Data Analysis: Models or Techniques?

In this commentary to Napoletani et al. (Found Sci 16:1–20, 2011), we argue that the approach the authors adopt suggests that neural nets are mathematical techniques rather than models of cognitive processing, that the general approach dates as far back as Ptolemy, and that applied mathematics is more than simply applying results from pure mathematics.     

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.     

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.     

What if Haecceity is not a Property?

In some sense, both ontological and epistemological problems related to individuation have been the focal issues in the philosophy of mathematics ever since Frege. However, such an interest becomes manifest in the rise of structuralism as one of the most promising positions in recent philosophy of mathematics. The most recent controversy between Keränen and Shapiro seems to be the culmination of this phenomenon. Rather than taking sides, in this paper, I propose to critically examine some common assumptions shared by both parties. In particular, I shall focus on their assumptions on (1) haecceity as an individual essence, (2) haecceity as a property, (3) the classification of properties, and thereby (4) the search for the principle of individuation in terms of properties. I shall argue that all these assumptions are mistaken and ungrounded from Scotus’ point of view. Further, I will fathom what consequences would follow, if we reject each of these assumptions.     

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.     

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.     

The Differential Method and the Causal Incompleteness of Programming Theory in Molecular Biology

The “DNA is a program” metaphor is still widely used in Molecular Biology and its popularization. There are good historical reasons for the use of such a metaphor or theoretical model. Yet we argue that both the metaphor and the model are essentially inadequate also from the point of view of Physics and Computer Science. Relevant work has already been done, in Biology, criticizing the programming paradigm. We will refer to empirical evidence and theoretical writings in Biology, although our arguments will be mostly based on a comparison with the use of differential methods (in Molecular Biology: a mutation or alike is observed or induced and its phenotypic consequences are observed) as applied in Computer Science and in Physics, where this fundamental tool for empirical investigation originated and acquired a well-justified status. In particular, as we will argue, the programming paradigm is not theoretically sound as a causal(as in Physics) or deductive(as in Programming) framework for relating the genome to the phenotype, in contrast to the physicalist and computational grounds that this paradigm claims to propose.

Symmetries and Symmetry-Breakings: The Fabric of Physical Interactions and the Flow of Time

This short note develops some ideas along the lines of the stimulating paper by Heylighen (Found Sci 15 4(3):345–356, 2010a). It summarizes a theme in several writings with Francis Bailly, downloadable from this author’s web page. The “geometrization” of time and causality is the common ground of the analysis hinted here and in Heylighen’s paper. Heylighen adds a logical notion, consistency, in order to understand a possible origin of the selective process that may have originated this organization of natural phenomena. We will join our perspectives by hinting to some gnoseological complexes, common to mathematics and physics, which may shed light on the issues raised by Heylighen.     

"数学常青"——从第22届国际科学史大会看数学史研究的特点与走向

通过对第22届国际科学史大会上数学史的特邀报告、主题讨论会和论文报告的全面分析，我们看到传统内史型的数学史研究依然具有重要的生命力，但新的“热点”更应值得关注，比如，“多元文化之间数学知识的交流与传播”就已成为新的主流。为了应对“全球化和文化多样性”带来的挑战，这次国际科学史大会上展现出的新思想、新视角和新进路，对于中国数学史是十分重要的。     

Bridging the Gap Between Argumentation Theory and the Philosophy of Mathematics

We argue that there are mutually beneficial connections to be made between ideas in argumentation theory and the philosophy of mathematics, and that these connections can be suggested via the process of producing computational models of theories in these domains. We discuss Lakatos’s work (Proofs and Refutations, 1976) in which he championed the informal nature of mathematics, and our computational representation of his theory. In particular, we outline our representation of Cauchy’s proof of Euler’s conjecture, in which we use work by Haggith on argumentation structures, and identify connections between these structures and Lakatos’s methods.     

Arithmetization and Rigor as Beliefs in the Development of Mathematics

With the arrival of the nineteenth century, a process of change guided the treatment of three basic elements in the development of mathematics: rigour, the arithmetization and the clarification of the concept of function, categorised as the most important tool in the development of the mathematical analysis. In this paper we will show how several prominent mathematicians contributed greatly to the development of these basic elements that allowed the solid underpinning of mathematics and the consideration of mathematics as an axiomatic way of thinking in which anyone can deduce valid conclusions from certain types of premises. This nineteenth century stage shares, possibly with the Heroic Age of Ancient Greece, the most revolutionary period in all history of mathematics.     

Optimization Strategies for Two-Mode Partitioning

Two-mode partitioning is a relatively new form of clustering that clusters both rows and columns of a data matrix. In this paper, we consider deterministic two-mode partitioning methods in which a criterion similar to k-means is optimized. A variety of optimization methods have been proposed for this type of problem. However, it is still unclear which method should be used, as various methods may lead to non-global optima. This paper reviews and compares several optimization methods for two-mode partitioning. Several known methods are discussed, and a new fuzzy steps method is introduced. The fuzzy steps method is based on the fuzzy c-means algorithm of Bezdek (1981) and the fuzzy steps approach of Heiser and Groenen (1997) and Groenen and Jajuga (2001). The performances of all methods are compared in a large simulation study. In our simulations, a two-mode k-means optimization method most often gives the best results. Finally, an empirical data set is used to give a practical example of two-mode partitioning. We would like to thank two anonymous referees whose comments have improved the quality of this paper. We are also grateful to Peter Verhoef for providing the data set used in this paper.     

Mathematical,Philosophical and Semantic Considerations on Infinity (I): General Concepts

In the Reality we know, we cannot say if something is infinite whether we are doing Physics, Biology, Sociology or Economics. This means we have to be careful using this concept. Infinite structures do not exist in the physical world as far as we know. So what do mathematicians mean when they assert the existence of ω (the mathematical symbol for the set of all integers)? There is no universally accepted philosophy of mathematics but the most common belief is that mathematics touches on another worldly absolute truth. Many mathematicians believe that mathematics involves a special perception of an idealized world of absolute truth. This comes in part from the recognition that our knowledge of the physical world is imperfect and falls short of what we can apprehend with mathematical thinking. The objective of this paper is to present an epistemological rather than an historical vision of the mathematical concept of infinity that examines the dialectic between the actual and potential infinity.     

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.     

华蘅芳《积较术》数学方法分析

在《积较术》中,基于独特的差分定义,华蘅芳构造了一个与Ｎｅｗｔｏｎ有限差分公式完全不同的差分体系,针对各种数表的使用,华蘅芳设计了一种”乘表相加“的计算方法。算理分析表明,这一算法与近代矩阵乘法一致。对《积较术》 中的数学思想与数学方法的分析,揭示了清末传统数学研究所的生长点及其在向近代数学转变过程的积极意义。     

The Emergence of Symbolic Algebra as a Shift in Predominant Models

Historians of science find it difficult to pinpoint to an exact period in which symbolic algebra came into existence. This can be explained partly because the historical process leading to this breakthrough in mathematics has been a complex and diffuse one. On the other hand, it might also be the case that in the early twentieth century, historians of mathematics over emphasized the achievements in algebraic procedures and underestimated the conceptual changes leading to symbolic algebra. This paper attempts to provide a more precise setting for the historical context in which this decisive step to symbolic reasoning took place. For that purpose we will consider algebraic problem solving as model-based reasoning and symbolic representation as a model. This allows us to characterize the emergence of symbolic algebra as a shift from a geometrical to a symbolic mode of representation. The use of the symbolic as a model will be situated in the context of mercantilism where merchant activity of exchange has led to reciprocal relations between money and wealth.

Forms of Causal Explanation

In the literature on scientific explanation two types of pluralism are very common. The first concerns the distinction between explanations of singular facts and explanations of laws: there is a consensus that they have a different structure. The second concerns the distinction between causal explanations and uni.cation explanations: most people agree that both are useful and that their structure is different. In this article we argue for pluralism within the area of causal explanations: we claim that the structure of a causal explanation depends on the causal structure of the relevant fragment of the world and on the interests of the explainer.