A discussion on Ontology of numbers

Summary of Collins A.: ON THE QUESTION ‘DO NUMBERS EXIST?’ in The Philosophical Quarterly, Vol. 48, No. 190 (1998). pp. 23-36

1.0 Introduction

In human thought, the numbers appear to have originated from the various physical situations encountered by man – the difference between a herd and a goat, the difference between 3 deers and 1 deer, the correspondence between number of shadows and number of people etc. At some point, the abstract property which is common to say two groups (ex. 5 arrows and 5 birds) was recognized and this is what is called a number. This is a primitive notion of a number, in contemporary times we have various kinds of numbers – the rationals, the irrationals, the imaginary numbers etc. But all the kinds of numbers are equally abstract. So the question arises, where does the number exist or do they exist at all. The paper is an attempt to this question. The author treats the problem as of having a linguistic origin. The author rejects the usual answers to the problem that is realism and nominalism, so first I will explain these usual answers in brief before presenting authors views.

2.0 Realism and Nominalism

“Realism is a philosophy of mathematics and an ontological commitment” (Collins, pp23). That is, realism is the belief that properties, usually called Universals, exist independently of the things that manifest them. Therefore, if we remove all the rectangular shaped objects from the universe, still the universal rectangle will exist. Thus it can be seen that there are two aspects to realism. First, there is a claim about existence. The billiard tables, the football grounds, the books exist and so does the rectangleness. The second aspect of realism concerns independence. The fact that the billiard table exists and is rectangular is independent of anything anyone happens to say or think about the matter. The question of the nature and plausibility of realism is a controversial issue (Miller, 2002).

Nominalism is an anti-realist stand. The doctrine holding that abstract concepts, general terms, or universals have no independent existence but exist only as names. The relation between universal and name is conventional.

But there is a problem with nominalism as pointed out by Prof. Gomatam during the class discussion; it is the fact that a universal say rectangleness cannot be attributed to arbitary things, so universals are not just names.

3.0 The short argument

It is a familiar thought that we might posit numbers to explain the known arithmetical truths and scientific truths the expression of which requires numerical representation (Collins, pp. 23). To explain this author gives an example,

There are four prime numbers smaller that 8.

If this fact is known, then we know the numbers exist. This is what author calls the short argument for the existence of numbers. Further, the author believes that if there were no debate between realism and nominalism the short argument would be satisfactory. I am unable to agree with this point. First of all, the condition of ‘primality’ comes into discussion when we talk about numbers. It seems that first we have numbers and then we define the condition of primality on them. Thus, the fact that “there are four prime numbers less that 8”, cannot be known without the prior knowledge of numbers. It seems that the short argument is trying to prove the existence of numbers in the hindsight i.e. by at first assuming them. I think Quine is also trying to point to this fact in the following quote, “…indispensability of mathematical entities and the intellectual dishonesty of denying the existence of what one daily presupposes”(Quine as quoted by Putnam in Collins,1998, pp. 26). I think it is right to say that if numbers exist it means that ‘there are four prime numbers smaller than 8’. Although, this does not answer the original question but it shows the problem in the short argument. Even if we consider the scientific truths, for example, the velocity of light is 3 X 10⁸, the same problem arises. The short argument says that since we know this fact, it implies the numbers exist. But the concept of velocity (=distance/time) comes into discussion only if we have prior knowledge of numbers. Author claims that the short argument leaves no space for positing of the numbers. I he knows that there are four primes less than 8 it entails that he already has the knowledge of numbers and he need not posit them, whereas, both realism and nominalism posit numbers.

Quine is a soft realist. As quoted by Putnam, Quine thinks it is intellectual dishonesty to use and talk about mathematical entities, yet deny their existence. Quine treats everything as a myth, but he treats the myth of physical objects as superior to say Homer’s gods. He says, “…The myth of physical objects is epistemologically superior to most in that it has proved more efficacious as a device for working a manageable structure into the flux of experience”. Hartry Field also has a similar view. If believes that if numbers can be shown to be a useful myth then it can be shown that they are fictional thus, they do not exist.

4.0 Linguistic solution

Author says that short argument is sufficient to show the existence of numbers but the question about the way they exist, should not be asked as it is inappropriate. He accepts that numbers do not exist as the physical objects do, still it is right to say that they exist. As pointed by Prof. Gomatam, it seems what author wants to say is that by asking the question about way the numbers exist, we are making a category mistake. For example, if pen is in the pocket, we can ask someone to pull it out. Whereas, if someone says, he has an idea in his head we do not ask him to pull it out.

Another problem while thinking about the existence of numbers is that they are defined by giving them negative attributes like they are non-physical, non-spatial, non-temporal, non-corrutible, non-contingent and they do not enter the causal relationships(Collins, pp. 30).

5.0 Conclusion

Although, I am unable to agree that the short argument is sufficient to establish the existence of numbers. But I am inclined to believe that the numbers exist based on Quine’s argument. Field also recognizes that Quine’s argument is the strongest argument for realism. Quine argues that it is indispensable to talk about mathematical entities yet their existence is denied which implies that there is an intellectual dishonesty. Yet, the question of about what is for them to exist is not answered. Author argues that the question is wrong because our discourse with numbers does not generate the question. Also, because of the fact that the numbers are given negative attributes, the question of what is for them to exist sounds wrong to me.

6.0 References:

1. Miller, A (2002), Stanford Encyclopedia of Philosophy, http://setis.library.usyd.edu.au/stanford/entries/realism/.

2. Collins, A.(1998, ON THE QUESTION ‘DO NUMBERS EXIST?’ in The Philosophical Quarterly, Vol. 48, No. 190 pp. 23-36.