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The author describes the old, pre-revolutionary way of doing mathematics as "relying on intuition and physical experience." In certain fields, such as algebra a
by thebear 14y ago
The author describes the old, pre-revolutionary way of doing mathematics as "relying on intuition and physical experience." In certain fields, such as algebra and number theory, the old way of doing it had another important trait: to prove the existence of something, such as the greatest common divisor or the unique factorization of a polynomial, one had to give an algorithm to construct it. The new, axiomatic way of thinking, by contrast, is content with showing that the assumption of non-existence leads to a contradiction. The rise of symbolic computation over the last few decades (Mathematica, Maple, etc.) can therefore be seen as a comeback of the old style of mathematics. I call it Kronecker's revenge. With respect to math education, well, considering the importance of computing, I am inclined to think that we should pay more attention to the 19th century way of thinking rather than less.
- cwzwarich 14y agoMost mathematicians are sympathetic to utilitarian constructive mathematics, i.e. finding constructive proofs for the sake of actually computing the results in applications, as opposed to some philosophical reason. However, nonconstructive mathematics (or at least mathematics that cares little for avoiding nonconstructive arguments) will probably continue to dominate mathematical research, since many major research areas now depend heavily on its results as a basis for a conceptual framework.
- defrost 14y agoYou can get a pretty good feel for the practical level of sympathy by popping down to your local math library and looking at the checkout histories of the works of Brouwer [1] and Bishop [2]. [1] http://plato.stanford.edu/entries/brouwer http://plato.stanford.edu/entries/brouwer [2] http://en.wikipedia.org/wiki/Errett_Bishop http://en.wikipedia.org/wiki/Errett_Bishop 'Sparse' would be an understatement & poor old Errett got such wonderfully backhanded reviews as: > "Even those who are not willing to accept Bishop's basic philosophy must be impressed with the great analytical power displayed in his work." and > Bishop's historical commentary is "more vigorous than accurate".
- cwzwarich 14y agoWell, both Brouwer and Bishop fall into the category of constructivism for philosophical reasons. As I mentioned, this doesn't garner much sympathy from ordinary mathematicians. Brouwer and Bishop wrote against the practice of classical mathematics with considerable vitriol, as opposed to proposing constructive mathematics as an additional lens for viewing classical results. In the era of widespread use of computers to perform calculations, the advantages of this latter point of view are obvious.
- defrost 14y agoThat's a fair assessment. Still it's a shame their work more or less petered out due to politics, personality and philosophical disdain, had they played better with others and taken a slightly different path they could well have been better regarded than they are now.
- Filter 14y agoI don't understand your comment. The entire article is about why "relying on intuition and physical experience" just wasn't good enough -- how the difficulties inherent in 19th-century style mathematics forced mathematicians to evolve towards a modern modern style. Algebra and number theory are both heavily reliant on modern mathematical thinking. Abstract algebra is practically the defining example of it. How does one create Mathematica and Maple without modern mathematical thinking? Indeed, formal logic is rather closely involved in computing generally. How does a precollege mathematics system that is based on a 19th century style of mathematics pay more attention to 19th century style mathematics? I find your confident dismissal of modern mathematics to be vexing and oddly misaimed.
- chipsy 14y agoI beg to differ. As software itself has grown, it's developed an enormous need for axiomatic methods because of the benefits it can confer for software reliability. Simply, we want to prove that a program is wrong at the earliest stage possible - "intuition and physical experience" is also known as "the program failed in production." Although we've invented a number of algorithmic methods to find bugs in software, the long-term trend is to find "better axioms" to define the program against, which reduce both the code size and complexity - that's the goal otherwise known as "programming language design." We don't want people to code in 19th century fashion, because it sucks.