On Fermat’s Last Theorem (Conjecture)

Fermat, a French mathematician of the late 17th century, came up with a conjecture that baffled other mathematicians for three and half centuries until Andrew Wiles published a proof in the mid-nineties. Most of you are familiar from high school geometry with the Pythagorean theorem, that the sum of two integers squared may be equal to another integer squared: a2 + b2 = c2, but Fermat imagined a more general problem for integers where an + bn ≠ cn where n>2: Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. That last bit is the mystery—that the margin was too small for his proof. Many mathematicians believe he did not have a proof, but all the same, he did throw down the gauntlet by making the conjecture. He just wrote the conjecture, that an + bn = cn, is not possible. Wiles’ proof is so complex and convoluted, however, that you have to be a brilliant mathematician to even begin to understand his arguments. For as simple as the Pythagorean theorem looks, Fermat’s conjecture is inversely complex, and complex in ways that not even a great mathematician can dream. The conjecture looks simple, but the answer seems to be one of the most complex ever proved in the history of mathematics. The proof, almost as elusive as the Holy Grail, is unintelligible to the average lay person, and difficult for even the gifted. What kind of mind does it take to fathom the dark and profound reaches of Fermat’s conjecture? This conjecture, according to a French academy of math, has the dubious honor of having the highest number of incorrect proofs written about it. In other words, many mathematicians have tried to conquer the proof, but died ignominiously on the battlefield without having succeeded. That fact that Wiles did his work in secret suggests that even he thought the little problem might be paradoxically unsolvable—a no-win scenario, as it were, and a career-ending catastrophe. That there is, after all, a solution to Fermat’s last theorem is of little consolation to all of that failure. (Sorry mathematicians,formatting limitations don’t allow for the little raised numbers in the equations.)

On Fermat’s Last Theorem (Conjecture)

Fermat, a French mathematician of the late 17th century, came up with a conjecture that baffled other mathematicians for three and half centuries until Andrew Wiles published a proof in the mid-nineties. Most of you are familiar from high school geometry with the Pythagorean theorem, that the sum of two integers squared may be equal to another integer squared: a2 + b2 = c2, but Fermat imagined a more general problem for integers where an + bn ≠ cn where n>2: Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. That last bit is the mystery—that the margin was too small for his proof. Many mathematicians believe he did not have a proof, but all the same, he did throw down the gauntlet by making the conjecture. He just wrote the conjecture, that an + bn = cn, is not possible. Wiles’ proof is so complex and convoluted, however, that you have to be a brilliant mathematician to even begin to understand his arguments. For as simple as the Pythagorean theorem looks, Fermat’s conjecture is inversely complex, and complex in ways that not even a great mathematician can dream. The conjecture looks simple, but the answer seems to be one of the most complex ever proved in the history of mathematics. The proof, almost as elusive as the Holy Grail, is unintelligible to the average lay person, and difficult for even the gifted. What kind of mind does it take to fathom the dark and profound reaches of Fermat’s conjecture? This conjecture, according to a French academy of math, has the dubious honor of having the highest number of incorrect proofs written about it. In other words, many mathematicians have tried to conquer the proof, but died ignominiously on the battlefield without having succeeded. That fact that Wiles did his work in secret suggests that even he thought the little problem might be paradoxically unsolvable—a no-win scenario, as it were, and a career-ending catastrophe. That there is, after all, a solution to Fermat’s last theorem is of little consolation to all of that failure. (Sorry mathematicians,formatting limitations don’t allow for the little raised numbers in the equations.)

On talking

One of my favorite activities is having a nice, long, drawn out conversation with another person about almost absolutely nothing, solving world peace, why people won’t signal a turn in Texas, the pro’s and con’s of gun control, the weather, food in the United States, why flying is never boring no matter how much you’ve done, the fear of flying, garbage, cell phone, sports, Italian politics, the Mona Lisa, Boccaccio, and a host of golden daffodils. Sip a double espresso and listen as a friend goes on about their day, their concerns, what they had for lunch. I don’t always have time to do this, but my day is always better when I give myself a little recess from the stress of the day and let someone else tell me how their day went. Of course, I do my share of talking as well. Perhaps it is more important to actually have friends with whom you share a certain intimacy who will sit and listen to you as well. I know I’m not the most enthralling or interesting speaker myself, so listening to me ramble about making bread or leading a search committee is not the most dynamic conversation in the world. What is probably more important than the topics being discussed is the time spent with the other person. American work ethics, however, do not lend themselves to taking a coffee break and just chatting about the world. I guess there may be something to “all work and no play make Jack a dull boy.” To have a good conversation and meaningful interaction, paradoxically one does not have to talk about anything profound or transcendental. In fact, perhaps it would be better if one is not talking about anything profound at all. A slow give-and-take is all a body really needs, but perhaps the conversation is better if it’s not a strenuous debate on presidential election politics. Talking about the weather in Texas is just about mundane enough to qualify for the perfect conversation between two people who really want to hang out together, but who also don’t want to complicate their lives by talking about something that stirs conflict. Intimacy is seldom about conflict and more often about subjects and beliefs held in common. I would suggest that most people do not base their intimate interactions on debate, conflict, or strife. In fact, most people need intimacy to reaffirm their own identities by seeing themselves in others. Talking, a coffee conversation fits the bill entirely. Sure, one might spend some time laughing at the latest political fiasco coming out of Washington, or why handguns on a college campus is insane, or whether one might boost, or not, a flagging economy with economic incentives or tax relief, but it will always be more interesting to discuss which drunk starlet has been sent back to jail for violating her parole or who is going to win the Oscar for best-supporting actor. Life cannot be just work. There has to be more. I find that while talking to another person, my mind tends to work on other problems which I might being trying to solve at any given moment. I’m sure my blood pressure goes does, as do my levels of stress. Having a nice long chat with a friendly person is like going out for recess and letting off some steam, and breaking the day’s routine can only be a good thing. We spend our whole lives wallowing in our daily routines, mindlessly bending our wills to schedules, time tables, and calendars. Now these aids help us to get our work done, which is good, but too much of anything can be a negative thing. In the isolation of work schedules, we eschew human interaction and robotically dedicate our time and energies to work, just work. Getting out and talking to another person may be just the ticket for breaking out of our zombie-like dedication and working on our overall good mental health.

On talking

One of my favorite activities is having a nice, long, drawn out conversation with another person about almost absolutely nothing, solving world peace, why people won’t signal a turn in Texas, the pro’s and con’s of gun control, the weather, food in the United States, why flying is never boring no matter how much you’ve done, the fear of flying, garbage, cell phone, sports, Italian politics, the Mona Lisa, Boccaccio, and a host of golden daffodils. Sip a double espresso and listen as a friend goes on about their day, their concerns, what they had for lunch. I don’t always have time to do this, but my day is always better when I give myself a little recess from the stress of the day and let someone else tell me how their day went. Of course, I do my share of talking as well. Perhaps it is more important to actually have friends with whom you share a certain intimacy who will sit and listen to you as well. I know I’m not the most enthralling or interesting speaker myself, so listening to me ramble about making bread or leading a search committee is not the most dynamic conversation in the world. What is probably more important than the topics being discussed is the time spent with the other person. American work ethics, however, do not lend themselves to taking a coffee break and just chatting about the world. I guess there may be something to “all work and no play make Jack a dull boy.” To have a good conversation and meaningful interaction, paradoxically one does not have to talk about anything profound or transcendental. In fact, perhaps it would be better if one is not talking about anything profound at all. A slow give-and-take is all a body really needs, but perhaps the conversation is better if it’s not a strenuous debate on presidential election politics. Talking about the weather in Texas is just about mundane enough to qualify for the perfect conversation between two people who really want to hang out together, but who also don’t want to complicate their lives by talking about something that stirs conflict. Intimacy is seldom about conflict and more often about subjects and beliefs held in common. I would suggest that most people do not base their intimate interactions on debate, conflict, or strife. In fact, most people need intimacy to reaffirm their own identities by seeing themselves in others. Talking, a coffee conversation fits the bill entirely. Sure, one might spend some time laughing at the latest political fiasco coming out of Washington, or why handguns on a college campus is insane, or whether one might boost, or not, a flagging economy with economic incentives or tax relief, but it will always be more interesting to discuss which drunk starlet has been sent back to jail for violating her parole or who is going to win the Oscar for best-supporting actor. Life cannot be just work. There has to be more. I find that while talking to another person, my mind tends to work on other problems which I might being trying to solve at any given moment. I’m sure my blood pressure goes does, as do my levels of stress. Having a nice long chat with a friendly person is like going out for recess and letting off some steam, and breaking the day’s routine can only be a good thing. We spend our whole lives wallowing in our daily routines, mindlessly bending our wills to schedules, time tables, and calendars. Now these aids help us to get our work done, which is good, but too much of anything can be a negative thing. In the isolation of work schedules, we eschew human interaction and robotically dedicate our time and energies to work, just work. Getting out and talking to another person may be just the ticket for breaking out of our zombie-like dedication and working on our overall good mental health.

On time and being

Time measured versus time perceived has always been an interesting problem. Time flies while you are having fun, but time slows to a crawl when you have to wait. You can look at a clock with a sweep-second hand and watch time go by, but that doesn’t tell you anything. Being objective about time is about as rational as being objective about thirst: objectivity has nothing to do with either. When a dentist is working on reshaping a broken tooth and the drill is whining, bits of tooth are flying everywhere, and you have eight things in your mouth at once, time stands still. When you wake up at six in the morning, you wonder where the entire night went. You turn fifty, and you have no idea what just happened to the last thirty years. A watched teapot will never boil, and that bagel in the toaster will only pop up after you sit down. Stopped at a red light, your entire life drags out before you, but time flies when you see a green light, which you will not reach before it turns red again. When you are in a hurry or late, time races like a scared jack rabbit. The fact that time is so malleable and dependent on our perception of it would suggest that time as a fixed rate of progression is an illusion dreamed up by watch and clock makers in the eighteenth century. Before that, time was a much harder thing to measure. The Illustration and its proponents thought that humanism, science and the corresponding empiricism could be used to lock up time, put it in a box, and regularize it. Much to our chagrin and illusory time ideology, time has never been a part of such a plan. It took Einstein and his “Theory of Special Relativity” in order to disavow such an idea that time is fixed, regular, predictable, but instead that time is totally dependent on one’s context, point-of-view, frame of reference. Though we will never have enough “time,” time is really all we have. What makes time so scarce is over-commitment, 80 hour-a-week jobs, and time poverty. What we do with our time is often a mystery, but we fight our calendars, arrive late everywhere, cut our rest short, skip time with our families, sell ourselves short. Being seems to come at a premium completely dependent on our inability to manage and distribute or time sensibly. Before we know it, we’re on the run, trying to make our next thing, or time is up, and we have to leave, get in the car, hop a train, take an airplane. The speed at which we live is geometrically proportional to the speed at which we travel, getting back to Einstein again. Perspective, frame of reference, context, dictate that the connection between being and time is contingent on how we perceive the passage of time. As we live, we create the illusion that time moves forward, especially given the structure of our verb systems, past, present, and future, language being our only mode for expressing what we experience. These are measly attempts at creating order within a structure that lies outside of senses, our perception of the universe, but because we must move within something, we call “time” that fourth dimension which surrounds our movements and gives meaning to our being. Beingness is necessarily a question of what we call time, but we only have the faintest notion of what “time” might really be. Our senses fall well short of understanding the impossibility of time, but our philosophy is equally deficient to even ask the correct questions concerning this enigmatic phenomenon. So we settle for a simple explanation of seconds, minutes, hours, and days because our little brains have no chance of understanding what is actually going on. We create simulacra to account for something that we not only don’t understand, but that we have absolutely no chance of understanding.

On the road

Today, I drove about sixty miles on the roads around this central Texas town, going to work, running errands, getting lunch, returning home. The road is a metaphor for life, obviously, this has been a literary commonplace for centuries whether we are with Chaucer and his pilgrims or Jack Kerouac surfing the highways of America with the Beat Generation mid-twentieth century. We hit the road to go places, and we do it more and more, which makes me wonder, what are we running to or from? Perhaps we live and work one place but our heart really belongs elsewhere. Or we love the mountains but live on the plains, or we live in the mountains but miss the ocean, the waves, the salty spray, pelicans. I doubt the grass is greener elsewhere, but sometimes I hope it is. I believe this yearning to travel, to be on the road, is a defining characteristic of the human psyche and that it is impossible to try and calm the unquiet spirit of the inner traveler who is always pushing us to hit the road. We cross oceans and seas in flimsy boats, trek across blazing deserts and frigid steppes, migrate on foot across thousands of miles of untamed wilderness because our inner gyroscopes are constantly striking out to find news spaces. We do it alone or in groups, two by two or three by three, making a pilgrimage to faraway places to see unfamiliar sites, speak foreign languages, eat strange food, sleep under unfamiliar stars, love new people, relax in new scenery. None of this is either safe or logical, sound or common sense. Wouldn’t it be more sensible to take a vacation at home where you might rest from the rigors of work and daily life instead of packing up the station wagon and striking out for territories unknown so you can risk your neck white-water rafting down a dangerous river in the middle of nowhere? Why go camping when you have a perfectly comfortable bed with clean sheets and nice pillows in your own house with no bed bugs or cockroaches or ants? The human spirit is not happy unless it is traveling, but I have no idea why. I must admit that visiting unknown sites and new people is rather thrilling. The same old thing all the time, even if it is comfortable, is boring, and if there is anything that will kill the human spirit in a big hurry it is boredom. Boredom is a killer, but again, this is a paradox because when we get out on the road in search of adventure, we are also putting ourselves in danger. Being on the road goes against self-preservation, and travelers have consistently put themselves at risk time and time again, and sometimes they never get home. Perhaps because the road is a metaphor for life, one really can’t live at all without participating in the journey, a journey that doesn’t end until they carry you out feet first and drop you down six feet deep. Sure, we can play it safe, but we won’t be happy unless we are on the road. So risk is a part of our character, but it is also a part of our collective success. Those that chose not to travel have been eliminated from the pack because they never passed on their cautionary genes, never wanting to risk anything. Ask yourself this the next time you hit the road: am I finally happy?

On the road

Today, I drove about sixty miles on the roads around this central Texas town, going to work, running errands, getting lunch, returning home. The road is a metaphor for life, obviously, this has been a literary commonplace for centuries whether we are with Chaucer and his pilgrims or Jack Kerouac surfing the highways of America with the Beat Generation mid-twentieth century. We hit the road to go places, and we do it more and more, which makes me wonder, what are we running to or from? Perhaps we live and work one place but our heart really belongs elsewhere. Or we love the mountains but live on the plains, or we live in the mountains but miss the ocean, the waves, the salty spray, pelicans. I doubt the grass is greener elsewhere, but sometimes I hope it is. I believe this yearning to travel, to be on the road, is a defining characteristic of the human psyche and that it is impossible to try and calm the unquiet spirit of the inner traveler who is always pushing us to hit the road. We cross oceans and seas in flimsy boats, trek across blazing deserts and frigid steppes, migrate on foot across thousands of miles of untamed wilderness because our inner gyroscopes are constantly striking out to find news spaces. We do it alone or in groups, two by two or three by three, making a pilgrimage to faraway places to see unfamiliar sites, speak foreign languages, eat strange food, sleep under unfamiliar stars, love new people, relax in new scenery. None of this is either safe or logical, sound or common sense. Wouldn’t it be more sensible to take a vacation at home where you might rest from the rigors of work and daily life instead of packing up the station wagon and striking out for territories unknown so you can risk your neck white-water rafting down a dangerous river in the middle of nowhere? Why go camping when you have a perfectly comfortable bed with clean sheets and nice pillows in your own house with no bed bugs or cockroaches or ants? The human spirit is not happy unless it is traveling, but I have no idea why. I must admit that visiting unknown sites and new people is rather thrilling. The same old thing all the time, even if it is comfortable, is boring, and if there is anything that will kill the human spirit in a big hurry it is boredom. Boredom is a killer, but again, this is a paradox because when we get out on the road in search of adventure, we are also putting ourselves in danger. Being on the road goes against self-preservation, and travelers have consistently put themselves at risk time and time again, and sometimes they never get home. Perhaps because the road is a metaphor for life, one really can’t live at all without participating in the journey, a journey that doesn’t end until they carry you out feet first and drop you down six feet deep. Sure, we can play it safe, but we won’t be happy unless we are on the road. So risk is a part of our character, but it is also a part of our collective success. Those that chose not to travel have been eliminated from the pack because they never passed on their cautionary genes, never wanting to risk anything. Ask yourself this the next time you hit the road: am I finally happy?

On a day off

Although most working folks work a lot of days, sometimes having a day off is not an entirely bad thing. I’m not talking about “mental health days”, which I do not recommend if you want to keep that job, but days in which the entire work force is taking off due to holiday or some such similar circumstance. Having a day off is a breath of fresh air. You don’t have to get up early and shave. You can get up late and make coffee, have breakfast, read the paper, take your time, and maybe not shave if you don’t feel like it. In fact, a day off is about not having to do anything you don’t feel like doing. You don’t have to climb into the the hustle and bustle of the mass transit system. You get a break from whatever it is that you do, and you must admit that no matter how much you love your job, sometimes it’s good to have a little break from the routine. You don’t have to be in charge, make decisions, get it done because the office (or whatever) is closed for the day. For a day, time stands still and doesn’t punish you into hitting your marks, sticking to a schedule, making sure that production doesn’t falter. Have a day off is like refilling your tanks–water, gas, air–and starting over. And when your day off falls on a Friday or a Monday your heart just dances with joy. You finally get a chance to break the daily routine and do something different: have a cook out, go to the cabin, fish, ski, have a picnic, visit somebody, go shopping for something other than groceries or underwear. A day off means never having to say you are sorry. Maybe you finally get to try out your new recipe for fish soup? Or you go hiking in the local state park, or maybe you sit by the fire and read a good book as it rains outside. A day off is about the freedom we willingly give up so we can pay our bills, mortgage, car. Perhaps what makes a day off so sweet is that you recuperate the independence that you had as a child to do whatever you want. A day off makes that next Monday morning sweeter still because at least for a moment you were free once again.

On a day off

Although most working folks work a lot of days, sometimes having a day off is not an entirely bad thing. I’m not talking about “mental health days”, which I do not recommend if you want to keep that job, but days in which the entire work force is taking off due to holiday or some such similar circumstance. Having a day off is a breath of fresh air. You don’t have to get up early and shave. You can get up late and make coffee, have breakfast, read the paper, take your time, and maybe not shave if you don’t feel like it. In fact, a day off is about not having to do anything you don’t feel like doing. You don’t have to climb into the the hustle and bustle of the mass transit system. You get a break from whatever it is that you do, and you must admit that no matter how much you love your job, sometimes it’s good to have a little break from the routine. You don’t have to be in charge, make decisions, get it done because the office (or whatever) is closed for the day. For a day, time stands still and doesn’t punish you into hitting your marks, sticking to a schedule, making sure that production doesn’t falter. Have a day off is like refilling your tanks–water, gas, air–and starting over. And when your day off falls on a Friday or a Monday your heart just dances with joy. You finally get a chance to break the daily routine and do something different: have a cook out, go to the cabin, fish, ski, have a picnic, visit somebody, go shopping for something other than groceries or underwear. A day off means never having to say you are sorry. Maybe you finally get to try out your new recipe for fish soup? Or you go hiking in the local state park, or maybe you sit by the fire and read a good book as it rains outside. A day off is about the freedom we willingly give up so we can pay our bills, mortgage, car. Perhaps what makes a day off so sweet is that you recuperate the independence that you had as a child to do whatever you want. A day off makes that next Monday morning sweeter still because at least for a moment you were free once again.

On relaxing

One would think that writing about relaxation would be a walk in the park, but I am so stressed out. My good Minnesota-Lutheran ethics just won’t allow me to relax because relaxing is the same as being a good-for-nothing bum. If you aren’t working, you are slacking! Yet, if I work all the time, I get to the point where I start forgetting to do things, get cranky and ornery, and I become ineffective. But if I relax for a moment I feel guilty, and I need to get back to work. What is wrong with me? Really, what is relaxing, anyway, but not working? I know that all work and no play make Jack a dull boy, but what does that really mean? Nothing but questions tonight, and I am afraid of the answers. For example, let’s just say that I want to sit and read a book that has nothing to do with research or teaching, drink something tasty, and put my feet up. I might even let my book down and doze for a few minutes, letting my glasses fall into my book and my head droops to one side. I might even get a snack that is not only not nutritious, it is downright bad for me. Potato chips would be a good choice. Maybe some salted peanuts. So I just let that pile of exams sit for a day and smoulder. The grass in the yard needs mowing, there are dirty dishes on the counter, the taxes need finishing. There is always something else that needs to be done. Now I don’t want to obsess over this issue of relaxing, but I still find the whole idea of sitting down and not doing something useful to be rather stressful. What if someone sees me relaxing, watching old re-runs of ancient black and white sitcoms? I have heard that relaxing can lower your blood pressure, brighten your outlook, and improve mental health. Who studies relaxing? Seems like a bit of a paradox: it’s as if that paper, by definition, would never get written. Maybe I’ll put on a little music to lighten the mood. Where are my slippers? Maybe I still have some caramel corn left over from Christmas. Should I start to read The Hunger Games tonight? Relaxing is going to damage my self-image, but I think it is something I would like to try, at least once or twice. And if I practice and I like it, I might try again, but without the guilt. Naw.