Non-Technical

R.J. Lipton:

• Interdisciplinary Research–Challenges (regarding the language barrier, I think the biggest problem isn’t new words (even new words for the same thing), the biggest problem is different meanings for the SAME word. For example, in mathematics, the notion of “topology” is pretty clear-cut. But I’ve never been able to figure out what the word is supposed to mean when used in, say, the social sciences, where it almost seems to be a meaningless buzzword). Sometimes vocabulary outside our own field is quite useful: the biological notion of a “clade” is used all the time in mathematics, but without ever being given such a short succinct name, and this results in a lot unnecessary verbiage.)
• The Singularity Is Here In Chess (Detecting cheating in chess using mathematics. Regarding the Open Problem at the end: don’t we already cheat in mathematics using computers? See e.g. the four color theorem)

Timothy Gowers:

• The Work of Endre Szemerédi (PDF)
• Elsevier’s open letter point by point, and some further arguments (world-class mathematician vs. mercenary Public Relations department)

Bradley Garrett:

• Place Hacking: Tales of Urban Exploration (This document will rightfully go down in history as the world’s first doctoral dissertation on Urban Exploration)

Cat Valente:

• #shitsiskosays (a 21st century look at the old show Deep Space Nine, which, in retrospect, got The Future pretty badly wrong)

Kas Thomas:

• How Google is Quietly Killing Firefox (I can personally confirm what Kas says about programmers developing a 6th sense for what software is doing)
• Splash screens == sloth

Tim Kastelle:

• You Are Responsible for Getting Your Ideas to Spread (“An artist who maintains that he has been misunderstood is almost always a bad artist”)

Patrick Rhone:

• TV Is Broken (I can haz 21st century?)

The Onion:

• This Article Generating Thousands Of Dollars In Ad Revenue Simply By Mentioning New iPad

Paul Graham:

• Writing and Speaking (this is why I prefer to read things rather than watch someone lecture about them)

John Armstrong:

• A Continued Rant on Electromagnetism Texts and the Pedagogy of Science

Rich Gibbs:

• Spinning the TSA (a murderer on trial can’t just order the judge not to admit certain witnesses, so why do big institutions get to do exactly that?)
• New Einstein Archive Site Launched (Awesome news. But why should these sites be limited to super-megastars like Einstein and Newton? I dream of a day when EVERY dead scientist has such a site. The internet might not be big enough to contain the Erdős archive, though…)
• Firefox to support SPDY Protocol (I could go on for days about the inefficiency of the HTTP protocol)
• Encyclopædia Britannica to Drop Print Edition (I love how twenty years ago, this would’ve made heads explode, but now it’s like “meh… Britannica hasn’t shuttered its doors yet?”)

Stuart Buck:

• A Trivia Quiz for Dogs (funny)

zhai2nan2:

• Ubisoft releases Anno 2070 game to showcase most annoying bugs (parody)
• If it’s not paying you anything and you know you’re not enjoying it, drop it (very obvious sounding, and yet I think we’ve all been guilty of disobeying this advice from time to time)

Alex Papadimoulis:

• The Thank-You Change (makes me appreciate how intelligent and great at communicating all my own colleagues are)

Bill Gasarch:

• “Math was a mistake- I made it too hard” (on an interesting dilemma in quoting people)
• How do legit fields of knowledge decide between competing theories? How does Astrology? (Relativistic Astrology. Hah!)

Thijs Markus:

• Why ACTA is so mercilessly pursued (when you stop producing anything of value and instead put everything into branding a product that slave workers overseas produce, and could sell themselves, suddenly you become extremely reliant on the government enforcing your monopoly with strongarm tactics)

Kresimir Josic:

• Thinking Fast and Slow (the social sciences still have a ways to go before they’re as rigorous as the science sciences)

Rick Falkvinge:

• Copyright Monopolies Stifle Innovation – Just Ask History (I’ll never understand how the decision to basically round up all the best things in society and bury them, is considered pro-innovation)

Casey Bergman:

• Did Finishing the Drosophila Genome Legitimize Open Access Publishing? (Going slightly off on a tangent, the other day, a PLoS ONE article on Drosophila was featured on the front page of Reddit, which almost certainly resulted in more people reading the article, in one single day, than would have read it in the entire lifetime of the universe if it had been published in one of the closed-access journals)

John Robb:

• Drones that Operate for Years on their Own

Jon Henley:

• Greece on the breadline: cashless currency takes off (I wonder how long until they start using Bitcoins)

Michael Lugo:

• Test Design and Grade Inflation (brilliant idea for how to design tests to be more easily graded, I wish more departments would use this as the standard)

Assistant Village Idiot:

• Not The Point (a better title would be, “how NOT to convert people to your political ideology”)

Simon Grey:

• Government Hypocrisy (more security theater from the TSA)

Dave Winer:

• Before I Use Branch (not so interesting for the specifics of one particular commenting company, but for more general remarks on writing)
• The bug in our process (it’s insane that California, Texas, and New York have the same number of senators as Montana, Wyoming, and Rhode Island)

Alexandre Borovik:

• Liar’s Paradox (more precisely, a very short nitpick about certain logic puzzles)

Technical

Jesse Johnson:

• After the VHC (It is an interesting paradox that a field can become LESS exciting when an open problem in that field is solved)

Jeffrey Morton:

• Cohomology, Groupoidification, and TQFT

Martin Davis:

• What are foundations for?

Joan Moschovakis and Garyfallia Vafeiadou:

• Some axioms for constructive analysis (PDF)

Tanya Khovanova:

• Guessing the Suit (beneath the fluff, there are very interesting questions about information coding here)

Terence Tao:

• Some ingredients in Szemerédi’s proof of Szemerédi’s theorem (includes an interesting comparison with Perelmen’s proof of the Poincaré conjecture)

Gil Kalai:

• Satoshi Murai and Eran Nevo proved the Generalized Lower Bound Conjecture

Burt Totaro:

• Why believe the Hodge Conjecture?

Russell O’Connor:

• Functor is to Lens as Applicative is to Biplate: Introducing Multiplate (See here for some scandal surrounding the ACM concerning this paper… we commend O’Connor for his courage and foresight in making the paper public domain)

Bill Gasarch:

• The Erdos- de Bruijn theorem (Be sure to check out the comments as well as the main body of the article)

Art/Photography

Telefunker:

• Puits SMN (France) (Ever wonder what it would look like if a mine had its own power plant, and then the mine and the plant were abandoned and allowed to decay? Look no further)
• Bärenquell Brauerei (abandoned brewery in Germany)

Bradley Garrett:

• Space Travellers: Barcelona (great look at Barcelona though a perspective only an Explorer gets to see)
• Bolt Action (get your daily fix of subway photography)

Zachary Abel:

• Many Morley Triangles

The General Antiderivative of 1/x

As you know, the antiderivative of an antidifferentiable function is not unique. Indeed, add any constant to a given antiderivative, and you get another. Thus, students are taught the following

• Misleading Fact: The “general” antiderivative of 1/x is
  • 

  where C is a constant of integration.

But as Tom Leinster pointed out, this indexed family of antiderivates is not exhaustive. There are antiderivatives of 1/x which do not have the above form for any value of C. The correct version is

• Fact: The truly most general antiderivative of 1/x is
  • 

  where C and D are constants of integration.

• Proof: We’ll use the following theorem from James Stewart’s Calculus: Early Transcendentals (6th edition, p.340):
  • If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F(x)+C, where C is an arbitrary constant.

  Note that the domain of 1/x, which is (−∞,0)∪(0,∞), is not an interval, and that’s why our Fact does not contradict the theorem from Stewart’s book. Now, the reader can check by direct differentiation that all functions of the form (*) are antiderivatives of 1/x on (−∞,0)∪(0,∞). We must show the converse: that if F is any antiderivative of 1/x on (−∞,0)∪(0,∞), then F has the form (*) for some values of C and D. So, suppose F is any antiderivative of 1/x on (−∞,0)∪(0,∞). Let F^R be the restriction of F to (0,∞) and let F^L be the restriction of F to (−∞,0). Since F^R is an antiderivative of 1/x on the interval (0,∞), by the theorem, F^R has the form ln|x|+C for some particular value of C. And since F^L is an antiderivative of 1/x on the interval (−∞,0), again, F^L has the form ln|x|+D for some particular value of D (we use D here because the variable C is already in use). This proves F has the form (*).

Generalizing from Leinster’s Example

More generally, if the domain of f consists of a disjoint union of n open intervals, where n is either a positive natural number or countable infinity, and if f has any antiderivative on its domain, then the set of antiderivatives of f is an n-dimensional space determined by n constants of integration.

In a comment at the Café, Mike Shulman pointed out that the function 1/cos(x) is an example of a function with an infinite-dimensional space of antiderivatives [Note: we use 1/cos(x) rather than the more esoteric "sec(x)" because sec(x) is an artifact from a more primitive age and its use should be absolutely proscribed in this era, along with its ugly siblings cot(x) and csc(x)]. The most general antiderivative of 1/cos(x) is

where …,C_-3,C_-2,C_-1,C₀,C₁,C₂,C₃,… are countably infinitely many constants of integration. (The above equation also doubles as a Two-Handed Sword of Science.)

Calculus textbooks include Tables of Integrals which, for seemingly unknown reasons, list certain one-dimensional subspaces of the spaces of antiderivatives of functions. For example, a Table of Integrals lists the one-dimensional space of functions of the form ln|x|+C as the antiderivatives of 1/x, even though we saw that the antiderivatives of 1/x form a 2-dimensional space. I say the reason for doing this is seemingly unknown, but there is some method to the textbook writers’ madness, as we’ll see in the next section.

Boundary Value Differential Equations Problems

As I pointed out in my comment on Leinster’s post, the whole reason we teach freshmen about constant of integration is to facilitate the solution of differential equations with boundary conditions. Normally– and this, I believe, explains why textbooks only list a one-dimensional space of antiderivatives– we only care about such questions when the known boundary conditions and the unknown values in question both lie in an interval where the functions in question are defined. This is rather subtle, so please consider the following three problems. On their face, they look identical, but you’ll quickly see that the first two are radically different than the third.

• Question 1. A car’s velocity function is v(t)=1/t.  At time t=1, the car has position 0.  Find its position at time t=2.
• Queston 2. A car’s velocity function is v(t)=1/t.  At time t=−2, the car has position 0.  Find its position at time t=−1.
• Question 3. A car’s velocity function is v(t)=1/t.  At time t=−1, the car has position 0.  Find its position at time t=1.

To solve Question 1, we antidifferentiate and obtain the position function y(t)=ln|t|+C for some constant C. Since y(1)=0, ln|1|+C=0, so C=0, therefore y(t)=ln|t|, and so the answer is y(2)=ln|2|≈0.693. Similarly, the answer to Question 2 is −ln|2|≈−0.693.

Now, let’s try to solve Question 3 using the usual technique. First we write y(t)=ln|t|+C. Since y(−1)=0, we get C=0. Thus, it seems, the answer is y(1)=ln|1|+0=0. But this can’t possibly be right. Consider a different car, call it Trollcar because its purpose is to troll us. Trollcar’s position function is defined to be

By differentiation, we see that Trollcar’s velocity function is… umm… 1/t. And Trollcar’s velocity at time t=−1 is… umm… ln|−1|=0. So Trollcar perfectly satisfies the differential equation and the initial conditions, but its position at time t=1 is ln|1|+1=1, contradicting the answer we computed for that initial value problem.

The reason the usual technique failed us in Question 3 is because we used the one-dimensional space of antiderivatives instead of the two-dimensional space. We did not need both dimensions for Questions 1 and 2, because both the initial condition and the unknown position were in the same interval in the domain of 1/x. In Question 3, the initial condition was in one interval, and the unknown position was in another. The correct answer to Question 3 is: “Not enough information.” The initial condition allows us to compute the constant C from equation (*) way up above, but we don’t have any information that would allow us to compute the constant D, which we need to compute y(1).

In general, in order to find y(x₀) in a differential equation boundary-value problem, where x₀ is a particular point in the domain of the position function, we need to be given a sufficient number of boundary conditions within the same interval as x₀ (when I say that a boundary condition is within the same interval as x₀, I mean that the position function and its derivatives are defined at every point in between). Boundary conditions separated from the unknown x-value by gaps in the position function’s domain, are useless.

A related problem is, rather than find a particular position, find the position function itself. In order to do this, in the case of an nth-order differential equation, it is necessary to have n boundary conditions on each interval of the position function’s domain. For example, if a car’s acceleration function is a(t)=1/t, and we want to find the car’s position function, we’re going to need at least 2 boundary values in (−∞,0), and 2 boundary values in (0,∞), a total of 4 boundary values. If a car’s velocity function is v(t)=1/cos(t), and we want its position function, we’re going to need one boundary condition in each interval (π/2+nπ,π/2+(n+1)π), a total of ∞ many boundary conditions.

The Pedagogical Problem

Tom Leinster’s survey focused not on the actual math, but on the pedagogy. This is, indeed, a rather itchy problem. Constants of integration already cause freshmen quite enough grief– are we really supposed to expect them to sit through things like the Two-Handed Sword of Science?

It’s a difficult problem. But I think it’s one which is solvable. Right now, calculus classes typically budget time for sections like “How to use Tables of Integrals”. In the 21st century, those sections are pretty outdated. If we threw them out, it would give us plenty of time to explain the subtler theory of calculus. And if I may quote Brendan Cordy, one of the commenters from Tom’s post: “A wise man once told me that students have no problems with abstraction, so long are they’re comfortable with the concrete cases they’re abstracting from.”

Introduction

In their paper (which is very interesting and which I highly recommend), Aloupis, Demaine, and Guo claim that the old Nintendo game Super Mario Brothers is NP-Complete. I immediately saw a (very minor) flaw in their proof, and noted it on Scott Aaronson’s blog [3]. A few hours later Ryan Landay confirmed [4] that he had independently noticed the same flaw. Without going into the specific details (see my comment on Scott’s blog for those), the authors assumed (and we can hardly blame them) that a solid wall is enough to stop a small Mario. But this assumption is false: Mario can actually jump through certain walls by exploiting glitches in the video game’s code.

To be clear, I am not claiming that (generalized) Super Mario is easier than NP-Complete. For all I know, Aloupis et al. are correct (indeed I’d be shocked if they aren’t). It’s just that the proof in their paper is a tiny bit flawed. I conjecture that the proof (at least for Super Mario Bros) could be patched up easily… maybe by placing a “checkerboard” of alternating solid blocks and spiky-shelled enemies behind the wall which is meant to be impassable– so that if Mario attempts to glitch his way through, the spiky-shelled enemies will skewer him. The point is that the episode places video game glitches in the spotlight.

Console games are extremely complicated, and very difficult to model game theoretically. Some would say chess is complicated, game theoretically. But the rules of chess are fairly simple, you can write them down on one page. You cannot do that for SMB, not if you want to accurately capture the real engine which the game uses. The true rules of SMB are coded in (I think) 6502 Assembly Code, and are about (I think) 40 kilobytes in their compiled form (so that in disassembled form they’d be considerably larger). That utterly dwarfs the rules of chess!

It would be completely impractical to prove things about SMB using its actual machine code. Instead, we simplify the game and try to intuitively explain it based on our experience playing it. This yields a model of the game which is enormously simpler. One such model is what Aloupis et al. study. In their simplified model of the game, Mario can’t jump through walls. But this does not accurately capture the actual rules of the actual console game, as measured by its assembly code (that is, as measured by what the game is capable of being twisted into doing, given a sufficiently superhuman player).

I find it kind of amusing and appropriate that this paper was brought up on Scott Aaronson’s blog, where issues of quantum complexity theory are so very expertly discussed. In the universe of Super Mario Bro’s, the model of physics which says “Mario can’t jump through solid walls” is kind of like Newtonian physics, and the (more accurate) model which says “Actually, sometimes he can” is kind of like quantum physics. Ryan Landay’s and my instructions for how Mario can quickly solve the Hard puzzles of Aloupis et al. might be viewed as some kind of strange analogy for hypothetical quantum computers which are sometimes hyped up as being capable of performing similar feats in our own world.

The Substantive Math, part 1: Game Theoretically Defining Glitches

I wouldn’t say I’ve exhausted the literature, but what searching I have done has unearthed very little about glitches through the game theoretical perspective. In fact, I’d go so far as to say that there is almost no literature about glitches in any of the current scientific literature. I could be mistaken here, it could just be that I’m no good at literature review. Of course, whole libraries have been written about software exploits, but I think there is something different about video game glitches.

I’ll focus on glitches which the player can exploit to win the game faster than intended. Call these speedrun glitches. These differ from the software exploits most studied in computer science because the end result of a software exploit is usually a state which was never meant to be reached: the programmers of Internet Explorer never intended for attackers to be able to execute arbitrary code, no matter what route they took. By contrast, a speedrun glitch aims for a state which is a perfectly normal part of the game– e.g., the “You Win” screen– but via a route which is unexpected.

It is very difficult to formally define what a speedrun glitch is. Objectively speaking– and this philosophy underlies the entire tool-assisted speedrun community– if it’s programmed in the game’s code, it’s part of the game. Thus, Mario being able to leap through solid walls is just simply a part of the Super Mario Brothers game.

We cannot simply choose glitches case-by-case on an ad hoc basis: they are subjective. To illustrate this: Aloupis, Demaine, and Guo make the “crouching slide” an integral part of their NP-Complete proof for Super Mario Brothers. This is a phenomenon where a large Mario can duck after building up speed and thereby slide through corridors normally too small for him to fit through. The question is: if jumping through walls should be ignored because it is “an obviously unintended glitch”, why doesn’t the same go for crouch sliding? I imagine the crouch slide is not officially documented in the Super Mario Brothers game manual (though I could be mistaken). The only way to know whether crouch slide was “intentional” would be to read the programmers’ minds as they were programming the game. It seems less glitchy only because it’s well-known by almost all players of the game and can be easily performed, whereas jumping through walls is more obscure and is almost impossible for a mortal human to perform. But these are not considerations we can very well articulate game-theoretically!

For the sake of putting some actual substantive math into this article, I very tentatively propose the following as a possible definition of glitch (or as a step toward such a definition):

• Tentative, sketchy, incomplete, possible definition of video game glitches. (This definition uses vague notions of largeness and therefore needs more work.) Let G be a formal one-player game (in the sense of game theory). Say that G is console-like if G contains a very large number of subgames which are nearly identical to each other, and each of those subgames contains a very large number of subsubgames which are nearly identical to each other, and so on for a large number of levels. A speedrun glitch is a winning strategy which avoids a large number of said subgames.

To understand the intuition here, imagine a winning play of Super Mario Brothers. Now imagine a play which is identical except that the player waits 1 second longer at the beginning before taking his first step forward. (This does not alter the winningness of the play, as long as the first play finished the first level with at least 1 second remaining on the clock.) In the formal game of Mario Brothers, both plays follow extremely similar-looking paths through extremely similar-looking subgames. Similarly, if two winning plays are identical except that one of them enters the coin-pipe in World 1 Level 2, and the other does not, again this gives two nearly identical paths through two subgames which are nearly identical except for a brief initial part.

Now, suppose I find a way to beat Super Mario through some obscure sequence of keymashings which takes me directly from World 1 Level 1 to the victory screen. This yields a path through the formal game which skips an extremely huge number of nearly-identical subgames and subsubgames and so on. (Note: this is not such a hypothetical situation! Masterjun did almost exactly this for Super Mario World, and MUGG and anymac did something similar in Super Mario Land 2. See [5],[6]

Of course, there is still a lot of thinking to be done to make this definition precise. As is, it merely pushes the vagueness bubble around under the carpet: instead of “what is a glitch?”, the question becomes “what is a large number of subgames and what does it mean for two subgames to be almost identical?” It might be necessary to bite the bullet and require that the formal games be extended to explicitly make precise which subgames are intended to be unavoidable. Especially considering that ideally the definition shouldn’t give a false positive for strategies which use the intentional so-called “warp zones”!

The Substantive Math, part 2: An Actual Theorem??

The following theorem sketch is so vague that I won’t even try to fully spell it out. I fully acknowledge that I am here committing horrendous crimes against mathematics. Please have patience with me. The theorem, vague as it is, attempts to identify the “trivial and silly” problem of speedrunning with the dead serious problem of software verification.

• Theorem: The following problems are equivalent:
  1. Finding the fastest speedrun of a game.
  2. Checking that a game has no major game-breaking glitches (at least none which allow the game to be won in a super fast time)
• Proof:
  • (2 is reducible to 1). To check that the game has no major game-breaking glitches, first find its fastest speedrun. Watch the speedrun. If it does anything that knocks your socks off, you’ve found your glitches.
  • (1 is reducible to 2). To find the fastest speedrun of the game, first disassemble the game into machine code. Using 2, check the code for game-breaking glitches. If you find any, pick the most egregious one and use it to make a speedrunning mockery of the game. If not, you may reduce the speedrunning problem to the much simpler problem of speedrunning the much simpler abstract model of the game which everyone actually thinks of when they play it. As Aloupis, Demaine, and Guo point out, this may still be NP-Hard, but the space of paths you have to check is extremely drastically reduced (to use [5] as an example, you may immediately rule out any paths which involve having your pet dinosaur repeatedly chew up and spit out P-switches for no apparent reason).

References

[1] TASVideos. Welcome To TAS Videos, 2010. Retrieved 15 Mar 2012.
[2] Greg Aloupis, Erik Demaine, and Alan Guo. Classic Nintendo Games are (NP-)Hard (PDF). Preprint.
[3] Scott Aaronson. Big News. Shtetl-Optimized, 2012.
[4] Ryan Landay. Comment on [3], 2012.
[5] Masterjun. SNES Super Mario World (USA) “glitched” in 02:36.4, 2012. Retrieved 15 Mar 2012.
[6] MUGG and andymac. GB Super Mario Land 2 “glitched” in 2:08.98
[7] Alexander the Great. Solution to the Gordian Knot.

The Usual Algorithm

Let’s look at how we evaluate summation notation. Define expressions inductively: for any number n, n is an expression; the variables x,y,z,… are expressions; if A and B are expressions, then (A)+(B) and (A)(B) are expressions; and if A and B are expressions and x is a variable, then Sum_(x=0)^(B)A is an expression.

A closed expression is an expression with no free variables (this can be rigorously defined by induction but I’ll omit that). For example, Sum_(x=0)^(5)x is a closed expression, but Sum_(x=0)^(y)x is not, because it contains the free variable y; you can’t calculate the latter sum down to a number if you don’t know what y is.

From basic algebra, we know how to evaluate a closed expression. We do it inductively (of course, it usually isn’t explained to us in such precise detail, but this is equivalent to what we are usually taught):

• The value of a number n, is n itself.
• The value of (A)+(B) is the sum of the values of A and B.
• The value of (A)(B) is the product of the values of A and B.
• The value of Sum_(x=0)^(B)A is the sum of the values of A(0),…,A(b), where b is the value of B and each A(i) is the closed expression obtained by replacing all instances of x in A by i.

The above definition is an algorithm. How do we know it converges to an answer? By induction on expression complexity. Why isn’t there a case listed for evaluating variables? Because the algorithm is only intended to evaluate *closed* expressions, and a variable expression is not closed. In fact, one very eccentric definition of “closed expression” could be “does not cause the above algorithm to enter an undefined case”.

Now let’s zero in on part of the above algorithm. What does it mean to “replace all instances of x in A by i”? Well, it has a very obvious meaning which we typically take for granted. You copy A down but write “i” wherever “x” was. But of course, a computer has neither eyes nor pencil, so we have to make this precise. So we define:

• For a number n, if you replace x by i in n, you get n itself.
• For a variable y, if you replace x by i in y, you get: i, if y=x; y, if y≠x.
• If you replace x by i in (A)+(B), you get (A(i))+(B(i)), where A(i) and B(i) are the results of replacing x by i in A and B.
• If you replace x by i in (A)(B), you get (A(i))(B(i)).
• If you replace x by i in Sum_(y=0)^(B)A, where y≠x, you get Sum_(y=0)^(B)A(i).
• If you replace x by i in Sum_(x=0)^(B)A, you get Sum_(x=0)^(B)A.

Now, the above operations are not without a price. In order to replace x by i in an expression, a computer must run through every sub-expression, checking which case it is and either copying it or copying an altered form of it. For example, consider the expression Sum_(x=0)^(9)1 (which should evaluate to 10). The evaluation algorithm instructs us to evaluate 1(0)+1(1)+1(2)+…+1(9), where 1(i) means “the expression obtained by replacing x by i in 1″. Each time we compute 1(i), we have to run the “replace” subroutine above. The subroutine tells us to check which type of expression 1 is; we see it’s a number, so we copy it verbatim. We do this 10 times, and store 10 separate copies of “1″ in memory. What a waste! Of course, in practice we would skip all this work and immediately see Sum_(x=0)^(9)1=10, but the means by which we do this are ad hoc. How would you program a computer to do something so clever? You could shoehorn this one special case into the algorithm, but I could list ten new special cases for each one you add.

The More Efficient Algorithm

The problem with the algorithm we’re taught for evaluating summation notation is it involves writing down lots of modified copies of subexpressions. If we follow the algorithm to the very letter (as a computer must), we end up doing more substitutions than actual computations, and all those copies-of-subexpressions take up a ton of space (on paper or in RAM).

A better way is to leave the expression and its sub-expressions unchanged, and instead keep track of the values of variables. In fact, we have one of those situations– so beautiful whenever they occur– where the problem is actually easier if you generalize it. In the previous section, we limited ourselves to evaluating *closed* expressions. Let us now turn to the more general problem of evaluating arbitrary expressions (which may include free variables), given the values of the variables. The algorithm writes itself:

• Input: an expression S and a list L of variable values, including values of all variables occurring free in S.
• Desired Output: the value of S relative to L.
• If S is n, a natural number, then output n.
• If S is x, a variable, then output whatever value L says x has.
• If S is (A)+(B), then evaluate A and B using L, and output their sum.
• If S is (A)(B), then evaluate A and B using L, and output their product.
• Suppose S is Sum_(x=0)^(B)A. Let b be the value of B using L. Let SUM=0. For i=0 to b:
  • Let L(x|i) be the list obtained by adding “x=i” to L (if L already had a value for x, do not include the old value in L(x|i))
  • Increment SUM by A(L(x|i)), the value of A relative to L(x|i).
• Output SUM.

As for closed terms, we have a quick little “wrapper” algorithm:

• Input: a *closed* expression S. Desired output: its value.
• Output the value (using the previous algorithm) of S relative to the empty list.

Note that on a computer, this method does not need to involve any copying at all of subexpressions, since instead of passing the (unmodified) subexpressions themselves, we can just pass pointers to them. We don’t even need to make copies of L, provided we’re sufficiently clever with singly-linked lists.

The Ambiguity

If the two algorithms aren’t equivalent, then we have something of a crisis for mathematics, because both methods seem equally valid as potential definitions for the value of summation notation; which one is the “correct” one? Mathematicians abhor nothing more than having to make an arbitrary choice. So the question is: are the two algorithms equivalent?

I love this question because it seems so obvious the answer is yes! But how the heck do you prove it? It’s actually a special case of a more general result in my paper, The First-Order Syntax of Variadic Functions (PDF), which I’m infinitely happy to say will appear in the very excellent Notre Dame Journal of Formal Logic. We can view “expressions”, as defined here, as terms in first-order logic in an appropriate language, except for one little catch: first-order logic does not come with machinery for summation notation! No problem, we can just add that machinery. And that’s precisely what I do in the paper.

In first-order logic, an assignment is a map which takes variables in the language into elements of the universe. This corresponds to the “lists” in the efficient algorithm above. If s is an assignment and t is a term, we usually write t^s for the interpretation (value) of t in a structure assuming variable values given by s. But, due to the above sections, there is an ambiguity in how this should be defined when the language contains something like summation notation. For simplicity, work in the language with + and *, constant symbols for the naturals, and summation notation, and let the universe be the natural numbers. At least until we resolve the ambiguity question, there are potentially two term interpretations, which I’ll write t^s and t_s:

• If n is (a constant symbol of) a natural then n^s=n_s=n.
• If x is a variable then x^s=x_s=s(x).
• If A and B are terms then ((A)+(B))^s = A^s + B^s and ((A)+(B))_s = A_s + B_s.
• If A and B are terms then ((A)(B))^s = (A^s)(B^s) and ((A)(B))_s = (A_s)(B_s).
• If A and B are terms and x is a variable, then (Sum_(x=0)^(B)A)^s = A(x|0)^s +…+ A(x|B^s)^s, where A(x|i) is the result of substituting (the constant symbol) i for x in A.
• If A and B are terms and x is a variable, then (Sum_(x=0)^(B)A)_s = A_(s(x|0)) +…+ A_(s(x|B_s)), where s(x|i) is the assignment which agrees with s except that it maps x to i.

Now, in order to show that the two algorithms in this article are equivalent, it is enough to show that for every term A and assignment s, A^s=A_s. This seems like it should be a trivial induction on the structure of A. All the cases except for summation are trivial. As for the summation case, we start computing:

• (Sum_(x=0)^(B)A)^s = A(x|0)^s +…+ A(x|B^s)^s (by definition)
• (Sum_(x=0)^(B)A)^s = A(x|0)_s +…+ A(x|B_s)_s (by induction)

Now, what we need to make the final step, is that for each i, A(x|i)_s = A_s(x|i). So we’re done by that basic result of first-order logic, the Substitution Lemma, which says precisely this! …right?

Well, not quite. The Substitution Lemma is a theorem of standard first-order logic, and we’re not working in standard first-order logic, we’re working in first-order logic with summation notation added. So, the final step is to re-prove the Substitution Lemma in this nonstandard logic. And I do that in my paper. The details become substantially more complicated than they are in ordinary first-order logic. But boy is it worth it, since now we can use whichever version of summation notation evaluation interchangeably, with no fear of ambiguity.

Acknowledgments

Thanks to the anonymous referees from NDJFL, without whose suggestions I never would’ve been led to notice this ambiguity. Thanks to Mike Fenwick for reading a rough draft of this article and for much feedback and discussion on it.

FURTHER READING

The First-Order Syntax of Variadic Functions (PDF)
Goodstein Sequences
The Illogician
Ambiguities in Mathematics
Obtaining Strong Recursion from Weak Recursion

Suppose you have an infinite number of Turing Machines T1, T2, … and you want to check whether any of them output “0″ when run on a blank tape. A naive solution would be to begin running T1 until you get an output, check whether it’s “0″, if not then begin running T2, and so on. The problem is, T1 might never halt, and there’s no way for you to determine whether or not it’s ever going to halt.

The Dovetailing solution is like this: you run T1 for one step. Then you run T1 for two steps, followed by T2 for one step. Next, run T1 for three steps, followed by T2 for two steps, followed by T3 for one step. And so on. In the kth iteration, you run T1 for k steps, you run T2 for k-1 steps, …, and you run Tk for one step. If any of the machines will ever output “0″, you’ll eventually see it. (More precisely, if Tk outputs 0 after N steps, you’ll see this on the (N+k-1)th iteration, assuming you don’t see a different 0 earlier.)

If the ordered sequence (T1,T2,…) is recursively enumerable, you can even go so far as to create a single machine which does all this checking for you. In fact, you can create a machine which accepts an arbitrary input N and then checks whether any of the machines listed by the Nth machine ever output “0″ on the blank tape. You can do all kinds of crazy stuff with dovetailing.

Application to Complexity Analysis

Suppose you’ve got two algorithms. Algorithm A runs in probabilistically-expected time O(n^2) but in the worst-case scenarios, can take as long as O(n^4). Algorithm B runs in time O(n^3). Can you get the best of both worlds? Yes! You can dovetail the two algorithms together to simulate running them in parallel. (The dovetailing described above should be altered slightly: rather than starting over from scratch each time you return to one of your algorithms, start it where you last left it off) Thus, run one step of A, followed by one step of B, one step of A, one of B, and so on until one of them finishes. Now you’ve got yourself an algorithm with probabilistic-expected runtime O(n^2), and you’ve whittled its worst-case runtime down to O(n^3)!

Of course, only a theorist would think of something like this, since in practice the dovetailed algorithm would tend to be twice as slow as algorithm A, even though they have the same probabilistic big-O runtime.

Dovetailing Research

In some sense, all academic research is dovetailed. We do not know what the optimal strategy is for attacking an open problem like the Riemann Hypothesis. We could choose one strategy and have the entire mathematics community pursue it single-mindedly. And if that strategy turned out to be the best one, or even within three or four orders of magnitude of the best one, this would get us a solution faster. However, it’s also possible that the strategy is a “non-halting Turing Machine”, that it never leads to the solution, and that pursuing it is a waste of time.

So instead we dovetail. We spend some time following one strategy, then we move to the next, and then the next… presumably, if a strategy hasn’t been completely exhausted, somebody will eventually get around to pushing it a little further… and we hope that at least one of the strategies works, and that we aren’t in the situation where NONE of the machines reaches a desired outcome!

FURTHER READING

Turing Machines
Levels of Infinity
Obtaining Strong Recursion from Weak Recursion

Marketing

If we look across the aisle toward book publishers, we see they add a tremendous value to what they print: they market it. Compare this with academic journals. These offer nothing in the way of marketing. The Elsevier outcry has focused on the fact that the community does the writing, the typesetting, the reviewing, the editing, and the judging all for free. One more task which the community does for free is the marketing. Who takes the paper on the road and presents it at conferences? The author. Who cites the paper? The author and other authors. Who hypes it up with blog posts and comments? The author, and if it’s a really fantastic paper, others in the community. Who issues press releases if the paper is groundbreaking? The author’s institution and the community.

I’ve felt hesitant about writing this. The reason I felt hesitant is because we cannot even imagine academic publishers marketing for us without screwing it up (we shudder to imagine Elsevier spamming math bloggers with form letters). But as I thought about it, I realized, that isn’t how it ought to be. In a perfect world a highly-paid publisher ought to be qualified to talk about what they publish. If an Elsevier journal publishes a paper about self-distributivity and braid ordering, they ought to have in their employ some scientists competent enough to understand the paper. Of course, that might be expensive, requiring the publisher to house entire academic departments within its walls. But we’re talking about a giant corporation here, and how that corporation could earn its (very expensive) keep. If we can’t even imagine handing marketing to the publisher for fear of them screwing it up, then why on Earth do we let them publish us in the first place?

Proof Verification

The technology is still learning to walk, but one way a publisher could add very cutting-edge value to the math community would be to hire a bunch of technicians in Coq and Isabelle and other automated proof software, and formally verify accepted math papers. This is something of a pipe dream, but if the publishers pulled it off, then nobody could deny the tremendous added value. I recently taught myself Coq when a referee suggested I partially mechanize one of my own papers. It’s a fun language. I bet a lot of math papers could be formally verified by technicians with master’s degree levels of training in mathematics; if a paper is unverifiable because of too much handwaving, then merely alerting the author to this fact would itself be value-adding. Inevitably, the verification process would catch subtle glitches that eluded even official peer review, and that too would add terrific value.

To facilitate this, authors could submit two different manuscripts to the publisher. One would be the “human” manuscript where certain proofs can be “Straightforward.” or “Left to the reader.” or “By induction on n.” The other would be a larger manuscript where those omissions are spelled out, and this, while not itself being published, would be used by the Coq/Isabelle technicians. This would require more work on the writer’s behalf, but on the other hand… … … you guys ARE already checking those details on your own before submitting them as “straightforward”, right?!? (Guilty glances shoot around the room)

Translation

Here’s yet another way the big publishers could add tons of value: translate the journals into all different languages. Again, this would require they employ competent scientists, but remember: with great profit comes great responsibility. I, for one, would be thrilled half to death to have my papers translated into Mandarin and French and German and Japanese and Hungarian and… Hey, you know how Elsevier always says, “Acceptance of the copyright agreement will ensure the widest possible dissemination of information”? (cough cough BS cough cough) That might actually mean something if they had a department of competent bilingual technicians paid to translate your paper into exotic foreign tongues!

Analytics

“Translation? Lame, that’s so twentieth century.” Alright, back to the future, here’s what else the publishers could provide: advanced analytics like the ones Google offers webmasters. What might those analytics look like? Here’s a very rudimentary idea…

Analytics for your paper, “Counterexample to the ‘Circle is Round’ Conjecture”

• Published in: Transactions of the Kurdish Mathematical Society
• Traffic Sources (click for more detailed breakdown):
  • MathSciNet: 27%.
  • Google Scholar: 13%.
  • Xamuel.com Eighty-fifth Linkfest: 11%.
  • Other Sources: 49%.
• How Far Readers Got (click for more detailed breakdown):
  • Abstract only: 43%.
  • Abstract and Introduction: 25%.
  • Up to first major theorem: 19%.
  • Entire paper (not including appendices): 6%.
  • Entire paper (including appendices): 7%.
• What Did Readers Spend Their Time On (click for more detailed breakdown):
  • Abstract: 1 minute 23 seconds on average
  • Introduction: 6 minutes 10 seconds on average
  • Lemma 1: 12 minutes 53 seconds on average
  • Proposition 2: 26 minutes 4 seconds on average
  • Main Theorem: 42 minutes 12 seconds on average

Have any other ideas how the big publishers could justify their continued existence? Post ‘em in the comments!

The previous Linkfest was Linkfest 13.

Technical

Harvey Friedman: Maximality, Choice, and Incompleteness

Asger Törnquist and William Weiss: The Σ¹₂ counterparts to statements that are equivalent to the Continuum Hypothesis

John Armstrong: Too many to list! A very small sample chosen mostly at random: Coulomb’s Law, The Biot-Savart Law, Currents, Gauss’ Law for Magnetism, Faraday’s Law, Conservation of Charge, Deriving Physics from Maxwell’s Equations

Yemon Choi: Primary literature? What’s that, Gramps? (on “proof by appeal to Fields-medal winning paper”)

Abhishek Parab: The Riemann zeta function

Carlos Matheus: Special curves in Hilbert modular surfaces and Lyapunov exponents of Prym eigenforms, SPCS 4

Terence Tao: Montgomery’s uncertainty principle, Random matrices: Sharp concentration of eigenvalues, Random matrices: The Universality phenomenon for Wigner ensembles, Every odd integer larger than 1 is the sum of at most five primes

JoshTAS: Potentially incoherent ramblings regarding Stun-Glitch (this should quiet the critics who complain the “Technical” sections of these Linkfests never contain anything besides math/physics)

Nathaniel Johnston: Counting and Solving Final Fantasy XIII-2′s Clock Puzzles (as long as we’re on the subject of video games…)

David Speyer: A way to discover the Gamma function

Qiaochu Yuan: A less biased definition of a group, A first blog post on noncommutative rings

Akhil Mathew: Delooping and the bar construction, Categories and cohomology theories, The Quillen-Suslin theorem, Homotopy is not concrete

Assaf Rinot: The order-type of clubs in a square sequence, Dushnik-Miller for regular cardinals (part 1)

Daniel Moskovich: Beyond the trivial connection

KW Regan: What makes a knot knotty?

Lance Fortnow: Being Random and Trivial in Dagstuhl

Ben Webster: What is a symplectic manifold, really?

Shreevatsa Rajagopalan: Are there Fibonacci numbers starting with 2012?

Benjamin Steinberg: A Semigroup Approach to Finite Markov Chains

Ngô Quốc Anh: A note on the maximum principle

Andrej Bauer and Peter Lumsdaine: On the Bourbaki-Witt Principle in Toposes (PDF)

Dan Brumleve, Joel David Hamkins, and Philipp Schlicht: The mate-in-n problem of infinite chess is decidable (PDF) (seems obvious, ’til you realize that rooks and bishops and queens can travel arbitrarily far in a single move)

Matthias Aschenbrenner, Lou van den Dries, and Joris van der Hoeven: Towards a Model Theory for Transseries (PDF)

Mathieu Hoyrup, Cristobal Rojas, and Klaus Weihrauch: Computability of the Radon-Nikodym derivative (PDF)

Richard Lipton: Lonely Runner Conjecture: An Attack

John Baez: Archimedean Tilings and Egyptian Fractions

Non-Technical

Timothy Gowers: Elsevier – my part in its downfall (the blog post that made the boycott explode)

Tyler Neylon and 3,763 others and counting: The Cost of Knowledge

John Baez: Ban Elsevier

Scott Aaronson: Boycott Elsevier!

Rob Ousbey: My Career Advice: Make Yourself Redundant

Frank Quinn: A Revolution in Mathematics? What Really Happened a Century Ago and Why It Matters Today (pdf)

Sebastian Delmont: Good bye, Google Maps… thanks for all the fish

Scott Aaronson: Review of The Access Principle by John Willinsky (PDF) (Interesting commentary on academic journals)

Scott Morrison: Journals and the arXiv

Spike Japan: Kiyosato: high plains drifter

Paul Graham: Schlep Blindness

Simone Santini: Famous rejects in computer science (These are fake, but are funny anyway)

Timmo: Truth: Scientific and Beyond

Rich Gibbs: Satellite Phone Encryption Cracked

Sonic Charmer: (On the name of) Facebook

Orchid64: Won’t Miss #414: Beating around the bush (I can definitely confirm this one)

Tyler Neylon: 97% of US schools cannot afford Elsevier journals

Mark Eichenlaub: Parables

Dave Winer: NYT Growing the Wrong Way, The internet is realistic, The bosses do everything better, Get the tech back in tech

Tae Kim: A gentle introduction to Kanji

Cosma Shalizi: Scientific Community to Elsevier: Drop Dead

Terence Tao: The cost of knowledge

Andrew Gelman: Suggested resolution of the Bem paradox

Michael Nielsen: On Elsevier

Jeffrey Shallit: In Memory of Sheng Yu (1950-2012)

Derek Jones: Relative spacing of operands affects perception of operator precedence, Type compatibility the hard way

Michel Bauwens (excerpting Barbara and John Ehrenreich): Occupy Wall Street and the Decline of the Professional Managerial Class

Giovanni Dannato: Longevity Outliers: People to Watch

Jonathan McCalmont: Hang All The Critics: Towards Useful Video Game Writing

Ilkka Kokkarinen: Effing ficient (on price efficiency of universities)

Remy Porter: The Shredder

Laura Zirbel: Productive Procrastination: The Importance of Having Several Projects

Delta: Calculator Equals

Roger Ebert: I’ll tell you why movie revenue is dropping… (and Ilkka’s discussion thereof)

Daniel Lau: Comics are Trash

JSE: Is commercial writing more honest than academia?

Izabella Laba: Let’s overhaul the seminar!, The state of the profession (of mathematician)

Andrés E. Caicedo: A letter (on the resignation of the editorial board of Topology)

Steve Blank: Why the movie industry can’t innovate and the result is SOPA

Simon Grey: Another Needless Enemy (on U.S. vs. North Korea)

Santo D’Agostino: Failing … To Learn, How Much Mathematics Should A Student Memorize? Part 5, The Multiplication Table

Jonathan Benson: Indian Government Files Biopiracy Lawsuit Against Monsanto

John Robb: Modern Darknets

Jon Purdy: The Web Is Wrong

Art and Photography

Bradley Garrett: Home Turf: Carving Places in Space

Ken Baker: Chains and Tangles

Telefunker: Centrale Thermique Terres Rouges (abandoned power station in Luxembourg), Military Hospital R (abandoned military hospital from East Germany from before the Iron Curtain fell), RAW Pankow (abandoned German roundhouse in Berlin), Manoir de la Chapelle (abandoned Belgian villa)

Zhai2nan2: Crafting in Skyrim and New Vegas versus design-as-gameplay in Minecraft and Dwarf Fortress, Attention all costumed do-gooders: you are NOT a superhero if your only power is super-cleavage

Alex Papadimoulis: Server Tent, and other photography

• “Remind me to pick up my package tomorrow.”

But maybe you’re not going in to the department tomorrow. Maybe you’re going in the day after tomorrow. It wouldn’t be appropriate to ask your sweetheart to remind you to pick up your package in two days. So you need to wait one more day before asking her to remind you. But what if you won’t remember to ask her to remind you? You’d better take care of that!

• “Remind me to ask you to remind me to pick up my package tomorrow tomorrow.”

But what if you aren’t going in to the department until the day after the day after tomorrow?

• “Remind me to ask you to remind me to ask you to remind me to pick up my package tomorrow tomorrow tomorrow.”

I’ll leave it to you to generalize The Reminder Protocol to an arbitrary number of days after tomorrow.

FURTHER READING

First World Problems (if this isn’t one, I don’t know what is!)
The Illogician
Troll Dad

(Continued: Morning, Sleepy Head! at ALIL)

FURTHER READING

First World Problems
Inglip Comics
Dictators: How to Immunize Yourself Against Revolution

For example, take The Daily Show with Jon Stewart, or The Colbert Report with Stephen Colbert. People are always linking me to clips from these shows, and I used to consider myself a fan, but nowadays I don’t follow those links unless I’m bored or there’s some compelling reason. Why? Because they haven’t changed in years, and I can almost always predict their punchlines in advance (or if not the exact punchline, the overall idea). This isn’t to say the shows are bad at comedy. They’re still quite funny! But I don’t need to view them. Even if it were imperative that I maintain a solid grasp on those shows (e.g., if I hang out with a lot of young white people), I still would not need to view individual clips.

Again, there are some bloggers whose every scribbling I used to devour, but nowadays I rapidly skim their posts or skip them entirely. Based on nothing but the post title, I can pretty reliably predict what they’re gonna say. In fact, I’ll go further than that: given just the title, if I were forced to, I could write an appropriate article myself. And I must emphasize, I’m still talking about some top-notch bloggers here. It’s just that I’ve reached a point where generic articles no longer offer anything unpredictable. To grow, we must expose ourselves to radically new material.

“The Onion” is another great example. It’s quite fun to browse The Onion just looking at headlines. Once you’ve read a few stories (or maybe even before you’ve read a few stories) you can begin absorbing all the humor through the headlines alone, no longer needing to actually open them and read the full articles. I was an Onion reader way back in the early days, before they became bajillionaires (yeah, yeah, call me a hipster). Back then, most the headlines weren’t even clickable. They didn’t even bother writing stories for them, but those headlines were still the best part of the site.

Any content whose cornerstone is parody, satire, or sarcasm, is bound to become predictable fast (unless it’s of the finest subtlety). My deepest respect goes out to writers who keep on reinventing themselves and reinventing themselves, writing from way out of left field, keeping me on the tips of my toes. And to writers whose work is stuffed full of solid content, whether it be mathematical theorems or cocktail recipes. If there’s one thing I’ve learned from all my blogging, it’s a deep appreciation for all things unpredictable.

FURTHER READING

Blogging vs. Academic Writing
Fun with Pop-Evolutionary Biology
Incentives for Creativity