THE MATHEMATICS-NATURAL SCIENCES ANALOGY AND THE UNDERLYING LOGIC -THE ROAD THROUGH THOUGHT EXPERIMENTS AND RELATED METHODS 1 penultimate draft (original) (raw)
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Croatian Journal of Philosophy, 2018
The aim of this paper is to point to the analogy between mathematical and physical thought experiments, and even more widely between the epistemic paths in both domains. Having accepted platonism as the underlying ontology as long as the platonistic path in asserting the possibility of gaining knowledge of abstract, mind-independent and causally inert objects, my widely taken goal is to show that there is no need to insist on the uniformity of picture and monopoly of certain epistemic paths in the epistemic descriptive context. And secondly, to show the analogy with the ways we come to know the truths of (natural) sciences.
The Applicability of Mathematics as a Scientific and a Logical Problem
Philosophia Mathematica, vol.18 (2010.6), no.2, p.144-165, 2010
This paper explores how to explain the applicability of classical mathematics to the physical world in a radically naturalistic and nominalistic philosophy of mathematics. The applicability claim is first formulated as an ordinary scientific assertion about natural regularity in a class of natural phenomena and then turned into a logical problem by some scientific simplification and abstraction. I argue that there are some genuine logical puzzles regarding applicability and no current philosophy of mathematics has resolved these puzzles. Then I introduce a plan for resolving the logical puzzles of applicability.
Review of Philosophy (Revista de Filosofie), 2014
Contemporary philosophical accounts of the applicability of mathematics in physical sciences and the empirical world are based on formalized relations between the mathematical structures and the physical systems they are supposed to represent within the models. Such relations were constructed both to ensure an adequate representation and to allow a justification of the validity of the mathematical models as means of scientific inference. This article puts in evidence the various circularities (logical, epistemic, and of definition) that are present in these formal constructions and discusses them as an argument for the alternative semantic and propositional-structure accounts of the applicability of mathematics. Keywords: philosophy of mathematics; applicability of mathematics; mathematical entities; mapping accounts; semantic accounts; circular definition; epistemic circularity; Frege; formal language; second-order logic; first-order logic; contingent truth; isomorphisms
La question épistémologique de l'applicabilité des mathématiques
2018
International audienceThe question of the applicability of mathematics is an epistemological issue that was explicitly raised by Kant, and which has played different roles in the works of neo-Kantian philosophers, before becoming an essential issue in early analytic philosophy. This paper will first distinguish three main issues that are related to the application of mathematics: (1) indispensability arguments that are aimed at justifying mathematics itself; (2) philosophical justifications of the successful application of mathematics to scientific theories; and (3) discussions on the application of real numbers to the measurement of physical magnitudes. A refinement of this tripartition is suggested and supported by a historical investigation of the differences between Kant's position on the problem, several neo-Kantian perspectives (Helmholtz and Cassirer in particular, but also Otto Hölder), early analytic philosophy (Frege), and late 19th century mathematicians (Grassmann, D...
A Defense of Scientific Platonism without Metaphysical Presuppositions
Abstract: From the Platonistic standpoint, mathematical edifices form an immaterial, unchanging, and eternal world that exists independently of human thought. By extension, “scientific Platonism” says that directly mathematizable physical phenomena – in other terms, the research field of physics – are governed by entities belonging to this objectively existing mathematical world. Platonism is a metaphysical theory. But since metaphysical theories, by definition, are neither provable nor refutable, anti-Platonistic approaches cannot be less metaphysical than Platonism itself. In other words, anti-Platonism is not “more scientific” than Platonism. All we can do is to compare Platonism and its negations under epistemological criteria such as simplicity, economy of hypotheses, or consistency with regard to their respective consequences. In this paper I intend to show that anti-Platonism claiming in a first approximation (i) that mathematical edifices consist of meaningless signs assembled according to arbitrary rules, and (ii) that the adequacy of mathematical entities and phenomena covered by physics results from idealization of these phenomena, is based as much as Platonism on metaphysical presuppositions. Thereafter, without directly taking position, I try to launch a debate focusing on the following questions: (i) To maintain its coherence, is anti-Platonism not constrained to adopt extremely complex assumptions, difficult to defend, and not always consistent with current realities or practices of scientific knowledge? (ii) Instead of supporting anti-Platonism whatever the cost, in particular by the formulation of implausible hypotheses, would it not be more adequate to accept the idea of a mathematical world existing objectively and governing certain aspects of the material world, just as we note the existence of the material world which could also not exist?
A Defense Of Platonic Realism In Mathematics: Problems About The …
The conflict between Platonic realism and Constructivism marks a watershed in philosophy of mathematics. Among other things, the controversy over the Axiom of Choice is typical of the conflict. Platonists accept the Axiom of Choice, which allows a set consisting of the members resulting from infinitely many arbitrary choices, while Constructivists reject the Axiom of Choice and confine themselves to sets consisting of effectively specifiable members. Indeed there are seemingly unpleasant consequences of the Axiom of Choice. The non-constructive nature of the Axiom of Choice leads to the existence of non-Lebesgue measurable sets, which in turn yields the Banach-Tarski Paradox. But the Banach-Tarski Paradox is so called in the sense that it is a counter-intuitive theorem. To corroborate my view that mathematical truths are of non-constructive nature, I shall draw upon Gödel's Incompleteness Theorems. This also shows the limitations inherent in formal methods. Indeed the Löwenheim-Skolem Theorem and the Skolem Paradox seem to pose a threat to υlatonists. In this light, Quine/υutnam's arguments come to take on a clear meaning. According to the model-theoretic arguments, the Axiom of Choice depends for its truth-value upon the model in which it is placed. In my view, however, this is another limitation inherent in formal methods, not a defect for Platonists. To see this, we shall examine how mathematical models have been developed in the actual practice of mathematics. I argue that most mathematicians accept the Axiom of Choice because the existence of non-Lebesgue measurable sets and the Well-Ordering of reals open the possibility of more fruitful mathematics. Finally, after responding to ψenacerraf's challenge to Platonism, I conclude that in mathematics, as distinct from natural sciences, there is a close connection between essence and existence. Actual mathematical theories are the parts of the maximally logically consistent theory that describes mathematical reality. 1 I will use the word-Constructivism‖ in a broader sense than ψrower's ωonstructivism. In ψrower's Constructivism mathematical entities are constructible in our mind. But I will use the word-Constructivism‖ in a narrower sense than Gödel's axiom of constructibility. Gödel's Axiom of Constructibility is a much stronger assumption than Constructivism as I call it. Choice that we couldn't otherwise. Platonists accept the Axiom of Choice, which allows a set consisting of the members resulting from infinitely many arbitrary choices, while Constructivists reject the Axiom of Choice and confine themselves to sets consisting of effectively specifiable members (Chapter 1). Lebesgue's theory of measure will set the stage for discussing the Banach-Tarski Paradox and the existence of measurable cardinals in later chapters. Also, since Lebesgue is one of the French Constructivists, it is interesting to see the non-constructive nature of Lebesgue measure creates an irreconcilable tension with Lebesgue's skeptical attitude toward the Axiom of Choice (Chapter 2). The Hausdorff Paradox is the prototype of the Banach-Tarski Paradox. Informally, the Hausdorff Paradox states that a sphere is decomposed into finite number of pieces and reassembled by rigid motions to form two copies of almost the same size as the original. Here-almost‖ means-except on a countable subset.‖ ψanach and Tarski made improvement on the Hausdorff Paradox by eliminating the need to exclude a countable subset from a sphere. Informally, the Banach-Tarski Paradox states that a sphere is decomposed into finite number of pieces and reassembled by rigid motions to form two copies of exactly the same size as the original. The Banach-Tarski Paradox deepened the skepticism about the Axiom of Choice. But the Banach-Tarski Paradox is so called in the sense that it is a counter-intuitive theorem, as distinct from a logical contradiction or a fallacious reasoning. I argue that we should accept the Banach-Tarski Paradox as a Platonic truth and rejects epistemology based on a mathematical intuition (Chapter 3). Next, from a slightly different perspective, I corroborate my view that mathematical truths are of non-constructive nature. Once we got the undecidability of Peano Arithmetic (PA), Gödel's First Incompleteness Theorem is immediate. The set of true sentences in PA is not recursively enumerable. But the set of theorems (provable sentences) in PA is recursively enumerable. So it is easy to see that there is a sentence that is true but unprovable. This implies that there are some arithmetical truths we cannot get access to in an effective way. We also have to note Gödel's Incompleteness Theorems show that there are limitations inherent in formal methods (Chapter 4).
Human Thought, Mathematics, and Physical Discovery - Final
Mathematical Knowledge, Objects and Applications: Essays in Memory of Mark Steiner, 2022
In this paper I discuss Mark Steiner's view of the contribution of mathematics to physics and take up some of the questions it raises. In particular, I take up the question of discovery and explore two aspects of this question ‒ a metaphysical aspect and a related epistemic aspect. The metaphysical aspect concerns the formal structure of the physical world. Does the physical world have mathematical or formal features or constituents, and what is the nature of these constituents? The related epistemic question concerns humans' cognitive ability to reach the formal structure of the physical world. Among other things, I explore the interaction of mathematical and non-mathematical cognition in physical discovery.
On the Relation between Mathematics, Natural Sciences, And Scientific Inquiry
In this article, we will shortly review a few old thoughts and recent thoughts on the relation between Mathematics and the Natural Sciences. Of course, the classic references to this open problem will include Wigner’s paper (1964); a more recent review article is Darvas (2008). But it appears that this issue is partly on the domain of natural philosophy and also philosophy of inquiry. Therefore we will begin with a review on some known thoughts of Kant, Bacon, Popper, etc.