This essay was long when originally published; now, on November 30, 2019, I have made it longer, in an attempt to clarify some points.
The essay begins with two brief quotations, from Collingwood and Pirsig respectively, about what it takes to know people.
Ideas that come up along the way include the following.

For C. S. Lewis, the reality of moral truth shows there is something beyond the scope of natural science.

I say the same for mathematical truth.

Truths we learn as children are open to question. In their educational childhoods, mathematicians have often learned wrongly the techniques of induction and recursion.

The philosophical thesis of physicalism is of doubtful value.

Mathematicians and philosophers who ape them (as in a particular definition of physicalism) use “iff” needlessly.

A pair of mathematicians who use “iff” needlessly seem also to misunderstand induction and recursion.

Their work is nonetheless admirable, like the famous expression of universal equality by the slavedriving Thomas Jefferson.

Mathematical truth is discovered and confirmed by thought.

Truth is a product of every kind of science; it is not an object of natural science.

The distinction between thinking and feeling is a theme of Collingwood.

In particular, thought is selfcritical: it judges whether itself is going well.

Students of mathematics must learn their right to judge what is correct, along with their responsibility to reach agreement with others about what is correct. I say this.

Students of English must learn not only to judge their own work, but even that they can judge it. Pirsig says this.

For Monk, Collingwood’s demise has meant Ryle’s rise: unfortunately so since, for one thing, Ryle has no interest in the past.

In a metaphor developed by Matthew Arnold, Collingwood and Pirsig are two of my touchstones.

Thoreau is another. He affects indifference to the past, but his real views are more subtle.

According to Monk, Collingwood could have been a professional violinist; Ryle had “no ear for tunes.”

For Collingwood, Victoria’s memorial to Albert was hideous; for Pirsig, Victorian America was the same.

Again according to Monk, some persons might mistake Collingwood for Wittgenstein.

My method of gathering together ideas, as outlined above, resembles Pirsig’s method, described in Lila, of collecting ideas on index cards.

Our problems are not vague, but precise.