3 Mathematical Objects
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In the philosophical framework of this paper there are three types
of mathematical `objects':
Finite sets are abstract idealizations of what can exist
Objective properties of finite sets, like being an even integer,
are logically determined human creations. Whether a particular
finite set has the property follows logically from the definition
of the property. However, the property can involve questions that
are logically determined but not determinable. One example is the
set of Gödel numbers of Turing Machines (TM)1
that do not halt. What the TM does at each time step is logically
determined and thus so is the question of whether it will halt,
but, if it does not halt, there is no general way to determine
this. The non-halting property is objectively determined but not in
- finite sets,
- properties of finite sets and
- properties of properties.
3.1 Properties of properties
The property of being a subset of the integers has led to the idea
that uncountable collecitons exist. No finite or countable formal
system can capture what mathematicians want to mean by the set of
all subsets of the integers. One can interpret this as the human
mind transcending the finite or that mathematics is a human
creation that can always be expanded. The inherent incompleteness
of any sufficiently strong formal system similarly suggests that
mathematics is creative.
3.2 Gödel and mathematical
Gödel proved that any formalization of mathematics strong
enough to embed the primitive recursive functions (or alternatively
a Universal Turing Machine) must be either incomplete or
inconsistent. In particular such a system, will be able to
model itself, and will not be able to decide if this model of
itself is consistent unless it is inconsistent.
This is often seen as putting a limit on mathematical knowledge. It
limits what we can be certain about in mathematics, but not what we
can explore. A divergent creative process can, in theory, pursue
every possible finite formalization of mathematics as long as it
does not have to choose which approach or approaches are correct.
Of course it can rule out those that are discovered to be
inconsistent or to have other provable flaws. This may seem to be
only of theoretical interest. However the mathematically capable
human mind is the product of just such a divergent creative process
known as biological evolution.
3.3 Cantor's incompleteness proof
One can think of Cantor's proof (when combined with the
Löwenheim-Skolem theorem) as the first great incompleteness
proof. He proved the properties defining real numbers can always be
expanded. The standard claim is that Cantor proved that there are
"more" reals than integers. That claim depends on each
real existing as a completed infinite totality. From an objective
standpoint, Cantor's proof shows that any formal system that meets
certain prerequisites can be expanded by diagonalizing the real
numbers provably definable within the system. Of course this can
only be done from outside the system.
Just as one can always define more real numbers one can always
create more objective mathematics. One wants to include as
objective all statements logically determined by a recursively
enumerable sequence of events, but that can only be precisely
defined relative to a particular formal system and will always be
incomplete and expandable just as the reals are.
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- Solmon Feferman, John W. Dawson Jr., Stephen C.
Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van
Heijenoort, editors. Publications 1929-1936, volume 1
of Kurt Gödel Collected Works. Oxford University
Press, New York, 1986.
1There is overwhelming evidence that
one can create a Universal Turing Machine that can simulate every
possible computer. Assuming this is true, one can assign a unique
integer to every possible program for this universal computer. This
is called a Gödel number because Gödel invented this idea
in a different, but related, context a few years before the concept
of a Universal Turing Machine was proposed.
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