Philosophy of Mathematics
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Philosophy of Mathematics

Øystein Linnebo

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eBook - ePub

Philosophy of Mathematics

Øystein Linnebo

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A sophisticated, original introduction to the philosophy of mathematics from one of its leading contemporary scholars Mathematics is one of humanity's most successful yet puzzling endeavors. It is a model of precision and objectivity, but appears distinct from the empirical sciences because it seems to deliver nonexperiential knowledge of a nonphysical reality of numbers, sets, and functions. How can these two aspects of mathematics be reconciled? This concise book provides a systematic yet accessible introduction to the field that is trying to answer that question: the philosophy of mathematics.Written by Øystein Linnebo, one of the world's leading scholars on the subject, the book introduces all of the classical approaches to the field, including logicism, formalism, intuitionism, empiricism, and structuralism. It also contains accessible introductions to some more specialized issues, such as mathematical intuition, potential infinity, the iterative conception of sets, and the search for new mathematical axioms. The groundbreaking work of German mathematician and philosopher Gottlob Frege, one of the founders of analytic philosophy, figures prominently throughout the book. Other important thinkers whose work is introduced and discussed include Immanuel Kant, John Stuart Mill, David Hilbert, Kurt Gödel, W. V. Quine, Paul Benacerraf, and Hartry H. Field.Sophisticated but clear and approachable, this is an essential introduction for all students and teachers of philosophy, as well as mathematicians and others who want to understand the foundations of mathematics.

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Mathematics as a Philosophical Challenge
Mathematics poses a daunting philosophical challenge, which has been with us ever since the beginning of Western philosophy.
To see why, imagine a community that claims to possess a wonderful kind of knowledge resulting from some discipline practiced there. Community members claim that this knowledge has three distinctive characteristics. First, it is a priori, in the sense that it doesn’t rely on sense experience or on experimentation. Truths are arrived at by reflection alone, without any sensory observation. Second, the knowledge is concerned with truths that are necessary, in the sense that things could not have been otherwise. It is therefore safe to appeal to these truths when reasoning not only about how the world actually is but also when reasoning about how it would have been had things been otherwise. Third, the knowledge is concerned with objects that are not located in space or time, and that don’t participate in causal relationships. Such objects are said to be abstract.
In fact, the knowledge that our imagined community claims to posses is rather like the knowledge promised by rational metaphysics, which for centuries professed to deliver insights into the ultimate nature of reality and ourselves, based solely on reason and without any reliance on sense experience. Many people today would dismiss such knowledge claims as incredible. And in fact, science and philosophy have developed in ways that now allow this dismissal to proceed fairly smoothly.
The philosophical challenge posed by mathematics is this. Mathematics seems to deliver knowledge with the three distinctive characteristics that are claimed by our imagined community. “The queen of the sciences”—as Gauss famously called mathematics, usurping a title previously reserved for rational metaphysics—seems to be practiced by means of reflection and proof alone, without any reliance on sense experience or experimentation; and it seems to deliver knowledge of necessary truths that are concerned with abstract things such as numbers, sets, and functions. But in stark contrast to rational metaphysics, mathematics is a paradigm of a solid and successful science. In short, by being so different from the ordinary empirical sciences, mathematics is philosophically puzzling; but simultaneously, it is rock solid.
This challenge obviously requires closer examination. Let us begin with mathematics’ strong credentials, before we return, in the sections that follow, to its three apparent characteristics. Mathematics is an extremely successful science, both in its own right and as a tool for the empirical sciences. There is (at least today) widespread agreement among mathematicians about the guiding problems of their field and about the kinds of methods that are permissible when attempting to solve these problems. By using these methods, mathematicians have made, and continue to make, great progress toward solving these guiding problems. Moreover, mathematics plays a pivotal role in many of the empirical sciences. The clearest example is physics, which would be unimaginable without the conceptual resources offered by modern mathematics; but other sciences too, such as biology and economics, are becoming increasingly dependent on mathematics. So a wholesale dismissal of mathematics on the grounds that it is philosophically puzzling would be sheer madness. Such a successful discipline cannot be rejected out of hand but needs to be accommodated within our philosophy in some way or other, albeit perhaps with changes to our pretheoretic conception of how the discipline works. Moreover, unlike rational metaphysics, mathematics permeates our current scientific world view and hence cannot be excised from it.
In sum, our challenge is to explain how we can make room within a broadly scientific world view for a science with features as puzzling as those of mathematics. We shall encounter two lines of response. One is to deny some or all of the distinctive features that appear to set mathematics apart from the ordinary empirical sciences and thus cause philosophical puzzlement. Another line of response is to accept that mathematics is more or less as it appears to be and to explain how this is possible. We shall see that the need for such explanations has profoundly shaped the philosophical outlooks of a number of great thinkers.
A good way to approach the seeming apriority of mathematics is to read Plato. And a good place to start is the dialogue Meno, where Plato describes a slave boy who has been taught no mathematics but is nevertheless able to discover “out of his own head” an interesting geometrical truth about squares, namely that the square of the diagonal is two times the square of each side. In the dialogue, Socrates asks the slave boy some carefully chosen questions, which prompt the boy to reflect on geometry and discover some simple geometrical truths and eventually reason his way to the mentioned fact about squares.
The story of the slave boy is meant to establish two things. First, that mathematical concepts are innate; that is, they are not acquired but form part of the mind’s inborn endowment. And second, that mathematical truths are a priori and can be known without relying on experience for one’s justification. It may be objected that the slave boy relies on experience in order to understand Socrates’ questions. Of course he does! But this experience serves only to trigger the process that results in geometrical knowledge and doesn’t itself constitute evidence for this knowledge.1
Suppose Plato is right that we possess innate mathematical concepts and a priori mathematical knowledge. How can this be? The usual answer from rationalistically inclined philosophers has been that our “faculty of reason” is the source of such concepts and knowledge. Until more has been said about this faculty and its workings, however, this answer is little more than a pompous redescription of what we set out to explain. Plato, to his credit, recognizes the need to say more. In the Meno, he therefore proposes—or at least entertains—an explanation.
The soul, then, as being immortal, and having been born again many times, and having seen all things that exist, whether in this world or in the world below, has knowledge of them all. (Meno, 81cd)
The envisaged explanation is as follows. The soul must have preexisted the body. In this disembodied existence, the soul has “seen all things”—including, crucially, the objects with which geometry is concerned—and acquired “knowledge of them all.” So when the slave boy—and the rest of us, for that matter—seem to acquire mathematical concepts and knowledge, this is in fact nothing but recollection of concepts and truths that our souls encountered when they existed in a purer, disembodied state and had direct access to the objects of mathematics (as well as to a range of abstract, but perfectly real, “forms” or “ideas” that Plato also postulated).
Of course, this explanation has little appeal today. Plato nevertheless deserves our highest admiration for identifying a deep philosophical problem, namely the seeming apriority of mathematics. The mark of philosophical greatness is as much to identify good questions as it is to answer them. And as we shall see throughout the book, Plato’s question has shaped the philosophical debate about mathematics right up until this day.
Consider any truth of pure mathematics, say 2 + 2 = 4. It is part of the traditional Platonistic conception of mathematics that this truth is not accidental—as it is accidental that you are currently reading this book—but that 2 + 2 = 4 is necessarily true, that is, true not only as things actually are, but true no matter how things might have been.
One might worry that this necessity claim is idle philosophical speculation and therefore dispensable. This worry is fueled in part by the philosophical controversy that the notion of necessity has generated and the skepticism it has encountered. The necessity claim has real significance, however, despite these genuine difficulties. Consider the role that mathematics plays in our reasoning. We often reason about scenarios that aren’t actual. Were we to build a bridge across this canyon, say, how strong would it have to be to withstand the powerful gusts of wind? Sadly, the previous bridge fell down. Would it have collapsed had its steel girders been twice as thick? This style of reasoning about counterfactual scenarios—or alternative “possible worlds,” as philosophers like to call them—is indispensable to our everyday and theoretical deliberations alike. Now, part of the cash value of the claim that the truths of pure mathematics are necessary is that such truths can freely be appealed throughout our reasoning about counterfactual scenarios. Had you not been reading this book, or had some girders been twice as thick, 2 + 2 would still have been 4. Indeed, the truths of pure mathematics can be trusted even in an investigation of how things would have been in scenarios where the laws of nature are different.
The great German mathematician and philosopher Gottlob Frege (1848–1925), who figures prominently in this book, liked to make a similar point in terms of the “domains” that various kinds of truth “govern.” The logical and arithmetical truths are said to govern “the widest domain of all; for to it belongs not only the actual, not only the intuitable, but everything thinkable” (Frege, 1953, §14).2 Presumably, the domain of “everything thinkable” includes everything that is possible in the sense explained above.
Let us pause to note an immediate but important consequence of the necessity of the truths of pure mathematics. Since such truths can freely be appealed to throughout our counterfactual reasoning, it follows that these truths are counterfactually independent of us humans, and all other intelligent life for that matter. That is, had there been no intelligent life, these truths would still have remained the same. Pure mathematics is in this respect very different from humdrum contingent truths. Had intelligent life never existed, you would obviously not have been reading this book. More interestingly, pure mathematics also contrasts with various social conventions and constructions, with which it is sometimes compared.3 Had intelligent life never existed, there would have been no laws, contracts, or marriages—yet the mathematical truths would have remained the same. These truths can thus be assumed by us actually existing intelligent agents when we reason about this sad intelligence-free scenario.
The final distinctive feature traditionally attributed to mathematics is a concern with abstract objects. An object is said to be abstract, we recall, if it lacks spatiotemporal location and is causally inefficacious; otherwise it is said to be concrete. While this distinction may not be entirely sharp, it suffices for our present purposes.4
Now, it certainly seems that mathematics is concerned with abstract objects. Mathematical texts brim with talk about numbers, sets, functions, and more exotic objects yet, and these objects seem nowhere to be found in space and time.5 It is useful to “factor” the third feature of mathematics into two distinct claims.
Object realism. There are mathematical objects.
Abstractness. Mathematical objects are abstract.
While object realism was endorsed already by Plato, the first clear defense of it was due to Frege. Consider the following sentences:6
(1) Evelyn is prim.
(2) Eleven is prime.
The two sentences seem to have the same logical structure, namely a simple predication based on a proper name, which refers to an object, and a predicate, which ascribes some property to this object. As Frege argued, for a sentence of this simple subject-predicate form to be true, the proper name must succeed in referring to an object, and this object must have the property ascribed by the predicate (cf. §2.3). Moreover, (2) is true, as anyone who possesses even basic arithmetical competence will confirm. It follows that ‘Eleven’ must succeed in referring to an object, and hence there are mathematical objects.
Of course, the argument is not beyond reproach. We shall encounter various challenges to it throughout the book. Perhaps the claims of mathematics cannot be taken at face value. Or perhaps they aren’t true after all. For now, however, it suffices to observe that the argument has sufficient force to shift the burden of proof onto opponents, who need to explain where they think the argument goes wrong.
The claim that mathematical objects are abstract has been less controversial. It is not hard to see why. If possible, our philosophical account of mathematics should avoid claims that would render our ordinary mathematical practice misguided or inadequate. But if mathematical objects had spatiotemporal location, then our ordinary mathematical practice would be misguided and inadequate. We would then expect mathematicians to take a professional interest in the location of their objects, just as zoologists are interested in the location of animals. By taking mathematical objects to be abstract, our actual practice becomes far more appropriate.
In contemporary philosophy, the word “platonism” (typically with a lowercase ‘p’) is often used in a more general sense than anything we can ascribe to Plato (and which thus counts as “Platonistic” with an uppercase ‘P’). The platonist conception of mathematics does not ...

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