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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.(More)
In his writings on Hilbert's foundational work, Paul Bernays sometimes contrasts Euclid's geometrical method in the Elements with Hilbert's in Foundations of Geometry, identifying two features that distinguished the latter from the former (notably, in [6] and in the introduction to [3]). First, Hilbert's theory is abstract. Though the primitives of the(More)
For more than two millennia, Euclid's Elements set the standard for rigorous mathematical reasoning. The reasoning practice the text embodied is essentially diagrammatic, and this aspect of it has been captured formally in a logical system termed Eu [2, 3]. In this paper, we review empirical and theoretical works in mathematical cognition and the psychology(More)
This paper demonstrates that the property of 'replication invariance', generally considered to be an innocuous requirement for the extension of fixed-population poverty comparisons to variable-population contexts, is incompatible with other plausible variable-and fixed-population axioms. The World Institute for Development Economics Research (WIDER) was(More)
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