Formalismis the idea that mathe-
matics is truly formal; therefore, it is only
concerned with the algorithmic manipu-
lation of symbols. In formalism, predicate
calculus does not denote predicates—or
anything else—meaning mathematical
objects do not exist at all. This definitely
fits into today’s world of computers, espe-
cially in the field of artificial intelligence.
But this philosophy does not take into
account human mathematical under-
standing, not to mention mathematical
applications in physics and engineering.Constructivismwas a “fringe” move-
ment at the turn of the 21st century. Con-
structionists believe that mathematical
knowledge is obtained by a series of pure-
ly mental constructions, with all mathe-
matical objects existing only in the mind
of the mathematician. But construc-
tivism does not take into account the
external world, and when it is taken to
extremes it can mean that there is no possibility of communication from one mind to
another. This philosophy also runs the risk of rejecting the basic laws of logic. For
example, if you have a mathematical problem with a yes or no nature, and the answer
is unknown, then neither “yes” nor “no” is in the mind of the mathematician. This
means that a disjunction is not a legitimate mathematical assumption, and, thus,
ideas such as Aristotle’s law of the excluded middle (“either or”) are cast aside. Not
many modern mathematicians want to throw out centuries of logic.Set-theoretical Platonismsounds as if mathematicians are regressing back to
Plato’s time. In reality, this philosophy is based on a variant of the Platonic doctrine of
recollection in which we are born possessing all knowledge, and our realization of that
knowledge is contingent on our discovery of it. In the set-theoretical Platonism, infi-
nite sets exist in a non-material, purely mathematical realm. By extending our intu-
itive understanding of this realm, we can cope with problems such as those encoun-
tered by the Gödel Incompleteness Theorem. But this philosophy, like the others, has
a seemingly infinite number of gaps, especially the question of how a theory of infinite
sets can be applied to a finite world.
What do these philosophies tell about the state of modern mathematics and logic?
Like many abstract and complex studies, philosophies come and go; some are good,
114 some seemingly on the mathematical fringe. But they also show us that there is cur-
The Greek philosopher Plato’s notion that people are
born possessing all knowledge inspired the mathe-
matical philosophy of Set-theoretical Platonism,
which deals with the concept of infinite sets. Library
of Congress.