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Organizers |
L. E. J. Brouwer: solipsist - why?
by
Dr Philip Catton
University of Canterbury (Philosophy / History and Philosophy of Science)
Consider (A1) intuitionistic logic, (A2) the Kantian pure intuition of time, (A3) constructive mathematics, (A4) experience as a state of the subject. Consider furthermore, towards making four parallel comparisons, (B1) classical logic, (B2) the Kantian pure intuition of space [with a modification], (B3) classical mathematics, (B4) experience as the mode of a subject’s connecting to its public, intersubjective world. Note that: 1. Intuitionistic logic, as Gödel firmly established, deserves our respect if classical logic does; but the converse is, as Gödel again firmly established, also true — classical logic deserves our respect if intuitionistic logic does. 2. Time is, in various ways that Kant points out to us, a deeper condition on experience than space; and yet in ways that Kant also points out the converse is also true — space is in various ways a deeper condition on experience than time is (a thesis that Kant establishes with arguments that are not touched by the usual criticism against Kant that his philosophy is embarrassed by non-Euclidean geometries). [My modification on B2 is to weaken Kant's assessment of the pure intuition of space, at least to the extent of its not encompassing Euclideanness of the metric, and perhaps so far as its not encompassing any more than topological structure.] 3. Constructive mathematics commands special respect as mathematics but the notion that it is uniquely worthy cannot be sustained; classical mathematics too has its place. 4. The private, what-it-is-like-for-the-subject aspect of experience commands special attention; but Kant established as well the converse point, that our right to view experience as a mode of the subject’s connecting to what it is not actually precedes our right to view experience as a state of the subject. I argue that the parallelism between my various A1-B1, A2-B2, A3-B3 and A4-B4 comparisons is both perfectly strong, and significant - it explains the inevitability of Brouwer's solipsistic tendency, identifies the special significance of mathematics that proceeds (according to Brouwer's prescription) out of constructions within the pure intuition of time, and at the same time identifies deep reasons to conclude that such a constructive approach cannot even potentially complete the whole of mathematics.
Date received: October 30, 2008
The platypus and the mathematician.
by
Hannes Diener
University of Canterbury
Mathematicians generally like to insist that their field is not science, since its method of gaining knowledge is deductive rather than inductive. In this talk we argue that, nevertheless, there is a healthy dose of inductive reasoning in the way mathematicians think.
Date received: October 30, 2008
Nature's drawing
by
Ofer Gal
Unit for History and Philosophy of Science, University of Sydney
The challenge of assigning mathematics an explanatory role in natural philosophy forced 17th century savants to accept a difficult conversion: turning local motion---the paradigm of change---into the carrier of order. Kepler and Galileo met the challenge by treating geometrical curves not as ideal representations of motion, but as traces left by nature itself, which the mathematician is called upon to analyse. In contrast to their Renaissance predecessors, for whom mathematical order was eternal and static, Kepler and Galileo construed nature in terms of pure motion, essentially mathematical. Their successors no longer conceived of the
application of mathematics to nature as requiring difficult metaphysical legitimation. For Huygens and Hooke the mathematical structure of
nature resided in its malleability to the mathematician.
Date received: November 16, 2008
Rules of engagement: conventions for medieval recreational problems
by
John Hannah
University of Canterbury
Ancient and medieval mathematical texts often discuss highly impractical problems disguised by realistic-sounding contexts. For example, men with unknown amounts of money might exchange known fractions of their holdings in order to buy a horse of unknown value, and you would have to find all the unknowns. In this talk I shall discuss some of the conventions which seem to have determined which problems you were allowed to set, and which solutions were deemed valid. Examples from the work of Leonardo of Pisa (also known as Fibonacci) will be used to illustrate these conventions.
Date received: October 29, 2008
A symbolic history of the derivative
by
Clemency Montelle
University of Canterbury
How many ways to symbolically represent the derivative can you call to mind? Are they really equivalent? Among the many disagreements Newton and Leibniz are remembered for, they had a big one over notation. They both independently developed distinctive notation for the derivative when they published their results in calculus. In turn, allegiance in both British and European mathematical communities to strictly one or the other persisted for almost half a century until Leibniz's notation finally prevailed. We will look at this scuffle and its consequences and reflect upon the ways in which notational considerations can affect mathematics.
Date received: October 30, 2008
Euclid and Aristotle, in Persian and in Sanskrit
by
Kim Plofker
Union College, Schenectady NY, USA
The Indo-Persian empires of the mid-second millennium CE in northern India fostered, both deliberately and accidentally, a great number of intellectual exchanges between Greco-Islamic science and the indigenous Sanskrit tradition. The anonymous Hayatagrantha ("Book on Spherical Astronomy"), a Sanskrit translation of the Ris¯ala dar hay'a ("Treatise on Spherical Astronomy") by the fifteenth-century Samarqand astronomer `Ali al-Q¯ushj¯, bears witness to some of the philosophical adjustments that were required to fit the Persian version of traditional Euclidean geometry and Aristotelian cosmology into the intellectual framework of Indian mathematics.
Date received: November 3, 2008
Pappus of Alexandria: analysis/synthesis outside of Book 7 in the Mathematical Collection
by
Bronwyn Rideout
University of Canterbury
Although Pappus of Alexandria is often identified with inspiring the early modern mathematics of Descartes, Desargues, and, to a lesser extent, Newton, it has only been in the past century that he has been the focus of interest and textual criticism. However, translations of his text into English have been few and far between and consequently certain books have risen to prominence while others, unfairly receive nominal attention. In this talk, I will discuss Pappus' infamous work on Analysis and Synthesis in Book 7 of the Mathematical Collection and the implications it has on Books 2 and 3 of the Collection. One will find that Book 7 is far from the final word Pappus has to say on the subject as I show how Pappus bring analysis and synthesis into play numerically and in different scenarios.
Date received: November 2, 2008