intuitionistic topology & foundations of constructive mathematics

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. 

my dear friend wim couwenberg and i have started a wonderful project, aiming to explain the ideas behind intuitionism in a simple and enjoyable way. the article-in-progress natural topology will show 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: 

• on the foundations of constructive mathematics - especially in relation to the theory of continuous functions
  
  (2001, revised 2003, published in Foundations of Science , Volume 10, Number 3, September 2005, pp. 249-324, Springer Verlag [.pdf, 400Kb])
  
  

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_11 for RUSS is introduced. This is necessitated by the fact that there can be no common axiom of choice for RUSS on the one hand and CLASS and INT on the other.

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_D combined with the bar-decidable-descent axiom BDD.

Remark: for the continuity problems in BISH, recent developments in formal topology suggest a partial solution in the field of constructive formal topology, not in normal BISH analysis - see Erik Palmgren's paper `Continuity on the real line and in formal spaces'. However, in my humble opinion the problem in general remains unresolved. Although it is shown that every continuous^BIS function from R to R is representable by a continuous mapping in the corresponding formal 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 between the corresponding formal topologies (which are indicated in the paper above), implies the Fan Theorem.

Without the Fan Theorem, the formal topology corresponding to R⁺ does not consistently match with R⁺ as a metric subspace of R in BISH, in my humble opinion. 

Although the continuity problem in BISH is now well-known, no reaction that I know of has been issued on how to deal with it in normal BISH analysis. 

 

 

• modern intuitionistic topology
  
  (phd thesis 1996, radboud university nijmegen [.pdf, 1Mb]) 

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: 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.