Do Numbers Exist?: A Debate about Abstract Objects

by Peter van Inwagen, William Lane Craig

In Do Numbers Exist? Peter van Inwagen and William Lane Craig take opposite sides on whether there are abstract objects, such as numbers and properties. Craig argues that there are no abstract objects, whereas Van Inwagen argues that there are. Their exchange explores various arguments about the existence and nature of abstract objects. They focus especially on whether our ordinary and scientific thought and talk commit us to abstract objects, surveying the options available to us and the objections each faces. The debate covers central problems and methods in metaphysics, and also delves into theological questions raised by abstract objects. Key

Genres: Philosophy

298 pages, Paperback

Published March 12, 2024

About the author

Peter van Inwagen is an American analytic philosopher and the John Cardinal O'Hara Professor of Philosophy at the University of Notre Dame. earned his PhD from the University of Rochester under the direction of Richard Taylor and Keith Lehrer.

Today, Van Inwagen is one of the leading figures in contemporary metaphysics, philosophy of religion, and philosophy of action. He has taught previously at Syracuse University and was the president of the Society of Christian Philosophers from 2010 to 2013. He was elected to the American Academy of Arts and Sciences in 2005 and was President of the Central Division of the American Philosophical Association in 2008-2009. Van Inwagen has also received an honorary doctorate from the University of Saint Andrews in Scotland.

Reviews

[Matthew Adelstein · Rating 4 out of 5]

It does get rather tedious, but interesting clash between these two thinkers with diametrically opposed views about numbers. I found PvI more convincing.

[Carson Phillips · Rating 5 out of 5]

I never thought a book about the philosophy of mathematics could be a page-turner, but this one is! This also serves as the best introductory book on the subject that I’ve read yet, while still adding more depth than similar books. Peter van Inwagen takes the position of the platonist (realist) in holding that abstract objects exist. William Lane Craig defends an anti-realist position that abstract objects do not exist. Van Inwagen opens with an articulation of what he takes to be the ontologically committing parts of speech from Premiss 1 of the neo-Quinean Indispensability Argument as formulated by Mark Balaguer. This is effectively the claim that all objects existentially quantified over in true, simple sentences are ontologically committing. William Lane Craig pushes back against this claim, as well as Premiss 2 of the Indispensability Argument, with a myriad of anti-realist positions that cut the nerve of the Indispensability Argument. The book also touches on whether the existence of abstract objects is a threat to Divine Aseity and the problem of the applicability of mathematics to the natural world. If you are interested at all in platonism and it’s counter-positions, this is a must read!

[Husk Jones · Rating 2 out of 5]

Here’s my very subjective and very opinionated opinion on this one.

The book is quite frustrating. Peter Van Inwagen (PVI) comes out the gate swinging — against works of William Lane Craig (WLC) that the reader has not yet even been sufficiently introduced to. He frequently assumes more background than one would anticipate given that this is clearly intended to be, at least in part, an introductory text on the topic. (WLC’s writing is more accessible.) PVI’s entries also frequently become technical, or their conclusions or steps get lost in the details. His portions feel more like drafts than finished works. Several of PVI’s arguments are interesting nevertheless.

WLC comes out of the gate bizarrely. Rather than develop and defend a specific view, WLC enumerates of a large number of, apparently, distinct views. As a non-specialist in this area, the move felt bizarre and unsatisfactory. It felt like the philosophical equivalent of Gish galloping (a debate tactic in which one speaks rapidly, dumping large amounts of information to overwhelm one’s opponent.) WLC’s entires felt undeveloped, almost insouciant at times. I assume this is the product of his choosing to enumerate a big swath of views rather than developing any single one.

Take Yablo’s Figuralism which, according to WLC, claims that mathematical statements are not literally true but rather figurative, make-believe statements. On this view, mathematical statements are literally false. However, taken non-literally, mathematical statements are made true by logic — not Platonic mathematical entities. Logic itself is not made true by Platonic logical entities.

I assume Yablo’s actual view is more sophisticated than this. Yet, the way WLC described it, the view suffers an obvious flaw: what makes the logical statements true? If nothing makes the logical statements true, then they’re false, and since they’re false, then the mathematical statements whose truth-value they ground are also false. So, by invoking logic, the view simply pushes the question back a step. Anyone with a serious undergraduate education would have spotted this immediately upon reading it. That WLC says nothing more here is bizarre and deeply unsatisfactory. (On another interpretation of WLC’s reading of Figuralism, mathematical statements are trivially true. Yet, if that is the case, then 2+2=5 is trivially true — which is obviously the wrong verdict.) I assume these are problems with WLC’s presentation rather than Yablo’s actual view.

Later on, while describing Williamson’s view Necessitism, WLC claims that Necessitism is a metaphysically heavy modal logic. On Necessitism, modal logic weighs in on substantive metaphysical and ontological truths. WLC complains, writing that “we want a modal logic that is neutral as to ontological commitments” (pg 128). But that’s not possible, and if he’d read Williamson’s book, he’d know that — or at least try to make arguments to the effect that it is possible.

Here’s the case that there can be no such thing as a metaphysically neutral modal logic. Suppose that a modal logic doesn’t weigh in on metaphysical truths. Well, that fact, the fact that this modal logic doesn’t weigh in on metaphysical truths, is ITSELF a metaphysical truth. That is, there is no METAPHYSICAL connection between logical truths and OTHER metaphysical truths — which is itself a truth about metaphysics. But, if that is a metaphysical truth, then there can’t be a modal logic that doesn’t weigh in on metaphysical truths. So, therefore, there can’t be a modal logic that doesn’t weigh in on metaphysical truths. Hence, the type of logic “we want” isn’t even possible and WLC has given no reason at all to think that it is.

Worse, even if there COULD be a modal logic that didn’t weigh in on at least some metaphysical truths, it’s not clear why we’d want that. We want the best theory overall — that theory will answer questions. A theory that doesn’t answer questions won’t be the best theory. So, ideally, we’d have a substantive logic, one that answers the questions.

WLC offers no reason to think either of these obvious lines of reasoning are false. Hence, what he says should be dismissed.

I could go on about the things WLC says about free logic or so forth, but I’ll stop. His submissions didn’t strike me as being serious philosophy.

Unfortunately, PVI didn’t give his best either — he didn’t engage with WLC as much as one would have liked, and often wrote in obscure, indirect ways that were unhelpful. PVI felt to be in his own world, doing his own thing, rather than a serious party to an important debate.

I’m sure others will get more out of this than I did, but I can’t say I benefited much from this, despite being interested in the topic.