Analysis 65 (285):18–24 (2005)
The following inference is valid: There are exactly 101 dalmatians, There are exactly 100 food bowls, Each dalmatian uses exactly one food bowl Hence, at least two dalmatians use the same food bowl. Here, “there are at least 101 dalmatians” is nominalized as, "x1"x2…."x100$y(Dy & y ¹ x1 & y ¹ x2 & … & y ¹ x100) and “there are exactly 101 dalmatians” is nominalized as, "x1"x2…."x100$y(Dy & y ¹ x1 & y ¹ x2 & … & y ¹ x100) & Ø"x1"x2…."x101$y(Dy & y ¹ x1 & y ¹ x2 & … & y ¹ x101). This is abbreviated $101xDx. The validity of the above inference corresponds to the valid formula, PHP(100): [$101xDx & $100xFx & "x(Dx ® Ff(x))] ® $x1$x2(Dx1 & Dx2 & x1 ¹ x2 & f(x1) = f(x2)). More generally, for variable n, the formula PHP(n) is PHP(n): [$n+1xDx & $nxFx & "x(Dx ® Ff(x))] ® $x1$x2(Dx1 & Dx2 & x1 ¹ x2 & f(x1) = f(x2)). A mathematical proof that PHP(n) is valid, for all n > 0, is quite short (less than a page), but refers to numbers, functions and sets. It uses the Pigeonhole Principle. This explains why PHP(n) is valid, for all n>0. However, I estimate that a predicate calculus derivation of PHP(100), using natural deduction, say, would require around 107 symbols. Unfeasibility Problem: nominalism is the radical anti-realist view that there are no numbers, functions or sets. So, how could a nominalist know that PHP(100) is valid, without directly performing the rather long derivation? Can the nominalist “ride piggyback” on the standard mathematical proof? If so, how is this justified?
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References found in this work BETA
Foundations Without Foundationalism: A Case for Second-Order Logic.Stewart Shapiro - 1991 - Oxford University Press.
On Gödel's Theorems on Lengths of Proofs I: Number of Lines and Speedup for Arithmetics.Samuel R. Buss - 1994 - Journal of Symbolic Logic 59 (3):737-756.
Polynomial Size Proofs of the Propositional Pigeonhole Principle.Samuel R. Buss - 1987 - Journal of Symbolic Logic 52 (4):916-927.
Computability, an Introduction to Recursive Function Theory.Nigel Cutland - 1980 - Cambridge University Press.
Citations of this work BETA
Naturalizing Indispensability: A Rejoinder to ‘The Varieties of Indispensability Arguments’.Henri Galinon - forthcoming - Synthese:1-14.
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