Rioux, Jean W. (Author)
University of St. Thomas
Lee, W. Patrick
Edition Details: Advisor: Lee, W. Patrick
Physical Details: 215 pp.
Language: English
Each of the major modern attempts to base the study of arithmetic upon a solid foundation has met with difficulty. Formalist systems, abstract structures without content, are subject to incompleteness and an inability to demonstrate their own consistency. Logicism, which reduces arithmetic to symbolic logic, falls prey to paradoxes of various sorts. And intuitionism, which allows only constructible numbers, is unable to deal effectively with negative propositions and the ultimately arbitrary character of intuition itself. To these approaches we bring a new perspective on arithmetical foundations: that of Aristotle and Thomas Aquinas. Beginning with the notion of truth itself, we have provided an analysis of their understanding of arithmetic _ as a science_. This has enabled us to argue, we think convincingly, that arithmetic truly _is_ a science, that its foundations are absolutely sound, and that the claims made within it are speculatively true. We have also taken up three fundamental difficulties which rise from this account of arithmetic. Are what we _presume_ to be the subject of arithmetic, numbers, actually real? Is it necessary to abstract the nature of any particular number (say, a trillion,) _from the physical world_, before we can determine its properties? And, does the nature of the unit, as this is understood by Aristotle and Aquinas, require us to exclude such 'numbers' as transfinites, irrationals, and even fractions and negative numbers from arithmetic's consideration? In answer to these questions, we have concluded that the numerical unit is the proper subject of arithmetic. Thus, though numbers do exist in physical things, it is not necessary that they be abstracted _from_ them in order to be studied. Furthermore, certain 'numbers' cannot be regarded as anything but the products of art, for it can be shown that they are incompatible with the nature of the unit and cannot be constructed from it.
...MoreDescription Cited in Diss. Abstr. Int. A 67/05 (2006): 1761. UMI pub. no. 3221864.
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