Norms and Mathematics

Norms in math and logic have played a special role to in philosophy about norms in general and moral philosophy in particular. To put in very crude terms a dialectic which I will spend much of this section analyzing, the norms arising from math get invoked as follows.

A naturalist (or, if you think that morality must by definition be what I will call ‘substantive’, a skeptic) puts forth a spare picture of morality as a matter of moral sentiments of pity, sympathy, fair play, praise, blame and linguistic practices and the like and challenges all those who hold there to be more robust moral facts which these practices track. They ask questions like: How are we supposed to know about these moral facts? Why think that there is anything beyond our own psychological state which our feelings of praise and blame allow us to perceive, or which determines the truth conditions of our moral talk? How could there be such things as moral facts which seem to straddle the gap between fact and value, by being external facts which at the same time have nessicary consequences for how we should act?

The moral realist/substantivist then, rather than answering these questions about access and external reality brings up math and logic. How do we know moral facts? Not by perception it’s true, but by the same kind of a priori access which gives us knowledge of mathematical facts. Isn’t it strange to think there are objective moral facts out there which our moral intuitions put us in contact with? It is no stranger than thinking that there are objective mathematical facts which our intuitions put us in contact with. How can moral facts necessitate that we should do certain things? In the same way that logical relations and mathematical facts necessitate that we should believe certain things.

The actual explanation of how these questions are to be answered in the mathematical case is generally omitted (perhaps because of facts about professional specialization). But the realists point is already interesting and somewhat forceful. As far substantivism about moral facts is subject to the epistemic, naturalistic and fact-value gap based objections just mentioned, so are mathematical facts. The prospect of skepticism about the objective truth of 2+2=4 or the fact that one oughtn’t believe that a cup is and isn’t red is much less appealing then that of skepticism about morality. 

Now I agree that substantive views of the norms of mathematics and morals face many of the same challenges, but I take this point in just the opposite way from the familiar realist defence just mentioned. I will argue that we should accept a naturalistic view of the norms of mathematics, and hence (if the symmetry which the realist has been pointing out holds) of morals as well.

My argument for naturalism about mathematical norms has a few different pieces. First, I aim to assuage a certain kind of reluctance to accept a naturalist view of mathematics by sharply distinguishing the question of naturalism vs. subtantivism about a given domain from that of internalism and externalism. Then I will mention some rather familiar reasons for thinking that if an adequate naturalist account of our mathematical practices can be given this should be preferred to the substantivist account. Next, in the main section I will describe a general naturalist picture of evaluative practices and then I will consider the special features which arise in the case of mathematics and morals. 

 

1.Internal vs. external and substantive vs. naturalistic

People who like to read moral philosophy are probably already pretty familiar with the contrast between internal and external reasons, and the debate about whether there are any. Internalists about morals say that all reasons for a given person to act come from and depend on facts about that person, their desires values and the like. So when we talk about X being a reason to Y, the underlying logical form is really that X is a reason for person P to Y. Since reasons depend on the above mentioned ‘internal’ facts about a person, the same fact X might be a reason for a different person Q not to Y. So it doesn’t really make sense (except as an abbreviation) to talk about a ‘X being a reason to Y’ on it’s own without specifying a person or class of persons. In contrast believers in external reasons think there are some facts which are or would be reasons for any person to act, and specifically that what makes these facts reasons is something external to the person’s desires.

It’s important, though tricky, to note here that the two criteria which I have muddled together in the above description (Do reasons come from people’s desires vales and the like? Vs. Is there anything which is nessicarally a reason for everyone) may come apart. This is because there are some reasons which can, grammatically speaking, only apply to people who have certain desires. So for example only greedy people could be in a situation where the following sentence is true ‘By restraining your greed locally you can get more money over all.’ since for non-greedy people the phrase ‘your greed’ wouldn’t refer, so the sentence would be meaningless not true. And greedy people have a desire for money which (on the internalist view) gives them a reason to restrain their greed in such a case. Thus even if you think that all reasons come from internal states like values and desires you might think that certain propositions with an indexical component are ‘reasons for everyone’ in the sense that any person of whom the proposition could be true would be someone whose internal states gave them reason to act a certain way. To put the same point differently ‘By restraining your greed locally you can get more money over all.’ is a reason for everyone to restrain their greed in the sense that it is a reason for everyone to whom this claim could apply. It’s not clear whether to count this as internalism, but since it’s also not very important I will just call this The Middle View.

Now as far as I know the next contrast has not been as much discussed but I suspect the essence of it will be familiar. By a naturalistic view of a certain kind of norms I mean a view which invokes certain kinds of facts about how people act and how they are inclined and disposed to act, their experiences, the ways they use language and the like and then claims that this is all there is to the story. In contrast a substantivist view of some kind of norms holds that something crucial is missing from the above kind of naturalist description, be it a role for objective external moral items or properties or some sense in which not only to people approve of and desire to act in certain ways but beyond this they ‘really ought to’. The naturalist tells his story and the substantivist complains that something vitally important is missing. (Hume and Plato, as they are usually represented in college courses are representatives of each of these views respectively, in the domain of moral normativity.)

So finally, what I want to say, and the reason I have been dragging you thorugh all this stuff about the distinctions is this. These two contrasts are independent of one another. A naturalist could be in internalist about morals claiming that our rules for applying moral terms always required that we only say that someone ought to do something if we have reason to believe that it would be supported by some of their values and desires. But he could also be an externalist who held that our practices of talking about what people ought to do were seldom or never scrupulous in this respect, and just required that we check a certain action against our own moral sentiments or some communally shared ones. Similarly a substantiveist could also be an internalist. The most likely case has to do with norms of rationality. They might think that rationality is an objective metaphysically real property of actions, over and above our tendency to praise and blame actions and also think that whether or not a given person’s action will have this property depends crucially on their desires. Or, of course you could be a substantivist and an externalist, thinking that an act’s possessing or lacking this objective property of rightness often has nothing to do with the agent’s internal states.

I’m emphasizing this distinction because I think it de-rails a possible, bad, reason for rejecting the naturalist account of the norms of mathematics. The right answer to 2+2 is definitely 4, and it’s 4 for everyone not 4 for me and 5 for others depending on their values or desires. The logical contradiction involved in the idea of a barber who shaves everyone who doesn’t shave themselves provides everyone with a reason not believe that there is such a barber, not just those who are ‘invested’ in avoiding contradiction. But this is a reason for rejecting internalism about the norms of math in favor of externalism or The Middle View not a reason to reject naturalism. A naturalist can very well say that our practices determine a single right answer for everyone, they only want to insist that the way this answer is determined invokes no non-natural facts and that the sense in which the answer is right can be cashed out without invoking such facts.

One might still reasonably hesitate about these latter two points (if so read on!) but I hope to have shown here that the idea that there is a single right answer, independent of people’s desires is, on its own, no reason to reject naturalism.

 

2.Why a natuaralistic account should be prefered if such an account can be given

In the introduction I described the contrast between the nominalist and the substantive about norms in terms of nominalist attempts to account for various aspects of normative talk about a certain domain and the role which this plays in our lives in contrast against a sucession of substantivist objections that something (e.g. the sense in which disputants about morality are talking about the same thing, our disapproval of the sensible knave) is missing. I say this partly to sneakily shift the burden of proof: my inclination is to say ask, given that we all agree that such-and-such processes are going on, what reason have we to suppose that something else is involved? But I also say it out of genuine confusion about what the ‘something more’ which those of realist intuitions feel must be involved could be. The very mysteriousness of REAL normaitivity (in distinction from the practices of normative talk which the naturalist recognizes), which makes the substantive conception so suspicious also makes it hard to prove that such a poltergeist isn’t hidden somewhere in the wings.

Thus my strategy will be to give a naturalist account of normativity in general with a focus on its application to math and morals and then put forward a challenge for the substantivist to show what is wrong or what is missing.

            Before plunging into said positive story though, I want to offer one further support for this way of casting the dialectic. Specifically I want to claim that at least for many objects and properties which we take to have real existence external to us this requirement (show me what’s wrong with an account that doesn’t acknowledge these entities/properties!) is not too hard to meet. Consider a logic-and-practice-projection account (like my view of mathematics) of our practice of saying that things are heavy. Suppose that some heaviness-anti realist wants to maintain that there is no such real external property as being heavy, and that our practice of saying that things aren’t heavy is just an isolated language game, entirely separate from facts about object’s mass and how it feels to pick them up. Echoing my description of the naturalist above this person might say: all there are is intuitions that certain things are heavy which are better understood as emotional reactions to conceiving of certain objects or extensions of claims which the subject has heard other speakers make rather than perceptions of some external facts about the world around us (or even positivistic sense data).

            Such a person would have a hard time accounting for a number of things. Why do a person’s heaviness ascriptions to an object often change when they pick that object up? Why do all the things which people call heavy turn out to have other physical similarities like mass? In contrast our present physical theory of heaviness, which posits that objects counting as heavy is best understood as their having an external objective property answers these questions. It posits that objects have masses which causally interact with our muscles and skin in such a way as to produce experiences, and that in learning words like heavy we are trained to classify objects by means of (in the simplest case) the characteristic modification in our experience which these objects’ weights cause. Positing heaviness as a real (if non-fundamental) property of external objects simply gives a better explanation of our claims that certain objects are heavy and others are not.

(Perhaps a similar argument can be made against sense-data theories.)

I am challenging the realist to similarly either provide a similar argument either showing explanatory or empirical flaw in the naturalist account outlined below, or showing that another account of the causes and significance of our normative talk is to be preferred.

 

3.Evaluative practices

I. A naturalistic account of epistemic reasons

It’s a descriptive fact about the world that doing certain things will make you money. Given this we can pick the class of possible actions for a person which will (or are likely to) lead to their making money. We can then say of these actions, that the person has a financial reason to perform them. Similarly it’s a descriptive fact that certain propositions are logical consequences of others i.e. that certain propositions couldn’t be true without other propositions being true. Thus we can talk descriptively about what someone has logical reason to believe: you have logical reason to believe that P if P is a consequence of things you already believe. Also we can pick out (roughly) a class of propositions which a person has self serving reasons to believe, things which it would make them feel better to believe (e.g. that their company isn’t seriously harming the environment). Then we can descriptively say that a person has self-serving reasons to believe something if having this belief would indeed make life more comfortable for them. [Here I am not talking about the further causal claim that they do believe the proposition out of this kind of motivation: one might praise a person for nobly questioning and rejecting views which they have self-serving reasons to believe]

Note that on its own this kind of descriptive talk about what a person has an X reason to do or believe is quite distinct from motivation. We can talk in this descriptive way about what a monk has financial reason to do even if the monk is quite indifferent to these worldly considerations. However, motivation does play a role in determining which of these descriptive reasons we bother to talk about. It’s interesting to talk and think about what people have financial reason to do because many people are to some extent motivated by making money. In contrast, its not surprising that we don’t talk about what people have blades-of-grass-counting reason to do (i.e. which of their possible actions would be likely to lead to their successfully counting blades of grass) because almost no one is at all motivated by the prospect of counting blades of grass. We only talk about reasons with respect to a certain kind of thing if some people are somewhat motivated by achieving that thing. 

Now, one thing that people are generally quite motivated by is purging falsehoods and adding truths to their beliefs. I should note here that by motivation I don’t mean the kind of conscious feeling of craving which one sometimes feels about eating certain foods or going to out to nightclubs. I doubt anyone (except maybe the historical Socrates) feels the same kind of conscious sensual relish at the prospect of believing more truths and less falsehoods. Rather, I take someone to have a desire/value/be motivated by a certain state when a) their experiences and behavioral dispositions can be well accounted for on the model of beliefs and desires and b) the best such account posits such a desire. Thus, I am saying that almost everyone is very motivated by truth because of rather mundane facts about how people tend to revise their beliefs. When you tell someone that it is raining and it is snowing, they will, in all but the most exceptional cases come to believe that it is raining. If you tell someone that all snakes of a certain species which have been encountered up to date are poisonous they will come to believe that the next, as yet unexamined snake of this species will be as well. We are tremendously inclined to add (simple) logical and inductive consequences of our beliefs.

Thus, in a thin sense at least, I claim that people are motivated by the prospect of believing truths. I don’t mean to quibble about the exact use of the word motivation here. There are likely some important differences between motivations to act and ‘motivations’ to believe. What is important is just the pretty undeniable generalization that people tend to add beliefs when presented with (logical or inductive) evidence that these beliefs are true. This fact is all we need to explain, in the above framework, why we should be interested in the logical, inductive, and truth based descriptive reason to believe.

We should note here that this core sense in which people are motivated by reasons to believe is also shared by some other kinds of descriptive reasons to believe which are less generally admired. If one part of a simple explanatory account of people’s tendencies to belief revision involves positing something like a motivation to believe truths, another part involves positing a motivation to believe things which would be comfortable to believe or would be pleasant if true. The paragraphs above are only intended to explain why we bother to talk about logical, evidenciary and self-serving reasons to believe and not about descriptive reasons arising from other properties of beliefs. (e.g. we don’t bother to pick out and talk about which propositions are such that believing them would let a person believe more falsehoods, or believe more objects to be yellow, or believe more things which are likely to get them burned at the stake “you have a getting-burned-at-the-stake reason to reject Christianity in favor of the celtic gods”).

Beyond this core sense in which it’s not surprising that we engage in descriptive reasons talk about logical and inductive reasons to believe there is a further motivational force which in contrast (for most people) the question of believing truths does not share with that of believing things which make one’s life comfortable to believe. As far as belief goes, the cases are similar: you are likely to believe things which you have evidence for, and you are likely to believe things which it would be comfortable to believe. But, for most people, a difference between these two properties of beliefs arises (being comfortable and being true) when one considers the indirect relationship between belief and action. For one of the things which can motivate a person to act is that of bringing it about that  their beliefs have certain properties. A cynical person might take classes because they wanted to bring it about that their beliefs about how to interpret certain literary texts to match what was commonly accepted in their culture. Or Pascal might have been motivated by his wager to attend the rights and ceremonies of Christianity so as to form a belief in the existence of God. Or, to choose a much more normal case, a person may choose to perform certain actions: to buy a telescope or talk to a critical friend, or examine the neighbor’s yard or read a book on higher mathematics with the aim of ending up with more true beliefs in a certain subject.

In some cases people seem to want true beliefs as an end in themselves, while in numerous other cases they want to believe truly about a certain areas as an instrumental means to achieving some other goal. This stands in contrast to the case of comforting beliefs or beliefs such that holding them will make one accepted in society. The cases of Pascal and the cynical person who wants to be cultured notwithstanding people generally aren’t motivated to act by the prospect of acquiring comfortable or socially acceptable beliefs. Thus, it is possible to give a double explanation for the practice of talking about what people have evidenciary (i.e. logical or inductive) reason to believe. First, people’s tendencies to form beliefs are well explained by invoking the aim of truth. If a person believes something and you point out a logical or inductive consequence to them they are quite likely to believe that as well. Thus talk about what people have descriptive logical and inductive reason to believe can help us predict and give simple explanatory descriptions of the epistemic behavior of our fellow humans. But secondly, many people are not only belief-motivated by truth, but motivated to take actions like attending classes and setting up prize contests by the prospect of forming true beliefs (at least about certain areas). For this reason, not only can you predict that someone would come to believe a logical consequence of their present beliefs if you pointed it out to them, but it is likely that they will be grateful and perhaps reward you for doing so. For many people are not only disposed to believe the evidenciary consequences of their beliefs but want to actually realize this dipsposition by getting into positions where they will end up forming many beliefs this way.

So far I have described the practice of talking about descriptively about which propositions provide evidenciary reasons for believing which others, and talked about why people might be so interested in these kinds of facts. In a sentence: we are disposed to believe what we have evidence for and to try to get in situations where actually do believe what we have evidence for. It seems to be that this account (or some sophistication of it which addresses questions in the philosophy of science about what exactly the descriptive evidence relation which belong to the philosophy of science) to explain our actual practice of talking about reasons to believe. Specifically I think that no “real”/“categorical”/non-natural normativity needs to be invoked to explain this practice. And given that there is no explanatory need for such normativity, it seems to me that there is no reason for us to believe it. At least, there is no epistemic reason – I make no claim to psychological insight as to whether the belief in real substantive norms will make your life inner life more comfortable or your social life more successful.

 

2. How epistemic norms are and aren’t like table manners

            Now I’m going to see how this account answers the interesting and classic question of whether our practices of talking about what people have epistemic reason to believe is or isn’t like the practice of talking about what is good table-manners. I will distinguish two different aspects of this question, and argue that epistemic reasons are like table manners in one significant sense but not in another.

            So, one notable thing about table manners is that people in different countries have different manners. This is not to say that there aren’t objective descriptive facts about the good and bad manners of various ways of eating. Rather, it means that the question ‘is X an instance of good or bad table manners?’ will be undefined unless context specifies which country’s manners are meant. If Jim the American executive on a trip to china cleans his plate at a formal business dinner it is an objective fact that he is exhibiting good American table manners but poor Chineese table manners.

To put the point differently, one interesting thing about table manners is that we can make out different, even conflicting practices of evaluating how a person eats all to be different kinds of or standards for table manners. Similarities in something like the role which whatever a given society’s table manners play in that society motivate us to consider these different evaluative practices as different instances of a common kind.

So one aspect of the question of whether there could be different practices of evaluating belief revisions which differed widely in their judgments but still all counted as practices of evaluating evidence relations. It seems to me that the answer to this is ‘no’. As I see it there are objective facts about which propositions entail which other ones (these facts arise out of our use of language, which is a *very* long story) and (though a precise answer to this would involve careful attention to the problem of induction and what kind of naturalized epistemology can be given for it) which propositions provide empirical evidence for which other ones. If we assume an inductive measure on the space of possible worlds then we can put the objective evidence relation like this ‘A is evidence for B for a subject with prior belief state S if most of the A worlds in S are B worlds’. I take this question of what must, or is likely to be true given that what else is true to articulate the essence of our talk about evidenciary reasons to believe. Thus I don’t think that there could be different ways of saying what is a reason to believe what which would still count as practices of evidenciary belief revision.

            There are, of course, other kinds of belief evaluation – some of which I have mentioned already. We can talk about what someone has self-interested reason to believe or what they have social reason to believe. But, as a rather trivial linguistic matter, I doubt that we would ever describe beings with very different practices as having different evidence relations so that A provided human evidence for B for subject S but not martian evidence. The idea of evidence is pretty well fixed by objective relations between the truth conditions of various propositions, and the tendency of various belief forming practices to lead to true beliefs. Thus its unlikely that we would ever consider any very different way of evaluating belief formation as involving a different kind of evidence rather than a way of evaluating belief with respect to a standard other than evidence.

            This is one half of the story then. I don’t think that we would ever say that other cultures or beings had radically different standards of evidence in the way that we do say that different people can have radically different standards for good table manners. They might have different kinds of standards but (this is just a trivial point about language) I think we would always understand these standards as articulating different virtues which other beings cared about their beliefs having, rather than as articulating a different concept of evidence.

            Now for the sense in which epistemic norms are like table manners. This has to do with motivation. I think a sensible (and I hope uncontroversial) account of the relationship between evaluations of certain actions as good and bad table manners and motivation goes like this. People often want to eat in ways that impresses, or at least avoids disgusting the people they are eating with. To this end they try to figure out (or at least remain unconsciously aware of) which ways of eating have certain descriptive properties: these are the normal ways of eating for a particular culture. They label the acts which this normal way of eating requires as those which they have manners reasons to perform. Then, seeing that has a manners reason to perform a certain act can help motivate them to act. But manners reasons are no source of intrinsic motivation. Rather we are antecedently motivated by certain goals and doing what we have manners reason to do is instrumental to achieving these goals.

            Generally: people care about and are motivated by certain things, and because doing a certain descriptive kind of thing (using the salad fork for salad etc) can help us achieve what we care about we invented talk about what manners give us reason to do. Similarly, people are motivated by the prospect of having true beliefs (both in terms of belief and action) and believing what one descriptively has evidence for is instrumental to this so we invented talk about what we have evidenciary reason for convenience. Thus the perception that one has evidenciary reason to believe something only motivates people to believe things by way of their antecedent motivation to believe truths. What’s important about evidenciary reasons is not that their nature as reasons but their relation to truth.

            This brings up the further question of whether there could be something like an epistemic version of the sensible knave. The view I sketched here is similar to the classic Humean view about morality. People care about the flourishing of themselves and others and have moral sentiments of praise and blame towards people or actions corresponding roughly to their tendency to promote or prevent human flourishing. So people have reason to do the kinds of things which they praise because these actions promote things which they care about. Hume points out that this account of human moral motivation means that if there was a person, the sensible knave, who did not have any moral sentiments then this person would have no reason to be morally good. Not caring about other people’s well being is base and morally objectionable, of course. But it is not, in any real sense, irrational.

            So, one might wonder whether my account of epistemic reasons leads to a similar result. What does it say about people (if there are any) who don’t care about having true beliefs, who are thus not motivated to believe by the descriptive relations of logical consequence and empirical evidence? I will ultimately argue that there is something analogous to the sensible knave in the moral case, though we will need to consider two little technical problems with the question of whether there could be an epistemic version of the sensible knave first.

            The first issue concerns the ‘sensibleness’ of the sensible knave. Hume makes out that the knave could be fully rational and also bad by contrasting rationality with moral goodness. The knave is sensible in that he finds out and takes the means nessicary to his ends but morally bad in that these ends are bad ends. It is a little bit tricky whether the same contrast can be drawn between rationality and epistemic rationality. It seems to me that to describe a belief revision as rational simply is to describe it as one which the subject has evidenciary reason to adopt. Thus rational and irrational are the equivalents for belief of terms like noble and base for action. Thus there couldn’t be a rational person who didn’t care about truth in the same sense as there couldn’t be a virtuous person who didn’t care about human flourishing.

            The second distracting issue concerns the nature of belief ascription. We don’t ascribe beliefs to most objects, or indeed to most living things (like plants or bacteria). In order to say that something has beliefs we require that it display a certain complex pattern of relationships between it’s observations and its actions of various kinds (any maybe also that have conscious experiences). So we can imagine, perhaps, a depressed person who is disposed to believe what they have evidence for but doesn’t care in the action-guiding (as opposed to belief-guiding) sense whether these epistemic tendencies which are so conducive to believing truths stay or go. But if we try to imagine a person who isn’t motivated by truth even in the belief-forming way, so that they are inclined to voice and act on propositions at random, or in some way that has nothing to do with the evidence presented to them, then I don’t think we would consider this person to have beliefs at all. So rather than having described someone who isn’t motivated by truth in their belief revision we have described someone/something that doesn’t have any beliefs to be rational or irrational in revising.

            I want to try however, to cut through these technical issues to get at an important underlying similarity between the moral and epistemic cases. As far as the first problem goes, I think there is a further standard which we can interestingly apply to someone whom we have stipulated isn’t rational in the sense of revising their beliefs according to evidence. This is the standard of means-ends rationality. And as far as the second problem goes we can make out the notion of a believer who isn’t motivated to believe by logical and evidenciary relations if we imagine someone who is usually inclined to believe what they have evidence for but doesn’t have this inclination in a certain circumscribed domain.

            So consider a person who is generally believes the (more obvious and direct) logical and inductive consequences of their beliefs, but lacks this disposition in regard to a certain domain, say celebrity weddings. And suppose that they don’t have either a general desire (in the action-guiding sense) to believe the truth on all subjects, or any other desires for the attainment of which it would be instrumentally useful for them to have true beliefs about which celebrities are married to each other. Here it seems to me that we can (just barely) make out the person to actually have beliefs about celebrity weddings, if we suppose there are reliable patterns in how they answer questions and how they would be disposed to act, because of the relation between these ‘beliefs’ and other beliefs which are appropriately connected up to evidence gathering. And, in light of the stipulations about the person’s desires/values/goals, I think that such a person would also count as being means-ends rational.

            Of course, their belief revisions aren’t rational insofar as belief rationality just means believing the logical and empirical consequences of one’s beliefs. But there is an important sense in which their actions (in not taking steps to bring it about that they believe what they have evidence for or avoid contradictions in their beliefs concerning a subject which they have no direct desire or instrumental reason to care about having true beliefs about) are means-ends rational. We might even criticize them for behaving otherwise. One indeed actively consider such a person irrational for listing out and axiomatizing their beliefs about celebrities’ marriages in order to spot contradictions and infer new truths, given that they care nothing for knowing the truth about this subject.