Ramble, Part 61: Take It to the Limit
Nov. 13th, 2008 10:59 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowThe job of repairing the shaky foundations of analysis fell to the hands of two mathematicians, Augustin Cauchy and Karl Weierstrass. It was an extended process, and it would take a long time to detail its erratic progress; here, I'll simply discuss in outline the final result.
At the inception of calculus, Newton wrote of "prime and ultimate ratios" - rates of change of a variable y at the very first or very last instant; but the notion of an instant is antithetical to the vision of smoothly flowing quantities ("fluents", as Newton called them). Leibniz thought in terms of infinitesimals: change the independent variable x by an infinitesimal amount dx, and the dependent y will change, likewise, by infinitesimal dy, and the ratio dy/dx turns out to be an ordinary real number. This, too, is rife with problems, as Leibniz treats infinitesimals as something hybrid, like zero (so small that their squares may be regarded as zero) but not like zero (since it is possible to divide by them)1.
It had long been realized that some sort of notion of limit was needed, and a vague formulation was already in use by the end of the eighteenth century. It seems to have been Cauchy who recognized a way of rigorizing the notion.
Let x and y be two variables, somehow related. We need to explain what it means to say that, as x approaches a particular value c, y must approach some limit L. Cauchy saw that what were needed were statements of the following form: if x is, say, within .01 of c, then y is within .001 of L. More generally, y can be made as close to L as desired by requiring x to be close enough to c. How do we make these terms precise? Thus.
(The preferred mathematical formulation of this last is the dreaded "epsilon-delta" definition:
Once this newly rigorous definition of limit was in place, almost everything else fell in line. The functions that had been discovered in the wake of Fourier's work could not be wished away, but it was now possible to describe precisely the ways in which they misbehaved, and clear and logical theorems could be established detailing exactly which functions satisfied the classical notions of Newton and Leibniz. It seemed that calculus was finally on a sound footing.
It wasn't. The ancient problems involving numbers (for counting) and magnitudes (for measuring) still lurked, and their final resolution still waited.
[1] I know of three ways of getting around this, all three a bit dicey. None of them, however, was available in Leibniz' day, or even Cauchy's. We'll look at a couple of them later.
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At the inception of calculus, Newton wrote of "prime and ultimate ratios" - rates of change of a variable y at the very first or very last instant; but the notion of an instant is antithetical to the vision of smoothly flowing quantities ("fluents", as Newton called them). Leibniz thought in terms of infinitesimals: change the independent variable x by an infinitesimal amount dx, and the dependent y will change, likewise, by infinitesimal dy, and the ratio dy/dx turns out to be an ordinary real number. This, too, is rife with problems, as Leibniz treats infinitesimals as something hybrid, like zero (so small that their squares may be regarded as zero) but not like zero (since it is possible to divide by them)1.
It had long been realized that some sort of notion of limit was needed, and a vague formulation was already in use by the end of the eighteenth century. It seems to have been Cauchy who recognized a way of rigorizing the notion.
Let x and y be two variables, somehow related. We need to explain what it means to say that, as x approaches a particular value c, y must approach some limit L. Cauchy saw that what were needed were statements of the following form: if x is, say, within .01 of c, then y is within .001 of L. More generally, y can be made as close to L as desired by requiring x to be close enough to c. How do we make these terms precise? Thus.
Any specified degree of closeness between y and L can be guaranteed by requiring some particular degree of closeness between x and c.You want y to be within one one-millionth of L? Insist that x be within one one-billionth of c. If every such challenge can be met, then the limit of y, as x approaches c, is L.
(The preferred mathematical formulation of this last is the dreaded "epsilon-delta" definition:
For every ε > 0, there is a δ > 0 such that, if 0 < |x-c| < δ, then |y-L| < ε.But the version above is more flexible; the notion of "closeness" can be redefined in any number of ways, in different contexts, and the notion of limit adapts.)
Once this newly rigorous definition of limit was in place, almost everything else fell in line. The functions that had been discovered in the wake of Fourier's work could not be wished away, but it was now possible to describe precisely the ways in which they misbehaved, and clear and logical theorems could be established detailing exactly which functions satisfied the classical notions of Newton and Leibniz. It seemed that calculus was finally on a sound footing.
It wasn't. The ancient problems involving numbers (for counting) and magnitudes (for measuring) still lurked, and their final resolution still waited.
[1] I know of three ways of getting around this, all three a bit dicey. None of them, however, was available in Leibniz' day, or even Cauchy's. We'll look at a couple of them later.
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no subject
Date: 2008-11-14 05:42 am (UTC)no subject
Date: 2008-11-15 07:19 pm (UTC)Yes. The notion is very clear. I don't know why it takes us so long to get our memories and tongues around the statement.
no subject
Date: 2008-11-15 11:32 pm (UTC)Then again, there's something that feels a little backwards about it. Ordinary English would have it that "if you get x close enough to c, you can get y as close as you want to L", putting the reference to x first and to y second. Rephrasing it to put y first is syntactically a bit clunky.