悖论的自指性与循环性

本文使用语义网分析悖论与自指性和循环性。主要结论是证明了有穷悖论都是自指的,同时其矛盾性必定基于循环性。我们还证明存在非自指但基于循环性的(无穷)悖论,比如亚布鲁悖论及其一般变形;又证明了存在自指但不基于循环性的(无穷)悖论,比如超穷赫兹伯格悖论和麦基悖论。这表明自指性与循环性对悖论而言是两个不同的概念。     

Finite mathematics

A system of finite mathematics is proposed that has all of the power of classical mathematics. I believe that finite mathematics is not committed to any form of infinity, actual or potential, either within its theories or in the metalanguage employed to specify them. I show in detail that its commitments to the infinite are no stronger than those of primitive recursive arithmetic. The finite mathematics of sets is comprehensible and usable on its own terms, without appeal to any form of the infinite. That makes it possible to, without circularity, obtain the axioms of full Zermelo-Fraenkel Set Theory with the Axiom of Choice (ZFC) by extrapolating (in a precisely defined technical sense) from natural principles concerning finite sets, including indefinitely large ones. The existence of such a method of extrapolation makes it possible to give a comparatively direct account of how we obtain knowledge of the mathematical infinite. The starting point for finite mathematics is Mycielski's work on locally finite theories.I would like to thank Jeff Barrett, Akeel Bilgrami, Leigh Cauman, John Collins, William Craig, Gary Feinberg, Haim Gaifman, Yair Guttmann, Hidé Ishiguro, Isaac Levi, James Lewis, Vann McGee, Sidney Morgenbesser, George Shiber, Sarah Stebbins, Mark Steiner, and an anonymous referee for encouragement and various useful suggestions. The research described in this article and the preparation of the article were supported in part by the Columbia University Council for Research in the Humanities.     

Two Ways Of Looking At A Newtonian Supertask

A supertask is a process in which an infinite number of individuated actions are performed in a finite time. A Newtonian supertask is one that obeys Newton's laws of motion. Such supertasks can violate energy and momentum conservation and can exhibit indeterministic behavior. Perez Laraudogoitia, who proposed several Newtonian supertasks, uses a local, i.e., particle-by-particle, analysis to obtain these and other paradoxical properties of Newtonian supertasks. Alper and Bridger use a global analysis, embedding the system of particles in a Banach space, to determine the origin of the strange behavior. This paper provides a common framework for the discussion of both the local and global methods of analysis. Using this single framework, the areas of disagreement and agreement are made explicit. Further examples of supertasks are proposed to illuminate various aspects of the discussion.     

Arrovian Aggregation of Generalised Expected-Utility Preferences: (Im)possibility Results by Means of Model Theory

Cerreia-Vioglio et al. (Econ Theory 48(2–3):341–375, 2011) have proposed a very general axiomatisation of preferences in the presence of ambiguity, viz. Monotonic Bernoullian Archimedean preference orderings. This paper investigates the problem of Arrovian aggregation of such preferences—and proves dictatorial impossibility results for both finite and infinite populations. Applications for the special case of aggregating expected-utility preferences are given. A novel proof methodology for special aggregation problems, based on model theory (in the sense of mathematical logic), is employed.     

The axiology of Robert S. Hartman: A critical study

Formal axiology is based on the logical nature of meaning, namely intension, and on the structure of intension as a set of predicates. It applies set theory to this set of predicates. Set theory is a certain kind of mathematics that deals with subsets in general, and of finite and infinite sets in particular. Since mathematics is objective and a priori, formal axiology is an objective and a priori science; and a test based on it is an objective test based on an objective standard.

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Marxist Axioms as Self-Contradictory Parsonian Statements in Sociology

This paper examines the implicit foundations of several theoretical propositions characteristic of the Marxist tradition in sociology. It argues that these propositions derive from self-contradictory critical premises which are paradoxes of Action Theory. Implicit in these premises is an ideal picture of social reality quite different from the one analytically described by Parsons. I suggest that Action Theory can provide conceptual tools needed to address some specific issues characteristic of the Marxist perspective and, moreover, offers a solution to some epistemological problems raised by the authors who endorse the postmodern position in social sciences.     

Thomson's Lamp is Dysfunctional

James Thomson envisaged a lamp which would be turned on for 1 minute, off for 1/2 minute, on for 1/4 minute, etc. ad infinitum. He asked whether the lamp would be on or off at the end of 2 minutes. Use of “internal set theory” (a version of nonstandard analysis), developed by Edward Nelson, shows Thomson's lamp is chimerical; its copy within set theory yields a contradiction. The demonstration extends to placing restrictions on other “infinite tasks” such as Zeno's paradoxes of motion and Kant's First Antinomy. Resolution of such logical-philosophical problems leads to some very general constraints which must be placed upon the syntax of physical theories. In particular, at some scale space and time would appear granular. The suitability of internal set theory for analyzing phenomena is examined, using a paper by Alper and Bridger (1997) to frame the discussion. This revised version was published online in June 2006 with corrections to the Cover Date.     

The three arrows of Zeno

We explore the better known paradoxes of Zeno including modern variants based on infinite processes, from the point of view of standard, classical analysis, from which there is still much to learn (especially concerning the paradox of division), and then from the viewpoints of non-standard and non-classical analysis (the logic of the latter being intuitionist).The standard, classical or Cantorian notion of the continuum, modeled on the real number line, is well known, as is the definition of motion as the time derivative of distance (we are not concerned with position and motion in more than one dimension, since Zeno wasn't). The real number line consists of its points, the distance between distinct points being positive and finite. The standard, classical derivative relies on the classical notion of limit, which does not use infinitesimals.In non-standard analysis, the real line is again the set of its points, but infinitesimal distances between points are allowed, while the derivative is defined in terms of the ratio of infinitesimals. This has the surprising consequence that there is a function of time giving positions, which is not constant, but whose derivative is everywhere zero, so that a particle whose position is given by this function can move a finite distance in a finite time, while being at rest all along. Such a function, of course is external — it can't be defined in the formal language. More suprising still, a model of the non-standard real line can be found which is internally finite — that is there is an injection from an initial segment of non-negative integers to the non-standard interval [0,1) whose range includes all the standard real numbers, and the ratio of the number of points in the range in any subset S of the interval to the number of those in the entire interval differs by an infinitesimal from the Lebesgue measure of the set of standard points in S. Of course, the formalism can't tell the difference between standard and non-standard integers or points — that is an external concept. Still, this allows a discrete model of the line with points an infinitesimal distance apart, without sacrificing any of the results of standard analysis, including measure theory.Non-classical analysis is based on topos theory in an intuitionist setting. All curves are piecewise linear, or straight over infinitesimal distances, and the non-classical derivative is defined as the slope in such intervals. So whereas the direction of a non-standard smooth curve changes infinitesimally over infinitesimal distances, the direction of a non-classical curve (all of them are smooth), doesn't change at all over infitesimal distances. And while a standard or non-standard line can be identified with the set of its points, the points on a non-classical line might be said not to occupy all positions, and the line is an object which is not the set of its points.We explore the implications of all this for the various paradoxes of Zeno, and some modern variants.     

The resolution of two paradoxes by approximate reasoning using a fuzzy logic

Summary The method of approximate reasoning using a fuzzy logic introduced by Baldwin (1978 a,b,c), is used to model human reasoning in the resolution of two well known paradoxes.It is shown how classical propositional logic fails to resolve the paradoxes, how multiple valued logic partially succeeds and that a satisfactory resolution is obtained with fuzzy logic.The problem of precise representation of vague concepts is considered in the light of the results obtained.     

Moving Without Being Where You’re Not; A Non-Bivalent Way

The classical response to Zeno’s paradoxes goes like this: ‘Motion cannot properly be defined within an instant. Only over a period’ (Vlastos.) I show that this ob-jection is exactly what it takes for Zeno to be right. If motion cannot be defined at an instant, even though the object is always moving at that instant, motion cannot be defined at all, for any longer period of time identical in content to that instant. The nonclassical response introduces discontinuity, to evade the paradox of infinite proximity of any point of a distance with any ‘next’. But it introduces the wrong sort of discontinuity because, rather than assuming the discontinuity of motion, as Quantum Theory does, it assumes the discontinuity of space. Due then to the resulting spacetime disorder, though all else is certainly lost, the Tortoise now turns up at least as fast as Achilles and hence not even this much is rescued. Zeno rejects motion because he shows that a moving object must be where it is not. Hence motion, if to occur, must violate the Law of Contradiction (LNC). Applying the concept of quantum discontinuity, I produce an alternative. If an object is to move discontinuously between two boundary points, A and B, what actually obtains is, rather, that it is nowhere at all in-between A and B. And cannot therefore be at two places in-between A and B. And cannot therefore be where it is not. Thus, LNC is conserved. However, in these conditions, the Law of the Excluded Middle (LEM) fails. To mitigate the undesirability of this effect, I show that LEM fails because LNC holds. Thus, the resulting nonbivalent logic, which is also appropriate for quantized transitions of all kinds, will always turn up nonbivalent, because consistent. And this is not too bad, considering.     

Surreal decisions

Although expected utility theory has proven a fruitful and elegant theory in the finite realm, attempts to generalize it to infinite values have resulted in many paradoxes. In this paper, we argue that the use of John Conway's surreal numbers shall provide a firm mathematical foundation for transfinite decision theory. To that end, we prove a surreal representation theorem and show that our surreal decision theory respects dominance reasoning even in the case of infinite values. We then bring our theory to bear on one of the more venerable decision problems in the literature: Pascal's Wager. Analyzing the wager showcases our theory's virtues and advantages. To that end, we analyze two objections against the wager: Mixed Strategies and Many Gods. After formulating the two objections in the framework of surreal utilities and probabilities, our theory correctly predicts that (1) the pure Pascalian strategy beats all mixed strategies, and (2) what one should do in a Pascalian decision problem depends on what one's credence function is like. Our analysis therefore suggests that although Pascal's Wager is mathematically coherent, it does not deliver what it purports to, a rationally compelling argument that people should lead a religious life regardless of how confident they are in theism and its alternatives.     

Fluids,Molecules and Paradoxes of Infinity

Several paradoxes of infinity have recently featured in this journal involving gases distributed in a denumerable infinite series of compartments. I shall demonstrate in this paper that:

a) None of these new paradoxes applies where the gases comply with both Boyle’s law and Avogadro’s law. As several of these new paradoxes expressly require compliance with Boyle’s law, it is unclear, in principle, as to whether there is a plausible model of gas that is able to uphold them all.

b) Notwithstanding a), any of the above paradoxes (and their variations) can be reinstated by acknowledging (contrary to what is widely assumed in the literature) that there are two distinct, non-equivalent concepts of ideal gas. Indeed, the various infinity puzzles actually enable a distinction to be made between the two concepts (which is a particularly elegant way of doing so).

     

Aggregate Theory Versus Set Theory

Maddy's (1990) arguments against Aggregate Theory were undermined by the shift in her position in 1997. The present paper considers Aggregate Theory in the light of this, and the recent search for `New Axioms for Mathematics'. If Set Theory is the part-whole theory of singletons, then identifying singletons with their single members collapses Set Theory into Aggregate Theory. But if singletons are not identical to their single members, then they are not extensional objects and so are not a basis for Science. Either way, the Continuum Hypothesis has no physical interest.     

Six Groups of Paradoxes in Ancient China From the Perspective of Comparative Philosophy

This paper divides the sophisms and paradoxes put forth by Chinese thinkers of the pre-Qin period of China (before 221 BCE) into six groups: paradoxes of motion and infinity, paradoxes of class membership, semantic paradoxes, epistemic paradoxes, paradoxes of relativization, other logical contradictions. It focuses on the comparison between the Chinese items and the counterparts of ancient Greek and even of contemporary Western philosophy, and concludes that there turn out to be many similar elements of philosophy and logic at the beginnings of Chinese and Greek civilizations.     

Hobson’s Conception of Definable Numbers

In this paper, I explore an intriguing view of definable numbers proposed by a Cambridge mathematician Ernest Hobson, and his solution to the paradoxes of definability. Reflecting on König’s paradox and Richard’s paradox, Hobson argues that an unacceptable consequence of the paradoxes of definability is that there are numbers that are inherently incapable of finite definition. Contrast to other interpreters, Hobson analyses the problem of the paradoxes of definability lies in a dichotomy between finitely definable numbers and not finitely definable numbers. To bypass this predicament, Hobson proposes a language dependent analysis of definable numbers, where the diagonal argument is employed as a means to generate more and more definable numbers. This paper examines Hobson’s work in its historical context, and articulates his argument in detail. It concludes with a remark on Hobson’s analysis of definability and Alan Turing’s analysis of computability.     

Logical, Semantic and Cultural Paradoxes

The property common to three kinds of paradoxes (logical, semantic, and cultural) is the underlying presence of an exclusive disjunction: even when it is put to a check by the paradox, it is still invoked at the level of implicit discourse. Hence the argumentative strength of paradoxical propositions is derived. Logical paradoxes (insolubilia) always involve two contradictory, mutually exclusive, truths. One truth is always perceived to the detriment of the other, in accordance with a succession which is endlessly repetitive. A check is put on the principle of the excluded middle by the logical paradoxes, because self-reference leads to an endlessly repeating circle, out of which no resolution is conceivable. Logical paradoxes are to be compared with the `objective ambiguity' prevalent in oracles (Gallet, 1990). Semantic paradoxes are contextually-determined occurrences, whose resolution at the metalinguistic level is made possible by the discovery of a middle term. They express a wilful ambiguity, in which the interlocutor is invited to take an active part in the construction of sense, since what must be found is the unexpected sense thanks to which A and not-A can be asserted simultaneously. Cultural paradoxes play about doxa (`common sense') and openly challenge common opinion because of their character as inopinata (`unexpected'). My aim is to show that even cultural paradoxes hide sometimes a flaw of argumentation similar to logical or semantic paradox; they too imply an exclusive disjunction leading to the disappearance of the middle terms. Finally, basing myself on the theory of topoi (Anscombre and Ducrot, 1983), a tentative resolution of the cultural paradoxes will be suggested.     

Oppy,infinity, and the neoclassical concept of God

In this article I concentrate on three issues. First, Graham Oppy’s treatment of the relationship between the concept of infinity and Zeno’s paradoxes lay bare several porblems that must be dealt with if the concept of infinity is to do any intellectual work in philosophy of religion. Here I will expand on some insightful remarks by Oppy in an effort ot adequately respond to these problems. Second, I will do the same regarding Oppy’s treatment of Kant’s first antinomy in the first critique, which deals in part with the question of whether the world had a beginning in time or if time extends infinitely into the past. And third, my examination of these two issues will inform what I have to say regarding a key topic in philosophy of religion: the question regarding the proper relationship between the infinite and the finite in the concept of God.     

Time intervals production in tapping and oscillatory motion

We applied spectral analysis on series of time intervals produced in a synchronization-continuation experiment. In the first condition intervals were produced by finger tapping, and in the second by an oscillatory motion of the hand. Results obtained in tapping were consistent with a discrete, event-based timing model. In the oscillatory condition, the spectra suggested a continuous, dynamic timing mechanism, based on the regulation of effector stiffness. It is concluded that the oscillatory character of movement can offer an important resource for timing control. The use of an event-based timing control such as postulated in the Wing-Kristoffersson model could be restricted to a quite limited class of rhythmic tasks, characterized by the concatenation of discrete events.     

An epistemological use of nonstandard analysis to answer Zeno's objections against motion

Three of Zeno's objections to motion are answered by utilizing a version of nonstandard analysis, internal set theory, interpreted within an empirical context. Two of the objections are without force because they rely upon infinite sets, which always contain nonstandard real numbers. These numbers are devoid of numerical meaning, and thus one cannot render the judgment that an object is, in fact, located at a point in spacetime for which they would serve as coordinates. The third objection, an arrow never appears to be moving, is answered by showing that it only applies to a finite number of instants of time. A theory of motion is also advanced; it consists of a finite series of contiguous infinitesimal steps. The theory is immune to Zeno's first two objections because the number of steps is finite and each lies outside the domain of observation. Present motion is hypothesized to be an unobservable process taking place within each step. The fact of motion is apparent through a summing (Riemann integration) of the steps.     

What the Liar Taught Achilles

Zeno's paradoxes of motion and the semantic paradoxes of the Liar have long been thought to have metaphorical affinities. There are, in fact, isomorphisms between variations of Zeno's paradoxes and variations of the Liar paradox in infinite-valued logic. Representing these paradoxes in dynamical systems theory reveals fractal images and provides other geometric ways of visualizing and conceptualizing the paradoxes.