On the Foundations of Constructive Mathematics – Especially in Relation to the Theory of Continuous Functions

We discuss the foundations of constructive mathematics, including recursive mathematics and intuitionism, in relation to classical mathematics. There are connections with the foundations of physics, due to the way in which the different branches of mathematics reflect reality. Many different axioms and their interrelationship are discussed. We show that there is a fundamental problem in BISH (Bishop’s school of constructive mathematics) with regard to its current definition of ‘continuous function’. This problem is closely related to the definition in BISH of ‘locally compact’. Possible approaches to this problem are discussed. Topology seems to be a key to understanding many issues. We offer several new simplifying axioms, which can form bridges between the various branches of constructive mathematics and classical mathematics (‘reuniting the antipodes’). We give a simplification of basic intuitionistic theory, especially with regard to so-called ‘bar induction’. We then plead for a limited number of axiomatic systems, which differentiate between the various branches of mathematics. Finally, in the appendix we offer BISH an elegant topological definition of ‘locally compact’, which unlike the current definition is equivalent to the usual classical and/or intuitionistic definition in classical and intuitionistic mathematics, respectively.     

Mathematics and Argumentation

Some authors have begun to appeal directly to studies of argumentation in their analyses of mathematical practice. These include researchers from an impressively diverse range of disciplines: not only philosophy of mathematics and argumentation theory, but also psychology, education, and computer science. This introduction provides some background to their work.     

浅析西方术语学发展历程——介绍“术语学是一种‘科学研究纲领’”的思想

文章介绍了加拿大学者安杰拉·坎波的长篇论文《对欧根·维斯特著作的接受和术语学的发展》的主要观点,及西方术语学领域的最新发展。在术语学的发展史上,欧根·维斯特的著作,一直是各国从事术语学工作和研究的学者们的灵感源泉。维斯特被公认为“现代术语学之父”。自20世纪90年代初,随着科技和相关领域的迅猛发展,术语学领域也出现了新的工作方法或理论导向,它们大都对传统术语学持批评态度,由此引起了激烈的学术争论。安杰拉·坎波以拉卡托斯提出的“科学研究纲领方法论”作为解释模型,有力地说明了现代术语学是一个在理论和方法论上更加强大的独立学科。     

Bolzano’s Infinite Quantities

In his Foundations of a General Theory of Manifolds, Georg Cantor praised Bernard Bolzano as a clear defender of actual infinity who had the courage to work with infinite numbers. At the same time, he sharply criticized the way Bolzano dealt with them. Cantor’s concept was based on the existence of a one-to-one correspondence, while Bolzano insisted on Euclid’s Axiom of the whole being greater than a part. Cantor’s set theory has eventually prevailed, and became a formal basis of contemporary mathematics, while Bolzano’s approach is generally considered a step in the wrong direction. In the present paper, we demonstrate that a fragment of Bolzano’s theory of infinite quantities retaining the part-whole principle can be extended to a consistent mathematical structure. It can be interpreted in several possible ways. We obtain either a linearly ordered ring of finite and infinitely great quantities, or a partially ordered ring containing infinitely small, finite and infinitely great quantities. These structures can be used as a basis of the infinitesimal calculus similarly as in non-standard analysis, whether in its full version employing ultrafilters due to Abraham Robinson, or in the recent “cheap version” avoiding ultrafilters due to Terence Tao.     

Symbolic Languages and Natural Structures a Mathematician’s Account of Empiricism

The ancient dualism of a sensible and an intelligible world important in Neoplatonic and medieval philosophy, down to Descartes and Kant, would seem to be supplanted today by a scientific view of mind-in-nature. Here, we revive the old dualism in a modified form, and describe mind as a symbolic language, founded in linguistic recursive computation according to the Church-Turing thesis, constituting a world L that serves the human organism as a map of the Universe U. This methodological distinction of L vs. U helps to understand how and why structures of phenomena come to be opposed to their nature in human thought, a central topic in Heideggerian philosophy. U is uncountable according to Georg Cantor’s set theory but Language L, based on the recursive function system, is countable, and anchored in a Gray Area within U of observable phenomena, typically symbols (or tokens), prelinguistic structures, genetic-historical records of their origins. Symbols, the phenomena most familiar to mathematicians, are capable of being addressed in L-processing. The Gray Area is the human Environment E, where we can live comfortably, that we manipulate to create our niche within hostile U, with L offering overall competence of the species to survive. The human being is seen in the light of his or her linguistic recursively computational (finite) mind. Nature U, by contrast, is the unfathomable abyss of being, infinite labyrinth of darkness, impenetrable and hostile to man. The U-man, biological organism, is a stranger in L-man, the mind-controlled rational person, as expounded by Saint Paul. Noumena can now be seen to reside in L, and are not fully supported by phenomena. Kant’s noumenal cause is the mental L-image of only partly phenomenal causation. Mathematics occurs naturally in pre-linguistic phenomena, including natural laws, which give rise to pure mathematical structures in the world of L. Mathematical foundation within philosophy is reversed to where natural mathematics in the Gray Area of pre-linguistic phenomena can be seen to be a prerequisite for intellectual discourse. Lesser, nonverbal versions of L based on images are shared with animals.     

Ambiguities of Fundamental Concepts in Mathematical Analysis During the Mid-nineteenth Century

In this paper we consider the major development of mathematical analysis during the mid-nineteenth century. On the basis of Jahnke’s (Hist Math 20(3):265–284, 1993) distinction between considering mathematics as an empirical science based on time and space and considering mathematics as a purely conceptual science we discuss the Swedish nineteenth century mathematician E.G. Bj?rling’s general view of real- and complexvalued functions. We argue that Bj?rling had a tendency to sometimes consider mathematical objects in a naturalistic way. One example is how Bj?rling interprets Cauchy’s definition of the logarithm function with respect to complex variables, which is investigated in the paper. Furthermore, in view of an article written by Bj?rling (Kongl Vetens Akad F?rh Stockholm 166–228, 1852) we consider Cauchy’s theorem on power series expansions of complex valued functions. We investigate Bj?rling’s, Cauchy’s and the Belgian mathematician Lamarle’s different conditions for expanding a complex function of a complex variable in a power series. We argue that one reason why Cauchy’s theorem was controversial could be the ambiguities of fundamental concepts in analysis that existed during the mid-nineteenth century. This problem is demonstrated with examples from Bj?rling, Cauchy and Lamarle.     

Non-Monotonic Set Theory as a Pragmatic Foundation of Mathematics

In this paper I propose a new approach to the foundation of mathematics: non-monotonic set theory. I present two completely different methods to develop set theories based on adaptive logics. For both theories there is a finitistic non-triviality proof and both theories contain (a subtle version of) the comprehension axiom schema. The first theory contains only a maximal selection of instances of the comprehension schema that do not lead to inconsistencies. The second allows for all the instances, also the inconsistent ones, but restricts the conclusions one can draw from them in order to avoid triviality. The theories have enough expressive power to form a justification/explication for most of the established results of classical mathematics. They are therefore not limited by Gödel’s incompleteness theorems. This remarkable result is possible because of the non-recursive character of the final proofs of theorems of non-monotonic theories. I shall argue that, precisely because of the computational complexity of these final proofs, we cannot claim that non-monotonic theories are ideal foundations for mathematics. Nevertheless, thanks to their strength, first order language and the recursive dynamic (defeasible) proofs of theorems of the theory, the non-monotonic theories form (what I call) interesting pragmatic foundations.     

Misrepresentation in Context

We can witness the recent surge of interest in the interaction between cognitive science, philosophy of science, and aesthetics on the problem of representation. This naturally leads us to rethinking the achievements of Goodman’s monumental book Languages of Art. For, there is no doubt that no one else contributed more than Goodman to throw a light on the cognitive function of art. Ironically, it could be also Goodman who has been the stumbling block for a unified theory of representation. In this paper, I shall contrast the ways how differently misrepresentation has been treated in cognitive science, aesthetics, and philosophy of science. And I shall show that it is Goodman’s unnecessary separation of resemblance and representation in art that made such a difference. As a conclusion, I will indicate some of the most promising projects toward the unified theory of representation the revolt against Goodman’s rejection of resemblance theories might promise to us.     

The Surveyability of Long Proofs

The specific characteristics of mathematical argumentation all depend on the centrality that writing has in the practice of mathematics, but blindness to this fact is near universal. What follows concerns just one of those characteristics, justification by proof. There is a prevalent view that long proofs pose a problem for the thesis that mathematical knowledge is justified by proof. I argue that there is no such problem: in fact, virtually all the justifications of mathematical knowledge are ‘long proofs’, but because these real justifications are distributed in the written archive of mathematics, proofs remain surveyable, hence good.     

数学中的经验:认识论上的紧张视角

就形式数学来说,它是脱离经验,在内涵上是一种社会建构。计算机技术提高了人们的计算能力,对数学的这一社会建构平添了紧张,加强了“作为计算的数学”与“作为证明的数学”之间的不平衡性。在历史上,相似的认识论上的争论表现在计算或实践的印度数学与精神或形式的西方数学之间。我们认为,认识论上的紧张,可通过以下方式得以消解:认同数学是基于经验、可错的观点,并据此引领数学课程标准的基本理念。     

Towards a Theory of Mathematical Argument

In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a full-fledged mathematical proof or merely some non-deductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent with argument assessment in non-mathematical contexts. I demonstrate this claim by considering the assessment of axiomatic or deductive proofs, probabilistic evidence, computer-aided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator’ view of proofs because it places derivations—which may be thought to invoke formal logic—at the center of mathematical justificatory practice. However, when the notion of ‘derivation’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument.     

Coal to Diamonds

In this commentary to Napoletani et al. (Foundations of Science 16:1–20, 2011), we put agnostic science in a wider historical context of philosophy of mathematics. Secondly, the parallel to Tukey’s “exploratory data analysis” will be discussed. Thirdly, it will be argued that what is new is the mutually interdependent dynamics of data (on which Napoletani et al. focus) and of computational modeling—which puts science closer to engineering and vice versa.     

The Epistemological Functions of Symbolization in Leibniz’s Universal Characteristic

Leibniz’s universal characteristic is a fundamental aspect of his theory of cognition. Without symbols or characters it would be difficult for the human mind to define several concepts and to achieve many demonstrations. In most disciplines, and particularly in mathematics, the mind must then focus on symbols and their combinatorial rules rather than on mental contents. For Leibniz, mental perception is most of the time too confused for attaining distinct notions and valid deductions. In this paper, I argue that the functions of symbolization differ depending upon the kind of concepts that are replaced with characters. In my view, most commentators did not sufficiently underline the distinction between two main functions of formal substitution in Leibniz’s characteristic: either increasing our knowledge or simply structuring it. In the first case, we complete our knowledge because formal substitution makes sensible and imaginary concepts more distinct. In the second case, symbolization helps to organize contents that are already conceived of by reason. Thus the process of substitution is not always identically applicable, for symbols replace different types of concepts.     

On the Persuasiveness of Visual Arguments in Mathematics

Two experiments are reported which investigate the factors that influence how persuaded mathematicians are by visual arguments. We demonstrate that if a visual argument is accompanied by a passage of text which describes the image, both research-active mathematicians and successful undergraduate mathematics students perceive it to be significantly more persuasive than if no text is given. We suggest that mathematicians’ epistemological concerns about supporting a claim using visual images are less prominent when the image is described in words. Finally we suggest that empirical studies can make a useful contribution to our understanding of mathematical practice.     

Computational and Biological Analogies for Understanding Fine-Tuned Parameters in Physics

In this philosophical paper, we explore computational and biological analogies to address the fine-tuning problem in cosmology. We first clarify what it means for physical constants or initial conditions to be fine-tuned. We review important distinctions such as the dimensionless and dimensional physical constants, and the classification of constants proposed by Lévy-Leblond. Then we explore how two great analogies, computational and biological, can give new insights into our problem. This paper includes a preliminary study to examine the two analogies. Importantly, analogies are both useful and fundamental cognitive tools, but can also be misused or misinterpreted. The idea that our universe might be modelled as a computational entity is analysed, and we discuss the distinction between physical laws and initial conditions using algorithmic information theory. Smolin introduced the theory of “Cosmological Natural Selection” with a biological analogy in mind. We examine an extension of this analogy involving intelligent life. We discuss if and how this extension could be legitimated.     

Representation and Truthlikeness

Woosuk Park’s paper “Misrepresentation in Context” is a useful plea for a theory of representation with promising interaction between cognitive science, philosophy of science, and aesthetics. In this paper, I argue that such a unified account is provided by Charles S. Peirce’s semiotics. This theory puts Park’s criticism of Nelson Goodman and Jerry Fodor in context. Some of Park’s pertinent remarks on the problem of misrepresentation can be illuminated by the account of truthlikeness and idealization developed by philosophers of science.     

Medieval Representations of Change and Their Early Modern Application

The article investigates the role of symbolic means of knowledge representation in concept development using the historical example of medieval diagrams of change employed in early modern work on the motion of fall. The parallel cases of Galileo Galilei, Thomas Harriot, and René Descartes and Isaac Beeckman are discussed. It is argued that the similarities concerning the achievements as well as the shortcomings of their respective work on the motion of fall can to a large extent be attributed to their shared use of means of knowledge representation handed down from antiquity and the Middle Ages. While the interpretation of medieval diagrams was unproblematic in the scholastic context from which they arose, in the early modern context, which was characterized by the confluence of natural philosophy and practical mathematics, it became ambiguous. It was the early modern mathematicians’ work within this contradictory framework that brought about a new conceptualization of motion which, in particular, eventually led to an infinitesimal concept of velocity. In this process, the diagrams themselves remained largely unchanged and thus functioned as a catalyst for concept development.     

从希尔伯特规划到数学的地图

希尔伯特规划的原初目的是为无穷数学辩护,然而为哥德尔不完全性定理所挫。反推数学的根本目标是为数学命题找寻能够证明它的下限公理,而其中相当一部分工作可以看作为对希尔伯特规划的部分实现。本文在梳理有关工作的基础上试图为希尔伯特规划提供一个新的视角,即在绕开哲学负担之后,希尔伯特规划或许可以推进为为数学绘制地图。     

A Theory of Scientific Model Construction: <Emphasis Type="Italic">The Conceptual Process of Abstraction and Concretisation</Emphasis>

The process of abstraction and concretisation is a label used for an explicative theory of scientific model-construction. In scientific theorising this process enters at various levels. We could identify two principal levels of abstraction that are useful to our understanding of theory-application. The first level is that of selecting a small number of variables and parameters abstracted from the universe of discourse and used to characterise the general laws of a theory. In classical mechanics, for example, we select position and momentum and establish a relation amongst the two variables, which we call Newton’s 2nd law. The specification of the unspecified elements of scientific laws, e.g. the force function in Newton’s 2nd law, is what would establish the link between the assertions of the theory and physical systems. In order to unravel how and with what conceptual resources scientific models are constructed, how they function and how they relate to theory, we need a view of theory-application that can accommodate our constructions of representation models. For this we need to expand our understanding of the process of abstraction to also explicate the process of specifying force functions etc. This is the second principal level at which abstraction enters in our theorising and in which I focus. In this paper, I attempt to elaborate a general analysis of the process of abstraction and concretisation involved in scientific- model construction, and argue why it provides an explication of the construction of models of the nuclear structure.     

Quantum Superpositions and the Representation of Physical Reality Beyond Measurement Outcomes and Mathematical Structures

In this paper we intend to discuss the importance of providing a physical representation of quantum superpositions which goes beyond the mere reference to mathematical structures and measurement outcomes. This proposal goes in the opposite direction to the project present in orthodox contemporary philosophy of physics which attempts to “bridge the gap” between the quantum formalism and common sense “classical reality”—precluding, right from the start, the possibility of interpreting quantum superpositions through non-classical notions. We will argue that in order to restate the problem of interpretation of quantum mechanics in truly ontological terms we require a radical revision of the problems and definitions addressed within the orthodox literature. On the one hand, we will discuss the need of providing a formal redefinition of superpositions which captures explicitly their contextual character. On the other hand, we will attempt to replace the focus on the measurement problem, which concentrates on the justification of measurement outcomes from “weird” superposed states, and introduce the superposition problem which focuses instead on the conceptual representation of superpositions themselves. In this respect, after presenting three necessary conditions for objective physical representation, we will provide arguments which show why the classical (actualist) representation of physics faces severe difficulties to solve the superposition problem. Finally, we will also argue that, if we are willing to abandon the (metaphysical) presupposition according to which ‘Actuality = Reality’, then there is plenty of room to construct a conceptual representation for quantum superpositions.