This discussion paper is a reply to Stathis Psillos’ paper “Induction and Natural Necessities” (2017), published in this journal. In that paper, he attempts to refute David Armstrong’s solution to the problem of induction. To accomplish this desideratum, he proposes that the best explanation for our observed regularities is a sort of “best before date” necessity. That is, necessary connections may break down and are not by default timeless. He develops arguments against my (author 1) defence of the necessitarian solution regarding a previous paper by Helen Beebee (2011). He alleges that a) best before date necessity is no worse than timeless necessity; b) his proposal does not imply any further inductive generalisation to timeless necessity; and c) inductive inferences are justified. In this discussion paper, I provide arguments against these three claims.

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.

Some recent literature [Hicks, M. T. and van Elswyk. P., (2015) pp. 433-443, 2015; Bhogal, H. (2017), pp. 447-460] has argued that the non-Humean conceptions of laws of nature have a same weakness as the Humean conceptions of laws of nature. That is, both conceptions face an explanatory circularity problem. The argument is as follows: the Humean and the non-Humean conceptions of laws of nature agree that the law statements are universal generalisations; thus, both conceptions are vulnerable to an explanatory circularity problem between the laws of nature and their instances. In this paper, I argue that Armstrong’s necessitarian view of laws of nature is invulnerable to this explanatory circularity problem.

O argumento melhorado da indispensabilidade matemática, proposto por Alan Baker (2005), defende que nos devemos comprometer com as entidades matemáticas, porque as entidades matemáticas desempenham um papel explicativo indispensável nas nossas melhores teorias científicas. Este artigo clarifica as doutrinas que suportam este argumento, nomeadamente, as doutrinas do naturalismo e do holismo da confirmação.