# Objectivity in Mathematical Knowledge

# Objectivity in Mathematical Knowledge

This chapter proposes an idea for reconciling the hypothetical conception of mathematics with the traditional idea of the objectivity of mathematical knowledge. The basic notion is that, because new hypotheses are embedded in the web of mathematical practices, they become systematically linked with previous strata of mathematical knowledge, and this forces upon us agents (for example, research mathematicians or students of math) certain results, be they principles or conclusions. The chapter first considers a simple case that illustrates objective features in the introduction of basic mathematical hypotheses. It then discusses Georg Cantor's “purely arithmetical” proofs of his set-theoretic results, along with the notion of arbitrary set in relation to the Axiom of Choice that has strong roots in the theory of real numbers. It also explores Cantor's ordinal numbers and the Continuum Hypothesis.

*Keywords:*
objectivity, mathematical knowledge, hypotheses, Georg Cantor, purely arithmetical proof, arbitrary set, Axiom of Choice, real numbers, ordinal numbers, Continuum Hypothesis

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