### Mathematics: Truth and Fiction?

- in Mathematics.’ Philosophia Mathematica
, 1999

"... n advanced against what he argues is the best version of platonism. More specifically, he defends what he calls full-blooded platonism (`FBP'), the view that every mathematical object that could possibly exist does exist. It is important to the conclusions later in the book that FBP is the only viab ..."

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n advanced against what he argues is the best version of platonism. More specifically, he defends what he calls full-blooded platonism (`FBP'), the view that every mathematical object that could possibly exist does exist. It is important to the conclusions later in the book that FBP is the only viable form of platonism, so in this first section Balaguer also attempts to demonstrate that all other platonist positions are indefensible. In the second part of the book, Balaguer tries to show that no good arguments have been advanced against (a broadly Fieldian kind of) fictionalism. Although it is fictionalism that Balaguer defends, he also makes it clear that other anti-realist positions, such as deductivism and formalism, are more or less equivalent to fictionalism and so he has no serious quarrel with them. He prefers fictionalism, however, because it "provides a standard semantics for the language of mathematics" (p. 104), whereas other anti-realist accounts (such as Chihara [1990], f

### VISUALIZATION OF ORDINALS ∗

"... We describe the pictorial representations of infinite ordinals used in teaching set theory, and discuss a possible use in naturalistic foundations of mathematics. 1 ..."

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We describe the pictorial representations of infinite ordinals used in teaching set theory, and discuss a possible use in naturalistic foundations of mathematics. 1

### From Hilbert’s Program to a Logic Tool Box

"... www.cs.technion.ac.il/∼janos Abstract. In this paper I discuss what, according to my long experience, every computer scientists should know from logic. We concentrate on issues of modeling, interpretability and levels of abstraction. We discuss what the minimal toolbox of logic tools should look lik ..."

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www.cs.technion.ac.il/∼janos Abstract. In this paper I discuss what, according to my long experience, every computer scientists should know from logic. We concentrate on issues of modeling, interpretability and levels of abstraction. We discuss what the minimal toolbox of logic tools should look like for a computer scientist who is involved in designing and analyzing reliable systems. We shall conclude that many classical topics dear to logicians are less important than usually presented, and that less known ideas from logic may be more useful for the working computer scientist. For Witek Marek, first mentor, then colleague and true friend, on the occasion of his 65th birthday.

### Naturalising Mathematics: A Critical Look at the Quine-Maddy Debate

"... This paper considers Maddy’s strategy for naturalising mathematics in the context of Quine’s scientific naturalism. The aim of this proposal is to account for the acceptability of mathematics on scientific grounds without committing to revisionism about mathematical practice entailed by the Quine-Pu ..."

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This paper considers Maddy’s strategy for naturalising mathematics in the context of Quine’s scientific naturalism. The aim of this proposal is to account for the acceptability of mathematics on scientific grounds without committing to revisionism about mathematical practice entailed by the Quine-Putnam indispensability argument. It has been argued that Maddy’s mathematical naturalism makes inconsistent assumptions on the role of mathematics in scientific explanations to the effect that it cannot distinguish mathematics from pseudo-science. I shall clarify Maddy’s arguments and show that the objection can be overcome. I shall then reformulate a novel version of the objection and consider a possible answer, and I shall conclude that mathematical naturalism does not ultimately provide a viable strategy for accommodating an anti-revisionary stance on mathematics within a Quinean naturalist framework.

### WE HOLD THESE TRUTHS TO BE SELF-EVIDENT: BUT WHAT DO WE MEAN BY THAT?

"... Mathematicians at first distrusting the new ideas (Cantor made his first discoveries in 1873), then got used to them;... Waismann (1982, p. 102) Abstract. At the beginning of Die Grundlagen der Arithmetik (§2) [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where pro ..."

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Mathematicians at first distrusting the new ideas (Cantor made his first discoveries in 1873), then got used to them;... Waismann (1982, p. 102) Abstract. At the beginning of Die Grundlagen der Arithmetik (§2) [1884], Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”. This, of course, is true, but thinkers differ on why it is that mathematicians prefer proof. And what of propositions for which no proof is possible? What of axioms? This talk explores various notions of self-evidence, and the role they play in various foundational systems, notably those of Frege and Zermelo. I argue that both programs are undermined at a crucial point, namely when self-evidence is supported by holistic and even pragmatic considerations. At the beginning of Die Grundlagen der Arithmetik (§2) (1884), Gottlob Frege observes that “it is in the nature of mathematics to prefer proof, where proof is possible”, noting that “Euclid gives proofs of many things which anyone would concede him without question”. Frege sets himself the task of providing proofs of such basic arithmetic propositions as

### ∗For encouragement or helpful criticism in this project thanks are

, 2007

"... responsible for all errors that remain. Unless otherwise indicated in the text, translations from German are by the author. The original German passages will be confined to footnotes wherever possible. 1 1 ..."

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responsible for all errors that remain. Unless otherwise indicated in the text, translations from German are by the author. The original German passages will be confined to footnotes wherever possible. 1 1

### Toward a topic-specific logicism? Russell’s theory of geometry in the Principles of Mathematics.

"... “The tragedy of Russell’s paradox was to obscure from Frege and from us the great interest of his actual positive accomplishment”. ([Boolos, 1998], p. 267) ..."

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“The tragedy of Russell’s paradox was to obscure from Frege and from us the great interest of his actual positive accomplishment”. ([Boolos, 1998], p. 267)

### The cognitive basis of arithmetic

"... Arithmetic is the theory of the natural numbers and one of the oldest areas of mathematics. Since almost all other mathematical theories make use of numbers in some way or other, arithmetic is also one of the most fundamental theories of mathematics. But numbers are not just abstract entities ..."

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Arithmetic is the theory of the natural numbers and one of the oldest areas of mathematics. Since almost all other mathematical theories make use of numbers in some way or other, arithmetic is also one of the most fundamental theories of mathematics. But numbers are not just abstract entities

### Philosophia Mathematica (III) 13 (2005), 115–134. doi:10.1093/philmat/nki010 ‘Mathematical Platonism ’ Versus Gathering the Dead: What Socrates teaches Glaucon †

"... Glaucon in Plato’s Republic fails to grasp intermediates. He confuses pursuing a goal with achieving it, and so he adopts ‘mathematical platonism’. He says mathematical objects are eternal. Socrates urges a seriously debatable, and seriously defensible, alternative centered on the destruction of hyp ..."

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Glaucon in Plato’s Republic fails to grasp intermediates. He confuses pursuing a goal with achieving it, and so he adopts ‘mathematical platonism’. He says mathematical objects are eternal. Socrates urges a seriously debatable, and seriously defensible, alternative centered on the destruction of hypotheses. He offers his version of geometry and astronomy as refuting the charge that he impiously ‘ponders things up in the sky and investigates things under the earth and makes the weaker argument the stronger’. We relate his account briefly to mathematical developments by Plato’s associates Theaetetus and Eudoxus, and then to the past 200 years ’ developments in geometry. Plato was much less prodigal of affirmation about metaphysical ultimates than interpreters who take his myths literally have supposed. (Paul Shorey [1935], p. 130) Mathematics views its most cherished answers only as springboards to deeper questions. (Barry Mazur [2003], p. 225)