A Counterfactual Approach to Explanation in Mathematics

2019 ◽  
Vol 28 (1) ◽  
pp. 1-34 ◽  
Author(s):  
Sam Baron ◽  
Mark Colyvan ◽  
David Ripley
Keyword(s):  
Scientific Explanation ◽  
Mathematical Explanation ◽  
Test Case ◽  
Mathematical Explanations ◽  
Counterfactual Theory ◽  
Explanatory Structure ◽  
Counterfactual Approach

ABSTRACT 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.

scholarly journals Counterfactual Scheming

Mind ◽  
2019 ◽  
Vol 129 (514) ◽  
pp. 535-562
Author(s):  
Sam Baron
Keyword(s):  
Mathematical Explanation ◽  
Explanatory Role ◽  
Unification Theory ◽  
Mathematical Explanations ◽  
Counterfactual Theory ◽  
Counterfactual Approach

Abstract Mathematics appears to play a genuine explanatory role in science. But how do mathematical explanations work? Recently, a counterfactual approach to mathematical explanation has been suggested. I argue that such a view fails to differentiate the explanatory uses of mathematics within science from the non-explanatory uses. I go on to offer a solution to this problem by combining elements of the counterfactual theory of explanation with elements of a unification theory of explanation. The result is a theory according to which a counterfactual is explanatory when it is an instance of a generalized counterfactual scheme.

Possible Worlds Counterfactual Theories of Causation

1980 ◽  
Vol 6 ◽  
pp. 119-138
Author(s):  
Richard Adler
Keyword(s):  
Alternative Treatment ◽  
Possible Worlds ◽  
Great Length ◽  
Current Interest ◽  
Possible Worlds Semantics ◽  
Causal Relations ◽  
Its Analysis ◽  
Counterfactual Theory ◽  
Causal Theories ◽  
Counterfactual Approach

The numerous difficulties facing the traditional Humean regularity approach to the problem of causation have been discussed in the literature at great length. In view of the current interest in possible worlds semantics, it is not surprising that the only serious alternative treatment of causation presently available, the counterfactual approach, has been explored recently as a means of circumventing the apparently unresolvable difficulties facing regularity causal theories. It is the purpose of this paper to suggest that such a strategy holds little promise. Specifically, I will argue that, in addition to giving rise to problems directly analogous to those facing regularity accounts, the counterfactual approach fails in principle to reflect important properties of causal relations as we understand them intuitively. David Lewis's possible worlds account, the most comprehensive counterfactual theory to date, is further criticized for implicit problems with natural lawhood even more serious than those typically raised for regularity accounts, for additional inadequacies in its analysis of causal relations, and for its failure to satisfy basic empiricist epistemological standards.

scholarly journals Regularity and counterfactuality in Hume's treatment of causation

2011 ◽  
Vol 52 (124) ◽  
pp. 355-364 ◽  
Author(s):  
José Oscar de Almeida Marques
Keyword(s):  
David Lewis ◽  
Regularity Theory ◽  
Causal Inferences ◽  
Counterfactual Theory ◽  
Definition Of ◽  
Counterfactual Approach ◽  
Conceptual Resources

Of the several theories of causation current in our days, Hume is said to be the inspiration of two of the most influential and accepted: the regularity theory, first clearly formulated by Thomas Brown in 1822, and the counterfactual theory, proposed by David Lewis in 1973. After a brief outline of the comparative merits and difficulties of these two views, I proceed to examine whether Hume's own treatment of causation actually corresponds to any of them. I will show that his first definition of cause, coupled with his rules by which to judge about causes and effects, contains elements that, properly developed, allow us to address successfully some traditional difficulties of the regularity view of causation, without resorting to the conceptual resources employed in the counterfactual approach. Therefore, we can properly classify Hume as an advocate of the conception of causation as regularity, noting however that his primary goal in his research and definitions of the concept was to provide not so much an analysis of causation as such, but of causation as we apprehend it, in the form of our ability to make causal inferences and refine them to reach the more sophisticated causal reasonings that are required in the theoretical and practical issues of life.

scholarly journals A deductive-nomological model for mathematical scientific explanation

2020 ◽  
Vol 24 (1) ◽  
pp. 1-27
Author(s):  
Eduardo Castro
Keyword(s):  
Hamiltonian Systems ◽  
North American ◽  
Scientific Explanation ◽  
Laws Of Nature ◽  
Mathematical Explanation ◽  
Causal Explanations ◽  
Scientific Explanations ◽  
The North

I propose a deductive-nomological model for mathematical scientific explanation. In this regard, I modify Hempel’s deductive-nomological model and test it against some of the following recent paradigmatic examples of the mathematical explanation of empirical facts: the seven bridges of Königsberg, the North American synchronized cicadas, and Hénon-Heiles Hamiltonian systems. I argue that mathematical scientific explanations that invoke laws of nature are qualitative explanations, and ordinary scientific explanations that employ mathematics are quantitative explanations. I analyse the repercussions of this deductivenomological model on causal explanations.

scholarly journals Unification and mathematical explanation

2021 ◽  
Author(s):  
Robert Knowles
Keyword(s):  
Divide And Conquer ◽  
Mathematical Explanation ◽  
Indispensability Argument ◽  
Mathematical Explanations ◽  
Enhanced Indispensability Argument ◽  
One Step ◽  
The Right

AbstractThis paper provides a sorely-needed evaluation of the view that mathematical explanations in science explain by unifying. Illustrating with some novel examples, I argue that the view is misguided. For believers in mathematical explanations in science, my discussion rules out one way of spelling out how they work, bringing us one step closer to the right way. For non-believers, it contributes to a divide-and-conquer strategy for showing that there are no such explanations in science. My discussion also undermines the appeal to unifying power in support of the enhanced indispensability argument.

scholarly journals Platonic Relations and Mathematical Explanations

2020 ◽  
Author(s):  
Robert Knowles
Keyword(s):  
Mathematical Explanation ◽  
Indispensability Argument ◽  
Scientific Explanations ◽  
Turn On ◽  
Mathematical Explanations ◽  
Mathematical Platonism ◽  
Adequate Understanding ◽  
Enhanced Indispensability Argument

Abstract Some scientific explanations appear to turn on pure mathematical claims. The enhanced indispensability argument appeals to these ‘mathematical explanations’ in support of mathematical platonism. I argue that the success of this argument rests on the claim that mathematical explanations locate pure mathematical facts on which their physical explananda depend, and that any account of mathematical explanation that supports this claim fails to provide an adequate understanding of mathematical explanation.

scholarly journals STUDENT EVALUATION MATHEMATICAL EXPLANATION IN DIFFERENTIAL CALCULUS CLASS

2021 ◽  
Vol 6 (1) ◽  
pp. 45-56
Author(s):  
Gabariela Purnama Ningsi ◽  
Fransiskus Nendy ◽  
Lana Sugiarti ◽  
Ferdinandus Ardian Ali
Keyword(s):  
Evaluation Method ◽  
Student Evaluation ◽  
Evaluation Methods ◽  
Mathematical Explanation ◽  
Sociomathematical Norms ◽  
Student Work ◽  
Video Recordings ◽  
Mathematical Explanations ◽  
Field Notes

This study aimed to determine that the failure of students to evaluate mathematical explanations based on mathematics is influenced by sociomathematical norms, teaching authority, and classroom mathematics practice. The research method used is the case study method. The research data were obtained from inside and outside the research class. The data in the research class were in the form of field notes, video recordings of the class, video recordings of student group work, and student work. Data outside the research class is the result of interviews with three interview subjects. By studying the three evaluation methods students used in evaluating explanations, it was found that each student applied a different evaluation method at different times. The three evaluation methods contributed to some of the difficulties students experience in evaluating their mathematical descriptions. The results indicate that the failure of students in evaluating explanations is not solely due to errors in choosing the method, approach, or learning model used but can be caused by sociomathematical norms, authority, and classroom mathematics practices applied in the classroom.

Test Validation Without Measurement

2016 ◽  
Vol 32 (3) ◽  
pp. 204-214 ◽  
Author(s):  
Emilie Lacot ◽  
Mohammad H. Afzali ◽  
Stéphane Vautier
Keyword(s):  
Test Data ◽  
Test Scores ◽  
Scientific Explanation ◽  
Test Validation ◽  
Clinical Tests ◽  
Ethical Perspective ◽  
Power Of Test ◽  
Memory Accuracy ◽  
Social Uses ◽  
Special Case

Abstract. Test validation based on usual statistical analyses is paradoxical, as, from a falsificationist perspective, they do not test that test data are ordinal measurements, and, from the ethical perspective, they do not justify the use of test scores. This paper (i) proposes some basic definitions, where measurement is a special case of scientific explanation; starting from the examples of memory accuracy and suicidality as scored by two widely used clinical tests/questionnaires. Moreover, it shows (ii) how to elicit the logic of the observable test events underlying the test scores, and (iii) how the measurability of the target theoretical quantities – memory accuracy and suicidality – can and should be tested at the respondent scale as opposed to the scale of aggregates of respondents. (iv) Criterion-related validity is revisited to stress that invoking the explanative power of test data should draw attention on counterexamples instead of statistical summarization. (v) Finally, it is argued that the justification of the use of test scores in specific settings should be part of the test validation task, because, as tests specialists, psychologists are responsible for proposing their tests for social uses.

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