A view of Newton’s mistakes
“The greatest of all mathematicians, those who have discovered the greatest quantities of mathematical truths, are also those who have published the greatest numbers of lacunary proofs, insufficiently qualified assertions, and flat mistakes. By attempting to make natural philosophy into a part of mathematics, Newton relinquished the diplomatic immunity granted to nonmathematical philsophers, chemists, psychologists, etc., and entered the area where an error is an error even if it is Newton’s error; in fact, all the more so because it is Newton’s error.
“The mistakes made by a great mathematician are of two kinds: first, trivial slips that anyone can correct, and, second, titanic failures reflecting the scale of the struggle wich the great mathematician waged. Failures of this latter kind are as important as successes, for they give rise to major discoveries by other mathematicians.”*
This is an interesting quotation unto itself, but it makes me think of how Eric Temple Bell characterized Leibniz’ vision of a calculus ratiocinator, a mathematical logic. Instead of arguing in words, we would say, “Gentlemen, let us calculate,” i.e., use mathematical logic to determine the truth or falsehood of what is under discussion.
Under Stephen Wolfram’s New Kind of Science empirical exploration of mathematics via computer, this translates into “Ladies and gentlemen, let us simulate.” In other words, decide truth or falsehood via a computer simulation.
*C. Truesdell, “Reactions of Late Baroque Mechanics to Success, Conjecture, Error, and Failure in Newton’s Principia,” in The Texas Quarterly, The Annus Mirabilis of Sir Isaac Newton Tricentennial Celebration, Vol X, No 3 (Autumn 1967).
© 2009 Kristen W. Carlson
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Who invented calculus?.Newton or Leibniz?.Did they reach their conclusions independently of each other?.
Comment by charleyjk4  September 18, 2009 
From my reading there is no doubt they were independent, but since Leibniz published first, Leibniz and his friends suspected Newton of poaching. The Dictionary of Scientific Biography articles on Newton and Leibniz give the details. Today we use Leibniz’ notation for the derivative and integral, which, aside from the method and proof of calculus, is itself a significant innovation. Interestingly, we use Newton’s notation for and interpretation of fractional and negative exponents (great primary source article in James R. Newman, The World of Mathematics The World of Mathematics
).
And that leads me back to Wolfram, whose abbreviated operators in the Mathematica programming language are a great modern innovation in mathematical notation. For those interested, my talk on a new Mathematica programming syntax is here: A New Mathematica Programming Style
Comment by kristencarlson  September 18, 2009 