A brief note on Renaissance algebra

While investigating the background to Cardano's use of ‘capitulum’ to refer to a type or category of algebraic equation, I came across an interesting passage from a letter written by Regiomontanus in 1471:

Sunt enim qui se iactant ampliorem habere artem algebricam quam in sex capitulis vulgatissimis traditur. Sed ipsi profecto ignorant hanc artem ad cubos, census censuum, atque ulteriores potentias extendi non posse nisi prius geometria solidorum equipollentium edatur. Quemadmodum enim tria capitula composita superficierum equipollentiis nituntur, ita novum artis additamentum ex commutatione solidorum hauriatur [? l. hauriri] necesse est.

Menso Folkerts, to whom all historians of medieval and Renaissance mathematics must be grateful, translates this as follows (1996):

Many flatter themselves that they understand the higher (ampliorem) algebra from the six standard forms. But they completely ignore the fact that this art cannot be extended to cubes or to fourth and higher powers, unless the geometry of solids of equal volume is first treated. Just as the three composed forms (of quadratic equations) are proved by means of figures of equal area, so the new extension of the art must be based upon the transformation of solids.

Which is puzzling, because the Latin is conspicuously rather different:

For there are those who boast that they have a more extensive algebraic art than is handed down in the six most commonly known capitula. But these people clearly do not know that this art cannot be extended to cubes, squares of squares, and further powers unless the geometry of equivalent solids is provided first. For just as the three compound capitula rely on equivalences between surfaces, so the new addition to the art must be drawn from the transformation of solids.

At any rate, the sex capitula in question are from al-Khwārizmī's Algebra, and it might be helpful to list them here in modern notation:

Simple	Compound
ax² = bx	ax² + bx = c
ax² = c	ax² + c = bx
bx = c	bx + c = ax²

Must have led a sheltered life

I have previously queried Peake's and Coleridge's use of ‘must have V-en’ in the main clause of past-tensed conditionals, e.g. ‘the Ledge at the bottom was [so] exceedingly narrow, that if I dropt down upon it I must of necessity have fallen backwards & of course killed myself.’

After a quick search for ‘must surely have’ in the British National Corpus, I'm reluctantly coming round to the idea that I have indeed just led a sheltered life when it comes to this construction.  Witness:

‘But for the Munitions of War Act of July 1915 which enabled the Board of Trade if necessary to impose arbitrated settlements on unwilling employers, the union's policy of patriotic co-operation must surely have failed’ – Arthur Marsh & Victoria Ryan, The Seamen: A History of the National Union of Seamen (1989).

‘Science and technology must surely have progressed in a different way if these principles had been embraced from the start’ – Storm Constantine, Hermetech (1991).

‘Had it not been for their masks, the Phantasms' faces must surely have blistered – a gulf of rising furnace-air yawned beyond that hatch’ – Ian Watson, Warhammer 40,000: Space Marine (1993).

‘Had it not been for the psychic tracer, they must surely have lost themselves in the labyrinthine entrails of what was not one vessel but many’ – Ian Watson, Warhammer 40,000: Inquisitor (1993).

I don't know whether to make anything of the fact that three of these are from the trashier end of sci-fi (and that two are even from the same author) even though the search was conducted over a wide variety of sources.  If I ever find the time, I'll try wading through the vastly more extensive search results for ‘must have’ simpliciter.

Samuel Taylor Coleridge's use of ‘must have’

In an earlier post I considered Mervyn Peake's use of the modal must in phase-modified clauses.  The ‘must have V-en’ construction usually encodes an inference as to fact (‘What happened to my agent?  Bastard must have died’), but Peake also uses it where I'd expect the modal would, e.g. ‘had he not made this same journey through the darkness a thousand times he must surely have lost himself in the night.’

Well, I've just stumbled across the same phenomenon in Coleridge's description of his death-defying descent of Broad Stand in 1802:

‘the Ledge at the bottom was [so] exceedingly narrow, that if I dropt down upon it I must of necessity have fallen backwards & of course killed myself.’

A rummage turns up another example from ch. 14 of his Biographia Literaria (1817), discussing the turn-of-the-century Lyrical Ballads:

‘Had Mr Wordsworth's poems been the silly, the childish things, which they were for a long time described as being; had they been really distinguished from the compositions of other poets merely by meanness of language and inanity of thought; had they indeed contained nothing more than what is found in the parodies and pretended imitations of them; they must have sunk at once, a dead weight, into the slough of oblivion, and have dragged the preface along with them.’

And finally, again from the Biographia, in Satyrane's Letters no. 1:

‘And after dinner, when he was again flushed with wine, every quarter of an hour or perhaps oftener he would shout out to the Swede, “Ho!  Nobility, go—do such a thing!  Mr Nobility!—tell the gentlemen such a story,” and so forth, with an insolence which must have excited disgust and detestation, if his vulgar rants on the sacred rights of equality, joined to his wild havoc of general grammar no less than of the English language, had not rendered it so irresistibly laughable.’

Maybe I've just led a sheltered life when it comes to this construction.

Codex on Aristotle's Doctrine of the Mean

When I first studied the Nicomachean Ethics, I did so with the help of Urmson's wonderful book Aristotle's Ethics (1988), which surprised me with its strictures against interpreting the doctrine of the mean as a simple thesis of moderation.  Surely, I thought, no one could have read that into II.6.  Now, six and a half years later, I've had a glimpse of what Urmson meant – in the pages of an Inspector Morse novel:

“Morse skipped his way along [the report].  ‘… would suggest a period of between 72–120 hours before the body was discovered.  Any greater precision about these time limits is precluded in this case…’  As in all cases you ever have, muttered Morse.  He had never ceased to wonder why, with the staggering advances in medical science, all pronouncements concerning times of death remained so disconcertingly vague.  For that was the real question: when had Quinn died?  If Aristotle could be believed (why not?) the truth would probably lie somewhere in the middle: 94 [sic] hours, say.”

By coincidence, I was only reading this because of a sloppy piece of academia.  In Twentieth-Century Crime Fiction (2005), Lee Horsley, comparing the relationship between the authors and readers of detective fiction to that between the setters and solvers of cryptic crosswords, promised me that “Colin Dexter also gestures towards parallels between the mystery story and the cryptic crossword in The Silent World of Nicholas Quinn (1977), which involves Inspector Morse with a suspect who is a crossword-setter called Daedalus.”  Alas, this turns out to be an embellishment from Julian Mitchell's TV adaptation (1987), which made Ogleby a more interesting character.

The Myth of the Tenseless (I)

Philosophers are prone to describe mathematical and conceptual statements (e.g. that two plus two makes four and that red is a colour) as ‘tenseless’ on the grounds that time is irrelevant to mathematics and conceptual analysis.  But of course our language does not know which topics are sensitive to time, so on this theory the English present verb form must be systematically ambiguous between present-tensed and tenseless interpretations.  The extravagance of such a theory is astounding.  A much simpler theory would be that English requires every message encoded with it to have a tense, even though sophisticated people understand that in mathematics and conceptual analysis the tense chosen is usually unimportant.  (Cf. Dudman, ‘Conditional Interpretations of If-Sentences’ (1984), §48.)

Sadly, cavalier attitudes towards language are endemic in modern philosophy.  This is the first in a series of posts exposing the problems that arise when people try to foist tenseless statements on English.

According to McArthur's introductory textbook Tense Logic (1976), the statement that it always rains in Boston ‘do[es] not convey any positional information concerning the temporal relation of the speaker and the event or state of affairs depicted’, and its truth-value is independent of the time of its utterance.  Sed contra: any speaker of English would understand the reply ‘I know, it's awful – it wasn't like this 50 years ago.’  This can only be because the habitual statement that it always rains in Boston is as present-tensed as it looks.

Confusion about the Future (II)

The lack of a future tense in Romance and Germanic languages is not widely known; on the tacit assumption that there must be one somewhere, people tend to seize on verb forms and modals whose function is to signal that the speaker is making not a statement of fact but a judgement in the absence of the relevant information – a function that makes them useful for discussing the future, of course, but also for making conjectures about present and past facts.

I shall not explain this further here; from the above remarks, it is obvious what is going on when, for instance, a Dutch blogger working in Cameroon writes of a friend whose time there has recently come to an end:  ‘Ze zal nu al thuis zijn, hopelijk heeft ze het niet te koud’ (‘she will already be home by now, hopefully she is not too cold’).

Instead, this is the second in a series of posts exposing the confusion that ensues when people fail to recognize that these Romance verb forms and Germanic modals do not intrinsically encode futurity.

This time my target is the common theory that the ‘future tense’, when not expressing the future, expresses probability as opposed to certainty.  This admits of a homely and a sophisticated refutation.

Firstly, there is no obvious uncertainty when a French couple, reporting on their round-the-world trip, write:  ‘On sera sans doute passé trop vite pour creuser et trouver des endroits plus sauvages, mais il y avait quand même un peu trop de tourisme a notre goût en Thailande’ (‘we will no doubt have passed through too quickly to delve deeper and find wilder places, but even so there was a bit too much tourism in Thailand for our tastes’).  Other examples abound.

Secondly, if non-future ‘will’ expressed probability, people would use it in situations where in fact they do not.  For instance, if you have just tossed two coins without looking at them, you won't assent to the claim that they won't both have come up heads – whereas you will assent to the claim that they probably won't both have come up heads.  What's worse, the same goes for any probability short of 1.

Both difficulties are avoided by the theory that what ‘zal’, ‘will’ and ‘sera’ have in common is that they signal a judgement in the absence of the relevant information – for it is of course possible to make a certain judgement even in the absence of concrete information.

Confusion about the Future (I)

The lack of a future tense in Romance and Germanic languages is not widely known; on the tacit assumption that there must be one somewhere, people tend to seize on verb forms and modals whose function is to signal that the speaker is making not a statement of fact but a judgement in the absence of the relevant information – a function that makes them useful for discussing the future, of course, but also for making conjectures about present and past facts.

I shall not explain this further here; from the above remarks, it is obvious what is going on when, for instance, an Italian, commenting on Obama’s denial of reports that he uses a Zune instead of an iPod, writes:  ‘E comunque lo Zune sarà già stato sepolto nell’Area 51’ (‘and anyway the Zune will already have been buried in Area 51’).

Instead, this is the first in a series of posts exposing the confusion that ensues when people fail to recognize that these Romance verb forms and Germanic modals do not intrinsically encode futurity.

My first exhibit is from the Bristol Classical Press edition of Borges’ Ficciones.  The third sentence of ‘Tlön, Uqbar, Orbis Tertius’ (1940) is ‘El hecho se produjo hará unos cinco años.’  Translating word for word, this reads: ‘The event took place it will make some five years.’  The editors explain that ‘hace [‘it makes’] would be more common but the future indicates a certain vagueness: ‘some four or five years ago’.’  Of course, the future is nowhere in sight; this is simply a judgement in the absence of the relevant information.  A more faithful translation would be ‘it will be some five years ago that the event took place’, or ‘the event will have taken place some five years ago’.

There's a close parallel in William Morris’ The Water of the Wondrous Isles (1897), part 6, ch. 7:  “But Birdalone spake, hardening her heart thereto for very need: ‘Belike then there is a change of days here, for when I last knew of the land there was little peace therein.’  ‘And that will not be so long agone,’ said a townsman, smiling, ‘for I doubt we should see no grey hair in thine head if thy sallet were off it.’  Birdalone reddened: ‘It will be some five years agone,’ said she.”

Here Borges’ editors would presumably explain that ‘it is’ would be more common but the future indicates a certain vagueness, so that what Birdalone meant was that it was some four or five years ago.  But while the latter is obviously something that she could have said if she had wanted to, it is equally obviously not what she actually said.

Dudman's Fourth Category

V. H. Dudman has identified three major categories of messages expressible in English if-sentences: (1) hypothetical, (2) habitual, and (3) conditional.  These categories may be exemplified in turn by the natural interpretations of the following sentences:

(i)  If Socrates was a man, he was mortal.
(ii)  If Baby cries, we beat him.
(iii)  If Oswald doesn't shoot Kennedy, someone else will.

In §4.7 of ‘Antecedents and Consequents’ (1986), Dudman discerns an egregious fourth category, (4), which he exemplifies with:

(iv)  If Tom was fat, his sister was immense.

Grammatically, the difference between (1) and (4) is supposed to be that (4) forces the if-string to prefix the independent sentence, whereas (1) allows freer amalgamation.  Semantically, in category (4) the ‘common trait seems to be apposition: while the independent message is affirmed outright, the 'dependent' message is there for the sake of analogy, pointing up a similarity or contrast, perhaps suggesting a comparison.’  In (1), by contrast, the independent message is normally affirmed only on the given hypothesis.

Dudman's sample sentence (iv) is unhappy, as it is easily interpreted as encoding a hypothetical.  (Compare: ‘If Kate Moss is fat, I'm morbidly obese.’)  To facilitate discussion of category (4), we need to use clearer examples.  I am compiling a list of them here.

Mervyn Peake's use of ‘must have’

I've just finished Titus Alone, the black sheep of the Gormenghast trilogy.  Peake's command of the language is impressive, despite the erratic punctuation, so I was especially intrigued by his peculiar use of the modal must in phase-modified clauses, that is, his use of ‘must have V-en’.  This form of words usually encodes an inference as to fact, as in Withnail's ‘What happened to my agent?  Bastard must have died.’  But now look at these (pagination from 1998 Vintage editions):

Titus Groan
‘The sky was overcast and had he not made this same journey through the darkness a thousand times he must surely have lost himself in the night.’  (The Library, p. 206)

‘Her fever had raged, and but for the care with which the old man watched over her she must surely have died.’  (Farewell, p. 357)

‘Had the flesh, the fibres, and the bones of the chef and those of Mr Flay been conjured away and away down that dark corridor leaving only their four eyes suspended in mid-air outside the Earl's door, then, surely, they must have reddened to the hue of Mars, reddened and smouldered, and at last broken into flame, so intense was their hatred – broken into flame and circled about one another in ever-narrowing gyres and in swifter and yet swifter flight until, merged into one sizzling globe of ire they must surely have fled, the four in one, leaving a trail of blood behind them in the cold grey air of the corridor, until, screaming as they fly beneath innumerable arches and down the endless passageways of Gormenghast, they found their eyeless bodies once again, and re-entrenched themselves in startled sockets.’  (Early One Morning, p. 365)

Gormenghast
‘Had he thought himself awake he must surely have pursued, however faint his hope of overtaking the slender creature.’  (§19, p. 127)

‘Had his exile in the woods not inured him to loneliness, then he must surely have found these long days insupportable.’  (§45, p. 292)

‘For Titus to have seen a tenth of it must have taken the edge, not off his wonder or speculation, but off the shock of pleasure that he was finally to receive when evening came.’  (§50.i, pp. 312f.)

‘Had she as a girl been naturally joyous yet all that had befallen her must surely have driven away the bright birds, one by one, from her breast.’  (§75, p. 445)

Titus Alone
‘This gun-boom had come just in time, for had it been delayed a moment longer Titus must surely have been grabbed and questioned.’  (§20, p. 35)

‘Some, in his place, must surely have seen battle or the great jaws of carnivores or landscapes of infinite mystery and invention complete with bridges and deep chasms, forests and craters.’  (§49, p. 113)

‘Had it not been that he took up the rearguard station his facial fatuities must surely have maddened his two companions.’  (§78, p. 180)

This list may not be comprehensive, but it'll do.  In each case I find that must jars, and my ear cries out for would instead.  I don't think that this is merely a matter of stylistic preference, but I can't be sure, so I would be interested to hear of similar examples from elsewhere.

Incipit

I've started this blog as a repository for notes on academic matters outside the remit of Speculum Stultorum.  I was pleasantly surprised to find another apt title going begging.