### WHITHER MATHEMATICS?

, 2004

"... whither10.tex We describe three successive crises faced by mathematicians during the twentieth century, and their implications for the nature of mathematics. 1 ..."

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whither10.tex We describe three successive crises faced by mathematicians during the twentieth century, and their implications for the nature of mathematics. 1

### Domain-Theoretic Methods for Program Synthesis

"... formal proofs. A recent outcome of this analysis is the development of computer systems for automated or interactive theorem proving that can for instance be used for computer aided program verication. An example of such a system is the interactive theorem prover Minlog developed by the logic group ..."

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formal proofs. A recent outcome of this analysis is the development of computer systems for automated or interactive theorem proving that can for instance be used for computer aided program verication. An example of such a system is the interactive theorem prover Minlog developed by the logic group at the University of Munich (7). As a former member of this group I was mainly involved in the theoretical background steering the implementation of the system. The system also exploits the so-called proofs-as-programs paradigm as a logical approach to correct software development: from a formal proof that a certain specication has a solution one fully automatically extracts a program that provably meets the specication. We carried out a number of extended case studies extracting programs from proofs in areas such as arithmetic (6), graph theory (7), innitary combinatorics (7), and lambda calculus (1,2). Special emphasis has been put on an ecient implemen

### Adjoints,absolutevaluesandpolardecompositions Douglas Bridges,Fred Richman,Peter Schuster

"... Abstract. Variousquestionsaboutadjoints,absolutevaluesandpolar decompositions ofoperators are addressed from a constructive pointofview. The focus is on bilinearforms. Conditions are given forthe existence ofan adjoint,andageneralnotionofapolardecompositionisdeveloped.TheRiesz representationtheoremi ..."

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Abstract. Variousquestionsaboutadjoints,absolutevaluesandpolar decompositions ofoperators are addressed from a constructive pointofview. The focus is on bilinearforms. Conditions are given forthe existence ofan adjoint,andageneralnotionofapolardecompositionisdeveloped.TheRiesz representationtheoremisprovedwithoutcountablechoice. 1. Introduction. Let

### U.U.D.M. Report 2008:42 Setoids and universes

"... Abstract. Setoids commonly take the place of sets when formalising mathematics inside type theory. In this note, the category of setoids is studied in type theory with as small universes as possible (and thus, the type theory as weak as possible). Particularly, we will consider epimorphisms and disj ..."

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Abstract. Setoids commonly take the place of sets when formalising mathematics inside type theory. In this note, the category of setoids is studied in type theory with as small universes as possible (and thus, the type theory as weak as possible). Particularly, we will consider epimorphisms and disjoint sums. It is shown that, given the minimal type universe, all epimorphisms are surjections, and disjoint sums exist. Further, without universes, there are countermodels for these statements, and if we use the Logical Framework formulation of type theory, these statements are provably non-derivable. 1.

### Approximate counting in bounded arithmetic

, 2007

"... We develop approximate counting of sets definable by Boolean circuits in bounded arithmetic using the dual weak pigeonhole principle (dWPHP(P V)), as a generalization of results from [15]. We discuss applications to formalization of randomized complexity classes (such as BPP, APP, MA, AM) in P V1 + ..."

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We develop approximate counting of sets definable by Boolean circuits in bounded arithmetic using the dual weak pigeonhole principle (dWPHP(P V)), as a generalization of results from [15]. We discuss applications to formalization of randomized complexity classes (such as BPP, APP, MA, AM) in P V1 + dWPHP(P V).

### Did Brouwer Really Believe That?

, 2007

"... This article is a commentary on remarks made in a recent book [12] that perpetuate several myths about Brouwer and intuitionism. The footnote on page 279 of [12] is an unfortunate, historically and factually inaccurate, blemish on an otherwise remarkable book. In that footnote, in which Ok discusses ..."

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This article is a commentary on remarks made in a recent book [12] that perpetuate several myths about Brouwer and intuitionism. The footnote on page 279 of [12] is an unfortunate, historically and factually inaccurate, blemish on an otherwise remarkable book. In that footnote, in which Ok discusses Brouwer (who, incidentally, was normally known not as “Jan ” but as “Bertus”, a shortening of his second name, Egbertus), 1 he says:...later in his career, he [Brouwer] became the most forceful proponent of the so-called intuitionist philosophy of mathematics, which not only forbids the use of the Axiom of Choice but also rejects the axiom that a proposition is either true or false (thereby disallowing the method of proof by contradiction). The consequences of taking this position are dire. For instance, an intuitionist would not accept the existence of an irrational number! In fact, in his later years, Brouwer did not view the Brouwer Fixed Point Theorem as a theorem. These sentences contain a number of outdated but still common misconceptions

### Separatedness in Constructive Topology

- DOCUMENTA MATH.
, 2003

"... We discuss three natural, classically equivalent, Hausdorff separation properties for topological spaces in constructive mathematics. Using Brouwerian examples, we show that our results are the best possible in our constructive framework. ..."

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We discuss three natural, classically equivalent, Hausdorff separation properties for topological spaces in constructive mathematics. Using Brouwerian examples, we show that our results are the best possible in our constructive framework.

### A Constructive Study of Landau's . . .

, 2009

"... A summability theorem of Landau, which classically is a simple consequence of the uniform boundedness theorem, is examined constructively. ..."

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A summability theorem of Landau, which classically is a simple consequence of the uniform boundedness theorem, is examined constructively.

### 1 Solving the Dirichlet problem constructively

"... Abstract: The Dirichlet problem is of central importance in both applied and abstract potential theory. We prove the (perhaps surprising) result that the existence of solutions in the general case is an essentially nonconstructive proposition: there is no algorithm which will actually compute soluti ..."

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Abstract: The Dirichlet problem is of central importance in both applied and abstract potential theory. We prove the (perhaps surprising) result that the existence of solutions in the general case is an essentially nonconstructive proposition: there is no algorithm which will actually compute solutions for arbitrary domains and boundary conditions. A corollary of our results is the non-existence of constructive solutions to the Navier-Stokes equations of fluid flow. But not all the news is bad: we provide reasonable conditions, omitted in the classical theory but easily satisfied, which ensure the computability of solutions.

### Documenta Math. 973 Inheriting the Anti-Specker Property

, 2009

"... Abstract. The antithesis of Specker’s theorem from recursive analysis is further examined from Bishop’s constructive viewpoint, with particular attention to its passage to subspaces and products. Ishihara’s principle BD-N comes into play in the discussion of products with the anti-Specker property. ..."

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Abstract. The antithesis of Specker’s theorem from recursive analysis is further examined from Bishop’s constructive viewpoint, with particular attention to its passage to subspaces and products. Ishihara’s principle BD-N comes into play in the discussion of products with the anti-Specker property.