ocho
infinito, co-design with wim couwenberg

(3 steps iteration of complex root
function transform of circle to lemniscate)

welcome to the math page of frank waaldijk. as you can guess from the title, my interests are mainly in intuitionism, topology and constructive mathematics.
some time ago my dear friend wim couwenberg and i started a wonderful project, aiming to explain the ideas behind intuitionism in a simple and enjoyable way. this resulted in the recent book natural topology. the book shows that intuitionism and topology are closely -and very naturally- related. here is where you can find wim's homepage for some interesting mathematical subjects (complex reflection groups, hypergeometric functions, theta functions, diophantine equations, recursive texture mapping, and more). |

Some of my research (4 publications, click on the blue for pdf files and a more detailed description): -
Natural Topology (2011, with big support from Wim Couwenberg) -
modern intuitionistic topology (my PhD thesis, 1996) -
Some elementary results in intuitionistic model theory (1996, with Wim Veldman)
Natural topology (click for .pdf) (2012 second edition, published by the Brouwer Society, .pdf 1.1 Mb, first edition 2011) also available on arXiv: http://arxiv.org/abs/1210.6288
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. ~ Addenda and errata can be found on my math & science & philosophy blog.
On
the foundations of constructive mathematics - especially in
relation to the theory of continuous functions (click for .pdf) This article describes many foundational issues concerning what is known as constructivism in mathematics. First of all a flaw in the foundations of Bishop-style constructive mathematics BISH is discussed. A main theorem shows that the two current BISH-definitions of `continuous function' are not equivalent within BISH, and that -together with the natural properties of `continuous function'- they imply the axiom FT (fan theorem). This problem is closely related to the non-topological definition in BISH of `locally compact'. The theorem sparks an investigation into the realm of topology, and the axioms underpinning intuitionism (INT), classical mathematics (CLASS), recursive mathematics (RUSS) and BISH. Some new elegant axioms are introduced, to prove theorems showing that CLASS and INT are far closer than usually believed (`reuniting the antipodes'). The distance to RUSS is seen to be far greater, due perhaps to a philosophical difference regarding `real world' phenomena. In fact this is seen to tie in with the old philosophical debate on determinism. And perhaps with the modern physics' debate as well? In section 7 a real-world experiment is described which could cast an alternative mathematical light on this matter. Axioms of choice are discussed (since the current BISH
treatment is unsatisfactory in the author's eyes) and a simple axiom of
choice CT The fundamental intuitionistic axiom BT (Bar Theorem or
Brouwer's Thesis; also true in CLASS) is presented in a simplified and
elegant version, and shown to be equivalent to Kleene's bar induction
axiom BI
Remarks: - Only as late as 2011 I came across the nice new
(2006) book
*Techniques of Constructive Analysis*by Douglas Bridges and Luminiţa Simona Vīţă. Unbeknownst to me earlier, it contains a modification of the definition of `continuous function' precisely along the lines of the above article, which is even kindly referenced. For this, I am truly grateful.
- Another way for solving continuity problems in BISH: recent
developments in formal topology have produced a nice partial solution
in the field of constructive formal topology - see
Erik Palmgren's paper `Continuity on
the real line and in formal spaces'.
- We refind this solution also in the contex of natural topology. In the meantime, in my
humble opinion I don't think that the problem in general is solved for
BISH. Although one can show that every continuous
^{BIS}function from R to R is representable by a continuous mapping (morphism) in the corresponding formal or natural topology (and vice versa), the same cannot be said of R^{+}. More specifically: the statement that every continuous^{BIS}function from [0,1] to R^{+}is representable by a continuous mapping (morphism) between the corresponding formal/natural topologies (which are indicated in the paper above and in the book `Natural topology'), implies the Fan Theorem.In the book `Natural topology' there is more discussion on this topic. [The issue of BISH continuity was already noted in my PhD-thesis in 1996, see below.]
modern
intuitionistic topology (click for .pdf)
This monograph holds a self-contained modern approach to intuitionistic topology. First, a foundational framework is built up, including a non-impredicative definition of `topological space', which is classically equivalent to the usual impredicative definition. Local properties, separation axioms, homeomorphisms, compactness, metrizability,... are defined in a natural way, closely paralleling the classical approach. Some interesting results are obtained. For instance the metrizability of star-finitary spaces, which closely resembles the classical metrization of paracompact spaces. The theory of star-finite refinements is developed to yield the tool of partitions of unity. From there on we prove a constructive version of the Dugundji extension theorem, and an intuitionistic version of the Michael selection theorem. A promising new concept (although I seem to stand alone in this!) is that of a metric subspace being (strongly) halflocated in its mother space. The beautiful properties of this concept are brought to the fore. A nice structure arising intuitionistically is an example of a pathwise connected compact space which is not arcwise connected. Many other positive examples are given, as well as counterexamples to natural conjectures. Errata: a) Remark 3.3.2 is wrong. The in 3.3.2 (by countable induction) defined `weakly stable closure' is not always weakly stable. For weak stability of the closure one needs to define the weakly stable closure with transfinite induction. Although this does not affect any of the main theorems in chapter 3, some theorems and examples which specifically are about the weakly stable closure should be examined closely to see what part still holds and what is false. The most important falsehood is thm. 3.3.9 that the weakly stable closure of a metric spread is always again spreadlike. This limits -imho- the usefulness of the concept `weakly stable closure'. `Weakly stable' remains an interesting topological property, in my humble opinion. Wim Veldman has studied a similar concept `perhapsity', with the added benefit of correcting the above errors. b) The proof of lemma 2.4.4. is
incorrect. It is
possible to repair this, by invoking AC
Some elementary results in intuitionistic model theory (click for .pdf) (together with Wim Veldman, the Journal of Symbolic Logic vol. 61, no. 3, pp 745-767, 1996; [.pdf, 1.3 Mb; this is a scanned document retrieved from the university library's webservice]) This paper contains most of the results found during my first two PhD-years on intuitionistic model theory. The rather incomplete abstract states: `We establish constructive refinements of several well-known theorems in elementary model theory. The additive group of the real numbers may be embedded elementarily into the additive group of pairs of real numbers, constructively as well as classically'. The paper's beautiful style is all Wim Veldman's, as he did the actual writing. Perhaps I will post some additional results which were too complicated to easily include in the above paper. |

if you think this site is worthwhile, please consider linking to it ~ all comments are welcome: info@fwaaldijk.nl ~ last modified 15 nov 2013 |

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