Mathematics and the definitions of religion

Abstract

In 2014, I published a proposal for a definition of “religion”. My goal was to offer a definition of this contentious term that would include Buddhism, Daoism, and other non-theistic forms of life widely considered religions in the contemporary world. That proposal suggested necessary and sufficient conditions for treating a form of life as a religious one. It was critiqued as too broad, however, on the grounds that it would include the study of math as a religion. How can one include forms of life based on non-theistic realities without including math? In this paper, I show the flaw in the previous definition and the weaknesses of two attempts to evade that flaw, before recommending a shift, first, to a Wittgenstein-inspired polythetic definition of “religion” and, second, to a certain kind of polythetic definition that I call “anchored”.

Introduction

In 2014, I wrote a manifesto for the philosophy of religion in which I argued that this academic discipline ought to be reconceived to include not only the philosophical questions traditionally raised by the practice of religion—whether belief in God is justifiable, for example—but also those newly raised by the study of religion.¹ On the view I recommend, philosophers of religion should see one of the tasks of their discipline to be participating in those contentious, multi-disciplinary conversations about what it means to interpret and to explain human behavior, how social realities are like and unlike natural realities, the history of the emergence of their own academic discipline, and the norms that should hold in academic work. I judge that it is important they address these questions qua philosophers, because all of these reflexive questions about the tools with which scholars study religions have philosophical aspects that are often ignored or treated poorly by those without philosophical training or interests.

The reflexive question in the academic study of religions that is perhaps the most central concerns how the term “religion” is best defined. How scholars should draw the boundaries of “religion” is a topic that one found at the beginning of many philosophy of religion textbooks in the 1960s and 1970s, but the topic largely fell out of favor as philosophers of religion in the postmodern era—since, say, the emergence of Reformed epistemology—retreated from the study of the full variety of religious beliefs and practices around the world and retrenched themselves, more narrowly, as critics or defenders of classical theism. As a contribution to the reflexive turn in philosophy of religion that I recommend, however, I made a proposal for my own answer to the question of how best to define religion, namely: a mixed or “dithetic” definition that combines both substantive and functional approaches to the concept in order to take into account both ontological and practical criteria for what would count as a religious belief, practice, institution, or experience. Andrea Sauchelli (2016) offers a critique of precisely that part of my proposal.² He claims that the definition of religion that I propose leads to a strange conclusion, namely, that the study of mathematics would be considered a religion.

My proposal has attracted other responses, but none that led me to change the proposals I make in that manifesto.³ Sauchelli’s critique, however, raises an objection regarding my proposed definition of religion, an objection that will concern any scholars who seek a definition of “religion” broader than theism. My hope is that other philosophers of religion who engage the reflexive question about the central concept in the field will be interested to consider a conceptual challenge to a contemporary definition of “religion” and the different possible ways that one might respond. After explaining the problem raised by mathematical truths, therefore, this paper considers two possible answers to Sauchelli’s critique that I judge are poor solutions and the one answer—namely, adopting a certain new kind of family resemblance definition of “religion”—which, I will argue, is better.

The problem of mathematical truths

Sauchelli argues that if one accepts my definition of “religion,” it follows that the study of mathematics would be counted as a religion. I agree that this is a counter-intuitive, unwanted result. If my proposal does in fact lead to this conclusion, then I agree with him that there is a problem with my proposal. Is he right?

In my original proposal, I noted that many of the debates about how best to define religion—debates that have been described as interminable or hopeless—fall into stalemates between those who take substantive approaches and those who take functionalist approaches to the question. The substantive approaches contend that beliefs are religious only when they involve reference to a certain kind of “substance,” that is, only when the beliefs refer to a particular kind of alleged reality. Practices, experiences, or institutions would then be religious when they are based on religious beliefs: for example, the practice of growing a beard is not inherently religious, but it can acquire a religious character when the person seeks to emulate Jesus or Muhammad or someone else about whom the person holds religious beliefs.⁴ To make such a definition work, scholars have to specify the substance or content whose presence makes a belief, experience, practice, or institution religious, and among academics the most common way of specifying it is the “Tylorean” position that to be religious, phenomena have to include “belief in spiritual beings” or (to use terms more common today) "spiritual beings" might be replaced by "superhuman" or "nonobvious" or "counter-intuitive beings." Given such a definition, intercessory prayers to saints or bodhisattvas, visions of a God or an angel, and the social roles, material artifacts, and physical spaces dedicated to such practices would be treated as religious because they are based on beliefs with this content. By contrast, functionalist approaches to defining religion contend that a belief, practice, experience, or institution is religious only if functions in one’s life in a certain way. This religious function has to be specified, and two common ways of specifying the characteristic function of religious phenomena are to say that they “unite a community” or that they “express one’s ultimate concern.” Given a functional definition like these, the beliefs, experiences, practices, and institutions that serve that function would be treated as religious, even if they lack the substance preferred in the other kind of definition.

Each of these two ways of defining “religion” identifies a certain range of phenomena as religious. To certain extent, the two approaches overlap: if a Muslim, for example, takes submission to God—a spiritual, superhuman, and nonobvious being—as her highest concern, then the beliefs, practices, experiences, and institutions that constitute that form of life would be considered religious according to both definitions. A devout Muslim is a paradigmatic example of a religious person. But the two approaches can diverge. A purely substantive definition would categorize “belief in God” as religious, even if that belief plays no function in a person’s life, that is, even if the belief is of no special concern to the person. For instance, imagine a person who admits that she does think that a “some being greater than us” exists, but who is not a member of any religious community and who does not pray, celebrate religious holidays, or consciously put her life in accord with that “something.” We might call her religiosity “mere belief,” and mere belief counts as religious according to a pure substantive definition. Analogously, a purely functional definition would categorize a person’s ultimate concern or their social glue as religious, even if that concern or that glue was not connected to any spiritual, superhuman, and nonobvious being. Given this approach, some functionalists have argued that nationalism or the pursuit of money is religious. In contrast to both pure definitions, my proposal was that the academic study of religion should not take as its object of study mere belief in spiritual beings that play no role in a person’s life nor the practices that function in a religious way but do not include the substantive element. Or, to put this negative point positively, my proposal is that best way to define religion is to require both features, so that for some form of life to count as a religion, it has to both have the substantive element and to play a functional role in the lives of its participants. In this way, a mixed or “dithetic” definition like this one includes both ontological and pragmatic criteria.⁵

My proposed definition includes a second novel element. As noted above, scholars who defend a substantive approach (whether in a pure way or in my mixed way) must answer the question about that defining substance: what kind of ontological commitment makes a belief religious? I argued that one can see in western modernity a four-stage evolution operating in the answers to this substantive question and, in that evolution, the circle of the content that is said to identify what counts as religion has grown at each stage. At the earliest stage, something counted as religion only if it was based on belief in the Christian God. At this stage, only Christians were understood to have a religion; others had preparations for or departures from the Christian religion, or they had no religion at all. At the second stage, intellectuals like Edward Herbert of Cherbury (1583–1648) developed the idea of “natural religion” which, on his account, recognized forms of life as religions if they were based on a belief in a Supreme Deity. The introduction of this monotheistic criterion makes the term “religion” into a genus of which there are non-Christian types. At the third stage, forms of life were counted as religions if they were based on belief in a Supreme Deity or other spiritual beings. This is the approach of Edward Tylor (1832–1917), mentioned above, and it is still popular in the field today precisely because it provides relatively clear boundaries on what is and is not to be counted as in the category. At the fourth stage, one counts as religion not only the Christian traditions of the first stage, the monotheisms of the second, and the polytheisms of the third, but also those forms of life based on belief in forces or powers that are not beings or agents. Because this fourth view includes both person-like and not-person-like religious realities, it is the first that would include Buddhism, the so-called litmus test of definitions of “religion,” as well as other non-theistic forms of life. There has thus been a growing circle of how the substance that makes something religious has been understood. This fourth view of the substance that make something religion is, I suspect, the one most widely held by the non-academic public today—call it “the colloquial use”—and it is the one I sought to defend.

The hurdle I saw in treating as religious both belief in God or other non-obvious beings and also belief in non-person-like forces like Buddha-nature or the Dao is that it is not clear that these disparate realities have some feature in common. By reference to what ontological feature could one draw a boundary here that would justify the colloquial use that treats these divergent traditions as members of a coherent set? Is there a feature that can serve as the defining substantive characteristic of religion when one includes beliefs that do not include reference to spiritual beings? To this question, some answer No. For example, Craig Martin makes strong claims about the impossibility of a coherent substantive definition of religion. He writes that “We can’t formulate a definition that fits the colloquial use … because the colloquial use groups together dissimilar things. Those things we call ‘religions’ do not share a set of core properties … [and so] we will not be able to make any generalizations about those cultural traditions we call religions” (2012, 4–5; emphasis removed). We might call Martin’s view “definitional pessimism.” But my optimistic working hypothesis was that there is some characteristic shared by the cultural traditions that people today call religion. There is something in common among Hinduism, Jainism, Buddhism, Daoism, Nuer religion, Judaism, Christianity, Lakota Sioux religion, Islam, and others that overcomes definitional pessimism. To get at this common feature, I introduced the idea of what I call “superempirical” realities.

This idea of superempirical realities is central to the definition of religion I propose and it is what opens the door to Sauchelli’s counter-example, and so it needs some explanation. Sauchelli correctly sees that the idea is introduced as an alternative to supernatural or transcendent in order to avoid the assumption that all cultures have imagined reality dualistically as having two “levels.” But he misses what is for me a crucial aspect of my neologism: the term “superempirical” does not simply mean non-empirical. For me, the label “superempirical” is meant to refer to only some of those things that are non-empirical, namely, those alleged realities that are both (1) not empirical and also (2) not the product of anything empirical.⁶ Thus, one cannot perceive a nation with the senses and I would treat “the nation” as a non-empirical reality, but if one takes the emergence of nation states in the modern world as the product of human efforts, then they would not be superempirical in my sense and nationalism would not be included as a religion. I would make an analogous argument against including capitalism. However, insofar as the Stoics taught that one should put one’s life in accord with Logos, a principle of rational order pervading and structuring the cosmos and antedating the existence of human and all other particular beings, Logos would be an alleged non-empirical reality that would also be superempirical in my sense. On my definition, then, Stoic practices predicated on the existence of Logos would be counted as religious.

Sauchelli treats superempirical as a synonym for non-empirical and he therefore misses what I hoped was the novel part of my proposal. Unfortunately for my proposal, however, the failure to grasp this distinction does not hurt his counter-example. According to mathematical realists, mathematical truths have to do with facts that obtain whether or not there are human beings. Mathematical facts do not depend for their existence on human knowers and, unlike nations or capital, they are not the product of any empirical realities. From this realist perspective, which is probably the dominant view among both professional mathematicians and lay people, the truths of math would be superempirical in my strict sense. For this reason, Sauchelli is absolutely right that they could qualify as the content that makes a belief or a practice religious. In fact, even if mathematical realists were wrong in the way they understand mathematical facts—that is, even if mathematical claims do not concern facts that are independent of human calculations or measurements and so they do not concern superempirical realities—this would not hurt his counter-example. The criterion I am recommending for what makes a belief religious does not depend on whether the alleged reality actually exists or whether the belief about it is true. A criterion like that would be unhelpful for the academic study of religion. My criterion only requires that some community take it to be real or true. Mathematical realism would qualify.

This conclusion that beliefs about mathematical facts would count as religious beliefs, that learning math would count as a religious practice, and that math departments in schools would count as religious institutions is counter-intuitive at best, and I agree that this definition does not exclude it. Since I want the definition to capture the colloquial use of the term, there appears to be a problem in the definition of religion I proposed.

Trying to deny the problem

A first response to the above result might be to grant that this definition includes math as a religion but to deny that this is a problem. That is, one might argue that the observation that mathematical facts are ontologically very much like other religious realities may seem counter-intuitive at first blush, but this observation does not undermine but actually better illuminates the value of this definition. One can base this argument on two reasons.

First, recall that the notion of superempirical realities was introduced to broaden the substantive aspect of religion to include the diverse forms of life of those who base their practices on some reality that is not like a person but is instead some fact about the nature of things. The idea of superempirical realities is intended precisely to offer a broader category that would include realities that are not agents. In addition to the Stoic teaching about Logos already mentioned, therefore, it includes Brahman, said by some Hindu teachers to be the unchanging substance of the apparently changing world, a reality that cannot be perceived but is that by which all things are perceived. It includes the Dao, said by some Daoist teachers to be the natural patterns of movement in the world evidenced in the creative rhythms of the seasons, of matter, and one’s body, taken as a guide for political and personal well-being. It includes the law of karma, said by some teachers in India to be the automatic mechanism by which the seeds of the beneficial and damaging energies that are present in good and bad actions come to flower in future appropriate consequences. It includes the claim that all things that exist are characterized by selflessness, the lack of independent origination said by some Buddhist teachers to be, along with impermanence and unsatisfactoriness, one of the three marks of existence. It includes a Great Principle, said by some neo-Confucian teachers to structure the energy and patterns of transformations of all entities. None of these alleged realities are “beings” or “agents;” none has a mind or a will, nor emotions toward or expectations regarding the things in the world. Nevertheless, each is central to what we might call an axiological cosmology and each is superempirical in the sense I specified. Mathematical truths are not unlike these realities. Mathematical truths concern features of the world that could be considered its order or structure, a structure that wise people should know and with which they should put their lives in accord. One might therefore argue that the inclusion of mathematical truths as religious seems odd only given the assumption—held over from the first three stages of substantive definitions—that religions must concern superhuman agents. From this perspective, it is actually an advantage of my proposed definition that it overcomes this assumption.

A second argument that math is not really a problematic counter-example for this definition is to note not only that mathematical truths resemble other non-person-like superempirical realities, but also that, historically speaking, mathematical truths have played a role in spiritual disciplines. Philosophy in both western and eastern antiquity was a spiritual exercise, a therapeutics of the passions and a method for training people how to live. Speaking of the philosophical texts composed in ancient Greece, Pierre Hadot describes them as “the products of a philosophical school, in the most concrete sense of the term, in which a master forms his disciples, trying to guide them to self-transformation and -realization” (Hadot 1995, 104–105). Pythagoras and those teachers called mathēmatikoi saw numbers as the keys to the cosmic unity that they called God, and it seems that they practiced celebratory rituals about the beauty and logic of these insights. Even more is known about Plato, who saw a curriculum in mathematics as an invaluable means of drawing the soul away from the sensible realm to the intellectual. The training of an ideal leader would begin with 10 years of mathematics and would be theoretical or contemplative and not simply practical (Republic Book 7; cf. Mueller 1992). In the Meno, Socrates uses geometry to show that a person knows more than one has learned in the present life, and this is presented as evidence of the existence of a pre-existing and reincarnating soul. Despite the tension between mathematical Platonists who see mathematical truths as eternal and Christian Platonists who see them as created, the notion that the study of math leads the mind beyond the physical world grew widespread in the common era. Given the use of math in spiritual disciplines, then, one can speak not only of beliefs in mathematical facts as kin to other religious beliefs that lack divine beings, but also of mathematical practices in antiquity as kin to contemplative religious disciplines. One might therefore argue that we should not let the bounded character of religious institutions in the modern western world blind us to the operations of religious aspects of culture when practical commitments to superempirical realities permeated education, politics, art, and so on before those boundaries emerged.⁷ From this perspective, Sauchelli’s counter-example does not draw attention to an absurdity implied by this definition, namely: that mathematics are religious. Rather, it provides a rich example that helps us think about the variety of religious forms of life and how their categorizations have changed over time. One might therefore deny that math presents a problem and leave the definition as it is.

However, I resist this proposal. It is true that some communities have used math as a therapy for the passions or as a means of elevating the mind to non-empirical realities, and these historical facts should lead scholars to recognize that mathematical beliefs are not inherently secular but can be put to use as part of contemplative religious disciplines. It is nevertheless implausible—or, at least, it contradicts the colloquial use—to say that, apart from any spiritual exercise, beliefs about mathematical facts should be counted as religious beliefs, that learning math should be counted as a religious practice, or that math departments in schools should be counted as religious institutions. If “religion” as a category of cultural forms is to have any analytic utility, then math classes should not be assimilated to religion. I would therefore agree with Sauchelli that the math counter-example presents a problem for my original definition, and that therefore the definition should be changed.

There are two ways to change the definition I originally proposed. My definition is a monothetic since it stipulates necessary and sufficient criteria for membership in the category. One solution is to keep this as a monothetic definition but to restrict its scope in some way that will exclude math. The other is to shift to a polythetic definition. I consider those two in order.

A stricter monothetic definition

Given that my “dithetic” definition includes two elements, there are two paths by which one might keep this as a monothetic definition but narrow it in such a way as to exclude the study of math: one can circumscribe either the substance or the function. If one chooses to specify further the substance, then one limits the range of superempirical realities that would be counted as religious. For example, one might limit the definitional substance to those superempirical realities that have causal power.⁸ Given this qualification, one can grant that mathematical facts are superempirical but point out that they are inert. They do not act on the world, whereas God(s) can act and intervene and create. Not unlike a God, the Dao is said to be the mother of all things. Brahman is said to be the power of sight, not what is seen. The law of karma is likened to a mechanism that decides one’s fate. With examples like these, one might make the case that the superempirical realities that should be counted as basic to religion are those that are agents or those that are not agents but still have effects on the world. One might say that one counts those that are not merely “realities” but also “forces” or “powers.” They seem to have some kind of causal efficacy.

The difficulty with this proposal is that the concept of cause here is ambiguous. Agents are causes of events when they initiate them. Non-agent-like realities are causes in a different sense, namely, in that their existence explains events. Take “natural selection.” Natural selection explains the evolution of species and could be called a cause in that sense, but the concept misleads if one takes it to mean that nature acts purposefully to select which organisms do and which do not survive. The same applies to Logos or the Dao or the law of karma: they are causes not in the sense of agents but rather in the sense of mechanisms, laws, and structures that explain why the world has the character it does. The Dao is the “mother” of the world only metaphorically. The difficulty, then is that mathematical laws also structure and explain the events in the world. For this reason, if one includes as religious non-agent-like superempirical realities that are causes in that they explain events, then it is not clear why math would not also slip back in. Even with this circumscribed understanding of the superempirical, this distinction does not seem to exclude mathematical truths.

The second way to circumscribe this dithetic definition is to limit the range of functions that one would count as religious. In my original proposal, as Sauchelli notes, I did not specify any particular function or kinds of function as definitive of religion. I recognized that some practices predicated on superempirical realities are performed to heal the body, others to reconcile people to each other, still others to control natural phenomena, and still others to provide an over-arching orientation of values, and I rejected the idea that some of these functions are “really” religious and others are not. Given such an inclusive understanding of religious functions, the only phenomena excluded by this approach would be individual mental states that involved no social practices. That is, the functional aspect of my definition was meant to exclude as religious “mere belief.” Since this part of the definition is so inclusive, however, this may be a fruitful place to put limits. For example, one might propose that religious practices are those predicated on superempirical realities that seek benefits. This is the approach taken by Martin Riesebrodt (2010) who defines religion as “interventionist practices” that establish contact between social actors and superhuman powers believed to have influence on those dimensions of human life that escape direct human control: “People turn to these powers for protection, help, and blessings” (2010, 94–95). Riesebrodt provides a tremendously rich list of calendrical, life-cycle, and other practices designed to avert disease, hunger, persecution, drought, and other misfortunes, or to provide health, abundance, peace, social harmony, and other blessings. His list also includes ethical behavior and the study of texts insofar as these practices are done as a form of religious service to those powers. But in all cases the practices seeks benefits. If the study of mathematics expects no interventions, however—if knowing mathematical truths confers no benefits until that knowledge is later applied—then math would not be counted as a religious. Religion is here defined as a practical pursuit, and mathematics bakes no bread.

To see the central difference between this narrower definition and mine, it helps to distinguish between practices pursued instrumentally as a means to obtain some benefit and practices whose ends are intrinsic to the practice. A distinction between actions that are done for the sake of something else and those that are not goes back at least to Aristotle’s Nicomachean Ethics. In a thoughtful discussion of this distinction, Christine Korsgaard (1986) gives the examples of walking-to-the-bank as an activity done as a means to achieve some benefit and taking-a-walk as an activity done without pursuing a benefit apart from the activity. Taking-a-walk is a practice whose value is internal to the practice or is the practice itself; practices like this are in Aristotle’s terms “final.” If one then focuses on the set of practices predicated on the existence of superempirical realities, one can distinguish between those performed to obtain some benefit like a good harvest or a healed child and those whose value is simply internal to the practice, performed with no desire for benefit. In this latter category would be those forms of worship, meditation, and rituals that seek no extrinsic good and are done simply because they are right or fitting or good to do. The difference between the two definitional approaches, then, is that on the narrower definition, practices that do not seek benefits are not religious and therefore, insofar as the study of math provides no benefit other than knowing mathematical truths, it would be excluded. On my wider definition, both kinds of practice are equally religious and math would still be included.

Which definition is better: one that narrows religious practices to those seeking benefits and thereby excludes the study of math or one that is broader and, counter-intuitively, includes math? I judge that it is analytically useful to recognize a distinction between the pursuit of instrumental and intrinsic goods. It seems plausible that most practices predicated on the existence of superempirical realities have been performed instrumentally for some external benefit. It also seems plausible that, historically speaking, the earliest ones were. Recognizing this distinction may help us to understand differences between the practices of lay people and the contemplative, meditative, or speculative practices of those considered virtuosi. Nevertheless, both are predicated on the existence of superempirical realities, and both are as woven together in religious people’s lives as are walking-to-the-bank and taking-a-walk. For this reason, though one might distinguish between primary and secondary religious practices, I would continue to define terms in such a way that neither kind of practice is non-religious. Moreover, to define “religious” in terms of those practices that seek divine intervention continues to trade on the idea that the practices that are religious are the ones predicated upon a reality that responds to entreaties because it is an agent. Like the proposal that one restrict the definition to include only those superempirical realities that are causes, then, the proposal that one restrict the function to seeking benefits seems to reflect the theism that defined “religion” until the twentieth century.

As I argue in the book, I do not think that one can adjudicate the question of how best to define “religion” by reference to the practices themselves. To adopt a restricted monothetic definition that only recognizes as religious the superempirical realities that are causes or that only recognizes as religious the practices that seek benefits (or that uses both restrictions) is coherent and workable. But in order to develop a definition of “religion” that is not privileged against the forms of life whose superempirical foci are not agents, I continue to prefer my more inclusive proposal. For that reason, I resist these proposals that seek to save a monothetic definition. I think that the best solution to Sauchelli’s challenge is to give up a monothetic definition altogether.

Polythetic definitions of religion

A monothetic definition is one that stipulates necessary and sufficient criteria for membership in the category. I called my proposed definition “dithetic” because it includes both ontological and pragmatic “theses,” but the pair of ontological-and-functional characteristics jointly constitute a single necessary and sufficient criterion that makes a social practice religious and so this is, properly speaking, nevertheless still a monothetic definition (Schilbrack 2014, 147). This is why I put “dithetic” in scare quotes. The problem that Sauchelli points to, then, and with which I now agree, is that a monothetic definition is not flexible enough to categorize as religious the forms of life colloquially called religious without also including the study of math. Others may want to defend a monothetic definition by using the strategies considered above, arguing either (1) that math does not present a problem to the definition or (2) that one can circumscribe the substance or function theses. But for the reasons given, I don’t see a good way to define “religion” in terms of practices predicated on superempirical realities unless I give up a monothetic definition.

The alternative is a polythetic definition. As it is usually understood, a polythetic definition identifies some class of things in terms of a number of features, no one of which is necessary or sufficient.⁹ For example, one might define the class in terms of five features (call them features A, B, C, D, and E), and stipulate that to be categorized as a member in the class, a thing has to have at least three of them. Thus, those nine kinds of things that have features ABC, ABD, ABE, ACD, ACE, ADE, BCD, BCE, or CDE would be counted in the class. This is also sometimes called a family resemblance definition, since, as Wittgenstein (2009) pointed out in his seminal discussion, this approach treats some words as referring to things that resemble each other even though they have no one feature in common.¹⁰ In a polythetic definition, none of the features is required and so any one of them might be absent without disqualifying class membership.

To define a category polythetically drops the assumption that the members of the category share some necessary feature, some essence. This approach to definitions therefore produces concepts with borders that are vague. Those things that have only two of the features would not be included in membership in the class, but they would be close. They would resemble the members of the class and would be borderline cases. And those things that have all five of the features would be prototypical or exemplary members of the class, and those with four would be more prototypical than those with only three.

One can develop a definition like this for the concept of “religion.”¹¹ Let me propose an arbitrary example, not as a final recommendation for how “religion” should be defined, but simply to illustrate how this approach differs from the monothetic one that Sauchelli critiqued. I assume that a form of life provides some function for those who participate in it, whether instrumental or intrinsic, and so any form of life will be more than mere belief and will meet the pragmatic aspect of my original “dithetic” definition. Consider then the following five possible religion-making features for a form of life. A form of life may:

• A. Be predicated on the existence of superempirical realities;

• B. Teach a moral code;

• C. Promise a path to overcoming human suffering;

• D. Distinguish those who participate as a community from those who do not;

• E. Rank all of one’s values by identifying one’s ultimate concern.

With a set of five religion-making characteristics like this and the rule that a form of life must have at least three of them, this definition would recognize nine different kinds of religion corresponding to the kinds of things above. This definition would also recognize borderline cases that are not religions but are close, like any forms of life that teach a moral code and that distinguish those who participate in the community from those who do not, but that lack any of the other religion-making characteristics. And it would also recognize prototypical or exemplary members of the class.

The anthropologist Benson Saler has long argued for the value of a polythetic definition of religion, noting that “religion” operates in the colloquial use with Christianity as a prototypical or exemplary member of the category (1993, ch. 6). Membership in the category then reflects how similar other forms of life are to the most prototypical member, the one that, as I mentioned above, was originally the only member of the set. Saler also argues persuasively that “it is family resemblances all the way down” (1999, 397) in that not only is “religion” best understood as a family resemblance concept, but so are (1) each of the denominated religions, like “Islam,” “Christianity,” and “Buddhism,” and (2) the elements that make up religions, like “sacrifice,” “pilgrimage,” or “theism.”

Shifting to a polythetic definition like this successfully responds to Sauchelli’s objection. If a form of life has to have three of the above features to be classified as a religion, then the study of math would not be included. Though the study of math includes feature A, it lacks any of the other features and so it would not be classified even as a borderline case.

My argument for a polythetic definition, however, is not that it helps scholars avoid the definitional problem raised by mathematical realism, but rather that the colloquial use of “religion” simply is polythetic. Precisely because the concept of “religion” has a history and the term was used first to label Christianity, then Christianity and other monotheisms, then Christianity, monotheisms, and polytheisms, and now Christianity, monotheisms, polytheisms, and nontheistic traditions, the word has been used to name a class of forms of life with a linked and expanding set of features. The word has been stretched from its original meaning when it was used for a unique case to its present contemporary colloquial use that recognizes that original case as prototypical but no longer exclusive.¹² In this way, the term “religion” is analogous to the term “politics.” Though the letter concept was originally used to describe governing a polis, and though antiquity had no conception of the emergence of the bounded institutions of the modern nation-state, it has been put to a linked but extended use in our contemporary context by those who do not know its genealogy. The shifts in the history of the uses of a term do not undermine its utility.

I see two objections to using a polythetic definition of “religion.” The first is that it makes class membership fuzzy or vague. As Wittgenstein insisted, we work with vague concepts all the time, as when one says “Stand over roughly here” (Section 71). But does a family resemblance approach make the concept of “religion” too vague? Timothy Fitzgerald, for example, an “abolitionist” who holds that scholars ought to drop the term “religion” as an analytic category, rejects the shift to a polythetic definition, insisting that “the concept of religion must have some essential characteristic, and if it does not, then the family of religion becomes so large as to be practically meaningless and analytically useless” (1996, 216). Like Fitzgerald, Craig Martin rejects the shift to polythetic definitions, saying, “If there are no common characteristics among those traditions colloquially called religions, it will be impossible to make relevant generalizations about them” (2009, 167).¹³ The second objection is that a polythetic definition, at least as described above, drops the requirement that to be a religion a form of life must be predicated on the existence of a superempirical reality. This is to say that a polythetic definition no longer requires any substantive element and becomes untethered from the history of the term. In fact, these two complaints are connected. It is because the polythetic definition described above does not include a substantive feature that its range of reference becomes so wide and disparate. And if one works with a polythetic definition that has more the five religion-making features [as with Alston (1967) who has nine or Edwards (Edwards 1972) who has fourteen¹⁴], then the term would be even vaguer.

The way to respond to both of these objections simultaneously is to develop a different kind of polythetic definition which requires a feature that is necessary but not sufficient. To distinguish this proposal from the “open” polythetic definition just described, I will call it an “anchored” polythetic definition. An anchored polythetic definition treats some feature as core or essential, but not sufficient, for class membership. This approach recognizes that the required feature is always combined with other features that will vary. To illustrate this idea with the five features above, one could say that to be classed as a religion, a form of life must be predicated on the existence of superempirical realities (that is, it must have feature A), plus at least any two other features. Feature A is then a kind of anchor shared by all members of the class. By requiring it as a necessary feature of any member of the class, one limits the size of the class.¹⁵ Like an open polythetic definition, an anchored definition also produces fuzzy or vague borders. A form of life that had feature A and one other feature would be borderline. A form of life that only had feature A in common with the class would resemble the other class members in that core respect, but would not join the class. And a form of life with feature A plus three other features, or plus all four, would be, as before, an exemplary or prototypical member of the set. With this distinction of kinds of polythetic definitions, one can recognize three general strategies for how one might define “religion.” A monothetic definition stipulates that some feature is both necessary and sufficient for membership in the category; an anchored polythetic definition stipulates that some feature is necessary but not sufficient for membership; and an open polythetic definition stipulates that no feature is either necessary or sufficient for membership.

Like an open polythetic definition, an anchored polythetic definition solves Sauchelli’s objection, and for the same reason: since the study of math is predicated on the existence of superempirical realities, it includes feature A, but it does not have the others and so it would not be counted as a religion. But I judge that if one adopts an anchored polythetic definition like this, one is able to conceptualize the relation of mathematical realism to religion with some sophistication. The dithetic element in the definition requires both that the reality on which the practice is predicated be superempirical and that it play some function in one’s life. Because of this pragmatic requirement, one can recognize that mathematical facts are superempirical but note that there are (presumably many) cases in which the study of mathematical facts does not come to play a pragmatic role in the student’s life and so would not be classified as religious. This would be true of any study of math that fosters “mere belief” in mathematical truths. In other cases, the study of math might both be predicated on the existence of an superempirical reality and also function in the student’s life in some way. This might characterize the practices of math professors, engineers, and others dedicated to understanding and applying mathematical truths to their lives. When the study of math includes both of the dithetic elements in this way, it would share the essential the necessary feature of religion, and this can explain why the study of math had the appeal it did to Plato and others who saw it as contributing to a religious discipline. But given an anchored polythetic definition, the study of math would still lack the other elements that typify religion and so here too it would not be classified as a religion.

Are there any cultural examples in which the study of math has the superempirical elements, the pragmatic function, and two or more of the other religion-making features? I expect not. The development of a religion of math is, on this definition, a logical possibility, however, and this explains why Sauchelli’s counter-example is so good.