The 1975 published version of a 1974 talk at a workshop by Errett Bishop contains the following comment:

"A more recent attempt at mathematics by formal finesse is non-standard analysis. I gather that it has met with some degree of success, whether at the expense of giving significantly less meaningful proofs I do not know. My interest in non-standard analysis is that attempts are being made to introduce it into calculus courses. It is difficult to believe that debasement of meaning could be carried so far."

This was published in

Bishop, E. "The crisis in contemporary mathematics." Proceedings of the American Academy Workshop on the Evolution of Modern Mathematics (Boston, Mass., 1974). Historia Math. 2 (1975), no. 4, 507--517.

Was anyone present at Bishop's lecture who can testify whether Bishop actually made those comments in his oral presentation?

Note 1. As per discussion in the comments: The reason I doubt Bishop actually said that in his oral presentation is because in the ensuing discussion, also published in Historia Math. along with the lecture, nobody challenged Bishop on his comments even though a number of logicians were present. It seems likely that, had he said that publicly, there would have been some reaction and ensuing discussion.

Note 2. This question is primarily concerned with what Bishop said, or more precisely did not say, at the workshop, but since in the comments editors have responded with remarks on the effectiveness of teaching calculus using infinitesimals I would mention that there are many approaches to teaching the calculus, some more effective than others, but one rarely finds people dismissing an approach right out of hand without looking at the details and whether it actually works in the classroom, etc.--that is, except when it comes to true infinitesimal calculus, where everything seems to be allowed. Such a situation, I would argue, is partly the result of an atmosphere of demonisation of Robinson's framework created by the likes of Bishop, Halmos, and Connes. For more details on this aspect of the story see the many published articles here.

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    $\begingroup$ The arxiv version is 28 pages long. It says "there was, in fact, nothing to challenge him on. Bishop did not say a word about non-standard analysis in his oral presentation, according to a workshop participant [40] who attended his talk." Here [40] is "Manning, Kenneth: private communication, july ’09." This seems to answer the question. $\endgroup$ Mar 21, 2017 at 13:39
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    $\begingroup$ the 46 participants of this 1974 meeting are listed here -- barring pseudonyms, none seem to be MO users... $\endgroup$ Mar 21, 2017 at 14:22
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    $\begingroup$ Per Note 1. It seems quite reasonable to me that a logician could agree that nonstandard analysis should not be introduced into calculus courses. $\endgroup$ Mar 21, 2017 at 17:42
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    $\begingroup$ I am a logician and I agree that non-standard analysis has no place in introductory calculus. What is most needed in introductory calculus is logical hygiene, i.e., preference for direct proofs to non-direct ones, and proper respect for the distinction between free and bound variables. That helps students when they need to do calculus on a computer. $\endgroup$ Mar 21, 2017 at 21:34
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    $\begingroup$ Perhaps I'm misunderstanding Mikhail Katz, but it seems to me that he's expecting logicians to object to the implicit assumption that non-standard analysis leads to "debasement of meaning," not that he's expecting logicians to object to introducing non-standard analysis in introductory calculus per se. $\endgroup$ Mar 22, 2017 at 0:16

1 Answer 1


We were able to obtain an audio file of Bishop's talk from the American Academy of Arts and Sciences. Our analysis will be published in Historia Mathematica and is available on the arxiv.

We examine the preparation and context of the paper "The Crisis in Contemporary Mathematics" by Errett Bishop, published 1975 in Historia Mathematica. Bishop tried to moderate the differences between Hilbert and Brouwer with respect to the interpretation of logical connectives and quantifiers. He also commented on Robinson's Non-standard Analysis, fearing that it might lead to what he referred to as 'a debasement of meaning.' The 'debasement' comment can already be found in a draft version of Bishop's lecture, but not in the audio file of the actual lecture of 1974. We elucidate the context of the 'debasement' comment and its relation to Bishop's position vis-a-vis the Law of Excluded Middle.

One can only speculate concerning the reasons that may have led Bishop to suppress the 'debasement' comment when faced with an actual audience on 9 august 1974, or what he meant exactly when he declared, at the exact spot of the omission, that "that is all I want to say about pure mathematics." However, a reader of the 1975 published version who may have been surprised or disappointed not to find any reaction to the 'debasement' comment on the part of the audience that included a number of logicians will now have an explanation for their silence.

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    $\begingroup$ 9 August 1974 was the day Nixon resigned. Nothing to do with Bishop's speech, just recognized the date & thought I'd mention it. $\endgroup$ Apr 10, 2018 at 9:38
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    $\begingroup$ There is a certain poetic justice in that :-) @GerryMyerson $\endgroup$ Apr 10, 2018 at 9:39
  • $\begingroup$ Which one said "I'm not a crook?" @GerryMyerson $\endgroup$ Apr 11, 2018 at 10:03
  • $\begingroup$ youtube.com/watch?v=sh163n1lJ4M 17 November 1973. $\endgroup$ Apr 11, 2018 at 10:05

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