References
- Gurwitsch constructed the higher order number systems in the style used by Friedrich Waismann, Introduction to Mathematical Thinking, tr. T. Benac, Frederick Ungar (N.Y., 1951).
- William McKenna, “Gurwitsch's Theory of the Constitution of the Ordinal Numbers”, Symposium in Memory of Aron Gurwitsch, Research in Phenomenology, Vol. V (1975).
- See note 1.
- Edmund Husserl, “On the Concept of Number: Psychological Analysis”, tr. Dallas Willard, Philosophia Mathematica, Vol. 10, No. 1, Summer, 1973, pp. 37–87.
- See Ernst Cassirer, The Problem of Knowledge, tr. W. Woglom and C. Hendel, Yale U. Press (New Haven, 1950), pp. 28–53.
- The Peano axioms are the following:
- Any version of the Peano axioms expressed in first order logic (unlike the axioms expressed here) has “non-standard” models, containing elements beyond the integers. E.g., see Herbert Enderton, A Mathematical Introduction to Logic, Academic Press (New York, 1972), pp. 178–80.
- See infra, §6, p. 170. while these two manifolds are not identical, either will do for defining the set of natural numbers. The primitive predicate constant (viz. the successor predicate) of the language of the Peano axioms can be defined in terms of the primitive predicate constant (viz. the less than predicate) of the language of the well-ordering axioms (see note 19). This can be accomplished within the object language as follows: The successor of x is the least item y such that x is less than y. The reverse (i.e., defining ‘less than’ in terms of 'successor of) cannot be done within the object language. For this reason we prefer to replace the Peano axioms with the well-ordering axioms.
- Jean Piaget, The Psychology of Intelligence, tr. M. Piercy and D. Berlyne, Littlefield, Adams, & Co. (Ottawa, 1950). pp. 36–78, 82, 112, 116, 119, 120, 122,140–153, 163–73.
- Jean Piaget, Genetic Epistemology, tr. E. Duckworth, Columbia U. Press (N.Y., 1970), pp. 24–57.
- Jean Piaget, The Principles of Genetic Epistemology, tr. W. Mays, Basic Books (N.Y., 1972), p. 91.
- Ernst Cassirer, op. cit., pp. 28–53.
- Ernst Cassirer, Substance and Function, tr. W. and M. Swabey, Dover (N.Y., 1923), pp. 88–91.
- Ernst Cassirer, The Philosophy of Symbolic Forms, Vol. 3, tr. R. Manheim, Yale U. Press (New Haven, 1957), pp. 352–56.
- Ernst Cassirer, “The Concept of Group and the Theory of Perception”, Philosophy and Phenomenological Research, Vol. V, No. 1, Sept., 1944, pp. 1–35.
- Gilbert Null and Roger Simons, “Manifolds, Abstract Moments, and Concepts”, in Parts and Moments, ed. Barry Smith, Philosophia Verlag (Munich, 1981).
- Bertrand Russell, Principles of Mathematics, Cambridge U. Press (Cambridge, 1903), p. 166.
- Guido Küng, Ontology and the Logistic Analysis of Language, tr. E. C. M. Mays, D. Reidel (Dordrecht-Holland, 1967), pp. 170–5.
- Waismann, op. cir., pp. 185–205.
- Throughout this discussion, the term ‘language’ denotes classical first- or second-order formal languages only.
- Scientific conceptualizations (which are always expressed in language) involve thematization of the kind of equivalence types which we call ‘manifolds’. Equivalence types may be dealt with within the general theory of intentionality as the types whose instances are encountered via prepredicative typification, and in particular as the limits to the variability of the open possibilities of the inner horizon of the nucleus of the perceptual noema. See Null, “Generalizing Abstraction and the Judgment of Subsumption in Aron Gurwitsch's Version of Husserl's Theory of Intentionality”, Philosophy and Phenomenological Research, Vol. XXXVIII, No. 4, June 1978. Manifolds (concepts and eidē constituted as themes via ideational abstractions) form a subclass of the class of equivalence types.
- Null and Simons, op. cit.
- See note 7.
- Raymond Wilder, Introduction to the Foundations of Mathematics, 2nd ed., John Wiley & Sons (N.Y., 1952, p. 65), Ch. V.
- The axioms of well-ordering are stated formally as follows:
- Null and Simons, op. cit.