[FOM] On the Nature of Mathematical Objects
Dmytro Taranovsky
dmytro at mit.edu
Sun Aug 8 01:31:26 EDT 2004
As Vladimir Sazonov has pointed out, the recent discussion about the scope of
and formalization in mathematics glossed over what is a mathematical structure.
I will try to answer that question.
The question is what are mathematical objects and what questions about them are
mathematical. Mathematical objects are objects whose intrinsic properties are
absolute, as opposed to being contingent on the state of the world. A
mathematical question is a rigorous and completely unambiguous question about
mathematical objects, where "about" is interpreted strictly to mean that the
question cannot invoke non-mathematical objects. A semimathematical question
is a question that is primarily about mathematics.
Questions about fictitious characters, and even about physical reality are
riddled with ambiguity. One can try to define what a star is by noting some
properties that all stars have (such as being heavy) and some properties that
no star has, but the definition will be in some cases ambiguous (for example,
are brown dwarfs stars? what about neutron stars?). By contrast, a
mathematical definition, such as that of an even integer, has no ambiguity.
The reach of particular physical objects is ambiguous (for example, is solar
corona a part of the sun?), but no such ambiguity exists, say, for the set of
prime numbers: The set reaches those only those natural numbers that have
exactly two different divisors. (Intrinsic) properties of mathematical objects
do not evolve with time.
An example of a semimathematical question that is not mathematical is, "What is
the best way to continue the sequence 1, 2, 3, 4?" "The best way" is not
rigorous. If we are observing a physical system that gives 1, 2, 3, and 4 as
its first four outputs, the question of what should come next is partially a
question about physics. If we try to decide what infinite sequence a person
had in mind, the question becomes partially that of psychology.
Related is the question of why mathematics is so useful and why formalization is
so succesful in mathematics. In studying behavior of a system, one wants to
model it by something certain and precise, so physical systems are modeled by
mathematical ones. The modern emphasis on precision (such as precision in
making microprocessors) corresponds to the need for mathematics. Because
mathematical objects and properties are unambiguous and precise, they can be
studied formally. Hence, formal systems are much more useful in mathematics
than in literary criticism.
A major development of modern mathematics is the recognition that all
mathematical questions that have been asked can be rephrased as questions about
sets, more precisely (if there are other types of sets), well-founded,
extensional sets that are built from the empty set. Thus, it seems likely that
without loss of generality, mathematics could be defined as set theory.
Philosophers disagree on whether every mathematical question can be answered by
a human a priori. The issue is complicated by the fact that people make
mistakes. However, a set of integers can be decided by a Turing machine in the
limit (the Turing machine makes and can change guesses on whether an integer is
in the set, but can change its guess about a particular integer only finitely
many times) iff it is \Delta-0-2, so if human civilization is correct in the
limit about all arithmetical statements, it is non-recursive.
Comments and corrections to the answer are welcome.
Dmytro Taranovsky
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