1.    A. Whitehead, Science and Philosophy (Paterson, N.J.: Littlefield, Adams & Co., 1964), 103.

2.    Quoted by E. Cassirer in The Philosophy of the Enlightenment (Boston: Beacon Press, 1961), 237.

3.    Dooyeweerd, New Critique, vol. 1, 223-61.

4.    Collected Works of John Stuart Mill, ed. J. Robson et al. (Toronto: University of Toronto Press, 1973), bk. 2, chaps. 5 and 6; and bk. 3, chap. 24.

5.    B. Russell, Principles of Mathematics (New York: W. W. Norton, 1938), xi. 

6. Ibid., 119:

1 + 1 is the number of a [logical] class — w — which is the logical sum of two classes — u and v — which have no common term and each have only one term. The chief point to be observed is that the logical addition of classes is the fundamental notion, while the arithmetical addition of numbers is wholly subsequent.

Formally, Russell’s proposal would read:

(∃u)(∃v)(∃w)( {[(u ∈ w) ∧ (v ∈ w)] ∧ (u /= v)}

∧ (∀z) {(z ∈ w) → [(z = u) ∨ (z = v)]} ).


It is hard to be sympathetic with Russell’s claim that no quantitative meaning is involved in this formula when the symbol means “is a member of,” which is no different from “is one member of.” Besides, the existential quantifier means “there exists at least one x such that. . . .” Thus quantity is unavoidably both presupposed and referred to by the meaning of the formula, even if the formula’s quantifiers range over only logical classes.

7.    B. Russell. “The Study of Mathematics,” reprinted in Mysticism and Logic (Garden City, N.Y.: Doubleday Anchor Books), 65.

8.    John Dewey, Reconstruction in Philosophy (Boston: Beacon Press, 1964), 156. 

9. Ibid., 149.

10. Ibid., 137.

11.      For the sake of accuracy, it should be noted that the biological perspective is not the final step of Dewey’s theory of reality. That is because he viewed the biological aspect as dependent upon (or included in) the physical. So in the final analysis, it is the physical (or the physico-biotic) aspects of creation which he takes to be the basic nature of reality.

12.     Morris Kline, Mathematical Thought from Ancient to Modern Times (New York: Oxford University Press, 1972), 32.

13. Ibid., 115.

14. Morris Kline, Mathematics, The Loss of Certainty (New York: Oxford University Press, 1980), 236.

15. Ibid., 237.

16. Ibid., 233.

17.     Ibid., 6.

18.     It should also be noted that many intuitionists, while declaring the independence of the mathematical from the other aspects of experience, still insist that the truths of math are also somehow dependent on the human mind. This is puzzling because it seems to require both that the truths of math reflect something self-existent and that they are dependent. One way of reconciling this conflict would be to say, with Kronecker, that “God created the natural numbers while all the rest is the work of man.” But in commenting on Brouwer’s version of intuitionism, Karl Popper has offered yet another interpretation to avoid inconsistency. He takes Brouwer’s theory to require what he (Popper) calls a “third world” of reality which includes (at least) mathematical and linguistic entities. Like Plato, Popper regards this world as self-existent (“ontologically autonomous”). But unlike Plato, he holds it to be a realm of possibilities needing human thought for their actualization. Thus there is a sense in which the third world is dependent on human thought even though it is divine in another sense. Popper’s position thus reflects a pagan religious belief. See his Objective Knowledge (Oxford: Clarendon Press, 1972), esp. 128-90.

19.     See the quote from Mill in note 32 to chapter 2.

20.      For example, W. V. O. Quine and Nelson Goodman developed a formal calculus of individuals in order to avoid treating predicates as representing really existing universals. See chap. 2 of The Structure of Appearance (Indianapolis: Bobbs, Merrill, 1966), 33 ff.

21.     For more on the non-reductive impact that belief in God makes upon theories in math, see Dooyeweerd’s New Critique, vol. 2, 55-93. This view has been given further development by the following thinkers (among others):

 D. H. T. Vollenhoven. De Wijsbegeerte der Wiskunde van Teistische Standpunt (Amsterdam: Wed G. Van Soest, 1918).

              . De Noodzakeljkhbeid eener Christelyjke Logica (Amsterdam: H. J. Paris, 1932).

              . “Problemen en Richtingen in de Wijsbegeerte der Wiskunde,” Philosophia Reformata 1 (1936).

              . “Hoofdlijnen der Logica,” Philosophia Reformata 13 (1948).

D. Strauss. “Number Concept and Number Idea,” Philosophia Reformata 35, no. 3 (1970) and 35, no. 4 (1971).

A. Tol. “Counting, Number Concept and Numerosity,” in Hearing and Doing: Philosophical Essays Dedicated to Evan Runner, ed. J. Kraay (Toronto: Wedge, 1979).

D. Strauss. “Infinity,” in Basic Concepts in Philosophy, ed. Z. Van Straaten (Oxford: Oxford University Press, 1981).

              . “Are the Natural Sciences Free from Philosophical Presuppositions?” Philosophia Reformata 46, no. 1 (1981).

              . “Dooyeweerd and Modern Mathematics,” Reformational Forum, no. 2, 1983, 40-55.

              . “The Nature of Mathematics and Its Supposed Arithmetization,” Proceedings of the Ninth National Congress on Mathematics Education, 1988, 10-31 (Mathematical Association of South Africa).

              . “The Uniqueness of Number and Space and the Relation between Realism and Nominalism,” Journal for Christian Scholarship, 1ste & 2de kwartaal, 1990, 104-25.

              . “A Historical Analysis of the Role of Beliefs in the Three Foundational Crises in Mathematics,” in Facets of Faith and Science, ed. J. van der Meer (Lanham, Md.: University Press of America, 1997), vol. 2, 217-30.

              . “Primitive Meaning in Mathematics: The Interaction between Commitment, Theoretical Worldview, and Axiomatic Set Theory,” in ibid., vol. 2, 231-56.

              . “Reductionism in Mathematics,” Journal for Christian Scholarship, Jaargang 37, 1ste & 2de kwartaal, 2001, 71-88.

              . Paradigms in Mathematics, Physics, and Biology (Bloemfontein: Teksor, 2001).

              . “Frege’s Attack on ‘Abstraction’ and His Defense of the ‘Applicability’ of Arithmetic as Part of Logic,” South African Journal of Philosophy 22(1), 2003, 63- 80.

              . “Is a Christian Mathematics Possible?” Journal for Christian Scholarship, 3de & 4de kwartaal, 2003, 31-49.


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