George Berkeley
Philosophy of mathematics

In addition to his contributions to philosophy, Berkeley was also very influential in the development of mathematics, although in a rather indirect sense. "Berkeley was concerned with mathematics and its philosophical interpretation from the earliest stages of his intellectual life." Berkeley's "Philosophical Commentaries" (1707–1708) witness to his interest in mathematics:

Axiom. No reasoning about things whereof we have no idea. Therefore no reasoning about Infinitesimals. (#354)

Take away the signs from Arithmetic & Algebra, & pray what remains? (#767)

These are sciences purely Verbal, & entirely useless but for Practise in Societys of Men. No speculative knowledge, no comparison of Ideas in them. (#768)

In 1707, Berkeley published two treatises on mathematics. In 1734, he published The Analyst, subtitled A DISCOURSE Addressed to an Infidel Mathematician, a critique of calculus. Florian Cajori called this treatise "the most spectacular event of the century in the history of British mathematics." However, a recent study suggests that Berkeley misunderstood Leibnizian calculus. The mathematician in question is believed to have been either Edmond Halley, or Isaac Newton himself—though if to the latter, then the discourse was posthumously addressed, as Newton died in 1727. The Analyst represented a direct attack on the foundations and principles of calculus and, in particular, the notion of fluxion or infinitesimal change, which Newton and Leibniz used to develop the calculus. In his critique, Berkeley coined the phrase " ghosts of departed quantities ", familiar to students of calculus. Ian Stewart 's book From Here to Infinity captures the gist of his criticism.

Berkeley regarded his criticism of calculus as part of his broader campaign against the religious implications of Newtonian mechanics – as a defence of traditional Christianity against deism, which tends to distance God from His worshipers. Specifically, he observed that both Newtonian and Leibnizian calculus employed infinitesimals sometimes as positive, nonzero quantities and other times as a number explicitly equal to zero. Berkeley's key point in "The Analyst" was that Newton's calculus (and the laws of motion based on calculus) lacked rigorous theoretical foundations. He claimed that:

In every other Science Men prove their Conclusions by their Principles, and not their Principles by the Conclusions. But if in yours you should allow your selves this unnatural way of proceeding, the Consequence would be that you must take up with Induction, and bid adieu to Demonstration. And if you submit to this, your Authority will no longer lead the way in Points of Reason and Science.

Berkeley did not doubt that calculus produced real-world truth; simple physics experiments could verify that Newton's method did what it claimed to do. "The cause of Fluxions cannot be defended by reason", but the results could be defended by empirical observation, Berkeley's preferred method of acquiring knowledge at any rate. Berkeley, however, found it paradoxical that "Mathematicians should deduce true Propositions from false Principles, be right in Conclusion, and yet err in the Premises." In The Analyst he endeavoured to show "how Error may bring forth Truth, though it cannot bring forth Science". Newton's science, therefore, could not on purely scientific grounds justify its conclusions, and the mechanical, deistic model of the universe could not be rationally justified.

The difficulties raised by Berkeley were still present in the work of Cauchy whose approach to calculus was a combination of infinitesimals and a notion of limit, and were eventually sidestepped by Weierstrass by means of his (ε, δ) approach, which eliminated infinitesimals altogether. More recently, Abraham Robinson restored infinitesimal methods in his 1966 book Non-standard analysis by showing that they can be used rigorously.
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