Explanatory appeals to abstract ‘objects’ (numbers, moral values/principles, possible worlds etc…) are ubiquitous in philosophy, science, and everyday reasoning. Cicadas emerge in prime-number cycles because of number-theoretic advantages, lying is wrong because it violates the categorical imperative, Celtic FC would have won yesterday’s match had the referee been unbiased. Many philosophers take such claims’ explanatory usefulness to justify ontological commitment to the abstracta involved. Yet, by definition, abstracta are spatially and causally removed from the concrete world we seek to explain, raising a fundamental question: when, if ever, does explanatory appeal to abstracta genuinely license belief in their existence?
To answer this, I propose a methodological framework which distinguishes merely representative/heuristic explanations from metaphysically substantive ones. Two criteria structure the framework: ‘Basis’, scrutinises the reality of explanans and explanandum (independent of their inclusion in a particular explanation); and ‘Relevance’, assessing whether the explanans stands in an appropriate ontic-explanatory relation to the explanandum.
Applying this framework to case studies in mathematics, morality, and modality, I argue that explanatory appeals to abstracta systematically fail both criteria. Abstraction may be an indispensable representational tool, but abstracta themselves are never adequate explanans for why the concrete world truly is as it is.
