Most physics theories are deterministic, with the notable exception of quantum mechanics which, however, comes plagued by the so-called measurement problem. This state of affairs might well be due to the inability of standard mathematics to "speak" of indeterminism, its inability to present us a worldview in which new information is created as time passes. In such a case, scientific determinism would only be an illusion due to the timeless mathematical language scientists use. To investigate this possibility it is necessary to develop an alternative mathematical language that is both powerful enough to allow scientists to compute predictions and compatible with indeterminism and the passage of time. We suggest that intuitionistic mathematics provides such a language and we illustrate it in simple terms.
Hence, the fact that the law of excluded middle is incompatible with Heyting’s intuitionistic … When a philosopher claims that the law of excluded middle is not valid, it is frequently … Light of the No-Class Theory 191 6. The Law of Excluded Middle and Constructivistic Interpre¬
Hence, the fact that the law of excluded middle is incompatible with Heyting’s intuitionistic … When a philosopher claims that the law of excluded middle is not valid, it is frequently … Light of the No-Class Theory 191 6. The Law of Excluded Middle and Constructivistic Interpre¬
...n arbitrary statement, or of that of its negation, the law of the excluded middle was subjected to criticism by representatives of the intuitionistic and con
The paper argues that a view entitled `constructive nominalism' shows that (at least ``moderate'' versions of) the traditional foundationalist schools in the philosophy of mathematics (intuitionism, logicism, platonism and nominalism) are compatible. Constructive nominalism identifies truth with what holds in a certain topos, essentially, the finite-type category generated by certain linguistic terms. (For details, see the author and \textit{P. J. Scott}: Introduction to higher order categorical logic (1986; Zbl 0596.03002).) The view is an interesting one in its own right, and certainly includes some of the aspects of each traditional view; but it would seem to sacrifice central planks of each (as the paper -- mostly -- points out), and so cannot demonstrate their compatibility. (i) Though the mathematics delivered is intuitionist, a good deal more is required for non-elementary intuitionist mathematics. (ii) Against logicism, the natural numbers cannot be defined, but numerals have to be taken as primitive. (iii) The central claim of platonism, that mathematical structures are mind-independent, hardly seems to be respected: such mind-dependence would seem to entail both the law of excluded middle and the omega completeness of arithmetic, both of which fail in the construction. (iv) Finally, though the ontology of the theory is linguistic, the items would appear to be types, not tokens, and so traditional nominalism is not respected.
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