by Carl Pierer

Among the many tools available to mathematicians attempting to prove a statement is something called “proof by contradiction” or reduction ad absurdum. The general method of the proof is a very smug one: Let the statement to be proved be Φ. The strategy, then, is to suppose that Φ is false and to consequently derive a contradiction. Now there are quite a few, infinitely many one might say, ways of going about this. This shan't be our concern. What is of interest here is the question as to what kind of contradiction forms the end of such a proof. Let us distinguish 2 cases:

• Internal contradiction. The proof takes the form of:

Suppose ¬Φ. Then γ.

Δ.

∴, ¬γ.

Contradiction! ∴ Φ.

Here, we deduce an immediate consequence (γ) from the assumption that ¬Φ and then proceed by a sequence of logical steps (Δ) to show that this leads to ¬γ – a blatant contradiction. For example, let us say our statement to prove is:

Φ: is not a rational number

Then, ¬Φ is ” is a rational number” and an immediate consequence of this would be:

γ: Then we can write √2 = m/n , where m,n are natural numbers, such that m,n are coprime. That is, m/n is an irreducible fraction.

One standard proof then goes as follows:

1. Then, 2 = m²/n²

2. So, 2n² = m²

3. This means, m is even.

4. So m can be written as m = 2k for some natural number k.

5. Thus: 2n² = (2k)²

6. Ergo, 2n² = 4k²

7. And hence: n² = 2k²

8. So n is even.

9. Then 2 divides both n and m.

10. Therefore, ¬γ.

Now, we have the desired contradiction, and therefore we conclude Φ.

• External contradiction. Here, the proofs take the following form:

Suppose ¬Φ. Then:

Δ.

∴ μ.

But ¬μ! Contradiction. Therefore, Φ.

Here, ¬μ is some generally accepted mathematical truth, such as 1 ≠ 2. In contrast to the internal contradiction, it need not be a statement that is a consequence of ¬Φ. ¬μ is some statement in the general corpus of mathematical truths (i.e. proven statements) and is not necessarily linked in its content to Φ.

Now, the astute reader might question the meaningfulness of the distinction between internal and external contradiction. In particular, since μ appeared in the proof of Φ, there must be a sequence of logical derivations that relate the two. So, one might ask, how much sense does it make to distinguish between a “direct” consequence of ¬Φ and one that is related by a longer sequence of logical steps? Certainly, formally speaking, such an objection is valid. But there seems to be an intuitive sense in which some consequences are more immediate than others, and this is all that is needed at this stage.

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by Charlie Huenemann

Plato, as we know, told tales of an abstract realm beyond the senses, a realm beyond the dim and dark cave we call “the world.” It was a realm of forms, first glimpsed through the discipline of mathematics, and more thoroughly known through philosophical cross-examination, or dialectic. It’s not clear just how much religion there was in Plato’s own philosophy, but that philosophy certainly was enlarged into mystical proportions by the time of Plotinus (204-270 c.e.).

We can get a richer sense of this notion – that the pure intellect can grasp divinity – by exploring the life of Hypatia, a mathematician, astronomer, and philosopher who lived in the great city of Alexandria about a century after Plotinus. Hypatia was brilliant and utterly dedicated to the life of the intellect. She was famous as a philosopher and mathematician, and a school formed around her. She was also beautiful (it is said), and attracted many suitors; but she resisted them all in deference to the requirements of her philosophy. She became caught up in a power struggle between the city's governor and its Christian bishop, and met a grisly death at the hands of the bishop's supporters.

Hypatia's life and death has been refashioned many times over the centuries, usually in the attempt either to attack or to defend organized religion. Just reading a pair of book titles is enough to give the general idea. In 1720, the infamous atheist John Toland published Hypatia, or the History of a Most Beautiful, Most Virtuous, Most Learned and in Every Way Accomplished Lady; Who Was Torn to Pieces by the Clergy of Alexandria, to Gratify the Pride, Emulation, and Cruelty of the Archbishop, Commonly but Undeservedly Titled St. Cyril. (Earlier times featured the most informative book titles!) Toland's book was answered promptly in a pamphlet by Thomas Lewis, entitled The History of Hypatia, a Most Impudent School-Mistress of Alexandria: In Defense of St. Cyril and the Alexandrian Clergy from the Aspersions of Mr. Toland. I know of these titles from a more recent work with the decidedly more neutral title, Hypatia of Alexandria, by Maria Dzielska (1995). Dzielska offers an overview of all the various uses in both fiction and scholarly literature to which Hypatia has been put to use, and then delivers a very plausible and thorough account of what we can plausibly put forward as the facts of the case.

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by Mara Naselli

One evening in February 2012, I was in a Chicago noodle shop looking for a table for one. The television was on—a news report from Syria. The Syrian Army had begun its attack on Homs. The frame of the screen, jostling in the confusion, captured the faces of a woman and a boy. The woman was distraught. The boy, bewildered. I watched agape, for an instant transposing myself in the place of the woman and my own sons in the place of the boy. Children cannot take in their shattering world. The slight young man waiting tables that evening must have seen something in my expression. He changed the channel to a soccer match.

The poet and artist David Jones was just nineteen years old when he enlisted in the Royal Welsh Fusiliers as an infantryman in the British army in January 1915. Later that year, just after his twentieth birthday, he was serving on the western front until he was wounded in July 1916.

For the next two decades Jones wrote In Parenthesis, his account of his experiences in the First World War. It is not an easy read. There are many different kinds of language at work in Jones’s modern epic—not just the Welsh, English, and Cockney of the infantrymen and officers, but also the military jargon and slang, rhymes, and popular songs. There are also the well-known allusions to Arthurian legend, Shakespeare, Lewis Carroll, Hopkins, Coleridge, and others. Jones’s language—its syntax, sound, and diction—was so foreign to me, I found myself enchanted and lost. I copied out long passages, including the footnotes, to track this myriad mind at work.

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by Tasneem Zehra Husain

The development of [my] thought-world is in a certain sense a continuous flight from wonder.

—Albert Einstein

We are marked in large part by our celebrations: what we celebrate, and how we choose to do so, says a lot about who we are. As a global society, we seem to be increasingly fascinated with genius, and almost sixty years after his death, Einstein continues to be emblematic of this phenomenon. Over time, he has become larger than life – more myth than man.

In the annals of physics, Einstein's footprints are everywhere; his contributions as various and scattered as if they too, were subject to the brownian motion he elucidated. Along most paths he trod, he left staggering achievements in his wake. Einstein made crucial contributions to a nascent quantum theory, his incisive explanation of the photoelectric effect was so brilliant, it won him the Nobel Prize, and yet, most physicists, if asked to name Einstein's definitive work, would unblinkingly pick general relativity. The theory celebrates its hundredth birthday in a couple of weeks, and festivities are underway across the globe.

Over four successive Thursdays in November 1915, Einstein presented his (still developing) theory to the Prussian Academy of Sciences in Berlin. He laid down The Formal Foundations on 4 November 1915, and worked feverishly every day, polishing and honing the theory, coaxing out some of the gems that lay hidden within, until finally, on 25 November he unveiled the spectacular Field Equations of Gravitation.

“Hardly anyone who truly understands it will be able to escape the charm of this theory,” wrote Einstein in this final paper, and his remark has stood the test of time, just as well as his equations have. The General Theory of Relativity is a work of unparalleled beauty; in fact, it exemplifies what it means for a physical theory to be beautiful, and is often quoted as the canonical example of such.

There is an air of inevitability about general relativity, which Nobel Laureate Steven Weinberg equates with beauty. “In listening to a piece of music or hearing a sonnet one sometimes feels an intense aesthetic pleasure at the sense that nothing in the work could be changed, that there is not one note or one word that you would want to have different,” he writes. It is so with general relativity. No idea or symbol seems extraneous or out of place.

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by Paul Braterman

For disrespectful blogging and criticism of the religious authorities, one thousand lashes, to be administered 50 at a time. A fine of one million Riyals (roughly £170,000). 10 years in jail. If, like me, you have been wondering what horrible crimes could merit so severe a punishment, now you can find out.

I across a selection of Raif Badawi's writings in my local Waterstone's, and see that it has been published in the US, UK, and Canada, and that it is also available in French, German, and Italian. I do not know if there is an Arabic version; if there is, it will certainly not be available in the author's native Saudi Arabia. However, the attempts to silence Badawi have ensured him a far wider audience than he could ever have thought possible.

Having read the offending blog posts, I am shocked. Not because they are strident, or violent, or opposed to religion, or subversive of government, but because they are none of these things, and yet have attracted so extreme a reaction.

A brief foreword to the book (see below) is followed by a short preface, by the bilingual TV journalist Constantin Schreiber. This places Badawi's writings in context, and describes how the hopes he expressed in the days of the Arab Spring have been dashed by events. Unlike Schreiber, I am neither an Arabic speaker nor an expert on events in the Middle East, so I am doing my best here using the English language translation and my own limited background knowledge. If I have been guilty of any mistakes or misinterpretations, I hope that better-informed readers will point these out.

The first piece is a plea for freedom of thought and expression, using a quotation from the Quran itself in support. The second, a complaint against censorship and the outrage synthesised to justify it, begins with the unconsciously prophetic words

Many of the Islamist activists of Saudi Arabia dream of the return of an era along gone: they fantasize about the times of the caliphs. Those caliphs were known to banish and murder their opponents.… The modern Islamists hope history will repeat itself.

Indeed, we now once again have a self-styled caliph, at the head of the entity known as Daesh, [1] that now of all times needs no further discussion by me.

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.
.
Fundamental Misunderstanding

— thoughts on Charlie Hebdo, and Kenya and Beirut …and Paris

everything ever written or said
everything drawn or played or sung
every headline that cried or bled
every fresco, every poem
everything wrung from our cranial sponge
every inky insult flung
every instrument ever made
every expletive blasted from lungs
every face on a canvas hung
every righteous canto prayed
…. that pounded the planks of heaven’s floor
every school Kalashnikov-sprayed
every smartass quote with bite
every thought of rich or poor
every Icarus grasping at height
…. whose waxy wings soon came apart
every joke and laugh and snort
every misbegotten poison dart
every sentiment or thing
…. that burst from brain’s well-tensioned spring
every sura, gospel or verse
every prayer that followed a hearse
every love, lost or won
every song and every hum

every murmuring merciful must
that reached the sky or bit the dust
are not of a glad or angry God
…………………..but of life that thrusts,
from inner to outer, the stuff of us

Jim Culleny
1/6/14
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by Matt McKenna

James Bond, like most action heroes, is a conservative protagonist. Even as the other characters in the film–both friendly and hostile–deride 007 for sticking with his outmoded methods of problem solving (blowing stuff up, shooting everybody, etc.), Bond stoically carries on, winning the day without the expectation of apologies from his doubters much less thanks from those he saves. Of course, Bond's attitude makes a lot of sense for an action movie hero. Instead of shooting down aircraft to stop nefarious organizations, can you imagine the snorefest that would ensue if Bond attempted to solve problems diplomatically? Thankfully, director Sam Mendes stays true to the franchise's legacy in his second Bond film, Spectre. While the movie drags towards the end, especially during the perfunctory scene in which the villain (Christoph Waltz) blathers exposition while torturing Bond (Daniel Craig), the opening action sequence alone warrants the price of admission. If you can manage it, the best viewing strategy is to buy a matinee ticket for Spectre then clandestinely head into another auditorium to see a better movie immediately after the helicopter fight scene ends. I realize this plan may go against movie theatre policy, but don't you think it's quite Bond-like?

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by Bill Benzon

No, I don’t think it is, morally repugnant; quite the contrary. But it IS controversial and problematic, and that’s what I want to deal with in this post. But I don’t want to come at it directly. I want to ease into it.

As some of you may have gathered, I have been trained as an academic literary critic, and academic literary criticism forswore value judgments in the mid-1950s, though surreptitious reneged on the deal in the 1980s. In consequence, overt ethical criticism is a bit strange to me. I’m not sure how to do it. This post is thus something of a trial run.

I take my remit as an ethical critic from “Literature as Equipment for Living” by the literary critic, Kenneth Burke [1]. Using words and phrases from several definitions of the term “strategy” (in quotes in the following passage), he asserts that (p. 298):

… surely, the most highly alembicated and sophisticated work of art, arising in complex civilizations, could be considered as designed to organize and command the army of one’s thoughts and images, and to so organize them that one “imposes upon the enemy the time and place and conditions for fighting preferred by oneself.” One seeks to “direct the larger movements and operations” in one’s campaign of living. One “maneuvers,” and the maneuvering is an “art.”

Given the subject matter of The Wind Rises, Burke’s military metaphors are oddly apt, but also incidental. The question he would have us put to Mizayaki’s film, then, might go something like this: For someone who is trying to make sense of the world, not as a mere object of thought, but as an arena in which they must act, what “equipment” does The Wind Rises afford them?

I note that it is one thing for the critic to answer the question for his or herself. The more important question, however, is the equipment the film affords to others. But how can any one critic answer that? I take it then that ethical criticism must necessarily be an open-ended conversation with others. In this case, I will be “conversing” with Miyazaki himself and with Inkoo Kang, a widely published film critic.

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by Eric Byrd

Cyril Connolly was depressed by biographies of unlucky poets. Reading yet another life of Baudelaire “we know, with each move into a cheap hotel, exactly how many cheap hotels lie ahead of him.” Mr. Lincoln's Army (1951) made me feel that way about armies – in this case the Army of the Potomac, the shield of Washington and the main army in the highly politicized, closely-covered Virginia theater of the American Civil War, in which the national and rebel capitals lay 100 miles apart. Catton at his best puts you in the field –

the skirmish lines went down the slope, each man in the line separated from his fellows by half a dozen paces, holding his musket as if he were a quail hunter with a shotgun, moving ahead step by step, dropping to one knee to shoot when he found a target, pausing to reload, and then moving on again, feeling the army's way into the danger zone

– but he never allows you to forget that the battle being recounted – a perfect apocalypse while you're reading – is but one of the early clashes of a long war. There will more dying. This battle will decide nothing; that general will blunder; these men will die in vain. Mr. Lincoln's Army ends in November 1862. Eighteen months later, in spring 1864, Sherman wrote his wife: “the worst of the war is not yet begun.”

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by Josh Yarden

there's that little boy

trekking to school

black galoshes and a yellow raincoat

stomping in puddles

imagining

he's an explorer

on a short cut

to an adventure

through the woods

down the hill into the trees

crossing the creek

one might meet Lewis and Clark

stop to catch a crayfish

a salamander

or hide out in the forrest

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By Scott F. Aikin and Robert B. Talisse

Conference season is drawing near for many academics. In our discipline, Philosophy, already the regional conferences are in full swing, and the American Philosophical Association will have its large Eastern Division meeting in early January. This has got us thinking about these conferences and the many papers that will be presented at them. The trouble, as we see it, is that the paper sessions are so often disappointing, and so frequently less fruitful than they otherwise might be.

It's not that the papers chosen for presentation are poorly written or intellectually inept. To the contrary, the content and even the style of the writing of the papers tends to be of very high quality. What makes conference sessions in Philosophy so frequently disappointing is that, for reasons we cannot fully grasp, the disciplinary norm still heavily favors reading one's paper to one's audience. That's right: At professional Philosophy conferences, it is most common for speakers to read to their audiences. Conference presentations tend to last 20-30 minutes; then there is often a second speaker who offers a critical comment on the first presenter's paper, and the commentary often runs for another 10-15 minutes. And sometimes there is yet a third recitation — the first presenter is given the opportunity to respond briefly to the commentator's critical remarks, and this, too, is often read from a prepared text. Then, with what time is left, the floor is open for questions from the audience. And even when a speaker elects to present her work using presentation technology, still the dominant tendency is to simply read from the projected slides.

Many Philosophy conferences run for two to three days. Imagine three full days of being read to in this way. Even under the best circumstances — with dynamic readers and exciting content — it's simply exhausting.

That philosophers should be in the habit of reading their papers out loud to each other at professional meetings strikes us as bizarre. Notice how the disciplinary norm differs when it comes to pedagogy. These days, it's almost unheard of for a professor of Philosophy to read her lectures to her students. It is far more common to speak extemporaneously from notes, which forces the instructor to devise fresh formulations and to think on her feet. After all, we are educators, and in our classes we often present to our students highly detailed and challenging ideas. And when teaching material in our own research areas, we commonly take ourselves to have no need for a prefabricated script. Moreover, as almost everyone in the profession will readily admit, the really exciting exchanges at Philosophy conferences occur in the informal setting of the conference reception, or, even more frequently, the hotel bar. Why, then, should we persist in reading to each other in the official conference sessions? Why not adopt a new practice of talking to the audience?

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by Jalees Rehman

Hearing about the HannoverGEN project made me feel envious and excited. Envious, because I wish my high school had offered the kind of hands-on molecular biology training provided to high school students in Hannover, the capital of the German state of Niedersachsen. Excited, because it reminded me of the joy I felt when I first isolated DNA and ran gels after restriction enzyme digests during my first year of university in Munich. I knew that many of the students at the HannoverGEN high schools would be thrilled by their laboratory experience and pursue careers as biologists or biochemists.

What did HannoverGEN entail? It was an optional pilot program initiated and funded by the state government of Niedersachsen at four high schools. Students enrolled in the HannoverGEN classes would learn to use molecular biology tools that are typically reserved for college-level or graduate school courses to study plant genetics. Some of the basic experiments involved isolating DNA from cabbage or how bacteria transfer genes to plants, more advanced experiments enabled the students to analyze whether or not the genome of a provided maize sample was genetically modified. Each experimental unit was accompanied by relevant theoretical instruction on the molecular mechanisms of gene expression and biotechnology as well as ethical discussions regarding the benefits and risks of generating genetically modified organisms (“GMOs”). You can only check out the details of the HannoverGEN program in the Wayback Machine Internet archive because the award-winning educational program and the associated website were shut down in 2013 at the behest of German anti-GMO activist groups, environmental activists, Greenpeace, the Niedersachsen Green Party and the German organic food industry.

Why did these activists and organic food industry lobbyists oppose a government-funded educational program which improved the molecular biology knowledge and expertise of high school students? A press release entitled “Keine Akzeptanzbeschaffung für Agro-Gentechnik an Schulen!” (“No Acceptance for Agricultural Gene Technology at Schools“) in 2012 by an alliance representing farmers growing natural or organic crops accompanied by the publication of a study with the same title (PDF), funded by this group as well as its anti-GMO partners, gives us some clues. They feared that the high school students might become too accepting of using biotechnology in agriculture and that the curriculum did not sufficiently highlight all the potential dangers of GMOs. By allowing the ethical discussions that were part of the HannoverGEN curriculum to not only discuss the risks but also mention the benefits of genetically modifying crops, students might walk away with the idea that GMOs may be a good thing. Taxpayer money should not be used to foster special interests such as those of the agricultural industry that may want to use GMOs, according to this group.

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by Misha Lepetic

“People for them were just sand, the fertilizer of history.”
~ Chernobyl interviewee VM Ivanov

For a few years, if you were on Twitter and you used the word “inconceivable” in a tweet, you would almost immediately receive an odd, unsolicited response. Hailing from the account of someone named @iaminigomontoya, it would announce “You keep using that word. I do not think it means what you think it means.” Whether you were just musing to the world in general, or engaging in the vague dissatisfaction of what passes for conversation on Twitter, this Inigo Montoya fellow would be summoned, like some digital djinn, merely by invoking this one word.

Now, those of us who possessed the correct slice of pop culture knowledge immediately recognized Inigo Montoya as one of the characters of the film “The Princess Bride”. Splendidly played by Mandy Patinkin, Montoya was a swashbuckling Spaniard, an expert swordsman and a drunk. Allied to the criminal mastermind Vizzini, played by Wallace Shawn, Montoya had to listen to Vizzini mumble “inconceivable” every time events in the film turned against him. Montoya was eventually exasperated enough to respond with the above phrase. Like many other quotes from the 1987 film, it is a bit of a staple, and has since been promoted to the hallowed status of meme for the Internet age.

Of course, it's fairly obvious that no human being could be so vigilant (let alone interested) in monitoring Twitter for every instance of “inconceivable” as it arises. What we have here is a bot: a few lines of code that sifts through some subset of Twitter messages, on the lookout for some pattern or other. Once the word is picked up, @iaminigomontoya does its thing. Now, and through absolutely no fault of their own, there will always be a substantial number of people not in on the joke. These unfortunates, assuming that they have just been trolled by some unreasonable fellow human being, will engage further, such as the guy who responded “Do you always begin conversations this way?”

So here we have an interesting example of contemporary digital life. In the (fairly) transparent world of Twitter, we can witness people talking to software in the belief that it is in fact other people, while the more informed among us already understand that this is not the case. Ironically, it is only thanks to the lumpy and arbitrary distribution of pop culture knowledge that we may at all have a chance to tell the difference, at least without finding ourselves involuntarily engaged in a somewhat embarassing mini-Turing Test. But these days, we pick up our street smarts where we can.

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