FOM: Successes of intuition over rigor

Gordon Fisher gfisher at
Thu Feb 14 16:54:07 EST 2002

Neil Tennant wrote:

> Here is a modest list of the successes of intuition over rigor.
> 1. Everything is composed of earth, air, fire and water.
> 2. The sun goes round the earth.
> 3. The earth is flat.
> 4. God exists.
> 5. There is an afterlife.
> 6. For every property F, there is a set of all Fs.
> 7. Peano arithmetic is complete.
> 8. Space is Euclidean.
> 9. There is only one kind of infinity.
> 10. Mind is distinct from matter.

I have been careful to use "intuition" in the sense
of German "Anschauung," and have indicated
that this invokes visualization, imaging or imagining.
Note that the title of Hilbert and Cohn-Vossen's
_Anschauliche Geometrie_ was translated into
English as _Geometry and the Imagination_.
The point H & C-V were making is that there
is something _anschaulich_ about geometry
which isn't present in, say, such structures as
found in Hilbert and Ackermann's _Grundzuege
der theoretischen Logik_.

Of course, the term "intuition" is often used to mean
"hypothesis," or even as "guess" or "hunch" or maybe
"feeling" (as in "I have a feeling that so and so").
Sometimes it is used pejoratively, as in the above
message, to refer to hypotheses which seemed
reasonable, but further investigation (empirical
and/or theoretical) showed to be untenable or
unsuitable, at least in some contexts where they
were thought to be tenable.  For some people, it is
pleasant to point out mistakes other people have
made.  It can, for example, give one a certain frisson
of superiority.

Incidentally, just to discuss one of your examples
of purported failed intuition, in the book by
Einstein and Infeld called _Evolution of Physics_,
you will find a remark to the effect that if general
relativity had also been formulated by or available
to Copernicus, then he could have realized that
saying the sun goes around the earth and the earth
goes around the sun are equivalent in the sense
that one can choose coordinates so that each of
these is tenable.  Thus, say E and I (in their own
words), a lot of the fuss that has been made over
the transition from Ptolemaic to Copernican
astronomy would have been unnecessary, or at
least transformed.

Note that navigators on or sufficiently near
the surface of the earth customarily work in a
kind of Ptolemaic framework, so to think in
terms of the sun going around the earth still has
practical applications.

Gordon Fisher     gfisher at

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