A Counterfactual Approach to Explanation in Mathematics
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Our goal in this paper is to extend counterfactual accounts of scientific explanation to mathematics. Our focus, in particular, is on intra-mathematical explanations: explanations of one mathematical fact in terms of another. We offer a basic counterfactual theory of intra-mathematical explanations, before modelling the explanatory structure of a test case using counterfactual machinery. We finish by considering the application of counterpossibles to mathematical explanation, and explore a second test case along these lines.
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Mothilal et al. are also concerned with counterfactual diversity.
“Mothilal et al. are also concerned with counterfactual diversity [105].”
From A Survey of Contrastive and Counterfactual Explanation Generation Methods for Explainable Artificial Intelligence (2021) · cites this paper as [105]
Mothilal et al. design a loss function with a diversity metric over the generated counterfactuals to provide end-users with multiple relevant counterfactual explanations.
“They design a loss function with a diversity metric over the generated counterfactuals to provide end-users with multiple relevant counterfactual explanations.”
From A Survey of Contrastive and Counterfactual Explanation Generation Methods for Explainable Artificial Intelligence (2021) · cites this paper as [105]
The most commonly acceptable definition of a counterfactual in the observed RQ3-related studies states that a counterfactual explanation is a set of minimal feature modifications that makes the model change the prediction.
“Nevertheless, the most commonly acceptable definition of a contfactual in the observed RQ 3 -related studies states that a contfactual explanation is a set of minimal feature modifications that makes the model change the prediction [81], [105], [122].”
From A Survey of Contrastive and Counterfactual Explanation Generation Methods for Explainable Artificial Intelligence (2021) · cites this paper as [105]
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Cites
CAUSALITY: MODELS, REASONING AND INFERENCE
Hitchcock, C., & Pearl, J. (2001) · The Philosophical Review · doi:10.2307/3182612 · Cited by 4,824 · see citing works
Making things happen: a theory of causal explanation
(2004) · Choice Reviews Online · doi:10.5860/choice.41-5840 · Cited by 3,383 · see citing works
New work for a theory of universals
Lewis, D. (1983) · Australasian Journal of Philosophy · doi:10.1080/00048408312341131 · Cited by 3,140 · see citing works
Stalnaker, R. C. (1968) · IFS · doi:10.1007/978-94-009-9117-0_2 · Cited by 2,220 · see citing works
Counterfactual Dependence and Time's Arrow
Lewis, D. (1979) · Noûs · doi:10.2307/2215339 · Cited by 1,514 · see citing works
Causality: Models, Reasoning, and Inference
Shanmugam, R. (2001) · Neurocomputing · doi:10.1016/s0925-2312(01)00330-7 · Cited by 1,377 · see citing works
Depth: An Account of Scientific Explanation
Gijsbers, V. (2009) · International Studies in the Philosophy of Science · doi:10.1080/02698590903007212 · Cited by 598 · see citing works
Impossible Worlds: A Modest Approach
Nolan, D. (1997) · Notre Dame Journal of Formal Logic · doi:10.1305/ndjfl/1039540769 · Cited by 577 · see citing works
Pringle-Pattison, A. S., & Dewey, J. (1904) · The Philosophical Review · doi:10.2307/2176309 · Cited by 543 · see citing works
The Indispensability of Mathematics
Colyvan, M. (2001) · doi:10.1093/019513754x.001.0001 · Cited by 520 · see citing works
Self-citationDepth: an account of scientific explanation
(2009) · Choice Reviews Online · doi:10.5860/choice.46-6127 · Cited by 430 · see citing works
Priest, G. (2002) · doi:10.1093/acprof:oso/9780199254057.001.0001 · Cited by 426 · see citing works
Are there Genuine Mathematical Explanations of Physical Phenomena?
Baker, A. (2005) · Mind · doi:10.1093/mind/fzi223 · Cited by 372 · see citing works
Explanatory Generalizations, Part I: A Counterfactual Account
Woodward, J., & Hitchcock, C. (2003) · Noûs · doi:10.1111/1468-0068.00426 · Cited by 317 · see citing works
Brogaard, B. “., & Salerno, J. (2012) · Synthese · doi:10.1007/s11229-012-0196-6 · Cited by 288 · see citing works
The indispensability of mathematics
Colyvan, M. (2002) · Choice Reviews Online · doi:10.5860/choice.39-2845 · Cited by 259 · see citing works
Self-citationWhat Makes a Scientific Explanation Distinctively Mathematical?
Lange, M. (2012) · The British Journal for the Philosophy of Science · doi:10.1093/bjps/axs012 · Cited by 226 · see citing works
Mortensen, C. (1995) · Springer eBooks · doi:10.1007/978-94-015-8453-1 · Cited by 205 · see citing works
Sober, E. R. (1983) · Philosophical Studies · doi:10.1007/bf00372383 · Cited by 183 · see citing works
(2018) · Oxford Scholarship Online · doi:10.1093/oso/9780198777946.001.0001 · Cited by 174 · see citing works
Massively collaborative mathematics
Gowers, W. T., & Nielsen, M. A. (2009) · Nature · doi:10.1038/461879a · Cited by 164 · see citing works
Lycan, W. G. (2001) · doi:10.1093/oso/9780199242078.001.0001 · Cited by 164 · see citing works
Williamson on Counterpossibles
Berto, F., French, R., Priest, G., et al. (2017) · Journal of Philosophical Logic · doi:10.1007/s10992-017-9446-x · Cited by 150 · see citing works
The Explanatory Power of Phase Spaces
Lyon, A., & Colyvan, M. (2007) · Philosophia Mathematica · doi:10.1093/philmat/nkm025 · Cited by 144 · see citing works
Self-citationAn Introduction to the Philosophy of Mathematics
Colyvan, M. (2012) · Cambridge University Press eBooks · doi:10.1017/cbo9781139033107 · Cited by 123 · see citing works
Self-citationOn the Ternary Relation and Conditionality
Beall, J., Brady, R. T., Dunn, J. M., et al. (2011) · Journal of Philosophical Logic · doi:10.1007/s10992-011-9191-5 · Cited by 122 · see citing works
How Mathematics Can Make a Difference
Baron, S., Colyvan, M., & Ripley, D. (2017) · Research Bank (Australian Catholic University) · Cited by 120 · see citing works
Self-citationBernstein, S. (2016) · Philosophical Studies · doi:10.1007/s11098-016-0672-9 · Cited by 116 · see citing works
Because Without Cause: Non-Causal Explanations in Science and Mathematics
Lange, M. (2016) · Cited by 110 · see citing works
Berto, F., & Jago, M. T. (2019) · Cited by 109 · see citing works
Mathematical Explanations Of Empirical Facts, And Mathematical Realism
Lyon, A. (2011) · Australasian Journal of Philosophy · doi:10.1080/00048402.2011.596216 · Cited by 104 · see citing works
Because without Cause: Non-Causal Explanations in Science and Mathematics
Povich, M., & Craver, C. F. (2018) · The Philosophical Review · doi:10.1215/00318108-6718870 · Cited by 98 · see citing works
Mathematical Explanation: Problems and Prospects
Mancosu, P. (2001) · Topoi · doi:10.1023/a:1010621314372 · Cited by 97 · see citing works
Mathematics and Aesthetic Considerations in Science
Colyvan, M. (2002) · Mind · doi:10.1093/mind/111.441.69 · Cited by 90 · see citing works
Self-citationTRANSFINITE CARDINALS IN PARACONSISTENT SET THEORY
Weber, Z. (2012) · The Review of Symbolic Logic · doi:10.1017/s1755020312000019 · Cited by 90 · see citing works
Who's Afraid of Impossible Worlds?
MARES, E. D. (1997) · Notre Dame Journal of Formal Logic · doi:10.1305/ndjfl/1039540767 · Cited by 85 · see citing works
Counterpossibles and Similarity
Laan, D. V. (2004) · doi:10.1093/oso/9780199274550.003.0021 · Cited by 83 · see citing works
The Ontology of Impossible Worlds
Laan, D. V. (1997) · Notre Dame Journal of Formal Logic · doi:10.1305/ndjfl/1039540772 · Cited by 75 · see citing works
Aspects of Mathematical Explanation: Symmetry, Unity, and Salience
Lange, M. (2014) · The Philosophical Review · doi:10.1215/00318108-2749730 · Cited by 74 · see citing works
A relevant theory of conditionals
MARES, E. D., & Fuhrmann, A. (1995) · Journal of Philosophical Logic · doi:10.1007/bf01306969 · Cited by 72 · see citing works
Asymmetries in Time Problems in the Philosophy of Science
Healey, R., & Horwich, P. (1987) · The Philosophical Review · doi:10.2307/2185522 · Cited by 66 · see citing works
Williamson, T. L. (2016) · Topoi · doi:10.1007/s11245-016-9424-x · Cited by 42 · see citing works
Baron, S., & Colyvan, M. (2016) · The Journal of Philosophy · doi:10.5840/jphil201611324 · Cited by 37 · see citing works
Self-citationMathematical Explanation beyond Explanatory Proof
D’Alessandro, W. (2018) · The British Journal for the Philosophy of Science · doi:10.1093/bjps/axy009 · Cited by 30 · see citing works
Optimisation and mathematical explanation: doing the Lévy Walk
Baron, S. (2013) · Synthese · doi:10.1007/s11229-013-0284-2 · Cited by 28 · see citing works
Self-citationCounterpossibles and Similarities
Laan, D. V. (2004) · Cited by 27 · see citing works
The explanatory dispensability of idealizations
Baron, S. (2014) · Synthese · doi:10.1007/s11229-014-0517-z · Cited by 19 · see citing works
Self-citationMathematical Explanations that are Not Proofs
Lange, M. (2017) · Erkenntnis · doi:10.1007/s10670-017-9941-z · Cited by 19 · see citing works
The Prospects for a Monist Theory of Non-causal Explanation in Science and Mathematics
Reutlinger, A., Colyvan, M., & Krzyżanowska, K. (2020) · Erkenntnis · doi:10.1007/s10670-020-00273-w · Cited by 15 · see citing works
Self-citationTao, T. (2016) · Texts and readings in mathematics · doi:10.1007/978-981-10-1789-6 · Cited by 15 · see citing works
Beauty Is Not All There Is to Aesthetics in Mathematics
Thomas, R. S. D. (2016) · Philosophia Mathematica · doi:10.1093/philmat/nkw019 · Cited by 13 · see citing works
Zardini, E. (2019) · Journal of Philosophical Logic · doi:10.1007/s10992-018-9483-0 · Cited by 13 · see citing works
Strong, universal and provably non-trivial set theory by means of adaptive logic
Verdée, P. (2012) · Logic Journal of IGPL · doi:10.1093/jigpal/jzs025 · Cited by 12 · see citing works
Beauty Is Not All There Is to Aesthetics in Mathematics
Thomas, R. S. D. (2018) · Princeton University Press eBooks · doi:10.2307/j.ctvc775mh.6 · Cited by 2 · see citing works
Flagged first, then by citations
Research works printed but not linked: 3
- [2][2] Sam Baron. Optimization and mathematical explanation: Doing the L´ evy walk. Synthese , 191(3):459-479, 2014. -
- [25][25] David Lewis. Counterfactuals . Blackwell, Oxford, 1973. -
- [42][42] Terrence Tao. Texts and Readings in Mathematics: Vol. 37. Analysis I . Springer, 2016. -
Cited by
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Stepin, I., Alonso, J. M., Catala, A., & Pereira-Farina, M. (2021) · IEEE Access · doi:10.1109/access.2021.3051315 · Cited by 376 · see citing works
- Authors
- Baron, S., Colyvan, M., & Ripley, D.
- Affiliations
- The University of Western Australia (AU)
- The University of Sydney (AU)
- Monash University (AU)
- Corresponding author
- Baron, Sam
- Venue
- Philosophia Mathematica
- Publisher
- Oxford University Press (OUP)
- Published
- 2019-12-02
- Volume
- 28
- Issue
- 1
- Pages
- 1-34
- ISSN
- 0031-8019 (ISSN-L)
- 1744-6406
- OpenAlex
- W2994360042
- Cited by
- 48
- References
- 54
- Citations by year
- 2026: 2 · 2025: 6 · 2024: 6 · 2023: 3 · 2022: 17 · 2021: 11 · 2020: 2 · 2017: 1
- FWCI
- 58.59
- Type
- journal article
- Language
- en
- Open access
- Licence
- Locations
- Philosophia Mathematica (publishedVersion)
- UWA Profiles and Research Repository (University of Western Australia) (submittedVersion, other-oa)
- Funders
- Australian Research Council Discovery Early Career Research (DE180100414, FT110100909)
- Carl Friedrich von Siemens Research Award of the Alexander von Humboldt Foundation
- Topics
- Physical Sciences › Mathematics › Theoretical Computer Science › History and Theory of Mathematics
- Social Sciences › Arts and Humanities › History and Philosophy of Science › Philosophy and History of Science
- Social Sciences › Arts and Humanities › Philosophy › Epistemology, Ethics, and Metaphysics
- Keywords
- mathematical explanation, counterpossibles, counterfactual reasoning, philosophy of mathematics, scientific explanation
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