Ramble, Part 60: Good Vibrations
Oct. 19th, 2008 04:41 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowJoseph Fourier was more a physicist than a mathematician (though the lines between the disciplines were less clear in his day than in ours); his mathematical papers, even his masterpiece Analytic Theory of Heat, were less than rigorous, and many of his claims proved to be false as stated. Nonetheless, with appropriate qualifications and restrictions, they could be made true, and their consequences have been enormous. Many important modern technologies rest on his work; fields of study ranging from physics to linguistics have drawn upon his insights; and - most importantly for our purposes - he had considerable, if indirect, impact on the foundations of mathematics.
There is a sense in which Fourier's ideas trace back to Pythagoras. According to legend, it was the Greek mathematician's discovery of the mathematical principles of harmony that launched him on the study of number theory. What he had found was that the sounds made by two vibrating strings sounded pleasant if the lengths of the strings were in the ratio of small integers: 2:1, 3:2, and so on. In mathematical terms, the sound waves produced are describable as, for example, sin x, sin 2x, sin 3x, and so on. Each of these is periodic; that is, they take the same values when x is increased by some specifiable amount - the function's period - and it is this rhythmic repetition that gives the sounds their purity of tone.
The functions sin x, sin 2x, and so on have different periods, but all of the periods divide the period of sin x, and so they repeat - possibly not for the first time - after that length of time. (Mathematicians usually scale things so that that period is 2π.) Fourier's investigations led him to consider all functions which repeat after an interval of 2π. He pointed out that, if you have any function defined on an interval of length 2π, you can turn it into a periodic function simply by fiat: define its value at x+2π to be the same as its value at x. He then came to the conclusion that any such function could be written in terms of the trigonometric functions; specifically, in the form
Series like these - trigonometric series - had been studied before; Fourier's boldness lay in declaring that every function could be written in this form. He turned out to be wrong on this, but a broad enough range of functions can be so written to make their study worthwhile. However, unlike power series, the functions definable by these "Fourier series" are not necessarily nice. They may be discontinuous; they may fail to be differentiable; they may misbehave in any of a variety of ways. This misbehavior forced mathematicians, once and for all, to confront the notion of "function" seriously, and generally to shore up the unsteady foundations on which calculus had been erected. I'll talk a bit about the ways this was done in the next Ramble or two.
I shouldn't end without mentioning the impact of Fourier's work on modern technology. Any sort of wave phenomenon - in particular, sound and light - is likely to be describable by a Fourier series, and it turns out to be relatively easy to find the coefficients an and bn. That makes it possible to describe a sound or a lightwave by a sequence of numbers, and this representation underlies most of our modern technology of audiovisual storage and reproduction - CDs, DVDs, and the like - as well as the enhancement techniques that allow us to "clean up" blurry or staticky recordings. For the purposes of the Ramble, the consequences for the foundations of mathematics are more important, but it's worth remembering that this isn't all ivory-tower stuff.
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There is a sense in which Fourier's ideas trace back to Pythagoras. According to legend, it was the Greek mathematician's discovery of the mathematical principles of harmony that launched him on the study of number theory. What he had found was that the sounds made by two vibrating strings sounded pleasant if the lengths of the strings were in the ratio of small integers: 2:1, 3:2, and so on. In mathematical terms, the sound waves produced are describable as, for example, sin x, sin 2x, sin 3x, and so on. Each of these is periodic; that is, they take the same values when x is increased by some specifiable amount - the function's period - and it is this rhythmic repetition that gives the sounds their purity of tone.
The functions sin x, sin 2x, and so on have different periods, but all of the periods divide the period of sin x, and so they repeat - possibly not for the first time - after that length of time. (Mathematicians usually scale things so that that period is 2π.) Fourier's investigations led him to consider all functions which repeat after an interval of 2π. He pointed out that, if you have any function defined on an interval of length 2π, you can turn it into a periodic function simply by fiat: define its value at x+2π to be the same as its value at x. He then came to the conclusion that any such function could be written in terms of the trigonometric functions; specifically, in the form
a0+a1 sin x+a2 sin 2x+...(The a0 is needed in case the function oscillates around some value other than 0; the cosine terms, in case the function isn't at that value when x=0.)
+b1 cos x+b2 cos 2x+....
Series like these - trigonometric series - had been studied before; Fourier's boldness lay in declaring that every function could be written in this form. He turned out to be wrong on this, but a broad enough range of functions can be so written to make their study worthwhile. However, unlike power series, the functions definable by these "Fourier series" are not necessarily nice. They may be discontinuous; they may fail to be differentiable; they may misbehave in any of a variety of ways. This misbehavior forced mathematicians, once and for all, to confront the notion of "function" seriously, and generally to shore up the unsteady foundations on which calculus had been erected. I'll talk a bit about the ways this was done in the next Ramble or two.
I shouldn't end without mentioning the impact of Fourier's work on modern technology. Any sort of wave phenomenon - in particular, sound and light - is likely to be describable by a Fourier series, and it turns out to be relatively easy to find the coefficients an and bn. That makes it possible to describe a sound or a lightwave by a sequence of numbers, and this representation underlies most of our modern technology of audiovisual storage and reproduction - CDs, DVDs, and the like - as well as the enhancement techniques that allow us to "clean up" blurry or staticky recordings. For the purposes of the Ramble, the consequences for the foundations of mathematics are more important, but it's worth remembering that this isn't all ivory-tower stuff.
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