Berkeley’s Analyst

George Berkeley The Analyst: A DISCOURSE Addressed to an Infidel MATHEMATICIAN. WHEREIN It is examined whether the Object, Principles, and Inferences of the modern Analysis are more distinctly conceived, or more evidently deduced, than Religious Mysteries and Points of Faith, London 1734

I haven’t read the original (yet), but Wikipedia seems credible. Berkeley was building on his A Treatise Concerning the Principles of Human Knowledge, Part I (1710), and might be best read in that context. A very odd view, that nothing exists unless it is observed, is sometimes attributed to Berkeley, but this attribution may be false. So don’t discount him on that basis.

Background and Purpose

Berkeley sought to take mathematics apart, claimed to uncover numerous gaps in proof, attacked the use of infinitesimals, the diagonal of the unit square, the very existence of numbers, etc. The general point was not so much to mock mathematics or mathematicians, but rather to show that mathematicians, like Christians, relied upon incomprehensible ‘mysteries’ in the foundations of their reasoning. Moreover, the existence of these ‘superstitions’ was not fatal to mathematical reasoning, indeed it was an aid.

Content

Berkeley did not dispute the results of calculus; he acknowledged the results were true. The thrust of his criticism was that Calculus was not more logically rigorous than religion.

Analysis

Berkeley’s criticisms of the rigor of the calculus were witty, unkind, and — with respect to the mathematical practices he was criticizing — essentially correct” (Grabiner 1997). While his critiques of the mathematical practices were sound, his essay has been criticized on logical and philosophical grounds.

Influence

[Calculus] continued to be developed using non-rigorous methods until around 1830 when Augustin Cauchy, and later Bernhard Riemann and Karl Weierstrass, redefined the derivative and integral using a rigorous definition of the concept of limit.

In 1966, Abraham Robinson introduced Non-standard Analysis, which provided a rigorous foundation for working with infinitely small quantities. This provided another way of putting calculus on a mathematically rigorous foundation that was in a similar spirit to the way calculus was done before the (ε, δ)-definition of limit had been fully developed.

Text and Commentary

Ewald [“From Kant to Hilbert: A Source Book in the Foundations of Mathematics”] concludes that Berkeley’s objections to the calculus of his day were mostly well taken at the time.

See Also

Wikipedia: “Berkeley, however, found it paradoxical that “Mathematicians should deduce true Propositions from false Principles, be right in Conclusion, and yet err in the Premises.”

Comment

Just to add:

  • What goes for calculus pretty much goes for the whole of mathematics (as it then was), including geometry and number theory (which is mentioned above) and probability. But it only took about 200 years to address Berkeley’s objections other than for calculus.
  • The question of which mathematics to apply to what is ultimately a matter for the relevant subject matter experts, not mathematicians. Calculus is still just as controversial as Quantum Physics.
  • However, Keynes (1919) presented an argument to the effect that some of mathematical physics and the social sciences was still open to Berkeley’s critique.
  • The debate continues. Physicists have at least attempted to address the issue, but economists, for example, largely seem to me to ignore them, possibly to ill effect.

Thus one might reasonably call Berkeley ‘the godfather of mathematics‘, as mathematicians currently think of it, with insights that might still be useful today.

Dave Marsay

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