Parker, Matthew W. (2008) Philosophical Method and Galileo's Paradox of Infinity. [Preprint]

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Abstract
We consider an approach to some philosophical problems that I call the Method of Conceptual Articulation: to recognize that a question may lack any determinate answer, and to reengineer concepts so that the question acquires a definite answer in such a way as to serve the epistemic motivations behind the question. As a case study we examine “Galileo’s Paradox”, that the perfect square numbers seem to be at once as numerous as the whole numbers, by onetoone correspondence, and yet less numerous, being a proper subset. I argue that Cantor resolved this paradox by a method at least close to that proposed—not by discovering the true nature of cardinal number, but by articulating several useful and appealing extensions of number to the infinite. Galileo was right to suggest that the concept of relative size did not apply to the infinite, for the concept he possessed did not. Nor was Bolzano simply wrong to reject Hume’s Principle (that onetoone correspondence implies equal number) in the infinitary case, in favor of Euclid’s Common Notion 5 (that the whole is greater than the part), for the concept of cardinal number (in the sense of “number of elements”) was not clearly defined for infinite collections. Order extension theorems now suggest that a theory of cardinality upholding Euclid’s principle instead of Hume’s is possible. Cantor’s refinements of number are not the only ones possible, and they appear to have been shaped by motivations and fruitfulness, for they evolved in discernible stages correlated with emerging applications and results. Galileo, Bolzano, and Cantor shared interests in the particulate analysis of the continuum and in physical applications. Cantor’s concepts proved fruitful for those pursuits. Finally, Gödel was mistaken to claim that Cantor’s concept of cardinality is forced on us; though Gödel gives an intuitively compelling argument, he ignores the fact that Euclid’s Common Notion is also intuitively compelling, and we are therefore forced to make a choice. The success of Cantor’s concept of cardinality lies not in its truth (for concepts are not true or false), nor its uniqueness (for it is not the only extension of number possible), but in its intuitive appeal, and most of all, its usefulness to the understanding.
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Item Type:  Preprint  

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Keywords:  Cantor Galileo Bolzano cardinal ordinal order type method methodology metaphilosophy metametaphysics metametaphysics conceptual change concept vagueness indeterminacy empty question set theory infinity infinite reference magnet semantic externalism realism Platonism Godel  
Subjects:  General Issues > Theory Change Specific Sciences > Mathematics General Issues > History of Science Case Studies General Issues > Conventionalism General Issues > Logical Positivism/Logical Empiricism 

Depositing User:  Matthew Parker  
Date Deposited:  02 Nov 2008  
Last Modified:  07 Oct 2010 15:17  
Item ID:  4276  
Subjects:  General Issues > Theory Change Specific Sciences > Mathematics General Issues > History of Science Case Studies General Issues > Conventionalism General Issues > Logical Positivism/Logical Empiricism 

Date:  September 2008  
URI:  http://philsciarchive.pitt.edu/id/eprint/4276 
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