[FOM] Natural Topology (book announcement)
fwaaldijk at gmail.com
Sun Jul 31 12:05:16 EDT 2011
some of you may know that Wim Couwenberg and I started a project called
`Natural topology' some years ago. I mentioned it earlier on FOM. The aim of
the project is a) to explain intuitionism to classical mathematicians b) to
simplify aspects of formal topology c) to show that intuitionistic topology
is elegant and constructive, and its results can be easily translated to
Bishop-style mathematics if one inductivizes the definitions d) to clarify
in-our-eyes-important aspects of the relation between foundations of
constructive mathematics and the foundations of physics.
Last January, we gave an informal talk about this project at the University
of Nijmegen for a general audience. Positive and sceptical reactions
together prompted me to work out the basic idea rigorously, all the time
linking it to other developments. I am happy to announce that
the long-promised book Natural Topology is now available online, from
my website at http://www.fwaaldijk.nl/natural-topology.pdf
An abstract of the book is given below. I hope it will provide food for
thought and discussion, also for the philosophy of mathematics and physics.
The book is essentially self-contained and on the advanced undergraduate
level (I believe), meaning that anyone with a knowledge of basic topology
and some perseverance should be able to read it. For a good appreciation of
the whole book, a certain knowledge of constructive mathematics probably is
necessary, I suspect. But enough interesting elements should be accessible
to anyone, I hope. (An interesting general-audience example concerns the
line-calling decision-support system Hawk-Eye used in professional tennis.)
Of course all comments and reactions are welcome.
We develop a simple framework called `natural topology', which can serve as
a theoretical and applicable basis for dealing with real-world phenomena.
Natural topology is tailored to make pointwise and pointfree notions go
together naturally. As a constructive theory in BISH, it gives a classical
mathematician a faithful idea of important concepts and results in
Natural topology is well-suited for practical and computational purposes. We
give several examples relevant for applied mathematics, such as the
decision-support system Hawk-Eye, and various real-number representations.
We compare classical mathematics (CLASS), intuitionism (INT), recursive
mathematics (RUSS), Bishop-style mathematics (BISH), formal topology and
applied mathematics, aiming to reduce the mutual differences to their
essence. To do so, our mathematical foundation must be precise and simple.
There are links with physics, regarding the topological character of our
Any natural space is isomorphic to a quotient space of Baire space, which
therefore is universal. We develop an elegant and concise `genetic
induction' scheme, and prove its equivalence on natural spaces to a
formal-topological induction style. The inductive Heine-
Borel property holds for `compact' or `fanlike' natural subspaces, including
the real interval [α,β]. Inductive morphisms preserve this
Heine-Borel property. This partly solves the continuous-function problem for
BISH, yet pointwise problems persist in the absence of
By inductivizing the definitions, a direct correspondence with INT
is obtained which allows for a translation of many intuitionistic results
into BISH. We thus prove a constructive star-finitary metrization theorem
which parallels the classical metrization theorem for strongly
paracompact spaces. We also obtain non-metrizable Silva spaces,
in infinite-dimensional topology. Natural topology gives a solid basis, we
think, for further constructive study of topological lattice theory,
algebraic topology and infinite-dimensional topology.
The final section reconsiders the question of which mathematics to choose
for physics. Compactness issues also play a role here, since the question
`can Nature produce a non-recursive sequence?' finds a negative answer in
CTphys. CTphys, if true, would seem at first glance to point to RUSS as the
mathematics of choice for physics. To discuss this issue, we wax more
philosophical. We also present a simple model of INT in RUSS, in the
two-player game LIfE (Limited Information for Earthlings).
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