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SCARCE_RIGOR

                                             September 30, 2019

Bertrand Russell (1919):

   "Mathematics is a deductive science: starting
   from certain premisses, it arrives, by a strict
   process of deduction, at the various theorems
   which constitute it. It is true that, in the       It's both a "scarcely
   past, mathematical deductions were often           attainable ideal" and the
   greatly lacking in rigour; it is true also that    only acceptable approach.
   perfect rigour is a scarcely attainable ideal.
   Nevertheless, in so far as rigour is lacking in
   a mathematical proof, the proof is defective;
   it is no defence to urge that common sense
   shows the result to be correct, for if we were    Common sense?
   to rely upon that, it would be better to          Intutition?  If you
   dispense with argument altogether, rather than    have to resort to
   bring fallacy to the rescue of common sense. No   things like *that*, why
   appeal to common sense, or 'intuition' or         bother?!  (But we *do*
   anything except strict deductive logic, ought     have to resort to them,
   to be needed in mathematics after the premisses   all the time... but
   have been laid down."                             math is *different*,
                                                     because...)
   "Kant, having observed that the geometers of
   his day could not prove their theorems by
   unaided argument, but required an appeal to the      A geometry based
   figure, invented a theory of mathematical            (partly) on empircal
   reasoning according to which the inference is        reasoning using
   never strictly logical, but always requires the      figure drawing does
   support of what is called 'intuition.' The           not strike me as
   whole trend of modern mathematics, with its          inherently useless.
   increased pursuit of rigour, has been against
   this Kantian theory."

   "The things in the mathematics of Kant's day
   which cannot be proved, cannot be known for
   example, the axiom of parallels.  What can be
   known, in mathematics and by mathematical
   methods, is what can be deduced from pure
   logic. What else is to belong to human
   knowledge must be ascertained other wise
   empirically, through the senses or through
   experience in some form, but not a priori."

   Bertrand Russell, "Incompatibility and the Theory of Deduction" p.144-145
   from _Introduction to the Philosophy of Mathematics_ (1919)


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