14,537 research outputs found
From Logical Calculus to Logical FormalityâWhat Kant Did with Eulerâs Circles
John Venn has the âuneasy suspicionâ that the stagnation in mathematical logic between J. H. Lambert and George Boole was due to Kantâs âdisastrous effect on logical method,â namely the âstrictest preservation [of logic] from mathematical encroachment.â Kantâs actual position is more nuanced, however. In this chapter, I tease out the nuances by examining his use of Leonhard Eulerâs circles and comparing it with Eulerâs own use. I do so in light of the developments in logical calculus from G. W. Leibniz to Lambert and Gottfried Ploucquet. While Kant is evidently open to using mathematical tools in logic, his main concern is to clarify what mathematical tools can be used to achieve. For without such clarification, all efforts at introducing mathematical tools into logic would be blind if not complete waste of time. In the end, Kant would stress, the means provided by formal logic at best help us to express and order what we already know in some sense. No matter how much mathematical notations may enhance the precision of this function of formal logic, it does not change the fact that no truths can, strictly speaking, be revealed or established by means of those notations
The Mathematical Background of Lomonosov's Contribution
This is a short overview of the influence of mathematicians and their ideas
on the creative contribution of Mikhailo Lomonosov on the occasion of the
tercentenary of his birth
The Emergence of Symbolic Algebra as a Shift in Predominant Models
Historians of science find it difficult to pinpoint to an exact period in which symbolic algebra came into existence. This can be explained partly because the historical process leading to this breakthrough in mathematics has been a complex and diffuse one. On the other hand, it might also be the case that in the early twentieth century, historians of mathematics over emphasized the achievements in algebraic procedures and underestimated the conceptual changes leading to symbolic algebra. This paper attempts to provide a more precise setting for the historical context in which this decisive step to symbolic reasoning took place. For that purpose we will consider algebraic problem solving as model-based reasoning and symbolic representation as a model. This allows us to characterize the emergence of symbolic algebra as a shift from a geometrical to a symbolic mode of representation. The use of the symbolic as a model will be situated in the context of mercantilism where merchant activity of exchange has led to reciprocal relations between money and wealth
On Kantâs Transcendental Argument(s)
Presented in the âCritique of Pure Reasonâ transcendental philosophy is the first theory of science,which seeks to identify and study the conditions of the possibility of cognition. Thus, Kant carries out a shift to the study of âmode of our cognitionâ and TP is a method, where transcendental argumentation acts as its essential basis. The article is devoted to the analysis of the transcendental arguments. In § 2 the background of йРâ transcendental method of Antiquity and Leibnizâs Principle of Sufficient Reason â are analyzed and their comparison with йРis given. § 3
is devoted to the analysis of TA in the broad and narrow senses; a formal propositional and presupposition models are proposed. In § 4 I discuss the difference between TA and metaphysicsâ modes of reasoning. It analyzes the Kantâs main limitations of the use TA shows its connection with the Modern Age and contemporary science
Historical objections against the number line
Historical studies on the development of mathematical concepts will help mathematics teachers to relate their studentsâ difficulties in understanding to conceptual problems in the history of mathematics. We argue that one popular tool for teaching about numbers, the number line, may not be fit for early teaching of operations involving negative numbers. Our arguments are drawn from the many discussions on negative numbers during the seventeenth and eighteenth centuries from philosophers and mathematicians such as Arnauld, Leibniz, Wallis, Euler and dâAlembert. Not only does division by negative numbers pose problems for the number line, but even the very idea of quantities smaller than nothing has been challenged. Drawing lessons from the history of mathematics, we argue for the introduction of negative numbers in education within the context of symbolic operations
On Kant's first insight into the problem of space dimensionality and its physical foundations
In this article it is shown that a careful analysis of Kant's "Thoughts on
the True Estimation of Living Forces" leads to a conclusion that does not match
the usually accepted interpretation of Kant's reasoning in 1747, according to
which the Young Kant supposedly establishes a relationship between the
tridimensionality of space and Newton's law of universal gravitation. Indeed,
it is argued that this text does not yield a satisfactory explanation of space
dimensionality, actually restricting itself to justify the tridimensionality of
extension.Comment: 14 page
Leibniz's Infinitesimals: Their Fictionality, Their Modern Implementations, And Their Foes From Berkeley To Russell And Beyond
Many historians of the calculus deny significant continuity between
infinitesimal calculus of the 17th century and 20th century developments such
as Robinson's theory. Robinson's hyperreals, while providing a consistent
theory of infinitesimals, require the resources of modern logic; thus many
commentators are comfortable denying a historical continuity. A notable
exception is Robinson himself, whose identification with the Leibnizian
tradition inspired Lakatos, Laugwitz, and others to consider the history of the
infinitesimal in a more favorable light. Inspite of his Leibnizian sympathies,
Robinson regards Berkeley's criticisms of the infinitesimal calculus as aptly
demonstrating the inconsistency of reasoning with historical infinitesimal
magnitudes. We argue that Robinson, among others, overestimates the force of
Berkeley's criticisms, by underestimating the mathematical and philosophical
resources available to Leibniz. Leibniz's infinitesimals are fictions, not
logical fictions, as Ishiguro proposed, but rather pure fictions, like
imaginaries, which are not eliminable by some syncategorematic paraphrase. We
argue that Leibniz's defense of infinitesimals is more firmly grounded than
Berkeley's criticism thereof. We show, moreover, that Leibniz's system for
differential calculus was free of logical fallacies. Our argument strengthens
the conception of modern infinitesimals as a development of Leibniz's strategy
of relating inassignable to assignable quantities by means of his
transcendental law of homogeneity.Comment: 69 pages, 3 figure
Possible Worlds in the Precipice: Why Leibniz Met Spinoza?
The main objective of the paper is to give initial answers to three important questions. Why did Leibniz visit Spinoza? Why did his preparation for this meeting include a modification of the ontological proof of God? What is the philosophical result of the meeting and what do possible worlds have to do with it? In order to provide answers, three closely related manuscripts by Leibniz from November 1676 have been compared and the slow conceptual change of his philosophical apparatus has been analyzed. The last of these manuscripts was presented and read in front of Spinoza. Around that time Leibniz abandoned the idea of plurality of worlds (cf. Tschirnhaus) and instead proposed the idea of possible worlds, thus introducing possibility into the (onto/theo)logical structure itself in order to avoid the âprecipiceâ of Spinozaâs necessity. What is interesting, however, is how exactly this conceptual change occurred at the end of 1676 and what its philosophical and methodological implications are
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