The Hypothetical Conception
This chapter proposes the idea that advanced mathematics is based on hypotheses—that far from being a priori, it is based on hypothetical assumptions. The concept of quasi-empiricism is often linked with the view that inductive methods are at play when the hypotheses are established. The presence of hypotheses at the very heart of mathematics establishes an important similitude with physical theory and undermines the simple distinction between “formal” and “empirical” sciences. The chapter first elaborates on a hypothetical conception of mathematics before discussing the ideas (and ideals) of certainty and objectivity in mathematics. It then considers the modern problems of the continuum that exist in ancient Greek geometry, along with the so-called methodological platonism of modern mathematics and its focus on mathematical objects. Finally, it describes the Axiom of Completeness and the Riemann Hypothesis.
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