Online research in philosophy

We present a formal system, E, which provides a faithful model of the proofs in Euclid's Elements, including the use of diagrammatic reasoning.

J. H. Lambert proved important results of what we now think of as non-Euclidean geometries, and gave examples of surfaces satisfying their theorems. I use his philosophical views to explain why he did not think the certainty of Euclidean geometry was threatened by the development of what we regard as alternatives to it. Lambert holds that theories other than Euclid’s fall prey to skeptical doubt. So despite their satisfiability, for him these theories are not equal to Euclid’s in justification. Contrary to recent interpretations, then, Lambert does not conceive of mathematical justification as semantic. According to Lambert, Euclid overcomes doubt by means of postulates. Euclid’s theory thus owes its justification not to the existence of the surfaces that satisfy it, but to the postulates according to which these “models” are constructed. To understand Lambert’s view of postulates and the doubt they answer, I examine his criticism of Christian Wolff’s views. I argue that Lambert’s view reflects insight into traditional mathematical practice and has value as a foil for contemporary, model-theoretic, views of justification.

Though pictures are often used to present mathematical arguments, they are not typically thought to be an acceptable means for presenting mathematical arguments rigorously . With respect to the proofs in the Elements in particular, the received view is that Euclid’s reliance on geometric diagrams undermines his efforts to develop a gap-free deductive theory. The central difficulty concerns the generality of the theory. How can inferences made from a particular diagrams license general mathematical results? After surveying the history behind the received view, this essay provides a contrary analysis by introducing a formal account of Euclid’s proofs, termed Eu . Eu solves the puzzle of generality surrounding Euclid’s arguments. It specifies what diagrams Euclid’s diagrams are, in a precise formal sense, and defines generality-preserving proof rules in terms of them. After the central principles behind the formalization are laid out, its implications with respect to the question of what does and does not constitute a genuine picture proof are explored.

The article evaluates the Domain Postulate of the Classical Model of Science and the related Aristotelian prohibition rule on kind-crossing as interpretative tools in the history of the development of mathematics into a general science of quantities. Special reference is made to Proclus’ commentary to Euclid’s first book of Elements , to the sixteenth century translations of Euclid’s work into Latin and to the works of Stevin, Wallis, Viète and Descartes. The prohibition rule on kind-crossing formulated by Aristotle in Posterior analytics is used to distinguish between conceptions that share the same name but are substantively different: for example the search for a broader genus including all mathematical objects; the search for a common character of different species of mathematical objects; and the effort to treat magnitudes as numbers.

In this system, the properties of space were believed to be in accord with the geometry of Euclid ; and one might have expected that the correctness of the ...

Students of geometry do not prove Euclid's first theorem by examining an accompanying diagram, or by visualizing the construction of a figure. The original proof of Euclid's first theorem is incomplete, and this gap in logic is undetected by visual imagination. While cognition involves truth values, vision does not: the notions of inference and proof are foreign to vision.

According to the standard analysis, humor theories can be classified into three neatly identifiable groups:incongruity, superiority, and relief theories. Incongruity theory is the leading approach and includes historical figures such as Immanuel Kant, Søren Kierkegaard, and perhaps has its origins in comments made by Aristotle in the Rhetoric. Primarily focusing on the object of humor, this school sees humor as a response to an incongruity, a term broadly used to include ambiguity, logical impossibility, irrelevance, and inappropriateness. The paradigmatic Superiority theorist is Thomas Hobbes, who said that humor arises from a “sudden glory” felt when we recognize our supremacy over others. Plato and Aristotle are generally considered superiority theorists, who emphasize the aggressive feelings that fuel humor. The third group, Relief theory, is typically associated with Sigmund Freud and Herbert Spencer, who saw humor as fundamentally a way to release or save energy generated by repression. In addition, this article will explore a fourth group of theories of humor: play theory. Play theorists are not so much listing necessary conditions for something’s counting as humor, as they are asking us to look at humor as an extension of animal play.