A Constructivist View of Newton’s Mechanics

In the present essay we attempt to reconstruct Newtonian mechanics under the guidance of logical principles and of a constructive approach related to the genetic epistemology of Piaget and García (Psychogenesis and the history of science, Columbia University Press, New York, 1989). Instead of addressing Newton’s equations as a set of axioms, ultimately given by the revelation of a prodigious mind, we search for the fundamental knowledge, beliefs and provisional assumptions that can produce classical mechanics. We start by developing our main tool: the no arbitrariness principle, that we present in a form that is apt for a mathematical theory as classical mechanics. Subsequently, we introduce the presence of the observer, analysing then the relation objective–subjective and seeking objectivity going across subjectivity. We take special care of establishing the precedence among all contributions to mechanics, something that can be better appreciated by considering the consequences of removing them: (a) the consequence of renouncing logic and the laws of understanding is not being able to understand the world, (b) renouncing the early elaborations of primary concepts such as time and space leads to a dissociation between everyday life and physics, the latter becoming entirely pragmatic and justified a-posteriori (because it is convenient), (c) changing our temporary beliefs has no real cost other than effort. Finally, we exemplify the present approach by reconsidering the constancy of the velocity of light. It is shown that it is a result of Newtonian mechanics, rather than being in contradiction with it. We also indicate the hidden assumption that leads to the (apparent) contradiction.Fil: Solari, Hernan Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Física de Buenos Aires. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Física de Buenos Aires; ArgentinaFil: Natiello, Mario Alberto. Lund University; Sueci

Logical foundations of physics. Resolution of classical and quantum paradoxes in the finitistic paraconsistent logic NAFL

Non-Aristotelian finitary logic (NAFL) is a paraconsistent logic that redefines finitism and correctly captures the notion of a potential infinity. It is argued that the existence of nonstandard models of arithmetic is an artifact of infinitary classical semantics, which must be rejected by the finitist, for whom the meaning of ``finite'' is not negotiable. A decisive critique of classical / intuitionistic propositional logic and classical infinitary reasoning, which require pre-existing truths, is given. This leads directly to the definition of NAFL truth as time-dependent axiomatic declarations of the human mind via provability in NAFL theories, with the consequent negative resolution of Hilbert's program. If the axioms of an NAFL theory T are pairwise consistent, then T is consistent. This metatheorem, which is the more restrictive counterpart of the compactness theorem of classical first-order logic, leads to the conclusion that T supports only constructive existence, and consequently, nonstandard models of T do not exist, which in turn implies that infinite sets cannot exist in consistent NAFL theories. It is shown that arithmetization of syntax, Godel's incompleteness theorems and Turing's undecidability of the halting problem, which lead classically to nonstandard models, cannot be formalized in NAFL theories. The NAFL theories of arithmetic and real numbers are defined. Several paradoxical phenomena in quantum mechanics, such as, quantum superposition, Wigner's friend paradox, entanglement, the quantum Zeno effect and wave-particle duality, are shown to be justifiable in NAFL, which provides a logical basis for the incompatibility of quantum mechanics and infinitary (by the NAFL yardstick) relativity theory. Zeno's dichotomy paradox and its variants, which lead to meta-inconsistencies in classical infinitary reasoning, are shown to be resolvable in NAFL

Directional Bias

There is almost a consensus among conditional experts that indicative conditionals are not material. Their thought hinges on the idea that if indicative conditionals were material, A → B could be vacuously true when A is false, even if B would be false in a context where A is true. But since this consequence is implausible, the material account is usually regarded as false. It is argued that this point of view is motivated by the grammatical form of conditional sentences and the symbols used to represent their logical form, which misleadingly suggest a one-way inferential direction from A to B. That conditional sentences mislead us into a directionality bias is a phenomenon that is well-documented in the literature about conditional reasoning. It is argued that this directional appearance is deceptive and does not reflect the underlying truth conditions of conditional sentences. This directional bias is responsible for both the unpopularity of the material account of conditionals and some of the main alternative principles and themes in conditional theory, including the Ramsey’s test, the Equation, Adams’ thesis, conditional-assertion and possible world theories. The directional mindset forgets a hard- earned lesson that made classical logic possible in the first place, namely, that grammatical form of sentences can mislead us about its truth conditions. There is a case to be made for a material account of indicative conditionals when we break the domination of words over the human mind

Buying Logical Principles with Ontological Coin: The Metaphysical Lessons of Adding epsilon to Intuitionistic Logic

We discuss the philosophical implications of formal results showing the con- sequences of adding the epsilon operator to intuitionistic predicate logic. These results are related to Diaconescu’s theorem, a result originating in topos theory that, translated to constructive set theory, says that the axiom of choice (an “existence principle”) implies the law of excluded middle (which purports to be a logical principle). As a logical choice principle, epsilon allows us to translate that result to a logical setting, where one can get an analogue of Diaconescu’s result, but also can disentangle the roles of certain other assumptions that are hidden in mathematical presentations. It is our view that these results have not received the attention they deserve: logicians are unlikely to read a discussion because the results considered are “already well known,” while the results are simultaneously unknown to philosophers who do not specialize in what most philosophers will regard as esoteric logics. This is a problem, since these results have important implications for and promise signif i cant illumination of contem- porary debates in metaphysics. The point of this paper is to make the nature of the results clear in a way accessible to philosophers who do not specialize in logic, and in a way that makes clear their implications for contemporary philo- sophical discussions. To make the latter point, we will focus on Dummettian discussions of realism and anti-realism. Keywords: epsilon, axiom of choice, metaphysics, intuitionistic logic, Dummett, realism, antirealis

Logical foundations of physics. Resolution of classical and quantum paradoxes in the finitistic paraconsistent logic NAFL

Non-Aristotelian finitary logic (NAFL) is a finitistic paraconsistent logic that redefines finitism and correctly captures the notion of a potential infinity. Classical infinitary reasoning is refuted in NAFL, with the consequent negative resolution of Hilbert's program. It is argued that the existence of nonstandard models of arithmetic is an artifact of infinitary classical semantics, which must be rejected by the finitist, for whom the meaning of ``finite'' is not negotiable. The main postulate of NAFL semantics defines formal truth as time-dependent axiomatic declarations of the human mind, a consequence of which is the following metatheorem. If the axioms of an NAFL theory T are pairwise consistent, then T is consistent. This metatheorem, which is the more restrictive counterpart of the compactness theorem of classical first-order logic, leads to the conclusion that T supports only constructive existence, and consequently, nonstandard models of T do not exist, which in turn implies that infinite sets cannot exist in consistent NAFL theories. It is shown that arithmetization of syntax, Godel's incompleteness theorems and Turing's undecidability of the halting problem, which lead classically to nonstandard models, cannot be formalized in NAFL theories. The NAFL theories of arithmetic and real numbers are defined. Several paradoxical phenomena in quantum mechanics, such as, quantum superposition, Wigner's friend paradox, entanglement, the quantum Zeno effect and wave-particle duality, are shown to be justifiable in NAFL, which provides a logical basis for the incompatibility of quantum mechanics and infinitary (by the NAFL yardstick) relativity theory. Finally, Zeno's dichotomy paradox and its variants, which lead to meta-inconsistencies in classical infinitary reasoning, are shown to be resolvable in NAFL

An Objection to Naturalism and Atheism from Logic

I proffer a success argument for classical logical consequence. I articulate in what sense that notion of consequence should be regarded as the privileged notion for metaphysical inquiry aimed at uncovering the fundamental nature of the world. Classical logic breeds necessitism. I use necessitism to produce problems for both ontological naturalism and atheism