Abstract
Introduction: where the general outline of the book is presented, together with a criticism of the traditional approaches to the philosophy of mathematics.
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Notes
- 1.
Tymoczko 1998, p. xiii.
- 2.
The apparently profound question “is mathematics invented or discovered?” has an obvious answer: both. Better, it is invented, even when it is discovered. By this, I mean that proto-mathematical entities can indeed be discovered, but they only become fully mathematical by some intervention on the part of mathematicians, i.e. by being somehow reinvented. As a rule, mathematicians create the objects with which they occupy their minds. This, of course, only makes the problem of the utility and wide applicability of mathematics more puzzling.
- 3.
The usual way of positing mathematical objects is the well-known fiat “let A be a domain of objects where such and such (relations, functions, etc.) are defined satisfying such and such properties”. Although the modern approach is to view these axiomatic stipulations as non-interpreted or formal systems open to different interpretations, they can also be viewed, as they, in fact, traditionally were, as object-positing stipulations. Objects so posited are purely formal, i.e. determined as to form but undetermined as to matter. These notions will be made more precise later.
- 4.
Hilbert’s formalist approach was originally devised for the sake of metamathematical investigations, not as a philosophy of mathematics, this came later.
- 5.
Lakatos 1977.
- 6.
Prolegomena to the Logical Investigations, which marked Husserl’s turning of the back to the philo-psychologism of the Philosophy of Arithmetic (which, by the way, as already sufficiently shown, owns nothing to the unfair and incompetent review of this work by Frege 1894).
- 7.
Rigorously speaking, logic is not content-free. Of course, the truth of logically true assertions does not depend of their particular contents, but depends on the sense of being of the domain to which they refer. In order for, say, either A or not-A to be valid, no matter which A, the domain where A is interpreted must be intentionally conceived in a certain way. It befalls on phenomenology the task of clarifying what this way of being is and why conceiving the domain of knowledge thus is justified in the overall schema of knowledge.
- 8.
“Those intuitions which we call Platonic are seldom scientific, they seldom explain the phenomena or hit upon the actual law of things, but they are often the highest expression of that activity which they fail to make comprehensible” (Santayana 1955, p. 7).
- 9.
References
Frege, G. (1894). Review of Dr. E. Husserl’s philosophy of arithmetic. Mind, 81(323), 321–337.
Husserl, E. (1954b). Der Unsprung der Geometrie als intentional-historisches Problem. Beilage III in Hua VI (pp. 365–386). Engl. trans. in Derrida 1989.
Husserl, E. (1962). Ideas – General introduction to pure phenomenology. New York: Colliers.
Husserl, E. (1965). Philosophy as a rigorous science. In Phenomenology and the crisis of philosophy (pp. 71–147). New York: Harper.
Husserl, E. (1970). Philosophie der Arithmetik, mit ergänzenden Texten (1890–1901). Husserliana XII, The Hague: M. Nijhoff. Engl. trans. Philosophy of arithmetic, psychological and logical investigations, with supplementary texts from 1887–1901 (Dallas Willard, Trans.). in Edmund Husserl Collected Works (Rudolf Bernet, ed.), Vol. X, Dordrecht: Springer, 2003.
Husserl, E. (2006). The basic problems of phenomenology – Lectures from the winter semester 1910–11. Dordrecht: Springer.
Lakatos, I. (1977). Proofs and refutations. Cambridge: Cambridge University Press.
Santayana, G. (1955). The sense of beauty. New York: Dover.
Tymoczko, T. (Ed.). (1998). New directions in the philosophy of mathematics. Princeton: Princeton University Press.
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da Silva, J.J. (2017). Introduction. In: Mathematics and Its Applications. Synthese Library, vol 385. Springer, Cham. https://doi.org/10.1007/978-3-319-63073-1_1
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