Their ideal is the experiment that, in one session, shoots down a claim cleanly and neatly. So let’s bring in dowsers who claim to be able to detect water flowing underground, set up control pipes and water-filled pipes, run them through their paces, and see if they meet reasonable statistical criteria. That’s science, it works, it effectively addresses an individual’s very specific claim, and I’m not saying that’s wrong; that’s a perfectly legitimate scientific experiment.

I’m saying that’s not the whole operating paradigm of all of science.

Plenty of scientific ideas are not immediately testable, or directly testable, or testable in isolation. For example: the planets in our solar system aren’t moving the way Newton’s laws say they should. Are Newton’s laws of gravity wrong, or are there other gravitational influences which satisfy the Newtonian equations but which we don’t know about? Once, it turned out to be the latter (the discovery of Neptune), and once, it turned out to be the former (the precession of Mercury’s orbit, which required Einstein’s general relativity to explain).

There are different mathematical formulations of the same subject which give the same predictions for the outcomes of experiments, but which suggest different new ideas for directions to explore. (E.g., Newtonian, Lagrangian and Hamiltonian mechanics; or density matrices and SIC-POVMs.) There are ideas which are proposed for good reason but hang around for decades awaiting a direct experimental test—perhaps one which could barely have been imagined when the idea first came up. Take directed percolation: a simple conceptual model for fluid flow through a randomized porous medium. It was first proposed in 1957. The mathematics necessary to treat it cleverly was invented (or, rather, adapted from a different area of physics) in the 1970s…and then forgotten…and then rediscovered by somebody else…connections with other subjects were made… Experiments were carried out on systems which almost behaved like the idealization, but always turned out to differ in some way… until 2007, when the behaviour was finally caught in the wild. And the experiment which finally observed a directed-percolation-class phase transition with quantitative exactness used a liquid crystal substance which wasn’t synthesized until 1969.

You don’t need to go dashing off to quantum gravity to find examples of ideas which are hard to test in the laboratory, or where mathematics long preceded experiment. (And if you do, don’t forget the other applications being developed for the mathematics invented in that search.) Just think very hard about the water dripping through coffee grounds to make your breakfast.

I’ve met, corresponded with and in a couple cases collaborated with authors of these papers, but I’ve had no input on writing or peer-reviewing any of them.

• B. Allen and M. A. Nowak (2013), “Cooperation and the fate of microbial societies” PLOS Biology 11, 4: e1001549.
  This is an introductory overview of what can happen when ecological and evolutionary processes occur on comparable timescales and feed back upon one another. It summarizes recent experimental and model-building work on the topic, as realised in Saccharomyces cerevisiae.
• J. A. Damore and J. Gore (2012), “Understanding microbial cooperation” Journal of Theoretical Biology 299: 31–41, DOI:10.1016/j.jtbi.2011.03.008 (PDF). PMID:21419783.
  Next, get a grounding in the sorts of complications which arise with real organisms, even tiny ones, and to see how old mathematics isn’t enough for new questions. In video-game terminology, this is where we defeat the Level 1 boss, “relatedness.” Recommended with the proviso that writing the Price Equation using covariance notation, though common, can be misleading. This is mentioned in the text, but it deserves special emphasis.
• B. Allen and C. E. Tarnita (2012), “Measures of success in a class of evolutionary models with fixed population size and structure” Journal of Mathematical Biology, DOI:10.1007/s00285-012-0622-x (PDF).
  This paper builds up in a nice way the treatment of evolution as a stochastic process. It is probably the most mathematical article on this list. The point made near the end about the way the Price Equation is often misleadingly written is an important one.
• M. Perc et al. (2013), “Evolutionary dynamics of group interactions on structured populations: a review” Journal of the Royal Society Interface 10, 80: 20120997; DOI:10.1098/rsif.2012.0997.
  This review article covers what’s known about evolutionary games on complex network substrates, including adaptive networks. It also touches on nonlinearity in payoff functions, which takes us beyond the idea that “assortment” is everything. To continue the gaming metaphor, this is the boss fight of Level 2.
• P. E. Smaldino, J. C. Schank and R. McElreath (2013), “Increased costs of cooperation help cooperators in the long run” American Naturalist 181 DOI:10.1086/669615 (PDF).
  One of many papers which points out the shortcomings of “moment closure” techniques, this also makes a key point about fitness being a question of timescale.
• S. B. Araujo, G. M. Viswanathan and M. A. de Aguiar (2010), “Home range evolution and its implication in population outbreaks” Philosophical Transactions of the Royal Society A 368, 1933: 5661–77. DOI:10.1098/rsta.2010.0270, PMID:21078641.
  Pair approximations done carefully, with acknowledgements of their limitations, too.
• B. Simon, J. A. Fletcher and M. Doebeli (2013), “Towards a general theory of group selection” Evolution (PDF).
  The distinction between type I and type II multilevel selection is crucial, and often crucially overlooked. Simon, Fletcher and Doebeli build quantitative models of an MLS-2 scenario: there are explicit group-level dynamics, but the set of groups is unstructured.
• B. Simon and A. Nielsen (2013), “Numerical solutions and animations of group selection dynamics” Evolutionary Ecology Research, 14: 757–68 (PDF).
  A follow-up to the previous paper.
• P. Bijma and M. J. Wade (2008), “The joint effects of kin, multilevel selection and indirect genetic effects on response to genetic selection” Journal of Evolutionary Biology 21: 1175–88, DOI:10.1111/j.1420-9101.2008.01550.x.
  In the context of MLS-1 scenarios, and considering only generation-to-generation changes, ideas which sound very different when said in words are mathematically interconvertible, with only linear transformations of coordinates. I put this down here because of caveats about the distinction between MLS-1 and MLS-2, the terminology of “relatedness” versus “assortment” and the solecism of writing the Price Equation with covariances.

Last updated 7 May 2013.

1. Hamlet couldn’t have said anything much before the play starts, because he was off at school in Wittenberg.

2. He sees the ghost on the night of the first day in the play where he appears. Not a long delay there. And his reaction to being told “The serpent that did sting thy father’s life now wears his crown” is, “O my prophetic soul!” Or, in a different idiom, “Called it!”

3. He has every reason not to act rashly, because (a) he wants to be King (Claudius “popp’d in between the election and my hopes”), and (b) he can’t trust that the ghost is really his father. “The devil hath power to assume a pleasing shape”, etc. Watch your Star Trek, people! Emo!Hamlet is a comparatively recent invention. Prior to the late 1700s, the standard was to play Hamlet as a chessmaster, a brilliant young man trying to turn a bad situation to his advantage, facing a shrewd opponent.

4. It’s the characters in the play who remark on Hamlet’s “transformation”. That’s why Claudius sends for Rosencrantz and Guildenstern.

Welcome, dear Rosencrantz and Guildenstern!
Moreover that we much did long to see you,
The need we have to use you did provoke
Our hasty sending. Something have you heard
Of Hamlet’s transformation; so call it,
Sith nor the exterior nor the inward man
Resembles that it was.

5. He’s so antisocial that he…has a girlfriend? And, as Claudius says, is beloved by the general populace of Denmark? Indeed, that’s a big part of why Claudius doesn’t have Hamlet killed for stabbing Polonius. As he tells Laertes, he doesn’t want to hurt Gertrude, and in addition…

The other motive,
Why to a public count I might not go,
Is the great love the general gender bear him;
Who, dipping all his faults in their affection,
Would, like the spring that turneth wood to stone,
Convert his gyves to graces; so that my arrows,
Too slightly timber’d for so loud a wind,
Would have reverted to my bow again,
And not where I had aim’d them.

6. He won’t kill his uncle first because he wants to be crowned, not executed; and second, because he wants Claudius damned, not just dead.

A villain kills my father; and for that,
I, his sole son, do this same villain send
To heaven.
O, this is hire and salary, not revenge.
He took my father grossly, full of bread;
With all his crimes broad blown, as flush as May;
And how his audit stands who knows save heaven?
But in our circumstance and course of thought,
‘Tis heavy with him: and am I then revenged,
To take him in the purging of his soul,
When he is fit and season’d for his passage?
No!
Up, sword; and know thou a more horrid hent:
When he is drunk asleep, or in his rage,
Or in the incestuous pleasure of his bed;
At gaming, swearing, or about some act
That has no relish of salvation in’t;
Then trip him, that his heels may kick at heaven,
And that his soul may be as damn’d and black
As hell, whereto it goes. My mother stays:
This physic but prolongs thy sickly days.

7. He jokes bawdily with Rosencrantz and Guildenstern when they appear. They sell him out and were escorting him to his execution.

8. Horatio, an emotionless stoic? The man tries to commit suicide when he sees Hamlet dying.

9. Hamlet is backed into the swordfight (Osric: “I mean, my lord, the opposition of your person in trial”).

10. And, by that point, Hamlet has found that things only go his way when he seizes whatever random opportunities which come by. Hamlet in Act 5 is basically, “Trying to be L in Death Note didn’t work out. Fighting pirates, though, was pretty wicked.”

11. “Lives with his parents his entire life”? Hamlet was away at university in another country.

[deep breath]

OK. Better now.

Abstract:

A stochastic version of the Brusselator model is proposed and studied via the system size expansion. The mean-field equations are derived and shown to yield to organized Turing patterns within a specific parameters region. When determining the Turing condition for instability, we pay particular attention to the role of cross-diffusive terms, often neglected in the heuristic derivation of reaction-diffusion schemes. Stochastic fluctuations are shown to give rise to spatially ordered solutions, sharing the same quantitative characteristic of the mean-field based Turing scenario, in term of excited wavelengths. Interestingly, the region of parameter yielding to the stochastic self-organization is wider than that determined via the conventional Turing approach, suggesting that the condition for spatial order to appear can be less stringent than customarily believed.

See also the commentary by Mehran Kardar.

“Well, there’s this guy who made a movie about a cute farmboy in the boondocks who never knew his real father, dreams of outer space, fights in a bar full of crazy aliens and then goes up against the evil overlord who killed his father—this really nasty guy with Roman Empire trappings, favorite color black, lots of glowy green energy—and who flies around in a giant ship bigger than anything else in space blowing up planets. He blasts the home planet of one of the heroes early on, so we know he’s serious, and at the end, it’s a race with the clock to stop him blowing up the planet that’s really important. But the good guys win and there’s a flashy award ceremony to wrap it all up.”

“Sounds great! Is there stuff which only makes sense if, like, Fate or Destiny is willing it?”

“Like you wouldn’t believe!”

“Perfect!”

Abstract: Shearing solutions of fibers or polymers tends to align fiber or polymers in the flow direction. Here, non-Brownian rods subjected to oscillatory shear align perpendicular to the flow while the system undergoes a nonequilibrium absorbing phase transition. The slow alignment of the fibers can drive the system through the critical point and thus promote the transition to an absorbing state. This picture is confirmed by a universal scaling relation that collapses the data with critical exponents that are consistent with conserved directed percolation.

I read this one a while ago in non-arXiv preprint form, but now it’s on the arXiv. M. Raghib et al. (2011), “A Multiscale maximum entropy moment closure for locally regulated space-time point process models of population dynamics”, Journal of Mathematical Biology 62, 5: 605–53. arXiv:1202.6092 [q-bio].

Abstract: The pervasive presence spatial and size structure in biological populations challenges fundamental assumptions at the heart of continuum models of population dynamics based on mean densities (local or global) only. Individual-based models (IBM’s) were introduced over the last decade in an attempt to overcome this limitation by following explicitly each individual in the population. Although the IBM approach has been quite insightful, the capability to follow each individual usually comes at the expense of analytical tractability, which limits the generality of the statements that can be made. For the specific case of spatial structure in populations of sessile (and identical) organisms, space-time point processes with local regulation seem to cover the middle ground between analytical tractability and a higher degree of biological realism. Continuum approximations of these stochastic processes distill their fundamental properties, but they often result in infinite hierarchies of moment equations. We use the principle of constrained maximum entropy to derive a closure relationship for one such hierarchy truncated at second order using normalization and the product densities of first and second orders as constraints. The resulting `maxent’ closure is similar to the Kirkwood superposition approximation, but it is complemented with previously unknown correction terms that depend on on the area for which third order correlations are irreducible. This region also serves as a validation check, since it can only be found if the assumptions of the closure are met. Comparisons between simulations of the point process, alternative heuristic closures, and the maxent closure show significant improvements in the ability of the maxent closure to predict equilibrium values for mildly aggregated spatial patterns.

J. A. Bonachela, M. A. Munoz and S. A. Levin (2012). “Patchiness and Demographic Noise in Three Ecological Examples” [arXiv:1205.3389].

Abstract: Understanding the causes and effects of spatial aggregation is one of the most fundamental problems in ecology. Aggregation is an emergent phenomenon arising from the interactions between the individuals of the population, able to sense only—at most—local densities of their cohorts. Thus, taking into account the individual-level interactions and fluctuations is essential to reach a correct description of the population. Classic deterministic equations are suitable to describe some aspects of the population, but leave out features related to the stochasticity inherent to the discreteness of the individuals.Stochastic equations for the population do account for these fluctuation-generated effects by means of demographic noise terms but, owing to their complexity, they can be difficult (or, at times, impossible) to deal with. Even when they can be written in a simple form, they are still difficult to numerically integrate due to the presence of the “square-root” intrinsic noise. In this paper, we discuss a simple way to add the effect of demographic stochasticity to three classic, deterministic ecological examples where aggregation plays an important role. We study the resulting equations using a recently-introduced integration scheme especially devised to integrate numerically stochastic equations with demographic noise. Aimed at scrutinizing the ability of these stochastic examples to show aggregation, we find that the three systems not only show patchy configurations, but also undergo a phase transition belonging to the directed percolation universality class.

A. F. Lütz et al. (2012), “Intransitivity and coexistence in four species cyclic games” [arXiv:1205.6411].

Abstract: Intransitivity is a property of connected, oriented graphs representing species interactions that may drive their coexistence even in the presence of competition, the standard example being the three species Rock-Paper-Scissors game. We consider here a generalization with four species, the minimum number of species that allows other interactions beyond the single loop (one predator, one prey). We show that, contrary to the mean field prediction, on a square lattice the model presents a transition, as the invasion rates change, from a coexistence to a state in which one species gets extinct. Such a dependence on the invasion rates shows that the interaction graph structure alone is not enough to predict the outcome of such models. In addition, different invasion rates permit to tune the level of transitiveness, indicating that for the coexistence of all species to persist, there must be a minimum amount of intransitivity.

Interesting things the authors observe about their model:

1. A phase transition as a function of invasion rate $\chi$ appears in the lattice simulations which the mean-field approximation doesn’t pick up at all.

2. The pair approximation (first-order correction to the mean field) fails too. I’d like to see this treated in more detail, as I’ve grown used to seeing mean-field approximations fail for spatial lattices, so the interesting question is whether the adjustments on top of them do any better.

U. Dobramysl and U. C. Täuber (2012), “Environmental vs. demographic variability in two-species predator-prey models” [arXiv:1206.0973].

Abstract: We investigate the competing effects and relative importance of intrinsic demographic and environmental variability on the evolutionary dynamics of a stochastic two-species Lotka-Volterra model by means of Monte Carlo simulations on a two-dimensional lattice. Individuals are assigned inheritable predation efficiencies; quenched randomness in the spatially varying reaction rates serves as environmental noise. We find that environmental variability enhances the population densities of both predators and prey while demographic variability leads to essentially neutral optimization.

X. Chen et al. (2012), “Impact of generalized benefit functions on the evolution of cooperation in spatial public goods games with continuous strategies” Physical Review E 85: 066133 [arXiv:1206.7119]. Interesting on a first skim (I’ve done some numerical work on a case analogous to their sharp-cutoff limit, in various population structures).

Abstract: Cooperation and defection may be considered as two extreme responses to a social dilemma. Yet the reality is much less clear-cut. Between the two extremes lies an interval of ambivalent choices, which may be captured theoretically by means of continuous strategies defining the extent of the contributions of each individual player to the common pool. If strategies are chosen from the unit interval, where 0 corresponds to pure defection and 1 corresponds to the maximal contribution, the question is what is the characteristic level of individual investments to the common pool that emerges if the evolution is guided by different benefit functions. Here we consider the steepness and the threshold as two parameters defining an array of generalized benefit functions, and we show that in a structured population there exist intermediate values of both at which the collective contributions are maximal. However, as the cost-to-benefit ratio of cooperation increases the characteristic threshold decreases, while the corresponding steepness increases. Our observations remain valid if more complex sigmoid functions are used, thus reenforcing the importance of carefully adjusted benefits for high levels of public cooperation.

L. D. Fernandes and M. A. M. de Aguiar (2012), “Turing patterns and apparent competition in predator-prey food webs on networks” [arXiv:1207.3424].

Abstract: Reaction-diffusion systems may lead to the formation of steady state heterogeneous spatial patterns, known as Turing patterns. Their mathematical formulation is important for the study of pattern formation in general and play central roles in many fields of biology, such as ecology and morphogenesis. In the present study we focus on the role of Turing patterns in describing the abundance distribution of predator and prey species distributed in patches in a scale free network structure. We extend the original model proposed by Nakao and Mikhailov by considering food chains with several interacting pairs of preys and predators. We identify patterns of species distribution displaying high degrees of apparent competition driven by Turing instabilities. Our results provide further indication that differences in abundance distribution among patches may be, at least in part, due to self organized Turing patterns, and not necessarily to intrinsic environmental heterogeneity.

A. Szolnoki et al. (2012), “Defense mechanisms of empathetic players in the spatial ultimatum game” Physical Review Letters 109: 078701 [arXiv:1207.4786].

Abstract: Experiments on the ultimatum game have revealed that humans are remarkably fond of fair play. When asked to share an amount of money, unfair offers are rare and their acceptance rate small. While empathy and spatiality may lead to the evolution of fairness, thus far considered continuous strategies have precluded the observation of solutions that would be driven by pattern formation. Here we introduce a spatial ultimatum game with discrete strategies, and we show that this simple alteration opens the gate to fascinatingly rich dynamical behavior. Besides mixed stationary states, we report the occurrence of traveling waves and cyclic dominance, where one strategy in the cycle can be an alliance of two strategies. The highly webbed phase diagram, entailing continuous and discontinuous phase transitions, reveals hidden complexity in the pursuit of human fair play.

S. M. Messinger and A. Ostling (2012), “The influence of host demography, pathogen virulence, and relationships with pathogen virulence on the evolution of pathogen transmission in a spatial context” Evolutionary Ecology DOI: 10.1007/s10682-012-9594-y.

Abstract: A major challenge in evolutionary ecology is to explain extensive natural variation in transmission rates and virulence across pathogens. Host and pathogen ecology is a potentially important source of that variation. Theory of its effects has been developed through the study of non-spatial models, but host population spatial structure has been shown to influence evolutionary outcomes. To date, the effects of basic host and pathogen demography on pathogen evolution have not been thoroughly explored in a spatial context. Here we use simulations to show that space produces novel predictions of the influence of the shape of the pathogen’s transmission–virulence tradeoff, as well as host reproduction and mortality, on the pathogen’s evolutionary stable transmission rate. Importantly, non-spatial models predict that neither the slope of linear transmission–virulence relationships, nor the host reproduction rate will influence pathogen evolution, and that host mortality will only influence it when there is a transmission–virulence tradeoff. We show that this is not the case in a spatial context, and identify the ecological conditions under which spatial effects are most influential. Thus, these results may help explain observed natural variation among pathogens unexplainable by non-spatial models, and provide guidance about when space should be considered. We additionally evaluate the ability of existing analytical approaches to predict the influence of ecology, namely spatial moment equations closed with an improved pair approximation (IPA). The IPA is known to have limited accuracy, but here we show that in the context of pathogens the limitations are substantial: in many cases, IPA incorrectly predicts evolution to pathogen-driven extinction. Despite these limitations, we suggest that the impact of ecology can still be understood within the conceptual framework arising from spatial moment equations, that of “self-shading’’, whereby the spread of highly transmissible pathogens is impeded by local depletion of susceptible hosts.

I think “descendant-shading” would be a better term than “self-shading” here, since it indicates more clearly the timescales involved.

S. Heilmann, K. Sneppen and S. Krishna (2012), “Coexistence of phage and bacteria on the boundary of self-organized refuges” PNAS 109, 31: 12828–33. DOI: 10.1073/pnas.1200771109.

Abstract: Bacteriophage are voracious predators of bacteria and a major determinant in shaping bacterial life strategies. Many phage species are virulent, meaning that infection leads to certain death of the host and immediate release of a large batch of phage progeny. Despite this apparent voraciousness, bacteria have stably coexisted with virulent phages for eons. Here, using individual-based stochastic spatial models, we study the conditions for achieving coexistence on the edge between two habitats, one of which is a bacterial refuge with conditions hostile to phage whereas the other is phage friendly. We show how bacterial density-dependent, or quorum-sensing, mechanisms such as the formation of biofilm can produce such refuges and edges in a self-organized manner. Coexistence on these edges exhibits the following properties, all of which are observed in real phage–bacteria ecosystems but difficult to achieve together in nonspatial ecosystem models: (i) highly efficient virulent phage with relatively long lifetimes, high infection rates and large burst sizes; (ii) large, stable, and high-density populations of phage and bacteria; (iii) a fast turnover of both phage and bacteria; and (iv) stability over evolutionary timescales despite imbalances in the rates of phage vs. bacterial evolution.

G. Demirel, F. Vazquez, G. A. Böhme and T. Gross (2012), “Moment-Closure Approximations for Discrete Adaptive Networks” [arXiv:1211.0449].

Moment-closure approximations are an important tool in the analysis of the dynamics on both static and adaptive networks. Here, we provide a broad survey over different approximation schemes by applying each of them to the adaptive voter model. While already the simplest schemes provide reasonable qualitative results, even very complex and sophisticated approximations fail to capture the dynamics quantitatively. We then perform a detailed analysis that identifies the emergence of specific correlations as the reason for the failure of established approaches, before presenting a simple approximation scheme that works best in the parameter range where all other approaches fail. By combining a focused review of published results with new analysis and illustrations, we seek to build up an intuition regarding the situations when existing approaches work, when they fail, and how new approaches can be tailored to specific problems.

C. Reigada and M. A. M. de Aguiar (2012), “Host-parasitoid persistence over variable spatio-temporally susceptible habitats: bottom-up effects of ephemeral resources” Oikos 121: 1665–79. DOI:10.1111/j.1600-0706.2011.20259.x.

We experimentally and theoretically investigated the persistence of hosts and parasitoids interacting in a Metapopulation structure consisting of Ephemeral Local Patches (MELPs). We used a host-parasitoid system consisting of necrophagous Diptera species and their pupal parasitoids. The basal resources used by the host species were assumed to be ephemeral, supporting only one generation of individuals before completely disappearing from the environment. We experimentally measured the host-parasitoid persistence and the effects of local demographic processes in two scenarios: (1) constant occurrence of basal resources at a single site (no dispersion or colonization of other sites) and (2) variable occurrence of basal resources between two sites (colonization of a new patch requiring species dispersal). The experimental setup and findings were then formalized into a mathematical model describing the interaction dynamics in a MELP structure. We evaluated the contribution of several factors to the host-parasitoid coexistence, such as resource allocation probability (probability of resource appearance in a site), variation in resource size and number of sites available to receive resources in the MELP. We found that demographic fluctuations and environmental stochasticity affected the density of migrants, patch habitat connectivity, persistence and spatial distribution of interacting species.

They did experiments! On nonsimulated life forms! With rotting meat and everything!

B. Allen, C. E. Tarnita (2012), “Measures of success in a class of evolutionary models with fixed population size and structure” Journal of Mathematical Biology online before print, DOI:10.1007/s00285-012-0622-x.

We investigate a class of evolutionary models, encompassing many established models of well-mixed and spatially structured populations. Models in this class have fixed population size and structure. Evolution proceeds as a Markov chain, with birth and death probabilities dependent on the current population state. Starting from basic assumptions, we show how the asymptotic (long-term) behavior of the evolutionary process can be characterized by probability distributions over the set of possible states. We then define and compare three quantities characterizing evolutionary success: fixation probability, expected frequency, and expected change due to selection. We show that these quantities yield the same conditions for success in the limit of low mutation rate, but may disagree when mutation is present. As part of our analysis, we derive versions of the Price equation and the replicator equation that describe the asymptotic behavior of the entire evolutionary process, rather than the change from a single state. We illustrate our results using the frequency-dependent Moran process and the birth–death process on graphs as examples. Our broader aim is to spearhead a new approach to evolutionary theory, in which general principles of evolution are proven as mathematical theorems from axioms.

(Last update: 2 December 2012.)

When I was in high school—at a pretty well-supported public school, out in the ‘burbs at the comparatively unimpoverished end of town—I took a “precalculus” class my eleventh-grade year. Most of the advanced-track students I knew did the same thing. (If you’d gotten yourself on the even-more-advanced track back in eigth grade, you took precalculus in tenth.) This was supposed to prepare us for taking the AP Calculus class our senior year, which would allow us to get college credit. Instead, it was a thoroughgoing waste of time. The content was a repeat of Algebra II/Trigonometry, which we’d taken the year before, with two exceptions thrown in. The first, probability, was a topic our teacher didn’t know how to teach. In fact, she admitted as much: “I don’t know how to teach probability, so you’re all going to read the book today.” The second, limits, served no purpose. I’ll explain why in a moment.

I suggest the following: scrap “precalculus” and replace it with a year-long statistics course. This plan has several advantages:

It takes care of the legitimate point raised by Hacker’s op-ed, what he called “citizen statistics.” If we need students to grow up being well-informed, numerate contributors to civilisation, let’s have a class on exactly that. Required reading: Darrell Huff’s How To Lie With Statistics (1954, but we could update it easily).

It’s easier to implement than demolishing all of post-arithmetic mathematics education and trying to invent a replacement. Just swap out one class! We could develop the necessary material comparatively quickly and try it out with a straightforward pilot program.

As an ancillary benefit, the course materials could be put online as OpenCourseWare for the general welfare.

Would replacing precalculus with statistics impair the students who go on to take AP Calculus? I doubt it. In fact, by teaching as much statistics as it’s possible to do without calculus, we’d prepare the way for improved calculus lessons. For example, when it comes time to show what you can actually use integration for, we could teach about probability distributions. Statistics class would, in part, be a precalculus course itself—just one with an actual raison d’être.

And as for the limits lesson, what serious harm could there be in missing one topic which had been shoved disjointedly into a clone of another class, and which everyone sees again the next year anyway? Personally, I’m of the mind that limits make more sense if you see them after the basic ideas of differential calculus. That way, you know what you’re aiming to do and why it matters. (This alternative also tracks better with the subject’s history: Newton and Leibniz came long before Weierstrass.) Otherwise, as a maths professor friend of mine told me, they look like a solution dying to meet a problem.

How did that precalculus class of mine teach us limits? Well, we had to do a lot of worksheets. Bear in mind, I’d already read about the subject in my father’s old calculus textbook (a doorstopper whose size and yellowing reminded me of a dictionary—far thicker than necessary for first learning the subject!). The book started with a healthy dose of examples, comparing a question one could answer with algebra or geometry and a corresponding question you’d need calculus for. Calculus, it said, was algebra and geometry “with the addition of the limit process.” What did “taking a limit” mean? If you had a function $f(x)$, then writing

$$ \lim_{x\rightarrow a} f(x) = L $$

means that you could make the value of $f(x)$ as close to $L$ as you wanted, by choosing $x$ to be sufficiently close to $a$.

Now, in my high-school precalculus class, this translated into plodding through several pages of Xeroxed problems. Almost all of them belonged to one of two types. Either you could get the answer by just plugging in the number $a$, so you weren’t learning anything new and couldn’t see any reason to care, or you had to find the limit as $x$ approached 0 of some hairy thing which looked like

$$ \frac{\sin(2y + x) – \sin(2y)}{x}. $$

If you already knew a little calculus, you could see that all the hairy problems were just taking the derivatives of functions. And knowing a bit about derivatives—I mean, knowing as much as a physics class teaches in half an hour—you could get the right answer every time in two lines of work. Those who didn’t have this pipeline of wisdom floundered for half a page of trigonometry, rewriting things back and forth and ending up with the wrong answer. Of course, for knowing what was going on and getting the right answer every time, we got the red pen: “That’s not how we learned to do this problem in class.”

Come to think of it, most of my fond memories of those years are about skipping class.

What would the curriculum for this proposed statistics class look like? We could do worse, I think, than cover the topics here: data description and summarization, means and standard deviations, measures of correlation, hypothesis testing… Flesh it out with examples of the “citizen statistics” variety—why was the SEC’s pilot study on repealing the “uptick rule” flawed?—and you’d have a full year’s worth. Alternatively, a one-semester introductory course with a strong “citizen statistics” flavour could be a warm-up for AP Statistics the following term. (In my experience, AP classes were so focused on taking the official exam at the end of the year that there was no time to spread out and look at anything which wasn’t easily testable in multiple-choice form.) The key point, assuming most school administrators are like the ones I had to deal with, would be revising the official tangled web of prerequisites, so that students could take AP Calculus without having taken precalculus first.

Incidentally, the professor friend I mentioned earlier is going to be teaching precalculus to college kids this fall, and is hoping to make it a better experience than the class I shuffled through in eleventh grade.

It’s been a while since I’ve felt riled enough to blog. But now, the spirit moves within me once more.

First, I encourage you to read Andrew Hacker’s op-ed in The New York Times, “Is Algebra Necessary?” Then, sample a few reactions:

• PZ Myers, “A modest proposal”
• Melanie Tannenbaum, “Algebra is Necessary: But what about how it’s taught?”
• Cathy O’Neil, “Does mathematics have a place in higher education?”
• Rob Knop, “When Andrew Hacker asks ‘Is Algebra Necessary?’, why doesn’t he just ask ‘Is High School Necessary?’”
• Mike the Mad Biologist, “It’s Not the Algebra, It’s the Arithmetic”
• Dana Goldstein, “On Algebra, High Expectations, and the Common Core” (Hacker conveniently avoids mentioning that “because 48 states and territories are planning to adopt the Common Core, the energy in school reform is tilting very much in favor of algebra”)
• Daniel Willingham, “Yes, algebra is necessary”
• Evelyn Lamb, “Abandoning Algebra is Not the Answer”
• RiShawn Biddle, “Why Algebra Matters (and Why Andrew Hacker is Off-Target)”
• Alexandra Petri, “The end of algebra”
• Mark C. Chu-Carroll, “Mathematical Illiteracy in the NYT” (good summary of Hacker: “There are no valid arguments to support the teaching of math, except for the valid ones, but I’m going to exclude those.”)

I will try to gather a few observations here which I haven’t seen made elsewhere, for the most part.

Towards the end, Hacker’s reasoning gets just bizarre. He keeps emphasising how important “citizen statistics” is. I’m baffled as to how one could teach statistics in any useful way without the material he wants to throw out! Prerequisites: we needz them. “Is Algebra Necessary?” If you want to do statistics or economics, yes, it is.

Quantitative literacy clearly is useful in weighing all manner of public policies, from the Affordable Care Act, to the costs and benefits of environmental regulation, to the impact of climate change. Being able to detect and identify ideology at work behind the numbers is of obvious use. Ours is fast becoming a statistical age, which raises the bar for informed citizenship. What is needed is not textbook formulas but greater understanding of where various numbers come from, and what they actually convey.

OK, so how are we supposed to teach where “numbers come from” and “what they actually convey” when the students can’t manipulate algebraic formulas? That’s like expecting to raise a generation of musicians who’ve never practised a scale.

It’s also nice how he skips past the serious problems we have with infrastructure. You know, schools not being able to buy books any more, and teachers who feel they aren’t capable of teaching even middle-school mathematics—because they aren’t. But saying that teachers need to be trained and books must be bought doesn’t make you a daring iconoclast worthy of a New York Times soapbox, now, does it?

Hacker’s standard for what counts as advanced—and, therefore, unnecessary—mathematics seems to drift from algebra to calculus:

Medical schools like Harvard and Johns Hopkins demand calculus of all their applicants, even if it doesn’t figure in the clinical curriculum, let alone in subsequent practice.

Maybe because doctors need to understand clinical trials, which requires understanding statistics, which in turn requires calculus? (I’m looking at The Cartoon Guide to Statistics, on page 66. Yup, integral sign.) Hey, here’s a radical thought: maybe not enough doctors really get what’s going on with clinical trials, and we’d be better off if more of them knew slightly higher mathematics, instead of fewer?

Of course, in Hacker’s world, everyone knows what they want to be when they grow up, and nobody needs training for anything else in case they want or need to switch to a new job.

It’s not hard to understand why Caltech and M.I.T. want everyone to be proficient in mathematics. But it’s not easy to see why potential poets and philosophers face a lofty mathematics bar.

Philosophers don’t need to know mathematics? Not even mathematics which is hundreds or thousands of years old? What. The. Fuck?

(by Oliver Byrne)

Oh, poor Euclid. Such a fate, and for you who alone have looked on beauty bare….

How many college graduates remember what Fermat’s dilemma was all about?

I’m guessing Hacker is referring to Fermat’s Last Theorem, but I don’t really know. (Nor does anyone else I know who read Hacker’s piece.) It doesn’t really fit the sense of “dilemma.” Fermat stated a problem and claimed that he had a solution. Perhaps the “dilemma” was that he felt compelled to write it down even though he didn’t have space to include his proof? Come on, I’m trying to be charitable here.

I might as well close with this bit from Hacker:

Yes, young people should learn to read and write and do long division, whether they want to or not. But there is no reason to force them to grasp vectorial angles and discontinuous functions.

So: the line is drawn wherever Hacker feels it should. Good to know. (I wonder: if long division is mandatory, what about, say, extracting square roots by hand? My mother learned how to do that in grade school, but I didn’t pick it up until I read the Feynman Lectures.) Incidentally, as a workaday scientist, I work with vectors and discontinuities far more often than I have to use long division. Honestly, I think “the line is broken in two pieces” is a significantly easier concept to get across than the rigmarole of pulling numbers down and multiplying them up and subtracting them down. But, of course, “vectorial angles” and “discontinuous functions” sound awfully intimidating to the proletariat, so let’s just hold them up as examples of what we don’t need to teach!

(by Eugene)

UPDATE (30 July 2012): I notice some people have been finding this blog entry by websearching for “Fermat’s dilemma,” so I figured I should say a bit more about that, following thoughts I developed on Twitter. As I said above, you could, if you really tried, interpret this odd choice of words as Hacker’s trying to be artful. But even if you try to read it that way, it’s sorry writing. In aiming to be evocative, Hacker just comes across ambiguous. You could say that Fermat’s little theorem—not to be confused with his Last Theorem!—embodies a dilemma. The little theorem gives us a test for saying whether a number is prime, but that test doesn’t always work: it can be fooled by a “pseudoprime.” Do we use a fallible test or not? We have a dilemma on our hands! Or, if that sounds like too much of a stretch, take the subject of probability theory, which Fermat and Blaise Pascal jumpstarted back in 1654. They wanted to answer a puzzle posed by Antoine Gombaud, Chevalier de Méré: how should two gamblers divvy up a pot of money if their game-playing is interrupted? Sorting out this kind of dilemma led Pascal and Fermat to probability theory.

See, we give things names in order to identify them, and if you make up your own names for things which already have them, you’re apt to confuse people. Sometimes, when the old terminology is muddled or unclear, a new turn of phrase is just what we need, but this is not one of those times. Hacker throws around big words like “polynomial functions and parametric equations” in order to make algebra sound intimidating, and when he makes up his own terms, he fumbles.

UPDATE (11 August 2012): Another item in the “oh, this is too fun” category. Hacker mocks the notion that being able to prove

$ (x^2 + y^2)^2 = (x^2 – y^2)^2 + (2xy)^2$

can lead to “more credible political opinions or social analysis.” I didn’t say anything about that until now, because I figured the rejoinder was implicit in what I said up top: if you can’t say that

$ (x^2 + y^2)^2 = (x^2)^2 + (y^2)^2 + 2(xy)^2 $

and

$ (x^2 – y^2)^2 = (x^2)^2 + (y^2)^2 – 2(xy)^2, $

and then combine the two and so deduce

$ (x^2 + y^2)^2 = (x^2 – y^2)^2 + 4(xy)^2, $

then actually learning or doing anything with “citizen statistics” will be beyond you. Them’s the breaks. However, there’s something else interesting here, which goes back to Hacker’s low expectations for poets and philosophers. To make things more clear, I’m going to replace $x^2$ with $a$ and $y^2$ with $b$. Then the Equation Mocked by Hacker becomes

$ (a + b)^2 = (a – b)^2 + 4ab. $

If we divide both sides of this equation by 4, we get a restatement of it which is just as true:

$ \frac{(a + b)^2}{4} = \frac{(a – b)^2}{4} + ab. $

And this is the algebraic statement of what Euclid said geometrically in Proposition II.5 of The Elements—which happens to be one of the oldest fragments of Euclid we have a surviving copy of. In fact, since this papyrus fragment was probably written between 75 and 125 CE, it’s likely older than any copy of any part of the Gospels. Rylands Library Papyrus P52, a scrap with a few verses from John 18, is the oldest one of those we’ve got, and it dates to about 125 CE itself.

It’s easy to say that not being willing to learn mathematics cuts one off from intellectual history. But damn, does Hacker ever put a point on it.

Congratulations, Professor Hacker. Your op-ed has earned a negative score on the Pritchard scale.

“Take the mathematical developments out of the history of science, and you suppress the skeleton which supported and kept together all the rest. Mathematics gives to science its innermost unity and cohesion, which can never be entirely replaced with props and buttresses or with roundabout connections, no matter how many of these may be introduced.”

—George Sarton, science historian

When talking of dynamical systems, our probability assignments really carry two time indices: one for the time our betting odds are chosen, and the other for the time the bet concerns.

A parenthesis in an appendix is already a pretty superfluous thing. Treating this as the jumping-off point for further discussion merits the degree of obscurity which only a lengthy post on a low-traffic blog can afford.

MY PRIMARY CONCEIT

Quantum mechanics provides statistical predictions for the results of measurements performed on physical systems that have been prepared in specified ways … I hope that everyone agrees at least with that statement. The only question here is whether there is more than that to say about quantum mechanics.
— Asher Peres

In this note, I shall take a rather strictly ascetic view of quantum physics. I’ll make the lifestyle choice that “quantum states” are encodings of probability assignments for the possible outcomes of as-yet unperformed experiments. Nothing more, but certainly nothing less. The image which floats hazily to mind is of the great unanalyzed diversity of the world teeming on, and when pieces of it come together, novelty happens: there takes place an act of creative generation which belongs to neither participant alone. Quantum theory is, following this lifestyle, a means of coping with and possibly even living well within a world having such a character. It studies the special case of novelty-generating interactions in which one participant is a scientific agent, a complex system capable of sustaining beliefs and entertaining them with varied degrees of fervency.

This leads me to another touchy question of lifestyle choice: personalist probability theory. The lifestyle starts with the idea of an agent which can believe things, and to these beliefs—which are arrayed in well-defined sets—we say the agent can attach a quantitative expression of its fervency in that direction. We impose the normative rule that the agent’s measures of credence be consistent with one another, and we make the dull matter of consistency more entertaining with stories about Dutch bookies or Ferengi bartenders. We say, and I think this is essentially a historical convention, that higher numbers should mean a stronger belief, even though we could just as well say that an agent writing a larger number means that the agent will be more surprised if that event turns out to happen. (Thanks to Claude Shannon, we know that the one convention is just the logarithm of the other.) Even the use of real numbers for degrees of fervency is, to my eye, a convention: if somebody wants to record credences using Conway’s surreal numbers, or with elements from some Lie group, all I can say is “peace be with you in your quest”. The normative standard of consistency will still yield constraints among credence assignments, the difference being that those credences won’t live in the same set as relative frequencies do. And, of course, there’s no guarantee that the exercise will lead to any useful novel structures.

The theory of subjective event weights built up in this way just is, in the same way that Euclidean geometry just is. It provides an imitation of Platonism in the way that all abstract constructions built up from axioms do. But we want to deal with the natural world our species evolved within! Ferengi-book coherence tells us that subjective event weights obey the axioms of “probability”. The natural question is, in those places where we scientists make use of “probability” talk, can we handle those tasks with SEWs? The claim of the Bayesian-statistics practitioner is that the answer is “yes”—and in those cases where it isn’t, the invocation of “probability” is what’s illegitimate, rather than the theory of SEWs.

In many circumstances of interest, quantum theory can be re-expressed solely in terms of SEWs. Though quantum theory is often (and validly) thought of as a generalization of ordinary probability theory to encompass a wider bestiary of mathematical structures, it can also be treated as a specialization of probability theory.

In quantum physics, we take what we think we know about a system, roll it into a density operator $\rho$, and use that density operator to make statistical predictions about what the system might do in particular experiments. But presenting that information as a matrix operator is not always the most illuminating choice. We can actually rewrite any finite-dimensional density matrix as a probability distribution, using the idea of informationally complete measurements. These are generalized measurement procedures (positive operator valued measures, or POVMs) which have an appealing ability: given a probability distribution over the possible outcomes of an informationally complete POVM, we can compute all the statistics which we could have gotten using the density matrix. Such POVMs can be constructed in any finite-dimensional Hilbert space. The nicest variety are the symmetric informationally complete POVMs, known familiarly as SICs, which are known to exist for many values of Hilbert-space dimension and are suspected to exist for the others. With these tools in hand, quantum theory becomes probability plus extra conditions: the bare bones of “SEW theory” are dressed with sinews whose anatomy depends on our doing quantum physics instead of some other theory.

With all that as prologue, then:

I have occasionally bumped into people who seemingly want to interpret all scientific discovery as Bayesian conditioning. I think it was Howard Barnum who said something about seeing science in “broadly Bayesian” terms, but judging from the “cocktail talk” and how people act in some corners of the Internet, not everyone would grant that “broadly”. New experiences always being weighed against our preconceived mesh of ideas? Yes. Holding to different ideas with varying degrees of tenacity? Yes. The clockwork ratcheting up and down of numerical fervencies defined over a “distinct number of consequences”? I doubt it. Even if such a story could be cooked up, involving a stupendously baroque and artificial sample space, what use would it have?

(A better term than “broadly Bayesian” might be “Darwinian”, or better yet just “evolutionary”. It’s been observed a few times that Bayesian updating is formally analogous to a formula in evolutionary theory called the discrete-time replicator equation. Prior probabilities map onto abundances of alleles in a gene pool, and the weight of new evidence maps onto biological fitness. The probability distribution after updating corresponds to the new gene pool composition after natural selection has operated. (Marc Harper describes it more fully in arXiv:0911.1763.) But the scenario modeled by the discrete-time replicator equation is just a tiny part of evolutionary phenomena. It’s even a tiny part of the mathematics developed to date for dealing with biological evolution. Other evolutionary processes could define other modes of inference whose mapping to Bayesian updating is contrived and awkward at best. For a relatively mundane example, I think Jeffrey conditioning is analogous to a discrete-time replicator equation with a mutation effect added.)

What this might mean for quantum theory is the following: were one enamored of a different physics-neutral mode of inference, the Born rule would be an empirical addition to that mathematics, phrased in its terms. (And, I’d guess, probably harder to translate into the vernacular of physics.) There is your realism for you: the extra addition to coherence due to the quantum character of the world must still be there. We’d just be writing down and trying to motivate a different equation for it.

TIME EVOLUTION

The Schrödinger equation in QM and the Liouville equation in classical mechanics are, I think, fundamentally synchronic statements about probability assignments. If I’m willing to gamble that a pendulum has position within some range $\Delta q$ and momentum within some interval $\Delta p$, and if I accept the mechanics lessons I had as a young’un, then my hands are forced: I must price lottery tickets referring to the pendulum at other times in a particular way. Each probability assignment concerning a dynamical system carries two time indices: one for the time the agent makes it, and one for the time of the event or proposition written on the ticket. We could write the former with a subscript and the latter in parentheses, for example:

$ \rho_\tau(q,p,t) $

would be a Liouville density for a one-particle system, a commitment made at time $\tau$ about the world’s affairs at time $t$. The Liouville equation connects $\rho_\tau(q,p,t_1)$ to $\rho_\tau(q,p,t_2)$. The subscripts are the same; it’s a synchronic statement. If I get new information at time $\tau_2$, then I can update the whole joint probability density for all times by conditioning on that new information.

If I keep gaining new information, say at times $\tau_i$, then the Shannon entropy of my Liouville density for the present time will keep decreasing. By evolving my Liouville density at time $\tau_i$ forwards and backwards, I can refine my distributions for what the pendulum will be doing in the future and what it had been doing in the past. Entropy decreasing over time? Yegads! We must be in contradiction with thermodynamics!

. . . except that by that reasoning, the thermodynamic entropy of any classical system we can characterize exactly must be zero, because the Shannon index of its Liouville density is nil! Indeed, the thermodynamic entropy of any simulated system must be exactly zero, because the computer doing the simulation knows where everything is and where all the pieces are going at all times!

Yeah, that’s pretty silly.

What it does mean is that I can extract energy more and more efficiently, and that I can make better and better predictions of how somebody else’s energy-extraction experiment will fare. (And a typical such experiment might well involve timescales significantly longer than those of the oscillations and vibrations within my system itself, so what matters isn’t where it can be in phase space at one particular time, but rather the possible variety in its trajectories over an interval of time—what I think in Jaynesian language is called its “caliber”.)

The way statistical-physics students are taught to relate Shannon entropy with thermodynamic entropy is through the “fundamental assumption of statistical mechanics”: we’re told to assign equal a priori probabilities to all points in phase space which have the same energy. From this starting point, one makes computations until relationships which look like the phenomenological equations of thermodynamics come out. But the starting point is one which should make a personalist Bayesian’s skin crawl! Who mandates a prior probability, and who died and made them king?

It’s only reasonable at all because of an even-more fundamental assumption: that we obtained all our information from highly coarse-grained measurements which could only access aggregate properties like the total energy. If we could make finer measurements in the first place, then we’d have no warrant to assign equal a priori probabilities across the constant-energy surfaces, and we’d have to rethink the “Shannon to thermodynamics” connection in a different way.

It [the Second Law of Thermodynamics] is not a law that dictates how things go by themselves, but rather how they go in response to particular experimental investigations.
— Campisi and Hänggi (2011)

HISTORICAL SCIENCE

In order to have any scientific weight, retrodictions have to yield predictions. Feynman has a bit in The Character of Physical Law where he explains how geologists “talk about the past by talking about the future”. If you dig in the ground where nobody has dug before—if you perform an as-yet-unperformed experiment—you’ll find fossil bones of the predicted kind. If a statement about the Earth’s geological past can’t be made to yield predictions about the consequences of new digging, we’re not talking about dinosaurs anymore— we’re talking about the invisible dragons which live in my garage. It’s fine to write a Bayesian probability $p(v)$ for the speed of the Chicxulub impactor, but if we can’t turn that into expectations for new investigations, why should anyone care?

So, what if we do commit ourselves to the idea that quantum uncertainty is uncertainty about future experiment outcomes? That measurements are not just disturbing, but generative? Then we must conclude that retrodictions are just a nostalgic kind of prediction, statements made thinking of the past which must concern the future to have any scientific content. If we disagree about things which happened in the past and stayed there—so what?! That’s like my housemate and I disagreeing today about the number of eggs laid yesterday by the invisible dragon living in our garage. Possible agreement in the future about experiments which can be done in the future—that’s the key.

Among other things, this affects, I believe, the criteria one uses to judge the compatibility of quantum probability assignments. I suspect that accepting agent experiences as the things beliefs are about changes significantly how important one feels various possible kinds of disagreement are. It appears possible in the quantum world that two physicists can be in sufficient discord that, as they perform experiments, their novel experiences bring them into agreement about the future but not about the past. More specifically, suppose Alice and her friend Bob are interested in a quantum system and each plan to receive word of an experiment performed on it. Prior to the experiment, Alice and Bob each make an assignment of probabilities to the possible experiment outcomes, in the form of a density matrix. When the measurement experiment intervenes on the system and coughs up a result, Alice and Bob update their “catalogues of knowledge” accordingly. It can transpire that the two physicists’ post-experiment density matrices agree, but the conditional density matrices which encode their beliefs about the past are incompatible. A classical analogue of this situation would be the following: suppose that Alice and Bob disagree about which side of a die is up. Alice says it’s a 6, Bob says it’s a 1. The die is rolled in a new experiment and comes up 6. Alice and her friend can come to agree about which side is up after the roll, but the new result changes nothing about their earlier disagreement.

I do not know if this is a serious inconvenience to doing science, or to the ascetic view that quantum states are catalogues of probabilities for possible future experimental outcomes. After all, if we hold fast to that position, then a post-measurement conditional density operator for the past spacetime region must be a probability catalogue for counterfactual experiments—interventions into nature which could have been done, but weren’t. Incompatible conditional density operators for spacetime regions in the past are, in this view, arguments over yesterday’s tomorrow, friction without heat. They’re what you get when “measurements” in your world are, not just disturbances, but acts of generation.

(Leifer and Spekkens write that “because nontrivial quantum measurements always entail a disturbance [...] coming to agreement about the state of the region after the measurement does not resolve a disagreement about the state of the region before the measurement.” To which a justifiable response might be, “Yes, and isn’t it wonderful?” More importantly, perhaps, I don’t like the phrasing of “a disagreement about the state of the region before the measurement.” The choice and arrangement of words seem wrong. I don’t know if it’s a matter of accident or of design towards a goal I disagree with. Doesn’t “disagreement about the state of the region” sound too much like, say, “disagreement about the calorie content of the region”—isn’t it just the phrasing one would choose if one believed that “the state” were a property of “the region”? This feels to me like bad language for any psi-epistemist, radically QBic or otherwise. “The states ascribed by two agents disagree” would be a more forceful and less muddling statement than “Two agents disagree about the state,” I think.)

A new slogan: Confusion about the past is the price we pay for a world of genuine novelty. The inability of Alice and her friend to come into agreement about retrodictions into the past beyond a measurement is, in microcosm, our inability to agree about what happened before the Big Bang.

(For a discussion of retrodictions-as-predictions in a cosmological context, see Schriffen and Wald (2012), section VI. I also think one could productively disagree with Schriffen and Wald’s discussion of thermal equilibration, at the beginning of Section III. To my eye, invocations of “ergodicity” and “mixing” do not resolve the problem of assigning probability distributions over microstates, but rather defer it. (Which is, to be sure, nonnegligible progress from a physicist’s perspective.) They speak of sampling a system at a “random time” during its dynamical time-evolution, which naturally provokes the question: what do you mean by “random time”? You still need some notion of what probability means to give the whole structure of concepts any content. It’s like the old circular, or rather downward-spiralling, conversation between a Bayesian and a frequentist: “If the probability of the coin landing heads is $p$, does that mean there will be exactly $1000p$ heads in 1000 flips?” “No, the number of heads will likely be close to $1000p$, but not exactly.” “What do you mean, likely? That’s the idea we’re trying to define!” And so forth. You could use large deviation theory to write a formula for how the probability of a deviation falls off with the entropy, but you still need to define probability eventually. That is, you’ve deferred the question—in a sophisticated, quantitative, maybe even useful way!—but not answered it.)

ORION DREAMING

The time-reverse of historical science is, in a sense, the issue of “Boltzmann Brains“—you know, complex structures arising from quantum vacuum fluctuations in the distant future of the universe. Supposedly, there should be stupendously more of them in the long (long, long) run than there are beings like we think we are, and from this the cosmologists deduce all sorts of things. E.g., that there exist an infinite number of beings who have all the memories of my brain up to 18 April 2012, including everything I’ve seen from Hubble and WMAP, but then in their memories they wake up from a long dream and return to being a green-skinned dancing girl from Orion.

(Alas, that is the fashion these days, populating the loneliness with shadow selves, frozen in branches of a stupefying state vector, floating in bubbles of spacetime frothed up by eternal inflation, or recorded in the memories of poor delusional Boltzmann Brains. Or, if you want to be particularly trendy, all of the above. Every variation of Hitler winning the war and von Stauffenberg’s bomb going off as planned, never the histories to meet. Every book in the Library of Babel a biography, innumerably many times over. Infinite iterations of Zhuangzi dreaming he is a butterfly; infinite butterflies dreaming they are Zhuangzi. Is this what we got into science for?)

But is that actually a meaningful number to compute, with quantum theory as it stands? What’s the point of asking, “What are the potential consequences to me of my experimental intervention into this phenomenon?” if the phenomenon in question is, by definition, inaccessible, both to myself and to my posterity?

I’ve read at least one cosmology person, Tom Banks, saying that Boltzmann-brainology is “silly.” His position was essentially that we can modify our physical theories in an infinite number of ways consistent with all available data and making the same predictions for all conceivable experiments, but with different numbers of Boltzmann Brains coming out. In addition, any detector sent out to gather data on Boltzmann Brains would disintegrate by quantum fluctuations itself long before it stood a chance of spotting any. . . but I think the issue is more fundamental than that. You’re asking a question which the theory is not prepared to answer! Sometimes, it’s obvious when that happens, like trying to calculate the self-energy of an ideal point electron and getting infinity, but here, people mostly don’t seem to be thinking of that possibility. If they do think the answer is absurd, they try to screw around with general relativity and invent a new cosmology that way.

Maybe physicists are generally accustomed to thinking about “limits to the validity of quantum mechanics”—if they believe any such exist at all—in an unproductive kind of way? Having prematurely excised the active agent, we naturally think “QM might fail for objects larger than the Planck mass” or something like that, rather than “QM is the wrong tool for answering questions divorced from agent experience”.

UPDATE (16 May 2012): I thank Howard Barnum for pointing out a misdirected hyperlink.

Style file:

• utphystw.bst

Example document:

\documentclass[aps,amsmath,amssymb]{revtex4}
\usepackage{amsmath,amssymb,hyperref}

\begin{document}
\bibliographystyle{utphystw}

\title{Test}
\author{Blake C. Stacey}
\date{\today}

\begin{abstract}
Only a test!
\end{abstract}

\maketitle

As indicated, this is only 
a test.\cite{stacey2011,sfi2011}

\bibliography{twtest.bib}

\end{document}

And the example bibliography file:

@TWEET{stacey2011,
       author={Blake Stacey},
       authorid={blakestacey},
       year={2011},
       month={July},
       day={25},
       tweetid={95521600597786624},
       tweetcontent={I find it hard to tell, in some 
                     areas of science, whether I am 
                     a radical or a curmudgeon.}}

@TWEET{sfi2011,
       author={anon},
       authorid={OverheardAtSFI},
       year={2011},
       month={June},
       day={23},
       tweetid={84018131441422336},
       tweetcontent={The brilliance of the word 
                     ``Complexity'' is that it 
                     means just about anything 
                     to anybody.}}

PDF output:

• twtest.pdf

One way to think about it is this: suppose you had to teach a physics class to first- or second-year undergraduates. Could you get all the textual materials you need from Open-Access sources on the Web? Would you know where to look?

What with Wikipedia, OpenCourseWare, review articles on the arXiv, science blogs, the Khaaaaaan! Academy and so forth, we probably already have a fair portion of this in various places. But the operative word there is various. I, at least, would like it gathered together so we can know what’s yet to be done. With a project like, say, Wikipedia, stuff gets filled in based on what people feel like writing about in their free time. So articles grow by the cumulative addition of small bits, and “boring” content — parts of the curriculum which need to be covered, but are seldom if ever “topical” — doesn’t get much attention.

I honestly don’t know how close we are to this ideal. And, I don’t know what would be the best infrastructure for bringing it about and maintaining it. Idle fantasies and pipe dreams!

I’d like to have this kind of resource, not just for the obvious practical reasons, but also because it would soothe my conscience. I’d like to be able to tell people, “Yes, physics and mathematics are difficult, technical subjects. The stuff we say often sounds like mystical arcana. But, if you want to know what we know, all we ask is time and thinking — we’ve removed every obstacle to your understanding which we possibly can.”

I don’t think this would really impact the physics cranks and crackpots that much, but that’s not the problem I’m aiming to (dreaming that we will) solve. Disdain for mathematics is one warning sign of a fractured ceramic, yes: I’ve lost count of the number of times I’ve seen websites claiming to debunk Einstein “using only high-school algebra!” We could make learning the mathematical meat of physics easier, but that won’t significantly affect the people whose crankishness is due to personality and temperament. Free calculus lessons, no matter how engaging, won’t help those who’ve dedicated themselves to fighting under the banner of Douche Physik.

Alchemists work for the people. —Edward Elric

The Who’s Who Among Executives and Professionals
400 West 19th St., New York City, NY 10001

It’s like being in high school and offered a place in an annual poetry anthology! all over again.

Yeah, no.

UPDATE (3 January 2012): I’ve received the same e-mail again, but this time headed by the subject line “Distinguished Women of The Year”.

It is a bit like the story of Niels Bohr’s horseshoe. Upon seeing it hanging over a doorway someone said, “But Niels, I thought you didn’t believe horseshoes could bring good luck.” Bohr replied, “They say it works even if you don’t believe.” [source]

I find it interesting that nobody seems to know where this story comes from. The place where I first read it was a jokebook: Asimov’s Treasury of Humor (1971), which happens to be three years older than the earliest appearance Wikiquote knows about. In this book, Isaac Asimov tells a lot of jokes and offers advice on how to deliver them. The Bohr horseshoe, told at slightly greater length, is joke #80. Asimov’s commentary points out a difficulty with telling it:

To a general audience, even one that is highly educated in the humanities, Bohr must be defined — and yet he was one of the greatest physicists of all time and died no longer ago than 1962. But defining Bohr isn’t that easy; if it isn’t done carefully, it will sound condescending, and even the suspicion of condescension will cool the laugh drastically.

Note the light dusting of C. P. Snow. Asimov proposes the following solution.

If you despair of getting the joke across by using Bohr, use Einstein. Everyone has heard of Einstein and anything can be attributed to him. Nevertheless, if you think you can get away with using Bohr, then by all means do so, for all things being equal, the joke will then sound more literate and more authentic. Unlike Einstein, Bohr hasn’t been overused.

I find this, except for the last sentence, strangely appropriate in the context of quantum-foundations arguments.

Gilbert N. Lewis appears to have introduced the word in “The Law of Physico-Chemical Change”, which appeared in the Proceedings of the American Academy of Arts and Sciences 37 (received 6 April 1901).

If any phase containing a given molecular species is brought in contact with any other phase not containing that species, a certain quantity will pass from the first phase to the second. Every molecular species may be considered, therefore, to have a tendency to escape from the phase in which it is. In order to express this tendency quantitatively for any particular state, an infinite number of quantities could be used, such, for example, as the thermodynamic potential of the species, its vapor pressure, its solubility in water, etc. The quantity which we shall choose is one which seems at first sight more abstruse than any of these, but is in fact simpler, more general, and easier to manipulate. It will be called the fugacity, represented by the symbol [tex]\psi[/tex] and defined by the following conditions: —

1. The fugacity of a molecular species is the same in two phases when these phases are in equilibrium as regards the distribution of that species.

2. The fugacity of a gas approaches the gas pressure as a limiting value if the gas is infinitely rarefied. In other words, the escaping tendency of a perfect gas is equal to its gas pressure.

By golly, I wish I’d had this book as an undergrad.

As it was, I had to wait until this past January, at the ScienceOnline 2011 conference. These annual meetings in Durham, North Carolina feature scientists, journalists, teachers and students, all blurring the lines between one specialization and another, trying to figure out how the Internet can help us do and talk science. Lots of the attendees had books recently published or soon forthcoming, and the organizers arranged a drawing. We could each pick a book from the table, with all the books anonymized in brown paper wrapping. Greg “Dr. Skyskull” Gbur had brought fresh review copies of his textbook. Talking it over, we realized that if somebody who wasn’t a physics person got a mathematical methods textbook, they’d probably be sad. So, we went to the table and hefted the offerings until we found one which weighed enough to be full of equations, and everyone walked away happy.

MMfOPE is, as the kids say, exactly what it says on the tin. It begins with vector calculus and concludes with asymptotic analysis, passing through matrices, infinite series, complex analysis, Fourierology and ordinary and partial differential equations along the way. Each subject is treated in a way which physicists will appreciate: mathematical rigour mortis is not stressed, but when more careful or Philadelphia-lawyerly treatments are possible, they are indicated, and the ways in which their subtleties can become relevant are pointed out. In addition, issues like the running time and convergence of numerical algorithms are, where appropriate, addressed.

The sticker put on by the intellectual toy store would read, I think, “Ages: sophomore and up”. The prerequisites which the text effectively expects would be covered by the first year or so of university mathematics. If you’ve made it through Silvanus P. Thompson’s Calculus Made Easy, you’d be pretty well set, though some experience with vectors, in the way which first-year physics courses grapple with them, would also be helpful. The physics content one ought to know before diving in is, likewise, pretty well encapsulated by first-year mechanics and electromagnetism (what I still think of as 8.01 and 8.02).

I wish I’d had this book when we got to 8.03, because it covers just about everything I should have learned then but ended up having to teach myself later. Rather than plodding through a stupefying number of simple examples to “make sure we get the point”, it builds at a reasonable pace so we can reach interesting things and learn material we can actually use to do stuff. (The sophomore-level classes were easily the worst of MIT’s physics curriculum. I don’t know why, but the sentiment was widely shared, and I appreciate an antidote like Gbur’s book, which makes the subject matter not just clear but also interesting.)

The content was chosen with optics in mind. This does not rule out the usefulness of the book for general physics majors, though an instructor would likely benefit by drawing problems from a supplemental source. Students aiming for statistical physics would enjoy applications of Fourier transforms and convolution to probability theory. Those whose interests lean to mechanical or electrical engineering might appreciate the poles-in-the-complex-plane technology being applied to control theory and the study of stability. This is just to say that a book written with a different goal in mind would have turned out different, and that I myself would like to see those topics explained with the same verve.

The level and style of the presentation befits a text written for students who are seeing most of the material for the first time. Fine books have been written which have the feel of a walk through a forest with a worldy-wise teacher, far from any blackboard. They can achieve a conceptual clarity, but the novice reader often wishes there were a few more stepping stones along the path between equations 17 and 18. MMfOPE generally assumes its readers do not need to be reminded “the derivative of the sine is the cosine”, but its paths are well furnished with intermediate steps.

The historical notes, which indicate who discovered what and where theorems came from, are much appreciated. Without making the book a history-of-science monograph, they correct some common misconceptions and convey the impression that history is richer than we often let on, which is a good thing to be aware of. (The author’s experience as a science blogger with an interest in historical quirks shines through here.) Many exercises also point into the literature. Homework problems which begin, “Read the paper by…” will help students learn what facing into The Literature requires (and that sometimes, it stares back into you).

When our paths have intersected, Prof. Gbur and I have got along in a singularly fabulous way. It is therefore with genuine regret that I note the presence of aberrations in the text which may interfere with its reading. Missing factors of [tex]\pi[/tex] are the bane of the working physicist; Zee advises us that the difference between a good theorist and a bad one is that the good one makes an even number of sign errors. None of the glitches I have found in MMfOPE seriously impede understanding. They are of the kind which can only get squeezed out when a text is used in a classroom a few times over, so its odd byways get explored by people who don’t already know what should be on the page. For example, on p. 147, Eq. (5.32) is missing an equals sign after the [tex]A^\dag A[/tex]. On p. 530, [tex]\alpha^2[/tex] is on the wrong side of Eq. (15.165), and this goof propagates for a few equations after that, though without affecting the conclusions of the section. In Figure 14.7 on p. 490, the two panels make sense on their own, but work oddly in juxtaposition; they would be more clear if the curve in part (b) were the derivative of that in part (a).

COI DISCLAIMER: As indicated, I got a copy of Mathematical Methods for Optical Physics and Engineering for free. That said, I have no financial stake in its success.

UPDATE (20 June 2012): Errata are now available at the author’s web page.