Though it is unclear after reading “Abstract Entities” which position (realist or anti-realist) Swoyer takes on abstract entities of any kind, the following is an attempt to capture some of the positive points he makes regarding their existence. Swoyer begins by placing abstract entities into a context. He contends that abstract entities are often postulated to solve certain philosophical problems, and it is these circumstances for which abstracta are given existence. We must, then, concentrate on the tasks these abstracta are given, and proceed to evaluate them accordingly.
Swoyer points out that the tasks assigned to abstracta are typically explanatory, and so we must analyze them in terms of what they are postulated to explain. In section 3.1 He presents 9 explanandum in number theory for which the existence (among other things) of numbers is supposed to be the explanans.
Swoyer’s main assertion is that we should view arguments for the existence of abstract entities as inferences to the best overall available ontological explanation. Here is where I would like to take some liberties in constructing a positive argument for the existence of abstract entities based on Swoyer’s approach and analysis.
My problem, as I shall outline next, is the conclusion Swoyer wishes to establish based on the inference to the best overall available ontological explanation. Now, let us give Swoyer the explanations he wishes numbers to achieve in his example in section 3.1. The 9 explanandum are accounted for by postulating the existence (among other things) of such abstract entities as Numbers. This is great. We had 9 things that are generally accepted in number theory, and to explain them, we have numbers, abstract entities. Hence, Numbers exist. Perhaps I am being too hasty to conclude their existence, for Swoyer used them in an example to show simply what role they would play in explaining those 9 points. But I wish to abstract from these considerations, and take his example as a paradigm of how things might look were we to conclude the existence of abstract entities from their being the best overall available ontological explanation. Let me formulate the situation as follows:
1 If numbers are the best overall available ontological explanation, then we may infer the existence of numbers.
2 If numbers can explain points 1-9, then numbers are the best overall available ontological explanation.
3 Numbers can explain points 1-9 (remember this is what I gave to Swoyer).
4 Numbers are the best overall available ontological explanation (from 2 and 3)
5 Therefore, we may infer the existence of numbers (from 1 and 4).
It appears that numbers do wonderful things. They explain 1-9 (so I suppose). Now it is what numbers do that I am focusing on, taking cue from Swoyer, that we should analyze abstract entities in terms of what they are postulated explain. However, my objection is that the success of Numbers in explaining 1-9, and being the best overall available ontological explanation does not bestow truth on the assertion that Numbers exist. What we have is success in explaining, and perhaps being the best explanation. But this alone does not establish the existence of numbers. Let us say that numbers successfully explain, and are the best explanation for, the 9 explanandum. But surely just because they have true consequences, i.e. the 9 explanandum, we should not conclude that Numbers exist. Upon reflection we can see that even a false postulates can have true consequences. Take for example Plato’s theory of Body and Soul. The soul, for Plato is what animates the body. Well, there are living bodies. But what is still in question is: Is there a soul? Citing it as the best overall available ontological explanation will not do. Hence, this type of realism falls into the following fallacy:
1 If there are numbers, then there will be true consequences.
2 There are true consequences.
3 Therefore, there are numbers.
Swoyer does not attempt to provide a deductively sound argument for the existence of abstract objects, but provides an alternative way of seeing the arguments for their existence. However, my point is that although it appears that no conclusive argument will establish their existence, it is impossible for the inference to the best overall available explanation to do so, for it involves the fallacy of begging the question.
(The above points on the fallacy of affirming the consequent are taken from Larry Laudan's A Confutation of Convergent Realism)