Gary Herstein's Open Salon Blog

Mathematics and the Beautiful

Sat, 20 Jun 2009 22:06:32 -0400

Mathematicians are not driven by truth so much as by beauty. There is nothing exceptional about this claim, and it is only surprising to those who are not themselves mathematicians. The driving force of beauty in mathematical research has occasionally been documented; G. Hardy's A Mathematician's Apology being one example, Davis and Hersch's The Mathematical Experience being another, more contemporary one. However, I thought I would mention another aspect of the connection between beauty and mathematics, one that is more personal and, as it happens, more relevant to Father's day. You see, many years ago my father gave to me his slide rule.

The particular instrument he gave me is a Keuffel and Esser Log Log Duplex Deci-Trig slide rule model 4081-3S. An excellent source for information on K&E instruments can be found at http://www.sphere.bc.ca/test/ke-sliderule.html. My camera is not good enough to pick out the close up details of my own slide rule, but links from the above site give good generic pictures. In the first one, the 4081-3x is shown at the top of four:

http://www.sphere.bc.ca/test/slide8/260-ke-slides.jpg

The second picture is of the complete 4081-3S "kit" including its characteristically orange leather sheath and manual (all of which I have):

http://www.sphere.bc.ca/test/slide7/233-ke-4081-3set.jpg

(I've provided the URL's to the original pic's.)

It was a source of some amusement to my father when I asked if I could keep his old slide rule. He had passed it along to me to entertain myself with since his work by that time had moved on to electronic calculators. Certainly it is the case that calculators offer a much higher level of numerical accuracy than a slide rule. However, unlike a calculator, a high-quality slide rule places you in a direct relationship with the numbers and equations you are working through that cannot be found with calculators. And this relationship is beautiful.

For one, there is a practical matter that tends to get overlooked. When working with calculators, numbers appear almost by magic, spewing forth in ridiculous decimal places that offer a pretense of accuracy without any presumption of understanding. That pretense of accuracy, I submit, is a significant problem. The accuracy of our measurements seldom goes beyond two decimal places, but the calculator will give an answer to 10 or more. This is an illusion, not a conclusion. With a slide rule, the limits of accuracy are already built into the calculating process itself. Thus the slide rule puts you in a truer relation to the practical facts in hand by denying you the illusion of unlimited accuracy. This truth has the rough-hewn beauty of a tree-limb found wild in the woods and hand polished to a personal walking stick over a perfectly synthesized, mass-produced crutch.

But there are quite real issues of theory here as well, both mathematical and aesthetic. Upon first examining the slide rule, the numbers and registers can appear to be scattered across its length in a haphazzard, even chaotic manner. But there is an order and symmetry to the slide rule that is its defining characteristic. The registers, far from being chaotic, are uniformly spaced according to their base 10 logarithm. ("Log" for short. The base 10 log of a number X is that number Y such that 10^Y = X. Thus the log of 2 (or just "log 2") is approximately .31.) Indeed, if one flips my slide rule over to look at the L register on the back, one will see that this is the only register that is evenly spaced; and, as one might then expect, this is the actual logarithm scale.

Multiplying and dividing on the slide rule is thus a matter of adding and subtracting logarithms. Since all numbers are evenly spaced according to their log, such activities are really just composition (plus or minus) of simple, linear lengths, like two simple rulers held up one edge against the other. In this way, fundamental mathematicla relationships reveal themselves to be of a kind with the most basic sorts of measurement acts: the slide rule is, after all, "simply" a multi-modal ruler. This reduction of basic mathematical and numerical relations to that of the comparison of extensive lengths is a rather exotic and difficult theoretical development. I hold it in my hands.

Further, these lengths compose end-to-end and wrap around, so that it is rather like the linear slide rule was actually circular. (Certain versions of slide rule actually are; but the precise, multifunction scientific instruments tended not to be so due to the requirements of accuracy for their mulit-layered and multi-modal registers and scales.)

These rules of composition on a slide rule -- that composition is well-defined, that it is self-contained and that it can always be reversed, are the foundational axioms of that mathematical structure known as a "group." Mathematical groups, in turn, are definitive of the formal study of symmetry. And symmetry, of course, has been recognized since the days of Aristotle and before as fundamental to the very notion of beauty. (No cheap URL for this one, I'm afraid; you may just have to read some Greek philosophy.)

Symmetry is about balance, proportion, distribution, etc. The Greek word "temperance" (roughly "sophrosyne") was originally an aesthetic ideal that came to be translated into a notion about a well lived human life. The very word "aesthetic" comes from the Greek word for "experience." Our notion of beauty is then in turn something that the Greek's held up as a fundamental ideal of human existence at its most basic, and most basically human. And such existence finds one of its most fundamental expressions in the group-theoretic expressions of unity and symmetry that I hold in my hand: this precision instrument, this lovingly crafted slide rule.

But the beauty of symmetry is not just to be found in those symmetries as they are manifested; it is there, too, to be found in those symmetries that are strategically broken. Some (mathematical) relations lead to zero or infinity; some to both. When that happens, the symmetry ends there -- the scales are shattered, the mirror crack'd.

But the slide rule enables and empowers these recognitions; the "LL" registers trail off from plus and minus zero; others track the dwindling of decimal places. Each such dwindling shadow of a number retains its original symmetry (one can invert the mirror, and return to the land of integers) even as it carries you over the falls toward a silent oblivion of non-mathematical answer.

But the balance is found even as it is lost, the tipping point ennunciates the level plane, the boundless is found in limitation. The infinite is discovered in the ruled line.

Obviously more could be said here, but I've said too much already. Aristotle gave the principle of the form and purpose of beauty to the idea of symmetry. Plato gave to the physician Eryximachus the poetic discourse on the connections between balance, harmony, life and love in The Symposium. My father gave me a slide rule.

We're even.

My Father's House

Tue, 9 Jun 2009 01:06:28 -0400

I have moved into my father's house.

The choice was between this and something much closer to homelessness.

My father will not be there, however. I have only just recently moved him into an assisted living center not too far away, because his dementia (which may be early onset Alzheimer's) had rendered living alone impossible. All the other options were even worse. As it happens, I will be living close at hand to my father and will be in a position to at least monitor his care and condition.

And make decisions, as the need arises. Yes, I have been elected the “decider.” It is mildly ironic that someone like myself, with a better than average knowledge of macro-economics but a pathetic inability to sensibly manage my own finances, was put in charge of managing my father's end-times. I suspect I'll be more of a pit-bull with his finances than I've ever been with my own. But my own finances (unlike my father's) are a train wreck. And so I am moving into my father's house. Not to be close to him but so I won't be begging quarters in the subway stations. And besides, the house has to be occupied; in this market (and in its particular location) it is unsellable.

But truth be told, I would not be homeless, even if this option were not on the table. For reasons that I struggle to understand, there are many people stepping forward looking to help me during my present “difficult” circumstances. I got this email the other day from someone I've known for over 45 years:

Date: Wed, 27 May 2009 11:03:37

I've had a chat with ________ and we would like to send you some money to help with your move. These are extraordinary times for you and we would like to do something to ease some of your worries. Its not much, but if you could help us find a way to wire you $_____, its yours today. We believe in your professional goals and it will happen someday. Actually I have always believed in your goals since we were little. You stand out in the world, you probably can't see it as yet, but I do.

best, __________

I could weep; I should weep, but just don't know how. I don't know how to respond to this, and the only way I can think to honor it is to do what I have been doing, to live in a way that might actually vindicate this kind of faith in me, a faith I wonder what I have done to deserve. I have various projects in mind, some of them book-length. A couple years in one place would be a good chance to get some writing done.

And now I have a place to pursue those projects. A place not my own: my father's house.

I am there now, but it is not enough to unpack my “stuff;” I've got to “unpack” (or, perhaps more accurately, “unload”) his. For example, until his old bed is gone, I am sleeping in the dinette (on my bed). (The facility where he was moved could not accommodate so large a bed as his King+, so it was not moved with him and remains in his old bedroom.) It is my father's house, not mine. Everything that matters most to me that is not physically defined by human DNA is here now. But I am still awaiting my arrival. Somehow, this is has to become my house.

I enviously note that my cats have already settled down. Even the feral one (they are all rescues of one form or another) has determined her spots and casually marches around demanding that I attend to her (when she is in the mood, of course). Rotten little bastards ...

So I care for my father while others care for me. These friendships (and there are more than just the one I mention above) did not happen by accident; yet it is almost galling to think that I somehow “deserve” this when so many others do not deserve what they get. And I struggle to find myself in my father's house.

A Right, A Left, another Right? Propertarians bitch slapped?

Fri, 8 May 2009 19:05:10 -0400

Andrew Leonard has a note over at Salon proper about the smack down Obama just levelled against the hedge funds that were trying to extort taxpayer money for their shares in Chrysler. I've nothing to add to Leonard's comments (here) , and merely observe that the group that was holding on to Chrysler assets has evidently dissolved and gone limping away. As Leonard notes, there is likely to be much puling and whining from the Right and the Libertarians about this; it is this that my insufferable habit to blather on the keyboard cannot resist insufferably blathering on the keyboard about.

In particular, the ideas of rights and property deserve some commentary.

There is an implicit thesis amongst both the conservatives and the libertarians that the "First" right is property. Certainly this is a derivable conclusion from the writings of John Locke, who insisted we had an inalienable right to "life, liberty and property." But note here it is John Locke that is being appealed to, and not Thomas Jefferson (who had the audacity to rephrase the above as "life, liberty, and the pursuit of happiness.") For all that Tom was merely a Founding Father of the United States rather than a Primary Source of Libertarian philosophy, he might have been on to something there.

Specifically, when property becomes the "First" right, it then becomes the only right. Because only people with property will ever be able to exercise their rights. And a right that cannot be exercised is a right which does not exist. Consider, for example, that as early as 1870 the 15^th Ammendment to the Constitution of the United States "guaranteed" the "right" to vote to every citizen of the United States, regardless of "race, color, or previous condition of servitude." Yet how many African-Americans were able to vote in Mississippi in 1963? A right that cannot be exercised is a flatus vocus devoid of any content.

The point being, of course, that property rights are parasitic upon other rights. Before there can be "property," ther must first be "persons." "Person" is the foundational concept of all rights -- inalienable or otherwise -- and therefore it cannot be trumped by some other putative claim, since that claim absolutely depends upon the existence of Persons before it even has the possibility of existence. Yet because of this apparent stomping by the current administration, we are apt to have armies of propertarians sanctimoniously wringing their hands about the infinitessimal hit to their unimaginable wealth, as though this were the foundation upon which human liberty was built.

(Along these lines, I submit that Libertarians are of a kind with the "white moderates" Martin Luther King so despaired of. For example, in the Letter from a Birmingham Jail, King laments that at least with the white supremicist who was spitting on you, you had an opponent you could directly confront with the claims of justice. But getting white moderates to honestly confront racial injustic was like trying to nail mercury to the wall. They would wring their hands about the injustice of the South, but insist that no meaningful action be taken because it might "break the law." Libertarians, I submit, are the same way. They speak for justice, for civil liberties, but act only in the support of unbridled power and the status quo.

Libertarians nominally support civil liberties, but their support is never more than nominal. This is because for Libertarians, property rights are always First rights. Thus, while they will proudly stand up for racial minorities, they will only do so for as long as it involves no actual impact on the wealth of the those who profit from institutionalized racism. Bear in mind, these are the folks who cheer Robert Nozick's position that people have a "right" to sell themselves into slavery (person is subordinate to proberty), a position which even Nozick ultimately realized was too monstrously irrational to defend.)

The anti-propertarian moment of the current administration is something of profound importance, because it is an official challenge to the idea that property is the First right. But to leverage this moment, we have to begin taking a more serious look at not only what is at stake, but the way in which those stakes have been framed by the propertarians.

Having Stuff

Tue, 5 May 2009 22:05:31 -0400

Earlier this year, Merchant Books republished Russell and Whitehead's three volume work the Principia Mathematica in paperback for about ¹/₁₁^th the price you'd have had to pony up for even a used set of the Cambridge edition (any year). I am the only person I know who could even possibly get excited about this. Granted, it means I can finally own a copy of this work (once I've settled down to a permanent location and can have stuff sent to me), but it is frankly of only marginal historical interest these days and no real formal interest at all. Stuff in logic has long since gone past the Principia, and while it was important in its day, formal logic does not linger over historical contributions once it has moved past them. Still, I was excited.

There are various extremities of the “having stuff” philosophy: at one end are the anarchist/communists who rally around the banner that “property is theft.” This thesis is generally nuanced in some way, and often enough in a form that is not entirely disingenuous, although intellectual pretzels are usually involved. At the other end are the anarchist/capitalists who insist that property is the First Right. It strains my imagination to figure who is stupider, but I think there is clear evidence regarding who is the more dangerous. Anarchist/communists are (for the most part) feckless twits who think a few rousing choruses of The Internationale will save the world. Anarchist/capitalists think government is evil (being the only entity of a size that even might oppose them), that property is the only right that matters and, since they actually have resources at their disposal, are prepared to employ more stringent methodologies than camp-fire songs to advance their viciously greedy and self-serving agendas. (But I editorialize ...)

Still, having “stuff” changes us when that stuff is ours. It is a remaking of the physical context in which we define our selves. We are not minds somehow accidentally attached to bodies; how we “body-forth” is irreducibly situated in the mise-en-scene of our lives. This can be good or bad, or good and bad. Too much bric-a-brac (of whatever conventional monetary value) and we drown our self in a tidal wave of dissipation and distraction. Things cease to have any relation to us, to our self, and become instead externalized opportunities for vapid and insensate self-negation. Too little and our physicality becomes unanchored and dis-affected by the reality we no longer know how to touch, or if we even do.

The right amount – and here I appeal to Aristotle rather than to Goldilocks – and we begin to find our-selves by making our-selves. What qualifies as the “right” amount can not be stipulated in advance by a mechanical formula, although careful human intelligence can find it if it is looking. But in any event, making requires tools, tools that we need to put our hands on. (Descartes was an idiot for imagining he could imagine a world without his hands.)

So while I envy people who, for example, work in wood, my hands reach for books as the tools of my trade, the tools of my making. They have a touch and sight; they make sounds when they are used; they even have a smell (not always pleasant). I use these instruments to mark out ideas and highlight turning-points of thoughts the way a carpenter will shift the orientation of a piece of lumber in order to properly employ and cooperate with the native grain of the wood.

But insofar, they have to be my books. What work could a carpenter do if he was only ever permitted to admire the wood, but never to use it or modify it. Abusing the metaphor a tad further, a library is like a lumberyard where you are only ever permitted to admire the wood; or a hardware store where you are invited to respect and admire the tools, but never actually use them. If the stuff does not belong to you, it is little more than a feast you can only watch without any hope of satiation.

So I ask myself, and I ask you: how much of our “stuff” is ours? How much of it contributes actively to our self-making, as opposed to the detritus that obstructs us from our-selves? Do you have your stuff, or does it have you?

In Vino Veritas (rum-inations)

Sat, 18 Apr 2009 22:04:22 -0400

When people ask, I tell them I saw Pirates of the Caribean and it changed by life. (I mean, seriously, why IS the rum gone?)

First, let's be clear that not everything I say -- or even most of what I say -- is presumed to have an lingering significance. But the whole point of blogging is to satisfy a kind of infantile egoism which is seldom satisfied by lingering significance. So I thought I'd muse briefly on some favored forms of booze.

First off, many people do not appreciate rum. Even bad rum can be pretty good all by itself. But insofar as people do anything with rum, they do it with Bacardi (generally over priced) or Captain Morgan (to which the thinking person can only respond, "why?") Bacardi's mainstream stuff is OK, but for OK rum you can do as well for considerably less. For high-end rums, I am still satisfied that nothing compares to Pampero Anniversario. This stuff is dark as molasses, and the smoothest thing you will ever drink. Anyone caught mixing this stuff needs to be bastanadoed.

I also like bourbon, but I've discovered an unpleasant effect it has on me. The day after a night when I've had a few drinks, I can still smell it everywhere. I walk down the street wondering if people passing by me suddenly think they're in Lynchburg, TN. It is seriously unpleasant.

Being me, I wonder after the nuances of the biochemical interactions that lead to this difference of mid-range effect. Both liquors are "sweet", and mix well with "tart." Ordinary rum with white grapefruit juice is my standard summer cooler, and of course the whiskey sour &/or Lynchburg Lemonade are classics for bourbon. Mixing either with a sweet second is not always a good idea. Fans of the Rum-and-Coke are notoriously ... er ... "youthful" in their manifestations. Neither rum nor bourbon go well with orange juice.

Anyway, drink responsibly, avoid anything with Red Bull or Jaeger Meister, etc.