Sunday, July 31, 2011
ocho infinito xxiii (cover drawing for the book `Natural Topology', own work, 2004-2011, click on the image for an enlargement)
Dear readers, some of you may know that Wim Couwenberg and I started a project called `Natural topology' some years ago. 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 intuitionism.
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 physical universe.
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 Brouwer's Thesis.
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).