Avner, Ash and Justin, ClarkeDoane (2023) Intuition and Observation. [Preprint]
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Abstract
The motivating question of this paper is: ‘How are our beliefs in the theorems of mathematics justified?’ This is distinguished from the question ‘How are our mathematical beliefs reliably true?’ We examine an influential answer, outlined by Russell, championed by Gödel, and developed by those searching for new axioms to settle undecidables, that our mathematical beliefs are justified by intuitions, as our scientific beliefs are justified by observations. On this view, axioms are analogous to laws of nature. They are postulated to best systematize the data to be explained. We argue that there is a decisive difference between the cases. There is agreement on the data to be systematized in the scientific case that has no analog in the mathematical one. There is virtual consensus on observations but conspicuous dispute over intuitions. In this respect, mathematics more closely resembles paradigmatic philosophy. We conclude by distinguishing two ideas that have long been associated  realism (the idea that there is an independent reality) and objectivity (the idea that in a disagreement, only one of us can be right). We argue that, while realism is true of mathematics and philosophy, these domains fail to be fully objective. One upshot of the discussion is a kind of pragmatism. Factual questions in mathematics, modality, logic, and evaluative areas go proxy for nonfactual practical ones.
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