FOM: reverse math: juicy quotes from Aristotle and David Ross

[Stephen G Simpson]

Recently I've been too busy to do more than read FOM.  I want to
rejoin the discussion of Aristotle, methodological purity, general
intellectual interest, reverse mathematics, Lakatos, subjectivism, and
other issues that have been aired recently.  Right now I only have
time to share the following:

Earlier I posted a juicy quote from Aristotle in which he discusses
reciprocation of premises and conclusions in a way that perhaps ties
in with reverse mathematics.  Here are some more juicy quotes on
"commensurate universality", i.e. the idea that reciprocation is a
feature of the best sort of scientific proof.

Posterior Analytics I.5 contains some mathematical examples.  At 74a13
we have:

  If a proof were given that perpendiculars to the same line are
  parallel, it might be supposed that lines thus perpendicular were
  the proper subject of the demonstration because being parallel is
  true of every instance of them.  But it is not so, for the
  parallelism depends not on these angles being equal to one another
  because each is a right angle, but simply on their being equal to
  one another.

and at 74a17:

  If isosceles were the only triangle, it would be thought to have its
  angles equal to two right angles qua isosceles.  ...  Even if one
  prove of each kind of triangle that its angles are equal to two
  right angles, whether by means of the same or different proofs;
  still, as long as one treats separately equilateral, scalene, and
  isosceles, one does not yet know, except sophistically, that
  triangle has its angles equal to two right angles, nor does one yet
  know that triangle has the property commensurately and universally,
  even if there is no other species of triangle but these.  Folr one
  does not know that triangle as such has this property, nor even that
  `all' triangles have it -- unless `all' means `each taken singly':
  if `all' means `as a whole class', then though there be none in
  which one does not recognize this property, one does not yet know it
  of `all triangles'. ...

and there is another example involving alternating proportionals, at
74a18:

  Alternation used to be demonstrated separately of numbers, lines,
  solids, and durations, though it could have been proved of them all
  by a single demonstration.  Because there ws no single name to
  denote that in which numbers, lengths, durations, and solids are
  identical, and because they differed specifically from one another,
  this property was proved of each of them separately. To-day,
  however, the proof is commensurately universal, for they do not
  possess this attribute qua lines or qua numbers, but qua manifesting
  this generic character which they are potulated as possessing
  universally.

At the end of Posterior Analytics I.9, 76a25, Aristotle says:

  It is hard to be sure whether one knows or not; for it is hard to be
  sure whether one's knowledge is based on the basic truths
  appropriate to each attribute -- the differentia of true knowledge.
  We think we have scientific knowledgeif we have reasoned from true
  and primary premisses.  But that is not so: the conclusion must be
  homogeneous with the basic facts of the science.

Sir David Ross, "Aristotle", University Paperbacks, xiv + 300 pages,
5th edition, 1949, says on page 46:

  From any proposed subject we must `strip off' all irrelevant
  differentiae till we come to that subject which is precisely
  commensurate with the predicate; the premises of science are
  reciprocating or simply convertible statements -- such alone have
  the elegance which the ideal of science requires.

and he refers to Posterior Analytics, I.4,5.  

(Oh, yes.  The David Ross above is W. D. Ross, editor in chief of the
Oxford translation of Aristotle; not David Ross of Hawaii who has
contributed to FOM.)

-- Steve

Stephen G. Simpson
Department of Mathematics, Pennsylvania State University
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