math in astro

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Mathematics as a Language

Blake wrote: "I have heard many People say, 'Give me the Ideas. It is no matter what Words you put them into.'" To this he replies, "Ideas cannot be Given but in their minutely Appropriate Words." - William Blake (quoted by J. Newman, The World of Mathematics, 1956)

This opinion is seconded by Bertrand Russell. Here is what he says in his Autobiography about meeting G.Peano at an International Congress on Philosophy in 1900:

The Congress was a turning point in my intellectual life, because I met there Peano. I already knew him by name and had seen some of his work, but had not taken the trouble to master his notation. In discussions at the Congress I observed that he was always more precise than anyone else, and that he invariably got the better of any argument upon which he embarked. As the days went by, I decided that this must be owing to his mathematical logic. I therefore got him give me all his works, and as soon as the Congress was over I retired to Fernhurst to study quietly every word written by him and his disciples. It became clear to me that his notation afforded an instrument of logical analysis such as I had been seeking for years, and that by studying him I was acquiring a new and powerful technique for the work that I had long wanted to do.

When I think of the development of Mathematics over the last 2500 years, I am less surprised that early mathematicians left lasting results than that, given the tools they possessed, they achieved anything at all that could have lived through centuries. Just think of it. Zero gained widespread use only in the last millennium. Systematic introduction of modern algebraic notations began only in the sixteenth century and is most often associated with the French mathematician François Viète (1540-1603). René Descartes (1596-1650) was first to use letters at the end of the alphabet for unknowns. He also introduced the power notations: x², x³. The sign of equality (two equal parallel strokes) has been invented by Robert Recorde (c. 1510-1558) in his The whetstone of witte (London, 1557):

I will sette as I doe often in worke use, a paire of paralleles, or Gemowe lines of one lengthe, thus: =, bicause noe.2. thynges, can be moare equalle.

To help you appreciate the expressive power of the modern mathematical language, and as a tribute to those who achieved so much without it, I collected a few samples of (original but translated) formulation of theorems and their equivalents in modern math language.