Nosimpler

The US DOJ.

Not sure that even the Bushies ever tried pulling the "modify the text of old archived speeches a year later" trick.
New version:

This landmark Initiative, spearheaded by the FBI, was launched to help streamline and advance investigations and prosecutions against fraudsters who allegedly targeted, and preyed upon, Americans struggling to keep their homes. And it’s been a model of success. Over the past 12 months, it has enabled the Justice Department and its partners to file federal criminal charges against 107 defendants for allegedly victimizing more than 17,185 American homeowners – and inflicting losses in excess of $95 million.

Original version:

This landmark Initiative, spearheaded by the FBI, was launched to help streamline and advance investigations and prosecutions against fraudsters who allegedly targeted, and preyed upon, Americans struggling to keep their homes. And it’s been a model of success. Over the past 12 months, it has enabled the Justice Department and its partners to file 285 federal criminal indictments and informations against 530 defendants for allegedly victimizing more than 73,000 American homeowners – and inflicting losses in excess of $1 billion.

We sketch the history of spectral ranking, a general umbrella name for techniques that apply the theory of linear maps (in particular, eigenvalues and eigenvectors) to matrices that do not represent geometric transformations, but rather some kind of relationship between entities. Albeit recently made famous by the ample press coverage of Google's PageRank algorithm, spectral ranking was devised more than a century ago, and has been studied in tournament ranking, psychology, social sciences, bibliometrics, economy and choice theory. We describe the contribution given by previous scholars in precise and modern mathematical terms: along the way, we show how to express in a general way damped rankings, such as Katz's index, as dominant eigenvectors of perturbed matrices, and then use results on the Drazin inverse to go back to the dominant eigenvectors by a limit process. The result suggests a regularized definition of spectral ranking that yields for a general matrix a unique vector depending on a boundary condition.

I have kept a log for many years that has convinced me that I have roughly monthly oscillations in motivation and libido, but I've not come across convincing evidence for roughly lunar or monthly cycles in men in literature searches. So, I perk up on seeing the examination by Cajochen et al. of sleep behavior under highly controlled conditions of a circadian laboratory study protocol without time cues. They find that subjective and objective measures of sleep vary according to lunar periodicity (~29.5 days). Subjects in the study were seventeen healthy young volunteers (nine women and eight men; age range, 20–31 years; mean, 25.0 ± 3.6 years [SD]) and 16 healthy older volunteers (eight women and eight men; age range, 57–74 years; mean, 65.0 ± 5.5 years) Here is their abstract:

Endogenous rhythms of circalunar periodicity (∼29.5 days) and their underlying molecular and genetic basis have been demonstrated in a number of marine species. In contrast, there is a great deal of folklore but no consistent association of moon cycles with human physiology and behavior. Here we show that subjective and objective measures of sleep vary according to lunar phase and thus may reflect circalunar rhythmicity in humans. To exclude confounders such as increased light at night or the potential bias in perception regarding a lunar influence on sleep, we retrospectively analyzed sleep structure, electroencephalographic activity during non-rapid-eye-movement (NREM) sleep, and secretion of the hormones melatonin and cortisol found under stringently controlled laboratory conditions in a cross-sectional setting. At no point during and after the study were volunteers or investigators aware of the a posteriori analysis relative to lunar phase. We found that around full moon, electroencephalogram (EEG) delta activity during NREM sleep, an indicator of deep sleep, decreased by 30%, time to fall asleep increased by 5 min, and EEG-assessed total sleep duration was reduced by 20 min. These changes were associated with a decrease in subjective sleep quality and diminished endogenous melatonin levels. This is the first reliable evidence that a lunar rhythm can modulate sleep structure in humans when measured under the highly controlled conditions of a circadian laboratory study protocol without time cues.

Hierarchical organization is widespread in the societies of humans and other animals, both in social structure and in decision-making contexts. In the case of collective motion, the majority of case studies report that dominant individuals lead group movements, in agreement with the common conflation of the terms “dominance” and “leadership.”...

Edgar J. Bermudez Contreras, Andrea Gomez Palacio Schjetnan, Arif Muhammad, Peter Bartho, Bruce L. McNaughton, Bryan Kolb, Aaron J. Gruber, Artur Luczak. Memory formation is hypothesized to involve the generation of event-specific neural activity patterns during learning and the subsequent spontaneous reactivation of these patterns. Here, we presen....

When light is chosen with a special frequency, its absorption by a cloud of atoms is independent of their internal orientation.

Published Thu Aug 01, 2013

Merriam-Webster's Word of the Day for July 16, 2013 is:

booboisie • \boob-wah-ZEE\  • noun
: the general public regarded as consisting of boobs

Examples:
"'Elitism' was always the sneer of the booboisie against opera…." — From an article by Stephanie Von Buchau in Opera News, July 1998

"[Simon] Doonan's silliness is a delight. This is not a rote throwback to the 'Queer Eye for the Straight Guy' advice of yore. There's no dumbing down for the booboisie." — From a book review by Christopher Muther in the Boston Globe, January 7, 2012

Did you know?
Journalist and critic H. L. Mencken is often credited with coining "booboisie," a blend of "boob," as it refers to someone who cares too much about things and too little about ideas and art, and "bourgeoisie," that French-origin term for the middle class. Mencken may have indeed coined the word, but it seems likely that he wasn't the word's only inventor. According to the 2006 Yale Book of Quotations, Mencken was quoted using "booboisie" in an August 1922 issue of The Dial before it appeared in any of his writings. But a February 22, 1922 Washington Post article opens with the following line: "A plot to mulct the 'booboisie' which might have been invented by an author of get-rich-quick fiction …." The booboisie, of course, couldn't care less.

Via Vintage Kids' Books My Kids Love (where there are several more).

Imagine that you are blindfolded inside an unknown room. You snap your fingers and listen to the room’s response. Can you hear the shape of the room? Some people can do it naturally, but can we design computer algorithms that hear rooms? We show how to compute the shape of...

Nature Neuroscience 16, 772 (2013). doi:10.1038/nn.3441

Author: Alfonso Renart

Chaotic networks produce rich temporal dynamics that could be useful for timing, but are extremely sensitive to perturbations. Work now shows that a learning rule for the weights of a chaotic recurrent network can stabilize time-varying activity patterns. This result can be used to train output units to produce generic timed responses.

PLoS One. 2013; 8(7): e67508
Leibon G, Rockmore DN

In this paper we show how the coupling of the notion of a network with directions with the adaptation of the four-point probe from materials testing gives rise to a natural geometry on such networks. This four-point probe geometry shares many of the properties of hyperbolic geometry wherein the network directions take the place of the sphere at infinity, enabling a navigation of the network in terms of pairs of directions: the geodesic through a pair of points is oriented from one direction to another direction, the pair of which are uniquely determined. We illustrate this in the interesting example of the pages of Wikipedia devoted to Mathematics, or "The MathWiki." The applicability of these ideas extends beyond Wikipedia to provide a natural framework for visual search and to prescribe a natural mode of navigation for any kind of "knowledge space" in which higher order concepts aggregate various instances of information. Other examples would include genre or author organization of cultural objects such as books, movies, documents or even merchandise in an online store.

Visual illusions and hallucinations are hallmarks of serotonergic hallucinogen-induced altered states of consciousness. Although the serotonergic hallucinogen psilocybin activates multiple serotonin (5-HT) receptors, recent evidence suggests that activation of 5-HT2A receptors may lead to the formation of visual hallucinations by increasing cortical excitability and altering visual-evoked cortical responses. To address this hypothesis, we assessed the effects of psilocybin (215 μg/kg vs placebo) on both α oscillations that regulate cortical excitability and early visual-evoked P1 and N170 potentials in healthy human subjects. To further disentangle the specific contributions of 5-HT2A receptors, subjects were additionally pretreated with the preferential 5-HT2A receptor antagonist ketanserin (50 mg vs placebo). We found that psilocybin strongly decreased prestimulus parieto-occipital α power values, thus precluding a subsequent stimulus-induced α power decrease. Furthermore, psilocybin strongly decreased N170 potentials associated with the appearance of visual perceptual alterations, including visual hallucinations. All of these effects were blocked by pretreatment with the 5-HT2A antagonist ketanserin, indicating that activation of 5-HT2A receptors by psilocybin profoundly modulates the neurophysiological and phenomenological indices of visual processing. Specifically, activation of 5-HT2A receptors may induce a processing mode in which stimulus-driven cortical excitation is overwhelmed by spontaneous neuronal excitation through the modulation of α oscillations. Furthermore, the observed reduction of N170 visual-evoked potentials may be a key mechanism underlying 5-HT2A receptor-mediated visual hallucinations. This change in N170 potentials may be important not only for psilocybin-induced states but also for understanding acute hallucinatory states seen in psychiatric disorders, such as schizophrenia and Parkinson's disease.

My friend Josh Vekhter sent me this blog post written by someone who calls herself celandine13 and tutors students with learning disabilities.

In the post, she reframes the concept of mistake or “being bad at something” as often stemming from some fundamental misunderstanding or poor procedure:

Once you move it to “you’re performing badly because you have the wrong fingerings,” or “you’re performing badly because you don’t understand what a limit is,” it’s no longer a vague personal failing but a causal necessity.  Anyone who never understood limits will flunk calculus.  It’s not you, it’s the bug.

This also applies to “lazy.”  Lazy just means “you’re not meeting your obligations and I don’t know why.”  If it turns out that you’ve been missing appointments because you don’t keep a calendar, then you’re not intrinsically “lazy,” you were just executing the wrong procedure.  And suddenly you stop wanting to call the person “lazy” when it makes more sense to say they need organizational tools.

And she wants us to stop with the labeling and get on with the understanding of why the mistake was made and addressing that, like she does when she tutors students. She even singles out certain approaches she considers to be flawed from the start:

This is part of why I think tools like Knewton, while they can be more effective than typical classroom instruction, aren’t the whole story.  The data they gather (at least so far) is statistical: how many questions did you get right, in which subjects, with what learning curve over time?  That’s important.  It allows them to do things that classroom teachers can’t always do, like estimate when it’s optimal to review old material to minimize forgetting.  But it’s still designed on the error model. It’s not approaching the most important job of teachers, which is to figure out why you’re getting things wrong — what conceptual misunderstanding, or what bad study habit, is behind your problems.  (Sometimes that can be a very hard and interesting problem.  For example: one teacher over many years figured out that the grammar of Black English was causing her students to make conceptual errors in math.)

On the one hand I like the reframing: it’s always good to see knee-jerk reactions become more contemplative, and it’s always good to see people trying to help rather than trying to blame. In fact, one of my tenets of real life is that mistakes will be made, and it’s not the mistake that we should be anxious about but how we act to fix the mistake that exposes who we are as people.

I would, however, like to take issue with her anti-example in the case of Knewton, which is an online adaptive learning company. Full disclosure: I interviewed with Knewton before I took my current job, and I like the guys who work there. But, I’d add, I like them partly because of the healthy degree of skepticism they take with them to their jobs.

What the blogwriter celandine13 is pointing out, correctly, is that understanding causality is pretty awesome when you can do it. If you can figure out why someone is having trouble learning something, and if you can address that underlying issue, then fixing the consequences of that issue get a ton easier. Agreed, but I have three points to make:

1. First, a non-causal data mining engine such as Knewton will also stumble upon a way to fix the underlying problem by dint of having a ton of data and noting that people who failed a calculus test, say, did much better after having limits explained to them in a certain way. This is much like the spellcheck engine of Google works by keeping track of previous spelling errors, and not by mind reading how people think about spelling wrong.
2. Second, it’s not always easy to find the underlying cause of bad testing performance, even if you’re looking for it directly. I’m not saying it’s fruitless – tutors I know are incredibly good at that – but there’s room for both “causality detectives” and tons of smart data mining in this field.
3. Third, it’s definitely not always easy to address the underlying cause of bad test performance. If you find out that the grammar of Black English affects students’ math test scores, what do you do about it?

Having said all that, I’d like to once more agree with the underlying message that a mistake is a first and foremost a signal rather than a reflection of someone’s internal thought processes. The more we think of mistakes as learning opportunities the faster we learn.

Boston Magazine has a fascinating article on the work of psychologist Lisa Feldman Barrett who has been leading the charge against the idea that we recognise the same facial expression of emotion across the world.

This was first suggested by Paul Ekman whose work suggested that humans can universally recognise six emotions: anger, disgust, fear, happiness, sadness, and surprise.

His research involved showing people from different cultures pictures of faces and asking them to label each expression from a choice of emotional words.

But Barrett has found a simple flaw in the procedure:

She returned to those famous cross-cultural studies that had launched Ekman’s career—and found that they were less than watertight. The problem was the options that Ekman had given his subjects when asking them to identify the emotions shown on the faces they were presented with. Those options, Barrett discovered, had limited the ways in which people allowed themselves to think.

Barrett explained the problem to me this way: “I can break that experiment really easily, just by removing the words. I can just show you a face and ask how this person feels. Or I can show you two faces, two scowling faces, and I can say, ‘Do these people feel the same thing?’ And agreement drops into the toilet.”

The article is on much more than this controversy in cognitive science and also tracks how research on emotion and facial expression is playing an increasing role in law enforcement – with not all of it well supported by evidence.

And if you want links to some of the scientific papers, the always interesting Neuroanthropology blog has more at the bottom of this post.
 

Link to Boston Magazine article ‘About Face’.

Weka is one of the most popular tools for data analysis.  But Weka takes 70 minutes to perform leave-one-out cross-validate using a simple naive bayes classifier on the census income data set, whereas Haskell’s HLearn library only takes 9 seconds.  Weka is 465x slower!

Code and instructions for reproducing these experiments are available on github.

Why is HLearn so much faster?

Well, it turns out that the bayesian classifier has the algebraic structure of a monoid, a group, and a vector space.  HLearn uses a new cross-validation algorithm that can exploit these algebraic structures.  The standard algorithm runs in time , where is the number of “folds” and is the number of data points.  The algebraic algorithms, however, run in time .  In other words, it doesn’t matter how many folds we do, the run time is constant!  And not only are we faster, but we get the exact same answer.  Algebraic cross-validation is not an approximation, it’s just fast.

Here’s some run times for k-fold cross-validation on the census income data set.  Notice that HLearn’s run time is constant as we add more folds.

And when we set k=n, we have leave-one-out cross-validation.  Notice that Weka’s cross-validation has quadratic run time, whereas HLearn has linear run time.

HLearn certainly isn’t going to replace Weka any time soon, but it’s got a number of cool tricks like this going on inside.  If you want to read more, you should check out these two recent papers:

• (ICML13) Algebraic Classifiers: a generic approach to fast cross-validation, online training, and parallel training

• (TFP13) HLearn: a machine learning library for Haskell

I’ll continue to write more about these tricks in future blog posts.

Subscribe to the RSS feed to stay tuned.

Occasionally I see mathematical questions that seem “grammatically incorrect” in some sense.

Example. “Is open or closed?”
Example. “Is a group?”
Example. “What’s the Fourier series of ?”

Here are some sillier examples.

Example. “Is a rectangle prime?”
Example. “Is ?”
Example. “What’s the Fourier series of the empty set?”

What all of these examples have in common is that they are type errors: they are attempts to apply some mathematical process to a kind of mathematical object it was never intended to take as input. If you tried to write a program in some highly mathematical programming language to answer these questions, it (hopefully!) wouldn’t compile.

Mathematical objects are usually not explicitly thought of as having types in the same way that objects in a programming language with a type system has types. Ordinary mathematics is supposed to be formalizable within Zermelo-Fraenkel (ZF) set theory, possibly with the axiom of choice, and in ZF every mathematical object is constructed as a set. In that sense they all have the same type. (In particular, the question “is ?” is perfectly meaningful in ZF! This is one reason not to like ZF as a foundation of mathematics.) However, I think that in practice mathematical objects are implicitly thought of as having types, and that this is a mental habit mathematicians pick up but don’t often talk about.

Instead of thinking in terms of set theory, thinking of mathematical objects as having types allows us to import various useful concepts into mathematics, such as the notions of type safety, typecasting, subtyping, and overloading, that help us make more precise what we mean by a mathematical sentence being “grammatically incorrect.” The rest of this post will be a leisurely discussion of these and other type-based concepts as applied to mathematics in general. There are various categorical ideas here, but for the sake of accessibility we will restrict them to parenthetical remarks.

An informal description of mathematical types

Informally, the type of a mathematical object describes what kind of object it is.

Example. The object has type (natural number).
Example. The object has type (integer).
Example. The object has type (rational number).
Example. The object has type (pair of natural numbers).
Example. The object has type (degree).
Example. The object has type (function from natural numbers to natural numbers).
Example. The object has type (Boolean).

Some less simple examples:

Example. The object has type (group).
Example. The object has type (topological space).
Example. The object has type (function from topological spaces to groups).

Types help you figure out what you can do with a collection of mathematical objects.

Example. You can take two objects of type and add or multiply them. In other words, there are functions .

Example. You can take two objects of type and, in addition to being able to add or multiply them, you can also subtract them. In other words, there are functions .

Example. You can take an object of type and apply a trigonometric function to it. In other words, there are functions .

Example. If are types, you can take an object of type and another object of type and apply the latter to the former. In other words, there is a function .

The type occupies a special place in this theory. There are only two objects of this type, namely and , and functions which output values can be used to check whether a property holds or doesn’t hold.

Example. You can take a natural number and ask whether it is prime. In other words, there is a function .

Example. If is a type, you can take two objects of type and ask whether they are equal. In other words, there is a function

,

which we might also write using the equals sign .

Example. You can take two integers and ask whether the first is greater than or equal to the second. In other words, there is a function

.

As the above examples suggest, there are various ways of combining types to form new types; these are called type constructors.

Example. An object of the product type is a pair of objects of type and respectively.

Example. An object of the sum type is either an object of type or an object of type .

Example. An object of the function type , also sometimes notated , is a function which takes as input objects of type and returns as output objects of type .

(These constructions should look familiar to fans of category theory: they are just the categorical product, categorical coproduct, and categorical exponential in a category of types. In other words, categories of types are bicartesian closed categories.)

Using the basic type constructors above we can build more complicated type constructors. For example, given a type we can form the list type , which is the type of (finite) lists of elements of type , as follows:

(where is a special type called the unit type; its defining feature is that there is exactly one object of unit type, which we will call a point). Heuristically the list type can be written as the “geometric series” .

Notation

Above we used the notation to denote that was an object of type . It makes sense to generalize this notation to arbitrary types, so that denotes that is an object of type . This is a type declaration.

Type safety and type errors

Besides helping you figure out what you can do with a collection of mathematical objects, types also help you figure out what you can’t do. Let’s go through the examples from the beginning of this post with this idea in mind.

Example. “Is open or closed?” The functions and don’t take as input a set or even a topological space; they take as input pairs where is a topological space and is a subspace of (and output whether or not is a closed or open subset of respectively). In other words, they don’t have type but type . The question of whether is open or closed depends on whether it’s being viewed as a subset of itself or, say, .

Example. “Is a group?” The function doesn’t take as input a set; it takes as input pairs where is a set and is a binary operation on (a magma). In other words, it doesn’t have type but type .

Example. “What’s the Fourier series of ?” The function doesn’t take as input a function on ; it takes as input a periodic function on , e.g. with period , or equivalently a function on , and is not periodic with any period. In other words, doesn’t have type but type .

The sillier examples can be analyzed similarly.

Looking for type errors, or type checking, can help you debug a piece of mathematics in the same way that a compiler looks for type errors to debug a piece of code. For example, whenever you see an expression of the form , you can check to see if it even makes sense before checking whether or not it’s true by checking whether and are of the same type. This is a very general version of dimensional analysis.

Type checking is also one way to check your understanding of a mathematical subject new to you. If you can’t say correctly typed sentences about the subject yet (as determined by other people who know the subject), then you haven’t yet understood the types of the main objects or functions in the subject. For example, if you say “fundamental group of a…” you’d better finish that sentence with either a pointed topological space or a path-connected topological space; otherwise, you haven’t understood an important aspect of the definition of the fundamental group, namely the role of basepoints.

Typecasting, subtyping, and overloading

One complication that makes type checking nontrivial is that most mathematical objects are naturally regarded as having more than one type. For example, we said above that has type , but it clearly also has type , and even . One way to describe this situation is that there are typecasting functions

that convert objects between types. If there is a typecasting operator , then we can use objects of type as inputs into functions that expect objects of type provided that we typecast first.

Another way to describe this situation is that some types are subtypes of other types; in other words, instead of being a collection of tyepcasting functions, the above describes a chain of subtype inclusions. Subtyping allows objects to have more than one type simultaneously, rather than to be a particular type at a particular point in some computation.

Subtypes have an interesting relationship to function types. If is a subtype of , then the function type is a subtype of , but if is a subtype of (a function which outputs objects of type also outputs objects of type ), then the function type is not a subtype but a supertype of (a function which accepts inputs of type also accepts inputs of type ). In other words, the construction of function types is covariant in output types but contravariant in input types with respect to subtypes.

(Again, this should be familiar to fans of category theory: this phenomenon reflects the fact that the construction of exponential objects is covariantly functorial in the target object but contravariantly functorial in the source object.)

A third way to describe the situation is that functions in mathematics are overloaded, and this is the description that is perhaps closest to mathematical practice. Overloading refers to the practice of defining functions which have the same name but which take different types of inputs. For example, the addition symbol refers not to one function but to many functions, all of which happen to have the same name, e.g.

and so forth. More generally we might use for the addition operation in any ring, and even more generally for the group operation in any abelian group. We even used it above to denote sum types!

Exponential notation is perhaps even worse. It might refer to functions of any of the following types:

,

or even more generally, might be an element of a group and might be an integer, or might be and might be an element of a topological ring, and we even used exponential notation above to denote function types! See also this math.SE question.

Overloading seems to confuse students, and I think part of the reason is that mathematicians rarely say explicitly that they are doing it. Overloading isn’t so bad when all the different instances of it are at least meaningfully related to each other, but sometimes concepts in mathematics are overloaded for no reason other than historical accident, e.g. words like “normal” and “regular.” This is even more confusing to students: sometimes the use of the same word in two different contexts is because of some meaningful relationship, e.g. normal extensions and normal subgroups, and sometimes it just isn’t. See also this MO question.

Recursive types

In type theory it is possible to define some types recursively, in terms of themselves, rather than directly in terms of other types. A surprising number of important kinds of mathematical objects can be defined this way.

Example. The type of natural numbers can be defined recursively as

;

that is, a natural number is either a point (namely ) or is another natural number (namely its predecessor).

Example. More generally, the list type can be defined recursively as

;

that is, a list of elements of type is either a point or it is an element of type (namely the beginning of the list) together with a list of elements of type (namely the rest of the list). In particular, is just the type of lists of points. Note that this is exactly the relationship one would expect to be satisfied by the “geometric series” .

Example. The type of (rooted, full binary) trees can be recursively defined as

;

that is, a tree is either a point or it is a pair of trees (namely the pair of trees obtained by removing its root).

Example. A version of the type of sets can be recursively defined as

;

that is, a set is a function on sets which returns either true (for the elements it contains) or false (for the elements it does not contain).

Example. The type of (not necessarily finite) combinatorial games can be recursively defined as

;

that is, a game is a pair of functions on games, which we might call and . Strictly speaking, a combinatorial game should really be called a game position, since it describes some particular moment in a game (e.g. the beginning of a game of chess) rather than what is colloquially referred to as a game (e.g. chess). We think of games as being played between two players, Left and Right, and at any game position Left has some possible options of game positions he can move the game to, namely the positions such that returns true, and Right has some possible options of game positions she can move to, namely the positions such that returns true.

Conway noticed that combinatorial games resemble a generalization of both ordinals (which can be described in terms of a set of ordinals) and Dedekind cuts (which are described by a pair of sets of rationals) and used this relationship to define a large class of numbers using games; see On Numbers and Games for details.

The type we defined above, , can also be thought of as the type of impartial games, which are games where the available moves don’t depend on which player is playing.

(For fans of category theory, recursive types are initial algebras relative to a suitable endofunctor of a category of types.)

Functions which take as input a recursive type can also be defined recursively. Examples involving the natural numbers should already be familiar; here are some other examples.

Example. The length of a list can be defined recursively as follows: the length of a point is . If the list is not a point, it consists of an element and a sublist, and then the length of the list is one greater than the length of the sublist. (I would write this down in pseudocode but I don’t know a clean way to display pseudocode on a WordPress site.)

Example. More generally, for any two types there is a function

called map which takes as input a function and returns as output a function defined recursively as follows: if the input list is empty, so is the output list. Otherwise, the input list consists of an object of type together with a sublist, and applies to the object and then to the sublist, and these together constitute the output list. The length of a list is the function obtained by applying to the unique function .

(Fans of category theory will recognize this as essentially the statement that the list constructor is a functor. There are even programming languages like Haskell which explicitly recognize this fact and put it to good use.)

Example. The height of a tree can be defined recursively as follows: the height of a point is . If the tree is not a point, it has two subtrees, and then the height of the tree is one greater than the maximum of the height of its subtrees.

Example. Games which are guaranteed to terminate can be organized into four disjoint classes: a game is

• positive if Left wins no matter who goes first,
• negative if Right wins no matter who goes first,
• zero if the second player to move wins,
• fuzzy if the first player to move wins.

Here “wins” means “wins according to the normal play condition,” where the first player who cannot make a move loses. The above classes correspond to four functions

which can be recursively defined in terms of each other as follows:

• A game is positive iff at least one of Left’s options is positive or zero and all of Right’s options are positive or fuzzy.
• A game is negative iff all of Left’s options are negative or fuzzy and at least one of Right’s options is negative or zero.
• A game is zero iff none of Left’s options are positive or zero and none of Right’s options are negative or zero.
• A game is fuzzy iff at least one of Left’s options is positive or zero and at least one of Right’s options is negative or zero.

(For fans of category theory, recursively defined functions are applications of the universal property of initial algebras.)

Recursive types are another reason to think of mathematics as having a type system: they allow for a much richer notion of recursion than is usually explicitly mentioned in mathematics, where we usually make do with induction (recursion on ) or sometimes transfinite induction (recursion on a modified version of ). Recursive types, by contrast, allow for a general notion of structural induction. Often we can reduce cases of structural induction we need to ordinary induction (e.g. if we want to perform structural induction over trees we can perform ordinary induction on their heights), but structural induction is conceptually cleaner (e.g. we can perform structural induction over trees without singling out a particular function called height).

Conclusion

Although mathematics is not often explicitly described as having a type system in theory, it seems both useful and accurate to describe mathematics as having a type system in practice. This gives us a language for understanding certain kinds of mathematical errors (type errors) as well as understanding certain implicit mathematical practices (overloading). Types also have an interesting mathematical structure and provide a richer language for defining and working with recursively defined mathematical objects than is commonly taught.

BMJ Case Reports has a paper that describes two patients with Parkinson’s disease who experienced hallucinations that transferred onto photos they took to try and prove they were real.

This is ‘Patient 1′ from the case report:

Patient 1 was first evaluated at age 66, having been diagnosed with PD [Parkinson's Disease] at age 58… She complained of daytime and night-time visual hallucinations for the past one year. Most of the time she did not have insight about them. She described seeing three children playing in her neighbour’s yard and a brunette woman sleeping under the covers in one of the beds in her house. She also saw images of different people sitting quietly in her living room. Most of her visual hallucinations subsided in open and brightly lit spaces but were, nevertheless, troublesome. In one instance, she saw a man covered in blood, holding a child and called 911.

Her husband, in an attempt to prove to her that these were hallucinations, took pictures of the neighbour’s yard and the bed in their house. Surprisingly, when shown these photos, the patient continued to identify the same children playing in the yard and the same brunette woman sleeping under the covers. This perception was present every time the patient looked at these photos. Within 6 months of stopping ropinirole and titrating quetiapine to 75 mg every night at bedtime the hallucinations were less severe and shorter in duration, but the patient continued to see them in the photos.

 

Link to locked article in BMJ Case Reports.

Since Einstein's fundamental paper of 1915, gravitation has been synonymous with General Relativitiy, while theories based on Lorentz invariance have been dismissed as formal analogies. Consequently modern cosmology focusses exclusively on Einstein's geometric theory, which is unnecessarily cumbersome in regions of weak fields, while Maxwell's equations are far more tractable.

It is shown here that Maxwell's equations are an {\it artifact} of Hodge theory, geometric in nature, independent of any specific physical mechanisms, and valid for any force field, attractive or repulsive, generated by a material density and current. In particular, Maxwell's equations apply to weak gravitational fields -- those in Minkowski space-time.

Classical analysis and linear partial differential equations therefore have a role to play in modern problems in cosmology. The gravitational field generated by the rotation of a spherical body is computed explicitly using a multipole expansion; and the method is compared with the Lense-Thirring theory of General Relativity. Maxwell's equations for gravitation are a suitable model for the dynamics of galaxies and galaxy clusters, and the problem of dark matter is discussed from the perspective of dynamic anomalies, which are not accounted for by Newtonian mechanics.

We show that a classical fluid mechanical system can violate Bell's inequality because the fluid motion is correlated over large distances.

It shows one half-second of trading in just one stock, boring old Johnson & Johnson, on May 2. The video slows down the trades so that the milliseconds -- thousandths of a second -- tick slowly by, and so that human eyes can comprehend what's happening.

What you see is trading gone haywire, hopelessly beyond the control of any regulators that might want to make sure all of these trades are legitimate. This flood of trading confuses even other machines, creating mismatches in orders that high-speed traders can exploit, millisecond by millisecond.

"These guys are not stealing dollars, they're stealing pennies," says Nanex founder Eric Hunsader, who presented the video at a recent Wired conference. "It's like paper cuts instead of first-degree murder."..

Inside of the one half-second of trading represented by the video, more than 1,200 orders and 215 actual trades occur, Hunsader says. (The colored boxes in the video represent exchanges, and the dots that go flying represent individual orders.)  

I fully understand that all of this is legal and I will readily concede that the process might "enhance liquidity in the market" (I know that when I place an order online at Schwab, it is completed as fast as I can click the "check order status" button).  But... this still scares the shit out of me.

Regulators are desperately trying to keep up. The European Union last year approved a new rule mandating that all trades must exist for at least a half-second, in order to try to minimize the kind of quote-stuffing that frightens and confuses markets. It turns out that half-second is an eternity. 

It is a financial Armageddon just waiting to happen.  Why doesn't anyone have the cojones to limit this activity?

Addendum:  A well-informed reader wrote to me privately re the content of this post, asking to remain anonymous, and offering the following counterpoints, which I thought were valid enough to bring above the fold to the content of the post -

That HuffPo article is kind of silly, what it shows and what it concludes don't follow at all. What the video seems to show is the pretty straightforward implementation of a regulation called RegNMS. If you send an order to one exchange but there's a better price at another one it's required that the exchanges get you that better price. Obviously that means the exchanges are going to have to talk to each other whenever an order comes in that might have a better price somewhere else. All you're seeing is those "hey can you beat this price" messages going out to all the other exchanges. Skip to 2:50 and you can see one or two happen all alone, the crazy visual is just a lot of that happening all at once. 

Notably there's no indication that anything in the video has anything at all to do with HFT. This is the exchanges and only the exchanges trying to clear trades as fast as they can while complying with the rules they operate under. The cacophony seems to be because there are a lot of orders and computers can execute the straightforward (and, again, legally mandated) arithmetic astonishingly quickly. All the underlying orders could very well be boring mutual funds trading with each other without it looking very different.

Which is not to say that there's no nefarious stuff going on in the markets just that this video doesn't show anything like that. I'm happy to be corrected if I've misinterpreted it of course; I'm not an expert in market microstructure.

(Nanex has an older but more detailed page about some of this at http://www.nanex.net/Research/IsNBBOIgnored.html that seems to be addressing a slightly different issue about which I know nothing)

As an aside, one reason your Schwab trades execute immediately may be that they never even make it to the public market. They're either matched inside Schwab by another customer doing the reverse trade or sent to a 'dark' venue that pays Schwab for doing so:

http://www.schwab.com/public/schwab/nn/legal_compliance/important_notices/order_routing.html

This is called 'payment for order flow'. Note that this still falls under RegNMS so Schwab (or the venue where your trade eventually ends up) must do the 'notify all exchanges' dance from the video. You can see a nice breakdown of where actual trading volume happens in the US at Bats' website: http://www.batstrading.com/market_summary/. The NASD and TRF volumes are things like Schwab filling the trade internally without ever reaching an exchange but also trades that happen on 'dark pools', like the ones that Schwab sends to. Note I'm not calling out Schwab in particular here, this is common practice. 

I will be the first to agree that the Huffington Post tends to be sensationalist in hyping mundane material to increase clicks, and while the comments above temper my angst a bit, I remain uncomfortable with the overall concept of HFT.  Re the latter, the reader who wrote to me also provided a link to A High Frequency Trader's Apology, which discusses the concept in some detail.

In ordinal symbolic dynamics, transcripts describe the algebraic relationship between ordinal patterns. Using the concept of transcript, we exploit the mathematical structure of the group of permutations to derive properties and relations among information measures of the symbolic representations of time series. These theoretical results are then applied for the assessment of coupling directionality in dynamical systems, where suitable coupling directionality measures are introduced depending only on transcripts. These novel measures estimate information flow in lower space dimension and reduce to well-established coupling directionality quantifiers when some general conditions are satisfied. Furthermore, by generalizing the definition of transcript to ordinal patterns of different lengths, several of the commonly used information directionality measures can be encompassed within the same framework.

This is a guest post by Josh Snodgrass.

As the Mathbabe noted recently, a lot of companies are collecting a lot of information about you. Thanks to two Firefox add-ons – Collusion (hat tip to Cathy) and NoScript — you can watch the process and even interfere with it to a degree.

Collusion is a beautiful app that creates a network graph of the various companies that have information about your web activity. Here is an example.

 

On this graph, I can see that nytimes.com has sent info on me to 2mdn.net, linkstorm.net, serving-sys.com, nyt.com and doubleclick.net. Who are these guys? All I know is that they know more about me than I know about them.

Doubleclick is particularly well-informed. They have gotten information on me from nytimes.com, yahoo.com and ft.com. You may not be able to see it on the picture but there are faint links between the nodes. Some (few) of the nodes are sites I have visited. Most of the nodes, especially some of the central ones are data collectors such as doubleclick and googleanalytics. They have gotten info from sites I’ve visited.

This graph is pretty sparse because I cleared all of my cookies recently. If I let it go for a week and the graph will be so crowded it won’t all fit on a screen.

Pretty much everyone is sharing info about me (and presumably you, too). And, I do mean everyone. Mathbabe is a dot near the top. Collusion tells me that mathbabe.org has shared info with google.com, wordpress.com, wp.com, 52shadesofgreed.com, youtube.com and quantserve.com. Google has passed the info on to googleusercontent.com and gstatic.com

I can understand why. WordPress and presumably wp.com are hosting her blog. Google is providing search capabilities. 52shadesofgreed has an ad posted (You can still buy the decks but even better, come to Alt-Banking meetings and get one free). Youtube is providing some content. It is all innocent enough in a way but it means my surfing is being tracked even on non-commercial sites.

These are the conveniences of modern life. Try blocking all cookies and you will find it pretty inconvenient to use the internet. It would be nice to be selective about cookies but that seems very hard. All of this is happening even though I’ve told my browser not to allow third-party cookies. If you look at cookie policies, it seems you have two alternatives:

• Block all cookies and the site won’t work very well
• Allow cookies and we will send your info to whomever we choose (within the law, of course).

So, it would be nice if there were a law that constrained what they do. My impression is that we Americans have virtually no protection. Europe is better from what I understand.

I’ve found another add-on called NoScript that is very helpful but also very disturbing. It tells you about JavaScripts that want to run when you visit a site.

I’m trying to access a site and there are scripts waiting to run from:

• Brightcove.com
• Quantserve.com
• Facebook.com
• Po.st Scorecard.com
• Wxug.com
• Admeld.com
• Googleadservices.com
• Legolas-media.com
• Criteo.com
• Crwdcntrl.com

Clearly a lot of those are about tracking me or showing me ads. As with cookies, if you block all the scripts, the site probably won’t function properly. But the great thing about NoScript is that is makes it easy to allow scripts one by one. So, you can allow the ones that look more legitimate until the site works well enough. Also, you can allow them temporarily.

NoScript and Collusion are great. But mostly they are making me more aware of all the tracking that is going on. And they are also making it clear how hard it is to keep your privacy.

This isn’t just on the internet. Years ago, an economist had an idea about having people put boxes on their cars that would track where they went and charge them for driving, particularly in high congestion times and places. The motivation was to reduce travel that causes a lot of pollution while no one is going anywhere. But people ridiculed the idea. Who would let themselves be tracked everywhere they went.

Well, 40 years later, nearly everyone who has a car has an EZ-pass. And, even if you don’t, they will take a picture of your license plate and keep it on file. All in the name of improving traffic flow.

And, if you use credit cards, there are some big companies that have records of your spending.

What to do about this?

I don’t know.

I like conveniences. Keeping your privacy is hard. DuckDuckGo is a search engine that doesn’t track you (another hat tip to Cathy). But their search results are not as good as Google’s.

Google has all these nice tools that are free. Even if you don’t use them, the web sites you visit surely do. And if they do, google is getting information from them, about you.

This experience has made me even more of a fan of Firefox and add-ons available in it. But what else should I use. And, none of these tools is going to be perfect.

What information gets tracked? A lot of privacy policies say they don’t give out identifying information. But how can we tell?

Just keeping on top of what is going on is hard. For example: what are LSOs? They seem to be a kind of “supercookies”. And Better Privacy seems to be an add-on to help with them.

FT.com’s cookie policy tells me that:

“Our emails may contain a single, campaign-unique “web beacon pixel” to tell us whether our emails are opened and verify any clicks through to links or advertisements within the email”

Who knew that a pixel could do so much?

The truth is, I want to see these sites. So I am enabling scripts (some of them, as few as I can). The question is how to make the tradeoff. Figuring that out is time consuming. I’ve got better things to do with my life.

I’m going to go read a book.

Front Comput Neurosci. 2013; 7: 34
Modolo J, Thomas AW, Legros A

Neural mass models are an appropriate framework to study brain activity, combining a high degree of biological realism while being mathematically tractable. These models have been used, with a certain success, to simulate brain electric (electroencephalography, EEG) and metabolic (functional magnetic resonance imaging, fMRI) activity. However, concrete applications of neural mass models have remained limited to date. Motivated by experimental results obtained in humans, we propose in this paper a neural mass model designed to study the interaction between power-line magnetic fields (MFs) (60 Hz in North America) and brain activity. The model includes pyramidal cells; dendrite-projecting, slow GABAergic neurons; soma-projecting, fast GABAergic neurons; and glutamatergic interneurons. A simple phenomenological model of interaction between the induced electric field and neuron membranes is also considered, along with a model of post-synaptic calcium concentration and associated changes in synaptic weights Simulated EEG signals are produced in a simple protocol, both in the absence and presence of a 60 Hz MF. These results are discussed based on results obtained previously in humans. Notably, results highlight that (1) EEG alpha (8-12 Hz) power can be modulated by weak membrane depolarizations induced by the exposure; (2) the level of input noise has a significant impact on EEG power modulation; and (3) the threshold value in MF flux density resulting in a significant effect on the EEG depends on the type of neuronal populations modulated by the MF exposure. Results obtained from the model shed new light on the effects of power-line MFs on brain activity, and will provide guidance in future human experiments. This may represent a valuable contribution to international regulation agencies setting guidelines on MF values to which the general public and workers can be exposed.

In the autumn of 1813, I left my house at Henderson, on the banks of the Ohio River, on my way to Louisville... The air was literally filled with pigeons; the light of noonday was obscured as by an eclipse; the dung fell in spots, not unlike melting flakes of snow, and the continued buzz of wings had a tendency to lull my senses to repose.

Before sunset I reached Louisville, distant from Hardinsburgh fifty-five miles. The pigeons were still passing in undiminished numbers and continued to do so for three days in succession. The people were all in arms. The banks of the Ohio were crowded with men and boys incessantly shooting at the pilgrims, which there flew lower as they passed the river. Multitudes were thus destroyed...

It may not, perhaps, be out of place to attempt an estimate of the number of pigeons contained in one of those mighty flocks, and of the quantity of food daily consumed by its members. The inquiry will tend to show the astonishing bounty of the great Author of Nature in providing for the wants of His creatures. Let us take a column of one mile in breadth, which is far below the average size, and suppose it passing over us without interruption for three hours, at the rate mentioned above of one mile in the minute. This will give us a parallelogram of 180 miles by one, covering 180 square miles. Allowing two pigeons to the square yard, we have 1,115,136,000 pigeons in one flock. As every pigeon daily consumes fully half a pint of food, the quantity necessary for supplying this vast multitude must be 8,712,000 bushels per day...

Let us now, kind reader, inspect their place of nightly rendezvous... The dung lay several inches deep, covering the whole extent of the roosting place, like a bed of snow... As the period of their arrival approached, their foes anxiously prepared to receive them. Some were furnished with iron pots containing sulphur, others with torches of pine knots, many with poles, and the rest with guns.. Suddenly there burst forth a general cry of “Here they come!” The noise which they made, though yet distant, reminded me of a hard gale at sea passing through the rigging of a close-reefed vessel. As the birds arrived and passed over me, I felt a current of air that surprised me. Thousands were soon knocked down by the pole men. The birds continued to pour in. The fires were lighted, and a magnificent as well as wonderful and almost terrifying sight presented itself. The pigeons, arriving by thousands, alighted everywhere, one above another, until solid masses as large as hogsheads were formed on the branches all round. Here and there the perches gave way under the weight with a crash and, falling to the ground, destroyed hundreds of the birds beneath, forcing down the dense groups with which every stick was loaded. It was a scene of uproar and confusion. I found it quite useless to speak, or even to shout to those persons who were nearest to me. Even the reports of the guns were seldom heard, and I was made aware of the firing only by seeing the shooters reloading.

No one dared venture within the line of devastation. The hogs had been penned up in due time, the picking up of the dead and wounded being left for the next morning’s employment. The pigeons were constantly coming, and it was past midnight before I perceived a decrease in the number of those that arrived... Toward the approach of day, the noise in some measure subsided, long before objects were distinguishable, the pigeons began to move off in a direction quite different from that in which they had arrived the evening before, and at sunrise all that were able to fly had disappeared. The howlings of the wolves now reached our ears, and the foxes, lynxes, cougars, bears, raccoons, opossums, and polecats were seen sneaking off, while eagles and hawks of different species, accompanied by a crowd of vultures, came to supplant them and enjoy their share of the spoil.

It was then that the authors of all this devastation began their entry among the dead, the dying, and the mangled. The pigeons were picked up and piled in heaps, until each had as many as he could possibly dispose of, when the hogs were let loose to feed on the remainder.

Persons unacquainted with these birds might naturally conclude that such dreadful havoc would soon put an end to the species. But I have satisfied myself by long observation that nothing but the gradual diminution of our forests can accomplish their decrease.

The latter proved to be a prescient comment.  The last known passenger pigeon died in 1914, as a result of combined predation and habitat loss.

Text excerpted from Audobon's Ornithological Biography, via Lapham's Quarterly.  Image source uncertain, via The Scientist.

Addendum: Reposted from 2013 to add some salient paragraphs from an essay in Harper's Magazine (November 2015) entitled "Rethinking Extinction: Toward a less gloomy environmentalism" -

The birds that most of us eat today are chickens — lots of them — and turkeys, with the occasional duck, quail, or pheasant thrown in. So it is something of a shock to remember that, not so long ago, Americans were happy to eat just about anything with wings. An 1867 inventory of fowl available in the game markets of New York City and Boston featured not only wild turkeys, partridges, and grouse but also robins, great blue herons, sandpipers, meadowlarks, blue jays, and snow buntings.

In season, passenger pigeons were especially plentiful. Alexander Wilson reported they were sometimes eaten for breakfast, lunch, and dinner. The pigeon potpie — sometimes garnished with pigeon feet stuck in the middle — was common fare in colonial America. Passenger pigeons were preserved for out-of-season consumption by being salted, pickled in apple cider, smoked to make jerky, or sealed in casks with molten fat.

According to Schorger, the birds were “a boon to the poor”: in 1754, a half dozen sold in New York for a penny, a sum equivalent to thirty cents today. In times of surplus, they were fed to hogs.

By the middle of the nineteenth century, railroads had connected the cities of the eastern seaboard to the great nesting colonies of the Midwest. Word of the flocks’ locations spread rapidly thanks to another new technology, the telegraph, which allowed professional market hunters, as well as local amateurs, to converge on a site.

The most common way to kill passenger pigeons was to shoot them. Because the birds clustered so densely, no great skill was required to blast them from trees or out of the sky with a shotgun. Nets were widely used as well. Trappers broadcast grain and deployed captive “stool pigeons” to attract the birds, enabling them to snare hundreds at once. Captured pigeons could be killed by crushing their skulls between the thumb and forefinger, though, as Schorger notes, “It was difficult to continue this method without fatigue when many birds were handled.” Some hunters used specially designed pliers to break the birds’ necks. Others used their teeth...

Continued at the link.

Since I already cracked open this door and was not punished by a flood of angry comments, I guess I’ll throw the door wide open and see what happens. This is one of my favorite non-math topics to read about, but I’ve always had a policy of keeping my posts as non-controversial as possible. I guess violating this rule once won’t be that bad. In honor of Easter, I’ll do a literary analysis of the passion narrative as presented in two of the Gospels.

The type of thing I’m going to do gets me in trouble with both the literalists and with fellow atheists. Most atheists believe that everyone should have a fairly good acquaintance with the Bible, and many think you should actually read the thing once. When I express that I think a lot can be gained by studying it as a work of literature, many scoff at that as nonsense. I aim to point out that by doing proper textual analysis you can learn a lot of interesting things.

Everything I write here should be attributed to either Bart Ehrman or Richard Carrier (and their many sources), unless you find some error. I’ve read several books by both of them and seen many debates and lectures they’ve given. I’m going to go almost entirely off of memory (and the text in the Bible itself) and hence will not be able to cite where I learned which interpretations. Both people mentioned are absolutely great Biblical scholars and if you find this post interesting you should look up their books which are excellent. I think essentially everything here should be in Jesus, Interrupted by Ehrman.

If you grew up in some Christian denomination, then you probably have this idea of what “the” passion narrative is. What I mean by this is that even if you’ve read all four accounts as given in the four Gospels, your brain has probably meshed them into one coherent piece. If asked to name any key difference between the four accounts you may have trouble coming up with any or might think they are all basically the same. This is not your fault, because it is probably what you were taught.

The reality is if you closely read any of the four accounts of any given particular aspect of Jesus’ life, you’ll find that there are not only discrepancies, but almost every single detail is in direct contradiction with some other detail of a different account. Ah. But the Christian retorts that even though the details are different the overall main point and content of the story is the same. The purpose of this post is to show you that not only is this false, but it is false for very good reasons.

The reason there are discrepancies even in the main overall points of the stories is that the authors want you to get different things out of it. It may be possible to shove interpretations together in an attempt to make some coherent overall story, but to do so is just wrong-headed. You’ll miss what the authors are trying to tell you by doing this.

Of course, Christians don’t like this idea because implicit in what I’m saying is that these are not meant to be read as historical documents describing what actually happened, but as long-form parables (or as I’ve heard Carrier say, “meta-parables”) with Jesus as the main character whose details are literary devices meant to have theological interpretation. Although this isn’t a well-known idea, it is in fact what the majority of historians (including Christians) believe (see The Homeric Epics and the Gospel of Mark by Dennis MacDonald for this idea fleshed out to a whole book).

Alright, on to the details (I can already tell this is going to be like 2000 words or something, but what else am I going to do while I sit in an airport for two hours). Let’s compare the death of Jesus as portrayed by Mark and Luke (I use the term “by” colloquially to mean “as written in the book of Mark and Luke respectively” because the authors were anonymous). The facts: In Mark the scene is solemn and tragic. Jesus says nothing and is mocked by everyone as he travels to his execution. The only words spoken by Jesus appear right before his death when he says “My God, my God, why have you forsaken me?” He dies. Then the veil at the temple rips. Then the centurion says that he must have been the son of God.

The two literary details to pay attention to are the following. First, Jesus seems to feel totally betrayed through the whole scene. He even asks why God has forsaken him. He doesn’t seem to realize what is happening. Right after his death the veil rips and the centurion tells you how to interpret this detail. The ripping of the veil (or curtain according to some translations) symbolizes that it was a Jewish ritual atonement. It seems clear that the whole scene is a literary construction to make a theological point. Jesus was an innocent lamb being lead unknowingly to a sacrificial atonement ritual.

Mark emphasizes the suffering and the feeling that God has left you. This was written to put the early persecution of Christians into context. It is those who suffer and are meek who will inherit the Earth. This theme is consistent throughout Mark. It is re-emphasized by the fact that he has women find the empty tomb later on as the last thing to happen in the story. It seems deliberate to show that women (who had less status) are the ones to learn of Jesus’ resurrection, and that this is the last event of the book (assuming that Mark actually ends at 16:8 which we won’t get into since books have been written on this debate, e.g. Perspectives on the Ending of Mark: Four Views).

In Luke we get a much different picture. Jesus is not silent while traveling to his execution. He comforts women on the way (tells them not to cry). He asks God to forgive the soldiers nailing him to the cross. He is not excessively mocked in the same way. Jesus tells one of the people being crucified with him that he will see him in heaven shortly. In other words, Jesus seems perfectly aware of his innocence and what is going on and why. He is not concerned that God has left him and seems to be in direct communion with God the whole time.

He even says “Into your hands I commend my spirit.” I repeat, there is no cry of being forsaken in this account at all. I was probably 20 before I realized this. I grew up believing (like most other Christians) that both are said. In reality one is said in one account for a very specific reason and the other is said in another account for a different reason. To combine these into one “historical” account is to completely miss the point of both authors.

In Luke, the temple curtain rips while Jesus is still alive. The centurion does not proclaim him the son of God in this version. Thus our earlier interpretation of this symbol is absolutely impossible here. A common interpretation of this is that the ripped curtain symbolizes God’s rejection of the Jewish system of worship. Sometimes said that the Old Covenant was repealed and Jesus brought the New Covenant.

Corroboration for this evidence is that Luke seems to be referencing Hebrews “By this the Holy Spirit indicates that the way into the holy places is not yet opened as long as the first section is still standing.” Or Jesus’ prophecy earlier in Luke that as long as the temple stands it signifies the continuation of the Old Covenant. Or even later in Acts (written by the same author as Luke) that God left the temple.

If again we take Luke’s account as a long-form parable, the moral does not seem to be that God is there even during great suffering and persecution and even when you feel forsaken by God. The moral in Luke seems to be that during times of suffering you should stay calm and confident of your future in heaven.

Of course the three hours of darkness occur in both, and the symbolism of that is pretty obvious. But remember, we can say with great certainty that this event never happened in history, so we are forced to think of it as a purely symbolic detail. A much more realistic interpretation of what the Gospels are, given this information, is that they are merely literary constructions and not historical accounts to teach the particular author’s theological point-of-view of this new emerging religion.

If you are a Christian and are persuaded of some level of historicity of the Gospels, the takeaway of this post should be the following. At very least, when you hear something in church you should go see what each of the four Gospels say about it separately by carefully comparing the details. Try to figure out the author’s intent rather than blindly taking some mixed interpretation that was presented to you. Question what they tell you and look it up for yourself. You’ll probably be surprised at what you find, and at very least you’ll find much richer theological interpretations of the authors.

Now to tie this back to the beginning. When I point out that almost every detail provided in the New Testament is contradicted somewhere else, the most common pushback I get is that “these are minor details, but the Bible is consistent on its main points and overall story it tells.” I guess you can decide that for yourself, but I’ll leave you with the this question. You could probably extract three major theological questions from the above analysis that get answered differently, but it seems to me that the most important theological question in all of Christianity is “Why did Jesus die?” If you read Mark you get one answer. If you read Luke you get a different answer. Should this be consider a minor detail?