Oxford Scholarship Online. Available in Oxford Scholarship Online - view abstracts and keywords at book and chapter level. Our much-valued mathematical knowledge rests on two supports: the logic of proof and the axioms from which those proofs begin. Naturalism in Mathematics investigates the status of the latter, the fundamental assumptions of mathematics. These were once held to be self-evident, but progress in work on the foundations of mathematics, especially in set theory, has rendered that comforting notion obsolete.
Given that candidates for axiomatic status cannot be proved, what sorts of considerations can be offered for or against them? That is the central question addressed in this book. One answer is that mathematics aims to describe an objective world of mathematical objects, and that axiom candidates should be judged by their truth or falsity in that world. This promising view—realism—is assessed and finally rejected in favour of another— naturalism—which attends less to metaphysical considerations of objective truth and falsity, and more to practical considerations drawn from within mathematics itself.
Penelope Maddy defines this naturalism, explains the motivation for it, and shows how it can be helpfully applied in the assessment of candidates for axiomatic status in set theory. Maddy's clear, original treatment of this fundamental issue is informed by current work in both philosophy and mathematics, and will be accessible and enlightening to readers from both disciplines. An excellent book The philosopher's task is not to provide external criticism, but rather to clarify debates in this so-called mathematical community. And Ms Maddy makes some fascinating and very technical steps in this direction.
The book is beautifully written, tightly argued and makes compelling reading. I believe the position Maddy introduces and defends - set theoretic naturalism - is a significant and original addition to the philosophy of mathematics landscape, and one that will certainly attract a great deal of attention. In sum, this is a very important book covering some fascinating terrain on the border between philosophy and mathematics.
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Academic Skip to main content. Of course it would be even more interesting to have a nominalistic reconstruction of a truly contemporary scientific theory such as Quantum Mechanics.
But given that the project can be carried out for Newtonian mechanics, some degree of initial optimism seems justified. This project clearly has its limitations.
It may be possible nominalistically to interpret theories of function spaces on the real numbers, say. But it seems far-fetched to think that along Fieldian lines a nominalistic interpretation of set theory can be found.
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For it would mean that, to some extent at least, mathematical entities appear to be dispensable after all. He would thereby have taken an important step towards undermining the indispensability argument for Quinean modest platonism in mathematics, for, to some extent, mathematical entities appear to be dispensable after all.
- Realism in Mathematics (Clarendon Paperbacks) by Penelope. Maddy
This leads to a position that has been called ultra-finitism Essenin-Volpin On most accounts, ultra-finitism leads, like intuitionism, to revisionism in mathematics. For it would seem that one would then have to say that there is a largest natural number, for instance. From the outside, a theory postulating only a finite mathematical universe appears proof-theoretically weak, and therefore very likely to be consistent. But Woodin has developed an argument that purports to show that from the ultra-finitist perspective, there are no grounds for asserting that the ultra-finitist theory is likely to be consistent Woodin Regardless of this argument the details of which are not discussed here , many already find the assertion that there is a largest number hard to swallow.
But Lavine has articulated a sophisticated form of set-theoretical ultra-finitism which is mathematically non-revisionist Lavine He has developed a detailed account of how the principles of ZFC can be taken to be principles that describe determinately finite sets, if these are taken to include indefinitely large ones. Admittedly it is not a simple task to give an account of how humans obtain knowledge of spacetime regions. But at least according to many but not all philosophers spacetime regions are physically real. So we are no longer required to explicate how flesh and blood mathematicians stand in contact with non-physical entities.
This leads to versions of nominalist structuralism , which can be outlined as follows. Let us focus on mathematical analysis. The nominalist structuralist denies that any concrete physical system is the unique intended interpretation of analysis. All concrete physical systems that satisfy the basic principles of Real Analysis RA would do equally well. This entails that, as with ante rem structuralism, only structural aspects are relevant to the truth or falsehood of mathematical statements. But unlike ante rem structuralism, no abstract structure is postulated above and beyond concrete systems.
According to in rebus structuralism, no abstract structures exist over and above the systems that instantiate them; structures exist only in the systems that instantiate them. Nominalist structuralism is a form of in rebus structuralism. But in rebus structuralism is not exhausted by nominalist structuralism. Even the version of platonism that takes mathematics to be about structures in the set-theoretic sense of the word can be viewed as a form of in rebus structuralism. So an existential assumption to the effect that there exist concrete physical systems that can serve as a model for RA is needed to back up the above analysis of the content of mathematical statements.
Putnam noticed early on that if the above explication of the content of mathematical sentences is modified somewhat, a substantially weaker background assumption is sufficient to obtain the correct truth conditions Putnam This is a stronger statement than the nonmodal rendering that was presented earlier. But it seems equally plausible. And an advantage of this rendering is that the following modal existential background assumption is sufficient to make the truth conditions of mathematical statements come out right:. It is possible that there exists a concrete physical system that can serve as a model for RA.
It is admittedly not easy to give a satisfying account of how we know that this modal existential assumption is fulfilled. But it may be hoped that the task is less daunting than the task of explaining how we succeed in knowing facts about abstract entities. Then a crude version of the relevant modal existential assumption becomes:. It is possible that there exist concrete physical systems that can serve as a model for ZFC.
Philosophy of Mathematics
Parsons has noted that when possible worlds are needed which contain collections of physical entities that have large transfinite cardinalities or perhaps are even too large to have a cardinal number, it becomes hard to see these as possible concrete or physical systems Parsons a. We seem to have no reason to believe that there could be physical worlds that contain highly transfinitely many entities. According to the previous proposals, the statements of ordinary mathematics are true when suitably, i.
The nominalistic account of mathematics that will now be discussed holds that all existential mathematical statements are false simply because there are no mathematical entities.
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For the same reason all universal mathematical statements will be trivially true. Fictionalism holds that mathematical theories are like fiction stories such as fairy tales and novels. Mathematical theories describe fictional entities, in the same way that literary fiction describes fictional characters.
This position was first articulated in the introductory chapter of Field , and has in recent years been gaining in popularity. This crude description of the fictionalist position immediately opens up the question what sort of entities fictional entities are. This appears to be a deep metaphysical ontological problem.
One way to avoid this question altogether is to deny that there exist fictional entities. Mathematical theories should be viewed as invitations to participate in games of pretence, in which we act as if certain mathematical entities exist. Pretence or make-believe operators shield their propositional objects from existential exportation Leng Nonetheless, mathematics is used to get truths across.
So we must subtract something from what is literally said when we assert a physical theory that involves mathematics, if we want to get at the truth. But this requires a theory of how this subtraction of content works. Such a theory has been developed in Yablo, If the fictionalist thesis is correct, then one demand that must be imposed on mathematical theories is surely consistency. Yet Field adds to this a second requirement: mathematics must be conservative over natural science. This means, roughly, that whenever a statement of an empirical theory can be derived using mathematics, it can in principle also be derived without using any mathematical theories.
If this were not the case, then an indispensability argument could be played out against fictionalism. Whether mathematics is in fact conservative over physics, for instance, is currently a matter of controversy.
Naturalism in the Philosophy of Mathematics
Fictionalism then shares this advantage over most forms of platonism with nominalistic reconstructions of mathematics. But the appeal to pretence operators entails that the logical form of mathematical sentences then differs somewhat from their surface form. If there are fictional objects, then the surface form of mathematical sentences can be taken to coincide with their logical form. In general, fictionalism is a non-reductionist account. Yet Burgess has rightly emphasized that mathematics differs from literary fiction in the fact that fictional characters are usually confined to one work of fiction, whereas the same mathematical entities turn up in diverse mathematical theories Burgess More determinate properties are ascribed to it than before, and this is all right as long as overall consistency is maintained.
The canonical objection to formalism seems also applicable to fictionalism. The fictionalists should find some explanation of the fact that extending a mathematical theory in one way, is often considered preferable over continuing it in a another way that is incompatible with the first. There is often at least an appearance that there is a right way to extend a mathematical theory. In recent years, subdisciplines of the philosophy of mathematics have started to arise. In this section, we look at a few of these disciplines. Many regard set theory as in some sense the foundation of mathematics.
It seems that just about any piece of mathematics can be carried out in set theory, even though it is sometimes an awkward setting for doing so.