That’s a fascinating exposition of Knapp’s argument for what I imagine is a quasi-fictionalist picture of the philosophy of mathematics! And indeed, not a Platonist picture as he denies the existence of mathematical abstracta (which I also do) but finds meaning in numbers through objects and their relations in the world.

There are a range of contentious premises here which could be interrogated to bring an alternative conclusion, amongst which the assumption that relational properties exist (and indeed, that properties exist! I’ve got a blog coming out soon about that, approached from a nominalist point of view this time) but this is certainly one route one could take to defend the view that mathematical claims at least have some basis in reality. I do think that this view (like my own, I hasten to add!) struggles to account for the sheer complexity and power of mathematics, and makes it difficult to account for non-integer truths such as “Pi is the ratio of a circle’s circumference to its diameter” – so there is more to do here.

Knapp’s conclusion is hard if not impossible for me to swallow as an Existence Monist (I subscribe to a hard fictionalist understanding of this topic, although I didn’t say as much in this post I don’t think!), since I personally reject objects never mind properties. An alternative for the proponent of Knapp’s version might be to make softer claims which don’t require the existence of relational properties but still make comparisons between existents. I did, of course, try to leave my personal ontological views out of this as I wanted to write a partisan post – but you’re absolutely correct that the first thing that needs to go is Platonism..!