Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond

Many historians of the calculus deny significant continuity between infinitesimal calculus of the 17th century and 20th century developments such as Robinson's theory. Robinson's hyperreals, while providing a consistent theory of infinitesimals, require the resources of modern logic; thus many commentators are comfortable denying a historical continuity. A notable exception is Robinson himself, whose identification with the Leibnizian tradition inspired Lakatos, Laugwitz, and others to consider the history of the infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies, Robinson regards Berkeley's criticisms of the infinitesimal calculus as aptly demonstrating the inconsistency of reasoning with historical infinitesimal magnitudes. We argue that Robinson, among others, overestimates the force of Berkeley's criticisms, by underestimating the mathematical and philosophical resources available to Leibniz. Leibniz's infinitesimals are fictions, not logical fictions, as Ishiguro proposed, but rather pure fictions, like imaginaries, which are not eliminable by some syncategorematic paraphrase. We argue that Leibniz's defense of infinitesimals is more firmly grounded than Berkeley's criticism thereof. We show, moreover, that Leibniz's system for differential calculus was free of logical fallacies. Our argument strengthens the conception of modern infinitesimals as a development of Leibniz's strategy of relating inassignable to assignable quantities by means of his transcendental law of homogeneity.Comment: 69 pages, 3 figure

Certainty Closure: Reliable Constraint Reasoning with Incomplete or Erroneous Data

Constraint Programming (CP) has proved an effective paradigm to model and solve difficult combinatorial satisfaction and optimisation problems from disparate domains. Many such problems arising from the commercial world are permeated by data uncertainty. Existing CP approaches that accommodate uncertainty are less suited to uncertainty arising due to incomplete and erroneous data, because they do not build reliable models and solutions guaranteed to address the user's genuine problem as she perceives it. Other fields such as reliable computation offer combinations of models and associated methods to handle these types of uncertain data, but lack an expressive framework characterising the resolution methodology independently of the model. We present a unifying framework that extends the CP formalism in both model and solutions, to tackle ill-defined combinatorial problems with incomplete or erroneous data. The certainty closure framework brings together modelling and solving methodologies from different fields into the CP paradigm to provide reliable and efficient approches for uncertain constraint problems. We demonstrate the applicability of the framework on a case study in network diagnosis. We define resolution forms that give generic templates, and their associated operational semantics, to derive practical solution methods for reliable solutions.Comment: Revised versio

Arguments for the existence of God in Anselm's Proslogion chapter II and III

Anselm's argument for the existence of God in Proslogion Chap.II starts from the contention that `lq when a Fool hears `something-than-which-nothing-greater-can-be-thought', he understands what he hears, and what he understands is in his mind. This is a special feature of the Pros.II argument which distinguishes the argument from other ontological arguments set up by, for example, Descartes and Leibniz. This is also the context which makes semantics necessary for evaluation of the argument. It is quite natural to ask `lq What is understood by the Fool, and what is in his mind? It is essential for a proper consideration of the argument to identify the object which is understood by the Fool, and so, is in his mind. A semantics gives answers to the questions of `lq What the Fool understands? and `lq What is in the Fool's mind? If we choose a semantics as a meta-theory to interpret the Pros.II argument, it makes an effective guide to identify the object. It is a necessary condition for a proper evaluation of the Pros.II argument to fix our universe of discourse, especially since, in the argument, we are involved in such talk about existing objects as Anselm's contention that `when a Fool hears `something-than-which-nothing-greater-can-be-thought', he understands what he hears, and what he understands is in his mind. The ontology to which a semantic theory commits us will be accepted as our scope of objects when we introduce our semantic theory to interpret the Pros.II argument, and this ontological boundary constrains us to identify the object in a certain way. Consistent application of an ontology, most of all, is needed for the evaluation of the logical validity of an argument. If we take Frege's three-level semantics, we are ontologically committed to intensional entities, like meaning, as well as extensional entities. Sluga contends that Frege's anti-psychologism for meanings should not be interpreted as vindicating reification of intensional entities in relation to Frege's contextualism, that Frege's anti-psychologism with his contextualism is nothing but a linguistic version of Kantian philosophy for the transcendental unity of a judgement. There is, however, another possible interpretation of Frege's contextualism. According to Dummett, the significance of Frege's contextualism must be understood as a way of explanation for a word's having meaning. If Dummett's view is cogent, we could say that Frege's contextualism does not prevent our interpreting his semantics as being committed to intensional entities. We need not worry that Frege's over all semantics, especially with his contextualism, would internally deny the ontological interpretation of his theory. We see Anselm's argument for the existence of God in Pros.II is an invalid argument if we introduce Frege's three-level semantics, i.e. if we acknowledge meanings of words as entities in our universe of discourse. We can also employ extensional semantics for the interpretation of the Pros.II argument. According to extensionalists, like Quine and Kripke, we need not assume intensional entities, like meaning, to be part of our ontological domain. They argue that we can employ our language well enough without assuming intensional entities. If we choose extensional semantics as a meta-theory to interpret the Pros.II argument, it commits us only to extensional entities as objects in the Universe of our interpretation. In Sections 1.4 and 1.5, I show that extensional semantics makes the Pros.II argument a valid argument for the existence of God. `lq Necessary existence is the central concept of Anselm's argument for the existence of God in Proslogion Chap.III. It has been said that, even if the argument is formally valid, it cannot stand as a valid argument for the existence of God, since `lq necessary existence is an absurd concept like `lq round square. And further that even if there is a meaningful combination of concepts for `lq necessary existence, it cannot quality as a subject of an a priori argument. As objections to the interpretations which make the Pros.III argument valid, it has been argued that even if there is a concept of `lq necessary existence which is meaningful and there is another concept of `lq necessary existence which is suitable as a subject of an a priori argument, there is no concept of `lq necessary existence which is meaningful and at the same time suitable as a subject of an a priori argument. In Chap.2 and Chap.3, I try to show that there can be concepts of `lq necessary existence which are proof against these objections. Anselm's arguments for the existence of God in Proslogian Chap.II and Chap.III are logically valid arguments on some logical principles. Some fideists, K. Barth, for example, argue that Anselm's arguments for the existence of God in Proslogion are not proofs for the existence of God even if they are logically valid arguments. I raise the question how this attitude could be possible, in Chap.4 and Chap.5. Barth's fideistic interpretation of Anselm's Proslogion arguments does not find any flaw in the validity of the arguments, and it accepts the meaningfulness and truth of the premises even to the fool in Proslogion. If this is the case, i.e. if Barth's interpretation accepts the validity of the arguments and the truth of the premises, I raise the question, how can the arguments not be interpreted as proofs for the existence of God? How is it possible that the function of the arguments is not that of proving the existence of God? According to Wittgensteinian fideism, premises in the arguments should not be intelligible to those who do not believe in God's existence already, and so the real function of the arguments is the elucidation, the understanding of believer's belief, rather than proving articles of belief to unbelievers. Barth's fideistic interpretation of the arguments, however, fully recognizes the meaningfulness and truth of the premises in the arguments as well as the validity of the arguments. I argue that there could be a justification for the Barthian fideism. As Malcolm notices, there are still atheists who understand Anselm's arguments as valid, but the only possibility for the people who recognize the validity of Anselm's arguments still to remain atheists has been thought to be to challenge the truth of premises employed in the arguments. Now, of the atheistic possibility, we can change the direction of our attention, that is, to the question about the function of a logically valid argument itself. What has not been thought of in relation to Anselm's arguments is the significance of logical truth or the logical validity of an argument. We have not asked such questions as `lq What does a logical truth say? and `lq What does a logically valid argument guarantee with true premises? Let us assume that even the premises are accepted by atheists. Do they all convert to theism? If that were so, the disagreement between atheist and believer over the ontological arguments should turn only on the truth of premises. If that is not so, there is some point in raising this other question. If there are people who, recognizing the premises and validity of an argument, are still reluctant to accept the conclusion, we have reason to question the function of a valid argument. I argue that there is a way of being consistently reasonable while accepting the premises and the validity of the ontological arguments and yet remaining an atheist or an agnostic

Representation and estimation of stochastic populations

This work is concerned with the representation and the estimation of populations composed of an uncertain and varying number of individuals which can randomly evolve in time. The existing solutions that address this type of problems make the assumption that all or none of the individuals are distinguishable. In other words, the focus is either on specific individuals or on the population as a whole. Theses approaches have complimentary advantages and drawbacks and the main objective in this work is to introduce a suitable representation for partially-indistinguishable populations. In order to fulfil this objective, a sufficiently versatile way of quantifying different types of uncertainties has to be studied. It is demonstrated that this can be achieved within a measure-theoretic Bayesian paradigm. The proposed representation of stochastic populations is then used for the introduction of various filtering algorithms from the most general to the most specific. The modelling possibilities and the accuracy of one of these filters are then demonstrated in different situations

Model Checking and Model-Based Testing : Improving Their Feasibility by Lazy Techniques, Parallelization, and Other Optimizations

This thesis focuses on the lightweight formal method of model-based testing for checking safety properties, and derives a new and more feasible approach. For liveness properties, dynamic testing is impossible, so feasibility is increased by specializing on an important class of properties, livelock freedom, and deriving a more feasible model checking algorithm for it. All mentioned improvements are substantiated by experiments

'God exists': meaning, reference and Anselm’s proslogion

Over the last century, philosophy has comprehensively criticised the 'common- sense' view of the proposition 'God exists' as being meaningfixl. The purpose of this thesis is therefore to show that instances of 'God exists' can be considered meaningful, whether or not God does in fact exist. From the intuitive premise of compositionality - that the meaning of a proposition is determined by the meaning of its parts - I ask what options 'God exists' presents. Its appearance is that of a simple subject-predicate sentence, restricting possible difficulties in interpreting compositionality; it appears to take a subject and attribute a property to that subject. However, several problems are apparent. The first is the concept of existence. The first chapter, therefore, compares the views of Bertrand Russell with recent work by Colin McGinn, arguing in favour of existence as a predicate. McGinn presents a challenge to allowing the predication of existence of 'God’, centred around the concepts by which ontological arguments characterise 'God'. The second chapter, as an historical-theological angle on the meaningfulness of 'God exists’, takes up this challenge in an attempt to resolve it using Anselm's Proslogion, which is traditionally thought to demonstrate the existence of God by using the idea of God. Analysis of the Proslogion and the thought underlying it do not provide an entirely acceptable resolution, but lay the foundations for the remainder of the thesis.The third chapter argues for the rejection of McGinn's challenge. Having provided arguments for seeing 'God exists' as a subject-predicate sentence, and noted the difficulties in conceiving adequately of God, I address the problem of what account to give of 'God’. Against a background of debate in the philosophy of language, I advocate understanding 'God' as a name in God exists', and argue for a view of the meaning and reference of 'God’ based upon the work of Jerome Gellman. Finally, I combine relevant elements from existence, reference and meaning - incorporating theological suggestions arising from Anselm - to provide a model for the meaningfulness of 'God exists' which, I argue, demonstrates God exists' to be a meaningful proposition if God does in fact exist or if God does not in fact exist

Chiral Rings and Anomalies in Supersymmetric Gauge Theory

Motivated by recent work of Dijkgraaf and Vafa, we study anomalies and the chiral ring structure in a supersymmetric U(N) gauge theory with an adjoint chiral superfield and an arbitrary superpotential. A certain generalization of the Konishi anomaly leads to an equation which is identical to the loop equation of a bosonic matrix model. This allows us to solve for the expectation values of the chiral operators as functions of a finite number of ``integration constants.'' From this, we can derive the Dijkgraaf-Vafa relation of the effective superpotential to a matrix model. Some of our results are applicable to more general theories. For example, we determine the classical relations and quantum deformations of the chiral ring of $\N=1$ super Yang-Mills theory with SU(N) gauge group, showing, as one consequence, that all supersymmetric vacua of this theory have a nonzero chiral condensate.Comment: 67 pages, minor change

Against `Realism'

We examine the prevalent use of the phrase ``local realism'' in the context of Bell's Theorem and associated experiments, with a focus on the question: what exactly is the ``realism'' in ``local realism'' supposed to mean? Carefully surveying several possible meanings, we argue that all of them are flawed in one way or another as attempts to point out a second premise (in addition to locality) on which the Bell inequalities rest, and (hence) which might be rejected in the face of empirical data violating the inequalities. We thus suggest that this vague and abused phrase ``local realism'' should be banned from future discussions of these issues, and urge physicists to revisit the foundational questions behind Bell's Theorem

On the possibility of stable regularities without fundamental laws

This doctoral dissertation investigates the notion of physical necessity. Specifically, it studies whether it is possible to account for non-accidental regularities without the standard assumption of a pre-existent set of governing laws. Thus, it takes side with the so called deflationist accounts of laws of nature, like the humean or the antirealist. The specific aim is to complement such accounts by providing a missing explanation of the appearance of physical necessity. In order to provide an explanation, I recur to fields that have not been appealed to so far in discussions about the metaphysics of laws. Namely, I recur to complex systems’ theory, and to the foundations of statistical mechanics. The explanation proposed is inspired by how complex systems’ theory has elucidated the way patterns emerge, and by the probabilistic explanations of the 2nd law of thermodynamics. More specifically, this thesis studies how some constraints that make no direct reference to the dynamics can be a sufficient condition for obtaining in the long run, with high probability, stable regular behavior. I hope to show how certain metaphysical accounts of laws might benefit from the insights achieved in these other fields. According to the proposal studied in this thesis, some regularities are not accidental not in virtue of an underlying physical necessity. The non-accidental character of certain regular behavior is only due to its overwhelming stability. Thus, from this point of view the goal becomes to explain the stability of temporal patterns without assuming a set of pre-existent guiding laws. It is argued that the stability can be the result of a process of convergence to simpler and stable regularities from a more complex lower level. According to this project, if successful, there would be no need to postulate a (mysterious) intermediate category between logical necessity and pure contingency. Similarly, there would be no need to postulate a (mysterious) set of pre-existent governing laws. Part I of the thesis motivates part II, mostly by arguing why further explanation of the notions of physical necessity and governing laws should be welcomed (chapter 1), and by studying the plausibility of a lawless fundamental level (chapters 2 and 3). Given so, part II develops the explanation of formation of simpler and stable behavior from a more complex underlying level

Improved Detection Criteria for the Multi-dimensional Optimal Order Detection (MOOD) on unstructured meshes with very high-order polynomials

This paper extends the MOOD method proposed by the authors in ["A high-order finite volume method for hyperbolic systems: Multidimensional Optimal Order Detection (MOOD)", J. Comput. Phys. 230, pp 4028-4050, (2011)], along two complementary axes: extension to very high-order polynomial reconstruction on non-conformal unstructured meshes and new Detection Criteria. The former is a natural extension of the previous cited work which confirms the good behavior of the MOOD method. The latter is a necessary brick to overcome limitations of the Discrete Maximum Principle used in the previous work. Numerical results on advection problems and hydrodynamics Euler equations are presented to show that the MOOD method is effectively high-order (up to sixth-order), intrinsically positivity-preserving on hydrodynamics test cases and computationally efficient