Sunday, January 7, 2018

The Double Truth

It is clear by now that the old Averroist doctrine of the “double truth” (veritas duplex)  must be revived:  this time, not along scientific-vs-theological lines, but political.  Radio talk-shows have long recognized the incommunicability of one set of truths, to partisans of the other, by offering separate Republican vs. Democrat call-in lines.

So, let us consider that simplest possible area of human belief and inquiry:  arithmetic, the discipline that offers up such hoary verities as “2 + 2 = 4”.  
In modern mathematics, the study of such numbers (called integers)  has exfoliated quite a bit, and indeed has bifurcated into named subdisciplines.   Thus we have, on the one hand, analytic number-theory,  which deals with such things as the Riemann hypothesis and the Goldbach conjecture, using techniques from complex analysis, and on the other, algebraic number-theory, involving extensions of the integers such as algebraic number-fields.   Both subfields have developed to such an extent, that specialists in the one will be scarcely able to understand, let alone prove, theorems in the other;  thus reproducing, unintentionally, a simulacrum of the current partisan Dialogue of the Deaf.

And now we have two new entrants to the disciplinary matrix:   Democratic number-theory and Republican number-theory.

Democratic number-theory is characterized (up to isomorphism) by its claim that all numbers are equal.   The denial of that doctrine is termed ‘fascism’.

Republican number-theory, by contrast, generalizes the ancient concept of a perfect number (equal to its aliquot sum) to that of the awesome numbers, which are greater than the sum of everybody else.  The leftovers are called the  pathetic numbers (also termed ‘losers’).  This theory focuses exclusively on the former subset (although the membership in this set  can vary capriciously  from day to day).

So far, neither body of theory has produced interesting theorems.

[Update 8 Jan 2018]   No sooner had I posted that would-be satirical thought-dream, than I happened upon this,  by the philosopher Stewart Shapiro:

Hartry Field takes mathematical language at face value.  Since he holds that mathematical objects do not exist, mathematical propositions have objective but vacuous truth-values.  For example, he maintains that ‘all natural numbers are prime’ is true, since there are no natural numbers.
-- Thinking about Mathematics (2000), p. 226

So once again, reality beggars satire.  (Apparently this Field fellow is some sort of an academic, rather than an inmate or homeless-person.)

Note, a propos, that the Fieldian assertion “all natural numbers are prime”  is actually an axiom of Democratic number-theory.   (“Everybody gets a trophy.  Everyone is special.”)

[Historical footnote]    If satire is the production of humor by clever exaggeration of a real situation, then here again the satire lacks its bite:  for such ideologically infected strains of the hard sciences  are already exemplified in history -- not merely in the musings of philosophical fantasists, but under dictators with the power of life or death.   Thus Hitlerian notions of Aryan science vs. Jewish science, or the various scientific orthodoxies under Stalin.    Under those two regimes, mathematics, fortunately, was relatively spared distortions of actual content (so long as the formulaic obeissances to ideology were packed into the preface),  though  in human terms  there were profound and pernicious effects on who was allowed to be mathematical practitioners.   The neo-Stalinoid zealots of “anti-racist mathematics” go further, prescribing severe limits to content and presentation as well.

No comments:

Post a Comment