Atdtda22: [43.8i] Crisis in European mathematics, 632 #2

[Paul Nightingale]

[632.26-27] As if to express the "imaginary" (or, as Clifford had termed it,
"invisible") realm of numbers ...

A branch of formalism called nominalism holds in its extreme form that a
theorem of mathematics is nothing more than a string of symbols, and has no
proper mathematical significance beyond this. 

A more liberal kind of formalism--and one that is particularly attractive
for the kind of issues considered in this book--considers the strings of
symbols to be initially uninterpreted, so that the "correctness" of a
mathematical theorem depends not on what interpretation is given to the
symbols, but on its method of derivation. Interpretations--if they are
needed or desired--can be given at later stages, and these may vary with
intended applications, from cases of no interpretation (extreme nominalism)
to cases that interpret the symbols in the same manner as ordinary
mathematics. As an example of a case between these extremes, consider a
formal version of axiomatic set theory. Suppose that there is reason for
considering the positive integers and certain of their subsets as "real, "
in the sense that they exist independently of human experience, and infinite
sets of infinite sets of positive integers as "fictitious, " in the sense
that they have no interpretation in reality. The fictions, although not
real, may nevertheless be useful--and even indispensable--for the derivation
of certain statements about real entities. Historically, various
mathematical concepts were handled this way, the imaginary numbers being a
particularly well-known instance. This kind of approach probably has even
more bite in certain scientific contexts where empirical reality can be
assigned to some of the mathematical concepts and no (empirical) reality to
others.
 
Formalism has also given rise to some interesting mathematics in its own
right. Because various parts of mathematics (and for that matter
mathematical science) can be looked at as formal manipulations of strings of
symbols by highly specified rules, such systems of symbol manipulations
themselves can be considered as mathematical structures, and thus have their
inathematical properties revealed by ordinary mathematical means. This sort
of mathematics of formal mathematical systems, called metamathematics by the
mathematician D. Hilbert (1862-1943), has proven in recent times to be a
fruitful source of ideas for the development of new techniques for proving
propositions of ordinary mathematics. 

From: Theories of Meaningfulness, 6