by Carl Pierer

In the first part of this essay, the axiom of choice was introduced and a rather counterintuitive consequence was shown: the Banach-Tarski Paradox. To recapitulate: the axiom of choice states that, given any collection of non-empty sets, it is possible to choose exactly one element from each of them. This is uncontroversial in the case where the collection is finite. Simply list all the sets and then pick an element from each. Yet, as soon as we consider infinite collections, matters get more complicated. We cannot explicitly write down which element to pick, so we need to give a principled method of choosing. In some cases, this might be straightforward. For example, take an infinite collection of non-empty subsets of the natural numbers. Any such set will contain a least element. Thus, if we pick the least element from each of these sets, we have given a principled method. However, with an infinite collection of non-empty subsets of the real numbers, this particular method does not work. Moreover, there is no obvious alternative principled method. The axiom of choice then states that nonetheless such a method exists, although we do not know it. The axiom of choice entails the Banach-Tarski Paradox, which states that we can break up a ball into 8 pieces, take 4 of them, rotate them around and put them back together to get back the original ball. We can do the same thing with the remaining 4 pieces and get another ball of exactly the same size. This allows us to duplicate the ball.

The second part of this essay demonstrated a useful consequence (or indeed, an equivalent) of the axiom of choice, known as Zorn’s Lemma and looked at a few applications of this Lemma. Two positions have been mapped out in the course of this essay. On the one hand, the axiom has very counterintuitive consequences, so much so that they’ve received the name of a paradox. On the other hand, the axiom proves to be very useful in deducing mathematical propositions. These considerations lead back to the question that had already been raised at the end of the first part: how are we to decide on the status of an axiom, on whether to accept it or reject it?

In this third and final part of the essay, we will take a more philosophical approach to this problem. In particular, we will look at a possible resolution offered by Penelope Maddy in her Defending the Axioms. The solution offered would lead onto further questions about the nature of mathematics: what is mathematics actually about? At the same time, Maddy’s view is based on a certain conception of proof that does not really reflect mathematical practice. The essay, due to limitations, only hints at a different perspective offered by looking at what mathematicians actually do and what role proofs play for them. Read more »

by Tamuira Reid

It’s nearing lunchtime when I make it over to Kevin’s, and beautiful out, but his window shades are still drawn closed, outside light on. I notice the porch slopes ever so slightly to the right, where a few forgotten footballs and beer bottles have now collected. I knock. Wait. Hear some movement and bustling. Then silence. I knock again. Silver masking tape covers large rips in the screen door, big enough for a head to push through. More movement. Finally Kevin emerges, a cigarette hanging from his lips.

He doesn’t say hi, but rather ushers me in, a quick gesture of his skinny body, a bony hand-to-back motion that says hurry.

I am used to this with Kevin. The hurry up and go of it all. When you’ve made the conscious decision to hangout with crystal meth addicts, life becomes a constant hurry-up-and- go, even if you’re only going to the bathroom.

I like Kevin. He’s thoughtful and smart and ridiculously resourceful. He’s also one of the worst addicts I’ve come across during my time in Clatsop County – or in my personal life – which is saying a lot. He will likely never get clean. He might commit more than a few crimes. And he will probably die too young. His life is already pedal to the metal, as he’d tell you. Pedal to the fucking metal. Read more »

by Michael Liss

He flew so fast and so close to the sun that it took an entire lifetime to fall back to Earth.

William Jennings Bryan was just 36 years old when, on July 9, 1896, he seized the Democratic Party’s Presidential nomination on the back of a single, electrifying speech, “Cross of Gold.”
Twenty-nine years later almost to the day, a haunted shell of his former self, he sat at the prosecution’s table, waiting for opening arguments in the Scopes Monkey Trial, unaware it would lead to his humiliation and ultimately hasten his tragic end.

In between, “The Great Commoner” was nominated twice more by his party, in 1900 and 1908, and served as Woodrow Wilson’s Secretary of State from 1913 to 1915. He then threw himself into efforts for causes as diverse as women’s suffrage, direct elections for Senators, and Prohibition. In the 1920s, he shifted his primary focus to his faith, but remained a prominent figure among Democrats through the 1924 Convention, when he was literally heckled off the stage in tears while trying to broker a compromise on an anti-KKK platform plank.

Bryan is an enigma. He failed frequently, but got multiple chances where abler men were passed over. Contemporaries questioned his intelligence and the scope of his interests, yet the exacting, often arrogant Wilson put him in his Cabinet and gave him a free hand with Latin American policy. His durability might best be ascribed to his possession of two tremendous assets: First, he was arguably the best orator of his time, compelling almost whenever and wherever he spoke, and, second, he seemed to have a psychic bond with his base. As the historian Richard Hofstadter noted, while other politicians of that era may have sensed the feelings of the people, Bryan embodied them. His people stayed with him through his successes and his disappointments. Read more »

by Anitra Pavlico

I recently read Simone Weil for the first time after having come across numerous references to her over the past year. I broke down and bought Waiting for God despite the intimidating and frankly confusing title.  I was not disappointed. One of her essays in particular, “Reflections on the Right Use of School Studies in View of the Love of God,” has opened and focused my thinking on education and learning in general, whether for children or later in life for the rest of us.

Weil writes that “prayer consists of attention. . . . Although today we seem ignorant [of] it, the formation of the faculty of attention is the true goal and unique interest of all studies.” She explains that by developing our capacity for attention, we can enhance our spiritual practice. Leaving that aside for the moment, it is nonetheless worth exploring what she means by attention. I am very interested (along with countless others) in how we in the internet era are maintaining our ability to focus given ever-multiplying distractions. As a mother of a school-age child, I also have a particular interest in how children are developing their ability to focus in this distracting climate.

Weil essentially promotes a meditative or mindful attitude for children facing challenging subject matter in school:

If someone searches with true attention for the solution to a geometric problem, and if after about an hour has advanced no further than from where they started, they nevertheless advance, during each minute of that hour, in another more mysterious dimension. Without sensing it, without knowing it, this effort that appeared sterile and fruitless has deposited more light in the soul.

Weil’s approach is timely because it makes learning less stressful and more enjoyable for students. Even if it does not seem as if the student is mastering the material, in Weil’s view she is coming closer to understanding by virtue of having focused her attention on it. In an age when students are sleep-deprived and unduly anxious about exams, college prep, and living up to parents’ lofty and usually unreasonable expectations, students may be comforted to hear from Weil that “we confuse attention with a kind of muscular effort. [. . .] Fatigue has no relationship to work. Work is useful effort, whether there is fatigue or not.” What is happening today in our schools is not your typical adolescent turmoil–it is a mental health epidemic. Suicide rates have surged; two-thirds of college students report “overwhelming anxiety.” [1] Clearly, merely applying more effort is backfiring. Read more »

by Bill Murray

It all started with zebras. 

Hard to believe, but sustained, hands-on field work in east Africa only has a sixty year history. Today Hans Klingel is an emeritus professor at the Braunschweig Zoological Institute, but when he arrived in Africa in 1962 Herr Klingel was one of only three scientists in the entire Serengeti.

Klingel and his wife made wildlife their career. Their first mission was to recognize individually and study ten percent of the 5500 zebras in the Ngorongoro Crater west of Mt. Kilimanjaro.

Zebra stripes are whole body fingerprints. The Klingels took photographs, taped the photos to file cards and carried them into the field. They came to recognize some 600 individuals.

Their file card technique caught on. In 1965 zoologist Bristol Foster studied giraffes at Nairobi National Park, photographing their left sides to memorize their unique patterns. He glued pictures onto file cards too. From 1969 a researcher named Carlos Mejia photographed and carried cards of giraffes in the Serengeti. Scientists swarmed into east Africa and the game was on.

On the open savanna, giraffes and zebras form a natural alliance. Zebras (and wildebeests, their fellow travelers) benefit from giraffes’ strong eyesight, elevated vantage point and superior field of view. Giraffes have the largest eyes among land animals and can see in color. Their peripheral vision allows them to just about see behind themselves. The next time your safari Land Cruiser rattles around the corner into view of a giraffe, you can bet the giraffe has already seen you. Read more »

by Brooks Riley

by Robert Fay

D.H. Lawrence had the goods on America. Like many foreign intellectuals and artists before and after, he was interested in the American “spirit of place” and its people’s curious experiment with displacement. He knew the “old American classics” stood toe-to-toe with the great Russian and French masterpieces of the 19^th Century, but he also knew there were a lot of bodies buried out back, and understanding America started from there.

In his masterful Studies in Classic American Literature (1923) he wrote, “at present the demon of the place and the unappeased ghosts of the dead Indians act with the unconscious and under-conscious soul of the white American, causing the great American grouch, the Orestes-like frenzy of restlessness of the Yankee soul, the inner-malaise that almost amounts to madness, sometimes.”

Lawrence was writing here about James Fenimore Cooper’s novels and the countries’ shameful relations with Native Americans, but it could equally apply to America’s ongoing treatment of African Americans. Lawrence goes on to observe, “America is tense with latent violence and resistance. The very common sense of white Americans has a tinge of helplessness in it.”

There’s a growing consciousness among (some) white Americans that racism is not simply a societal ill creeping toward extinction (“Look, we just had a black president!), but more like a malarial parasite that cleverly adapts itself to new circumstances, new opportunities. Read more »

by Sarah Firisen

Many years ago in 1991, in my first job out of college, I worked for a small investment bank. By 1994, I was working in its IT department. One of my tasks was PC support and I had a modem attached to my computer so that I could connect to Compuserve  for research on technical issues. Yes, this was the heydey of Compuserve, the year that the first web browser came out and a time when most people had very little idea, if any, what this Internet thing was.

As a tech geek, I signed up for one of the early, local Internet Service Providers and had an email account on their Unix based system. I actually met my now ex-husband through that email account, which is a whole other story. During this period, the ex and I were just starting our email correspondence and I would dial into my ISP at work to check my email. At some point, these minimal phone charges came to the attention of the firm’s Managing Director who took me aside and asked what I was doing. I told him about this wonderful new thing, the Internet! He told me to stop using the company’s modem to connect to anything but Compuserve. I protested, somewhat, and tried to tell him what a wonderful innovation the Internet was (and bear in mind, at the time, there weren’t a lot of websites and they loaded incredibly slowly, so even a geek had to use some imagination to see the future possibilities). He told me that the company would not be doing anything with the Internet anytime in the future. And by the way, this is a company who had already made a lot of its money from deals and IPOs in the entertainment and technology sector, so that they might have been interested in what I had to say wasn’t an outrageous idea.

Suffice it to say, that Managing Director was wrong and over the years that investment bank has been involved in many of the most significant deals with some of the biggest Internet-related companies. So what was the missed opportunity there? Clearly, that Managing Director was no visionary but my old company also ended up doing just fine and caught onto the Internet early enough to make a lot of money anyway. But, how much more money could they have made if someone had listened to me back then? I was young and very junior at the company and felt ashamed to have been “caught” and told off. But in hindsight, what I could have done was tell him a better story about this new, disruptive technology. Read more »

by Bill Benzon

Seen on Google search, Friday morning, September 9, 2018:

I’m sure you’ve heard about it. Elon Musk went on Joe Rogan’s podcast, Rogan lit up a blunt, and Musk took a toke. The next day Tesla’s stock tanked. Well, not exactly tanked, but it was down seven points, and the drop can’t be attributed entirely to that toke–there’s been some turmoil in the executive ranks–but that made for a good lede.

Not to mention the image! Billionaire inventor, boy wonder, real-life Iron Man, with his “Occupy Mars” T-shirt, head cloaked in a cloud of smoke­. Get it? Share-holder value, up in smoke?

It’s the stuff of mythology, of realityTVnews.

But that wasn’t the most interesting thing in the interview by a long shot. Read more »

by Timothy Don

Speak the word “discovery” and familiar images of explorers, scientists, ships, and treasure chests come to mind. To look into the visual record of “discovery” over the past 50,000 years, however, is to witness the concept expand, swell, and overwhelm the imagination. There is a wonder that arises in the wake of one’s research with the realization that the deeper and closer one looks, the wider, richer, and more capacious the topic at hand becomes. Consider the egg.

A simple egg. The familiarity of its shape is unnerving, even disarming. Hovering there, alone in space, it has the weight of a moon or a planet. A red planet. A symbol of discovery in its expansive, exploratory sense: to discover is to reach out into space, to land on the moon, to plan for Mars. But…this object is not Mars. It is an egg. It is an ostrich egg, more than 5,000 years old, from the predynastic period of Northern Upper Egypt, dug out of a tomb. Mars remains undiscovered. This is an artifact from an era now lost to us, uncovered by a forgotten Egyptologist from the 19th century. It belongs to the past. This is an actual discovery. And it is wonderful. It is pure potential. It has been discovered, but it remains uncracked. Full of mystery, this old egg from the past, it fills you with wonder. At that moment a gestalt switch gets thrown, and one realizes that discovery’s arrow points not only forward and outward to unexplored planets, but backward and inward to things lost, buried, and forgotten. To discover a thing is also to discover the past, and the act of discovery is about the recovery of the past just as much as it is about the probing of the future. And so a planet (symbol of the undiscovered future) becomes an artifact (material expression of the discovered past), and the artifact (the egg) becomes a mystery, a wonder, a promise. To be an egg is to promise discovery. Every egg, from this one (c. 3400bc) to the one you opened into a frying pan this morning, contains and shelters something utterly familiar and utterly unique, something waiting for you to find it. Every egg is a discovery.

There are certain artists who seem to have something to say about everything and whose work as a result appears regularly in the pages of the journal at which I serve as visual curator. Hieronymus Bosch is one. Caravaggio is another. Paul Klee and William Kentridge make the list. The genius of these artists (and many others) makes their work ever-contemporary, as immediate and compelling today as it was when they made it. The density of their work allows it to absorb the assault of time, so that its meaning can shift and apply itself to any period without diluting the purity of its original conception and execution. Read more »

Little Miracles 2:

—Cloudmaker

you, generator of clouds,
are indispensable. you
fling them up as if
they were mere vapor
your creativity is unsurpassed
much like cloud yourself
you may be dark or bright
free and light or stretched
like a cowl under stars,
in daylight white grey saffron
pink

at night you draw curtains
cross a moon
but you are not mere vapor
as you stride
and dance with wind
waving arms to fan the air
you cool it down to give
us drink

Jim Culleny
8/2/18

Drawing, Cloudmaker
Jim C. 1997

by Jonathan Kujawa

Despite what he may wish, the President of the United States is not a king. We have Congress to act as a check and to ensure the varied opinions of the citizens are represented [1]. In principle, a representative democracy is straight-forward: the voters vote, select their representatives, and the legislature gets down to the business of running the country.

The devil, of course, is in details. The framers of the Constitution had knock-down, drag-out fights over basic questions like: does the legislature represent individual citizens or the states? On one side you had those who saw the new country as a joining together of independent, co-equal states. William Patterson of New Jersey compared a large state having more votes than a small one, to the idea “that a rich individual citizen should have more votes than an indigent one”. Those on the other side took the view if this is to be a true common enterprise, then every voter should be treated equally regardless of where they happened to reside. In what can only be a coincidence, a founder’s position on the issue almost invariably matched whether they came from a large state or a small one.

They finally agreed the Senate would have two representatives from each state, regardless of size, while the House of Representatives would have its members allocated to the states according to their population. Even this reasonable compromise nearly failed. Read more »