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.     

大数据分析技术在文化资源管理中的应用

文化资源是体现一个国家文化实力的核心要素，也是国家文化及文化产业发展的基础和源头。我国对各类物质和非物质文化资源数字化工作的开展，为我们利用大数据分析等先进技术，加强对中华文化的充分认知和深入挖掘利用提供了前所未有的契机和条件。本文利用大数据分析等技术手段，对我国如何加强文化资源管理的总体思路、技术框架和有关对策措施提出了建议。     

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。     

Mathematics,Philosophical and Semantic Considerations on Infinity (II): Dialectical Vision

Human language has the characteristic of being open and in some cases polysemic. The word “infinite” is used often in common speech and more frequently in literary language, but rarely with its precise meaning. In this way the concepts can be used in a vague way but an argument can still be structured so that the central idea is understood and is shared with to the partners. At the same time no precise definition is given to the concepts used and each partner makes his own reading of the text based on previous experience and cultural background. In a language dictionary the first meaning of “infinite” agrees with the etymology: what has no end. We apply the word infinite most often and incorrectly as a synonym for “very large” or something that we do not perceive its completion. In this context, the infinite mentioned in dictionaries refers to the idea or notion of the “immeasurably large” although this is open to what the individual’s means by “immeasurably great.” Based on this linguistic imprecision, the authors present a non Cantorian theory of the potential and actual infinite. For this we have introduced a new concept: the homogon that is the whole set that does not fall within the definition of sets established by Cantor.     

The Foundational Role of Ergodic Theory

The foundation of statistical mechanics and the explanation of the success of its methods rest on the fact that the theoretical values of physical quantities (phase averages) may be compared with the results of experimental measurements (infinite time averages). In the 1930s, this problem, called the ergodic problem, was dealt with by ergodic theory that tried to resolve the problem by making reference above all to considerations of a dynamic nature. In the present paper, this solution will be analyzed first, highlighting the fact that its very general nature does not duly consider the specificities of the systems of statistical mechanics. Second, Khinchin’s approach will be presented, that starting with more specific assumptions about the nature of systems, achieves an asymptotic version of the result obtained with ergodic theory. Third, the statistical meaning of Khinchin’s approach will be analyzed and a comparison between this and the point of view of ergodic theory is proposed. It will be demonstrated that the difference consists principally of two different perspectives on the ergodic problem: that of ergodic theory puts the state of equilibrium at the center, while Khinchin’s attempts to generalize the result to non-equilibrium states.     

有没有“其它情况均同”定律?

本文论证指出:(1)CP可能包括无限多的条件,因此即使是用科学语言,仍然是不可消去的;(2)我们可以检验CP定律的逆否命题,从而检验CP定律本身;(3)Earman把CP陈述看成展开式微分方程而非定律,与他的MRL定律观不一致,展开式微分方程也应是定律。笔者还提出思想实验表明,Earman对真值条件和应用有效性条件的区分,及其依附性论旨,可能得出自相矛盾的结论。     

Space-conserving agglomerative algorithms

This paper evaluates a general, infinite family of clustering algorithms, called the Lance and Williams algorithms, with respect to the space-conserving criterion. An admissible clustering criterion is defined using the space conserving idea. Necessary and sufficient conditions for Lance and Williams clustering algorithms to satisfy space-conserving admissibility are provided. Space-dilating, space-contracting, and well-structured clustering algorithms are also discussed.The work of J. Van Ness was supported by NSF Grant #DMS 9201075.     

Gödel's Incompleteness Theorems and Computer Science

In the paper some applications of Gödel's incompleteness theorems to discussions of problems of computer science are presented. In particular the problem of relations between the mind and machine (arguments by J.J.C. Smart and J.R. Lucas) is discussed. Next Gödel's opinion on this issue is studied. Finally some interpretations of Gödel's incompleteness theorems from the point of view of the information theory are presented.     

Covariance/Invariance: A Cognitive Heuristic in Einstein's Relativity Theory Formation

Relativity Theory by Albert Einstein has been so far littleconsidered by cognitive scientists, notwithstanding its undisputedscientific and philosophical moment. Unfortunately, we don't have adiary or notebook as cognitively useful as Faraday's. But physicshistorians and philosophers have done a great job that is relevant bothfor the study of the scientist's reasoning and the philosophy ofscience. I will try here to highlight the fertility of a `triangulation'using cognitive psychology, history of science and philosophy of sciencein starting answering a clearly very complex question:why did Einstein discover Relativity Theory? Here we arenot much concerned with the unending question of precisely whatEinstein discovered, that still remains unanswered, for we have noconsensus over the exact nature of the theory's foundations(Norton 1993). We are mainly interested in starting to answer the`how question', and especially the following sub-question: what(presumably) were his goals and strategies in hissearch? I will base my argument on fundamental publications ofEinstein, aiming at pointing out a theory-specific heuristic, settingboth a goal and a strategy: covariance/invariance.The result has significance in theory formation in science, especiallyin concept and model building. It also raises other questions that gobeyond the aim of this paper: why was he so confident in suchheuristic? Why didn't many other scientists use it? Where did he keep ? such a heuristic? Do we have any other examples ofsimilar heuristic search in other scientific problemsolving?     

Globally Optimal Clusterwise Regression By Column Generation Enhanced with Heuristics,Sequencing and Ending Subset Optimization

A column generation based approach is proposed for solving the cluster-wise regression problem. The proposed strategy relies firstly on several efficient heuristic strategies to insert columns into the restricted master problem. If these heuristics fail to identify an improving column, an exhaustive search is performed starting with incrementally larger ending subsets, all the while iteratively performing heuristic optimization to ensure a proper balance of exact and heuristic optimization. Additionally, observations are sequenced by their dual variables and by their inclusion in joint pair branching rules. The proposed strategy is shown to outperform the best known alternative (BBHSE) when the number of clusters is greater than three. Additionally, the current work further demonstrates and expands the successful use of the new paradigm of using incrementally larger ending subsets to strengthen the lower bounds of a branch and bound search as pioneered by Brusco's Repetitive Branch and Bound Algorithm (RBBA).     

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.     

Unification of Mathematical Theories

In this article the problem of unification of mathematical theories is discussed. We argue, that specific problems arise here, which are quite different than the problems in the case of empirical sciences. In particular, the notion of unification depends on the philosophical standpoint. We give an analysis of the notion of unification from the point of view of formalism, Gödel's platonism and Quine's realism. In particular we show, that the concept of “having the same object of study” should be made precise in the case of mathematical theories. In the appendix we give a working proposal of a certain understanding of this notion.     

Physics, Theoretical Knowledge and Weinberg's Grand Reductionism

The two main points of this contribution are the following: (1) Applied mathematical theories might complement physical theories in an essential way; some applied mathematical theories allow us to understand phenomena we are unable to explain by resorting to physical theories alone, (2) In the case of social sciences it might be necessary to account for examined phenomena by resorting to the idea of goal-oriented activity (the causal approach typical for natural science might be unsatisfactory). Weinberg's idea of grand reductionism ignores the two above mentioned facts and hence overestimates the foundational role of physics and its methodology.     

A Cauchy-Dirac Delta Function

The Dirac δ function has solid roots in nineteenth century work in Fourier analysis and singular integrals by Cauchy and others, anticipating Dirac’s discovery by over a century, and illuminating the nature of Cauchy’s infinitesimals and his infinitesimal definition of δ.     

Releasement and Nihilism in the Art of Living with Technology

In this contribution the author tries to formulate an approach to the art of living with technology based on Heidegger’s The Principle of Reason, a work often overlooked by contemporary commentators in the philosophy of technology. This approach couples the concept of releasement to insights hailing from Wolfgang Schirmacher concerning Heidegger’s nihilism.     

A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography

We analyze the developments in mathematical rigor from the viewpoint of a Burgessian critique of nominalistic reconstructions. We apply such a critique to the reconstruction of infinitesimal analysis accomplished through the efforts of Cantor, Dedekind, and Weierstrass; to the reconstruction of Cauchy’s foundational work associated with the work of Boyer and Grabiner; and to Bishop’s constructivist reconstruction of classical analysis. We examine the effects of a nominalist disposition on historiography, teaching, and research.     

A Further Review of the Incompatibility between Classical Principles and Quantum Postulates

The traditional “realist” conception of physics, according to which human concepts, laws and theories can grasp the essence of a reality in our absence, seems incompatible with quantum formalism and it most fruitful interpretation. The proof rests on the violation by quantum mechanical formalism of some fundamental principles of the classical ontology. We discuss if the conception behind Einstein’s idea of a reality in our absence, could be still maintained and at which price. We conclude that quantum mechanical formalism is not formulated on those terms, leaving for a separated paper the discussion about the terms in which it could be formulated and the onto-epistemological implications it might have.     

Algebraic Collisions

Algebraic equations in the tradition of Descartes and Frans Van Schooten accompany Christiaan Huygens’s early work on collision, which later would be reorganized and presented as De motu corporum ex percussione. Huygens produced the equations at the same time as his announcement of his rejection of Descartes’s rules of collision. Never intended for publication, the equations appear to have been used as preliminary scaffolding on which to build his critiques of Descartes’s physics. Additionally, Huygens used algebraic equations of this form to accurately predict the speeds of bodies after collision in experiments carried out at the Royal Society. Despite their deceptive simplicity, Huygens’s algebraic equations pose significant conceptual problems both mathematically and for their physical interpretation especially for negative speeds; they may very well have been the source of a new principle, the conservation of quantity of motion with direction.