The ugly truth about your Facebook friends

by Sarah Firisen

3quarksThe world seems a very depressing, scary place these days. Maybe it always was. I remember being 12 years old and driving with my father and expressing to him how terrified I was by the possibility of nuclear war with the Soviet Union. He talked to me about mutual assured destruction and the deterrent this was going to provide. Those fears seem almost quaint now; our current enemies don’t seem to play by the same rational rules of self-interest. Another thing that has changed is our exposure to just how much other people in our lives don’t share our values and opinions on these, and other issues. I always knew that I was somewhat at odds with elements of my family about Judaism and Israel’s relationship the Palestinian people. But for the most part, as we probably all do, I lived in a bubble where most of the people around me pretty much shared my political and social views. I’ve always had friends who vote Republican, but they’re all on the fiscal rather than social conservative spectrum; lower taxes but prochoice. I have no problem with people whose views differ from mine in these ways. Yes, we can debate the merits of trickledown economics, but as long as we all are in favor of gay marriage, a woman’s right to choose to have an abortion, and the normal laundry list of social items that US liberals care about, the friendship won’t be tarnished by the things we don’t agree on.

But social media has changed all this. The views that our acquaintances hold are often now fully in our faces, good, bad and sometimes very ugly. Reconnecting with your best friend from kindergarten now often brings with it the horrible realization that she’s grown into a narrow minded bigot. Yes, you can unfriend and unfollow, and we often do or have it done to us. But what about when that’s not viable option? And should it be our first reaction?

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Grant, Fuller, and Fascism

by Eric Byrd

116937John Keegan and Geoffrey Perret have repackaged the essential arguments of The Generalship of Ulysses S. Grant, first published in 1929, in more politically palatable prose. But I was interested by the book's datedness, the view it offers of the odd personality and ominous historical situation from which the reevaluation of Grant was launched. Major-General J.F.C. Fuller (1878 – 1966) is a somewhat sinister and repellent figure – a disciple of Aleister Crowley; a mystic whose Futurism graded into Fascism; the maverick mastermind of British tank operations in the Great War (and his skill at drawing elaborate occult symbols came in handy when the Tank Corps needed an insignia) whose theories of mobile armored warfare were ignored in interwar Britain but eagerly studied in Nazi Germany and Stalinist Russia, the rival tyrannies destined to build thousand-tank armies and smash them together on the burning steppes of the East. Fuller attended Hitler's fiftieth birthday party, in April, 1939, a celebration capped by a three-hour parade of tanks and motorized infantry. Afterwards Hitler asked Fuller if he was pleased with his “children.” “Your excellency,” Fuller replied, “they have grown up so quickly that I no longer recognize them.”

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Monday, December 7, 2015

Welders and Philosophers

by Scott F. Aikin and Robert B. Talisse

W6a00d8341c562c53ef01b7c7f5cbdb970b-800wiAt the fourth Republican Presidential Debate, Senator Marco Rubio asserted that the country “needs more welders and less philosophers.” The small corner of the internet where professional philosophers reside promptly was awash with repudiation, criticism, and outrage (for example, here and here). We were, we have to admit, puzzled by it all – both by Rubio's statement and by the philosophers' outrage.

Senator Rubio's remarks were patently silly. First, the rationale Rubio offered, that “welders make more money than philosophers” is false. Moreover, the proposed reason, even were it true, is irrelevant – the social value of a profession is not a matter of the income paid to those who practice it. Surely no one would argue that hedge fund managers and reality TV stars are more socially valuable than nurses and carpenters simply on the basis of the difference between their respective paychecks. Finally, Rubio's reasoning is self-defeating, as there is no better way to decrease the earning power of a welder than by flooding the labor market with more competitors. For sure, Rubio's case was more rhetorical flourish than serious reasoning; he intended to draw a line in the sand between two perspectives on the role of education in society. His welders-versus-philosophers line was just window dressing for his view that education should aim to produce serviceable labor, not people who think.

Now, given these easy criticisms of Rubio's remarks, one might think that we were outraged by his claims. Many of our colleagues certainly were, and our inboxes and social media feeds quickly filled with missives of all shapes and sorts. But, we admit again, we found this phenomenon even more curious than Rubio's statement.

Here is why. There is nary a day that goes by without someone making a joke or remark to us about philosophy's alleged uselessness. Philosophy's oldest story highlights this. Thales of Miletus, who Aristotle counts as the first of the philosophers, apparently was walking one night and gazing up at the stars, contemplating their eternal motion. And then he fell in a well. A Thracian serving woman witnessed the fall, and laughed at him, urging that he should think about where he was putting his feet. Philosophy starts with a pratfall, and everyone, even those who have never taken a philosophy course or read any of the great books, gets the joke: It's not just that somebody fell in a well, it's that it was a philosopher.

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Torturing the Other: Who is the Barbarian?

by Claire Chambers

Towards the end of J. M. Coetzee's Waiting for the Barbarians, his protagonist the Magistrate speculates about how much pain he, Coetzeean ageing, out-of-shape man, will be able to withstand. In this elliptical novel, which owes a debt to Kafka's 'The Penal Colony', the Magistrate is about to be tortured at the hands of the Empire. Despite years of loyal service, his antagonist Colonel Joll believes that the Magistrate has betrayed the Empire because of his romantic entanglement with a girl from the enemy 'barbarian' community.

The passage encapsulates many of torture's most important features. While the Magistrate's anxieties revolve around what degree of pain he can tolerate, that is not the purpose of torture. Instead, the Magistrate's tormentors reduce him to a body or a thing that is incapable of thought or political ideals. Coetzee conveys this in part through the use of the third person singular gender neutral pronoun 'it':

its head is gripped and a pipe is pushed down its gullet and pints of salt water are poured into it till it coughs and retches and flails and voids itself.

The diction here also exposes the elaborate, quasi-medical, inventive methods that torturers use on their victims. Finally, Coetzee emphasizes that the central event of torture, the interrogation of the prisoner, is in fact a cover story: a huge lie. The InterrogationMagistrate has prepared 'high-sounding words' with which to answer the interrogator's questions about his dealings with the barbarians. But there is no conversation, no questions, and no single interrogator; instead 'they came to my cell to show me the meaning of humanity, and in the space of an hour they showed me a great deal'. What 'they' demonstrate to the Magistrate is that when his body is in severe pain, he is incapable of thought, language, or ethics. As Coetzee puts it, he learns 'what it meant to live in a body, as a body, a body which can entertain notions of justice only as long as it is whole and well'.

Five years after the publication of Coetzee's novel, in 1985, the literary critic Elaine Scarry published The Body in Pain. At the risk of stating the obvious, in this seminal book she explores what happens to people when their bodies are in pain. And in the most important chapter for our purposes, 'The Structure of Torture', Scarry examines what the consequences are of inflicting pain on others – both for the inflictor and the afflicted. She argues that torture pivots on a display of agency, which often involves the victim being confronted with or 'being made to stare at' an outlandish and often outsized weapon.

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The Dire State of Science in the Muslim World

by Jalees Rehman

Habib_University_4

Habib University via Wikimedia Commons (by Samarhashmi)

Universities and the scientific infrastructures in Muslim-majority countries need to undergo radical reforms if they want to avoid falling by the wayside in a world characterized by major scientific and technological innovations. This is the conclusion reached by Nidhal Guessoum and Athar Osama in their recent commentary “Institutions: Revive universities of the Muslim world“, published in the scientific journal Nature. The physics and astronomy professor Guessoum (American University of Sharjah, United Arab Emirates) and Osama, who is the founder of the Muslim World Science Initiative, use the commentary to summarize the key findings of the report “Science at Universities of the Muslim World” (PDF), which was released in October 2015 by a task force of policymakers, academic vice-chancellors, deans, professors and science communicators. This report is one of the most comprehensive analyses of the state of scientific education and research in the 57 countries with a Muslim-majority population, which are members of the Organisation of Islamic Cooperation (OIC).

Here are some of the key findings:

1. Lower scientific productivity in the Muslim world: The 57 Muslim-majority countries constitute 25% of the world's population, yet they only generate 6% of the world's scientific publications and 1.6% of the world's patents.

2. Lower scientific impact of papers published in the OIC countries: Not only are Muslim-majority countries severely under-represented in terms of the numbers of publications, the papers which do get published are cited far less than the papers stemming from non-Muslim countries. One illustrative example is that of Iran and Switzerland. In the 2014 SCImago ranking of publications by country, Iran was the highest-ranked Muslim-majority country with nearly 40,000 publications, just slightly ahead of Switzerland with 38,000 publications – even though Iran's population of 77 million is nearly ten times larger than that of Switzerland. However, the average Swiss publication was more than twice as likely to garner a citation by scientific colleagues than an Iranian publication, thus indicating that the actual scientific impact of research in Switzerland was far greater than that of Iran.

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Lifeline

by Tamuira Reid

ScreenHunter_1537 Dec. 07 10.22Recently I was interviewed for a college podcast on the craft of writing. I dread this type of thing, mostly because no matter how hard I try to not sound like a complete asshole, I end-up sounding like one. Write about what you know? I guess. Write everyday? I sure don't, but okay. The truth is this: most writers I know are just trying to survive. Financially, yes. But mentally even more so.

Then there's always that question – when did you know you were a writer?

I was a weird fucking kid. I know everyone says that but it's very true in my case. In every class picture, my hairstyle is a couple years behind. The gap between my front teeth a little wider than it should be. Eyes kind of glazed over. I tap danced in my spare time, made wedding gowns out of paper towels that I'd put on spoons for their weddings to forks, played ice hockey down our long marble-floored hallway with a toilet plunger and a severed doll head. It was all just a tad off: my timing, my style, my eye-hand coordination.

When I discovered in grade school that I hated people, myself included, I decided to become a writer. I needed to leave something concrete for the aliens who would eventually come to take over Earth. If I was dead by then, how would they know the truth about humans? How would they know how much empathy and intellect our species truly lacked?

So it was with an altruistic spirit that I began to write. About my family. About my slutty, teenaged dance teacher with all the hickies on her neck. About the boy across the street who had two fathers and no mom. About the voices in my head that only seemed to go away when I wrote about them.

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Some Are Born To Sweet Delight

by Misha Lepetic

“Except for a wig of algorithms, and tears and automation.”
~Noah Raford,
Silicon Howl

Blake_01Last month I attempted to set up two conflicting frames. On the one hand, there is the advance of technology in its myriad forms, eg: social media, artificial intelligence, robotics. This may seem like an arbitrary selection. For example, why exclude fields of medicine, or energy production, or infrastructure? Of course, all technologies are intrinsically social, especially given the complexities required to design, develop, disseminate and maintain them on a global scale. But my concern here are those technologies that are explicitly social in nature: those inventions, whether hardware or software, that intervene in our lives to enable or enhance communications, experiences, or that provide services along such lines.

On the other hand, these technologies are laid over a long-established matrix of social differentiations. Categories that have traditionally motivated the investigations of social scientists, such as class, race, culture, religion, education, gender and age, form the inescapable substrate upon which technology is seeded and elaborates itself, or withers and dies. As I showed, and contrary to most writing about technology in the mainstream media, these boundaries are not magically dissolved by technology, and in many cases they may be further exacerbated. They are certainly not elided, which seems to be the most common attitude. Instead, those occupying the more privileged ends of these spectra of difference benefit more greatly from each advance, and the underprivileged are further shunted to the side. It is the technological equivalent of income inequality, except it is subtler, since we lack the pithiness of a single number, such as the Gini coefficient, to use as a signpost. (Incidentally, even this metric has of late become increasingly less useful as global inequality ascends to hyperbolic levels.)

Thus the object of our scrutiny should really be the ways in which technology further complicates a landscape that is already extremely difficult to parse. In this sense, these two frames are not really in conflict, but at least from a critical point of view, are rather insufficiently engaged with one another. Furthermore, and perhaps even more importantly, the inquiry should not have as its final destination any hope that technology will ultimately dissolve these differences. This is where efforts to bridge the so-called “digital divide” fall short for me: the idea of a level playing field has always been a fiction. Why should we aspire to it? Isn't it more compelling to understand what difference a difference makes? Conversely, if technology really does succeed in eroding all these categories of difference, we will have to scramble for another definition of what it means to be human. Given the difficulty we have with the current state of the definition, I somehow doubt that a tabula rasa approach would be at all helpful.

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Mutant Nature

by Dwight Furrow

Mutant natureNature is not disappearing; it's just hiding in your salad bowl.

Throughout most of human history human beings were utterly dependent on nature and everything about human life was determined by it. Adapt or die was the imperative that governed all life and so nature seemed infinite and without measure, a fact recognized by 18th century theories of the sublime. Yet, throughout most of that history, we refused to acknowledge this dependence striving to see ourselves as ultimately separate from nature. The separation of mind and body, of earth and heaven, the opposition of nature and culture, were taken to be simply obvious.

But today we have reversed that equation. Inexorably, we have learned to control nature through technologies which have reached such a critical mass that nature has been reduced to a mere instrument to be carved up and used as we see fit—a “standing reserve” as Heidegger called it. Even our biological make up will soon be subject to fundamental manipulation as gene editing comes online. The result is that nature now seems finite and fragile, disappearing under the deluge of techno-science and mass industrialization.

Paradoxically, as we gain more control over nature we have begun to acknowledge our dependence on it, as the Paris climate talks get underway amidst a deepening sense of crisis. The consequences of ignoring our dependence on nature are all too evident. For us today nature is both an instrument to be used up and a center of independent power, a Janus-faced phenomenon, on the one hand limited and circumscribed by human activity but on the other hand generating effluvia that create a devilishly devious constraint on human activity. The resistance of nature yields to our technology in countless ways but leaves behind a residue of pollution and devastation that threatens to undermine that hard won human control.

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Newtonianism for Ladies

by Jonathan Kujawa

6a01a510678336970c01bb089a19ee970d-320wiThis spring I had the pleasure of spending several months as a visitor at the Mittag-Leffler Institute in Sweden. Hanging on the wall above my desk was a copy of this print:

The Mittag-Leffler Institute has two patron saints: Gösta Mittag-Leffler and Sofia Kovalevskaya. The Institute is located in Mittag-Leffler's home just outside Stockholm. He donated his home and its extensive library in 1916 with the goal of establishing a place for mathematicians to visit and collaborate. Both were built using Mittag-Leffler's personal wealth (which he obtained the old fashioned way: he married into it). Nowadays there are a number of such institutes, but at the time it was the first of its kind.

Over my desk, however, was the watchful eye of Kovalevskaya. She was a truly remarkable woman who obtained significant results on differential equations during the second half the nineteenth century. Those being the times, she struggled to find opportunities to study mathematics. When she finally earned her PhD in 1874 she was the first woman in Europe to do so. And even then it was only thanks to rule-bending by the famous Weierstrauss. In 1883 Mittag-Leffler used his considerable influence to procure Kovalevskaya a position at the University of Stockholm. In addition to mathematics, Kovalevskaya wrote several books. Sadly, she died in 1891 at the young age of 41. For a taste of her life, I recommend Alice Munro's short story “Too Much Happiness”.

Whenever I had a bad math day [1], Kovalevskaya's portrait gently and kindly reproached me. At my age she had done all of the remarkable things mentioned above. Almost to the day, when I arrived at the Institute I was the age she was when she passed away. Every day when I arrived it was yet another day more than she had, and as a white, middle-class, American male I couldn't hardly claim any disadvantages to Kovalevskaya! As I'd leave for the day, she would tsk, tsk over all my dead ends and miscalculations.

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San Bernadino Terror Attack

by Omar Ali

POn December 2, 2015 Syed Farooq Malik, a young American of Pakistani origin (born in Illinois) was attending his workplace holiday party in San Bernadino. He left the party early (it is not clear if there was an argument of some sort before he left) and then returned with his wife, Pakistani-American Tashfeen Malik, and the couple opened fire on his coworkers and left after 4 minutes. Fourteen people were killed, 21 injured. It has since emerged that the couple had 2 assault rifles, thousands of rounds of ammo and several pipe bombs. They had also rented a Ford Expedition SUV a few days before the attack and used it for the attack as well as in the subsequent chase and confrontation with the police. Though they managed to escape the scene of the crime, they were eventually shot dead after an exchange of fire with the police. They had left their 6 month old baby girl with her grandmother on the morning of the attack. Sometime after the shooting, Tashfeen Malik also reportedly posted a “pledge of allegiance to ISIS” on her facebook page.

It has since emerged that Farooq Malik had a normally religious upbringing but had become “more religious” in the last two years. According to his (estranged) dad, he was obsessed with Israel and “shared the ideology of ISIS leader Abu Bakr al-Baghdadi”. And it seems that his wife was brought up in far more Islamist fashion than he was. Her father is a Pakistani who works in Saudi Arabia and supposedly became “more religious” there. She lived in both Pakistan and Saudi Arabia and was a full-time niqabi when she attended Bahauddin Zakariya University’s pharmacy department. She also attended classes at Al-Huda, an Islamist organization that runs schools to teach “pure Islam” in many countries. After marriage, she did not show her face even to her father-in-law and her brother-in-law and stayed in seclusion in her California apartment. She did not attend the baby shower thrown by her husband’s coworkers (the same people the couple later went to shoot) and it is very likely that she was more “radical” than her husband. It seems likely that the two of them decided to kill people because they wanted to strike a blow for their version of Islam, but the actual choice of target (i.e. where a group of people would be murdered) may still have involved some “workplace grievance” (though no convincing grievance has yet been revealed).

Post-Script: it is now clear that perpetrators were jihadists, had been turned down by some jihadist organizations, may have thought of bigger targets, and that one friend may have had some prior knowledge of their intentions.

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Monday, November 30, 2015

Monday Poem

How could something so beautiful not be right

Margaret Wertheim, on Einstein’s equations
……………….. for his General Theory of Relativity

The elegance of the simplest things
makes them right. The shape of a smooth stone
cannot be argued against —one touch
is testimony of its rightness. Its small heft
says, I'm here. Its mass, snapped by a spinning tire
shattering a windshield is evidence
of the absoluteness of its being.
Its adherence to universal laws says, I belong.
Its pleasing roundness
rolling in the cup of your palm
proves its truth. The way it rests in light,
glowing amber in harmony
with the color of the rising sun
is as much a claim to rightness
as the perfection of equations
or the presence of love.
Its contours, quanta, its silence,
strange and familiar as they are,
are as correct and beautiful
as this fleeting breath.
How in truth could anything
so beautiful not be right?
.

by Jim Culleny
11/28/15

THE BANKER WHO LOST HIS HEAD

by Paul Braterman

If Isaac Newton is the father of modern physics, then Antoine Lavoisier is the father of modern chemistry. Newton was knighted, and died in his bed at age 84. Lavoisier died at age 50, on the guillotine.

LavoisierAndWifeA civil servant scientist

Lavoisier originally trained as a lawyer, but studied science at the same time, and set about earning admission to the Academy of Sciences. This he achieved at the remarkably young age of 25, with a combination of pure science (composition of gypsum), and applications (problems of street lighting and water supply). He invested his inherited fortune in membership of a curious body called the Company of Tax Farmers. This was involved in the collection of indirect taxes throughout the whole of France, while its members individually lent money to the Crown, thus simultaneously taking on the roles of bankers, administrative civil servants, and investors in government securities.

Lavoisier's administrative responsibilities included supervising the Gunpowder Administration. Gunpowder is a mixture of sulphur, charcoal, and saltpetre (potassium nitrate). At the time, this last was obtained from fermenting organic matter with human and animal manure. French production had become haphazard, and lack of gunpowder was one reason for France's defeat in the Seven Years War/French and Indian War of 1754 – 1763. The Dutch had developed a system using beds of manure mixed with rotting vegetation, which Lavoisier copied, to such good effect that within a few years France was able to supply the material to its allies. French exports of saltpetre played an essential role in the American Revolution, and Lavoisier was able to write “One can truly say that North America owes its independence to French gunpowder”.

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Poem

Withered Rose

A version after Iqbal

By what words can I deem you
desire of the nightingale's heart?
The morning breeze was your cradle,
the garden a tray of perfumes.

My tears rain like dew,
and in my barren heart your ruin
an emblem of mine,
my life a dream of roses.

By Rafiq Kathwari, whose debut collection, In Another Country, is available here.

Feeling the Love

aby Maniza Naqvi

MargaretmarcusThis is about the biography of an American woman who was the author of key militant interpretations and texts which are followed by extremists today. She left New York in the 1960s and went to Egypt and then from there came to live in Lahore with the founder of the Jamaat e Islami Maulana Maududi. She lived in Lahore till she died in 2012 at the age of seventy- eight. This is also about an Austro-Hungarian man who became the spiritual advisor of the House of Saud. It is about Maryam Jameela, an American woman whose given name by her parents was Margaret Marcus and an Austro Hungarian man— Muhammad Asad whose given name was Leopold Weiss. It is about them then, and it is about us now.

The first time I saw razor wire was along the checkpoint at Eretz in Gaza, a dozen years and change ago. And then more recently I'd seen it coiled atop the compound walls of the offices and homes of donor agencies in Addis Ababa, a city changing fast with shanty towns being mowed down to create overpasses and underpasses, Malls and residential paradises for the purchasing powerful foreigners and visiting and returning diaspora. And I'd seen it at the peripheries of a holiday lakeside resort in Malawi built on land grabbed from fishermen, razor wire, presumably to keep the animals out. Now I stared at the knife's edge of gobs of razor wire at ground level in Lahore. And the last time I'd seen such an array of foreign corporate journalists passing through the barricades to speak to the local citizens as the notable and primary experts on that country, on just about every aspect of it, was well—in Bosnia. Now here they were, ‘conflict' experts, some the same, doing the same thing in Lahore.

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Monday, November 23, 2015

War on a New Planet: Reimagining Conflict and Leadership in the Time of ISIS

by Ali Minai

Network1The terrible terrorist attacks by ISIS in Paris on November 13 have understandably generated a great surge of opinion and analysis – some of it insightful and some just opportunistic. It is precisely at times like these that the volume of immediate response threatens to obscure deeper issues, and for a problem as deep as the threat of jihadi extremism, this is truly dangerous. While people are still reeling from the actual attacks and decision-makers are reaching for the most obvious – and frequently bad – choices, it is critical that policy-makers move towards a more realistic understanding of the conflict they face, and not make things worse than they are. Of course, history suggests that this likely to be a vain hope – especially since the proper course is far from clear. This motivation behind this article is not to prescribe specific actions, but to provide a general perspective that may trigger further thinking.

Following the Paris attacks, President Hollande of France declared, “France is at war!” Similar pronouncements have been made by world leaders, analysts and pundits since 9/11. Some see the conflict with jihadi terrorists as a “clash of civilizations”; others as a “battle of ideas”, pitting modern liberal democracy against a regressive ideology. Yet others have declared it to be a “battle for the soul of Islam.” Those wedded to conventional geopolitics see it in terms of military engagements, covert operations and counterinsurgency. There is some element of truth to all these characterizations, but only in the sense that the five blind men of India had some part of the truth about the elephant. What has remained largely unacknowledged is the terrible truth that this is the first war of its kind – a brand new thing never before seen in history, and therefore one for which there is no prior wisdom. It is the first great conflict of the age of globalization, and its phenomenology reflects that of a complex, nonlinear, self-organizing networked world. To make an imperfect analogy, it is to ordinary warfare what quantum physics is to Newtonian physics. It is a war where things don't add up normally, where distant events can be strangely entangled, where common sense may be a liability, and where the very geometry of comprehension is distorted.

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Monday Poem

Aging Face

With mirrors the aging face became personal.
It hung before only on the heads of others,
but with realization that the still surface
of a pond returned the image of the seer,
when polished metals revealed a clear and troubling truth,
when a silver-backed square of glass
served up serial images of hard fact so precisely
denial was impossible,
the aging face became a self portrait
in intimate time, like a film frame
on a reel of a fresh spring field
which, between glimpses,
had been raked by a ruthless gardener
determined to turn new life into that
which can only be remembered

Jim Culleny
11/16/15

Thoughts On Proofs by Contradiction

by Carl Pierer

DuhemAmong 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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