All Upon a Throw

'Tis So Much Joy
by Emily Dickinson

'Tis so much joy! 'Tis so much joy!
If I should fail, what poverty!
And yet, as poor as I
Have ventured all upon a throw;
Have gained! Yes! Hesitated so
This side the victory!

Life is but life, and death but death!
Bliss is but bliss, and breath but breath!
And if, indeed, I fail,
At least to know the worst is sweet.
Defeat means nothing but defeat,
No drearier can prevail!

And if I gain, – oh, gun at sea,
Oh, bells that in the steeples be,
At first repeat it slow!
For heaven is a different thing
Conjectured, and waked sudden in,
And might o’erwhelm me so!

Always reminds me a bit of Pascal's Wager.

Boole and Uninterpretable Terms

George Boole's The Laws of Thought is one of the key early works in the development of mathematical logic. From the beginning, however, there have been a number of objections to Boole's own system. I think some of these objections are due to reading the book as a book on mathematical logic. Despite the book's importance to the early history of mathematical logic, however, The Laws of Thought is something else entirely. It is what Boole explicitly says it is: an investigation of the fundamental laws of thought as part of "the philosophy of intellectual powers", insofar as they can be expressed by algebraic operations.

One of the longstanding puzzles and criticisms of Boole is his permissive attitude toward uninterpretable symbols. In Boole's use of algebra, some expressions admit of interpretation. For instance, this is the Boolean form of the principle of noncontradiction:

x(1-x)=0

In this expression we multiply x times (1-x), which is interpreted as saying "both x and not-x". This is equal to 0, which is interpreted as Nothing (the opposite of 1, which is Universe of Discourse, or, in other words, everything we are talking about at the moment). However, it quickly becomes clear when we solve any problems other than the most simple that there are many steps in our transformation that have no interpretation at all. For instance, both of these are uninterpretable in Boole's system:

x(1-x)(1+x)=0
x(1-x)(-1-x)=0

Both of these could be elements in the analysis of a particular logical argument, but neither of them has any logical interpretation: 1+x is uninterpretable because 1 is everything in the Universe of Discourse, so there can't be something more than 1, while -1-x is uninterpretable because -1 has no logical meaning, violating as it does one of the cornerstones of Boole's entire approach, the index law, x²=x. (I think there's an argument that Boole could have relented on 1+x, interpreting it as 'irrelevant', but that would still leave -1-x, and there's nothing that strictly requires that he interpret it that way, anyway.)

Boole has no problem with these uninterpretable steps at all, citing the fact that mathematicians use imaginary numbers without requiring an interpretation (they are, it turns out, interpretable, as coordinates for rotations, for instance, and Boole in fact recognizes that, but he points out that the use of imaginary numbers does not require mathematicians to have any interpretation of what they stand for). The basic principles of the system don't depend on the uninterpretable steps; whether an operation is used depends only on whether it applies to interpretable expressions, and, if applied to interpretable expressions, yields correct interpretable expressions as a result.

And as Boole notes, even when the expressions are interpretable, we must keep in mind that we are doing "Algebra of Logic", not algebra simply speaking. Take the following expression:

x + y + z = 1

This is a perfectly intelligible expression in Boole's system. It tells us that the Universe of Discourse is divided into three parts, x, y, and z. But since Boole's algebra is an algebra of 0's and 1's, there is no numerical solution if we assume that we really are talking about three classes of things; none of the three can be 0 without being nothing, and none of the three can be 1 without being everything. The x, y, and z aren't numbers; they are signs of concepts that have certain algebraic features.

Thus Boole is not building a 'logical system' in our sense. He's investigating the laws of thought as a physicist investigates the laws of motion. Nothing about the way a physicist proceeds requires that every mathematical symbol he uses has a physical interpretation; he's just interested in kinds of calculation that yield the right conclusions. Boole himself insists that his investigation is a posteriori, based on observation of what thinking fundamentally requires, not a priori. He also occasionally uses the analogy to physics. Any mathematics used is tailored to fit the elements of logical reasoning. If algebraic operations have properties that don't fit these elements, they are ignored; if an assumption has to be made to get a logical interpretation of an equation, it is made; and if an expression has no logical interpretation, it is still used if it gives workable results. When Boole talks about "laws of thought", he means "laws" in pretty much exactly the same sense as a physicist does when he talks about "laws". In a sense, Boole's work is an attempt to build a mathematical theory of intellectual motion. It is therefore not surprising that he has no problem with uninterpretable terms; they are even, perhaps, to be expected.

Dear Mother of Fresh Thoughts

To Sleep
by William Wordsworth

A flock of sheep that leisurely pass by,
One after one; the sound of rain, and bees
Murmuring; the fall of rivers, winds and seas,
Smooth fields, white sheets of water, and pure sky;
I have thought of all by turns, and yet do lie
Sleepless! and soon the small birds' melodies
Must hear, first uttered from my orchard trees;
And the first cuckoo's melancholy cry.
Even thus last night, and two nights more, I lay,
And could not win thee, Sleep! by any stealth:
So do not let me wear to-night away:
Without Thee what is all the morning's wealth?
Come, blessed barrier between day and day,
Dear mother of fresh thoughts and joyous health!

Inclemency

The U.S. Constitution gives the President the power “to grant Reprieves and Pardons for Offences against the United States, except in Cases of Impeachment,” and at least several of the Founding Fathers thought that such a power was important to the usual and normal order of government. Laws tend naturally to severity because they cannot take into account circumstances. A government deriving its authority from the people, however, needs to have a built-in recognition that circumstances do sometimes matter, even if the law does not account for them. In the words of James Wilson, “Citizens, even condemned citizens, may be unfortunate in a higher degree, than that, in which they are criminal.” Taking this into account is one of the ways in which a government can make clear that law exists for the good of the people rather than the reverse.

President Obama has to date issued 22 pardons and one commutation of sentence. This is extraordinarily stingy. There was not a single pardon or commutation for the 2012 year, unless you count the Thanksgiving turkeys. The President has only issued pardons on three occasions. His clemency grant rates are extraordinarily low.

Read the rest of this post at the First Thoughts blog.

The Reynolds Diagram

This interesting diagram is from Frances Reynolds's An enquiry concerning the principles of taste, and of the origin of our ideas of beauty, &c. (1789). Frances Reynolds was the sister of the painter Sir Joshua Reynolds, and a talented painter in her own right, although she's usually seen as lacking the development and experience of her brother. She was also a close friend of Samuel Johnson, who called her 'Renny'.

In this diagram (you can click the image for a closer view) we see the human mind placed at the center, just as each introspecting person sees himself or herself as being in some way at the center of the world. The mind springs from "inscrutable principles of nature". We have a direction of progress, finding as we do sources of illumination in the world, in which we come to partake ourselves. Thus we have a natural tendency to the boundary of nature, which brings us to "demonstrable beauty and truth, and the utmost power of rules" -- in other words, the things that order nature itself and can be represented as laws. However, chance and fortune often impede us in our progress, and most of us come to rest at a mid-point between completely uncultivated powers of mind and the natural boundary of our excellence. This is 'common sense'. Common sense is to our natural mental excellence as common form is to beautiful form or beauty. With common sense we have something more like excellence than our rude natural beginnings, but it is mixed with a sort of mediocrity. If they attain to demonstrable beauty and truth, however, the intellectual powers may yet continue to rise upward -- Reynolds says they undulate like a flame -- through grace, to the upmost pinnacle of sublimity. In a sense we aren't going beyond beauty and truth, just beyond demonstrable beauty and truth, flickering upwards into regions that we can't perfectly capture in deductive systems and rigorous maxims. Demonstrable beauty and truth is in our power, achievable by sheer skill. But we know that there is something beyond the limits of skill alone, something we can't reach in a systematic way, but sometimes do attain.

Thus we rise up from our uncultivated powers of mind, through The Common, The Beautiful, and The Graceful, to our highest excellence, The Sublime. From demonstrable beauty and truth up to sublimity we are in the region of genius and taste. Most great genius and fine taste reaches up into the realm of grace, but very few reach the heights of sublimity, which Reynolds calls the "ne plus ultra of human conception".

Well aware that people might think the diagram a bit fanciful, she takes the trouble to point out that it is just a graphical representation of a philosophical account that has close analogues in the works of John Locke and Sir Francis Bacon, although the account itself was developed independently of their works.

Fortnightly Book, January 13

I spent second grade through seventh grade (and senior year of high school) in Carlsbad, NM, which is on the Pecos River. The Pecos River is an extraordinary breeding ground for carp, which are already a hardy and fertile fish species. Carp can be a pretty destructive species, crowding out other fish and unbalancing ecosystems, so it helps to have something in place to keep their numbers down. Every so often there would be a carp tournament in the Carlsbad Municipal portion of the river. You'd go down for the day with rod and reel and lots of white bread (easily the best bait for carp, rolled into little balls and put on the hook) and pull carp up like you wouldn't believe. I don't remember much in the way of detail, although since they were fishing tournaments there probably wasn't all that much to remember. But one year (fifth or sixth grade, perhaps) we managed to get a prime location near a water outlet, and just kept pulling them up without stop. I remember a little bit more about that year because I won a place in some contest or other (sponsored by a radio station, I think). I think it was third place. And the prize was an ice chest and a choice between a book and something else, I forget which. For a bookish kid like myself, there was no contest there, and so I came into possession of a copy of The Teka Stone: A Science Fiction Story by George B. Markle IV. Since school is starting up this week again and this term I am behind on preparation due to some sort of sinus/cough thing that won't entirely go away, I needed the fortnightly boook to be a light burden -- a re-read, not too difficult. And so I've decided to revisit this one.

Almost certainly the reason the book was one of the prizes was that it was written by a resident of Carlsbad. George B. Markle IV is actually better known for his writings on health; his most cited work, at least at first glance, seems to be a work published in the November 1988 American Medical News satirizing the push for organ donations among executed prisoners, and several other works I've come across (usually letters to editors) show the same interest in general medical ethics. He was an M.D. and general surgeon -- is still listed as practicing, although I don't know how extensively. According to the blurb on the back of the book, he was born in Hazelton, PA and graduated from Yale and the University of Pennsylvania. The Markles are a very old, and historically very wealthy, family in that area; all sorts of things are named after them.

The book is an adventure story of a deliberately old-fashioned kind. The blurb on the back gives this explanation:

George B. Markle IV is a doctor of medicine, a general surgeon, an artist, philosopher, lecturer, columnist and writer. He was raised on good literature and deplores the dearth of writers today who can hold the attention of all ages as could H. Ryder [sic] Haggard, Robert Lewis [sic] Stevenson, H. G. Wells and others of the past who told marvelous stories without filth, explicit sex or gratuitous violence.

Here is an adventure story - call it a science fiction novel, if you will - that harks back to the days when a book could be spellbinding without being degrading or sordid.

The publisher, as you might guess from a blurb like that, seems to be a small private publishing firm; I'm not even sure if it still exists. Nonetheless, the description is really rather accurate. There is something rather H. Rider Haggard-ish about the story, there is no explicit sex, and there is no gratuitous violence. Rather than sex and violence, the story relies on suspense, and does so, if my memory is correct, quite as well as one could wish.

It all starts when an archeology professor joins an expedition to the Yucatan. Things go horribly wrong when their plane is forced down in a little-known and dangerous part of the jungle, where they discover the ruins of the long-lost Teka civilization and learn the terrible secret of their downfall....

Reflex Senses

A fine taste is neither wholly the gift of nature, nor wholly the effect of art. It derives its origin from certain powers natural to the mind; but these powers cannot attain their full perfection, unless they be assisted by proper culture. Taste consists chiefly in the improvement of those principles which are commonly called the powers of imagination, and are considered by modern philosophers as internal or reflex senses supplying us with finer and more delicate perceptions, than any which can be properly referred to our external organs. These are reducible to the following principles; the senses of novelty, of sublimity, of beauty, of imitation, of harmony, of ridicule, and of virtue.

Alexander Gerard, An Essay on Taste. An interesting question is whether this list of 'senses' of taste is exhaustive. The basic terminology is from Francis Hutcheson; the use of the word 'sense' is closely related to our use of the same term in the phrase 'sense of humor'.