(Article II.1.) Existence, Substantiality, and Counterfactuality Observations on the Status of Mathematics according to Aristotle, Euclid, and Others

The article tries to answer the (arguably anachronistic) question whether ancient philosophers and mathematicians held mathematical objects or mathematical truth as a whole to exist. Aristotle being the philosopher who more than anybody else takes up the general question of ‘‘existence” with all its inherent ambiguities, the first and major section analyses what he has to say about mathematical objects being in some sense substances, that is, existing entities. The second section looks at Euclid’s postulates (literally ‘‘requests”), three of which are simply false according to the standard cosmologies of the time. The likely answer of geometricians, actually offered by Cicero, is that these postulates have to be granted if geometry is to be practised – that is, so to speak, that they provide the foundation for what Wittgenstein would call a ‘‘language game”. The last section looks at the question whether mathematics as a whole is a free construction or somehow determined, from the viewpoints of Pythagorean, Platonic and post-Platonic, Aristotelian and other philosophies.