[FOM] Re: Question on the Scope of Mathematics
Dmytro Taranovsky
dmytro at mit.edu
Thu Jul 29 15:40:16 EDT 2004
In writing,
>> However, unless stated or clearly understood otherwise, a claim that a
>> mathematical statement is true includes and should continue to include an
>> implicit claim that enough is known so that it is "routine" to produce a
>> formal proof of the statement in the appropriate formal system
>>(typically, ZFC).
I only meant to say that, for example, if you announce that the Riemann
Hypothesis is true, then unless you have a proof of the Riemann Hypothesis, you
should qualify your announcement by stating that you know the truthfulnes of
the Riemann Hypothesis based on philosophical and empirical reasons rather than
provability in ZFC.
The replies of Aatu Koskensilta and Henrik Nordmark are, of course, relevant to
the main question about the scope of mathematics.
Here is a mathematical statement whose truthfulness we know based on empirical
rather than formal grounds: In the modification of the game of chess where the
black side is left without the rook at A8, the white side has a winning
strategy.
Also, in the nineteenth century, it was disputed whether set theory is
mathematics. The agreement that set theory is mathematics was established
because of formalization of set theory.
My preference is to define mathematics broadly; and use the word "formal
mathematics" for the narrow notion of mathematics. However, some believe that
all mathematics is formal mathematics, and in any case, to avoid uncertainty
and controversy, as much as reasonable of one's mathematical work should be
valid as formal mathematics.
Dmytro Taranovsky
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