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Eine psychologische Studie. As many other students of Brentano, Ehrenfels engaged repeatedly with the philosophy of mathematics, but until now his dissertation remained nearly completely unknown. Alexius Meinong in 19th Century Philosophy. Brentano School in 19th Century Philosophy. Christian von Ehrenfels in 19th Century Philosophy. Growing evidence indicates that a Brentanist philosophy of mathematics was already in place before Husserl. Rather than an original combination at the confluence of two different streams, his early writings represent an elaboration of topics and problems that were already being discussed in the School of Brentano within a pre-existing framework.

Brentano and Other Philosophers in 19th Century Philosophy. Husserl: Development and Influences in Continental Philosophy. Husserl: Philosophy of Mathematics in Continental Philosophy. The concept of a Mannigfaltigkeit in Husserl has been given various interpretations, due to its shifting role in his works. Mannigfaltigkeitslehre would then not be a Cantorian set-theory, but come rather closer to topology. Then, in the Prolegomena, Husserl introduces the idea of a pure Mannigfaltigkeitslehre as a meta-theoretical enterprise which studies the relations among theories, e.

When Husserl announces that in fact the best example of such a pure theory of manifolds is what is actually practiced in mathematics, this sounds slightly misleading. The pure theory of theories cannot simply be the mathematics underlying topology, but should rather be considered as a mathesis universalis. The mathesis universalis in this sense is formal, a priori and analytic, as theory of theory in general. It is an analysis of the highest categories of meaning and their correlative categories of objects.

Areas of Mathematics in Philosophy of Mathematics. Quantity is the first category that Aristotle lists after substance.

### About An Historical Introduction to the Philosophy of Mathematics: A Reader

Continuous quantities such as the ratio of circumference to diameter of a circle are as clearly known as discrete ones. The theory that mathematics was "the science of quantity" was once the leading philosophy of mathematics. The article looks at puzzles in the classification and epistemology of quantity. Epistemology of Mathematics, Misc in Philosophy of Mathematics. Theories of Mathematics, Misc in Philosophy of Mathematics. The leaders of the Scientific Revolution were not Baconian in temperament, in trying to build up theories from data.

Their project was that same as in Aristotle's Posterior Analytics: they hoped to find necessary principles that would show why the observations must be as they are. Their use of mathematics to do so expanded the Aristotelian project beyond the qualitative methods used by Aristotle and the scholastics. In many cases they succeeded. History of Physics in Philosophy of Physical Science. Scientific Revolutions in General Philosophy of Science.

The Application of Mathematics in Philosophy of Mathematics. Joseph Agassi is an Israeli scholar born in Jerusalem on May 7, He has many books and articles published contributing to the fields of logic, scientific method, foundations of sciences, epistemology and, most importantly for this Journal, in the historiography of science. He studied with Karl Popper, who was definitely his biggest influence.

He taught around the world in different universities. He currently lives in Herzliya, Israel. For his important contribution to the historiography of science, we chose to open Babbage wrote two relatively detailed, yet significantly incongruous, autobiographical accounts of his pre-Cambridge and Cambridge days. He published one in and in it advertised the existence of the other, which he carefully retained in manuscript form. The aim of this paper is to chart in some detail for the first time the discrepancies between the two accounts, to compare and assess their relative credibility, and to explain their author's possible reasons for knowingly fabricating the less credible of the two.

I will focus on and give relevance to the Lectures on the Foundations of Mathematics, plus the Remarks on the Foundations of Mathematics. Rather, contradictions are problematic when we do not know what to infer from them.

## The Philosophy of Mathematics Education

Once a meaning is established through a new rule of inference, the contradiction becomes a usable expression like many others in our inferential apparatus. Thus, the apparent problem is dissolved. Finally, I will take three examples of dissolved contradictions from Wittgenstein to clarify further his notion. I will conclude considering why his position on contradictions led him to clash with Alan Turing, and whether the latter was convinced by the Wittgensteinian proposal. History of Logic in Logic and Philosophy of Logic.

Logical Pluralism in Logic and Philosophy of Logic.

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Ludwig Wittgenstein in 20th Century Philosophy. Its editor, Sofia Alexandrovna Yanovskaya — , was a renowned Soviet mathematician, whose significant studies on the foundations of mathematics and mathematical logic, as well as on the history and philosophy of mathematics are unduly neglected nowadays.

Yanovskaya, as a militant Marxist, was actively engaged in the ideological confrontation with idealism and its influence on modern mathematics and their interpretation. Concomitantly, she was one of the pioneers of mathematical logic in the Soviet Union, in an era of fierce disputes on its compatibility with Marxist philosophy. Yanovskaya managed to embrace in an originally Marxist spirit the contemporary level of logico-philosophical research of her time.

Due to her highly esteemed status within Soviet academia, she became one of the most significant pillars for the culmination of modern mathematics in the Soviet Union. Among the several issues I discuss, her encounter with L. Wittgenstein is striking. Mathematical Logic in Formal Sciences. This is the first complete English translation of Gottlob Frege's Grundgesetze der Arithmetik and , with introduction and annotation. As the culmination of his ground-breaking work in the philosophy of logic and mathematics, Frege here tried to show how the fundamental laws of arithmetic could be derived from purely logical principles.

Logical Empiricism in 20th Century Philosophy. Review of Douglas Patterson. Alfred Tarski: Philosophy of Language and Logic. Alfred Tarski in 20th Century Philosophy.

History of Western Philosophy. It is argued that geometrical intuition, as conceived in Kant, is still crucial to the epistemological foundations of mathematics. For this purpose, I have chosen to target one of the most sympathetic interpreters of Kant's philosophy of mathematics — Michael Friedman — because he has formulated the possible historical limitations of Kant's views most sharply. I claim that there are important insights in Kant's theory that have survived the developments of modern mathematics, and thus, that they are not so intrinsically These insights include the idea that mathematical knowledge relies on the manipulation of objects given in quasi-perceptual intuition, as Charles Parsons has argued, and that pure intuition is a source of knowledge of space itself that cannot be replaced by mere propositional knowledge.

In particular, it is pointed out that it is the isomorphism between Kantian intuition and a spatial manifold that underlies both the epistemic intimacy of the most fundamental type of geometrical intuition as well as that of perceptual acquaintance. Charles Sanders Peirce in 19th Century Philosophy.

Cardinals and Ordinals in Philosophy of Mathematics. Philosophy of Mathematics. Philosophers unacquainted with the workings of actual scientific practice are prone to imagine that our best scientific theories deliver univocal representations of the physical world that we can use to calibrate our metaphysics and epistemology. Those few philosophers who are also scientists, like Heinrich Hertz, tend to contest this assumption.

In his preface he motivates his book by noting that most of the work on Hertz's philosophy of science fails to engage with what Hertz does after the Introduction of his Principles. Austrian Philosophy in European Philosophy. Geometry in Philosophy of Mathematics. Thought Experiments in Metaphilosophy. Science, Logic, and Mathematics. This book is a welcome contribution to the literature on Kant's philosophy of mathematics in two particular respects.

First, the author systematically traces the development of Kant's thought on mathematics from the very early pre-Critical writings through to the Critical philosophy.

Secondly, it puts forward a challenge to contemporary Anglo-Saxon commentators on Kant's philosophy of mathematics which merits consideration. A central theme of the book is that an adequate understanding of Kant's pronouncements on mathematics must begin with the recognition that mathematics For Kant, Euclidean geometry, with its heavy reliance on the geometric image, was the paradigm of certainty. The algebraic revolution of the nineteenth century replaced that paradigm with an algebraic formalism, thereby freeing mathematics from any connection to the geometric image, and also severing the link to intuition.

So great was the shift, Pierobon suggests, that, after the developments of the nineteenth century, it became difficult to find any sense in Kant's conception of mathematics as sensible knowledge.

This book attempts to offer a corrective to that position by offering a Kantian conception of mathematics …. Notre but consiste au contraire On the contrary, this article aims at reconstructing the coherence and originality of the Vorlesungen. More importantly, we will insist on the remarkable logical procedures worked out by Pasch in order to adapt his mathematical development to the strictures of his broad philosophical position. His friend from their days in Vienna, Rudolf Carnap, was in the audience, and afterward wrote a note to himself in which he raised a number of questions on incompleteness.

In Helmholtz discussed the foundations of measurement in science as a last contribution to his philosophy of knowledge. This essay borrowed from earlier debates on the foundations of mathematics, on the possibility of quantitative psychology, and on the meaning of temperature measurement. Yet it inspired two mathematicians with an eye on physics, and a few philosopher-physicists.

Du Bois-Reymond; H.

Grassmann; J. Measurement in Science in General Philosophy of Science. I resist this.

## Montreal Inter-University Workshop on History and Philosophy of Mathematics

Simples and Gunk in Metaphysics. This paper argues that kant's general epistemology incorporates a theory of algebra which entails a less constricted view of kant's philosophy of mathematics than is sometimes given. To extract a plausible theory of algebra from the "critique of pure reason", It is necessary to link kant's doctrine of mathematical construction to the idea of the "schematism".