Plato Was Not a Mathematical Platonist

Although it is easy to use mathematical objects in daily life and in the mathematics classes, it is quite a difficult duty to give an account of the status of mathematical entities in a metaphysical level. And it is the duty of the philosopher of mathematics to think about those entities. Throughout history, certain mathematicians and mostly philosophers have tried to give accounts of the mathematical objects which are used in the works of mathematics. In the present paper, the main aim will be to describe a rather mainstream view of mathematical objects called 'Mathematical Platonism' and then to provide the reasons why this popular view of mathematical entities is not reliable by giving three different challenges.

Manuscrito, 2005

In this paper I examine arguments by Benacerraf and by Chihara against Gödel's platonistic philosophy of mathematics. The paper derives from my response to Benacerraf (1973) in the American Philosophical Association symposium Mathematical Truth. I briefly commented on Benacerraf’s paper at the symposium, and then developed my response in more detail, presenting versions of it at Cornell, Princeton, and Rockefeller in 1974-75, and later at other places as well. It was an important part of the background motivation for my book Logical Forms, where a more sustained examination of the issues raised by Benacerraf begins to be articulated.

2014

4 Introduction 5 Aims of the thesis 7 Statement of Terminology 9 Section One: The Republic 31 Chapter One: Introducing the Sun, Line and Cave 33 Chapter Two: Readings of the Allegories in Context 44 i. Knowledge, Belief and Gail Fine 44 ii. Propositions or Objects: Gonzalez on Fine 56 Chapter Three: My Reading 68 i. How Seriously Should We Take the Allegories? 68 ii. Ascending the Scale 78 iii. Noēsis and the Role Definition in Plato’s Epistemology 109 Chapter Four: A Closer Look at dianoia 112 i. The Dianoectic Image 113 ii. The Hypothesis 134 iii. Theaetetus: Hypothesis and Image in the Search for a Definition 146 Section Two: The Meno 152 Chapter One: Definition in the Meno 155 a. Meno’s Definitions 156 b. Socrates’ Definitions 159 c. What is Plato’s preferred answer? 162 ii. What role do definitions play in Plato’s epistemological scale? 169 a. Definition and Essence 170 b. Gail Fine and the Meno 171 Chapter Two: Aporia and the Psychology of Mathematics 181 i. What is aporia? 18...

2019

In this article I analyze the issue of many levels of reality that are studied by natural sciences. Particularly interesting is the level of mathematics and the question of the relationship between mathematics and the structure of the real world. The mathematical nature of the world has been considered since ancient times and is the subject of ongoing research for philosophers of science to this day. One of the viewpoints in this field is mathematical Platonism. In contemporary philosophy it is widely accepted that according to Plato mathematics is the domain of ideal beings (ideas) that are eternal and unalterable and exist independently from the subject’s beliefs and decisions. Two issues seem to be important here. The first issue concerns the question: was Plato really a proponent of present-day mathematical Platonism? The second one is of greater importance: how mathematics influences our understanding of the nature of the world on its many ontological levels? In the article I c...

Where Plato had robustly conceived of numbers as Mathematical Ideas generated by the supreme Principles and multiply instantiated in numerically distinct sensible objects, Aristotle rejects Mathematical Ideas and thinly re-conceives of numbers as no more than abstract concepts generalized by the intellect from quantities of numerical distinct sensible substances. Aristotle’s many criticisms of Plato’s theory of Mathematical Ideas are, however, an ignorant argument (ignorationes elenchi) that, not only disregards the eidetic generation of numbers from the supreme Principles, but may only plausibly succeed against his own forced re-conception of eidetic numbers as mathematical numbers. The many absurdities that Aristotle purports to derive from Plato's theory of Mathematical Ideas are thus the consequence of his own, rather than Plato's, conception of mathematical objects. The following commentary will (§I) describe how Aristotle re-conceives of Plato's Mathematical Ideas of eidetic numbers; (§II) defend Plato's theory of Mathematical Ideas against Aristotle's criticisms in Metaphysics XIII 6-8; and (§III) prosecute the case for Plato's transcendental argument for eidetic numbers against Aristotle's abstraction theory for mathematical numbers.

2004

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).

Philosophia Mathematica, 2008

Philosophers of mathematics agree that the only interpretation of arithmetic that takes that discourse at ‘face value’ is one on which the expressions ‘N’, ‘0’, ‘1’, ‘+’, and ‘×’ are treated as proper names. I argue that the interpretation on which these expressions are treated as akin to free variables has an equal claim to be the default interpretation of arithmetic. I show that no purely syntactic test can distinguish proper names from free variables, and I observe that any semantic test that can must beg the question. I draw the same conclusion concerning areas of mathematics beyond arithmetic.

2024

This thesis examines the nature and purpose of the Greek sciences ἀριθμητική, λογιστική and γεωμετρία in the texts of Plato. The statements of some other ancient authors are also mentioned, and the relevant modern research is consulted. ἀριθμός is at any instance, as Klein has already noted, 'a definite number of definite objects'. In Plato's philosophical ἀριθμητική, ἀριθμός seems to always consist of 'the odd and even', or it is the 'multitude of the μονάδων/units', just as in Euclid. Many of the key concepts of Plato's mathematics appear to have a hierarchical order, or a duality (perhaps later called 'προποδισμός' process, progression). Plato seems to employ a peculiar 'oracular/religious vocabulary' which is only recognized in the original Greek sources. There is an obvious form of 'spirituality' in the entire philosophy of Plato's mathematics. The source of the mathematical concepts, and of the 'Forms', is from a god (Prometheus?). The concept of the soul's purification and 'σωτηρία' (salvation) is probably one of the ultimate purposes of Plato's mathematics, along with the aim of reaching the 'Good' and 'Being'. ἀριθμητική, λογιστική and γεωμετρία draw the soul towards 'Truth' ("πρὸς ἀλήθειαν"), and this is one of their purposes and an oft-mentioned theme by Plato. It is concluded that Plato's mathematics is in its broadest extent an all-encompassing study of the very things (τῶν ὄντων) of nature and existence, in the background of a spiritual philosophy.