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Conceptual (and Hence Mathematical) Explanation, Conceptual Grounding and Proof

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Abstract

This paper studies the notions of conceptual grounding and conceptual explanation (which includes the notion of mathematical explanation), with an aim of clarifying the links between them. On the one hand, it analyses complex examples of these two notions that bring to the fore features that are easily overlooked otherwise. On the other hand, it provides a formal framework for modeling both conceptual grounding and conceptual explanation, based on the concept of proof. Inspiration and analogies are drawn with the recent research in metaphysics on the pair metaphysical grounding–metaphysical explanation, and especially with the literature in philosophy of science on the pair causality-causal explanation.

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Notes

  1. E.g. see Correia and Schnieder (2012).

  2. E.g. see Menzies and Beebee (2019) and Woodward (2019).

  3. E.g. see Glaizer (2020) and Maurin (2019).

  4. See Raven (2012).

  5. See Hempel (1942) and Woodward (2003).

  6. E.g. see Betti (2010), Chalmers (2012), Schnieder (2006) and Smithson (2020).

  7. On the links between conceptual grounding and proof-theory see Poggiolesi (2018), Prawitz (2019) and Rumberg (2013). On the links between mathematical explanations and the notion of proof see Mancosu (1999, 2001).

  8. See Smithson (2020), p. 4.

  9. E.g. Fine (2012) and Raven (2015).

  10. E.g. see Audi (2012), Betti (2010), Hofweber (2009), Merlo (2020) and Smithson (2020).

  11. This has already been observed by Correia (2014).

  12. Mathematical grounding as a subtype of conceptual grounding is also considered in Carrara and De Florio (2020).

  13. See for example Bolzano (2004), §13.

  14. The notion of conceptual grounding in Bolzano’s philosophy involves differences with the modern conception, e.g. see Roski and Schnieder (2019) and Rumberg (2013).

  15. E.g. see Tatzel (2002).

  16. See Schnieder (2006), p. 405.

  17. See Schnieder (2006, p. 404).

  18. External mathematical explanations are explanations in which non-mathematical phenomena are partially explained by mathematical findings e.g. the explanation of why hive-bee honeycombs have a hexagonal structure in virtue of the fact that any partition of the plane into regions of equal area has as perimeter at least that of the regular hexagonal honeycomb tiling.

  19. According to Betti, even external mathematical explanations should count as conceptual explanations. However since in this case the link might be more controversial and Schnieder does not mention it, we leave this type of explanations aside for future research.

  20. E.g. see Audi (2012) and Raven (2015).

  21. A detailed reconstruction of Bolzano’s notion of conceptual complexity can be found in Roski and Rumberg (2016) and Roski and Schnieder (2019).

  22. What we call here fundamentality corresponds to what Tahko (2018) calls relative fundamentality. As he says “Given its importance, it is perhaps surprising that there are relatively few explicit accounts of relative fundamentality in the literature so far.”

  23. See Poggiolesi (2016).

  24. See Ginammi et al. (2020).

  25. In Sect. 4, we will put forward an alternative conceptual complexity scale for the colors red and crimson.

  26. See Ginammi et al. (2020).

  27. This seems a widespread conception behind explanations, see Woodward (2003), p. 153.

  28. E.g. see Hempel (1942), Jansson (2017), Salmon (1984), Strevens (2008) and Woodward (2003).

  29. We have slightly changed Bolzano (2004)’s example to make it more adequate for our study.

  30. E.g. see Poggiolesi (2016).

  31. E.g. see also Ginammi et al. (2020).

  32. E.g. see Mancosu (1999) and Steiner (1978).

  33. E.g. see Schaffer (2016).

  34. To take another example, consider the colors red and crimson. We can think of a theory T where the color red is defined as the set of all types of red, namely scarlet, crimson, ...: in T the color red will count as conceptually more complex than the color crimson. However, we can equally think of a theory \(T^{\prime }\) where the color red is primitive and crimson is defined as a particular type of red, e.g. red plus violet: in \(T^{\prime }\) crimson will count as conceptually more complex than red. Hence, this approach seems to systematize an intricate debate concerning grounding sentences that convey colors, e.g. see Koslicki (2015), p. 7.

  35. E.g. see Correia (2014), Fine (2012) and Smithson (2020).

  36. See for example Jansson (2015).

  37. See Winther (2016).

  38. See also Rumberg (2013), p. 443.

  39. E.g. see de Jongh and Betti (2010) or Muddy (2011).

  40. Note that a similar remark applies to the example of causality advocated in Sect. 2. “There is a fire in the forest because a cigarette was lit in the forest” is a sentence expressing a causal relation. For such a sentence to be a bone fide causal explanation, the specification that it is the law of combustion that makes it possible for the cigarette lit in the forest to start the fire, is needed.

  41. While in the philosophy of science literature, the distinction causality-causal explanations relies on the use of generalizations and laws, in the metaphysical literature, as far we know, the distinction metaphysical grounding–metaphysical explanation does not mention such a feature (an exception is Wilsch (2016)). A conjecture to explain this difference is that the latter literature only considers simple cases and hence the need of the use of generalizations, in metaphysical explanations, does not emerge.

  42. See also Schaffer (2016), p. 51.

  43. See Buhl (1961), Rumberg (2013) and Tatzel (2002).

  44. We remind the reader that a cut-free (logical variant of the) sequent calculus only contains introduction logical rules that introduce formulas either on the right or on the left of the sequent. Hence in a proof p of a cut-free sequent calculus where only logical rules are used, the complexity of the formulas involved in p cannot but increase. For further details see Poggiolesi (2010, 2020b).

  45. E.g. see Gentzen (1935) and Troelstra and Schwichtenberg (1996).

  46. Because our approach is based on Poggiolesi’s formal results (e.g. see Poggiolesi 2016, 2018) and these results exploit the natural deduction calculus, we prefer presenting derivations in logic by using the natural deduction logic. However, as underlined before, it is also plausible to use the resources of the sequent calculus.

  47. E.g. see Francez (2019) and Schroeder-Heister (1991, 2018).

  48. E.g. Sambin et al. (2000).

  49. On the logical difference between introduction and elimination rules see Troelstra and Schwichtenberg (1996).

  50. For an accurate examination of this point see Poggiolesi (2020a).

  51. See Poggiolesi (2020b).

  52. In the interests of readability, we use the following abbreviations:

    • T for triangle,

    • Q for quadrangle,

    • 2RA for two right angles,

    • 4RA for four right angles.

  53. In case the conclusion has the form of an universal quantifier, an implication or a negative formula, the situation is slightly more complicated than the one described above, see Poggiolesi (2020a).

  54. Here conceptual complexity should be understood in terms of composition of concepts. As shown by Ginammi et al. (2020), if conceptual complexity is understood in terms of generality, then it is a requirement compatible with the presence of generalizations as premisses in explanations.

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  1. CNRS, UMR 8590 IHPST - Institut d’Histoire et de Philosophie des Sciences et des Techniques, Université Paris 1 Panthéon-Sorbonne, Paris, France

    Francesca Poggiolesi & Francesco Genco

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Poggiolesi, F., Genco, F. Conceptual (and Hence Mathematical) Explanation, Conceptual Grounding and Proof. Erkenn 88, 1481–1507 (2023). https://doi.org/10.1007/s10670-021-00412-x

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