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Recursion, Induction

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A Mathematical Prelude to the Philosophy of Mathematics

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Abstract

This chapter introduces an informal version of primitive recursive arithmetic.

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Notes

  1. 1.

    I am not going to supply rigorous definitions of composition and recursion. A good source for a more thorough treatment is the WikipediA article on primitive recursive functions (http://en.wikipedia.org/wiki/Primitive_recursive_function). For an especially meticulous (though perhaps not so readable) presentation, see Curry [3] (http://www.jstor.org/stable/2371522).

  2. 2.

    See Hilbert [5], p. 163; English translation in Mancosu [6], p. 202.

  3. 3.

    See Hilbert [5], p. 162; Mancosu [6], p. 202.

  4. 4.

    See Hilbert [5], p. 163, footnote 1; Mancosu [6], p. 214.

  5. 5.

    For an especially clear discussion see Burgess [2], pp. 157–161.

  6. 6.

    But is sameness-of-shape the right sort of relation? Equivalence relations of the sort required are transitive: if \(a\) relates to \(b\) and \(b\) relates to \(c\), then \(a\) relates to \(c\). However, the relation appears-to-be-the-same-shape is, notoriously, not transitive because imperceptible differences in shape can add up to perceptible ones. Does this consideration not apply to our numeral-tokens?

  7. 7.

    A case of the sort I have in mind is discussed in Chap. 3: when we craft a set theory just strong enough to supply an interpretation of arithmetic, it turns out that arithmetic supplies an interpretation of our set theory. Positive integers are exceptionally powerful devices for coding up all sorts of information—information that may or may not have obvious connections to arithmetic. (We got a taste of this in the previous section when we used individual numbers to code ordered pairs of numbers. There will be more such examples in Chaps. 2 and 3.) It would be surprising if the \(\varPi ^{0}_{1}\) part of arithmetic were too weak to code up the mathematically salient claims about numeral-tokens that we need for our reinterpretation of \(\varPi ^{0}_{1}\) sentences. Granted, interpretability does not always work both ways. There are theories that interpret arithmetic but are not interpretable in arithmetic. Our project, however, does not seem to require us to formulate such a theory.

  8. 8.

    Section 5.4 of Chap. 5 may help you see how this works.

  9. 9.

    To take an extreme example of the sort of thing I have in mind, suppose we try to imagine what it would be like for an elephant to be a pencil. Will we then be thinking about elephants and pencils or about things we just call “elephants” and “pencils”? For a real-life example, consider what Geraldine Ferraro said when Barack Obama became the front-runner in the race for the Democratic presidential nomination. “If Obama was a white man, he would not be in this position. And if he was a woman of any color, he would not be in this position. He happens to be very lucky to be who he is. And the country is caught up in the concept.” Can we really think clearly about a situation in which Barack Obama (not someone like Obama, but Obama himself) is, say, a Polynesian woman? As our imaginary excursions become more and more fanciful, could we not reach a point where we are not really talking about Obama?

  10. 10.

    See Bernays [1], pp. 338–341; English translation in Mancosu [6], pp. 243–245.

  11. 11.

    See Gödel [4], p. 272, footnote b.

References

  1. Bernays, P. (1930–1931). Die Philosophie der Mathematik und die Hilbertische Beweistheorie. Blätter für deutsche Philosophie, 4, 326–367.

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  2. Burgess, J. P. (2005). Fixing frege. Princeton NJ: Princeton University Press.

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  3. Curry, H. B. (1941). A formalization of recursive arithmetic. American Journal of Mathematics, 63, 263–282.

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  4. Gödel, K. (1990). Collected works (Vol. 2). New York: Oxford University Press.

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  5. Hilbert, D. (1922). Neubegründung der Mathematik. Abhandlungen aus dem Mathematischen Seminar der Hamburgischen Universität, 1, 157–177.

    Article  Google Scholar 

  6. Mancosu, P. (Ed.). (1998). From Brouwer to Hilbert. New York: Oxford University Press.

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Authors and Affiliations

  1. Department of Philosophy and Religion, Truman State University, Kirksville, MO, USA

    Stephen Pollard

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  1. Stephen Pollard
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Correspondence to Stephen Pollard .

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Pollard, S. (2014). Recursion, Induction. In: A Mathematical Prelude to the Philosophy of Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-05816-0_1

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