# Some Research Issues

The following research issues or questions are among those that arise naturally from, or around, the mathematical and logical work that Bolzano published or prepared for publication.

(i) Investigation and assessment of Bolzano’s theory of measurable numbers in his *Reine Zahlenlehre*. Do they have the expected completeness, order and metric properties? Can they accommodate consistently standard and non-standard analysis? Do they offer an intuitive, but rigorous, treatment of infinitesimals?

(ii) Bolzano takes over and elaborates throughout his mathematical work a notion of ‘determination’ which derives from Leibniz and Wolff. For example, some characteristics of a geometric object determine the object (and therefore its other characteristics). He uses the notion to good effect to avoid problems of motion and empirical elements in geometry. A Cauchy sequence ‘determines’ its limit. He writes about ‘determinable’ functions, and the ‘determining grounds’ (specifications?) of a set. What does he mean by this notion? Does it mean the ‘same thing’ in different parts of mathematics? Can it be axiomatised? Can we give a good philosophical account of the notion? How does it relate to functional dependency, or cause, or evaluation?

(iii) The notion of a variable (and that of a function) in mathematics and logic was undergoing profound changes during Bolzano’s lifetime. What exactly were the ‘arbitrarily small quantities’ (my phrase not his) introduced in his *Der binomische Lehrsatz* in 1816? What was the nature of the changes in the concept of variable and how did they affect mathematics and logic?

(iv) Mathematicians and philosophers alike have been intrigued and puzzled by Bolzano’s notion of ‘ground-consequence’ [*Abfolge*]: a kind of objective grounding holding in the realm of truths and being a pattern for the laying out proofs and the organisation of knowledge in a domain. Can we make mathematical or philosophical sense of such a notion and has it anything useful to offer us still in the philosophy of mathematics?

(v) Bolzano distinguished more clearly than his predecessors the content of a proposition or idea, from its linguistic expression, or the thinking of it by somebody, or its realisation, if any, in the world. Such meanings, or logical objects, in some ways played a similar role for Bolzano as syntactic objects do within modern logic. Bolzano had no inkling of formal languages or systems but he went a long way towards preparing the ground for such systems. An interesting project would be to examine the very varied role and significance that formal systems have had within mathematics, logic and computing over the past 150 years.