A Quote by Stephen Smale
An announcement of [Christopher] Zeeman's lecture at Northwestern University in the spring of 1977 contains a quote describing catastrophe theory as the most important development in mathematics since the invention of calculus 300 years ago.
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Arithmetic starts with the integers and proceeds by successively enlarging the number system by rational and negative numbers, irrational numbers, etc... But the next quite logical step after the reals, namely the introduction of infinitesimals, has simply been omitted. I think, in coming centuries it will be considered a great oddity in the history of mathematics that the first exact theory of infinitesimals was developed 300 years after the invention of the differential calculus.
Even the simplest calculation in the purest mathematics can have terrible consequences. Without the invention of the infinitesimal calculus most of our technology would have been impossible. Should we say therefore that calculus is bad?
The most important single thing about string theory is that it's a highly mathematical theory, and the mathematics holds together in a very tight and consistent way. It contains in its basic structure both quantum mechanics and the theory of gravity. That's big news.
Catastrophe Theory is-quite likely-the first coherent attempt (since Aristotelian logic) to give a theory on analogy. When narrow-minded scientists object to Catastrophe Theory that it gives no more than analogies, or metaphors, they do not realise that they are stating the proper aim of Catastrophe Theory, which is to classify all possible types of analogous situations.
The Internet is the most important single development in the history of human communication since the invention of call waiting.
I became a member of the faculty at Northwestern University in 1965 but did not complete my thesis until two years later at a graduate ceremony at which Carnegie Institute of Technology became Carnegie-Mellon University. At Northwestern, I was mentored by the 'three Bobs:' Robert Eisner, Robert Strotz and Robert Clower.
With an absurd oversimplification, the "invention" of calculus [method in mathematics] is sometimes ascribed to two men, Newton and Leibniz.
I happen to think that computers are the most important thing to happen to musicians since the invention of cat-gut which was a long time ago.
The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics; and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.
Adjusted for inflation, somebody going to college today to a state university, is paying about 300 percent of what her mom or dad did just 30 years ago.
Mathematics is ordinarily considered as producing precise and dependable results; but in the stock market the more elaborate and abstruse the mathematics the more uncertain and speculative are the conclusions we draw there from. Whenever calculus is brought in, or higher algebra, you could take it as a warning that the operator was trying to substitute theory for experience, and usually also to give to speculation the deceptive guise of investment.
We have all been brought up with an ethical system of 2,000 years ago, an industrial-managerial system of 200-300 years ago, a statecraft system of 200 years ago, and so on. None of this is working very well for the requirements of a time as complex and variegated as our own. So we stand shuttering at the threshold, with no clear map.
After years of finding mathematics easy, I finally reached integral calculus and came up against a barrier. I realized that this was as far as I could go, and to this day I have never successfully gone beyond it in any but the most superficial way.
Yes, business really does change. 400 years ago, corporations were formed by royal decree. 300 years ago, many countries were powered by slave labour, or its closest moral equivalent. 200 years ago, debtors didn't go bankrupt, they went to prison. 100 years ago - well, business is largely the same as it was a century ago. And that's exactly the problem. Business hasn't changed, but today's array of tectonic global shocks demands a different, radically better kind of business. Yesterday's corporations visibly cannot meet today's economic challenges.
I attended schools in Seattle through the University of Washington, from which I was graduated in 1931. I spent the next year at Northwestern University.
I have been personally victimized by organized disruption of a public lecture on a university campus - at the University of North Carolina at Chapel Hill, Michigan State University, and Rhode Island's Providence College, to name only a few.
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