Bernard Bolzano: Philosophy of Mathematical Knowledge

Bernard Placidus Johann Nepomuk Bolzano was born on 5 October 1781 in Prague. He was the son of an Italian art merchant and of a German-speaking Czech mother. His early schooling was unexceptional: private tutors and education at the lyceum. In the second half of the 1790s, he studied philosophy and mathematics at the Charles-Ferdinand University. He began his theology studies in the Fall of 1800 and simultaneously wrote his first mathematical treatise. When he completed his studies in 1804, two university positions were open in Prague, one in mathematics, the other one in the “Sciences of the Catholic Religion.” 

In Bernard Bolzano’s theory of mathematical knowledge, properties such as analyticity and logical consequence are defined on the basis of a substitutional procedure that comes with a conception of logical form that prefigured contemporary treatments such as those of Quine and Tarski. Three results are particularly interesting: the elaboration of a calculus of probability, the definition of (narrow and broad) analyticity, and the definition of what it is for a set of propositions to stand in a relation of deducibility (Ableitbarkeit) with another. 

The main problem with assessing Bolzano's notions of analyticity and deducibility is that, although they offer a genuinely original treatment of certain kinds of semantic regularities, contrary to what one might expect they do not deliver an account of either epistemic or modal necessity. This failure suggests that Bolzano does not have a workable account of either deductive knowledge or demonstration. Yet, Bolzano’s views on deductive knowledge rest on a theory of grounding (Abfolge) and justification whose role in his theory is to provide the basis for a theory of mathematical demonstration and explanation whose historical interest is undeniable. 

The importance of Bolzano’s contribution to semantics can hardly be overestimated. The same holds for his contribution to the theoretical basis of mathematical practice. Far from ignoring epistemic and pragmatic constraint, Bolzano discusses them in detail, thus providing a comprehensive basis for a theory of mathematical knowledge that was aimed at supporting work in the discipline. As a mathematician, Bolzano was attuned to philosophical concerns that escaped the attention of most of his contemporaries and many of his successors. His theory is historically and philosophically interesting, and it deserves to be investigated further.