The King of Infinite Space Read Free

recondite, invisible for this reason to all of the ancient civilizations but the Greek.
    The men of the ancient Near East no doubt knew what arguments were. They had so many of them. What they knew, they knew imperfectly. They lacked words to make clear the distinctions that they sensed. Why assess an argument when it was so much easier to end it by either violence or indifference? This point of view has never completely lost favor. It was the Greeks who did the assessment and forced the very idea of an inference into consciousness, asking patiently for an account of its nature, the way it controlled the movement of the mind, and where in the catalog of human powers it belonged.
    At roughly the same time that Euclid composed the Elements , Aristotle provided a subtle and refined analysis of syllogistic inference, the pattern in argument that takes Socrates—and the rest of us, alas—to his death by virtue of the fact that he is a man and we are mortal. Born in 384 BC and dying in 322 BC (another victim of his own syllogism), Aristotle might conceivably have known Euclid when Euclid was still a young man, perhaps even palpating his togaed shoulder. Far too little is known about the circumstances of Euclid’s life to say just whose hand he might have shaken. The two men worked hand in hand all the same.
    A RGUMENTS , A RISTOTLE ARGUED , may be divided into those that are good and those that are not. In the syllogism, two premises resolve themselves in one conclusion:
    All dogs are mammals.
    All mammals are animals.
    All dogs are animals.
    Good.
    No fish are dogs.
    No dogs can fly.
    All fish can fly.
    Bad.
    Any dog who has not lost something still has it.
    No dog has lost a fifth foot.
    All dogs have five feet.
    Shame.
    As these examples might indicate, good arguments are good by virtue of their form and not their content. The logician is indifferent to the distinction between all dogs are mammals and all men are mortal ; both cases are swallowed whole by all A’s are B. This is the Aristotelian insight, and logicians have accepted it ever since. The conclusion of a valid argument is entrained by its premises. Truth plays an ancillary role. If the premises of a valid argument are true, then their conclusion must be true, but whether they are true is a matter on which the logician has little to say; an argument may be good even though its premises are false, and bad even though its premises are true.
    It is tempting to imagine a fraternal give-and-take between Euclid and Aristotle, Euclid taking, Aristotle giving, with Euclid advancing proofs and arguments that Aristotle had antecedently assessed and classified. This is not quite so. The Elements is a work of great logical sophistication, but it is not a work of logical self-consciousness. Euclid’s subject is geometry, his business is proof, and Euclid was not a mathematician disposed to step back to catch himself in the act of stepping back. That his arguments were valid, he had no doubt, but in the question of what made them valid, he had no interest.
    Euclid often made use of arguments that Aristotle had not analyzed properly or analyzed at all. If the natural numbers progress by 1, then there is no natural number between 3 and 4. The natural numbers do progress by 1. So there is no natural number between 3 and 4. The inference proceeds by the stately music of modus ponens. No syllogism is involved, just the straightforward play between propositions and their particles— if, then, and. Euclid is especially fond of reaching his conclusions by demonstrating that a given proposition leads to a contradiction, and so must be rejected. In Euclid’s hands, this style of reasoning often becomes a torpedo.
    There remains the matter of the distinction between an axiomatic system and an argument.
    There is none.
    An argument is a small axiomatic system, and an axiomatic system is a large argument.

Chapter III
COMMON BELIEFS
    La dernière démarche de la