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Making and Breaking Mathematical Sense
Histories and Philosophies of Mathematical Practice
By Roi Wagner PRINCETON UNIVERSITY PRESS
Copyright © 2017 Princeton University Press
All rights reserved.
ISBN: 978-1-4008-8378-3
CHAPTER 1
Histories of Philosophies of Mathematics
This chapter sets this book against the background of canonical philosophies of mathematics. It will present several historical narratives that follow some major trends in the philosophy of mathematics. All these narratives are going to be highly selective and superficial. I allow myself this bird's-eye view precisely because these histories are presented in parallel, emphasizing the partiality of each particular narrative. But this does not mean that together they presume to exhaust the historico-philosophical landscape. There's much more that can be added to these stories, and many ways in which they can be retold.
I should emphasize that these histories do not presume to explain or analyze the philosophical positions that they bring up. The point is rather to highlight some issues at stake in the debate among philosophers — issues that were not always explicitly highlighted by those philosophers. I will therefore not go into the details of each philosophical position or even provide a decent survey of their arguments. Doing that would force me to deviate too far from my main argument and reproduce some well-known discussions that are covered in many other introductory works (among the most recent are Bostock 2009; Celluci 2007; Friend 2007; George and Velleman 2001; Hacking 2014; Murawski 2010; Shapiro 2005; and the Stanford and Routledge online encyclopedias of philosophy). Since I only want to thread together the various debates to make salient some questions that underlie philosophical debates, many deep philosophical points that were very important to the various quoted philosophers will be glossed over very quickly and superficially, so as to pick up the problems that are important for the argument of this book.
More specifically, instead of philosophical questions such as "Are mathematical objects real?," "Is mathematics reducible to logic, or intuitions, or axioms?," or "Is mathematics a priori?," I will narrate the history of some philosophical discussions as revolving around meta-philosophical tensions: tensions between natural order and conceptual freedom; between mathematics as originally constitutive versus reducible to reason and/or nature; between reining in conceptual monsters and attempting to let them roam free; and between different ways to distribute authority over mathematical norms and standards.
History 1: On What There Is, Which Is a Tension between Natural Order and Conceptual Freedom
Our first narrative of philosophies of mathematics will start from W. V. Quine's (1948) "On What There Is." This paper raises the question of whether mathematical entities really exist or are figments of the imagination. This question relates to a common sentiment that says that mathematics is good because it provides good descriptions or predictions of empirical or idealized realities.
Quine's paper draws an analogy between, on the one hand, scholastic realism, conceptualism, and nominalism and, on the other hand, modern logicism, intuitionism, and formalism respectively (we're going to introduce the former and latter in a few paragraphs). In order to tease out the tension between natural order and conceptual freedom in the context of mathematics, we'll need to take a quick (and superficial) look at these schools of thought.
According to Quine's analogy, medieval realists and twentieth-century logicists were committed to the existence of all kinds of abstract objects; medieval conceptualists and modern intuitionists were committed only to those that could be admitted via a restricted mental construction; and nominalists and formalists were committed only to names and inscribed marks, without requiring objects to which these names and marks would refer beyond specific examples.
Quine's essay marks a pivotal moment in the history of twentieth-century philosophy of mathematics — so much so that Hilary Putnam stated that in the analytic tradition "[i]t was Quine who single-handedly made Ontology a respectable subject" (2004, 78–79). Quine's historical narrative came up at a time when logicism, intuitionism, and formalism have exhausted much of their drive as formative philosophical projects, and became absorbed into more technical work in logic. His pivotal statement set the scene for the contemporary realist-nominalist discussion that would become a leading concern in contemporary philosophy of mathematics.
To see what Quine's historiography entails, let's pause briefly to consider one dimension of the scholastic debate. But note that reducing a debate that spanned hundreds of years necessarily verges on a caricature and involves gross overgeneralizations. A properly detailed historical account of the fluctuating debate concerning universals in scholastic thought is provided by de Libera (1996).
Scholastic realism involved an elaborate scheme of entities that mediated human perception and divine knowledge, exceeding the bounds of time and space. It depended on a complex division of labor between the objects, the senses, the passive and active human intellect (which mediate between the senses and mental concepts), and divine illumination (that provides nonsensory input and direction). These elements combined to give rise to subjective and objective concepts (those constructed by a single individual and those necessarily shared by all) and to individualized and absolute essences (the essence of an object as understood by an individual and as it really is from a divine point of view). Universal concepts in general (for example, good, blue), and mathematical concepts in particular (numbers, geometrical forms), were to be abstracted from the sensed world by human intellects and divine illumination, and held together objectively across different minds and different individual instances by their common essences. This architecture forced the work of God and man into a rational Aristotelian hierarchical and purposeful frame integrated with a Platonic reification of abstract concepts as independent and eternal.
Nominalists reacted against this elaborate divinely rational world by doing away with many of the pillars that held the realist conceptual architecture together. Instead of abstractions of the essences from similar individuals, concepts or common names became just means of grouping individuals together. No universal concepts or essences held these different names or groups together — they were individual or collective engagements with the world. Divine will was no longer tied down to the heavy ontological burden of Aristotelian hierarchical and purposeful classifications and to Platonic eternal forms. The link between man and God depended less on reason and more on faith. Focusing on the cultural impact of this position (specifically, Ockham's fourteenth-century position), Sheila Delany suggests that
Since God is bound by neither natural law nor his own promises, the universe becomes profoundly contingent and, in an absolute sense, unpredictable. Neither nature, society, nor the human mind are necessarily permanent or static in their structure; all are open to change and plurality, none can be fully understood by reference to an abstract a priori scheme. (Delany 1990, 48)
Divine and human wills became much more autonomous with respect to the nature of the created world. The cost, however, was that philosophical guarantees of communication, correct representation of the world, and a general sense of purpose could be lost.
Fitting this division with the early twentieth-century foundational trio of logicism, intuitionism, and formalism is tricky. First one has to bring in conceptualism to complete the scholastic duo into a triad (since the scholastic division was never as clearly defined as contemporary presentations pretend it to be, some of the more moderate forms of nominalism, which allow the mind to abstract some essence-ersatz from individuals, are termed "conceptualist"). Even then, if we consider the big picture, the early twentieth-century foundational positions don't have too much in common with the scholastic debate.
But Quine focused on quantification as a form of ontological commitment, not on the entire scholastic construction. Quantification is our use of the term "all ..." or "there are ..." in our scientific language. For Quine, using these terms meant that we acknowledge the existence of the entities that they refer to. So if we say "all numbers are ...," we commit to the existence of numbers. Now, if we restrict our attention to quantification alone, Quine's analogy works to an extent. Let's review this analogy.
Frege's and Russell's "logicism" advocated the reduction of mathematics to logic and pure reason. This seemed to be a tenable project, because by the end of the nineteenth century logic had become much more rich and expressive than it had ever been before. Their attempt was to present numbers and other mathematical entities as sets subject to the logical laws of some basic set theory. Logicists, as commonly interpreted, do quantify over abstract terms such as "sets," "numbers," and "geometrical shapes," as did scholastic realists (assuming we can retroject the term "quantify" a few hundred years into the past), and considered them to be real existing entities. As the early Bertrand Russell put it (and later retracted):
All knowledge must be recognition, on pain of being mere delusion; Arithmetic must be discovered in just the same way in which Columbus discovered the West Indies, and we no more created numbers than he created the Indians. The number 2 is not purely mental, but is an entity which may be thought of. Whatever can be thought of has being, and its being is a precondition, not a result, of its being thought of, since it certainly does not exist in the thought which thinks of it. (Russell 1938, ch. 51, §427)
Brouwer's "intuitionism" (preceded in some ways by such figures as Poincaré and Kronecker) questioned any mathematics that could not be finitely constructed starting with counting a sequence of moments (in a Kant-like framework of temporality, to be discussed in the next section). In Brouwer's view, actual infinities could not be contained by such constructions and were therefore rejected.
Nonconstructive proofs were suspect as well. For instance, it's clear that either [MATHEMATICAL EXPRESSION OMITTED] is an example of two irrational numbers yielding a rational power, or, if their power turns out to be irrational, then [MATHEMATICAL EXPRESSION OMITTED] is an example of two irrational numbers yielding a rational power. But as long as we cannot decide which alternative holds, we have failed to construct a rational power of two irrational numbers, and so radical intuitionists would say that we have not proved the existence of two irrationals yielding a rational power. This rejection of classical concepts and arguments required a thorough review of a lot of classical mathematics.
Intuitionists are harder to fit into Quine's analogy. They quantify over mental constructions based on the elementary intuition of temporal succession (counting the moments 1, 2, 3, ...), not over mere common (nominalist) names or over abstractions of fully fledged (realist) essences. One could therefore say that intuitionist and conceptualist objects exist in the mind. But what "mind" means, and how things come to exist in the mind works rather differently in the scholastic and intuitionist systems. The conceptualist scholastic mind collects individuals under concepts by analogies. The intuitionist mind constructs on its own by following the basic internal experience of the advance of time.
Hilbert's "formalism" tried to see mathematics as analyzing which combinations of signs can be obtained when following strict syntactic rules, without pretending to anchor these signs to any reference or meaning (think for example about algebraic equations with rules for simplifying them, but without deciding what kind of numbers the unknowns may stand for, or if they stand for numbers at all). But in Hilbert's system, such formal languages of signs and rules were to be analyzed and compared inside some system (a meta-language), and Hilbert determined this system to be the elementary core of finite arithmetic subject to constructive and logical restrictions that would be legitimate in the eyes of all philosophical parties involved.
Formalists make things more complicated for Quine's analogy, because of their double articulation of mathematical discourse into the language of formal proofs (meaningless signs following syntactic rules) and the meta-language that applies finite, constructive reasoning to the first-level languages.
Indeed, the formalist meta-language quantifies over numbers (it counts signs and makes claims that depend on the length of bits of text), but Poincaré and Hilbert were at odds as to whether the numbers of the meta-language were mere collective names of empirical marks on paper or full-blown mathematical mental constructions, especially where induction was used in the meta-language to make general statements about proofs (Ewald 1996, 1021–51).
As for the level of formal languages, in which mathematical proofs were written, according to the Hilbertian view these languages were collections of marks and rules that may discard any denotation. This is his famous statement that "point," "line," and "plane" in Euclidean geometry may as well be replaced by "tables," "chairs," and "beer mugs"; what matters is only the rules of the language (Euclidean axioms and postulates, as well as those that Euclid used implicitly), and we do not decide in advance what the Euclidean terms designate. (Hilbert actually constructed different kinds of geometry by offering different designations to the same Euclidean terms, so as to make them obey different sets of axioms.) The marks of formal language are therefore not names that group individuals together, as we would expect following Quine's analogy between formalists and nominalists.
So the analogy between the scholastic and early twentieth-century trichotomies is shaky at best. To find a more adequate modern version of scholastic nominalism, we should probably try John Stuart Mill, who stated that
All numbers must be numbers of something: there are no such things as numbers in the abstract. Ten must mean ten bodies, or ten sounds, or ten beatings of the pulse. But though numbers must be numbers of something, they may be numbers of anything. (Mill 1843, book II, ch. 6, §2)
This means that when we say that "2 + 3 = 5," we make an empirical statement, but one that is collectively true for apples, chairs, and so on. It seems therefore that when Mill quantifies over numbers, he quantifies, like scholastic nominalists and like Hilbert in his meta-language treatment of numbers, over collections of individual instances.
If we proceed in time, and try to carry Quine's analogy over to contemporary realists and nominalists, we might find more substantial analogies. These analogies are not restricted to a concern with quantification. They even go beyond the borrowing of scholastic terms for realism and nominalism such as in rebus (existing in empirical things) and ante rem (existing a priori, independently of empirical things; see Reck and Price 2000 for details). I believe that the most substantial analogy between scholastics and modern philosophers of mathematics concerns the commitment to natural order versus conceptual freedom.
The following paragraph from Burgess and Rosen (1997, 241) — a critical study of contemporary nominalism in mathematics — is telling:
the most fashionable figures in the history and sociology and anthropology of science deny not only that there is a ready-made theory of the world, but even there is any ready-made world. They maintain not just that theories about life and matter and number are constructs of human history and society and culture, but that number and matter and life themselves are such constructs [ ... and] that mathematical and physical and biological facts, being created by us when we create mathematical and physical and biological theories, cannot impose any prior constraint on how we go about shaping those theories, leaving only constraints from our side — assumed to be social and political and economic — rather than the world's side.
The drums of the Science Wars can be clearly heard beating in the background, and contemporary forms of realism still seem concerned with the risk of relativism costing us our grip of reality. They sometimes even carry religious overtones. Mark Steiner's view (1998) is that some sort of anthropomorphic design is required to make the universe so highly accessible to human mathematics, and that if we disagree, then "[n]ominalism, like atheism, and for similar reasons, is a philosophical position that recommends itself to many modern philosophers" (2001, 73). Putnam's Ethics without Ontology (2004) is worried that losing grip of reality in the context of hard sciences is related to a dangerous descent into ethical relativity.
(Continues...)
Excerpted from Making and Breaking Mathematical Sense by Roi Wagner. Copyright © 2017 Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS.
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