[FOM] mathematics and ordinary language

[Martin Davis]

Replying to Harvey Friedman, Dean Buckner writes:

<<So one "substantive issue" is set theory itself, which postulates the
existence of things (sets) which seem unnecessary to explain the logic of
our ordinary numerical statements such as "if there exists one thing and
another thing, there exist two things, and if there is a third thing, there
are three things". I don't see why we need entities like sets to explain
these sorts of statements.

It has been argued here that formal discourse is a wholly different from
ordinary discourse. But the philosophy of language is not concerned with
symbols or utterances, it is concerned with their meaning. And I don't see
that what ordinary people mean by their numerical discourse is any different
from what mathematicians mean. And if it is wholly different, what are
mathematicians talking about?>>

Dean Buckner is entirely correct in everything he says in these two 
paragraphs. But he seems not to understand or perhaps to remember what it 
is that mathematics is about. Yes indeed, the numbers ordinary folk use in 
counting are the same entities as those with which mathematicians deal when 
they speak of "natural numbers". But does he really imagine that it is 
simple minded statements about how many apples Johnny has if he picks 5 and 
eats 2 with which mathematicians are concerned. Mr. Buckner admits not 
being a mathematician. But presumably he did study elementary algebra as a 
school boy. Perhaps he even recalls the derivation of the formula for the 
roots of a general quadratic equation by "completing the square". Let him 
carry out this demonstration in "ordinary language". This is a piece of 
mathematics beyond which the mathematicians of the Italian Renaissance were 
already advancing.

Let Mr. Buckner pick up a scholarly journal devoted to any of the sciences 
or even economics. Does he believe that these scientists deluded in their 
evident belief that the use of technical mathematical language is needed to 
properly deal with their concerns? Will he show us the power of ordinary 
language by writing the equations of general relativity defining the 
gravitational field of the universe or Schrödinger's equation for the 
evolution of the wave forms of quantum mechanics in those terms.

Martin



                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
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                        http://www.eipye.com