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Intuition and Observation

Avner, Ash and Justin, Clarke-Doane (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 non-factual practical ones.


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Item Type: Preprint
Creators:
CreatorsEmailORCID
Avner, Ash
Justin, Clarke-Doanejc4345@columbia.edu
Keywords: intuition, observation, axioms, set theory, laws, replacement, choice, logic, LEM, paraconsistent, intuitionistic, paracomplete, modal, pluralism, kripke, benacerraf, godel, russell, quantum mechanics, realism, objectivity, platonism, open question argument, is/ought, hume, morality, normative, evaluative, rawls, powerset, induction, least upper bound, formal proof, informal proof, independence, incompleteness, disagreement, relativism, foundation, theory laden, epistemology of disagreement, philosophy vs. science, state space, mackie, armchair pluralism
Subjects: Specific Sciences > Mathematics > Epistemology
Specific Sciences > Mathematics > Foundations
Specific Sciences > Mathematics > Logic
Specific Sciences > Mathematics > Practice
Specific Sciences > Mathematics > Proof
Specific Sciences > Mathematics > Values
General Issues > Scientific Metaphysics
General Issues > Conventionalism
Specific Sciences > Physics > Cosmology
General Issues > Determinism/Indeterminism
General Issues > Logical Positivism/Logical Empiricism
Specific Sciences > Mathematics
General Issues > Operationalism/Instrumentalism
Specific Sciences > Physics
Specific Sciences > Physics > Quantum Mechanics
General Issues > Realism/Anti-realism
Specific Sciences > Physics > Relativity Theory
General Issues > Theory/Observation
General Issues > Values In Science
Depositing User: Justin Clarke-Doane
Date Deposited: 01 Oct 2023 14:56
Last Modified: 01 Oct 2023 14:56
Item ID: 22612
Subjects: Specific Sciences > Mathematics > Epistemology
Specific Sciences > Mathematics > Foundations
Specific Sciences > Mathematics > Logic
Specific Sciences > Mathematics > Practice
Specific Sciences > Mathematics > Proof
Specific Sciences > Mathematics > Values
General Issues > Scientific Metaphysics
General Issues > Conventionalism
Specific Sciences > Physics > Cosmology
General Issues > Determinism/Indeterminism
General Issues > Logical Positivism/Logical Empiricism
Specific Sciences > Mathematics
General Issues > Operationalism/Instrumentalism
Specific Sciences > Physics
Specific Sciences > Physics > Quantum Mechanics
General Issues > Realism/Anti-realism
Specific Sciences > Physics > Relativity Theory
General Issues > Theory/Observation
General Issues > Values In Science
Date: 2023
URI: https://philsci-archive-dev.library.pitt.edu/id/eprint/22612

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