What are the natural numbers?

The discussion of intuition in the last chapter naturally leads to two further inquiries. Perhaps the most obvious one is about natural numbers: How are natural numbers given to us, and, in particular, are they objects of an intuition of the kind described in §28 or something similar to it? The second concerns intuitive knowledge: We gave, particularly in §29, some examples of propositions about the strings of our little language that are intuitively known. Since strings are evidently a model of arithmetic, the question arises how far intuitive knowledge in arithmetic extends, when we understand arithmetic by reference to this model. That formulation of the question sidesteps the first question, whether numbers properly speaking are objects of intuition.

Because our first question is probably the more urgent for most readers, I will take it up first, in this chapter. The other question will be the subject of Chapter 7. The question: “What are the natural numbers?” is motivated independently of questions about intuition, and we will examine it in general as well as answering the question about intuition of numbers. But both chapters will explore the limits of intuition as understood in Chapter 5, first by considering in this chapter whether intuition extends beyond quasi-concrete objects to some pure abstract objects such as numbers, and second by inquiring in Chapter 7 how far intuitive knowledge extends, referring to a domain that we have taken to be intuitive.