Semantic Foundations of Mathematics and other Sciences: Bolzano explains to Exner

Semantic Foundations of Mathematics and other Sciences: Bolzano explains to Exner

Peter Simons

From his first publication, on geometry, to his final incomplete encyclopedia of mathematics, Bernard Bolzano, like those other pioneering analytic philosophers, Frege and Russell, strove to place mathematics on a firm logical foundation. In Bolzano this foundation is determined by his semantics of propositions and ideas in themselves, “das logische An-sich”. We now recognise its remarkable affinity with Frege’s later “third realm”, but when it first appeared, readers found it extraordinarily difficult to grasp. None made greater efforts to overcome his misunderstandings than Bolzano’s Herbartian Prague contemporary, Franz Serafin Exner, who engaged in a semi-public correspondence with Bolzano based mainly on the compressed methodological introduction to Größenlehre. Bolzano was forced to explain himself more clearly and the exchange is a model of constructive criticism and explanation. With the benefit of many decades of excellent scholarship, we now find Bolzano’s views quite easy to understand. But that does not mean he is right. Perhaps Exner’s scepticism about the third realm has a point.