Abstract
In this chapter, the reader is taken on a journey from Maddy’s naturalist position to a more pluralist position. The pluralist is inspired by Maddy’s mathematical naturalism in the following respect. With Maddy, the pluralist is very interested in the practice of mathematics, and is quite willing to let mathematical practice delimit what is to count as ‘mathematics’. The pluralist parts company with Maddy over the data concerning the philosophical inclinations of mathematicians. Given different data, the pluralist finds himself driven towards a more pluralist conception of mathematics than Maddy’s.
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Notes
- 1.
A lot of my reading of Quine is indirect and comes from Maddy. I deliberately take Maddy’s Quine in order to emphasise her developments of naturalism. I am not so concerned with loyalty to Quine. There are less and more sophisticated (and flexible) interpretations of Quine. It turns out that the very sophisticated readings approach my overall point of view quite well.
- 2.
Quine understood that some metaphysics is inevitable in science. This was his critique against the positivists (who thought at one extreme that it is possible to do all science without metaphysics). When more moderate, the positivists wanted to rid science of as much metaphysics as possible. What Quine, like Popper, was against was a priori metaphysics, where we construct irrefutable theories.
- 3.
Quine had a preference for scientific investigations, but his meta-philosophical pragmatism is what Maddy draws on. She acknowledges her preference for investigating pure mathematics, as a science in its own right. I should like to thank Zawidzki for helping me appreciate the distinction between Quine’s preferences and his meta-philosophical pragmatic naturalism, which allows him to accommodate modifications to his views on the basis of different preferences.
- 4.
There are two distinct aspects to taking heed of the scientist. One is to take what practicing scientists do and discover seriously, the other is to take what they say about the philosophy of science seriously. The first is less controversial, since it will be at least a starting point for any philosophy of science. The second is taken seriously, especially when scientists are also philosophers. In fact, some would say that to be a good scientist, one has to be a bit of a philosopher too. But this is thought of as the exception by some philosophers who pay no attention to philosophical remarks made by scientists – since they are not qualified to make them. This dismissal relies on an easy partitioning of philosophy and science, but such partitioning is a little strained. This is plain when one considers that this sort of distinction only starts to make sense when we think of the modern education system where people are asked to specialise in one area early, and this prevents them from spending a lot of time in another. This is a very recent phenomenon. Newton and Leibniz, for example, were hardly ‘trained in science’ to the exclusion of other forms of ‘training’ or enquiry.
- 5.
This is especially true of the ‘early Quine’. The ‘later Quine’ accepted much more of mathematics and logic. This is done mainly through a fairly extensive ‘rounding out’ of the part of mathematics needed for science. Nevertheless, even the later Quine’s starting point is science, not mathematics itself.
- 6.
The scope of ‘recreational mathematics’ is a matter of debate. Quine was in favour of first-order Zermelo-Fraenkel set theory. Colyvan argues that if we take a ‘holistic approach’ then pretty much all of present day practiced mathematics is in the ‘real’ part, since there is some link between the immediately applied (to science) parts of mathematics and the ‘nether reaches’ (Colyvan 2001, 107 footnote 23). Insofar as Colyvan’s holistic attitude is convincing the recovery of most of mathematics is an artefact of the vast development of crosschecking in mathematics, the application of one mathematical theory, to check another. I discuss this especially in Chaps. 7, 8, 9 and 14. The difference between the pluralist and Maddy, on the one hand, and Colyvan and Quine on the other, lies in the presumed reason why mathematics is good science. For the pluralist and Maddy, the reason is that mathematics is rigorous, has a perfectly good methodology. For Quine and Colyvan, the physical world, prediction and causation are the ultimate reasons we can seriously engage in mathematics.
- 7.
We also have to be careful about the relationship between the rate of increase of the scope of real mathematics and the rate of increase in practiced mathematics. If the rate of increase of the first is far inferior to the second, then more and more of mathematics will be considered to be recreational by the Quinean naturalist. There is no guarantee about the relative rates of increase of ‘real’ and ‘recreational’ mathematics.
- 8.
Arguably, we do not need the full set of reals for science. We actually only ever (will) use a finite (and very small) number of reals. In fact, we do not need the real numbers at all. All we need is an approximation of some of them. Thus, strictly speaking, we only need the rational numbers for science, and not even all of those. Even if the mathematicians have an algorithm for producing the expansion of Π, for example, the scientists only need it to be expanded finitely, not infinitely. So they do not need the full mathematical theory of the reals. Regardless of whether or not the reader agrees, the cut off point between real and recreational mathematics is not important for the argument here. The cut off could be higher. Even if we consider a very high cardinal axiom as important for science, we would still be missing quite a lot of other high cardinal numbers, an infinite number at least!
- 9.
As noted, Colyvan makes a case for most, if not all, of mathematics being necessary for science. We shall look at objections to Colyvan’s naturalism in Sect. 3.4 of this chapter.
- 10.
If we were very keen on positing as few mathematical objects as ‘possible’, and if we are willing to entertain the idea that inconsistent objects are possible, then if we first note that we only ever use a finite number of predicates in science, then “Any mathematical theory presented in first-order logic has a finite paraconsistent model.” (Bremer 2010, 35). So, provided we agreed that a first-order language was enough for science (which Quine would agree to), and we allow the existence of inconsistent mathematical objects, then we only need a finite theory: one with a largest finite number. Since allowing inconsistent mathematical objects is not to everyone’s taste, we then engage in a negotiation between our metaphysical taste and our keenness to reduce our mathematical ontology.
- 11.
The notion of ‘increasing’ mathematics is at least ambiguous. It is not clear if we are counting theories, theorems, objects or what. Nevertheless, at an intuitive level we can see the point. No doubt, there will be some measures of ‘quantity of mathematics’ where the ‘rate of growth’ of mathematics outstrips scientific applications significantly – say by adding sets of large cardinal numbers; and yet other measures will show a slower ‘rate of growth’.
- 12.
The mismatch can also be identified with what Buldt and Schlimm call an Aristotelian conception of mathematics and a non-Aristotelian conception (Buldt and Schlimm 2010, 40). Roughly, twentieth century mathematics takes a more top-down, structuralist approach to mathematics, so applications are almost accidents; whereas an Aristotelian approach is one of abstraction from the observable world. If Buldt and Schlimm are correct in their diagnosis of the change in mathematical conception, then, in this respect, we might see Quine as still being entrenched in an Aristotelian conception of mathematics, (at least in this respect) whereas the practice of mathematics today is non-Aristotelian.
- 13.
Arguably, this criticism of Quine relies on taking too seriously Quine’s taste for science, and attributing this to the naturalist position. Arguably, this is a distortion of Quine, and if we look at his meta-philosophical pragmatism, then we see that he would, if pressed, endorse taking mathematics seriously in just the way Maddy does, and he would simply acknowledge that she has different concerns from his. I do not mind if one takes this reading of Quine. If one does, then one says that Maddy just extends Quine’s naturalist programme to include mathematics. For this reason I sometimes call her position ‘mathematical naturalism’. Under this reading of Quine, Maddy is Quinean, and both resonate with the pluralist. The reason for not treating this interpretation of Quine in the main text is that Maddy sees herself as departing from Quine on this issue, and I am really concerned with introducing the reader to pluralism, not in making an accurate critique of Quine. I apologise for any clumsiness in representing Quine’s position. More sophisticated interpretations of Quine approach pluralism.
- 14.
Her more recent book Second Philosophy, Maddy (2007) gives motivation for and develops the work done in Naturalism (Maddy 1997), but in it, too, Maddy does not take her own philosophical directives far enough, according to the pluralist. In some ways, for the pluralist, her more interesting work is done in her (1997). We shall see this in Chap. 14.
- 15.
This is not central, in the sense of what it is that most mathematicians do most of the time, but in the sense of is a recognised foundation of mathematics. Of course, one wants to specify in what respect a theory is a foundation, but that does not have to be done for these purposes.
- 16.
A short anecdote: at a conference on Brouwer and Intuitionism, Martin-Löf was asked by John Thomas “how much of mathematics is really needed for science?” Martin-Löf’s reply was: “a very tiny amount”. As has already been remarked, how one measures ‘a lot of mathematics’ and ‘a little mathematics’ is simply not clear. Nevertheless, we can say that if we imagined physicists, chemists and biologists being asked to decide on the basis of ‘usefulness for their science’ which mathematicians in a mathematics department to keep in employment, the great majority of mathematicians would be fired. In fact, they might all be fired, since we already know quite a lot about the mathematics that is already used, and how to use it, and so further investigation of the mathematical theory might be thought to be quite useless. For other, philosophically quite different sources which explore what is the minimum mathematics needed for science see Field (1980) or Bendaniel (2012).
- 17.
A lot of philosophers would think that it is to get the order wrong to think that physical ‘reality’ can act as a test for mathematics. At best it can only be used to test a particular application of mathematics. Such philosophers are the ones who think that there is a hierarchy of knowledge with logic at the top, then mathematics, then physics, then chemistry then biology, then the ‘softer’ human sciences. The relationship between the levels of the hierarchy concern necessity or laws (of a discipline) so, for example, biology is responsible to chemistry, i.e., biology has all of the laws of logic, mathematics, physics and chemistry. Biology cannot violate those, and has a few extra laws, which do not pertain to chemistry, or any discipline above it in the hierarchy. Thus, moving up the hierarchy, chemistry contains laws. Those are all of the laws of logic, mathematics and physics. None of these can be violated by chemical reactions. Also chemistry has its own laws not found in physics or any area higher in the hierarchy (Fig. 3.2).
We are not concerned with philosophers who have this view of natural or scientific laws here, and as a reader might well suspect, the pluralist has a rather more complicated picture of laws. Nevertheless, the pluralist will maintain that holding mathematics hostage to the physical world makes no sense. What does make sense for the pluralist is to use ‘physical reality’ to judge the fit of a mathematical theory as a model for some part of physical reality, and nothing else.
- 18.
This is meant in the sense of epistemic likelihood, that is ‘given what we know’ or what we ‘are certain of’. In these senses, ZFC+ is less likely true than ZFC.
- 19.
‘Topological’ is chosen here just to refer to the notion of centrality and of fruitfulness together.
- 20.
There is a lot to say about this argument, but it would be distracting here. One function of a footnote is to discuss issues, which would interrupt the flow of the main text. Using the footnote to this end: briefly, the first premise is only a description of a recent fact in the history of mathematics. It could be dismissed as an historical accident, having to do with the greater communication between mathematicians in the twentieth century etc. The second premise makes a lot of assumptions, and these can be questioned. For example, it holds sway if we assume that triviality is the only alternative to consistency. The third premise I like to call ‘the argument from fruit’, i.e., etcetera that if practice or assumption or theory X bears fruit, it follows that it must be true, correct etc. The argument from fruit is undermined by under-determination of truth by practice, assumption or theory. There are a number of similar arguments from other virtues: simplicity, beauty, symmetry, parsimony etc. The topological argument is not strong, when properly analysed. For some reason, it still persuades. We might learn more, if we give it a more sensitive treatment, as we do in the main text.
- 21.
It is quite interesting that in her more recent book: Second Philosophy, she practices first philosophy to argue for second philosophy, and only practices second philosophy starting on page 246, in a 411 page book, excluding index, bibliography etc. And even then, she does not spend the rest of the book practicing second philosophy. Rather, she deftly moves from first to second and back. So, looking at her own practice, she cannot really object that much to first philosophy.
- 22.
In her book Second Philosophy, Maddy devotes a whole section to the question of the philosophical role left for the second philosopher. She is both well aware of the problem, and does answer it in a way that is similar to what I say here.
- 23.
We shall encounter these again soon, so do not forget them.
- 24.
In this case, appearances are deceptive, as we shall see shortly.
- 25.
As has been mentioned, this is ‘back to front’ for some mathematicians and even some computer scientists. The borderline between science and mathematics, and the relationship between them is quite intricate. For example, if we consider simple scientific experiments to be algorithms (they are just finite procedures), See (Beggs et al. 2010). What we discover is that the notion of physical experiment carries not only imprecision in measurement, but also a type of uncertainty. We can use mathematical techniques to detect and measure the uncertainty of the data obtained through such simple experiments! This points to an inadequacy in physical science, vis-à-vis computational science, and this inadequacy cannot be recognised by a Quinean naturalist, since science sets the highest standard (for truth, measurement etc.) not mathematics.
- 26.
Colyvan argues that the ‘rounding out’ process re-captures a lot of mathematics. This is a move made to please the mathematicians who, prima facie, cannot recognise their mathematics when seen through the naturalist lens. But his project can be reversed. Rather than show that it is ‘reasonable’ for the naturalist to recapture a lot of mathematics, we could just as well stick to our suspicion of mathematical ontology. If we do this, then the ‘rounding out’ need capture very little, only a finite number of numbers.
- 27.
Careful. The finite mathematics is an inconsistent one. Discussion of paraconsistent mathematics will be resumed in Chap. 6.
- 28.
We could test rigour according to some fairly rigid, nay formal rules, but these, in turn would have to resonate with a pre-formal sense of rigour. Thus, at best we could engage in a dialectic between attempts to give a formal or very precise definition of rigour, and the intuitive idea.
- 29.
An anonymous reviewer to a paper where I develop these ideas commented that “Maddy accepts the reduced role of the philosopher.” The pluralist follows her naturalist arguments, but assumes a very important philosophical role. The full pluralist does not assume a ‘reduced’ role at all.
- 30.
Since she is careful to acknowledge that these are maxims, and not guarantors of truth or correct real ontology, we can take the liberty to call these ‘principles’.
- 31.
We assume here the following default relationship between ontology and truth. The ontology of a theory is the ‘truth-makers’ of the theory. For example, what makes it true that 2 + 2 = 4 is that in Dedekind-Peano arithmetic, the entities 2 and 4 exist, and the function + and the relation = all conspire to make the formula: ‘2 + 2 = 4’ true. There are ways of separating truth values from ontology, but they will be ignored here. If we do separate them, then the above argument will work for at least one of truth or ontology, but not necessarily both, in which case a second argument would have to be given.
- 32.
The relationship between ontological pluralism and fictionalism is interesting, but will not be developed in this book. Some things, even some obvious things, have to be omitted.
- 33.
The details about how this is done will be given in Part II of the book.
- 34.
I have not supplied evidence for this here, but there will be a lot surfacing in different parts of the book.
- 35.
Examples include embeddings, reductions, modeling or equi-consistency proofs.
- 36.
For a more extended discussion see: Goethe and Friend (2010).
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Friend, M. (2014). Motivating Maddy’s Naturalist to Adopt Pluralism. In: Pluralism in Mathematics: A New Position in Philosophy of Mathematics. Logic, Epistemology, and the Unity of Science, vol 32. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7058-4_3
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