It is folklore that if a continuous function on a complete metric space has approximate roots and in a uniform manner at most one root, then it actually has a root, which of course is uniquely determined. Also in Bishop's constructive mathematics with countable choice, the general setting of the present note, there is a simple method to validate this heuristic principle. The unique solution even becomes a continuous function in the parameters by a mild modification of the uniqueness hypothesis. (...) Moreover, Brouwer's fan theorem for decidable bars turns out to be equivalent to the statement that, for uniformly continuous functions on a compact metric space, the crucial uniform “at most one” condition follows from its non-uniform counterpart. This classification in the spirit of the constructive reverse mathematics, as propagated by Ishihara and others, sharpens an earlier result obtained jointly with Berger and Bridges. (shrink)

In his remarkable paper Formalism 64, Robinson defends his eponymous position concerning the foundations of mathematics, as follows:Any mention of infinite totalities is literally meaningless.We should act as if infinite totalities really existed. Being the originator of Nonstandard Analysis, it stands to reason that Robinson would have often been faced with the opposing position that ‘some infinite totalities are more meaningful than others’, the textbook example being that of infinitesimals. For instance, Bishop and Connes have made such claims regarding infinitesimals, (...) and Nonstandard Analysis in general, going as far as calling the latter respectively a debasement of meaning and virtual, while accepting as meaningful other infinite totalities and the associated mathematical framework. We shall study the critique of Nonstandard Analysis by Bishop and Connes, and observe that these authors equate ‘meaning’ and ‘computational content’, though their interpretations of said content vary. As we will see, Bishop and Connes claim that the presence of ideal objects in Nonstandard Analysis yields the absence of meaning. We will debunk the Bishop–Connes critique by establishing the contrary, namely that the presence of ideal objects in Nonstandard Analysis yields the ubiquitous presence of computational content. In particular, infinitesimals provide an elegant shorthand for expressing computational content. To this end, we introduce a direct translation between a large class of theorems of Nonstandard Analysis and theorems rich in computational content, similar to the ‘reversals’ from the foundational program Reverse Mathematics. The latter also plays an important role in gauging the scope of this translation. (shrink)

Elementary Recursive Nonstandard Analysis, in short ERNA, is a constructive system of nonstandard analysis with a PRA consistency proof, proposed around 1995 by Patrick Suppes and Richard Sommer. Recently, the author showed the consistency of ERNA with several transfer principles and proved results of nonstandard analysis in the resulting theories (see [12] and [13]). Here, we show that Weak König's lemma (WKL) and many of its equivalent formulations over RCA₀ from Reverse Mathematics (see [21] and [22]) can be 'pushed down' (...) into the weak theory ERNA, while preserving the equivalences, but at the price of replacing equality with equality 'up to infinitesimals'. It turns out that ERNA plays the role of RCA₀ and that transfer for universal formulas corresponds to WKL. (shrink)

We investigate which part of Brouwer’s Intuitionistic Mathematics is finitistically justifiable or guaranteed in Hilbert’s Finitism, in the same way as similar investigations on Classical Mathematics already done quite extensively in proof theory and reverse mathematics. While we already knew a contrast from the classical situation concerning the continuity principle, more contrasts turn out: we show that several principles are finitistically justifiable or guaranteed which are classically not. Among them are: fan theorem for decidable fans but arbitrary bars; continuity principle (...) and the axiom of choice both for arbitrary formulae; and $\Sigma _2$ induction and dependent choice. We also show that Markov’s principle MP does not change this situation; that neither does lesser limited principle of omniscience LLPO ; but that limited principle of omniscience LPO makes the situation completely classical. (shrink)

A form of Kripke's schema turns out to be equivalent to each of the following two statements from metric topology: every open subspace of a separable metric space is separable; every open subset of a separable metric space is a countable union of open balls. Thus Kripke's schema serves as a point of reference for classifying theorems of classical mathematics within Bishop-style constructive reverse mathematics.

The strong continuity principle reads “every pointwise continuous function from a complete separable metric space to a metric space is uniformly continuous near each compact image.” We show that this principle is equivalent to the fan theorem for monotone \ bars. We work in the context of constructive reverse mathematics.

Dini's theorem says that compactness of the domain, a metric space, ensures the uniform convergence of every simply convergent monotone sequence of real-valued continuous functions whose limit is continuous. By showing that Dini's theorem is equivalent to Brouwer's fan theorem for detachable bars, we provide Dini's theorem with a classification in the recently established constructive reverse mathematics propagated by Ishihara. As a complement, Dini's theorem is proved to be equivalent to the analogue of the fan theorem, weak König's lemma, in (...) the original classical setting of reverse mathematics started by Friedman and Simpson. (shrink)

Recently, the second author, Briseid, and Safarik introduced nonstandard Dialectica, a functional interpretation capable of eliminating instances of familiar principles of nonstandard arithmetic—including overspill, underspill, and generalizations to higher types—from proofs. We show that the properties of this interpretation are mirrored by first-order logic in a constructive sheaf model of nonstandard arithmetic due to Moerdijk, later developed by Palmgren, and draw some new connections between nonstandard principles and principles that are rejected by strict constructivism. Furthermore, we introduce a variant of (...) the Diller–Nahm interpretation with two different kinds of quantifiers, similar to Hernest’s light Dialectica interpretation, and show that one can obtain nonstandard Dialectica by weakening the computational content of the existential quantifiers—a process called herbrandization. We also define a constructive sheaf model mirroring this new functional interpretation, and show that the process of herbrandization has a clear meaning in terms of these sheaf models. (shrink)

The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years, such as relevance logics, paraconsistent logics, fuzzy logics and so on. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, and several (...) weaker principles, turn out to be distinguishable, giving perhaps supporting motivation for adopting minimal logic as the ambient logic for reasoning in the possible presence of inconsistency. (shrink)

Working within Bishop-style constructive mathematics, we examine some of the consequences of the anti-Specker property, known to be equivalent to a version of Brouwer's fan theorem. The work is a contribution to constructive reverse mathematics.

The glueing of (sequentially, pointwise, or uniformly) continuous functions that coincide on the intersection of their closed domains is examined in the light of Bishop-style constructive analysis. This requires us to pay attention to the way that the two domains intersect.

Working within Bishop’s constructive framework, we examine the connection between a weak version of the Heine–Borel property, a property antithetical to that in Specker’s theorem in recursive analysis, and the uniform continuity theorem for integer-valued functions. The paper is a contribution to the ongoing programme of constructive reverse mathematics.

Errett Bishop's work in constructive mathematics is overwhelmingly regarded as a turning point for mathematics based on intuitionistic logic. It brought new life to this form of mathematics and prompted the development of new areas of research that witness today's depth and breadth of constructive mathematics. Surprisingly, notwithstanding the extensive mathematical progress since the publication in 1967 of Errett Bishop's Foundations of Constructive Analysis, there has been no corresponding advances in the philosophy of constructive mathematics Bishop style. The aim of (...) this paper is to foster the philosophical debate about this form of mathematics. I begin by considering key elements of philosophical remarks by Bishop, especially focusing on Bishop's assessment of Brouwer. I then compare these remarks with ``traditional'' philosophical arguments for intuitionistic logic and argue that the latter are in tension with Bishop's views. ``Traditional'' arguments for intuitionistic logic turn out to be also in conflict with significant recent developments in constructive mathematics. This rises pressing questions for the philosopher of mathematics, especially with regard to the possibility of offering alternative philosophical arguments for constructive mathematics. I conclude with the suggestion to look anew at Bishop's own remarks for inspiration. (shrink)