FOM: naive or brainwashed? (objective vs. subjective in math.)

Torkel Franzen torkel at
Sat Mar 14 05:29:45 EST 1998

  Validimir Sazonov says:

   >By the way, can anybody here explain what is this fully 
   >"determinate" "standard model"?

  Certainly. The natural numbers are 0,s(0),s(s(0)), and so on. Or,
if you like, a natural number is anything obtainable from 0 by
iterating the successor operation.

  Now, of course you already know this, but you don't think this is
a "clear explanation". Here I'm sure you're right, in the sense that
no "clear explanation" such as you ask for can be given. Where I
disagree with you is in the conclusion that there is therefore some
indeterminacy or unclarity about the notion "natural number".

  >I have presented in this FOM list another 
  >*experimental arithmetical low* that base two logarithm function 
  >of arbitrary integer argument in unary notation is *bounded* by 
  >the number one thousand, and that logarithm of logarithm is 
  >bounded by ten.

  I don't think Martin Davis meant to suggest that the justification
for Lagrange's theorem has anything to do with experiments. What
degree of support experimental observations give to aritmetical
conjectures is, in general, a tricky and debated question. However,
your suggestion, that "log log x < 10" is an "arithmetical law"
justified by experimental findings simply lacks any obvious
justification. Why should we regard it as an arithmetical law that log
log x < 10 on the basis of the experience you refer to?

Torkel Franzen

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