Galileo’s Aggregate Infinity

by Corry Shores
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Galileo shifts the understanding of the infinite away from the Medieval Aristotelian notion to a more Platonic sort. He considers the infinite not in terms of magnitude, but of aggregate or multiplicity, and hence claimed that “the attributes ‘larger,’ ‘smaller,’ and ‘equal’ have no place either in comparing infinite quantities with each other or in comparing infinite with finite quantities (Opere, VIII, 82ff.)” (115d). If they were magnitudes, infinite quantities might be said to have larger or smaller (or equal) magnitude to each other. However, as aggregates or sets, these terms lose their applicability. He believed, for example, that the infinite class of positive integers can be set into one-to-one relations with one of its subclasses' terms, for example, with the set of all perfect squares. Later this approach to infinity was taken-up in developments of the calculus that established theories of infinite assemblages (116a).

Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover Publications, 1949.