Probabilities as Analogies

All that follows is just crude jotting, quite incomplete, and barely a start at anything.

C. S. Peirce remarks somewhere that every probability is in fact the ratio of a species to its genus. Since we usually think of probabilities as numbers, we have to say that we have a probability when the relation of a species to its genus corresponds in some way to the ratio of one number to another. Or, to put it in other words, probabilities are analogies, in the sense of 'A is to B as C is to D'.

If you think about it, mathematical analogies of this sort are not uncommon. To say that there are twelve inches to a foot, for instance, is to say:

inch : foot :: 12 : 1

Probabilities are of exactly this sort, except that to be probabilities they have to be analogies of a certain kind. To say that the probability of rolling a 1 on a six-sided die is to say something like:

rolling a one : dice rolls :: 1 : 6

If we take 's' to be whatever number we associate with species S and 'g' to be whatever number we associate with genus G, then, all probabilities have the following analogical form:

S: G :: s : g

We take the relation of a species of event to its genus and say that this relation corresponds to the relation between one number and another. Note that I say 'all probabilities have this analogical form', not 'all things of this analogical form are probabilities'; the analogy is constrained by the axioms of probability. Rather, my point is that every probability can be put in the form of an analogy.

If A and B are species, and MRG is the 'maximal relevant genus' including them both (in what follows I am treating each line as simply separate from the others):

P(A) = [A : MRG :: a : mrg]
P(B) = [B : MRG :: b : mrg]
P(A&B) = [AB : MRG :: a&b : mrg]
P(A|B) = [(AB : MRG :: a&b : mrg) ::: (B : MRG :: b : mrg)]

(Having both the expanding colons ::: and the parentheses is redundant; they're just ways to group which analogies are being analogized to which.)

Bayes Theorem is then the following analogy (assuming I haven't made any errors, which I may have):

(((AB : MRG :: a&b : mrg) ::: (B : MRG :: b : mrg)) :::: ((AB : MRG :: a&b : mrg) ::: (A : MRG :: a : mrg))) ::::: (((A : MRG :: a : mrg) :::: (B : MRG :: b : mrg))

Or, as an alternative (again assuming that I haven't made any stupid mistakes, for which I will not vouch):

(((AB : MRG :: B : MRG) ::: (AB : MRG :: A : MRG)) :::: (A: MRG :: B : MRG))) ::::: (((a&b : mrg :: b : mrg) ::: (a&b : mrg :: a : mrg)) :::: (a : mrg :: b : mrg)))

By assuming the MRG as always given, and mrg thus = 1, this particular analogy could be simplified, of course, as something like

((AB : B :: AB : A) ::: (A : B)) :::: ((a&b : b :: a&b : a) ::: (a : b))

None of this is particularly interesting in itself. But if probabilities are analogies, then probabilistic reasoning can be treated as a form of analogical reasoning, which would have implications for how we understand analogical reasoning.

Poem a Day Bonus

Nursery Rhyme

Beat-up pick-up on the road,
Why'd you make the traffic slow?
The sign says seventy we may go,
but you go fifty and break the flow.

Poem a Day XVIII

Caught in the Rain

It rains. The puddles my poor shoes
cover with squish and Sunday blues,
and on my head drops dripping down
make merry with my freezing grief.
Pelted by both dark cloud and leaf,
I am a sailor who is drowned
and now has fish for lively crew.

The Polemicist Who Wouldn't Hurt a Fly

Today is the feast of St. Roberto Bellarmino, polemical theologian and Doctor of the Church. Even if you didn't know he was a Jesuit you could figure it out by the fact that so many legends, apocryphal stories, and outright falsehoods have collected around him.

David Hume has an interesting comment about Bellarmine in The Natural History of Religion:

Bellarmine patiently and humbly allowed the fleas and other odious vermin to prey upon him. "We shall have heaven," said he, "to reward us for our sufferings; but these poor creatures have nothing but the enjoyment of the present life."

Like a number of Hume's weirder claims this comes from Pierre Bayle, who as a Calvinist is not wholly the most reliable source for information about Catholic saints. The advantage of Bayle, though, is that he tells us where he's getting these things. Bayle is drawing the story from Jacopo Fuligatti's life of Bellarmine. Fuligatti gives three stories about insects and Bellarmine's patience: first, it's said he counted gnats and other minor vexations to be from God, so he wouldn't drive them away, but simply endured them patiently; second, that he was once bitten by insects at Mass and turned to the statue of Christ to pray; and third, that Cardinal Crescentius told the story that in Rome, where flies are a problem, he wouldn't drive them away, and when asked why said, "It is unjust to disturb those little creatures whose only paradise is the freedom of flying and landing where they wish." Bayle sums it up by saying that he was said to be so patient that he allowed flies and other small insects to be troublesome to him; and being a sarcastic Calvinist, he insists on explicitly saying that shooing away bugs is consistent with the teachings of Christ. It should not need to be said, but unfortunately probably does, that there is nothing whatsoever in Bellarmine to suggest that he would not agree to this; that the stories are second-hand (third-hand by the time we get to Bayle) and, except for the third, which alone is sourced, they are vague and generic moral tales about patience and suffering which have parallels elsewhere in hagiographical convention (the first story explicitly links the bugs to minor vexations of life; the second links them to hell); that if Bellarmine actually did make the comment in the third story, it might well have been at least half-joking, because Bellarmine was notoriously fond of jokes and puns; and that, if true, the whole thing may for all we know have grown out of one single incident in the Cardinal's nearly eighty years of life. Bayle is telling the story from a popular source, and he is telling it as scholarly gossip and with a little malice.

It's noteworthy how Hume, who now gives the story at fourth-hand, embellishes on Bayle's story. We move from gnats and flies to "fleas and other odious vermin." Conceivably the vague reference to biting insects in the second story could be taken as meaning fleas; but it's a different story than the one that involves the comment. Hume takes the figurative reference to paradise and turns the whole saying into a comment about the afterlife, which is actually crucial to how it is used in context. Either he is telling the story from memory -- which is possible -- or he is reworking it to make it fit its context, which is the difference between the nobility of Greek heroes (representing polytheism) and the abasement of Catholic saints (representing monotheism).

Many more people in the English-speaking world read Hume now than read Bayle, so it's Hume's story that has spread since; and you can regularly find people stating as fact that Bellarmine let fleas drink his blood because they can't go to heaven. The thing of it is, this is hardly unusual with Bellarmine; half the stories you come across about him are just like this one. Actually, this one is better than a lot of the stories that are told about him, since Bayle's version, at least, is not provably false and has (unlike most such stories) some sort of provenance.

Poem a Day XVII

Drinking Tea

Drinking tea
you find the way,
world within,
enlightenment,
since you stop,
savor moments,
let the world
go its own way,
with calm grace
fall into a warm cup
and let the taste
carry you.

Scent and taste
join together
to make peace,
to touch with joy,
to give hope;
the mind is calmed
in the cup
and is restored.
A small thing,
but the world is in it.
Here you may glimpse
happiness.

Drinking tea
you find the way,
journey home,
come to insight,
for the cup
reflects the mind
in stillness,
the liquid soothes,
steam rises
with the freshness of dawn:
for one moment
you are here.

Fortnightly Book, September 16

Cooper's art has some defects. In one place in 'Deerslayer,' and in the restricted space of two-thirds of a page, Cooper has scored 114 offences against literary art out of a possible 115. It breaks the record.

There are nineteen rules governing literary art in the domain of romantic fiction—some say twenty-two. In Deerslayer Cooper violated eighteen of them.

So says Mark Twain in Fenimore Cooper's Literary Offenses, which is in great measure an attack on James Fenimore Cooper's The Deerslayer, or The First Warpath, which is the next fortnightly book.

The Deerslayer is part of Cooper's Leatherstocking Pentalogy. Putting the books in the order published, they are:

The Pioneers (1823)
The Last of the Mohicans (1826)
The Prairie (1827)
The Pathfinder (1840)
The Deerslayer (1841)

However, in the order of internal chronology:

The Deerslayer (up to 1745)
The Last of the Mohicans (1757)
The Pathfinder (late 1750s)
The Pioneers (1793)
The Prairie (1804)

(There seems to be some doubt about the date of The Pathfinder. It's almost always listed as number three, but at least one source I've seen puts its internal date at 1756, which puts it before The Last of the Mohicans, which certainly occurs in 1757.) So The Deerslayer was the last book published, but is the prequel of them all. By this point, Cooper's reputation was solid, and it was an instant hit; two editions appeared virtually simultaneously both in the United States and in England, both claimed by their respective publishing houses as the actual first edition. A French and a German version followed shortly.

Cooper himself was an extaordinarily combative person. He wrote his first novel, Precaution, simply to prove that he could write one. He made many enemies in his lifetime, and was famously litigious, slapping a libel lawsuit on practically anyone who criticized him (he generally won).

He became known as America's version of Sir Walter Scott, which is probably what made him a target for Twain, and was one of the things that James Russell Lowell tweaked Cooper about in his notoroius A Fable for Critics:

But he need take no pains to convince us he's not
(As his enemies say) the American Scott.

His characterization also comes in for sharp criticism by Lowell:

All his other men-figures are clothes upon sticks
The derniere chemise of a man in a fix,
(As a captain besieged, when his garrison's small,
bets up caps upon poles to be seen o'er the wall;)
And the women he draws from one model don't vary,
All sappy as maples and flat as a prairie.

Nonetheless, this should not be the last word, because part of all this is that Lowell and Twain had to deal with the extraordinary length of Cooper's shadow. For many European authors and critics, British and continental alike, James Fenimore Cooper was the serious American author, the one on whom American claim to have literary talent primarily rested. Nor is the list of his fans small or slight in name: Balzac, Victor Hugo, Rudolf Drescher, Wilkie Collins, and more. With Cooper, American literature entered the world stage as a major contender. And, perhaps more importantly, the Leatherstocking Tales, of which The Deerslayer is the first, have continued to be read and enjoyed through generations.

Poem a Day XVI

Fruitions

Shall life be but a minor thing? Let it never be!
The world is not so mean, nor shall it ever be.

The fruit is one moment on the vine, then fades;
to seize it one must be keen, and thus shall it ever be.

The days go by with swiftness, but do not fail:
their fulfillments are always seen, and they shall forever be.

One must grasp them with full heart, hold them fast,
and all their glory glean; thus must it ever be.

In life is always some glorious end, like a dawn:
it has never been capped with a keen, nor shall it ever be.