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Monday, August 9th, 2010

tante suhrkamp lässt erzählen

“Aus der Sicht des Gehirns”, Gerhard Roth, suhrkamp taschenbuch wissenschaft, S. 40ff.:

So haben wir bis zu einer Distanz von ca. 6 Metern ein direktes räumliches (stereoskopisches) Sehen, also eine echte Dreidimensionalität. […] Die stereoskopische Tiefenwahrnehmung ist sehr präzise, und deshalb können wir mit ruhiger Hand fast auf den Millimeter genau nach nahe gelegenen Gegenständen greifen. Das hochpräzise räumliche Sehen ist aber auf den Nahraum beschränkt, was natürlich Sinn macht. Mit zunehmender Entfernung wird die Disparität der beiden retinalen Bilder immer kleiner, und ganz andere Hilfsmittel zur Entfernungsschätzung kommen zum Einsatz, die auch mit einem Auge funktionieren. […] Eines dieser Hilfsmittel heisst Bewegungsparallaxe und nutzt die Tatsache aus, dass bei seitlichen Kopfbewegungen nahe Gegenstände sich stärker bewegen als etwas entferntere, und sich jene vor diesen hin und her zu bewegen scheinen.

Sehen wir mal davon ab, dass hier ein Wissenschaftler Stammeldeutsch schreibt – “was natürlich Sinn macht”, Lektor hin, Kommunikationswille her – dann bleiben immer noch die faktischen Gurken in diesem Kopfsalat: Disparitätswinkel nehmen mit der Entfernung ab, soweit schon richtig, aber das bedeutet nur, dass in grösserer Entfernung grössere Abstandsunterschiede nötig sind, um denselben qualitativen Tiefeneindruck zu gewinnen. Wer schon mal im Wald gestanden hat, weiss, dass binokulares Tiefensehen durchaus nicht auf die nächstgelegenen sechs Meter beschränkt ist; es sei denn, er war vor lauter Bäumen zu abgelenkt, genauer hinzugucken.

Die von Roth zur Behebung des von ihm erfundenen 3D-Notstands dann ins Feld geführte Bewegungsparallaxe ist drolligerweise geometrisch mit Disparitäten vollständig identisch: Ob ich mein Auge sechs Zentimeter seitwärts verschiebe oder mir stattdessen dort einfach ein zweites wachsen lasse, perspektivisch ist das ein und dasselbe.

Und “echte Dreidimensionalität”: auch Quatsch. Selbst mit fünf Augen sähe man nicht hinter die Dinge, und wäre also noch immer von einer einzelnen visuellen Oberfläche unterschiedlichen Abstands umgeben. Und in den Ausnahmefällen, in denen man zwei Dinge hintereinander sehen könnte, im Fall schmaler Objekte etwa, oder bei transparenten Folien, sorgt die Disparitätsgradientenschwelle meist dafür, dass nicht beide gleichzeitig voll sichtbar sein können. Halten Sie doch mal zwei Finger hintereinander, einen nahe, einen weiter weg. Ist der Abstandsunterschied gross genug, sieht zwar jedes Auge beide Finger, aber binokular oder “echt dreidimensional” sieht man nur den, den man direkt ansieht. Der andere Finger wird doppelt gesehen, sein Abstand ist unsicher. David Marr nannte das visuelle Ergebnis deshalb den “two-and-a-half-D sketch”. Zweieinhalb-D, Herr Roth, “echt”.

Ich freu mich schon aufs vollständig überarbeitete Kapitel zum freien Willen. Das wird bestimmt toll.

Saturday, August 7th, 2010

Cheap Dreams

Inception blatantly invites comparisons. The zero-g simulation and the center room of the dream heist evoke the mystery of 2001, the cityscape of Limbo wants to look like Brazil, and the gimmicky plot screams “this is what Matrix 2 and 3 should have been like”. Failed ambition.

If I were in want of stunning visuals lacking in emotional and logical depth, Avatar would be a much better fit than this and do the job much more nicely and honestly, without leaving the tinny taste of being taken for a sucker in the mouth. For all its technical glory, Inception makes almost no sense as story or metaphor, and it’s sad to imagine the movie this could have been, in the hands of a writer/director taking the mind seriously. Diving down into somebodies subconscious ought to be unsettling and intense, but instead there is just stuff being blown up and secret agents shooting blanks: mind as submachine gun. Yawn.

Additionally, the only way Inception can keep you from falling through the cracks in the plot is by keeping everything in frenzied motion. But once the action stops, everything crumbles. The crumbling is pretty to look at, no doubt, but what remains in the end is still only technorubble.

Wednesday, June 23rd, 2010

Shermer’s Folly

Figure 6: Some nonsense (not to scale)This plot is from chapter 4 of Michael Shermer’s book “Why People Believe Weird Things”, and it’s quite amusing.

The section of the book this is from concerns ESP experiments. Participants in these experiments have to predict one of five symbols on an unseen card, and the plot is meant to show the chance of a participant getting x of the answers right when being asked 25 times. Predictably, getting 5 right out of 25 is the most likely, but higher numbers of hits are entirely expected, and their occurence alone does not mean much. Shermer stops with the negative statement – high performance doesn’t mean ESP – and doesn’t spell out what a correct analysis would look like – not looking at a single result, but comparing a distribution of results to that predicted from chance – but this unfortunate focus on gleeful debunking rather than the education of the reader isn’t my concern. It’s the plot itself. Look at it.

Its x axis runs from -2.5 to 12.5. Bars of unit width indicate the probabilities of getting a certain number of answers right, and a Gaussian curve (Shermer calls it a Bell curve in the legend, and a normal distribution in the chapter text) to fit the bar data. Pretty much all of this is awful. First of all, fractions of x make no sense. The number of correct answers can only be a whole number, therefore probabilities should be plotted for whole numbers alone. If you want to use bars, make them discontinuous to emphasize their discrete character. Additionally, both the Gaussian and the leftmost bar extend into negative values, which also do not make sense for x. Negative values shouldn’t be on this plot at all. All this could be simply due to sloppy graphical design, of course, but the Gaussian extending into the negative is in fact a hint of the biggest error here: this distribution is not a normal distribution at all. To show a Gaussian here is a blatant statistical and conceptual error.

Random answering in a task like this in fact follows a binomial distribution. Interestingly, the numbers on the bars are from the correct binomial (except for 0.0238, which should have read 0.0236), meaning that whoever prepared the data knew what they were doing. But Shermer, fitting a Bell curve to them, clearly does not. You can even see it’s a bad fit. The bars don’t look symmetrical at all. Furthermore, the claim in the legend that “in a group of 25 several scores [above 7] will always occur purely by chance” is also quite false. He himself provides the probability for scoring below 8, which is 89.1%. Thus, in a group of 25, the chance of everybody scoring below 8 is 89.9% to the 25th power, or 5.56%. In other words, roughly one in eighteen such groups will have nobody scoring 8 or higher. There are lies, damn lies, and statistics done by fools.

Claims of the paranormal are emotionally and spiritually appealing, and it is important to counter them both on their own grounds by providing equally appealing stories about the world, and also by clearly showing where their purported proofs are untenable. Michael Shermer is the founder of the Skeptics Society, who claims this as their task, and he adopts a smug tone of intellectual superiority over the misled and uneducated foolish masses throughout this book. Yet apparently he doesn’t know what he’s talking about himself. He even told me so at the end of his foreword: “why should [you] believe anything [I] say? […] You shouldn’t.”

In the end, the whole thing is probably just a clever lesson in scepticisim. Or is it?

Thursday, April 29th, 2010

MADDness

Reasons to keep the drinking age at 21: Since the drinking age in New Jersey was raised to 21, the number of young people killed in drunk driving crashes has dropped nearly 78%. Need we say more…

I’ve stared at this claim numerous times, while being carried hither and tither by PATH. In it, MADD, the non-profit that took its name from Alhazred’s infamous Acronomicon (A Complete Reference Of Nerdy Or Maximally Impossibly Convoluted Organization Names) is trying to rally support for their cause, and they’re doing it in a way that makes my number sense go off. To the Mathcave!

First: they don’t tell us what they base their numbers, excuse me, what they base their number on. 78% of what, taken from which source, and calculated how? I realize it’s just a small subway ad, but it does manage to mimic a statistical claim quite well. Which it frankly isn’t. It’s an unfounded and barely even meaningful assertion.

Second, they don’t mention when exactly this raising of the drinking age happened, do they? It happened way back in 1982. In the 28 years since, the number of fatal accidents overall might have dropped considerably. Given the safety advances since then, it’s a fair bet it has. If it had dropped by as much as 80%, the number MADD gives us for drunksters would be merely the average drop. The same were true if just the number of young people on the road, or the number of young involved in any kind of crash, had dropped by 80%. In fact, there is a whole host of variables that a claim like this one needs to be controlled against for it to have meaning.

Third, the drop is in “young people killed in drunk driving crashes”. Sounds like that also includes crashes caused by drunk adults. Which are irrelevant to the question of drinking age.

And fourth, how many saved lives do those 78% actually correspond to, and what fraction are they of the total number of young people killed in traffic accidents? If both were small numbers, would the good of the few really outweigh the good of the many here? While this argument assumes there is a net benefit from getting drunk, which itself may seem debatable, there is no foregone conclusion either way. Not allowing people under 21 to drive at all would make their fatalities drop even further, yet I don’t envision that implemented any time soon. The cost would be too high.

Just to be clear: I think drunk driving is irresponsible and stupid at any age, and young people, especially males, are much more likely than the average to do it. They can’t help it, their frontal lobes are hormonal mush. But whether or not raising the drinking age lowers the risk is an empirical question that deserves proper treatment. Mothers, do not mislead us! It makes us SADD (Scientists Against Data Distortion).

Thursday, April 1st, 2010

welt des wissens

Durable relationships

Diese Grafik trägt den Titel “durable relationships” und stammt aus dem gestern erschienenen Aufsatz A Mathematical Model of Sentimental Dynamics Accounting for Marital Dissolution von José-Manuel Rey. Sie trägt Anstrengung gegen Gefühl auf, enthält zwei Variablen die zur herzten Potenz erhoben wurden, zahlreiche Pfeile und das sentimentale Gleichgewicht E.

Mehr ist dazu nicht zu sagen.

Saturday, September 19th, 2009

inkling of jest

More evidence is accumulating, and so to dispose some of the anxiety of soon having to leave the dear world of tennis, addiction and myths of feral babies, I take some time off the story and its footnoted branches, and instead review some new occurrences of numeral confusion, and a possible explanation, to boot.

In the odd insert of Himself’s fond look back upon his younger years, the event of a sheared off doorknob leads him into his future work in annulation, or at least is presented as a foundation for this future work. The description of the rotation of the circle on a stalk on the hard floor is faultless, and the mathematical discussion of the cycloid that ensues, is quite flawless until this bit of information intrudes: “But since here, on the bedroom’s floor, a circle was rolling around what was itself the circumference of a circle, the cycloid’s standard parametric equations were no longer apposite, those equations’ trigonometric expressions here becoming themselves first-order differential equations.”

So we have a parametric description of the curve a point on a ring will go through as the ring rolls in a straight line. This is the cycloid. And since here a ring is really rolling in a circle itself, being restricted by the attached stalk, something is different. All of this is true. But why would the same parametric equations not be applicable along the circular path of the doorknob? As far as I can see, they would, and Himself telling us this story should be well aware of that.

The second new piece of evidence comes from one of the end notes, where Pemulis briefs Hal on how to obtain derivatives. x to the power of n derives to, he claims, nx+x^(n-1), which is finally absurd enough to turn suspicion into certainty: this is not noise, it is signal. This signal, however, not carrying simple narrative, because once again the person speaking both clearly knows the correct answer and also has no reason to deceive, or to expect a deception to be undetected, derivation of a polynomial being your pretty basic analytical bread and butter for high school kids.

And then, a few notes down, we witness Pemulis say this to a snivelling Possalthwaite: “Todd, trust math. As in Matics. Math E. First order predicate logic. Never fail you. […] The vital statistics of God or equivalent. When all else fails […] You can fall back and regroup around math.” This is followed by some lemma on line integrals that is indeed (and somewhat confusingly) true.

So if math is considered the one unwavering basal truth, yet is bent out of shape almost every time it comes up in the fabric of the novel, along with the laws of optics in the case of the shadows in the desert, which please recall is also a field of expertise of Himself, as is annulation, the weirdly absurd process of waste negating waste, it all seems to point to the fact that James Incandenza maybe found a way to infect everything he touched with some sort of existential rot, eroding the foundations of truth itself, just like any good father will.

Thursday, September 17th, 2009

oodles of puns

Accretion is the better part of vapor.

Wie man sich betet, so lügt man.

Wednesday, August 19th, 2009

finite complaint

I am reading David Foster Wallace’s Infinite Jest, and I need to complain. If you haven’t read it, and plan to, don’t worry, I won’t spoil it for you. If you hate nitpicking, on the other hand, maybe do worry a bit, because I will pick nits. Three of them.

Overall, the book is fabulous so far (page 100), and I love the combination of low brow comments with language so erudite it often borders on the pretentious. This is a great stylistic game to play, and it is enhanced by esoteric factoids, partly made up, partly accurate. But for it to work it needs to stay on the light side of the pretentiousness boundary, and the author needs to be in control of the pieces of knowledge he’s throwing around. Now lets pick some nits.

Nit the first. In a discussion of the philosophical aspects of tennis, Hal’s father is cited, obliquely, as telling people about how a tennis move affords n responses, and how there then are 2^n responses to that move, soon spiraling into “a Cantorian continuum of infinities”. Now, firstly, the correct formula for this is n^2. Wallace might have done this on purpose – somewhat suggestive is the fact that the date of Cantor’s diagonal argument is given in the endnote as 1905 (Einstein’s two seminal papers came out that year) instead of the actual 1891. This is probably a purposeful mixup put into the mouth of the fictitious narrator to highlight the distant past-ness of it all – but if so, I fail to see the point of giving the wrong formula. On top of that, however, talking about a continuum of infinities is not quite accurate. The number of replies approaches the number of elements in the continuum only for games with countably infinite many moves, and any transfinity beyond that (the actual family of infinities Cantor discovered and Wallace alludes to) is unreachable by any tennis match. Overall, the passage sounds like a bit of finitely hot air to me, unlikely to come out of the mouth of an MIT educated polymath with an actual interest in Cantor.

Nit the second. When we first meet the assassin Marathe on the mountainside, he watches his shadow wander out in the setting sun toward the city of Tucson and, as the sun gets low enough, eventually reach it across the plains. This can not happen. A circular object of 20 cm diameter (a human head, say), has the same angular size as the sun (about half a degree) at a distance of roughly 22 meters (or yards). At distances greater than that, the object can no longer fully block the sun and casts no total shadow, or umbra. Moreover, the darkest point of the penumbra actually cast rapidly loses contrast and definition. At 44 meters, the darkest point will have 75% of the luminance of the surround, at 88 it’s up to close to 94%. A shadow of a human or anything smaller, cast by the sun and reaching out visibly for kilometers, is a physical impossibility.

Nit the third. Again concerning shadows, in that same scene: when the sun finally sets, Marathe sees his shadow return to him up the incline. This would happen with a rising sun, not a setting, and of course he’d have to face west, not east. If the sun were a point, the shadow cast out by a sunset would retain definition across large distances and grow longer and longer until it hits the advancing terminator and fuses with it. It would not shorten back toward the object casting it.

Is it sloppy writing and editing? Or a deliberate ruse to tick off obsessive compulsive physicists like me? It doesn’t really matter, because the book is still great, and these are tiny complaints.

Later Addition: The term Bröckengespenst Wallace uses for his fictional shadow is a) inappropriate, since a Brockengespenst is actually a shadow cast into fog or clouds, and b) has false diacritical marks on the o. This German spelling SNÄFÜ is in tune however with his german character later on calling his students “mein kinder”, which sports the wrong numerus of the possessive. “Meine Kinder” would have been correct. I’m leaning toward the sloppy writing hypothesis.

Thursday, November 20th, 2008

Anfang und Ende

Am Ende, wenn es von den Stühlen klatscht und aus den Gängen fragt, stehen immer ein paar auf und gehen, schnell, und ohne zurück zu blicken, woandershin, wo es interessanter ist. Manchmal kommt auch jemand, setzt sich einen Vortrag lang, steht auf, geht wieder. Der Laserpointer wird warm und feucht, der Punkt möchte ein Diagramm entlangwandern auf der Leinwand oder auf der Stirn der Moderatorin funkeln einen anarchischen Moment lang, aber ich lass ihn nicht. Jetzt hat jemand Gesichtern den Mund übermalt, durch einen verwaschenen hautfarbenen Fleck, und gleichzeitig noch die Stimmen verrauscht. Wenn man sich das ansieht, sagt er, als Film, dann guckt man den Gesichtern mit dem Mundfleck und den verrauschten Stimmen direkt unter die Nasenspitze – und wirklich wachsen da abwechselnd blaue Bällchen in den Nasenlöchern – weil man die Lippenbewegungen noch ein bisschen sehen kann zwischen Nase und Mundfleck. Ich höre aber kaum hin, gleich bin ich selber dran, es klatscht, es fragt, ich gehe die drei Treppen nach oben und überblicke meine Untertanen. Zwanzig sind noch da, nein, achtzehn, grade gehen noch zwei. Ich drücke auf das Slideshowsymbol und hebe den Pointer, der jetzt kühl ist und trocken. „This”, sage ich, “is what we wanted to tell you about today.” And so it begins.

Wednesday, November 19th, 2008

Memento

Wie das wäre, wenn man nicht wüsste, wie es gestern war, hat man ja ganz gut in diesem Film da gesehen: man liefe mit Tätowierungen durch die Gegend und wäre an allem selbst schuld (vereinfacht dargestellt). So ging es dem Patienten N.M. in den fünfziger Jahren, nachdem das Rauszupfen des hinteren Teils seines Temporallappens die Epilepsie nicht heilte und das Doktorenteam dann auch noch den Hippocampus, das sogenannte Seepferdchengehirn, rauszupfte. Das Seepferdchengehirn ist wichtig fürs Erinnern, und N.M. konnte von diesem Moment ab keine neuen Erinnerungen mehr anlegen. Das ist vermutlich ungefähr so furchtbar, wie es klingt, zum Beispiel, weil er sich Preisänderungen nicht merken konnte, und die Inflation ihm vermutlich bei jedem Einkauf von neuem das Gefühl gab, fürchterlich betrogen zu werden. Immerhin sind die endlosen Versuche, die mit ihm unternommen wurden, sicher besser zu ertragen, wenn jeder Tag der erste ist. Ganz schön ausserdem die kleine Geschichte, in der N.M. über Tage hinweg lernte, einen Stern im Spiegel nachzuzeichnen. Er habe erwartet, sagte er nach drei Tagen üben, dass dieses Spiegelzeichnen schwieriger sei – er erinnerte sich nicht mehr, das gelernt zu haben.