This paper discerns two types of mathematization, a foundational and an explorative one. The foundational perspective is well-established, but we argue that the explorative type is essential when approaching the problem of applicability and how it influences our conception of mathematics. The first part of the paper argues that a philosophical transformation made explorative mathematization possible. This transformation took place in early modernity when sense acquired partial independence from reference. The second part of the paper discusses a series of examples (...) from the history of mathematics that highlight the complementary nature of the foundational and exploratory types of mathematization. (shrink)
During the first phase of Greek mathematics a proof consisted in showing or making visible the truth of a statement. This was the epagogic method. This first phase was followed by an apagogic or deductive phase. During this phase visual evidence was rejected and Greek mathematics became a deductive system. Now epagoge and apagoge, apart from being distinguished, roughly according to the modern distinction between inductive and deductive procedures, were also identified on account of the conception of generality as continuity. (...) Epistemology of mathematics today only remembers the distinction, forgetting where they agreed, in this manner not only destroying the unity of the perceptual and conceptual but also forgetting what could be gained from Aristotelian demonstrative science. (shrink)
Anstelle einer Einleitung: ein Thema und eine Sichtweise desselben -- Analytische Philsophie zwischen Sprache, Logik und Mathematik -- Die analytische Philosophie und das Phänomenon der Komplementarität -- Kant, Bolzano und Peirce: die Unterschedung des Analytischen und Synthetischen, oder: von der Erkenntnistheorie zur Semantik und Zeichentheorie -- Ernst Cassirer und die Entwicklung von Analyse und Synthese seit Descartes und Leibniz -- Bertrand Russell (1872-1970) -- Die naturalisierte Erkenntnistheorie zwischen Wiener Kries und Pragmatismus: Willard Van Orward [sic] Quine -- Richard Rorty: der (...) Spiegel der Natur -- Die analytische Philosophie, der Mensch, die Kunst und das Denken der Maschinen. (shrink)
In the development of pure mathematics during the nineteenth and twentieth centuries, two very different movements had prevailed. The so-called rigor movement of arithmetization, which turned into set theoretical foundationalism, on the one hand, and the axiomatic movement, which originated in Poncelet's or Peirce's emphasis on the continuity principle, on the other hand. Axiomatical mathematics or mathematics as diagrammatic reasoning represents a genetic perspective aiming at generalization, whereas mathematics as arithmetic or set theory is mainly concerned with foundation and separation. (...) We may thus conclude when Peirce defines mathematics in terms of diagrammatic reasoning, that this implies some very profound distinctions in the epistemology of mathematics. (shrink)
The dynamics of the historical growth of mathematics as well as natural science is reflected by the interaction of the continuity principle and the principle of the identity of indiscernibles. Aristotle, for example on the one side is responsible for introducing the continuity principle into natural history. On the other hand he is regarded as the great representative of a logic which rests upon the assumption of the possibility of clear divisions and rigorous classification. In modern times the Aristotelian antithesis (...) presents itself in the problem of the concept of mechanical motion, and its mathematical formulation by means of the function concept. Leibniz' philosophy provides a differentiated and broad picture of the problem in which both principles play a major role. To talk of these means, according to Gueroult, "to take a central perspective from which one can perceive the unity of this colossal world of ideas as well as the sometimes contradictory components inherent in it". The contradictory elements alluded to here are I believe strongly related to the difference between philosophical and positive thought. Within positive scientific thought the continuity principle ranks very prominently, whereas the real importance of the principium indiscernibilium gets into view only from a philosophical perspective. (shrink)
A historical case which is most illuminating with respect to the interrelation between philosophy and mathematics is provided by Leibniz’ work. Two principles are fundamental in Leibniz’ philosophy and his philosophy of mathematics: the principle of identity of indiscernibles and the continuity principle. As Gueroult says, to speak about these two principles amounts “to assuming a central perspective from which both the enormity of his conceptual world and the contradictory components it contains become visible”.
The ArgumentIn writing the history of science, the fluctuations between two meanings of the concept of style are of special interest: a simple or direct meaning of this concept referring to a means of expression and of presentation, and a philosophical interpretation of this term referring to “a world of objective spiritual order.” The last two chapters of this paper consider the perspective of the simple meaning of the concept, the first two chapters take the philosophical meaning as their starting (...) point.The concept of style in its general epistemological meaning emerges within a conceptual space that becomes effective as a totality at the end of the eighteenth century and which is built up of further notions such as: individual, genius, expression, symbol, education, creativity, and others.The individual and, as believed, the nevertheless infinitely creative subject has taken the place that the concept of god had occupied within rationalism. But it is not only the subject as construction and will, but also the subject who reflected in a new way about the objective foundations of his conscience and tried to bring the object and the means of knowledge into a new relation. (shrink)