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How would you try to deal with the cicada case if you were a nominalist?

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If they exist, abstract objects are very unlike physical objects. They will exist outside space and time. They will not cause events to occur. Given these peculiar features, we would need to have a good reason to think that they do exist. One such reason might come in the form of (what’s known as the enhanced version of) the indispensability argument. The idea, roughly, is that some abstract objects will be mathematical objects – numbers being a good example. For numbers, if they exist, will be unchanging and will not obviously exist in space and time (I’ve never seen them, though of course I’ve seen the figures that stand for them). Thus,

• We ought rationally to believe in the existence of any entity that plays an indispensable explanatory role in our best scientific theories.

• Mathematical objects play an indispensable explanatory role in science. • Hence, we ought rationally to believe in the existence of mathematical objects.

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Proponents of this argument-form then typically try to give some reason to think that mathematical objects do indeed play an indispensable role in our best scientific theories. One such attempt has roughly the following structure: the life-cycle of the North American periodical cicada is 17 years in the north of the territory it inhabits, and 13 years in the south. This life-cycle has evolved, the evolutionary biologists tell us, because it is advantageous. It is advantageous because, 17 and 13 both being prime, having a 17-year or 13-year life-cycle minimizes intersection with other lifecycles – including those of predators. It minimizes this intersection because it is prime. Primeness is a feature of numbers, and nothing else. Thus, numbers feature as an indispensable part of our best scientific theories. The position that there really are such things as numbers, motivated by the above argument, can be challenged. The following thought experiment outlines one such way in which to develop that challenge

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Let’s allow that there are numbers and that numbers are abstract objects. Imagine a way that the world could be. Include in such a possible scenario all of the physical objects that actually exist. Arrange those physical objects exactly as they are now. However, from this scenario, excise any numbers and any abstract objects. Such a scenario seems conceivable. We normally take the conceivability of some scenario to indicate its possibility. Thus, it seems possible that the world might be exactly as it is, even if there were no numbers. And, in that case, it seems as if numbers aren’t really explaining how the physical world is the way it is. It therefore appears that we should not believe that they exist.

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If successful, this argument undermines the indispensability argument, given above. And if the argument is successful, then presumably we ought to give up on the existence of abstract objects. It would then behove us to say something about the truths of mathematics. If there are no such things as numbers, then why is it true that 2 + 7 = 9? Why is it true that there is a number between 3 and 5 – if there are no numbers?

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Two obvious strategies come to mind. The first strategy – a view known as mathematical nominalism – is to try to understand mathematical claims as about something other than mathematical objects. To illustrate, to say that ‘the number of electrons is 2’ is better interpreted as saying: ‘there is an electron; there is another electron’; to say that ‘the number of electrons is 3’ is better interpreted as saying: ‘there is an electron; there is another electron; there is a further electron’. This process can simply be iterated. The other obvious strategy is to deny that mathematical talk is true, though to agree that it’s very useful. Call this view ‘mathematical fictionalism’. The thought might go like this. Mathematical discourse seems to be about numbers. But there are no numbers. (Abstract objects are just too strange for us to countenance.) In that case, it’s straightforward to see that all of our mathematical discourse is false. However, mathematical discourse is still extremely useful. But we shouldn’t confuse its usefulness for its truth.

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What are the strengths and weaknesses of the enhanced indispensability argument?

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Do you think that the thought experiment shows that numbers cannot be playing a genuine explanatory role? Why?

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How do you think that the proponent of the existence of numbers should respond to the thought experiment? How plausible do you find that response?

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How plausible do you find the idea that abstract objects exist?

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How plausible did you find the view that mathematical claims could all be false but useful?

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How would you try to deal with the cicada case if you were a fictionalist?

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Of those outlined, which view do you find most plausible?