Numbers—in which we abandon logic to achieve understanding, then use logic to deepen understanding
How and why do we abandon logic while we learn about numbers? Just think back to your first meeting with ‘one’, ‘two’, ‘three’…. These characters were probably of very minor importance initially, being no more than labels to distinguish the verses of ‘one, two, buckle my shoe’, etc. Then they began to have relationships with one another— ‘one’ for some reason always came before ‘two’; and relationships with the outside world— ‘one’, ‘two’, ‘three’ were names attached to small collections of things, but ‘ten’, ‘eleven’, ‘twelve’ were associated with bigger collections. So, even at pre-school age, we had experience of numbers being used in at least three ways—as arbitrary labels, as a means of ordering events (Monday first, Tuesday second …) and as measures (of height, weight, more numerous, less numerous…)—and these interpretations seem to have very little to do with one another. If we imagine pre-school children sufficiently precocious to ask ‘What is a number?’ it would be extremely difficult to give them a sensible answer.
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Suggestions for Further Reading
- L. W. Cohen and G. Ehrlich, The Structure of the Real Number System, Van Nostrand (1962). Rather formal, but thorough. Fills the gaps in the path, ℕ→ℤ→ℚ→ℝ, and also does not, as we have done, take ℕ for granted, but derives its properties from Peano’s axioms.Google Scholar