Oct 13, 2012
Highlighting Thompson’s Notions of the Stretch of Mathematics from Theory to Philosophy
Yogyakarta State University
Thompson, P.,1993, perceived that philosophers of mathematics have, for thousands of years, repeatedly been engaged in debates over paradoxes and difficulties they have seen emerging from the midst of their strongest and most intuitive convictions; from the rise of non-Euclidean geometry, to present-day problems in the analytic theory of the continuum, and from Cantor's discovery of a transfinite hierarchy to the fall of
Frege's system, mathematicians have also voiced their concern at how we blindly cash our naïve everyday intuitions in unfamiliar domains, and wildly extend our mathematics where intuition either has given out, or becomes prone to new and hitherto unforeseen pitfalls, or outright contradiction. Thompson 1 indicated that at the heart of these debates lies the task of isolating precisely what it is that our intuition provides us with, and deciding when we should be particularly circumspect about applying it; nevertheless, those who seek an epistemologically satisfying account of the role of intuition in mathematics are often faced with an unappealing choice, between the smoky metaphysics of Brouwer, and the mystical affidavit of Gödel and the Platonists that we can intuitively discern the realm of mathematical truth.
It was indicated 2 that, in term of foundations, mathematics is perceived as logical science, cleanly structured, and well-founded or in short mathematics is a highly structured logical science; however if we dig deep enough and in depth investigation, we still find some sand that makes the discursion involves philosophy. It is the fact that 3, in term of the history of foundations, an assortment of historical came, starting in ancient Greece, running through the turbulent present into an existing future; while in term of logical foundation systems, the methods of mathematics are deductive, and logic therefore has a fundamental role in the development of mathematics. Suitable logical frameworks 4 in which mathematics can be conducted can therefore be called logical foundation systems for mathematics. Some problems still arises 5: in term of meaning, we are wondered about the use of special languages for talking about mathematics, whether they strange things or out of this world and what does it all mean?; and then, in the sense of ontology, we may wonder whether mathematicians talk about strange thing, whether they really exist, and how they can we tell or does it matter?. Epistemologically 6, mathematics has often been presented as a paradigm of precision and certainty, but some writers have suggested that this is an illusion. How can we know the truth of mathematical propositions?; and in term of application, how can knowledge of abstract mathematics be applied in the real world?; what are the implications for mathematics of the information revolution?; and what can mathematics contribute?. 7 Thompson, P.,1993, insisted that the analysis combines a cognitive, psychological account of the great "intuitions" which are fundamental to conjecture and discovery in mathematics, with an epistemic account of what role the intuitiveness of mathematical propositions should play in their justification. He examined that the extent to which our intuitive conjectures are limited both by the nature of our sense-experience, and by our capacity for conceptualization
1 Thompson, P.,1993, The Nature And Role Of Intuition In Mathematical Epistemology, University College, Oxford University
2-----,1997,The Philosophy of Mathematics, RBJ, http://www.rbjones.com/rbjpub/rbj.htm