[FOM] On Epistemology of Mathematics
Dmytro Taranovsky
dmytro at MIT.EDU
Thu May 28 16:11:04 EDT 2009
In deciding which axioms to use in mathematics, we should reflect on not merely
why they are necessary and true, but also what causes humans to believe their
truthfulness. The later is a scientific question. For example, it is
scientifically testable whether a computer science background increases the
probability of having a belief that all real numbers are recursive -- and
whether a background of meditation and reflection increases human belief in
large cardinal axioms. More important than such correlations is understanding
of the mechanisms by which, for example, the human race adopted ZFC as true.
Understanding of these mechanisms will in turn lead to a better understanding
of which undecidable propositions are true.
Below, I outline and explore the naturalist view -- the view that mathematical
insights are the product of the human brain, which is controlled by ordinary
physical laws, and that the human brain is a product of evolution and of the
real world experiences. While I do not fully endorse the naturalist view, the
view is useful and insightful here and in other scientific questions.
Our experiences have created intuitions about truth, and about naturalness and
beauty. These intuitions arise because they are useful, and in turn, they are
used to find truth in set theory, where experimentation and formal reasoning
alone are insufficient.
Things we consider obvious are obvious because they are thoroughly confirmed by
experience. For example, the axioms of number theory are obvious because of our
experience with numbers, and even first-order logic is based on our experience
with reasoning. First-order logic is not obvious to persons with limited
experience, such as young children.
To arrive at ZFC, humans have transferred their understanding of the finite sets
to the infinite and resolved the contradictions in the intuitions. The relevant
skills -- transferring knowledge to a different context and resolving the
resulting inconsistencies -- exist because of their usefulness in the real
world. To go beyond ZFC, humans use additional intuitions about finite sets
(such as existence of perfect play in definable games), as well as intuitions
about symmetry, set-theoretical reflection, the notion of all possible sets,
and other intuitions.
For an example of intuition, consider an alteration of set theory in which the
universe V is built as usual, except that no set can have exactly 13 elements.
Such a set theory would inter-interpretable with ordinary set theory, but it
would feel unnatural, and with the lack of 13 acutely felt. I think that in
the future, a set theory without a well-ordering of the reals of length omega_1
will be found to be similarly lacking, but we are not there today.
For evolutionary reasons, humans have a predisposition to attribute their
experiences (such as sight and hearing) to real objects. In mathematics, the
same tendency causes humans to believe that mathematical objects exist based on
the experience of reasoning about them. Of course, there are significant
differences from ordinary sensory experiences, and there is a substantial
controversy about existence of mathematical objects. There is also a greater
uncertainty about existence of uncountable sets because the experiences are
less vivid, with basic questions like the Continuum Hypothesis remaining
unanswered.
Like other knowledge, mathematical knowledge exists in a gradation of
confidences. We are more confident about some axioms than others. Our proofs
generally depend on pattern-recognition and memory circuits, which occasionally
misfire, temporarily resulting in false theorems.
In the naturalist view, human evolution can in principle be simulated on a
supercomputer, and mathematical insight is therefore a recursive process (but
likely with some randomness). It is likely that we will eventually discover an
algorithm by which such a supercomputer can be run. Understanding of the
algorithm will allow an unprecedented understanding of our limitations, and
perhaps marking of some arithmetical statements as permanently unknowable.
However, the understanding will not allow us to transcend the algorithm (or
violate the incompleteness theorems) because it is an algorithm for how the
human civilization develops, and not an algorithm for acquiring infallible
knowledge.
That was the naturalist view. My view is that humans have a universal capacity
to think that allows us to go beyond the recursive limitation.
Dmytro Taranovsky
http://web.mit.edu/dmytro/www/main.htm
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