Saturday, May 30, 2009

Hear, Hear

Peter Smith on the old adage "You can't prove a negative":

I'm reminded of the exasperated Bertrand Russell faced with the young Wittgenstein: "He thinks that nothing empirical is knowable. I asked him to admit that there was not a rhinoceros in the room, but he wouldn't. I looked under all the desks without finding one but Wittgenstein remained unconvinced." It is Wittgenstein here who is being obtuse and in the grip of a silly theory. Of course we can establish empirical propositions both positive and negative – for example, that there are five desks in the room and no rhinoceroses.

By any sane standard, it is just plain false that you can't prove a negative, and that supposed "discussion-killer" should itself be promptly killed off.

I've discussed this before myself; and even a very basic amount of research will uncover the fact that philosophers have been pointing out for at least two hundred years now that the adage is not really true. Of course, there are contexts where you can't prove a negative to a desired standard or level of proof. But it's hardly a general truth.

Friday, May 29, 2009

Ten Great Hymns

Sherry is collecting a list of great hymns, and asked everyone to contribute; I thought I'd follow Rebecca's list and put them up. There are a lot of hymns I like, so this list is a little bit arbitrary. Noticeably there are no hymns by Frances Havergal on the list, which is a serious omission and enough to prove that this list, while a list of great hymns can't presume to be a list of the greatest hymns in English, because something by Havergal would certainly be on that list. But this does give a rough idea of the hymns I like.

(1) When I Survey the Wondrous Cross (Isaac Watts): Like Rebecca, I think the words of this hymn are exquisite:

When I survey the wondrous cross
On which the Prince of glory died,
My richest gain I count but loss,
And pour contempt on all my pride.

I like the standard tune for it, "Hamburg," by Lowell Mason. The single best hymn originally written for the English language.

(2) O Come, O Come, Emmanuel (John Mason Neale, et al.): This is a hymn with a rich history. The original words are Latin, the so-called "O Antiphons" which are used in Latin-based liturgies during Advent, building on a large number of Scriptural images. They are spread out over several Sundays, but someone at some point had the brilliant idea of putting them all together into a single song, Veni, Veni, Emmanuel. John Mason Neale translated it brilliantly in the nineteenth century; the version we use today is not exactly Neale's, but Neale's with some modifications (e.g., he translated the first line as "Draw nigh, draw nigh, Emmanuel." The tune is likewise an adapted form of a fifteenth century French tune, which in turn was adapting chant to more popular singing. It always sums up for me the Church itself: no single human hand sat down and wrote it, and it has been sung by countless people across the centuries and the continents, its format adapted and re-adapted many times, and yet the message is still crystal clear and the hymn itself still exquisite:

O come, Thou Day-spring, come and cheer
Our spirits by Thine advent here;
Disperse the gloomy clouds of night,
And death’s dark shadows put to flight.

(3) Crown Him With Many Crowns (Matthew Bridges and Godfrey Thring): I'm actually torn about the tune; I like both the usual "Diademata" (by George Elvey) and the "Oliva Speciosa" (an eighteenth century Italian folk tune) arrangements. Sung by a good Baptist choir, this hymn will overwhelm; my paradigm case of exaltation is such a choir singing a rousing version of this:

Crown Him the Lord of life, who triumphed over the grave,
And rose victorious in the strife for those He came to save.
His glories now we sing, who died, and rose on high,
Who died eternal life to bring, and lives that death may die.

You get a sense of what it means for the heavenly anthem to drown all music but its own.

(4) How Great Thou Art (Stuart Hine): This is the most recent of the hymns on the list; the full lyrics are still copyrighted. The usual tune is a Swedish folk tune, usually called "O Store Gud," after the best-known Swedish hymn that uses it:

O Lord my God, when I in awesome wonder
Consider all the works Thy hands have made,
I see the stars, I hear the rolling thunder,
Thy pow'r throughout the universe displayed!
Then sings my soul, my Savior God, to Thee:
How great Thou art, how great Thou art!

This has been a favorite of mine since high school.

(5) Amazing Grace (John Newton): The Hymn that Needs No Introduction needs no introduction:

The earth shall soon dissolve like snow,
The sun forbear to shine;
But God, who called me here below,
Will be forever mine.

(6) Day of Wrath, O Day of Mourning (William Irons): This is Irons's version of the old Latin hymn, "Dies Irae" (the version of the Latin hymn that floors you with a sense of Judgment Day is always going to be Mozart's take on the first two stanzas). The Irons version was written in the wake of a murder; in 1848 the French had one of their regular revolutions of government, and the Archbishop of Paris was murdered. At his funeral the mourners walked by his casket murmuring the the Latin hymn written by Thomas Celano in the 13th century. Irons was there, and went home and wrote an English version of the hymn:

Day of wrath, O day of mourning!
See fulfilled the prophet’s warning,
Heaven and earth in ashes burning.
Oh, what fear man’s bosom rendeth
When from Heav’n the Judge descendeth
On Whose sentence all dependeth!

The tune is by John Dykes.

(7) Battle Hymn of the Republic (Julia Ward Howe): The tune, "Canaan's Happy Shore" by William Steffe, was originally used for a campfire song. It became popular in the Civil War using different lyrics, "John Brown's Body," and thus the tune is usually given that title. Howe heard it in camp one day, thought it catchy, but thought that the lyrics were horrible. That night she wrote new ones, about the inexorable providence of God:

I have read a fiery gospel writ in burnished rows of steel:
"As ye deal with my contemners, so with you my grace shall deal;
Let the Hero, born of woman, crush the serpent with his heel,
Since God is marching on."

(8) My Jesus, I Love Thee (William Featherston): This is a simple hymn, and the words were written by a teenager. The tune is "Gordon," by Adoniram Jordan. Because of the repeating last line, it's a sadder hymn that it looks at first glance, and is in fact usually sung in times of loss.

My Jesus, I love Thee, I know Thou art mine;
For Thee all the follies of sin I resign.
My gracious Redeemer, my Savior art Thou;
If ever I loved Thee, my Jesus, ’tis now.

(9) Dark Is the Night (Fanny Crosby): "Blessed Assurance" is the usual favorite when Crosby's name comes up, but of the Crosby hymns I've heard, I like this one the best. The tune is by Theodore Perkins.

With His loving hand to guide, let the clouds above me roll,
And the billows in their fury dash around me.
I can brave the wildest storm, with His glory in my soul,
I can sing amidst the tempest—Praise the Lord!

(10) O Perfect Love (Dorothy Gurney): The lyrics were written for the tune "Strength and Stay" by John Dykes, which were written for the hymn "O Strength and Stay," which was John El­ler­ton's and Fen­ton J. A. Hort's translation of Ambrose of Milan's "Rerum Deus tenax rigor"; Gurney liked the tune, but wanted the words to be more suitable to a wedding. Joseph Barnby took Gurney's lyrics and wrote a new tune for it, to serve as the wedding hymn for the wedding of the Princess of Wales to the Earl of Fife in 1889. That's the tune usually used. But the tune I like is William Monk's "Life and Love". As for why a wedding hymn makes it into my list of ten great hymns, what can I say? I'm a hopeless romantic:

O perfect Love, all human thought transcending,
Lowly we kneel in prayer before Thy throne,
That theirs may be the love which knows no ending,
Whom Thou forevermore dost join in one.

Ad Verecundiam

Wikipedia on Argument from Authority:

Argument from authority or appeal to authority is a logical fallacy, where it is argued that a statement is correct because the statement is made by a person or source that is commonly regarded as authoritative. The most general structure of this argument is:

Source A says that p.
Source A is authoritative.
Therefore, p is true.

This is a fallacy because the truth or falsity of the claim is not necessarily related to the personal qualities of the claimant, and because the premisses can be true, and the conclusion false (an authoritative claim can turn out to be false). It is also known as argumentum ad verecundiam (Latin: argument to respect) or ipse dixit (Latin: he himself said it).

There are some things wrong here. For one thing, argument from authority is not a fallacy but a rhetorical strategy; it can be entirely nonfallacious if the source really is authoritative. And not every argument that is not logically valid is a fallacy, as everyone learns in a good philosophy class. But it's interesting to think about 'argumentum ad verecundiam' (which in context actually means an appeal to the sort of shame associated with modesty), and why appeal to modesty (1) became discussed in this context at all; and (2) became regarded as a fallacy. The two points need to be separated because they are not the same.

Locke introduces the phrase into discussion of arguments in the Essay Concerning Human Understanding, IV.xvii.19. He is talking about "four sorts of arguments, that men, in their reasonings with others, do ordinarily make use of to prevail on their assent; or at least so to awe them as to silence their opposition." The first of these is described in these terms:

The first is, to allege the opinions of men, whose parts, learning, eminency, power, or some other cause has gained a name, and settled their reputation in the common esteem with some kind of authority. When men are established in any kind of dignity, it is thought a breach of modesty for others to derogate any way from it, and question the authority of men who are in possession of it. This is apt to be censured, as carrying with it too much pride, when a man does not readily yield to the determination of approved authors, which is wont to be received with respect and submission by others: and it is looked upon as insolence, for a man to set up and adhere to his own opinion against the current stream of antiquity; or to put it in the balance against that of some learned doctor, or otherwise approved writer. Whoever backs his tenets with such authorities, thinks he ought thereby to carry the cause, and is ready to style it impudence in any one who shall stand out against them. This I think may be called argumentum ad verecundiam.

Locke is not talking about fallacies here. He does deny that an argument ad verecundiam "brings true instruction" or "advances us on the way to knowledge". That is only possible with ad judicium, which appeals to probable and demonstrative reasons. But he does allow that the other three sorts of argument, including ad verecundiam might dispose you to knowledge. These four sorts of arguments are really rhetorical strategies: appeal to the modesty of your interlocutor, appeal to the fact that your interlocutor doesn't seem to be able to find a better argument, appeal to claims that your interlocutor has already conceded, and appeal to "proofs and arguments" about "the nature of things themselves".

This idea for classification seems to have caught on, and throughout the eighteenth century we see authors of logic textbooks expanding Locke's list. Isaac Watts, who wrote one of the important logic textbooks of the eighteenth century (he is the same Isaac Watts who wrote so many classic hymns), has a whole list, including ad verecundiam, that expands considerably on Locke's four. Watts considers ad verecundiam and its kindred labels to be topics, loci communes: they are a way of classifying arguments (whether good or bad) by their middle terms. This was a very clever idea, worth following up on, and, of course, largely ignored. Other authors also expanded Locke's list, so much so that Laurence Sterne takes time to make fun of them in Tristram Shandy, by proposing a new kind of argument, the argumentum ad fistulatorium, which he notes is an almost unanswerable argument: you argue ad fistulatorium when you answer your opponent by whistling a tune.

It was Whately who seems to have made popular the use of the label in discussions of fallacies, although he does not discuss it quite as such. In his Elements of Logic, when talking about ignoratio elenchi (fallacies of irrelevance) he says that "[t]here are certain kinds of argument recounted and named by Logical writers, which we should by no means universally call Fallacies; but which when unfairly used, and so far as they are fallacious, may very well be referred to the present head"; ad verecundiam is one of the ones listed. On Whately's view, ad verecundiam (like ad hominem) can be a perfectly good argument; it only becomes fallacious insofar as you fail to recognize the limitations. Arguments ad verecundiam only establish conclusions insofar as they relate to your opponent; they become fallacious when you take them to establish their conclusion absolutely.

As near as I can tell, however, the treatment of ad verecundiam as fallacious in itself seems to be due to Bentham, who, however, was clear that he was doing something slightly different from Locke. Bentham uses it to help sort out his classification of fallacies whenever he discusses arguments that tend to lead to erroneous conclusions in political contexts (which for Bentham really means the kinds of arguments that people make to argue that we should not accept utilitarian policies). Bentham has some fairly baroque fallacy classifications, but ad verecundiam seems to come up quite a bit. This use seems to predate Whately's usage; Whately, in fact, seems to be trying to get back to a more Lockean usage.

But the history of our mish-mash, crazy-quilt philosophical folklore about fallacies is tricky to follow, so there may be avenues I have not considered.

UPDATE: Whoops! Forgot the link to the article in question. Also fixed some typos.

Signs of Divine Wisdom

Whoever learns from man does not receive knowledge immediately from the intelligible species which are in his mind, but through sensible words, which are signs of intelligible concepts. Now as words formed by a man are signs of his intellectual knowledge; so are creatures, formed by God, signs of His wisdom. Hence it is written (Sirach 1:10) that God "poured" wisdom "out upon all His works." Hence, just as it is better to be taught by God than by man, so it is better to receive our knowledge from sensible creatures and not by man's teaching.

Thomas Aquinas, ST III.12.3 ad 2

Wednesday, May 27, 2009

Notes and Links

* Carnivalesque #50 is up at "Mercurius Politicus".

* The 91st Philosophers' Carnival is up at "sevenlayercake". None of the posts were particularly striking this time around, at least to my taste, but the host blog is a fairly interesting philosophy blog I hadn't come across before.

* Scott Gilbreath has a nice post on St. Bede, patron saint of historians.

* Very cool: the mathematics of language change:

Lieberman, Michel, and colleagues built upon previous study of seven competing rules for verb conjugation in Old English, six of which have gradually faded from use over time. They found that the one surviving rule, which adds an "-ed" suffix to simple past and past participle forms, contributes to the evolutionary decay of irregular English verbs according to a specific mathematical function: It regularizes them at a rate that is inversely proportional to the square root of their usage frequency.

In other words, a verb used 100 times less frequently will evolve 10 times as fast.

* Also cool: the Espresso Book Machine. It will probably make less of a change than one might think, but it's still an awesome bit of technology.

* I've never been much of a reader of graphic novels, but I've read a few onlines recently. Girl Genius I've already recommended; another good one is The Phoenix Requiem, by Sarah Ellerton; it's a much quieter and slower story than GG, although it makes up for the latter by having a much tighter story and shorter arcs of plot. It's a fantasy with broadly Victorian inspirations -- in effect, it's a sort of modern Victorian ghost story, although the events don't actually take place in Victorian Britain. Some of the artwork is exquisite.

* Harvet Ismuth's 42 Essential 3rd-Act Twists.

* I think much of the hubbub over Sotomayor has been blown out of proportion, for reasons I won't go into here. However, I haven't seen any discussion of the fact that Sotomayor, if her nomination gets approved, would bring the Catholics on the Supreme Court to six out of nine. (All the evidence is that Sotomayor is a pretty minimal Catholic, but still attends Catholic church on major holidays and when visiting family.) That will leave Stevens as the only Justice who is neither Catholic nor Jewish. This strikes me as an odd situation; do Catholics really so overwhelmingly dominate high-level judicial circles? Even as recently as Alito's nomination there were concerns about the Court tipping Catholic; given the political lay of the land, I doubt that this will be a major concern this time around, but it's curious that Catholics are so massively overrepresented on the Court in comparison to the general population. Steven Waldman notes that on the big political issue that always comes up we don't actually know anything about Sotomayor: that is, her record does not give any indication of whether she is pro-choice or pro-life.

* Here and there over the past few years I've seen a great many Christians who are of the opinion that argument with the so-called New Atheists should be a major priority among Christians, and I recently saw another instance of this. They don't generally ask my advice, but whenever people do, I always suggest that this is exactly the wrong way to go. The fact of the matter is, however important they may seem to themselves, and however visible they may be, they are of extraordinarily minute importance in the vast concerns of the Church. Our relations with Jews, Muslims, Sikhs, Hindus, and Buddhists are all vastly more important, and our relations with our fellow Christians more important still. And of all the foes we fight in our fight against the World, the darkness of the Zeitgeist, the New Atheists are puppy dogs; it is foolish to spend our time focusing so much on the little pups that we ignore the wolves. And of all the problems we face, we ourselves are more of a problem for us than they are; particularly the absurd ease with which we all are distracted from what is truly important by the fact of who happens to have made it to the bestseller list recently, or by some other utterly frivolous thing. And what is truly important, of course, is clear: Love God and neighbor, and when we somehow fail to do so, set out again and again until with God's grace we succeed. Everything else is hobby.

Tuesday, May 26, 2009

The Age of Enlightenment, the Century of Frederick

According to Kant, the two go together. As Kant sees it, the heart of Enlightenment is the freedom to use reason in public in every matter -- that is, everyone can reason however they will and talk about their reasoning without constraint. How do you guarantee this process of Englightenment? Through the despotism of a benevolent monarch:

But only a ruler who is himself enlightened and has no dread of shadows, yet who likewise has a well-disciplined, numerous army to guarantee public peace, can say what no republic may dare, namely: "Argue as much as you want and about what you want, but obey!" Here as elsewhere, when things are considered in broad perspective, a strange, unexpected pattern in human affairs reveals itself, one in which almost everything is paradoxical. A greater degree of civil freedom seems advantageous to a people's spiritual freedom; yet the former established impassable boundaries for the latter; conversely, a lesser degree of civil freedom provides enough room for all fully to expand their abilities. Thus, once nature has removed the hard shell from this kernel for which she has most fondly cared, namely, the inclination to and vocation for free thinking, the kernel gradually reacts on a people's mentality (whereby they become increasingly able to act freely), and it finally even influences the principles of government, which finds that it can profit by treating men, who are now more than machines, in accord with their dignity.

So, in other words, you increase people's freedom to reason by reducing their range of action. Argue as you please -- as long as you obey. You can disagree as much as you please, but you will be put in your place if you disrupt the peace. And slowly, over time, everyone will reason in public autonomously rather than merely following the public reasoning of others.

Of course, as is often the case with Kant it's difficult to say how much of this is intended to be serious and how much is merely Prussian flattery. But the view that Enlightenment is consistent with despotism was very widespread in the actual time period to which we usually attribute the Enlightenment. In the age of Frederick and Catherine and Joseph it was hardly possible to think of them as opposed. People like Diderot might insist that despotism was always wrong, but even Diderot took it for granted that Enlightenment was consistent with despotism -- he just thought that even benevolent and enlightened despotism led in the long run to a debased populace. As he put it in a memorandum to Catherine, three successive sovereigns like Elizabeth would have made slaves of the English.

We have a tendency to look at Enlightenment movements through the blurring lens of time; but continental theorists of the Enlightenment living in the period itself did not generally think of the Enlightenment as anything more than a transitional state; they did not regard it as holding the full answer to moral and social progress; they saw quite well that major elements of what they saw as the Enlightenment were occurring only under the patronage of authoritarian monarchs who did as they saw fit; and they formed no general consensus about how far this alliance was a good thing, although they all recognized the dangers of an unenlightened despot. And it could hardly have been possible otherwise in Europe at the time; it was the despots who made Enlightenment reforms possible, and it was the general mass of men who resisted. Some kinds of things considered to be progress had to be imposed from above by someone who merely had to say: "Obey!" And, of course, such a position naturally induces counterpositions, which arose soon enough.

Cartesian Consciousness and Other Minds

An odd claim in an IEP article on the problem of other minds:

There are, however, fundamental difficulties with the argument from analogy. First, if one accepts the Cartesian account of consciousness, one must, in all consistency, accept its implications. One of these implications, as we have seen above, is that there is no logically necessary connection between the concepts of "mind" and "body;" my mind may be lodged in my body now, but this is a matter of sheer contingency. Mind need not become located in body. Its nature will not be affected in any way by the death of this body and there is no reason in principle why it should not have been located in a body radically different from a human one. By exactly the same token, any correlation that exists between bodily behavior and mental states must also be entirely contingent; there can be no conceptual connections between the contents of a mind at a given time and the nature and/or behavior of the body in which it is located at that time.

What makes this odd is that this is not consistent with the Cartesian account of consciousness at all; Descartes explicitly rejects this view. Compare chapter 5 of the Discourse on Method:

And here I specially stayed to show that, were there such machines exactly resembling organs and outward form an ape or any other irrational animal, we could have no means of knowing that they were in any respect of a different nature from these animals; but if there were machines bearing the image of our bodies, and capable of imitating our actions as far as it is morally possible, there would still remain two most certain tests whereby to know that they were not therefore really men. Of these the first is that they could never use words or other signs arranged in such a manner as is competent to us in order to declare our thoughts to others: for we may easily conceive a machine to be so constructed that it emits vocables, and even that it emits some correspondent to the action upon it of external objects which cause a change in its organs; for example, if touched in a particular place it may demand what we wish to say to it; if in another it may cry out that it is hurt, and such like; but not that it should arrange them variously so as appositely to reply to what is said in its presence, as men of the lowest grade of intellect can do. The second test is, that although such machines might execute many things with equal or perhaps greater perfection than any of us, they would, without doubt, fail in certain others from which it could be discovered that they did not act from knowledge, but solely from the disposition of their organs: for while reason is an universal instrument that is alike available on every occasion, these organs, on the contrary, need a particular arrangement for each particular action; whence it must be morally impossible that there should exist in any machine a diversity of organs sufficient to enable it to act in all the occurrences of life, in the way in which our reason enables us to act. Again, by means of these two tests we may likewise know the difference between men and brutes.

Thus from a genuinely Cartesian view the lack of a logically necessary connection between mind and body in general is not relevant to the question. What is relevant is the very different issue of whether there are certain behaviors so constituted that they can reasonably be seen as the effects of a "universal instrument," which (the argument goes) cannot be had in an extended thing. For a Cartesian an unextended universal instrument is a mind; so you can conclude that someone else has a mind by learning what a mind is in one's own case and looking for effects that require it in someone else. The symmetry suggested in the article doesn't exist in the Cartesian view; mind and body have two different ways of operating on their own, and therefore we can tell (at least to some extent) the influence of mind on a body by recognizing cases where bodies act in ways going beyond what a body would do on its own. The Cartesians disagreed about how strong this argument was -- Arnauld, for instance, argued that it could be made demonstrative, and Malebranche argued that it could never be more than a 'conjecture', i.e., a probable argument. But this line of thought was a pretty important part of the Cartesian view of consciousness.

Monday, May 25, 2009

All Alone in the Night

I have recently finished re-watching Babylon 5 on Hulu. They only have the first two seasons, which means that the episodes they have leave off right about where I think the series began to become really interesting. But one thing I have noticed even in just watching those two seasons is how well the series holds up. Some of the effects would be done differently today, but the series still works, and for the reason it struck science fiction fans as so good originally.

First, the alien life is richly imagined. This is actually fairly rare in science fiction for TV, which tends to be humans with alien dressing. In Babylon 5 the alien races and alien politics are often more interesting than the politics of Earth, which is just one strand in the tapestry. You get a rich sense of the Centauri, of the Narns, of the Minbari, and even of the mysterious Vorlons, much richer than you get of aliens in other series, even in just the first two seasons.

Second, the plot is truly character-driven. Already in the first two seasons you get Londo, likable but driven by ambition, already pushed by his ambitions into going past the point of no return on the path of monstrous atrocity, already beginning to realize that he has let himself be caught up in a process that will inevitably turn him into a lonely monster who has done deeds even he loathes. He is one of the few villains science fiction for which one can feel a truly whole-hearted sympathy and pity. And, while G'Kar will have a long ways to go we already see him beginning to be tried in the fires of tribulation that will, once he finds his way past his thirst for vengeance, make him the sage he will eventually become. And there are plenty of other characters who are fascinating in their own right.

And third, while the profundity of science fiction is often exaggerated, its potential for profundity is not, and while Babylon 5's own depth, like the depth of any other franchise, is limited by the depth of thought of those who made and wrote it, by both a happy choice of ideas and a sort of dogged insistence in dealing with them, B5 really does do well in the deep ideas department, continually exploring those two most dangerous questions: Who are you? What do you want?

Feuerbach on the Power of Speech

A word is an abstract image, the imaginary thing, or, in so far as everything is ultimately an object of the thinking power, it is the imagined thought: hence men, when they know the word, the name for a thing, fancy that they know the thing also. Words are a result of the imagination. Sleepers who dream vividly and invalids who are delirious speak. The power of speech is a poetic talent. Brutes do not speak because they have no poetic faculty. Thought expresses itself only by images; the power by which thought expresses itself is the imagination; the imagination expressing itself is speech. He who speaks, lays under a spell, fascinates those to whom he speaks; but the power of words is the power of the imagination....

The word has power to redeem, to reconcile, to bless, to make free. The sins which we confess are forgiven us by virtue of the divine power of the word. The dying man who gives forth in speech his long-concealed sins departs reconciled. The forgiveness of sins lies in the confession of sins. The sorrows which we confide to our friend are already half healed. Whenever we speak of a subject, the passions which it has excited in us are allayed; we see more clearly; the object of anger, of vexation, of sorrow, appears to us in a light in which we perceive the unworthiness of those passions. If we are in darkness and doubt on any matter, we need only speak of it; – often in the very moment in which we open our lips to consult a friend, the doubts and difficulties disappear. The word makes man free. He who cannot express himself is a slave. Hence, excessive passion, excessive joy, excessive grief, are speechless. To speak is an act of freedom; the word is freedom. Justly therefore is language held to be the root of culture; where language is cultivated, man is cultivated.

Ludwig Feuerbach, The Essence of Christianity, Part I, Chapter VII

Sunday, May 24, 2009

Aquinas and Malebranche on Mathematical Infinites and God

Thomas Aquinas (Summa Contra Gentiles 1.43):

When our intellect understands something, it extends to infinity. A sign of this is that when any finite quantity is given, our intellect can think a greater one. But this order of our intellect to an infinite would be in vain were there not some infinite intelligible. And so it is necessary that some infinite intelligible exists, which must be the greatest of things. And this we call God.

Nicolas Malebranche (Search after Truth, Elucidation Ten, LO 614):

The mind clearly sees that the number which when multiplied by itself produces 5, or any of the numbers between 4 and 9, 9 and 16, 16 and 25, and so on, is a magnitude, a proportion, a fraction whose terms have more numbers than could stretch from one of the earth's poles to the other. The mind sees clearly that this proportion is such that only God could comprehend it, and that it cannot be expressed exactly, because to do so, a fraction both of whose terms were infinite would be required. I could relate man such examples demonstrating not only that the mind of man is limited but also that the Reason he consults is infinite. For, in short, the mind clearly sees the infinite in this Sovereign Reason, although he does not comprehend it. In a word, the Reason man consults must be infinite because it cannot be exhausted, and because it always has an answer for whatever is asked of it....No creature is infinite; infinite reason, therefore, is not a creature.

Even though both of these arguments begin with our ability to think about the mathematical infinite and conclude to the existence of God, they are radically different kinds of argument. Malebranche's argument is, we might say, an argument that thinks in terms of formal causality: the mathematical infinites we can reason about are thought to be mathematical infinites in God Himself. When a geometer works up a proof in geometry, Malebranche thinks he is studying God -- in particular, he is studying the archetype of space as it is found in God, or, to put it in other terms, he is studying the limits that divine Reason places on God's power to create space. The mathematician literally and directly studies God, albeit in a very limited way; mathematical reason is divine Reason.

St. Thomas's argument, on the other hand, is purely in terms of final causality. The mathematical infinite with which the mathematician is concerned is not the divine infinite in any way, shape, or form, and the reason he consults is not the divine Reason. But the infinite is still an issue requiring explanation: mathematical reason would be, as we might say, massive overkill if there were no actual infinite intelligible to which the human mind is somehow suited. And thus the intellect's ability to think of the infinites in mathematics is a sign of the fact that it is disposed to know God.

A major difference between the two is that St. Thomas the Aristotelian has no problem with the idea that created substances may be in some real way infinite. And, indeed, the Aristotelian account of the human intellect requires that human beings be infinite in a certain respect. Actually, we can probably be much stronger: on Aquinas's view, everything is infinite in some sense. Some of these ways of being infinite are not particularly interesting for the purposes discussed here, but the relative infinity of the human intellect means that there is no fundamental problem with mathematical infinites being drawn from sense experience and being contemplated in our own minds. But Malebranche does not allow for this possibility; indeed, he thinks it manifestly absurd. On his view we are obviously and completely finite creatures with completely finite intellects having access only to completely finite sensible experience; and thus there is no way we could even recognize mathematical infinites as infinite unless our minds already had access to something that actually was infinite. Every attempt to explain how we have an idea of the infinite either collapses, so that we are forced to regard mathematical discussions of the infinite as utterly unintelligible, or relies on a pre-existing infinite idea, either openly or by smuggling it in through the back door. You can't get the idea of the infinite from sensible things, you can't get it from yourself, and thus you must be consulting some actual infinite accessible to the mind.

So they are actually very different arguments, despite some similarities. In a sense one might say that they are as far apart as Aristotle and Plato.

The Sun and Every Vassal Star

Ascension Day
by John Keble

Why stand ye gazing up into Heaven? this same Jesus, which is taken up from you into Heaven, shall so come in like manner as ye have seen Him go into Heaven. Acts i. 11

Soft cloud, that while the breeze of May
Chants her glad matins in the leafy arch,
Draw’st thy bright veil across the heavenly way
Meet pavement for an angel’s glorious march:

My soul is envious of mine eye,
That it should soar and glide with thee so fast,
The while my grovelling thoughts half buried lie,
Or lawless roam around this earthly waste.

Chains of my heart, avaunt I say—
I will arise, and in the strength of love
Pursue the bright track ere it fade away,
My Saviour’s pathway to His home above.

Sure, when I reach the point where earth
Melts into nothing from th’ uncumbered sight,
Heaven will o’ercome th’ attraction of my birth.
And I shall sink in yonder sea of light:

Till resting by th’ incarnate LORD,
Once bleeding, now triumphant for my sake,
I mark Him, how by seraph hosts adored,
He to earth’s lowest cares is still awake.

The sun and every vassal star,
All space, beyond the soar of angel wings,
Wait on His word: and yet He stays His car
For every sigh a contrite suppliant brings.

He listens to the silent tear
For all the anthems of the boundless sky—
And shall our dreams of music bar our ear
To His soul-piercing voice for ever nigh?

Nay, gracious Saviour—but as now
Our thoughts have traced Thee to Thy glory-throne
So help us evermore with thee to bow
Where human sorrow breathes her lowly moan.

We must not stand to gaze too long,
Though on unfolding Heaven our gaze we bend
Where lost behind the bright angelic throng
We see CHRIST’S entering triumph slow ascend.

No fear but we shall soon behold,
Faster than now it fades, that gleam revive,
When issuing from his cloud of fiery gold
Our wasted frames feel the true sun, and live.

Then shall we see Thee as Thou art,
For ever fixed in no unfruitful gaze,
But such as lifts the new-created heart,
Age after age, in worthier love and praise.