Constructivism
argues that all of mathematics must be explicitly computable, and
finitism
argues that all of mathematics must be done with finite objects, which usually
means finite sets, rational numbers, and things that can always be written on a
page in canonical form. Both of these schools of thought raise many
controversial questions about how valid or meaningful certain areas of
mathematics can be, especially in topics like
set theory,
model theory,
and
analysis.
While these positions are more extreme than what I hold, I have personally
gained a lot from thinking about mathematics in more constructive ways, and
this part of my site exists to share some of the novel conclusions I have
reached as a result.
My Philosophy
Despite what these schools of thought offer, my position isn't one of hard-line
constructivism or finitism. When it comes to constructivism, I claim
that functions and proofs should be presented in concrete, constructive ways
whenever possible,
that this leads to the most useful and easy to understand theories,
that some ideas of set theory are revealed to be partly meaningless when
described constructively,
but also that all propositions are either true or false, and inferences that
follow from this dichotomy are valid.
Similarly, when it comes to finitism and rational numbers, I claim
that geometric problems should be approached using exact, finite methods
whenever possible,
that this leads to the most useful and easy to understand theories,
that calculus concepts like limits and transcendentals are really describing
properties of approximate solutions to problems, rather than exact solutions
to problems,
but that these approximate solutions are important nonetheless, and worth
developing rigorous theories of convergence around in order to understand.
Topics
These articles, sketches, and titles all cover places where I have tried to
find a clear explanation of the constructive content of a topic in mathematics,
and reached a novel conclusion about that topic as a result. These topics
cluster into subjects, and while I personally know which order you would need
to go through these subjects in order to derive things in a non-circular way, I
have presented the subjects in order of interest, rather than by which theories
are prior to which.