The Mathematical Experience |
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Page 235
... properties that it shares with almost all the other sets in M. It turns out to be possible in a precise way to make this distinction between special and generic properties per- fectly explicit and formal . Then when we choose a to be a ...
... properties that it shares with almost all the other sets in M. It turns out to be possible in a precise way to make this distinction between special and generic properties per- fectly explicit and formal . Then when we choose a to be a ...
Page 246
... properties " as the ordinary numbers of mathematics . On its face the idea seems self - contradictory . If infinitesimals have the same " properties " as ordinary numbers , how can they have the " property " of being positive yet ...
... properties " as the ordinary numbers of mathematics . On its face the idea seems self - contradictory . If infinitesimals have the same " properties " as ordinary numbers , how can they have the " property " of being positive yet ...
Page 250
... properties shared by all the standard real numbers may not apply to the nonstandard pseudonumbers , if these properties cannot be expressed in the formal language L. The Archimedean property ( nonexistence of infinitesi- mals ) of R can ...
... properties shared by all the standard real numbers may not apply to the nonstandard pseudonumbers , if these properties cannot be expressed in the formal language L. The Archimedean property ( nonexistence of infinitesi- mals ) of R can ...
Other editions - View all
The Mathematical Experience, Study Edition Philip Davis,Reuben Hersh,Elena Anne Marchisotto Limited preview - 2011 |
The Mathematical Experience, Study Edition Philip Davis,Reuben Hersh,Elena Anne Marchisotto Limited preview - 2011 |
The Mathematical Experience: Study Edition Philip J. Davis,Reuben Hersh,Elena Anne Marchisotto Limited preview - 1995 |
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abstract aesthetic algebra algorithmic analysis analytic answer applications argument arithmetic asserts axiom of choice Bibliography calculus called century circle complex conjecture construct continuum hypothesis course definition differential equations elements ematics Euclid Euclidean geometry Euler example existence experience fact figure finite formal language formalist formula Fourier Fourier series function Further Readings G. H. Hardy Hilbert human hypercube hypersquares idea ideal infinite set infinitesimal infinity integers intuition knowledge Lakatos logic mathe mathematical objects mathematical proof mathematicians matics means ment method natural numbers non-Euclidean geometry non-Riemannian nonstandard notion number theory parallel postulate philosophy of mathematics physical Platonism Platonist possible postulate prime number prime number theorem problem proof proved question real numbers reason restricted set theory result rigorous sense solution square statement straight line symbols theorem thing tion triangle true truth universe words zero