Reasoning with the Infinite: From the Closed World to the Mathematical
Universe by Michel Blay, translated by M.B. Debevoise, University of Chicago
Press, 拢23.95/$48.00, ISBN 0226058344
WE CAN measure the velocity of a planet through the Solar System or of a
cannon ball across a battlefield. But we can only do so because the concept of
velocity was 鈥渋nvented鈥�. And this happened in the 17th century, when scientists
such as Galileo and Descartes gradually developed a new mechanistic philosophy
of nature.
One of Michael Blay鈥檚 many fine achievements in Reasoning with the
Infiniteis to make us realise how velocity, and later instantaneous
velocity, came to play a vital part in the development of a rigorous
mathematical science of motion鈥攖he major triumph of the scientific
revolution. From this gradual mathematisation of motion grew the belief that the
whole of physical reality could be subdued under a mathematical umbrella.
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Central to Blay鈥檚 tale is the problem of infinity鈥攐r more particularly,
of the infinitesimal. From the time of the Greek philosopher Zeno, the issue of
infinities and infinitesimals had been central to the discourse about motion.
But the paradoxes raised by the philosophical logic which showed that Zeno鈥檚
arrow could never reach its target stymied a mathematical science of motion.
Theologians joined philosophers in believing the infinite was
inexplicable鈥攚hich proved an immense psychological barrier to new ideas.
Blay shows how a barrage of new mathematical techniques, culminating with
Leibnitz鈥檚 and/or Newton鈥檚 invention of the differential calculus, overcame this
resistance.
This stunning success came at a price. With the new techniques, the Galilean
project of mathematising the world moved from the firm foundations of geometry
to the dubious sands of differential calculus鈥攚hose meaning remained
shrouded in mystery. For while infinitesimals made calculus work, their
ontological status remained dubious at best: no-one understood what they
represented. But, as Blay notes, this has set the tone for physics ever
since.