In preparation of an upcoming talk on Spinoza and his critic, Bernard Nieuwentyt, at a conference at Max Planck, I am frantically reading Nieuwentyt's *Gronden van zekerheid of De regte betoogwyse der wiskundigen so in het denkbeeldige, als in het zakelyke: Ter wederlegging van Spinosaas denkbeeldig samenstel, en ter aanleiding van eene sekere sakelyke wysbegeerte*. Nieuwentyt wants to refute (what he takes to be) the pseudo-mathematical character of Spinoza's Ethics. (Here I am going to leave aside how cogent Nieuwentyt's claims about Spinoza are.) One of Nieuywentyt's arguments is that Spinoza ignores a crucial distinction between pure and applied mathematics. According to Nieuwentyt the former (including geometry) can be true but physically (or empirically) impossible. That is, something can be true, yet not exist (a view recently made popular by Jody Azzouni). (Nieuwentyt would have been unmoved by the so-called Quine-Putnam indispensibility argument.) Interestingly, Nieuwentyt is -- as far as I can tell -- not a Platonist about mathematics, but a conventionalist! In doing so, Nieuwentyt not only shifts early modern debate away from the thought that geometry is an empirical science (something to be found in Barrow and Newton), but also toward Humean and Kantian positions on the nature of mathematics.
Update 11 December after discussion especially with Gabor Zemplen: Nieuwentyt's view is Aristotelian in many ways (including his definition of truth, and the tendency to to treat mathematical objects as mental entities). The crucial question (from the vantage point if there is anything new-ish in Nieuwentyt) is to what degree Aristotle's view of mathematical objects allows their truth to be disconnected from existence (and real possibility) entirely.
Recent Comments