Mark Meckes responded to my statement

But it seems to me that this sort of thinking has mostly resulted in people thinking philosophy of math is merely a matter of logic and set theory.  That point of view has been ruinous to the practice of math.

with this comment:

I may be misreading your analysis of the second straw man, but you seem to imply that “people thinking philosophy of math is merely a matter of logic and set theory” has done great damage to mathematics. I think that’s quite an overstatement. It means that in practice, mathematicians find philosophy of mathematics to be irrelevant and useless. Perhaps philosophers of mathematics could in principle have something to say that mathematicians would find helpful but in practice they don’t; however, we’re getting along quite well without their help.

On the other hand, maybe you only meant that people who think “philosophy of math is merely a matter of logic and set theory” are handicapped in their own ability to do mathematics. Again, I think most mathematicians get along fine just not thinking about philosophy.

Mark is right that at least this aspect of philosophy of math is irrelevant and useless to mathematicians.  But my remark that the attitude that “philosophy of math is merely a matter of logic and set theory” is ruinous to math was sloppy, it was not what I should have said.    I was thinking of a related phenomenon which was ruinous to math communication and teaching.

By the 1950’s many mathematicians adopted the attitude that all math is is theorem and proof.  Images, metaphors and the like were regarded as misleading and resulting in incorrect proofs.  (I am not going to get into how this attitude came about).     Teachers and colloquium lecturers suppressed intuitive insights and motivations in their talks and just stated the theorem and went through the proof.

I believe both expository and research papers were affected by this as well, but I would not be able to defend that with citations.

I was a math student 1959 through 1965.  My undergraduate calculus (and advanced calculus) teacher was a very good teacher but he was affected by this tendency.  He knew he had to give us intuitive insights but he would say things like “close the door” and “don’t tell anyone I said this” before he did.  His attitude seemed to be that that was not real math and was slightly shameful to talk about.  Most of my other undergrad teachers simply did not give us insights.

In graduate school I had courses in Lie Algebra and Mathematical Logic from the same teacher.   He was excellent at giving us theorem-proof lectures, much better than most teachers, but he never gave us any geometric insights into Lie Algebra (I never heard him say anything about differential equations!) or any idea of the significance of mathematical logic.  We went through Killing’s classification theorem and Gödel’s incompleteness theorem in a very thorough way and I came out of his courses pleased with my understanding of the subject matter.  But I had no idea what either one of them had to do with any other part of math.

I had another teacher for several courses in algebra and various levels of number theory.   He was not much for insights, metaphors, etc, but he did do well in explaining how you come up with a proof.  My teacher in point set topology was absolutely awful and turned me off the Moore Method forever.   The Moore method seems to be based on: don’t give the student any insights whatever. I have to say that one of my fellow students thought the Moore method was the best thing since sliced bread and went on to get a degree from this teacher.

These dismal years in math teaching lasted through the seventies and perhaps into the eighties.  Apparently now younger professors are much more into insights, images and metaphors and to some extent into pointing out connections with the rest of math and science.  Since I have been retired since 1999 I don’t have much exposure to the newer generation and I am not sure how thoroughly things have changed.

One noticeable phenomenon was that category theorists (I got into category theory in the mid seventies) were very assiduous in lectures and to some extent in papers in giving motivation and insight.  It may be that attitudes varied a lot between different disciplines.

This Dark Ages of math teaching was one of the motivations for abstractmath.org.  My belief is that not only should we give the students insights, images and metaphors to think about objects, and so on, but that we should be upfront about it:   Tell them what we are doing (don’t just mutter the word “intuitive”) and point out that these insights are necessary for understanding but are dangerous when used in proofs.  Tell them these things with examples. In every class.

My other main motivation for abstractmath.org was the way math language causes difficulties.  But that is another story.

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