With this chapter we see Craig examine the argument in favor of the existence of abstract objects. While I am offering a brief summary of the chapter, you can listen to William Lane Craig deliver a more in depth lecture on this exact topic in a video on his YouTube channel here.

The Indespesability Argument

The contemporary debate over the existence of abstract objects is centered in the philosophy of mathematics. When it come to formulating an argument for grounding belief in abstract objects, the foremost argument would be the Indespensability argument. What this argument seeks to establish “is that we are committed to the reality of abstract objects by many of the statements like ‘1+1=2.’”[1] The formulation for the argument, which Craig takes from Balaquer, is as follows:

(1) If a simple sentence (i.e., a sentence of the form ‘a is F’, or ‘a is R-related to b’, or…) is literally true, then the objects that its singular terms denote exist. Likewise, if an existential sentence is literally true, then there exists objects of the relevant kinds; e.g., if ‘There is an F’ is true, then there exist some Fs. 

(2) There are literally true simple sentences containing singular terms that refer to things that could only be abstract objects. Likewise, there are literally true existential statements whose existential quantifies range over things that could only be abstract objects.

(3) Therefore, abstract objects exist. [2]

With the argument laid out, Craig then looks into each premiss more closely.

Premiss (1): A Criterion of Ontological Commitment

What premiss (1) seeks to establish is a criterion of ontological commitment. It does not seek to establish what actually exists, but rather seeks to establish what must exist if a sentence that is asserted is to be true. In other words, it shows what our discourse commits us to ontologically. According to this premiss, ontological commitments are made in two ways; through the use of singular terms, and through existential quantification. 

Singular Terms

Premiss (1) states that we make ontological commitments by the use of singular terms. Singular terms are words or phrases which are used to single out something. This would include  proper names like ‘John’, ‘HMS Bounty’, and so on; definite descriptions like ‘the man in the grey suit’, ‘your sister-in-law’, ‘my worst nightmare’, and so on; and demonstrative terms like ‘this pancake’, ‘that boy’, and so on.

In premiss (1), Balaquer uses ‘a’ as a logical constant for which we may substitute a singular term to form a simple sentence. ‘F’ stands for any property which is predicated upon the individual or item identified by the singular term, and ‘R’  any relation in which that entity may be said to stand. As an example, the sentence “The Queen Marry II is a huge oceanliner” has the form ‘a is F’. Likewise, the sentence “The Queen Mary II docks in South Hampton” has the form ‘a is R-related to b’. (For the remainder of the discussion, Craig chooses to ignore discussion related to simple sentences involving relations.)

Recall that the claim of premiss (1) is that if a simple sentence is literally true, then the objects denoted by the singular terms exists. Thus, if the sentence “The Queen Mary II is a huge oceanliner” is literally true, the Queen Mary II exists. Craig then draws attention to two aspects of this premiss that need to be examined: why a sentence must be literally true; and why the sentence must be simple.

Concerning the literal truth of a sentence, Balaquer argues that sentences that employ metaphors and other figures of speech do not ontologically commit the speaker to the reality of the singular terms. For example, if someone says “It is raining cats and dogs,” it is understood that the speaker is remarking that it is raining hard, but not that animals are literally falling from the sky. Therefore, non-literal sentences are excluded from the arguments consideration.

As for simple sentences, Balaquer is not trying to avoid complexity, but rather sentences involving intensional contexts. These are contexts that are not extensional contexts. “Extensional contexts are sentence phrases which have two characteristics. (i) Singular terms referring to the same entity can be switched without affecting the senence’s truth value… (ii) One can quantify into such contexts from outside the context.” [3]

Existential Quantification

The second part of (1) claims that we also make ontological commitments by way of existential sentences. These are sentences which involve existential quantifiers. Here Craig draws a distinction between universally quantified statements, and existentially quantified statements. Universally quantified statements are true with respect to all the members of the domain of quantification. While existentially quantified statements are true with respect to some of the members of quantification. 

There are numerous terms in the English language used to denote existential quantification, such as ‘some’, ‘at least one’, ‘there is/are’, and ‘there exists’. All of these are informal expressions that can be notated in formal logic by the formal quantifier ‘∃’ (turned E). Thus, a statement like “some bears live near the North Pole” has the form (∃x) (where x is a bear & x lives near the North Pole). This expression is to be read “There is some x such that x is a bear and x lives near the North Pole”. In order for this statement to be true, there must be at least one thing in the domain that can fulfill the value of ‘x’.

With that being said, the claim of premiss (1) is that the literal truth of simple sentences involving existential quantification of the form “There is an F” commits us to the reality of the object which is an F. So if we substitute something for F, like an apple for example, the statement “there is an apple” can be notated as (∃x) (x is an apple). The existential quantifier ‘∃’ is used to express existence, not in the metaphysically light sense, but in the heavy weight sense. The example above ((∃x) (x is an apple)), then means “there exists an apple.” As Craig puts it, “the person who makes assertions involving informal quantifiers like ‘some’ and ‘there is/are’ thereby commits himself to the reality of the things in the domain of quantification.”[4] 

What we can say in summary of premiss (1) is that it does not tell us what actually exists, but rather it demonstrates how our language ontologically commits us to the reality of the objects referred to by our statements. 

Premiss (2): Abstract Objects

Premiss (2) builds upon (1). As explained above, when we use literally true simple sentences where the singular terms refer to concrete objects like apples, bears, or cars, we are committed to the ontological existence of those referents. Premiss (2) then states that just like there are literally true simple sentences concerning concrete objects, there are also literally true simple sentences where the only possible referents for the singular terms are abstract objects. This includes statements like ‘2+2=4”. The referents of mathematical terms like ‘2’ and ‘4’ are undeniably abstract objects. The second part of premiss (2) likewise claims that there are literally true existentially quantified statements involving quantification over abstract objects. Statements like this would be: “There is a prime number between 2 and 4” or “There are prime numbers greater than 100”. Clearly these are statements that are not meant to be taken metaphorically. So then, premiss (2) asserts that existentially quantified abstract discourse commits the user to the existence of abstract objects. 

Craig then concludes his examination of the two premises of the indispensability argument by conceding that from premises (1) and (2), the conclusion (3) that abstract objects exists logically follows. So if the classical theist wishes to deny the existence of abstract objects, he must deny either (1) or (2).

Responses to the Indespensability Argument

Craig concludes the chapter by showing that various responses to the Indespensability argument and the challenge of Platonism. He starts by listing the three major umbrella categories: the realist views (mathematical objects exist), anti-realist views (mathematical objects do not exist), and arealist views (there just is no fact of the matter concerning the existence of mathematical objects). 

Realist responses divide into two further categories which label mathematical objects as either abstract or concrete. In the abstract camp, the view of “absolute creationism” affirms the reality of abstract objects, but holds a position that is modified from Platonism that has these objects being created by God. As for the concretist views of mathematical objects, proponents of this view argue that these objects are either physical or mental (mental as in the mind of God or humans). “Divine conceptualisim” is perhaps the foremost view in this camp and is related to the views of the church fathers and Philo. 

As for anti-realist views of mathematical objects, there are quite a large number of views offered. I will summarize them here:

Nuetralism – the use of singular terms and existential quantification is neutral when it comes to ontological commitment. Thus, this view denies the criterion of ontological commitment as presented in premiss (1).

Neo-Meinongianism – denies that existential quantification is ontologically commuting. 

Free Logic – holds that existential quantification is ontologically committing, but denies that the use of singular terms is. 

Fictionalism – accepts the Platonist’s criterion of ontological commitment, but denies that mathematical statements are true. 

Pretence Theory – mathematical discourse is a type of make believe. Mathematical objects then are like fictional characters. 

Paraphrastic strategies – holds that paraphrases of mathematical statements can be made to preserve the truth value of the statement, but without the ontological commitment to abstract objects.  

The final umbrella category was that of arealism. This is the view that asking wether or not abstract objects exists is a meaningless question and there just is no answer. The foremost view here was the “conventionalism of Rudolf Carnap.” Craig quickly dismisses this option by pointing out that for the classical theist, this view can’t even be considered an option. “For given God’s metaphysical necessity and essential aseity, there just is no possible world in which uncreated mathematical objects exist. Hence, there most certainly is a fact of the matter whether uncreated, abstract objects exist: they do not and cannot exist. Therefore, arealism is necessarily false, as is conventianalism about existence statements concerning mathematical and other abstract objects.” [5] In other words, given the theists understanding of God existing a se, the statement “mathematical objects exist” is either necessarily true, or necessarily false. There is no middle ground. 

 

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[1] Craig, “God Over All,” 45.

[2] Balaguer, ‘Platonism in Metaphysics”. qtd. in Craig, 45-46. 

[3] Craig, 47-48. (On a personal note, I really struggled with the way Craig tried to delineate the differences between intentional and extensional contexts. I still am not sure wether the difficulty is just that I am not really grasping the idea presented, or if Craig just poorly explained the difference.)

[4] ibid. 49.

[5] ibid. 53.