Is God a Mathematician_ - Mario Livio [70]
Imagine then a business talk between a merchant greedy for foreign trade but desperately afraid of being shipwrecked or eaten by savages, and a skipper greedy for cargo and passengers. The captain answers the merchant that his goods will be perfectly safe, and himself equally so if he accompanies them. But the merchant, with his head full of the adventures of Jonah, St. Paul, Odysseus, and Robinson Crusoe, dares not venture. Their conversation will be like this:
Captain: Come! I will bet you umpteen pounds that if you sail with me you will be alive and well this day a year.
Merchant: But if I take the bet I shall be betting you that sum that I shall die within the year.
Captain: Why not if you lose the bet, as you certainly will?
Merchant: But if I am drowned you will be drowned too; and then what becomes of our bet?
Captain: True. But I will find you a landsman who will make the bet with your wife and family.
Merchant: That alters the case of course; but what about my cargo?
Captain: Pooh! The bet can be on the cargo as well. Or two bets: one on your life, the other on the cargo. Both will be safe, I assure you. Nothing will happen; and you will see all the wonders that are to be seen abroad.
Merchant: But if I and my goods get through safely I shall have to pay you the value of my life and of the goods into the bargain. If I am not drowned I shall be ruined.
Captain: That also is very true. But there is not so much for me in it as you think. If you are drowned I shall be drowned first; for I must be the last man to leave the sinking ship. Still, let me persuade you to venture. I will make the bet ten to one. Will that tempt you?
Merchant: Oh, in that case—
The captain has discovered insurance just as the goldsmiths discovered banking.
For someone such as Shaw, who complained that during his education “not a word was said to us about the meaning or utility of mathematics,” this humorous account of the “history” of the mathematics of insurance is quite remarkable.
With the exception of Shaw’s text, we have so far followed the development of some branches of mathematics more or less through the eyes of practicing mathematicians. To these individuals, and indeed to many rationalist philosophers such as Spinoza, Platonism was obvious. There was no question that mathematical truths existed in their own world and that the human mind could access these verities without any observation, solely through the faculty of reason. The first signs of a potential gap between the perception of Euclidean geometry as a collection of universal truths and other branches of mathematics were uncovered by the Irish philosopher George Berkeley, Bishop of Cloyne (1685–1753). In a pamphlet entitled The Analyst; Or a Discourse Addressed to An Infidel Mathematician (the latter presumed to be Edmond Halley), Berkeley criticized the very foundations of the fields of calculus and analysis, as introduced by Newton (in Principia) and Leibniz. In particular, Berkeley demonstrated that Newton’s concept of “fluxions,” or instantaneous rates of change, was far from being rigorously defined, which in Berkeley’s mind was sufficient to cast doubt on the entire discipline:
The method of fluxions is the general key, by help whereof the modern Mathematicians unlock the secrets of Geometry, and consequently of Nature…But whether this Method be clear or obscure, consistent or repugnant, demonstrative or precarious, as I shall inquire with the utmost impartiality, so I submit my inquiry to your own Judgement, and that of every candid Reader.
Berkeley certainly had a point, and the fact is that a fully consistent theory of analysis was only formulated in the 1960s. But mathematics was about to experience a more dramatic crisis in the nineteenth century.
CHAPTER 6
GEOMETERS: