Understanding the infinite II: Coalgebra

In this paper we give an account of the rise and development of coalgebraic thinking in mathematics and computer science as an illustration of the way mathematical frameworks may be transformed. Originating in a foundational dispute as to the correct way to characterise sets, logicians and computer scientists came to see maximizing and minimizing extremal axiomatisations as a dual pair, each necessary to represent entities of interest. In particular, many important infinitely large entities can be characterised in terms of such axiomatisations. We consider reasons for the delay in arriving at the coalgebraic framework, despite many unrecognised manifestations occurring years earlier, and discuss an apparent asymmetry in the relationship between algebra and coalgebra.     

The infinite and infinitesimal quantities of du Bois-Reymond and their reception

The mixed fortunes of Paul du Bois-Reymond's infinitary calculus and ideal boundary between convergence and divergence are traced from 1870 to 1914. Cantor, Dedekind, Peano, Russell, Pringsheim and others objected. Stolz, Borel, Hardy and others accepted, at least in part, and built further. Hausdorff to some extent effected a compromise. The differing attitudes of the different participants toward infinitesimals and infinitely large quantities are described. In addition to shedding some light on the status of infinites, the whole story serves as a study of mathematical change, and shows that the proof-refutation process described by Lakatos is only one aspect of mathematical development. The story also sheds some light on the rise of 20^th century functional analysis, algebra and topology.Wer je den grossen Bau der Welt bedacht und fühlte nicht, wie Gottes hoher Geist noch über den Gesetzen wacht und kreist-wie blind erscheint, wer Schöpfertum verlacht! Wir kennen kaum den kleinsten Teil davon: Gesetz ist Wunder, Zahl ist Weltenton. Albrecht Haushofer, Moabit Sonnets, 1944, LXIX, Kosmos     

What conceptual spaces can do for Carnap's late inductive logic

In the last published account of his late inductive logic, the Basic System of Inductive Logic, Rudolf Carnap introduced a new element to the systems of inductive logic, namely the so-called attribute spaces. These geometrical structures model the meanings of the predicates of the object language and have a similar structure as the conceptual spaces employed by cognitive scientists like Peter Gärdenfors. I show how the development of the theory of conceptual spaces helps us to see the addition of attribute spaces as a step forward in explicating the concept of confirmation. I discuss the differences and similarities of the two theories and investigate the possibilities for developing further connections.     

Bootstrap prediction intervals for autoregressive models of unknown or infinite lag order

Recent studies on bootstrap prediction intervals for autoregressive (AR) model provide simulation findings when the lag order is known. In practical applications, however, the AR lag order is unknown or can even be infinite. This paper is concerned with prediction intervals for AR models of unknown or infinite lag order. Akaike's information criterion is used to estimate (approximate) the unknown (infinite) AR lag order. Small‐sample properties of bootstrap and asymptotic prediction intervals are compared under both normal and non‐normal innovations. Bootstrap prediction intervals are constructed based on the percentile and percentile‐t methods, using the standard bootstrap as well as the bootstrap‐after‐bootstrap. It is found that bootstrap‐after‐bootstrap prediction intervals show small‐sample properties substantially better than other alternatives, especially when the sample size is small and the model has a unit root or near‐unit root. Copyright © 2002 John Wiley & Sons, Ltd.     

Taking times out: Tense logic as a theory of time

Ulrich Meyer’s book The Nature of Time uses tense logic to argue for a ‘modal’ view of time, which replaces substantial times (as in Newton’s Absolute Time) with ‘ersatz times’ constructed using conceptually basic tense operators. He also argues against Bertrand Russell’s relationist theory, in which times are classes of events, and against the idea that relativity compels the integration of time and space (called by Meyer the Inseparability Argument). I find fault with each of these negative arguments, as well as with Meyer’s purported reconstruction of empty spacetime from tense operators and substantial spatial points. I suggest that Meyer’s positive project is best conceived as an elimination of time in the mode of Julian Barbour's The End of Time.     

The logic of experimental tests,particularly of Everettian quantum theory

Claims that the standard procedure for testing scientific theories is inapplicable to Everettian quantum theory, and hence that the theory is untestable, are due to misconceptions about probability and about the logic of experimental testing. Refuting those claims by correcting those misconceptions leads to an improved theory of scientific methodology (based on Popper׳s) and testing, which allows various simplifications, notably the elimination of everything probabilistic from the methodology (‘Bayesian’ credences) and from fundamental physics (stochastic processes).     

Three challenges to the complementarity of the logic and the pragmatics of science

The bipartite metatheory thesis attributes to Rudolf Carnap, Philipp Frank and Otto Neurath a conception of the nature of post-metaphysical philosophy of science that sees the purely formal-logical analyses of the logic of science as complemented by empirical inquiries into the psychology, sociology and history of science. Three challenges to this thesis are considered in this paper: that Carnap did not share this conception of the nature of philosophy of science even on a programmatic level, that Carnap's detailed analysis of the language of science is incompatible with one developed by Neurath for the pursuit of empirical studies of science, and, finally, that Neurath himself was confused about the programme of which the bipartite metatheory thesis makes him a representative. I argue that all three challenges can be met and refuted.     

一种基于一阶逻辑的软件代码安全性缺陷静态检测技术

软件代码安全性缺陷是可能引发软件系统高危后果的一类重要缺陷,针对该类缺陷的自动化检测和定位技术在软件维护和演化研究领域具有重要意义.本文提出并实现了一种形式化检测方法——基于一阶逻辑的软件代码安全性缺陷静态检测方法,利用命题逻辑和谓词逻辑定义模式路径公式,引入多个与依赖关系相关的谓词构造逻辑函数表达式,作为模式路径节点产生的制导条件,实现了对多类软件代码安全性缺陷的形式化描述,把安全性缺陷检测问题转化成在中间代码对应的有限状态空间中是否存在相应模式路径公式的判定问题.实验结果表明,该方法能适用于大多数类型的软件代码安全性缺陷检测,在对openssl,wu-ftpd等13个开源程序的测试中,准确重现了10个已公开安全漏洞,发现2个未公开安全漏洞.并且,与现有的模型检验等形式化静态分析方法相比,该方法的测试时间和代码规模成渐近线性关系.