Clune, Danesh Tarapore & Jean-Baptiste Mouret

Robots have transformed many industries, most notably manufacturing, and have the power to deliver tremendous benefits to society, such as in search and rescue, disaster response, health care and transportation. They are also invaluable tools for scientific exploration in environments inaccessible to humans, from distant planets to deep oceans. A major obstacle to their widespread adoption in more complex environments outside factories is their fragility. Whereas animals can quickly adapt to injuries, current robots cannot ‘think outside the box’ to find a compensatory behaviour when they are damaged: they are limited to their pre-specified self-sensing abilities, can diagnose only anticipated failure modes, and require a pre-programmed contingency plan for every type of potential damage, an impracticality for complex robots. A promising approach to reducing robot fragility involves having robots learn appropriate behaviours in response to damage, but current techniques are slow even with small, constrained search spaces. Here we introduce an intelligent trial-and-error algorithm that allows robots to adapt to damage in less than two minutes in large search spaces without requiring self-diagnosis or pre-specified contingency plans. Before the robot is deployed, it uses a novel technique to create a detailed map of the space of high-performing behaviours. This map represents the robot’s prior knowledge about what behaviours it can perform and their value. When the robot is damaged, it uses this prior knowledge to guide a trial-and-error learning algorithm that conducts intelligent experiments to rapidly discover a behaviour that compensates for the damage. Experiments reveal successful adaptations for a legged robot injured in five different ways, including damaged, broken, and missing legs, and for a robotic arm with joints broken in 14 different ways. This new algorithm will enable more robust, effective, autonomous robots, and may shed light on the principles that animals use to adapt to injury.

guest post by Mike Stay

Greg Meredith and I have a short paper that’s been accepted for Higher-Dimensional Rewriting and Applications (HDRA) 2015 on modeling the asynchronous polyadic pi calculus with 2-categories. We avoid domain theory entirely and model the operational semantics directly; full abstraction is almost trivial. As a nice side-effect, we get a new tool for reasoning about consumption of resources during a computation.

It’s a small piece of a much larger project, which I’d like to describe here in a series of posts. This post will talk about lambda calculus for a few reasons. First, lambda calculus is simpler, but complex enough to illustrate one of our fundamental insights. Lambda calculus is to serial computation what pi calculus is to concurrent computation; lambda calculus talks about a single machine doing a computation, while pi calculus talks about a network of machines communicating over a network with potentially random delays. There is at most one possible outcome for a computation in the lambda calculus, while there are many possible outcomes in a computation in the pi calculus. Both the lazy lambda calculus and the pi calculus, however, have as an integral part of their semantics the notion of waiting for a sub-computation to complete before moving onto another one. Second, the denotational semantics of lambda calculus in Set is well understood, as is its generalization to cartesian closed categories; this semantics is far simpler than the denotational semantics of pi calculus and serves as a good introduction. The operational semantics of lambda calculus is also simpler than that of pi calculus and there is previous work on modeling it using higher categories.

History

Alonzo Church invented the lambda calculus as part of his attack on Hilbert’s third problem, also known as the Entscheidungsproblem, which asked for an algorithm to solve any mathematical problem. Church published his proof that no such algorithm exists in 1936. Turing invented his eponymous machines, also to solve the Entscheidungsproblem, and published his independent proof a few months after Church. When he discovered that Church had beaten him to it, Turing proved in 1937 that the two approaches were equivalent in power. Since Turing machines were much more “mechanical” than the lambda calculus, the development of computing machines relied far more on Turing’s approach, and it was only decades later that people started writing compilers for more friendly programming languages. I’ve heard it quipped that “the history of programming languages is the piecemeal rediscovery of the lambda calculus by computer scientists.”

The lambda calculus consists of a set of “terms” together with some relations on the terms that tell how to “run the program”. Terms are built up out of “term constructors”; in the lambda calculus there are three: one for variables, one for defining functions (Church denoted this operation with the Greek letter lambda, hence the name of the calculus), and one for applying those functions to inputs. I’ll talk about these constructors and the relations more below.

Church introduced the notion of “types” to avoid programs that never stop. Modern programming languages also use types to avoid programmer mistakes and encode properties about the program, like proving that secret data is inaccessible outside certain parts of the program. The “simply-typed” lambda calculus starts with a set of base types and takes the closure under the binary operation →\to to get a set of types. Each term is assigned a type; from this one can deduce the types of the variables used in the term. An assignment of types to variables is called a typing context.

The search for a semantics for variants of the lambda calculus has typically been concerned with finding sets or “domains” such that the interpretation of each lambda term is a function between domains. Scott worked out a domain DD such that the continuous functions from DD to itself are precisely the computable ones. Lambek and Scott generalized the category where we look for semantics from Set to arbitrary cartesian closed categories (CCCs).

Lambek and Scott constructed a CCC out of lambda terms; we call this category the syntactical category. Then a structure-preserving functor from the syntactical category to Set or some other CCC would provide the semantics. The syntactical category has types as objects and equivalence classes of certain terms as morphisms. A morphism in the syntactical category goes from a typing context to the type of the term.

John Baez has a set of lecture notes from Fall 2006 through Spring 2007 describing Lambek and Scott’s approach to the category theory of lambda calculus and generalizing it from cartesian closed categories to symmetric monoidal closed categories so it can apply to quantum computation as well: rather than taking a functor from the syntactical category into Set, we can take a functor into Hilb instead. He and I also have a “Rosetta stone” paper summarizing the ideas and connecting them with the corresponding generalization of the Curry-Howard isomorphism.

The Curry-Howard isomorphism says that types are to propositions as programs are to proofs. In practice, types are used in two different ways: one as propositions about data and the other as propositions about code. Programming languages like C, Java, Haskell, and even dynamically typed languages like JavaScript and Python use types to talk about propositions that data satisfies: is it a date or a name? In these languages, equivalence classes of programs constitute constructive proofs. Concurrent calculi are far more concerned about propositions that the code satisfies: can it reach a deadlocked state? In these languages, it is the rewrite rules taking one term to another that behave like proofs. Melliès and Zeilberger’s excellent paper “Functors are Type Refinement Systems” relates these two approaches to typing to each other.

Note that Lambek and Scott’s approach does not have the sets of terms or variables as objects! The algebra that defines the set of terms plays only a minor role in the category; there’s no morphism in the CCC, for instance, that takes a term tt and a variable xx to produce the term λx.t\lambda x.t. This failure to capture the structure of the term in the morphism wasn’t a big deal for lambda calculus because of “confluence” (see below), but it turns out to matter a lot more in calculi like Milner’s pi calculus that describe communicating over a network, where messages can be delayed and arrival times matter for the end result (consider, for instance, two people trying to buy online the last ticket to a concert).

The last few decades have seen domains becoming more and more complicated in order to try to “unerase” the information about the structure of terms that gets lost in the domain theory approach and recover the operational semantics. Fiore, Moggi, and Sangiorgi, Stark and Cattani, Stark, and Winskel all present domain models of the pi calculus that recursively involve the power set in order to talk about all the possible futures for a term. Industry has never cared much about denotational semantics: the Java Virtual Machine is an operational semantics for the Java language.

What we did

Greg Meredith and I set out to model the operational semantics of the pi calculus directly in a higher category rather than using domain theory. An obvious first question is, “What about types?” I was particularly worried about how to relate this approach to the kind of thing Scott and Lambek did. Though it didn’t make it into the HDRA paper and the details won’t make it into this post, we found that we’re able to use the “type-refinement-as-a-functor” idea of Melliés and Zeilberger to show how the algebraic term-constructor functions relate to the morphisms in the syntactical category.

We’re hoping that this categorical approach to modeling process calculi will help with reasoning about practical situations where we want to compose calculi; for instance, we’d like to put a hundred pi calculus engines around the edges of a chip and some ambient calculus engines, which have nice features for managing the location of data, in the middle to distribute work among them.

Lambda calculus

The lambda calculus consists of a set of “terms” together with some relations on the terms. The set TT of terms is defined recursively, parametric in a countably infinite set VV of variables. The base terms are the variables: if xx is an element of VV, then xx is a term in TT. Next, given any two terms t,t′∈Tt, t' \in T, we can apply one to the other to get t(t′)t(t'). We say that tt is in the head position of the application and t′t' in the tail position. (When the associativity of application is unclear, we’ll also use parentheses around subterms.) Finally, we can abstract out a variable from a term: given a variable xx and a term t,t, we get a term λx.t\lambda x.t.

The term constructors define an algebra, a functor LCLC from Set to Set that takes any set of variables VV to the set of terms T=LC(V)T = LC(V). The term constructors themselves become functions: −: V→T mboxvariable −(−): T×T→T mboxapplication λ: V×T→T mboxabstraction \begin{array}{rll} -\colon & V \to T &\mbox{variable}\\ -(-)\colon & T \times T \to T &\mbox{application}\\ \lambda\colon & V \times T \to T &\mbox{abstraction} \end{array}

Church described three relations on terms. The first relation, alpha, relates any two lambda abstractions that differ only in the variable name. This is exactly the same as when we consider the function f(x)=x 2f(x) = x^2 to be identical to the function f(y)=y 2f(y) = y^2. The third relation, eta, says that there’s no difference between a function ff and a “middle-man” function that gets an input xx and applies the function ff to it: λx.f(x)=f\lambda x.f(x) = f. Both alpha and eta are equivalences.

The really important relation is the second one, “beta reduction”. In order to define beta reduction, we have to define the free variables of a term: a variable occurring by itself is free; the set of free variables in an application is the union of the free variables in its subterms; and the free variables in a lambda abstraction are the free variables of the subterm except for the abstracted variable. FV(x)= {x} FV(t(t′))= FV(t)∪FV(t′) FV(λx.t)= FV(t)/{x} \begin{array}{rl} \mathrm{FV}(x) = & \{x\} \\ \mathrm{FV}(t(t')) = & \mathrm{FV}(t) \cup \mathrm{FV}(t') \\ \mathrm{FV}(\lambda x.t) = & \mathrm{FV}(t) / \{x\} \\ \end{array}

Beta reduction says that when we have a lambda abstraction λx.t\lambda x.t applied to a term t′t', then we replace every free occurrence of xx in tt by t′t': (λx.t)(t′)↓ βt{t′/x}, (\lambda x.t)(t') \downarrow_\beta t\{t' / x\}, where we read the right hand side as “tt with t′t' replacing xx.” We see a similar replacement of yy in action when we compose the following functions: f(x)= x+1 g(y)= y 2 g(f(x))= (x+1) 2 \begin{array}{rl} f(x) = & x + 1 \\ g(y) = & y^2 \\ g(f(x)) = & (x + 1)^2 \\ \end{array}

We say a term has a normal form if there’s some sequence of beta reductions that leads to a term where no beta reduction is possible. When the beta rule applies in more than one place in a term, it doesn’t matter which one you choose to do first: any sequence of betas that leads to a normal form will lead to the same normal form. This property of beta reduction is called confluence. Confluence means that the order of performing various subcomputations doesn’t matter so long as they all finish: in the expression (2+5)*(3+6)(2 + 5) * (3 + 6) it doesn’t matter which addition you do first or whether you distribute the expressions over each other; the answer is the same.

“Running” a program in the lambda calculus is the process of computing the normal form by repeated application of beta reduction, and the normal form itself is the result of the computation. Confluence, however, does not mean that when there is more than one place we could apply beta reduction, we can choose any beta reduction and be guaranteed to reach a normal form. The following lambda term, customarily denoted ω\omega, takes an input and applies it to itself: ω=λx.x(x)\omega = \lambda x.x(x) If we apply ω\omega to itself, then beta reduction produces the same term, customarily called Ω\Omega: Ω=ω(ω)\Omega = \omega(\omega) Ω↓ βΩ.\Omega \downarrow_\beta \Omega. It’s an infinite loop! Now consider this lambda term that has Ω\Omega as a subterm: (λx.λy.x)(λx.x)(Ω)(\lambda x.\lambda y.x)(\lambda x.x)(\Omega) It says, “Return the first element of the pair (identity function, Ω\Omega)”. If it has an answer at all, the answer should be “the identity function”. The question of whether it has an answer becomes, “Do we try to calculate the elements of the pair before applying the projection to it?”

Lazy lambda calculus

Many programming languages, like Java, C, JavaScript, Perl, Python, and Lisp are “eager”: they calculate the normal form of inputs to a function before calculating the result of the function on the inputs; the expression above, implemented in any of these languages, would be an infinite loop. Other languages, like Miranda, Lispkit, Lazy ML, and Haskell and its predecessor Orwell are “lazy” and only apply beta reduction to inputs when they are needed to complete the computation; in these languages, the result is the identity function. Abramsky wrote a 48-page paper about constructing a domain that captures the operational semantics of lazy lambda calculus.

The idea of representing operational semantics directly with higher categories originated with R. A. G. Seely, who suggested that beta reduction should be a 2-morphism; Barney Hilken and Tom Hirschowitz have also contributed to looking at lambda calculus from this perspective. In the “Rosetta stone” paper that John Baez and I wrote, we made an analogy between programs and Feynman diagrams. The analogy is precise as far as it goes, but it’s unsatisfactory in the sense that Feynman diagrams describe processes happening over time, while Lambek and Scott mod out by the process of computation that occurs over time. If we use 2-categories that explicitly model rewrites between terms, we get something that could potentially be interpreted with concepts from physics: types would become analogous to strings, terms would become analogous to space, and rewrites would happen over time. The idea from the “algebra of terms” perspective is that we have objects VV and TT for variables and terms, term constructors as 1-morphisms, and the nontrivial 2-morphisms generated by beta reduction. Seely showed that this approach works fine when you’re unconcerned with the context in which reduction can occur.

This approach, however, doesn’t work for lazy lambda calculus! Horizontal composition in a 2-category is a functor, so if a term tt reduces to a term t′t', then by functoriality, λx.t\lambda x.t must reduce to λx.t′\lambda x.t'—but this is forbidden in the lazy lambda calculus! Functoriality of horizontal composition is a “relativity principle” in the sense that reductions in one context are the same as reductions in any other context. In lazy programming languages, on the other hand, the “head” context is privileged: reductions only happen here. It’s somewhat like believing that measuring differences in temperature is like measuring differences in space, that only the difference is meaningful—and then discovering absolute zero. When beta reduction can happen anywhere in a term, there are too many 2-morphisms to model lazy lambda calculus.

In order to model this special context, we reify it: we add a special unary term constructor [−]:T→T[-]\colon T \to T that marks contexts where reduction is allowed, then redefine beta reduction so that the term constructor [−][-] behaves like a catalyst that enables the beta reduction to occur. This lets us cut down the set of 2-morphisms to exactly those that are allowed in the lazy lambda calculus; Greg and I did essentially the same thing in the pi calculus.

More concretely, we have two generating rewrite rules. The first propagates the reduction context to the head position of the term; the second is beta reduction restricted to a reduction context. [t(t′)]↓ ctx[[t](t′)] [t(t')]\, \downarrow_{ctx}\, [[t](t')] [[λx.t](t′)]↓ β[t]{t′/x} [[\lambda x.t](t')]\, \downarrow_\beta\, [t]\, \{t'/x\} When we surround the example term from the previous section with a reduction context marker, we get the following sequence of reductions: [(λx.λy.x)(λx.x)(Ω)] ↓ ctx [[(λx.λy.x)(λx.x)](Ω)] ↓ ctx [[[λx.λy.x](λx.x)](Ω)] ↓ β [[λy.(λx.x)](Ω)] ↓ β [λx.x] \begin{array}{rl} & [(\lambda x.\lambda y.x)(\lambda x.x)(\Omega)] \\ \downarrow_{ctx}& [[(\lambda x.\lambda y.x)(\lambda x.x)](\Omega)] \\ \downarrow_{ctx}& [[[\lambda x.\lambda y.x](\lambda x.x)](\Omega)] \\ \downarrow_{\beta}& [[\lambda y.(\lambda x.x)](\Omega)]\\ \downarrow_{\beta}& [\lambda x.x] \\ \end{array} At the start, none of the subterms were of the right shape for beta reduction to apply. The first two reductions propagated the reduction context down to the projection in head position. At that point, the only reduction that could occur was at the application of the projection to the first element of the pair, and after that to the second element. At no point was Ω\Omega ever in a reduction context.

Compute resources

In order to run a program that does anything practical, you need a processor, time, memory, and perhaps disk space or a network connection or a display. All of these resources have a cost, and it would be nice to keep track of them. One side-effect of reifying the context is that we can use it as a resource.

The rewrite rule ↓ ctx\downarrow_{ctx} increases the number of occurrences of [−][-] in a term while ↓ β\downarrow_\beta decreases the number. If we replace ↓ ctx\downarrow_{ctx} by the rule [t(t′)]↓ ctx′[t](t′) [t(t')]\, \downarrow_{ctx'}\, [t](t') then the number of occurences of [−][-] can never increase. By forming the term [[⋯[t]⋯]][[\cdots[t]\cdots]], we can bound the number of beta reductions that can occur in the computation of tt.

If we have a nullary constructor c:1→Tc\colon 1 \to T, then we can define [t]=c(t)[t] = c(t) and let the program dynamically decide whether to evaluate an expression eagerly or lazily.

In the pi calculus, we have the ability to run multiple processes at the same time; each [−][-] in that situation represents a core in a processor or computer in a network.

These are just the first things that come to mind; we’re experimenting with variations.

Conclusion

We figured out how to model the operational semantics of a term calculus directly in a 2-category by requiring a catalyst to carry out a rewrite, which gave us full abstraction without needing a domain based on representing all the possible futures of a term. As a side-effect, it also gave us a new tool for modeling resource consumption in the process of computation. Though I haven’t explained how yet, there’s a nice connection between the “algebra-of-terms” approach that uses VV and TT as objects and Lambek and Scott’s approach that uses types as objects, based on Melliès and Zeilberger’s ideas about type refinement. Next time, I’ll talk about the pi calculus and types.

Take a look at this article (hat tip Felix Salmon), which has me absolutely raging this morning, about new legislation in Kansas that prevents poor people on welfare from taking out more than $25 per day using their state-issued debit cards.

To be clear, you have to round up to the nearest $20 if you want to take out money from an ATM, so that’s really the limit.

And to be clear, there’s a $1 fee to take out money, and then typically an extra $2.50 fee if you don’t have a bank account, which many of the affected people do not.

So altogether, they’re giving $3.50 for every $20 of their welfare benefits, which I’d characterize as a bank tax of 17.5%. Because poor people don’t need that money, never mind the convenience of paying their actual bills.

For fuck’s sake, Kansas.

Author(s): Tim Wogan

Rotating bacterial cells suck one another into a 2D crystal structure, an unprecedented pattern for living organisms.

[Physics 8, 35] Published Fri Apr 17, 2015

Like hallucinogenic drugs, full-field flickering visual stimulation produces regular, geometric hallucinations such as radial or spiral patterns. Computational and theoretical models have revealed that the geometry of these hallucinations can be related to functional neuro-anatomy. However, while experimental evidence links both visual flicker and hallucinogenic drugs to upward and downward modulations of brain oscillatory activity, the exact relation between brain oscillations and geometric hallucinations remains a mystery. Here we demonstrate that, in human observers, this link is bidirectional. The same flicker frequencies that preferentially induced radial (<10 Hz) or spiral (10–20 Hz) hallucinations in a behavioral experiment involving full-field uniform flicker without any actual shape displayed, also showed selective oscillatory EEG enhancement when observers viewed a genuine static image of a radial or spiral pattern without any flicker. This bidirectional property constrains the possible neuronal events at the origin of visual hallucinations, and further suggests that brain oscillations, which are strictly temporal in nature, could nonetheless act as preferential channels for spatial information.

Dr. Greene, working with a student, has also found that “squirrels understand ‘bird-ese,’ and birds understand ‘squirrel-ese.’ ” When red squirrels hear a call announcing a dangerous raptor in the air, or they see such a raptor, they will give calls that are acoustically “almost identical” to the birds, Dr. Greene said. (Researchers have found that eastern chipmunks are attuned to mobbing calls by the eastern tufted titmouse, a cousin of the chickadee.)

The titmice are in on it too.  The article has numerous further points of interest.

It's a familiar scene to most Americans. The poignant moment when a soldier is honored for his or her service before a cheering crowd during halftime of an NFL game.  It turns out, however, that at least some of these patriotic displays are not what they seem.

A New Jersey-based website, NJ.com, has a detailed report that reveals the Department of Defense is paying millions of dollars to many NFL teams in what are essentially paid promotions to honor America's heroes...

This does not mean, of course, that all halftime events featuring troops or veterans are paid promotions. However, the fact that many are could undermine such efforts and "leaves a bad taste in your mouth" one lawmaker said.

"Those of us go to sporting events and see them honoring the heroes," said Arizona Sen. Jeff Flake in an interview with NJ.com. "You get a good feeling in your heart. Then to find out they're doing it because they're compensated for it, it leaves you underwhelmed. It seems a little unseemly."

It's hardly a secret that the NFL is one of the leading recruitment vehicles for the U.S. military. The problem, Flake implies, is that these events are portrayed as genuine moments of gratitude expressed to America's servicemen, not advertisements.

More at Scout and NJ.com, with a discussion at Reddit.

Comparing mathematics with poetry is an infinitely rich game. For every opinion you express, there is an equally valid counter-opinion. Contrasted to Hilbert’s dismissal of a student who had left mathematics for poetry, “I always thought he didn’t have enough imagination for mathematics”, someone said to me recently that the early death of Schubert was a greater tragedy than that of Galois, since what Galois could have achieved would sooner or later be done by someone else, whereas Schubert’s potential was lost forever.

So it isn’t so surprising that a book by Ron Aharoni, newly translated into English, doesn’t come to a definite conclusion one way or the other. The best we can do in a book entitled Mathematics, Poetry and Beauty is to give many examples of beautiful mathematics and beautiful poetry and discuss what the similarities and differences are.

Ron Aharoni is a mathematician whose field is combinatorics. He has collaboration distance 2 from me (we are both co-authors of Paul Erdős). I hadn’t heard from him for a while. In the book he explains that he made a deliberate move from university to elementary school.

Many, though not all, of his examples of poetry are taken from Israeli poets. I don’t know whether Hebrew is a particularly good language for poetry, but some of these poets pack many layers of meaning into a few words. But other poets appear, including Johann Wolfgang von Goethe, John Donne, Emily Dickinson, Federico Garcia Lorca, Constantine Cafavy, William Carlos Williams, and Matsuo Basho.

Here is an example of the argument. Displacement is a mechanism which, according to Freud, occurs in almost every area of human thought. By focussing on a subsidiary idea, the main message slips through almost unnoticed, although it may be too painful to face directly. Aharoni suggests that this is a technique used by poets for diving inside themselves, and for mathematicians stuck on a problem who look at a seemingly irrelevant detail in the hope of a breakthrough. He gives several examples, both poetic and mathematical.

One of his telling comparisons, expanded over four chapters, is that both a poem and a mathematical proof constitute a game of ping-pong between the abstract and the concrete. A poem can have several such switches in a few lines, as in this example “Written in pencil in the sealed railway-car” by Dan Pagis (translated by Stephen Mitchell):

Here in this carload
I am Eve
with my son Abel
if you see my older boy
Cain son of Adam
tell him that I …

In mathematics, both finding a proof and (if you are kind to your readers) presenting it involve frequent shifts of focus between logical argument and examples. But Aharoni’s conclusion is

… the heart of the poem is given to the concrete, and it is in this direction that the poem goes. This is the diametric opposite of the ping-pong of mathematics, in which the last shot is always towards the abstract.

It is also true that, as he remarks, in many published proofs (most notoriously, those of Gauss, the “fox who effaces his tracks in the sand with his tail”, according to Abel), all traces of the concrete are covered and only the shots to the abstract remain.

This points to another important difference. A finished poem can convey its beauty to any open-minded reader; knowledge of the poet’s biography often gets in the way. But the beauty in mathematics lies in the experience of the discovery of the proof; this can be reproduced to some extent in a reader who follows the argument carefully, but does not reside in the published proof, and still less in the statement of the theorem (in most cases).

And on the same theme, Aharoni invites us to watch the mathematician and the poet at work. The striking secret he reveals is that the mathematician spends most of his time staring into space.

Both poets (as many thinkers have observed) and mathematicians are concerned, not with ever more florid invention, but with the truth. It seems like a different kind of truth. Poets remind us of things we already know. But almost every mathematician, no matter which side they take on the “discovered or invented” question, are in their ordinary work Platonists, and act as if that their mental constructions are “out there”. However, the means they have for diving inside themselves, described in detail by Hadamard in his book, and divided into four stages (preparation, incubation, illumination, and verification) go well beyond the subjective ways that poets operate.

There is much more thought-provoking material in the book, many more dimensions on which mathematics and poetry can be compared. The chapter titles give some indication, including “The miracle of order”, “The power of the oblique”, “Reality or imagination”, “Unexpected combinations”, “Symmetry”, “Content and husk”, and “Change”.

Ug99 is a strain of wheat stem rust that was first identified in Uganda in 1998 (and named in 1999). By 2001, Ug99 began appearing in fields in Kenya; in Ethiopia by 2003; Sudan and Yemen by 2006; and Iran a year later. It now plagues wheat plants in nine African and Middle Eastern countries. Should the pathogen establish a global presence, 90 percent of wheat varieties could succumb, with whole crops flopping over and rotting within weeks or months of infection. The annual global harvest of some 700 million tons of wheat would be decimated...

University of Minnesota’s Anderson says GM corn and soybeans may have received more acceptance because they’re often processed or fed to animals, whereas most wheat enters the human food stream. Some consumers fear that tinkering with crop genomes could reduce the nutritive value of food, introduce toxins, increase the use of pesticides, or propel our already heavily-processed diets further away from what nature provided...

Nevertheless, GM wheat may still make its debut. In 2010, Monsanto announced that it was re-entering the field of biotech wheat and would work to genetically engineer crops that are higher yielding, stress tolerant, or herbicide resistant... Wheat growers are also starting to come around to the idea of cultivating GM plants. Several years ago, the National Association of Wheat Growers and US Wheat Associates expressed their support of biotech wheat. “We just have to prepare ourselves for a future where GM crops are more accepted,” Wulff says.

Much more info at the link.

There is some excitement in the net about some news of Harold White’s experiment at NASA. I have uncovered it by chance at a forum. This is a well-frequented site with people at NASA posting on it and regularly updating about the work that they are carrying out. You can also have noticed some activity in the Wikipedia’s pages about it (see here at the section on EmDrive and here). Wikipedia’s section on EmDrive explains in a few lines what is going on. Running a laser inside the RF cavity of the device they observed an unusual effect. They do not know yet if this could be better explained by more mundane reasons like air heating inside the cavity itself. They will repeat the measurements in a vacuum chamber to exclude such a possibility. I present here some of the slides used by White to recount about this This is the current take by Dr. White as reported by one of his colleagues too prone to leak on nasaspaceflight forum:

 …to be more careful in declaring we’ve observed the first lab based space-time warp signal and rather say we have observed another non-negative results in regards to the current still in-air WFI tests, even though they are the best signals we’ve seen to date. It appears that whenever we talk about warp-drives in our work in a positive way, the general populace and the press reads way too much into our technical disclosures and progress.

I would like to remember that White is not using exotic matter at all. Rather, he is working with strong RF fields to try to develop a warp bubble. This was stated here even if implicitly. Finally, an EmDrive device has been properly described here. Using strong external fields to modify locally a space-time has been described here. If this will be confirmed in the next few months, it will represent a major breakthrough in experimental general relativity since Eddington  confirmed the bending of light near the sun. Applications would follow if this idea will appear scalable but it will be a shocking result anyway. We look forward to hear from White very soon.

Marco Frasca (2005). Strong coupling expansion for general relativity Int.J.Mod.Phys.D15:1373-1386,2006 arXiv: hep-th/0508246v3

Filed under: Astronautics, General Relativity, Mathematical Physics, News, Physics, Rumors Tagged: Alcubierre drive, General relativity, Harold White, NASA, Warp drive

History-dependent processes are ubiquitous in natural and social systems. Many such stochastic processes, especially those that are associated with complex systems, become more constrained as they unfold, meaning that their sample space, or their set of possible outcomes, reduces as they age. We demonstrate that these sample-space-reducing (SSR) processes necessarily...

Physicists have mapped out how to create quantum music, an experience that will be profoundly different for every member of the audience, they say.

One of the features of 20^th century art is its increasing level of abstraction from cubism and surrealism in the early years to abstract expressionism and mathematical photography later. So an interesting question is what further abstractions can we look forward to in the 21^th century?

I’m helping run a workshop on Information and Entropy in Biological Systems at NIMBioS, the National Institute of Mathematical and Biological Synthesis, which is in Knoxville Tennessee.

I think you’ll be able to watch live streaming video of this workshop while it’s taking place from Wednesday April 8th to Friday April 10th. Later, videos will be made available in a permanent location.

To watch the workshop live, go here. Go down to where it says

Investigative Workshop: Information and Entropy in Biological Systems

Then click where it says live link. There’s nothing there now, but I’m hoping there will be when the show starts!

Below you can see the schedule of talks and a list of participants. The hours are in Eastern Daylight Time: add 4 hours to get Greenwich Mean Time. The talks start at 10 am EDT, which is 2 pm GMT.

Schedule

There will be 1½ hours of talks in the morning and 1½ hours in the afternoon for each of the 3 days, Wednesday April 8th to Friday April 10th. The rest of the time will be for discussions on different topics. We’ll break up into groups, based on what people want to discuss.

Each invited speaker will give a 30-minute talk summarizing the key ideas in some area, not their latest research so much as what everyone should know to start interesting conversations. After that, 15 minutes for questions and/or coffee.

Here’s the schedule. You can already see slides or other material for the talks with links!

Wednesday April 8

• 9:45-10:00 — the usual introductory fussing around.

• 10:00-10:30 — John Baez, Information and entropy in biological systems.

• 10:30-11:00 — questions, coffee.

• 11:00-11:30 — Chris Lee, Empirical information, potential information and disinformation.

• 11:30-11:45 — questions.

• 11:45-1:30 — lunch, conversations.

• 1:30-2:00 — John Harte, Maximum entropy as a foundation for theory building in ecology.

• 2:00-2:15 — questions, coffee.

• 2:15-2:45 — Annette Ostling, The neutral theory of biodiversity and other competitors to the principle of maximum entropy.

• 2:45-3:00 — questions, coffee.

• 3:00-5:30 — break up into groups for discussions.

• 5:30 — reception.

Thursday April 9

• 10:00-10:30 — David Wolpert, The Landauer limit and thermodynamics of biological organisms.

• 10:30-11:00 — questions, coffee.

• 11:00-11:30 — Susanne Still, Efficient computation and data modeling.

• 11:30-11:45 — questions.

• 11:45-1:30 — lunch, conversations.

• 1:30-2:00 — Matina Donaldson-Matasci, The fitness value of information in an uncertain environment.

• 2:00-2:15 — questions, coffee.

• 2:15-2:45 — Roderick Dewar, Maximum entropy and maximum entropy production in biological systems: survival of the likeliest?

• 2:45-3:00 — questions, coffee.

• 3:00-6:00 — break up into groups for discussions.

Friday April 10

• 10:00-10:30 — Marc Harper, Information transport and evolutionary dynamics.

• 10:30-11:00 — questions, coffee.

• 11:00-11:30 — Tobias Fritz, Characterizations of Shannon and Rényi entropy.

• 11:30-11:45 — questions.

• 11:45-1:30 — lunch, conversations.

• 1:30-2:00 — Christina Cobbold, Biodiversity measures and the role of species similarity.

• 2:00-2:15 — questions, coffee.

• 2:15-2:45 — Tom Leinster, Maximizing biological diversity.

• 2:45-3:00 — questions, coffee.

• 3:00-6:00 — break up into groups for discussions.

Participants

Here are the confirmed participants, just so you can get a sense of who is involved:

• John Baez - mathematical physicist.

• Romain Brasselet - postdoc in cognitive neuroscience knowledgeable about information-theoretic methods and methods of estimating entropy from samples of probability distributions.

• Katharina Brinck - grad student at Centre for Complexity Science at Imperial College; did masters at John Harte’s lab, where she extended his Maximum Entropy Theory of Ecology (METE) to trophic food webs, to study how entropy maximization on the macro scale together with MEP on the scale of individuals drive the structural development of model ecosystems.

• Christina Cobbold - mathematical biologist, has studied the role of species similarity in measuring biodiversity.

• Troy Day - mathematical biologist, works with population dynamics, host-parasite dynamics, etc.; influential and could help move population dynamics to a more information-theoretic foundation.

• Roderick Dewar - physicist who studies the principle of maximal entropy production.

• Barrett Deris - MIT postdoc studying the studying the factors that influence evolvability of drug resistance in bacteria.

• Charlotte de Vries - a biology master’s student who studied particle physics to the master’s level at Oxford and the Perimeter Institute. Interested in information theory.

• Matina Donaldson-Matasci - a biologist who studies information, uncertainty and collective behavior.

• Chris Ellison - a postdoc who worked with James Crutchfield on “information-theoretic measures of structure and memory in stationary, stochastic systems - primarily, finite state hidden Markov models”. He coauthored Intersection information based on common randomness. The idea: “The introduction of the partial information decomposition generated a flurry of proposals for defining an intersection information that quantifies how much of “the same information” two or more random variables specify about a target random variable. As of yet, none is wholly satisfactory.” Works on mutual information between organisms and environment (along with David Krakauer and Jessica Flack), and also entropy rates.

• Cameron Freer - MIT postdoc in Brain and Cognitive Sciences working on maximum entropy production principles, algorithmic entropy etc.

• Tobias Fritz - a physicist who has worked on “resource theories” and haracterizations of Shannon and Rényi entropy and on resource theories.

• Dashiell Fryer - works with Marc Harper on information geometry and evolutionary game theory.

• Michael Gilchrist - an evolutionary biologist studying how errors and costs of protein translation affect the codon usage observed within a genome. Works at NIMBioS.

• Manoj Gopalkrishnan - an expert on chemical reaction networks who understands entropy-like Lyapunov functions for these systems.

• Marc Harper - works on evolutionary game theory using ideas from information theory, information geometry, etc.

• John Harte - an ecologist who uses the maximum entropy method to predict the structure of ecosystems.

• Ellen Hines - studies habitat modeling and mapping for marine endangered species and ecosystems, sea level change scenarios, documenting of human use and values. Her lab has used MaxEnt methods.

• Elizabeth Hobson - behavior ecology postdoc developing methods to quantify social complexity in animals. Works at NIMBioS.

• John Jungk - works on graph theory and biology.

• Chris Lee - in bioinformatics and genomics; applies information theory to experiment design and evolutionary biology.

• Maria Leites - works on dynamics, bifurcations and applications of coupled systems of non-linear ordinary differential equations with applications to ecology, epidemiology, and transcriptional regulatory networks. Interested in information theory.

• Tom Leinster - a mathematician who applies category theory to study various concepts of ‘magnitude’, including biodiversity and entropy.

• Timothy Lezon - a systems biologist in the Drug Discovery Institute at Pitt, who has used entropy to characterize phenotypic heterogeneity in populations of cultured cells.

• Maria Ortiz Mancera - statistician working at CONABIO, the National Commission for Knowledge and Use of Biodiversity, in Mexico.

• Yajun Mei - statistician who uses Kullback-Leibler divergence and how to efficiently compute entropy for the two-state hidden Markov models.

• Robert Molzon - mathematical economist who has studied deterministic approximation of stochastic evolutionary dynamics.

• David Murrugarra - works on discrete models in mathematical biology; interested in learning about information theory.

• Annette Ostling - studies community ecology, focusing on the influence of interspecific competition on community structure, and what insights patterns of community structure might provide about the mechanisms by which competing species coexist.

• Connie Phong - grad student at Chicago’s Institute of Genomics and System biology, working on how “certain biochemical network motifs are more attuned than others at maintaining strong input to output relationships under fluctuating conditions.”

• Petr Plechak - works on information-theoretic tools for estimating and minimizing errors in coarse-graining stochastic systems. Wrote “Information-theoretic tools for parametrized coarse-graining of non-equilibrium extended systems”.

• Blake Polllard - physics grad student working with John Baez on various generalizations of Shannon and Renyi entropy, and how these entropies change with time in Markov processes and open Markov processes.

• Timothee Poisot - works on species interaction networks; developed a “new suite of tools for probabilistic interaction networks”.

• Richard Reeve - works on biodiversity studies and the spread of antibiotic resistance. Ran a program on entropy-based biodiversity measures at a mathematics institute in Barcelona.

• Rob Shaw - works on entropy and information in biotic and pre-biotic systems.

• Matteo Smerlak - postdoc working on nonequilibrium thermodynamics and its applications to biology, especially population biology and cell replication.

• Susanne Still - a computer scientist who studies the role of thermodynamics and information theory in prediction.

• Alexander Wissner-Gross - Institute Fellow at the Harvard University Institute for Applied Computational Science and Research Affiliate at the MIT Media Laboratory, interested in lots of things.

• David Wolpert - works at the Santa Fe Institute on i) information theory and game theory, ii) the second law of thermodynamics and dynamics of complexity, iii) multi-information source optimization, iv) the mathematical underpinnings of reality, v) evolution of organizations.

• Matthew Zefferman - works on evolutionary game theory, institutional economics and models of gene-culture co-evolution. No work on information, but a postdoc at NIMBioS.

Zoology: Here be dragons

Nature 520, 7545 (2015). doi:10.1038/520042a

Authors: Andrew J. Hamilton, Robert M. May & Edward K. Waters

Emerging evidence indicates that dragons can no longer be dismissed as creatures of legend and fantasy, and that anthropogenic effects on the world's climate may inadvertently be paving the way for the resurgence of these beasts.

From Vice's Motherboard:

That means when you search for “cold sores,” for instance, and click the highly ranked “Cold Sores Topic Overview WebMD” link, the website is passing your request for information about the disease along to one or more (and often many, many more) other corporations...

Thus, Libert has discovered that the vast majority of health sites, from the for-profit WebMD.com to the government-run CDC.gov, are loaded with tracking elements that are sending records of your health inquiries to the likes of web giants like Google, Facebook, and Pinterest, and data brokers like Experian and Acxiom.

From there, it becomes relatively easy for the companies receiving the requests, many of which are collecting other kinds of data (in cookies, say) about your browsing as well, to identify you and your illness...

WebMD, for instance, is the 106th most-visited site in the US, according to Alexa, and figures prominently in search results for most commonly searched diseases. It sends third party requests to a whopping 34 separate domains, including the data brokers Experian and Acxiom.“WebMD is basically calling up everybody in town and telling them that’s what you’re looking at..."

With nonprofit sites like the CDC and the Mayo Clinic, again, it’s not due to any insidious intent; it’s simply because developers are installing “free” tools like Google Analytics and social media “share” buttons on their sites, and most users have no idea that means information about their searches is being shared with third parties. “The problem is that using these 'free' third-party tools is really easy for web developers. What developers don’t consider is, why are these tools free?”

More at the link.   Not an April Fool's joke.

As academics we live and die by our research. While our proofs are either correct or not, the import of our work has a far more subjective feel. One can see where the work is published or how many citations it gets and we often say that we care most about the true intrinsic or extrinsic value of the research. But the measure of success of a research that we truly care most about is how it is viewed within the community. Such measures can have a real value in terms of hiring, tenure, promotion, raises and grants but it goes deeper, filling some internal need to have our research matter to our peers.

So even little things can bother you. Not being cited when you think your work should be. Not being mentioned during a talk. Seeing a review that questions the relevance of your model. Nobody following up on your open questions. Difficulty in finding excitement in others about your work. We tend to keep these feelings bottled up since we feel we shouldn't be bragging about own work.

If you feel this way a few things to keep in mind. It happens to all of us even though we rarely talk about it. You are not alone. Try not to obsess, it's counterproductive and just makes you feel even worse. If appropriate let the authors know that your work is relevant to theirs, the authors truly may have been unaware. Sometimes it is just best to acknowledge to yourself that while you think the work is good, you can't always convince the rest of the world and just move on.

More importantly remember the golden rule, and try to cite all relevant research and show interest in other people's work as well as your own.

A cloud of atoms trapped close to a superconducting ring can detect the magnetic field inside the ring with single-quantum sensitivity.

Published Tue Mar 17, 2015

I first posted this video several years ago, but I find it to be endlessly fascinating.  A hat tip to 22 words for reminding me.

Addendum:  And a hat tip to several readers for pointing out that this diving skill has some intriguing scientific roots, nicely summarized at one of my favorite science blogs, Not Exactly Rocket Science:

Jaroslav Červený has found that when red foxes pounce, they mostly jump in a north-easterly direction. He thinks that they’re using the Earth’s magnetic field to hunt.

Červený spent over two years studying wild red foxes in the Czech Republic, with the help of a 23-strong team of wildlife biologists and experienced hunters. The team recorded almost 600 mousing jumps, performed by 84 foxes at a wide variety of locations and times.

They found that foxes strongly prefer to jump in a north-easterly direction, around 20 degrees off from magnetic north. This fixed heading was important for their success as hunters. They were more likely to make a kill if they jumped along their preferred axis, particularly if their prey was hidden by high cover or snow. If they pounced to the north-east, they killed on 73% of their attacks; if they jumped in the opposite direction, they success rate stayed at 60%. In all other directions, only 18% of their pounces were successful...

Červený suggests that a red fox could use the Earth’s magnetic field as a “rangefinder”, to estimate the distance to its prey and make a more accurate pounce. This targeting system works because the Earth’s magnetic field tilts downward in the northern hemisphere, at an angle of 60-70 degrees below the horizontal. As the fox creeps forward, it listens for the sound of a mouse. It’s searching for that sweet spot where the angle of the sound hitting its ears matches the slope of the Earth’s magnetic field. At that spot, the fox knows that it’s a fixed distance away from its prey, and it knows exactly how far to jump to land upon it...

This totally fascinating ability is discussed in more detail at the link, and illustrated here -


- from an article in Nature.  Evolution is so amazing.

Reposted from 2011 to add this video of a domestic dog pouncing the shadow of a bouncing baby:

From an article explaining how the prison system in the United States has been corporatized into profitable ventures:

The profits generated by the corrections economy have not been definitively calculated, and a comprehensive audit would be a staggering accounting task. The figure would have to include the cost of private-prison real estate, mandatory drug testing, electronic monitoring anklets, prison-factory labor, prison-farm labor, prison-phone contracts, and the service fees charged to prisoners’ families when they wire money for supplies from the prison commissary. Contracted commercial activity flows in and out of every city jail, rural prison, suburban probation office, and immigration detention center. For stakeholders in the largest peacetime carceral apparatus in the history of the world, the opportunities for profit add up. For analysts like Sommer, the system also offers a safe, government-secured investment...

Consider the prison-phone industry. For inmates, especially urban felons shipped to far-off rural sites, calls to the outside are a social lifeline and a proven method for reducing recidivism. But here, too, Wall Street has identified a high-demand, low-supply commodity. Other government contractors, be they food suppliers or dentists, collect fees paid out by the state. Prison-phone companies, and the prison-wire-transfer companies that are following their model, extract revenues directly from inmates and their families. (Fifteen dollars for a fifteen-minute phone call is not uncommon.)

As with partnership corrections, profits are largely determined by contracts, but phone and money-transfer companies sweeten the deals for their public partners with profit-sharing perks. These commissions kick back anywhere from 40 to 60 percent of revenue to the contracting government agency. According to a study by Prison Legal News, a publication of the Human Rights Defense Center, about 85 percent of non-federal jails sign up for commission-added contracts, and because commissions increase in proportion to the total contract value, cash-strapped public officials are motivated to choose the most expensive contract available. Prison Legal News found that when Louisiana put out a public request for proposals for phone services in 2001, the agency stated the wish explicitly: “The state desires that the bidder’s compensation percentages . . . be as high as possible.”

Addendum/update:  I posted the above in March of 2015.  Now in October comes a report that the FCC is going to mandate a cap of 11 cents per minute:

The FCC said its vote yesterday "lower[s] the cap to 11 cents per minute for all local and long distance calls from state and federal prisons, while providing tiered rates for jails to account for the higher costs of serving jails and smaller institutions."

Part of the problem is that jails and prisons have been charging phone companies big commissions in exchange for exclusive contracts. These commission payments are passed on to prisoners.

The FCC did not outlaw commissions but said that it "strongly encourages parties to move away from site commissions and urges states to take action on this issue." Clyburn said that "states must do their part and take a hard look at their site commission practices and how such payments impact prices, service, and the reverberating impact on the community."

Möbius strips are three-dimensional geometrical structures, fascinating for their peculiar property of being surfaces with only one “side”—or, more technically, being “nonorientable” surfaces. Despite being easily realized artificially, the spontaneous emergence of these structures in nature is exceedingly rare. Here, we generate Möbius strips of optical polarization by tightly focusing the light beam emerging from a q-plate, a liquid crystal device that modifies the polarization of light in a space-variant manner. Using a recently developed method for the three-dimensional nanotomography of optical vector fields, we fully reconstruct the light polarization structure in the focal region, confirming the appearance of Möbius polarization structures. The preparation of such structured light modes may be important for complex light beam engineering and optical micro- and nanofabrication. Authors: Thomas Bauer, Peter Banzer, Ebrahim Karimi, Sergej Orlov, Andrea Rubano, Lorenzo Marrucci, Enrico Santamato, Robert W. Boyd, Gerd Leuchs

The best of the rest from the Physics arXiv preprint server.

Computing Real Numbers using DNA Self-Assembly

This time I’d like to think about three different approaches to ‘defining equality’, or more generally, introducing equality in formal systems of mathematics.

These will be taken from old-fashioned logic — before computer science, category theory or homotopy theory started exerting their influence. Eventually I want to compare these to more modern treatments.

If you know other interesting ‘old-fashioned’ approaches to equality, please tell me!

The equals sign is surprisingly new. It was never used by the ancient Babylonians, Egyptians or Greeks. It seems to originate in 1557, in Robert Recorde’s book The Whetstone of Witte. If so, we actually know what the first equation looked like:

As you can see, the equals sign was much longer back then! He used parallel lines “because no two things can be more equal.”

Formalizing the concept of equality has raised many questions. Bertrand Russell published The Principles of Mathematics [R] in 1903. Not to be confused with the Principia Mathematica, this is where he introduced Russell’s paradox. In it, he wrote:

identity, an objector may urge, cannot be anything at all: two terms plainly are not identical, and one term cannot be, for what is it identical with?

In his Tractatus, Wittgenstein [W] voiced a similar concern:

Roughly speaking: to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing.

These may seem like silly objections, since equations obviously do something useful. The question is: precisely what?

Instead of tackling that head-on, I’ll start by recalling three related approaches to equality in the pre-categorical mathematical literature.

The indiscernibility of identicals

The principle of indiscernibility of identicals says that equal things have the same properties. We can formulate it as an axiom in second-order logic, where we’re allowed to quantify over predicates PP:

∀x∀y[x=y⇒∀P[P(x)⇔P(y)]] \forall x \forall y [x = y \; \implies \; \forall P \, [P(x) \; \iff \; P(y)] ]

We can also formulate it as an axiom schema in 1st-order logic, where it’s sometimes called substitution for formulas. This is sometimes written as follows:

For any variables x,yx, y and any formula ϕ\phi, if ϕ′\phi' is obtained by replacing any number of free occurrences of xx in ϕ\phi with yy, such that these remain free occurrences of yy, then

x=y⇒[ϕ⇒ϕ′] x = y \;\implies\; [\phi \;\implies\; \phi' ]

I think we can replace this with the prettier

x=y⇒[ϕ⇔ϕ′] x = y \;\implies\; [\phi \;\iff \; \phi']

without changing the strength of the schema. Right?

We cannot derive reflexivity, symmetry and transitivity of equality from the indiscernibility of identicals. So, this principle does not capture all our usual ideas about equality. However, as shown last time, we can derive symmetry and transitivity from this principle together with reflexivity. This uses an interesting form of argument where take “being equal to zz” as one of the predicates (or formulas) to which we apply the principle. There’s something curiously self-referential about this. It’s not illegitimate, but it’s curious.

The identity of indiscernibles

Leibniz [L] is often credited with formulating a converse principle, the identity of indiscernibles. This says that things with all the same properties are equal. Again we can write it as a second-order axiom:

∀x∀y[∀P[P(x)⇔P(y)]⇒x=y] \forall x \forall y [ \forall P [ P(x) \; \iff \; P(y)] \; \implies \; x = y ]

or a first-order axiom schema.

We can go further if we take the indiscernibility of identicals and identity of indiscernibles together as a package:

∀x∀y[∀P[P(x)⇔P(y)]⇔x=y] \forall x \forall y [ \forall P [ P(x) \; \iff \; P(y)] \; \iff \; x = y ]

This is often called the Leibniz law. It says an entity is determined by the collection of predicates that hold of that entity. Entities don’t have mysterious ‘essences’ that determine their individuality: they are completely known by their properties, so if two entities have all the same properties they must be the same.

This principle does imply reflexivity, symmetry and transitivity of equality. They follow from the corresponding properties of ⇔\iff in a satisfying way. Of course, if we were wondering why equality has these three properties, we are now led to wonder the same thing about the biconditional ⇔\iff. But this counts as progress: it’s a step toward ‘logicizing’ mathematics, or at least connecting == firmly to ⇔\iff.

Apparently Russell and Whitehead used a second-order version of the Leibniz law to define equality in the Principia Mathematica [RW], while Kalish and Montague [KL] present it as a first-order schema. I don’t know the whole history of such attempts.

When you actually look to see where Leibniz formulated this principle, it’s a bit surprising. He formulated it in the contrapositive form, he described it as a ‘paradox’, and most surprisingly, it’s embedded as a brief remark in a passage that would be hair-curling for many contemporary rationalists. It’s in his Discourse on Metaphysics, a treatise written in 1686:

Thus Alexander the Great’s kinghood is an abstraction from the subject, and so is not determinate enough to pick out an individual, and doesn’t involve the other qualities of Alexander or everything that the notion of that prince includes; whereas God, who sees the individual notion or ‘thisness’ of Alexander, sees in it at the same time the basis and the reason for all the predicates that can truly be said to belong to him, such as for example that he would conquer Darius and Porus, even to the extent of knowing a priori (and not by experience) whether he died a natural death or by poison — which we can know only from history. Furthermore, if we bear in mind the interconnectedness of things, we can say that Alexander’s soul contains for all time traces of everything that did and signs of everything that will happen to him — and even marks of everything that happens in the universe, although it is only God who can recognise them all.

Several considerable paradoxes follow from this, amongst others that it is never true that two substances are entirely alike, differing only in being two rather than one. It also follows that a substance cannot begin except by creation, nor come to an end except by annihilation; and because one substance can’t be destroyed by being split up, or brought into existence by the assembling of parts, in the natural course of events the number of substances remains the same, although substances are often transformed. Moreover, each substance is like a whole world, and like a mirror of God, or indeed of the whole universe, which each substance expresses in its own fashion — rather as the same town looks different according to the position from which it is viewed. In a way, then, the universe is multiplied as many times as there are substances, and in the same way the glory of God is magnified by so many quite different representations of his work.

(Emphasis mine — you have to look closely to find the principle of identity of indiscernibles, because it goes by so quickly!)

There have been a number of objections to the Leibniz law over the years. I want to mention one that might best be handled using some category theory. In 1952, Max Black [B] claimed that in a symmetrical universe with empty space containing only two symmetrical spheres of the same size, the two spheres are two distinct objects even though they have all their properties in common.

As Black admits, this problem only shows up in a ‘relational’ theory of geometry, where we can’t say that the spheres have different positions — e.g., one centered at the points (x,y,z)(x,y,z), the other centered at (−x,−y,−z)(-x,-y,-z) — but only speak of their position relative to one another. This sort of theory is certainly possible, and it seems to be important in physics. But I believe it can be adequately formulated only with the help of some category theory. In the situation described by Black, I think we should say the spheres are not equal but isomorphic.

As widely noted, general relativity also pushes for a relational approach to geometry. Gauge theory, also, raises the issue of whether indistinguishable physical situations should be treated as equal or merely isomorphic. I believe the mathematics points us strongly in the latter direction.

A related issue shows up in quantum mechanics, where electrons are considered indistinguishable (in a certain sense), yet there can be a number of electrons in a box — not just one.

But I will discuss such issues later.

Extensionality

In traditional set theory we try to use sets as a substitute for predicates, saying x∈Sx \in S as a substitute for P(x)P(x). This lets us keep our logic first-order and quantify over sets — often in a universe where everything is a set — as a substitute for quantifying over predicates. Of course there’s a glitch: Russell’s paradox shows we get in trouble if we try to treat every predicate as defining a set! Nonetheless it is a powerful strategy.

If we apply this strategy to reformulate the Leibniz law in a universe where everything is a set, we obtain:

∀S∀T[S=T⇔∀R[S∈R⇔T∈R]] \forall S \forall T [ S = T \; \iff \; \forall R [ S \in R \; \iff \; T \in R]]

While this is true in Zermelo-Fraenkel set theory, it is not taken as an axiom. Instead, people turn the idea around and use the axiom of extensionality:

∀S∀T[S=T⇔∀R[R∈S⇔R∈T]] \forall S \forall T [ S = T \; \iff \; \forall R [ R \in S \; \iff \; R \in T]]

Instead of saying two sets are equal if they’re in all the same sets, this says two sets are equal if all the same sets are in them. This leads to a view where the ‘contents’ of an entity as its defining feature, rather than the predicates that hold of it.

We could, in fact, send this idea back to second-order logic and say that predicates are equal if and only if they hold for the same entities:

∀P∀Q[∀x[P(x)⇔Q(x)]⇔P=Q] \forall P \forall Q [\forall x [P(x) \; \iff \; Q(x)] \; \iff P = Q ]

as a kind of ‘dual’ of the Leibniz law:

∀x∀y[∀P[P(x)⇔P(y)]⇔x=y] \forall x \forall y [ \forall P [ P(x) \; \iff \; P(y)] \; \iff \; x = y ]

I don’t know if this has been remarked on in the foundational literature, but it’s a close relative of a phenomenon that occurs in other forms of duality. For example, continuous real-valued functions F,GF, G on a topological space obey

∀F∀G[∀x[F(x)=G(x)]⇔F=G] \forall F \forall G [\forall x [F(x) \; = \; G(x)] \; \iff F = G ]

but if the space is nice enough, continuous functions ‘separate points’, which means we also have

∀x∀y[∀F[F(x)=F(y)]⇔x=y] \forall x \forall y [ \forall F [ F(x) \; = \; F(y)] \; \iff \; x = y ]

Notes

• [B] Max Black, The identity of indiscernibles, Mind, New Series 61 (1952), 153–164.

• [F] Peter Forrest, The identity of indiscernibles, Stanford Encyclopedia of Philosophy.

• [KL] Donald Kalish and Richard Montague, Logic: Techniques of Formal Reasoning, Oxford U. Press, Oxford, 1964, pp. 220-221.

• [L] Gottfried Wilhelm Leibniz, Discourse on Metaphysics, trans. Jonathan Bennett.

• [R] Bertrand Russell, Principles of Mathematics § 64, 1903.

• [RW] Bertrand Russell and Alfred North Whitehead, Principia Mathematica, ☆13, 1910.

• Wikipedia, Identity of indiscernibles.

• [W] Ludwig Wittgenstein, Tractatus Logico-Philosophicus 5.5303, ed. C. K. Ogden, Routledge & Kegan Paul, 1922.

Longtime friend and colleague Boaz Barak sent me a fascinating New York Times Magazine article that profiles people who lost their jobs or otherwise had their lives ruined, because of a single remark that then got amplified a trillionfold in importance by social media.  (The author, Jon Ronson, also has a forthcoming book on the topic.)  The article opens with Justine Sacco: a woman who, about to board a flight to Cape Town, tweeted “Going to Africa.  Hope I don’t get AIDS.  Just kidding.  I’m white!”

To the few friends who read Sacco’s Twitter feed, it would’ve been obvious that she was trying to mock the belief of many well-off white people that they live in a bubble, insulated from the problems of the Third World; she wasn’t actually mocking black Africans who suffer from AIDS.  In a just world, maybe Sacco deserved someone to take her aside and quietly explain that her tweet might be read the wrong way, that she should be more careful next time.  Instead, by the time she landed in Cape Town, she learned that she’d become the #1 worldwide Twitter trend and a global symbol of racism.  She lost her career, she lost her entire previous life, and tens of thousands of people expressed glee about it.  The article rather heartbreakingly describes Sacco’s attempts to start over.

There are many more stories like the above.  Some I’d already heard about: the father of three who lost his job after he whispered a silly joke involving “dongles” to the person next to him at a conference, whereupon Adria Richards, a woman in front of him, snapped his photo and posted it to social media, to make an example of him as a sexist pig.  (Afterwards, a counter-reaction formed, which successfully got Richards fired from her job: justice??)  Other stories I hadn’t heard.

Reading this article made it clear to me just how easily I got off, in my own recent brush with the online shaming-mobs.  Yes, I made the ‘mistake’ of writing too openly about my experiences as a nerdy male teenager, and the impact that one specific aspect of feminist thought (not all of feminism!) had had on me.  Within the context of the conversation that a few nerdy men and women were having on this blog, my opening up led to exactly the results I was hoping for: readers thoughtfully sharing their own experiences, a meaningful exchange of ideas, even (dare I say it?) glimmers of understanding and empathy.

Alas, once the comment was wrested from its original setting into the clickbait bazaar, the story became “MIT professor explains: the real oppression is having to learn to talk to women” (the title of Amanda Marcotte’s hit-piece, something even some in Marcotte’s ideological camp called sickeningly cruel).  My photo was on the front page of Salon, next to the headline “The plight of the bitter nerd.”  I was subjected to hostile psychoanalysis not once but twice on ‘Dr. Nerdlove,’ a nerd-bashing site whose very name drips with irony, rather like the ‘Democratic People’s Republic of Korea.’  There were tweets and blog comments that urged MIT to fire me, that compared me to a mass-murderer, and that “deduced” (from first principles!) all the ways in which my parents screwed up in raising me and my female students cower in fear of me.   And yes, when you Google me, this affair now more-or-less overshadows everything else I’ve done in my life.

But then … there were also hundreds of men and women who rose to my defense, and they were heavily concentrated among the people I most admire and respect.  My supporters ranged from the actual female students who took my classes or worked with me or who I encouraged in their careers, from whom there was only kindness, not a single negative word; to the shy nerds who thanked me for being one of the only people to acknowledge their reality; to the lesbians and bisexual women who told me my experience also resonated with them; to the female friends and colleagues who sent me notes urging me to ignore the nonsense.  In the end, not only have I not lost any friends over this, I’ve gained new ones, and I’ve learned new sides of the friends I had.

Oh, and I didn’t get any death threats: I guess that’s good!  (Once in my life I did get death threats—graphic, explicit threats, about which I had to contact the police—but it was because I refused to publicize someone’s P=NP proof.)

Since I was away from campus when this blew up, I did feel some fear about the professional backlash that would await me on my return.  Would my office be vandalized?  Would activist groups be protesting my classes?  Would MIT police be there to escort me from campus?

Well, you want to know what happened instead?  Students and colleagues have stopped me in the hall, or come by my office, just to say they support me.  My class has record enrollment this term.  I was invited to participate in MIT’s Diversity Summit, since the organizers felt it would mean a lot to the students to see someone there who had opened up about diversity issues in STEM in such a powerful way.  (I regretfully had to decline, since the summit conflicted with a trip to Stanford.)  And an MIT graduate women’s reading group invited me for a dinner discussion (at my suggestion, Laurie Penny participated as well).  Imagine that: not only are MIT’s women’s groups not picketing me, they’re inviting me over for dinner!  Is there any better answer to the claim, urged on me by some of my overzealous supporters, that the bile of Amanda Marcotte represents all of feminism these days?

Speaking of which, I met Laurie Penny for coffee last month, and she and I quickly hit it off.  We’ve even agreed to write a joint blog post about our advice for shy nerds.  (In my What I Believe post, I had promised a post of advice for shy female nerds—but at Laurie’s urging, we’re broadening the focus to shy nerds of both sexes.)  Even though Laurie’s essay is the thing that brought me to the attention of the Twitter-mobs (which wasn’t Laurie’s intent!), and even though I disagreed with several points in her essay, I knew on reading it that Laurie was someone I’d enjoy talking to.  Unlike so much writing by online social justice activists, which tends to be encrusted with the specialized technical terms of that field—you know, terms like “asshat,” “shitlord,” “douchecanoe,” and “precious feefees of entitled white dudes”—Laurie’s prose shone with humanity and vulnerability: her own, which she freely shared, and mine, which she generously acknowledged.

Overall, the response to my comment has never made me happier or more grateful to be part of the STEM community (I never liked the bureaucratic acronym “STEM,” but fine, I’ll own it).  To many outsiders, we STEM nerds are a sorry lot: we’re “sperglords” (yes, slurs are fine, as long as they’re directed against the right targets!) who might be competent in certain narrow domains, but who lack empathy and emotional depth, and are basically narcissistic children.  Yet somehow when the chips were down, it’s my fellow STEM nerds, and people who hang out with STEM nerds a lot, who showed me far more empathy and compassion than many of the “normals” did.  So if STEM nerds are psychologically broken, then I say: may I surround myself, for the rest of my life, with men and women who are psychologically broken like I am.  May I raise Lily, and any future children I have, to be as psychologically broken as they can be.  And may I stay as far as possible from anyone who’s too well-adjusted.

I reserve my ultimate gratitude for the many women in STEM, friends and strangers alike, who sent me messages of support these past two months.  I’m not ashamed to say it: witnessing how so many STEM women stood up for me has made me want to stand up for them, even more than I did before.  If they’re not called on often enough in class, I’ll call on them more.  If they’re subtly discouraged from careers in science, I’ll blatantly encourage them back.  If they’re sexually harassed, I’ll confront their harassers myself (well, if asked to).  I will listen to them, and I will try to improve.

Is it selfish that I want to help female STEM nerds partly because they helped me?  Here’s the thing: one of my deepest moral beliefs is in the obligation to fight for those among the disadvantaged who don’t despise you, and who wouldn’t gladly rid the planet of everyone like you if they could.  (As I’ve written before, on issue after issue, this belief makes me a left-winger by American standards, and a right-winger by academic ones.)  In the present context, I’d say I have a massive moral obligation toward female STEM nerds and toward Laurie Penny’s version of feminism, and none at all toward Marcotte’s version.

All this is just to say that I’m unbelievably lucky—privileged (!)—to have had so many at MIT and elsewhere willing to stand up for me, and to have reached in a stage in life where I’m strong enough to say what I think and to weather anything the Internet says back.  What worries me is that others, more vulnerable, didn’t and won’t have it as easy when the Twitter hate-machine turns its barrel on them.  So in the rest of this post, I’d like to discuss the problem of what to do about social-media shaming campaigns that aim to, and do, destroy the lives of individuals.  I’m convinced that this is a phenomenon that’s only going to get more and more common: something sprung on us faster than our social norms have evolved to deal with it.  And it would be nice if we could solve it without having to wait for a few high-profile suicides.

But first, let me address a few obvious questions about why this problem is even a problem at all.

Isn’t social shaming as old as society itself—and permanent records of the shaming as old as print media?

Yes, but there’s also something fundamentally new about the problem of the Twitter-mobs.  Before, it would take someone—say, a newspaper editor—to make a conscious decision to the effect, “this comment is worth destroying someone’s life over.”  Today, there might be such an individual, but it’s also possible for lives to be destroyed in a decentralized, distributed fashion, with thousands of Twitterers collaborating to push a non-story past the point of no return.  And among the people who “break” the story, not one has to intend to ruin the victim’s life, or accept responsibility for it afterward: after all, each one made the story only ε bigger than it already was.  (Incidentally, this is one reason why I haven’t gotten a Twitter account: while it has many worthwhile uses, it’s also a medium that might as well have been designed for mobs, for ganging up, for status-seeking among allies stripped of rational arguments.  It’s like the world’s biggest high school.)

Don’t some targets of online shaming campaigns, y’know, deserve it?

Of course!  Some are genuine racists or misogynists or homophobes, who once would’ve been able to inflict hatred their entire lives without consequence, and were only brought down thanks to social media.  The trouble is, the participants in online shaming campaigns will always think they’re meting out righteous justice, whether they are or aren’t.  But there’s an excellent reason why we’ve learned in modern societies not to avenge even the worst crimes via lynch mobs.  There’s a reason why we have trials and lawyers and the opportunity for the accused to show their innocence.

Some might say that no safeguards are possible or necessary here, since we’re not talking about state violence, just individuals exercising their free speech right to vilify someone, demand their firing, that sort of thing.  Yet in today’s world, trial-by-Internet can be more consequential than the old kind of trial: would you rather spend a year in jail, but then be free to move to another town where no one knew about it, or have your Google search results tarnished with lurid accusations (let’s say, that you molested children) for the rest of your life—to have that forever prevent you from getting a job or a relationship, and have no way to correct the record?  With trial by Twitter, there’s no presumption of innocence, no requirement to prove that any other party was harmed, just the law of the schoolyard.

Whether shaming is justified in a particular case is a complicated question, but for whatever it’s worth, here are a few of the questions I would ask:

• Did the person express a wish for anyone (or any group of people) to come to harm, or for anyone’s rights to be infringed?
• Did the person express glee or mockery about anyone else’s suffering?
• Did the person perpetrate a grievous factual falsehood—like, something one could prove was a falsehood in a court of law?
• Did the person violate anyone else’s confidence?
• How much does the speaker’s identity matter?  If it had been a man rather than a woman (or vice versa) saying parallel things, would we have taken equal offense?
• Does the comment have what obscenity law calls “redeeming social value”?  E.g., does it express an unusual viewpoint, or lead to an interesting discussion?

Of course, even in those cases where shaming campaigns are justified, they’ll sometimes be unproductive and ill-advised.

Aren’t society’s most powerful fair targets for public criticism, even mocking or vicious criticism?

Of course.  Few would claim, for example, that we have an ethical obligation to ease up on Todd Akin over his “legitimate rape” remarks, since all the rage might give Akin an anxiety attack.  Completely apart from the (de)merits of the remarks, we accept that, when you become (let’s say) an elected official, a CEO, or a university president, part of the bargain is that you no longer get to complain if people organize to express their hatred of you.

But what’s striking about the cases in the NYT article is that it’s not public figures being gleefully destroyed: just ordinary people who in most cases, made one ill-advised joke or tweet, no worse than countless things you or I have probably said in private among friends.  The social justice warriors try to justify what would otherwise look like bullying by shifting attention away from individuals: sure, Justine Sacco might be a decent person, but she stands for the entire category of upper-middle-class, entitled white women, a powerful structural force against whom the underclass is engaged in a righteous struggle.  Like in a war, the enemy must be fought by any means necessary, even if it means picking off one hapless enemy foot-soldier to make an example to the rest.  And anyway, why do you care more about this one professional white woman, than about the millions of victims of racism?  Is it because you’re a racist yourself?

I find this line of thinking repugnant.  For it perverts worthy struggles for social equality into something callous and inhuman, and thereby undermines the struggles themselves.  It seems to me to have roughly the same relation to real human rights activism as the Inquisition did to the ethical teachings of Jesus.  It’s also repugnant because of its massive chilling effect: watching a few shaming campaigns is enough to make even the most well-intentioned writer want to hide behind a pseudonym, or only offer those ideas and experiences that are sure to win approval.  And the chilling effect is not some accidental byproduct; it’s the goal.  This negates what, for me, is a large part of the promise of the Internet: that if people from all walks of life can just communicate openly, everything made common knowledge, nothing whispered or secondhand, then all the well-intentioned people will eventually come to understand each other.

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If I’m right that online shaming of decent people is a real problem that’s only going to get worse, what’s the solution?  Let’s examine five possibilities.

(1) Libel law.  For generations, libel has been recognized as one of the rare types of speech that even a liberal, democratic society can legitimately censor (along with fraud, incitement to imminent violence, national secrets, child porn, and a few others).  That libel is illegal reflects a realistic understanding of the importance of reputation: if, for example, CNN falsely reports that you raped your children, then it doesn’t really matter if MSNBC later corrects the record; your life as you knew it is done.

The trouble is, it’s not clear how to apply libel law in the age of social media.  In the cases we’re talking about, an innocent person’s life gets ruined because of the collective effect of thousands of people piling on to make nasty comments, and it’s neither possible nor desirable to prosecute all of them.  Furthermore, in many cases the problem is not that the shamers said anything untrue: rather, it’s that they “merely” took something true and spitefully misunderstood it, or blew it wildly, viciously, astronomically out of proportion.  I don’t see any legal remedies here.

(2) “Shame the shamers.”  Some people will say the only answer is to hit the shamers with their own weapons.  If an overzealous activist gets an innocent jokester fired from his job, shame the activist until she’s fired from her job.  If vigilantes post the jokester’s home address on the Internet with crosshairs overlaid, find the vigilantes’ home addresses and post those.  It probably won’t surprise many people that I’m not a fan of this solution.  For it only exacerbates the real problem: that of mob justice overwhelming reasoned debate.  The most I can say in favor of vigilantism is this: you probably don’t get to complain about online shaming, if what you’re being shamed for is itself a shaming campaign that you prosecuted against a specific person.

(In a decade writing this blog, I can think of exactly one case where I engaged in what might be called a shaming campaign: namely, against the Bell’s inequality denier Joy Christian.  Christian had provoked me over six years, not merely by being forehead-bangingly wrong about Bell’s theorem, but by insulting me and others when we tried to reason with him, and by demanding prize money from me because he had ‘proved’ that quantum computing was a fraud.  Despite that, I still regret the shaming aspects of my Joy Christian posts, and will strive not to repeat them.)

(3) Technological solutions.  We could try to change the functioning of the Internet, to make it harder to use it to ruin people’s lives.  This, more-or-less, is what the European Court of Justice was going for, with its much-discussed recent ruling upholding a “right to be forgotten” (more precisely, a right for individuals to petition for embarrassing information about them to be de-listed from search engines).  Alas, I fear that the Streisand effect, the Internet’s eternal memory, and the existence of different countries with different legal systems will forever make a mockery of all such technological solutions.  But, OK, given that Google is constantly tweaking its ranking algorithms anyway, maybe it could give less weight to cruel attacks against non-public-figures?  Or more weight (or even special placement) to sites explaining how the individual was cleared of the accusations?  There might be scope for such things, but I have the strong feeling that they should be done, if at all, on a voluntary basis.

(4) Self-censorship.  We could simply train people not to express any views online that might jeopardize their lives or careers, or at any rate, not to express those views under their real names.  Many people I’ve talked to seem to favor this solution, but I can’t get behind it.  For it effectively cedes to the most militant activists the right to decide what is or isn’t acceptable online discourse.  It tells them that they can use social shame as a weapon to get what they want.  When women are ridiculed for sharing stories of anorexia or being sexually assaulted or being discouraged from careers in science, it’s reprehensible to say that the solution is to teach those women to shut up about it.  I not only agree with that but go further: privacy is sometimes important, but is also an overrated value.  The respect that one rational person affords another for openly sharing the truth (or his or her understanding of the truth), in a spirit of sympathy and goodwill, is a higher value than privacy.  And the Internet’s ability to foster that respect (sometimes!) is worth defending.

(5) Standing up.  And so we come to the only solution that I can wholeheartedly stand behind.  This is for people who abhor shaming campaigns to speak out, loudly, for those who are unfairly shamed.

At the nadir of my own Twitter episode, when it felt like my life was now finished, throw in the towel, the psychiatrist Scott Alexander wrote a 10,000-word essay in my defense, which also ranged controversially into numerous other issues.  In a comment on his girlfriend Ozy’s blog, Alexander now says that he regrets aspects of Untitled (then again, it was already tagged “Things I Will Regret Writing” when he posted it!).  In particular, he now feels that the piece was too broad in its critique of feminism.  However, he then explains as follows what motivated him to write it:

Scott Aaronson is one of the nicest and most decent people in the world, who does nothing but try to expand human knowledge and support and mentor other people working on the same in a bunch of incredible ways. After a lot of prompting he exposed his deepest personal insecurities, something I as a psychiatrist have to really respect. Amanda Marcotte tried to use that to make mincemeat of him, casually, as if destroying him was barely worth her time. She did it on a site where she gets more pageviews than he ever will, among people who don’t know him, and probably stained his reputation among nonphysicists permanently. I know I have weird moral intuitions, but this is about as close to pure evil punching pure good in the face just because it can as I’ve ever seen in my life. It made me physically ill, and I mentioned the comments of the post that I lost a couple pounds pacing back and forth and shaking and not sleeping after I read it. That was the place I was writing from. And it was part of what seemed to me to be an obvious trend, and although “feminists vs. nerds” is a really crude way of framing it, I couldn’t think of a better one in that mental state and I couldn’t let it pass.

I had three reactions on reading this.  First, if there is a Scott in this discussion who’s “pure good,” then it’s not I.  Second, maybe the ultimate solution to the problem of online shaming mobs is to make a thousand copies of Alexander, and give each one a laptop with an Internet connection.  But third, as long as we have only one of him, the rest of us have a lot of work cut out for us.  I know, without having to ask, that the only real way I can thank Alexander for coming to my defense, is to use this blog to defend other people (anywhere on the ideological spectrum) who are attacked online for sharing in a spirit of honesty and goodwill.  So if you encounter such a person, let me know—I’d much prefer that to letting me know about the latest attempt to solve NP-complete problems in polynomial time with some analog contraption.

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Unrelated Update: Since I started this post with Boaz Barak, let me also point to his recent blog post on why theoretical computer scientists care so much about asymptotics, despite understanding full well that the constants can overwhelm them in practice.  Boaz articulates something that I’ve tried to say many times, but he’s crisper and more eloquent.

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Update (Feb. 27): Since a couple people asked, I explain here what I see as the basic problems with the “Dr. Nerdlove” site.

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Update (Feb. 28): In the middle of this affair, perhaps the one thing that depressed me the most was Salon‘s “Plight of the bitter nerd” headline. Random idiots on the Internet were one thing, but how could a “serious,” “respectable” magazine lend its legitimacy to such casual meanness? I’ve now figured out the answer: I used to read Salon sometimes in the late 90s and early 2000s, but not since then, and I simply hadn’t appreciated how far the magazine had descended into clickbait trash. There’s an amusing fake Salon Twitter account that skewers the magazine with made-up headlines (“Ten signs your cat might be racist” / “Nerd supremacism: should we have affirmative action to get cool people into engineering?”), mixed with actual Salon headlines, in such a way that it would be difficult to tell many of them apart were they not marked. (Indeed, someone should write a web app where you get quizzed to see how well you can distinguish them.) “The plight of the bitter nerd” is offered there as one of the real headlines that’s indistinguishable from the parodies.

I’m writing about ‘concepts of sameness’ for Elaine Landry’s book Category Theory for the Working Philosopher. After an initial section on a passage by Heraclitus, I had planned to write a bit about Gongsun Long’s white horse paradox — or more precisely, his dialog When a White Horse is Not a Horse.

However, this is turning out to be harder than I thought, and more of a digression than I want. So I’ll probably drop this plan. But I have a few preliminary notes, and I might as well share them.

Gongsun Long

Gongsun Long was a Chinese philosopher who lived from around 325 to 250 BC. Besides the better-known Confucian and Taoist schools of Chinese philosophy, another important school at this time was the Mohists, who were more interested in science and logic. Gongsun Long is considered a member of the Mohist-influenced ‘School of Names’: a loose group of logicians, not really a school in any real sense. They are remembered largely for their paradoxes: for example, they independently invented a version of Zeno’s paradox.

As with Heraclitus, most of Gongsun Long’s writings are lost. Joseph Needham [N] has written that this is one of the worst losses of ancient Chinese texts, which in general have survived much better than the Greek ones. The Gongsun Longzi is a text that originally contained 14 of his essays. Now only six survive. The second essay discusses the question “when is a white horse not a horse?”

The White Horse Paradox

When I first heard this ‘paradox’ I didn’t get it: it just seemed strange and silly, not a real paradox. I’m still not sure I get it. But I’ve decided that’s what makes it interesting: it seems to rely on modes of thought, or speech, that are quite alien to me. What counts as a ‘paradox’ is more culturally specific than you might realize.

If a few weeks ago you’d asked me how the paradox goes, I might have said something like this:

A white horse is not a horse, because where there is whiteness, there cannot be horseness, and where there is horseness there cannot be whiteness.

However this is inaccurate because there was no word like ‘whiteness’ (let alone ‘horseness’) in classical Chinese.

Realizing that classical Chinese does not have nouns and adjectives as separate parts of speech may help explain what’s going on here. To get into the mood for this paradox, we shouldn’t think of a horse as a thing to which the predicate ‘whiteness’ applies. We shouldn’t think of the world as consisting of things xx and, separately, predicates PP, which combine to form assertions P(x)P(x). Instead, both ‘white’ and ‘horse’ are on more of an equal footing.

I like this idea because it suggests that predicate logic arose in the West thanks to peculiarities of Indo-European grammar that aren’t shared by all languages. This could free us up to have some new ideas.

Here’s how the dialog actually goes. I’ll use Angus Graham’s translation because it tries hard not to wash away the peculiar qualities of classical Chinese:

Is it admissible that white horse is not-horse?

S. It is admissible.

O. Why?

S. ‘Horse’ is used to name the shape; ‘white’ is used to name the color. What names the color is not what names the shape. Therefore I say white horse is not horse.

O. If we take horses having color as nonhorse, since there is no colorless horse in the world, can we say there is no horse in the world?

S. Horse obviously has color, which is why there is white horse. Suppose horse had no color, then there would just be horse, and where would you find white horse. The white is not horse. White horse is white and horse combined. Horse and white is horse, therefore I say white horse is non-horse.

(Chad Hansen writes: “Most commentaries have trouble with the sentence before the conclusion in F-8, “horse and white is horse,” since it appears to contradict the sophist’s intended conclusion. But recall the Mohists asserted that ox-horse both is and is not ox.” I’m not sure if that helps me, but anyway….)

O. If it is horse not yet combined with white which you deem horse, and white not yet combined with horse which you deem white, to compound the name ‘white horse’ for horse and white combined together is to give them when combined their names when uncombined, which is inadmissible. Therefore, I say, it is inadmissible that white horse is not horse.

S. ‘White’ does not fix anything as white; that may be left out of account. ‘White horse’ has ‘white’ fixing something as white; what fixes something as white is not ‘white’. ‘Horse’ neither selects nor excludes any colors, and therefore it can be answered with either yellow or black. ‘White horse’ selects some color and excludes others, and the yellow and the black are both excluded on grounds of color; therefore one may answer it only with white horse. What excludes none is not what excludes some. Therefore I say: white horse is not horse.

One possible anachronistic interpretation of the last passage is

The set of white horses is not equal to the set of horses, so “white horse” is not “horse”.

This makes sense, but it seems like a way of saying we can have S⊆TS \subseteq T while also S≠TS \ne T. That would be a worthwhile observation around 300 BC — and it would even be worth trying to get people upset about this, back then! But it doesn’t seem very interesting today.

A more interesting interpretation of the overall dialog is given by Chad Hansen [H]. He argues that to understand it, we should think of both ‘white’ and ‘horse’ as mass nouns or ‘kinds of stuff’.

The issue of how two kinds of stuff can be present in the same place at the same time is a bit challenging — we see Plato battling with it in the Parmenides — and in some sense western mathematics deals with it by switching to a different setup, where we have a universe of entities xx of which predicates PP can be asserted. If xx is a horse and PP is ‘being white’, then P(x)P(x) says the horse is white.

However, then we get Leibniz’s principle of the ‘indistinguishability of indiscernibles’, which is a way of defining equality. This says that x=yx = y if and only if P(x)⇔P(y)P(x) \iff P(y) for all predicates PP. By this account, an entity really amounts to nothing more than the predicates it satisfies!

This is where equality comes in — but as I said, all of this is seeming like too much of a distraction from my overall goals for this essay right now.

Notes

• [N] Joseph Needham, Science and Civilisation in China vol. 2: History of Scientific Thought, Cambridge U. Press, Cambridge, 1956, p. 185.

• [H] Chad Hansen, Mass nouns and “A white horse is not a horse”, Philosophy East and West 26 (1976), 189–209.

That the speed of light in free space is constant is a cornerstone of modern physics. However, light beams have finite transverse size, which leads to a modification of their wave vectors resulting in a change to their phase and group velocities. We study the group velocity of single photons by measuring a change in their arrival time that results from changing the beam’s transverse spatial structure. Using time-correlated photon pairs, we show a reduction in the group velocity of photons in both a Bessel beam and photons in a focused Gaussian beam. In both cases, the delay is several micrometers over a propagation distance of ~1 meter. Our work highlights that, even in free space, the invariance of the speed of light only applies to plane waves. Authors: Daniel Giovannini, Jacquiline Romero, Václav Potoček, Gergely Ferenczi, Fiona Speirits, Stephen M. Barnett, Daniele Faccio, Miles J. Padgett