Meaningful Distinctions

:: history, constructions

By: Ben Greenman

“Meaningful distinctions deserve to be maintained.” — Errett A. Bishop

Likewise, memorable quotations deserve to be read in context. In this spirit, I am happy to present the above “basic principle” in its context: Schizophrenia in contemporary mathematics (pdf)

Read on for a brief summary.

I first read the above quotation in Michael Beeson’s introduction to the 2012 edition of Bishop’s Foundations of Constructive Analysis. That was two years ago.

Last month, I tried to find its context. Many other uses of the quote cited a Schizophrenia in comtemporary mathematics, but I could not find an electronic copy. (It turns out, the AMS Bookstore page for Erret Bishop: Reflections on Him and His Research includes a facsimile.)

Lest anyone else be tempted to conjure the ancient magic of inter-library loan, here is a scan of the pages I borrowed. Thanks to the University of Toledo for supplying the hard copy.

The document is Bishop’s “feeling for the philosophical issues involved” in constructive mathematics. First, Bishop lists “schizophrenic attributes” (trouble spots) of contemporary mathematics. Next, he gives basic principles of constructivism and Brouwer’s interpretation of the logical quantifiers. Along the way, and as a source of examples, Bishop describes integers, sets, and real numbers. The emphasis is always on common-sense meaning and finite constructions.

After a brief summary and reflection, the last ten pages list recent advances in constructive mathematics and upcoming tasks. The open tasks are particularly interesting:

  • systematically develop (constructive) algebra
  • give a constructive foundation for general topology
  • engage with the deeper “meaning of mathematics”

The popular quote on “Meaningful Distinctions” appears early in the paper, as one of Bishop’s four principles that “stand out as basic” to the philosophy of constructivism:

A. Mathematics is common sense.

B. Do not ask whether a statement is true until you know what it means.

C. A proof is any completely convincing argument.

D. Meaningful distinctions deserve to be maintained.

I had no idea that D was “a principle”, or that it had three siblings.

To further tempt you into reading the whole truth, here are some of my favorite phrases:

  • One suspects that the majority of pure mathematicians … ignore as much content as they possibly can.
  • We have geared ourselves to producing research mathematicians who will begin to write papers as soon as possible. This anti-social and anti-intellectual process defeats even its own narrow ends.
  • … truth is not a source of trouble to the constructivist, because of his emphasis on meaning.
  • … guided primarily by considerations of content rather than elegance and formal attractiveness …
  • Let me tell you what a smart sequence will do.
  • Classical mathematics fails to observe meaningful distinctions having to do with integers.
  • Constructive mathematics does not postulate a pre-existent universe, with objects lying around waiting to be collected and grouped into sets, like shells on a beach.
  • It might be worthwhile to investigate the possibility that constructive mathematics would afford a solid philosophical basis for the theory of computation …
  • … if the product of two real numbers is 0, we are not entitled to conclude that one of them is 0.
  • It is fair to say that almost nobody finds his proof intelligible.
  • Mathematics is such a complicated activity that disagreements are bound to arise.
  • Algebraic topology, at least on the elementary level, should not be too difficult to constructivize.
  • I hope all this accords with your common sense, as it does with mine.

Now go find their context!