Foundations of Calculus in the 1700’s

“Ghosts of departed quantities” Nagging Doubts

• Calculus worked, and the practitioners, including Newton, Leibniz, the Bernoullis, Euler, and others, rarely made mistakes or got into trouble in using it. • Still, the explanation of why it all worked was not logically satisfactory –it was not as logically put together as Euclid, for example. • Of particular concern were these pesky infinitesimals that were sometimes ignored as being essentially 0, and sometimes not – the “abominable little zeros,” as your book says. Bishop Berkeley

• The most important criticism of the foundations of early calculus came from George Berkeley, an Irish philosopher who was appointed a Bishop in the church. • His tract, The Analyst: A DISCOURSE Addressed to an Infidel Mathematician, was a response to recent attempts of the part of some philosophers and scientists to argue that Newton’s physics obviated the need for a personal God who was involved in the daily workings of the universe. Bishop Berkeley

• Thus he attempted to show that the mathematics upon which Newtonian physics was based was not at all certain. • It was probably addressed to the English astronomer Edmond Halley, who expressed the opinion that religious arguments were mysterious and illogical. • So he was returning the favor. Bishop Berkeley

• He didn’t deny the usefulness or validity of the results of calculus, but wanted to show that mathematicians had no valid justification for the methods they used (and thus their arguments were no more logical or valid than those of religion). From The Analyst

• Berkeley pointed out “…the fallacious way of proceeding to a certain Point on the Supposition of an Increment, and then at once shifting your Supposition to that of no Increment . . . Since if this second Supposition had been made before the common Division by 0, all had vanished at once, and you must have got nothing by your Supposition.” From The Analyst

• “Whereas by this Artifice of first dividing, and then changing your Supposition, you retain 1 and nxn‐1. But, notwithstanding all this address to cover it, the fallacy is still the same.”

• “The minutest errors are not to be neglected in mathematics.” ‐‐ Berkeley used a quote from Newton. From The Analyst

• “And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?” From The Analyst

• “… he who can digest a second or third fluxion, a second or third difference, need not, methinks, be squeamish about any point in divinity.”

• So There. Judith Grabiner on Berkeley

• “Berkeley’s criticisms of the rigor of the calculus were witty, unkind, and—with respect to the mathematical practices he was criticizing—essentially correct”

From Grabiner, Judith (1997), "Was Newton's Calculus a Dead End? The Continental Influence of Maclaurin's Treatise of Fluxions", The American Mathematical Monthly (Mathematical Association of America) 104 (5): 393–410 Responses to Berkeley

• MacLaurin’s response was perhaps the best conceived in that it did answer Berkeley’s objections and showed a real understanding of the calculus. • However, it relied heavily on classical geometry, was 754 pages, and was rather difficult to read. MacLaurin’s Response

• The processes of Newton’s calculus could be made rigorous by proving that, for example, if the rate of change of y is R, R cannot be less than 2x and cannot be more than 2x, so R must be exactly 2x. This method is similar to that used by Archimedes prove that the area of the circle couldn’t be less than 2π, and could not be greater than 2π; in other words, the good old method of exhaustion, coupled with the double reductio. MacLaurin’s Response

• The problem was that, although this might work as a proof method, still the only method of finding the rates of change involved the suspect methods of infinitesimals, fluxions, etc. In addition, the proofs in general cases were quite difficult. MacLaurin’s Response

• He also introduced the notion of instantaneous velocity as part of his explanatory framework. • He did not recognize that these could be measured by using limits, but his work did leave that door open. Responses to Berkeley

• d’Alembert used the idea of finding the limit of the ratio as particular magnitudes vanished. In fact, d’Alembert went farther by giving a definition of limit that sounds somewhat like ours: • “One magnitude is said to be the limit of another magnitude when the second may approach the first within any given magnitude, however small, though the second magnitude may never exceed the magnitude it approaches.” Responses to Berkeley

• In summary, the need for some idea of limit began to be obvious, but had its own conceptual problems. • The final appearance of a rigorous definition of limit would have to wait until Cauchy in the next century. And the next chapter.