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Loss of vision: How mathematics turned blind while it learned to see more clearly
 In B. Löwe and T. Müller (Eds.), Philosophy of Mathematics: Sociological Aspects and Mathematical Practice
, 2010
"... The aim of this paper is to provide a framework for the discussion of mathematical ontology that is rooted in actual mathematical practice, i.e., the way in which mathematicians have introduced and dealt with mathematical ..."
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The aim of this paper is to provide a framework for the discussion of mathematical ontology that is rooted in actual mathematical practice, i.e., the way in which mathematicians have introduced and dealt with mathematical
Proofs, Pictures, and Euclid
, 2007
"... The prevailing conception of mathematical proof, or at least the conception which has been developed most thoroughly, is logical. A proof, accordingly, is a sequence of sentences. Each sentence is either an assumption of the proof, or is derived via sound inference rules from sentences preceding it. ..."
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The prevailing conception of mathematical proof, or at least the conception which has been developed most thoroughly, is logical. A proof, accordingly, is a sequence of sentences. Each sentence is either an assumption of the proof, or is derived via sound inference rules from sentences preceding it. The sentence appearing at the end of the sequence is what has been proved. This conception has been enormously fruitful and illuminating. Yet its great success in giving a precise account of mathematical reasoning does not imply that all mathematical proofs are, in essence, a sequence of sentences. My aim in this paper is to consider data which do not sit comfortably with the standard logical conception: proofs in which pictures seem to be instrumental in establishing a result. I focus, in particular, on a famous collection of picture proofs—Euclid’s diagrammatic arguments in the early books of the Elements. The familiar sentential model of proof portrays inferences as transitions between sentences. And so, by the familiar model, Euclid’s diagrams would at best serve as a
VISUALIZATION OF ORDINALS ∗
"... We describe the pictorial representations of infinite ordinals used in teaching set theory, and discuss a possible use in naturalistic foundations of mathematics. 1 ..."
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We describe the pictorial representations of infinite ordinals used in teaching set theory, and discuss a possible use in naturalistic foundations of mathematics. 1