It has repeatedly been argued that nominalistic programmes in the philosophy of mathematics fail, since they will at some point or other involve the notion of logical consequence which is unavailable to the nominalist. In this paper we will argue that this is not the case. Using an idea of Nelson Goodman andW.V. Quine’s which they developed in Goodman and Quine (1947) and supplementing it with means that should be nominalistically acceptable, we present a way to explicate logical consequence in a nominalistically acceptable way.

This paper compares Kotarbiński’s reism and naturalism. It argues that basic ontological and epistemological reistic principles fit naturalism very well. In particular, the thesis claiming that there are only spatiotemporal things (bodies) gives a very simple naturalistic account of reality. Radical realism defended by Kotarbiński is a version of direct realism, a view about perception which is very accurate for naturalism. On the other hand, since difficulties of reism are also problems for naturalism, the former illuminates typical challenges for the latter.

Erik Wielenberg is the most important contemporary critic of theistic metaethics. Wielenberg maintains that God is unnecessary for objective morality because moral truths exist as brute facts of the universe that have no, and need no, foundation. At times his description of these brute facts make them sound like abstract objects or Platonic forms. At the American Academy of Religion’s annual meeting in Boston in November of 2017, we organized an Evangelical Philosophical Society panel to discuss Erik Wielenberg’s book Robust Ethics: The Metaphysics and Epistemology of Godless Normative Realism. All five papers presented there are included in this journal.

I argue that materialism is unable to account for knowledge deriving from such abstracta as rules of inference, algorithms, and the ideals of infinity, perfection, and eternity. Both reductive and nonreductive materialism subscribe to the causal closure of the physical world, which implies that a creature’s concepts derive exclusively from the interactions of brains with the physical environment. These resources do not explain the acquisition of abstract concepts or the successful use of these concepts in gaining important knowledge about the world. By contrast, if both God and souls exist, we can understand how knowledge based on abstracta is possible.

One problem for Molinism that critics of the view have pressed, and which Molinists have so far done little to address, is that even if there are true coun­terfactuals of freedom, it is puzzling how God could possibly know them. I defuse this worry by sketching a plausible model of the mechanics of middle knowledge which draws on William Alston’s direct acquaintance account of divine knowledge.

Two conceptions of the nature of mathematical objects are contrasted: the conception of mathematical objects as preconceived objects (Yablo 2010), and heavy duty platonism (Knowles 2015). It is argued that some theses defended by friends of the indispensability argument are in harmony with heavy duty platonism and in tension with the conception of mathematical objects as preconceived objects.

William Lane Craig maintains that propositions and properties are not real. Yet, if we examine his proposed nominalism and his appeal to Rudolf Carnap’s linguistic frameworks, we can find that his view depends upon their reality, even as abstract objects. By drawing upon phenomenological insights, I argue that if we pay close attention to what can be before our minds in conscious awareness, we can become aware that there is more to what is real than simple, concrete particulars, even in his linguistic examples. We can become aware of the reality of Platonic, ante rem universals, including propositions and properties.