Mathematical Intuition pp 92-118 | Cite as

# Natural Numbers I

## Abstract

We shall now take up the question of what it means to say that there is intuition of the natural number n, where we take n to be any particular number. There are many aspects to this question and we shall not be able to treat all of them fully. The account to be discussed is perhaps best be viewed in terms of what I have called the Kantian strategy; that is, in terms of the question how arithmetical knowledge is possible, how in particular we become aware of number. We shall be considering several cognitive processes which are conditions necessary for the awareness of number, and which therefore begin to provide an answer to the Kantian question. The processes to be considered include collecting, reflecting, abstracting, and comparing. In discussing these processes we shall be investigating what Husserl calls the “origin” of the awareness of number. This chapter and the next will be devoted to developing the account in some detail. Objections will be treated in both chapters and in Chapter 8.

## Keywords

Natural Number Ordinal Number Cardinal Number Order Arrangement Formal Abstraction## Preview

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## Notes

- 2.
*PA*, pp. 119–120. See also p. 96.Google Scholar - 3.See also Resnik’s discussion in
*Frege and the Philosophy of Mathematics*[116], pp. 43–47.Google Scholar - 4.PA, p. 116.Google Scholar
- 6.Cf. Brouwer, L.E.J.,
*Collected Works*, Vol. 1, [14], pp. 15,417,480,509–510,523. It seems that Brouwer made no explicit effort to avoid the charge of psychologism concerning his view of number but intuitionistic ideas concerning “idealized mathematicians” might be construed as having the same effectGoogle Scholar - 9.See [71], A103-A104.Google Scholar
- 10.In [71]. A142/B182-A144/B183.Google Scholar
- 13.See Husserl’s
*PITC*[67]. Also, I. Miller [94]. The expectation involved in pretention, which concerns only immediate awareness, is part of the immediate horizon of a given act. It is to be distinguished from*acts*of expectation, of planning, etc. As we said earlier, a distinction should be drawn between the immediate horizon of an act and the global, or general horizon of an actGoogle Scholar - 18.PA, p. 95.Google Scholar
- 19.
*PA*, p. 102.Google Scholar - 21.PA, p.82.Google Scholar
- 22.Bernays “Review of Wittgenstein” [10].Google Scholar