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Publication Supported by The John Templeton Foundation Grant “The Limits of Scientific Explanation”

© Copyright by Bartosz Brożek & Copernicus Center Press, 2013

ISBN 978-83-7886-041-9

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PREFACE

The problem of rule-following has been present in philosophical literature at least since Wittgenstein’s Philosophical Investigations. The number of contributions to the topic has been enormous, and their proper analysis would require several thick volumes. In face of this fact, an apologia seems in place: why have I decided to write yet another work on the subject? My defence is twofold. First, I believe that much of the existing literature pertaining to rule-following is centred around the problem of language rules, while other kinds of rule-following are ignored. This is quite understandable given the focus of Wittgenstein’s analysis, as well as the popularity of Kripke’s influential book, Wittgenstein on Rules and Private Language. However, I would suggest that the prevailing reading of Wittgenstein is too limiting: his insights are relevant for all types of rule-following. Second, there exists a cluster of ideas that have originated in various disciplines and which seem surprisingly coherent: Wittgenstein’s analysis of rule-following; the mirror neuron hypothesis in neuroscience; the importance of imitation in recently proposed evolutionary scenarios; the ‘embodied mind’ paradigm as developed by Lakoff and his collaborators; and finally some aspects of Popper’s long-neglected ontology of three worlds. It has been difficult to resist the temptation to describe the relationships between those ideas and it is my hope that the result of this attempt – the theory of rules developed in this book – offers some fresh insights regarding the nature of rule-following.

This book has benefited greatly from discussions with, and comments from, many people. I express my gratitude to Michael Heller and Jerzy Stelmach, who not only created the intellectually stimulating environment that I have the privilege to work in, but also shared their ideas pertaining to the problems I discuss in the study. I thank Robert Audi, Mateusz Hohol and Łukasz Kurek, who read the entire manuscript and provided me with insightful comments. My gratitude also goes to (in alphabetical order) Anna Brożek, Yadin Dudai, Marcin Gorazda, Jaap Hage, Maria Karolczak, Bartłomiej Kucharzyk, Nino Rotolo, Corrado Roversi, Magdalena Senderecka, Aeddan Shaw, Piotr Urbańczyk, Wojciech Załuski, and Radosław Zyzik, who read and commented upon parts of the text or discussed the ideas I deal with in the book. Finally, this book was written within The Limits of Scientific Explanation research project, carried out at the Copernicus Center for Interdisciplinary Studies in Kraków and sponsored by the John Templeton Foundation. I kindly thank both institutions for their support.

As usually, my deepest gratitude goes to Atka, for making my world richer than ever expected, and for showing me that whether you break rules or follow them, it is always better to do it together.

Bartosz Brożek

Kraków, July 2012

Introduction

Of rules

1. The many faces of rule-following

There are many, often quite different situations in which we speak of rule-following, observing rules, or of rule-guided behaviour. Examples of rules include: “One ought to maximize one’s utility function”, “One ought to file tax returns by April 30 each year”, “Speed limit should be observed”, “To inscribe a square in a circle, draw two diameters, A B, C D, intersecting each other at right angles; join their extremities, A C B D”, “When entering a church, one should remain silent”, “Soldiers should follow orders”, “The word ‘cheers’ should not be used in official correspondence”, or “A popular book should be no longer than 400 pages”. This shows that rules can be found in each and every sphere of human experience: in morality, law, mathematics, language, games, etiquette, etc.

Let us consider in more detail the following three examples:

(1) One ought not to steal from other people.

(2) One should apply the expression ‘green’ only to green objects.

(3) a*b = b*a (the commutativity of multiplication).

The first rule is a moral one, the second – linguistic, while the third may be described as a rule of mathematics. Now, one can ask whether they have anything in common. Is the fact that they are (or at least: may be) classified under the same heading just a linguistic coincidence, or do they rather share some basic properties in virtue of which we call them rules?

One can argue that they have little – if anything – in common. The moral rule pertains to our behaviour: it prescribes abstaining from acting in a particular way. Thus, the rule (1) provides justification for choosing a course of action. The justificatory force of the rule does not entail, of course, that people should never, under any circumstances, steal from other people. One can imagine cases in which our rule is in conflict with some other rule (e.g., one should not let one’s children starve), which – all things considered – would justify stealing. However, the rule (1) is not merely an instruction of how one may act if one chooses or merely wishes so: it does not succumb to pure prudential (egoistic) motives. In this sense it is categorical: while remaining prima facie (i.e., being prone to a defeat by some other moral rule), its normative force is not conditioned by some external normative criterion such as the precept to maximize one’s gains and minimize one’s losses.

The linguistic rule (2), in turn, seems not to possess a categorical character. It is rather hypothetical and may be rendered as saying that if one has (an external) obligation to speak correctly, or merely wishes so, one should use ‘green’ in relation to green objects only. As such, linguistic rules have no justificatory power; they cannot back the choice of the given course of action. One can even go as far as to claim that they are not action-guiding at all. Speaking (describing, requesting, asking, etc.) are actions only in a very broad sense of the word, and anyway have little in common with actions prescribed by moral rules. A linguistic rule can never defeat a moral precept, or better still, they can never be compared or weighed against one another. On the other hand, to follow a linguistic rule may be an element of a morally prescribed course of action (i.e., when one is morally obliged to tell the truth one should follow linguistic rules).

Finally, the rules of mathematics – such as the rule (3) – are best referred to as descriptions of mathematical facts, and not rules proper. When one speaks of rule-following in mathematics one seems to refer to a completely different phenomenon than observing a moral norm or even applying a linguistic rule. Mathematical rules can hardly be described as being categorical (i.e., generating an obligation independent of any external normative criteria) or even hypothetical (acquiring their normative force from some non-mathematical normative considerations, such as prudential or moral).

Yet, despite the plausibility of the above analysis, it is quite easy – at least in philosophical discourse – to speak of moral rules as hypothetical or even ‘hidden’ descriptions; to consider linguistic rules as describing the correct use of language or, to the contrary, as providing one with an imperative to speak in a given way; and to discuss mathematical rules as genuine prescriptions of what to do, for instance, in order to arrive at the correct result of multiplication, prove a theorem or solve an equation. For the representatives of utilitarianism, moral principles are justified by recourse to the utility-maximization criterion, and hence are hypothetical. Expressivists, on the other hand, believe that moral rules are only expressions of human emotions, that when I say “One ought not to steal from other people” I am trying to convey the message that I find stealing displeasing or repulsing. In language, the rule “One should apply the expression ‘green’ only to green objects” may be deemed categorical in the sense that even if – due to some moral or prudential considerations – I do not follow it, I still break it. In other words, linguistic rules do have some autonomous standing vis a vis moral or prudential ones. At the same time, dictionaries may be looked at as providing us not with norms, but rather with descriptions of how words are used, or what is the statistically prevalent way of speaking. The mathematical expression a*b = b*a represents a rule as it says that each time one multiplies two numbers, an arbitrary a and an arbitrary b, one is entitled to reverse their order. Such a rule may be deemed categorical (i.e. such that its normative force lies within mathematics) or hypothetical (once one – for any reason – accepts axioms defining multiplication, one is entitled to reverse the order of the multiplied numbers).

The above, of course, is just an overview, not an exhaustive analysis. I only intended to register some intuitions and show that, despite appearances, there are arguably no sharp theoretical distinctions between types of rule-following. The very fact that a given category of rules (e.g., moral or linguistic) may be characterized in various incompatible ways, is interesting and in need of explanation. Our language pertaining to rule-following practices is rather vague and imprecise. It may seem more natural to speak of mathematical rules as descriptions of necessary facts rather than hypothetical rules; or to speak of moral rules as action-guiding categorical principles rather than ways of describing social or psychological reality. However, none of the alternatives may be rejected outright: the philosophical discussion concerning the status of moral, linguistic or mathematical rules is not far-fetched, it is a genuine dispute. Still, I am not trying to play down the differences between kinds of rules or suggest that they all belong to the same ontic category. I am rather arguing that one should recognize that there are no sharp criteria for distinguishing kinds of rules, and that this is an interesting insight which calls for further analysis.

To begin such an analysis, it is worth noticing that there are two features that various rules seem to have in common; I will refer to them as pattern and normative conditions. The pattern condition says that rules constitute patterns of behaviour, or enable one to distinguish actions which are rule-conforming from those which are rule-violating. Our moral rule, “One ought not to steal from other people”, somehow divides the universe of all actions into those which constitute instances of prohibited stealing, and those which do not. The linguistic rule “One should apply the expression ‘green’ only to green objects” somehow applies to any situation in which one is referring to an object of a particular colour. The mathematical rule “a*b = b*a” somehow singles out a particular subset of mathematical operations. There are three intriguing aspects of the pattern condition. First, rules – as constituting patterns of behaviour – must be based on a mechanism of recognizing the similarity between objects or states of affairs. To apply the word ‘green’ to the given object correctly, one needs to have the capacity to recognize ‘greenhood’ in any object, and so to establish that those objects are similar relative to the feature of being green. Second, in order to fulfil the pattern condition, rules must be applicable to a potentially infinite number of cases. Put differently, any rule must somehow ‘contain’ a potentially infinite number of its applications. For our mathematical rule “a*b = b*a” it means that it somehow relates to an infinite number of cases in which one could reverse the order of multiplied numbers. Finally, rules as patterns of conduct must be stable or projectible: they pick out as right or correct not only past or present, but also future courses of action.

Rules must also meet the normative condition: some of them may possess normative force. This metaphorical expression can be analyzed into the following two aspects: a rule that has normative force is objective (i.e., independent of an individual’s beliefs) and may serve to justify an action or a belief according to some normative criterion and given some facts. For instance, given the fact that Tom’s mother has been hospitalized, the rule prescribing that one visits one’s mother when she is seriously ill provides the justification for Tom’s visiting his mother, according to, say, the Kantian criterion of moral action. Thus, in what follows, I shall say that rules are normative when they – under some factual circumstances – justify a course of action against some selected normative criterion, and that any normative criterion determines which rules have justificatory force. [1]

Still, there are two things that must be underlined. Firstly, there are different, ‘stronger’ and ‘weaker’, normative criteria. A moral rule may justify an action if the prescribed action is considered just; a linguistic rule may prescribe that it is correct to use an expression in some particular way; a mathematical rule may determine that some mathematical operation is warranted by some necessary connection between mathematical structures. Secondly, the same rule (say, a moral norm) may justify an action according to one normative criterion, but fail to do so according to some other. For instance, the norm “whosoever shall smite thee on thy right cheek, turn to him the other also”, has justificatory force in Christian ethics but, arguably, it does not according to utilitarianism. Normative variety, i.e., the fact that there are different normative criteria, and normative relativity, i.e. the fact that the same rule may have justificatory force according to one criterion, but not according to another, should be accounted for in any successful theory of rules.

2. At the crossroads of philosophy

It is my claim that through the pattern and normative conditions, the puzzle of rule-following is intimately connected with two of the central issues of Western philosophy: the problem of universalia and particularia, and the problem of normativity. Let us begin with the former by illustrating it by means of three examples.

In Ancient philosophy, the relation between universalia and particularia was discussed under the heading of the ‘one and many’ controversy. The question was, how is it possible that different objects, say different great things, may be referred to correctly with one concept, ‘great thing’. The famous answer provided by Plato was his theory of ideas or forms: although no two great things are identical, they all participate in the same idea of ‘greatness’, and it is in the virtue of this fact that one can correctly call them great things. However, Plato was well aware of the problems this answer generates. In one of his more intriguing dialogues, Parmenides, a conversation between Parmenides and young Socrates is recorded, in which the older philosopher raises doubts as to the merits of the theory of ideas. He observes:

And if you go on and allow your mind in like manner to embrace in one view the idea of greatness and of great things which are not the idea, and to compare them, will not another greatness arise, which will appear to be the source of all these?

Socrates replies:

It would seem so.

And Parmenides concludes:

Then another idea of greatness now comes into view over and above absolute greatness, and the individuals which partake of it; and then another, over and above all these, by virtue of which they will all be great, and so each idea instead of being one will be infinitely multiplied. [2]

The problem with which Plato deals here is that of similarity: when one claims that two things may be called ‘great’ because they are similar in virtue of a certain standard or measuring rod (the idea of greatness), one is forced to ask, what is the criterion for establishing similarity between any great thing and the idea of greatness. The natural answer is that there must be another standard, over and above the idea itself, and so a kind of meta-idea. This generates an infinite regress – no matter what is our standard of comparison (idea, meta-idea, meta-meta-idea), there always is a tertius homo or third man, a higher criterion which enables one to establish that the given two objects are similar. The same problem of similarity is pervasive in the context of rules. Indeed, Plato’s question may be considered as the rule-following puzzle in (Ancient) disguise. One should use the word ‘great’ only in relation to great objects. But how do we know that any two objects share the feature of greatness, or are similar in this respect?

Another example comes from Kant’s great critical project. In Critique of Pure Reason he attempts, inter alia, to explain why mathematical judgments are synthetic (they expand our knowledge), yet a priori (independent of sensual experience). Kant’s famous answer is that mathematical judgments are based on the intuitions of space and time, which, in turn, are not aspects of reality, but of the knowing subject. We are so constructed that our perception is always a perception in time and space, as both are elements of our cognitive structure. In this way, mathematical theorems are independent of experience, but not merely analytical or devoid of meaning. [3]

For Kant, the key problem of this account lies in the fact that any representation in intuition is a concrete object. When you see or merely imagine a triangle, it is always a concrete triangle: right-angled, acute-angled or obtuse-angled. Yet, mathematical knowledge is expressed in concepts which are universally applicable: that the angles of a triangle amount to 180 degrees is true of any triangle, not just a particular one represented in an intuition. So, the question reads, how is it possible that intuitive representations of particular mathematical objects may serve to justify universal mathematical knowledge.

Kant’s answer in Critique of Pure Reason is as follows:

The individual drawn figure is empirical, and nevertheless serves to express the concept without damage to its universality, for in the case of this empirical intuition we have taken account only of the action of constructing the concept, to which many determinations, e.g., those of the magnitude of the sides and the angles, are entirely indifferent. [4]

He believes that the tertius homo, or the intermediary between concrete representations and general concepts, is the procedure of construction, which he calls a transcendental schema:

Now it is clear that there must be a third thing, which must stand in homogeneity with the category on the one hand and the appearance on the other, and makes possible the application of the former to the latter. This mediating representation must be pure (without anything empirical) and yet intellectual on the one hand, and sensible on the other. Such a representation is the transcendental schema. [5]

A transcendental schema of a triangle is, therefore, a procedure or a rule that enables one to construct a particular triangle in intuition, and the same holds for all mathematical objects. A schema “can never exist anywhere but in thought, and it signifies a rule of synthesis of the imagination with respect to pure figures in space”, [6] making it possible to exhibit universal features inherent in particular instantiations of mathematical objects:

We say that we cognize the object if we have effected synthetic unity in the manifold of intuition. But this is impossible if the intuition could not have been produced through a function of synthesis in accordance with a rule that makes the reproduction of the manifold necessary a priori and a concept in which this manifold is unified possible. Thus we think of a triangle as an object by being conscious of the composition of three straight lines in accordance with a rule according to which such an intuition can always be exhibited. [7]

This is Kant’s solution to the problem of the relationship between particular mathematical objects and general mathematical concepts. It is noteworthy that Kant is dealing here with all three aspects of the pattern condition: similarity, infinity and projectibility. He explains how we can perceive similarities between concrete triangles, as all of them are constructed with the use of the same transcendental schema; furthermore, his theory also accounts for the fact that one concept refers to potentially infinitely many instantiations, for transcendental schemata are at work in each and every act of cognition; finally, mathematical knowledge is stable and mathematical concepts projectible, again because of the transcendental character of the schemata.

It is easy to see that the problem Kant addresses, namely the nature of mathematical knowledge and mathematical concepts, may be rephrased so as to concern mathematical rules. Instead of asking how the concept of a triangle corresponds to infinitely many intuitive representations of triangles, one may ask, how does a mathematical rule such as, for example, “In order to calculate the area of a triangle one should multiply the length of the base of the triangle by its height by ½”, refer to an infinite number of triangles. More importantly, in his solution Kant suggests yet another, foundational role for rules in mathematics. His schemata are procedures, or sets of rules. In other words, Kant resolves the ‘one and many’ controversy in mathematics by recourse to the concept of a rule; he fails short, however, of explaining how those rules meet the pattern condition. [8]

The final example also begins with Kant, but this time with his moral philosophy. Kant believed that moral norms should be rational, and his conception of practical rationality is encapsulated in the categorical imperative: “Act only according to that maxim through which you can at the same time will that it should become a universal law”. [9] Thus, Kant claims that moral rules are justified if they are universalizable: in morals – or, more generally, in all practical considerations – there is no justification without universalization. However, on more than rare occasions this ideal seems unachievable. Suppose I have promised my uncle to help him paint his apartment, but my mother has had an accident and she is seriously injured. I have, therefore, a strong moral justification to visit her in the hospital, which is in conflict with my obligation to help my uncle. What would be the justified moral decision in this case? Let us agree that it would be to go to the hospital. Is there any principle that could be the base for this decision? It is, obviously, the following: “You should keep your promise to your uncle unless your mother has had an accident and you have to visit her in the hospital”. But this poses a threat to the idea of justification or universalizability:

There is no finite set of finite principles that serves to axiomatize [...] evaluation: that is, no finite set of finite principles, such that, given any action fully described in non-evaluative terms, the principles and the description entail a given evaluative verdict if and only if it is true. [10]

It follows that it is impossible to construct a complete system of moral principles. Moral decisions are not, ultimately, justified by a universal principle; they always depend on the features of particular cases, or at least this is the position defended by moral particularists. [11]

Observe, that this may be construed as a problem which does not pertain to the pattern condition. Particularists claim that even if we had a perfect grasp of the obligatory patterns of behaviour encoded in moral rules, there would be circumstances when it would be debatable which of the two applicable patterns to follow. Their argument is even stronger: such conflicts between moral obligations are so typical that there is nothing which warrants our speaking of moral rules at all – what is obligatory is primarily determined by the particular features of the case at hand. But this line of reasoning may also be so rephrased as to pertain directly to the pattern condition. What the particularists’ argument shows is that it is impossible to capture our obligations in a system of mutually independent (isolated) patterns of conduct; under particular circumstances, they will usually ‘interfere’ with one another. This fact calls for an explanation.

Let us turn now to the normative condition. The story of normativity begins with the collapse of the classical worldview, or so I will argue. [12] By the classical worldview I understand the fundamental theses that served as the building-blocks for the most influential philosophies of Antiquity and the Middle Ages. The first such thesis is that the universe has a rational structure or a logos. [13] Importantly, logos embraces not only what we call the laws of physics, but also moral principles. It is nicely illustrated in the following passage from Summa theologiae:

We must speak otherwise of the law of man, than of the eternal law which is the law of God. For the law of man extends only to rational creatures subject to man. (...) Now just as man, by such pronouncement, impresses a kind of inward principle of action on the man that is subject to him, so God imprints on the whole of nature the principles of its proper actions. And so, in this way, God is said to command the whole of nature, according to Psalm 148,6: ‘He hath made a decree, and it shall not pass away.’ And thus all actions and movements of the whole of nature are subject to the eternal law. Consequently irrational creatures are subject to the eternal law, through being moved by Divine providence; but not, as rational creatures are, through understanding the Divine commandment. [14]

This means that moral principles and laws of physics have the same ontological status: they are both elements of the structure of the universe.

The second, more specific aspect of the classical worldview, was that among the ‘principles’ governing the universe, a designated place was occupied by the final or teleological principle: things exist not only because they were generated by something else, but also towards an end. Telos explains many things, from gravity (solid bodies fall down as they have a tendency to occupy their ‘natural place’, which is the centre of Earth) to morality (men should act according to reason, as reasonable life is the human ontological end). To quote Aquinas again:

Now the rule and measure of human acts is the reason, which is the first principle of human acts (...); since it belongs to the reason to direct to the end, which is the first principle in all matters of action, according to the Philosopher. Now that which is the principle in any genus, is the rule and measure of that genus: for instance, unity in the genus of numbers, and the first movement in the genus of movements. Consequently it follows that law is something pertaining to reason. [15]

Within such a framework, normativity is not problematic. We ought to act in a certain way, because the right course of action is determined by our telos, which in turn is inscribed into our ontological nature. Moreover, from the most general perspective there is no substantial difference between human beings and any other entity: everything that exists does so for an end, and it is this end that determines the common course of action in nature (communum cursus naturae). The only thing particular to human beings is free will: we can choose to act in violation of the rules of logos, while the rest of nature follows the natural law by necessity. Nevertheless, both are explainable within the same ontological scheme.

This general picture changed with the advent of Renessaince and, later, with the birth of modern science. Compare the preceding passage with the following one, taken from Montaigne’s Essays:

Philosophers can hardly be serious when they try to introduce certainty into Law by asserting that there are so-called Natural Laws, perpetual and immutable, whose essential characteristic consist in their being imprinted upon the human race. There are said to be three such laws; or four; some say less, some say more: a sign that the mark they bear is as dubious as all the rest. [16]

This observation leads Montaigne to his famous dismissal of the transcendent sources of law and normativity, as when he says: “Laws are often made by fools, and even more often by men who fail in equity because they hate equality: but always by men, vain authorities who can resolve nothing”. [17]

The rejection of the classical conception of the natural law, generated by the Renaissance ‘turn towards humanity’, was strengthened by the subsequent arrival of modern science, which offered a new method of uncovering the laws of physics through constructing mathematical models and experimentation, but which was unable to tackle the problems of morality. This was a decisive point: the claim that there is a specific method of investigating physical reality led firmly to the belief that there are two ontologically separate spheres or domains: those of facts and those of norms and values. This development was vividly stressed by Hume in his observation that there are no logical connections between the statements of facts and obligations. It was further elaborated by Kant in his distinction of pure (theoretical) and practical reason, and paved the way for the strict separation of science and morality in the 19th and 20th centuries.

The signalling out of the normative sphere generated the question of what are the sources of normativity. Modern philosophy has provided various, often opposite answers, which may roughly be divided into two categories: the monistic and the dualistic. According to the monists, who embraced naturalism, the distinction between the normative and the factual is an illusion. Although we do engage in normative discourse, it is ultimately reducible to the theoretical or factual one. The reduction in question took different forms (from voluntarism through refined psychologism to sociological accounts), but in all its incarnations it is implied that normative notions are factual notions in disguise, and that the normative sphere is simply a fiction. It follows that morality, the law and other apparently ‘normative’ domains should be investigated with the same methods which are utilized in the sciences. Dualists, in turn, were realists regarding norms and values. They claimed that there must exist intrinsically normative entities, such as states of affairs which are ‘good in themselves’, moral principles, etc. But if so, then there exists an unbridgeable gap between Is and Ought, a gap that allows no kind of reduction. [18]

This received ontological framework, which leaves us with the choice from among two options – monism or dualism – has daring consequences for the discussion pertaining to the status of rules. On the monistic account, rules are only hidden descriptions of psychological or sociological facts. Thus, they justify no action and fail to satisfy the normative condition. The dualistic ontology, in turn, sees rules in the sphere of Ought. They justify action as if ex definitione, but exactly this is problematic: such rules are ‘queer’ entities, “unlike anything in the universe”. [19] Moreover, instead of explaining where the normative force of rules comes from, dualists only claim that rules do indeed possess such a force, which is inherent in their ontological nature. This is no explanation, but rather an abdication of philosophy in face of the normativity puzzle. Furthermore, when dualism may be defended in relation to the ontological status of moral or legal norms, it seems futile as an attempt to account for the normativity of language, logic or mathematics.

The above remarks concerning the pattern and normative conditions warrant the thesis that the problem of rules lies at the crossroads of philosophy. Rule-following is a phenomenon pertaining to two fundamental philosophical debates, which makes it a welcome candidate for detailed analysis.

* * *

In the following chapters I venture to answer a number of questions. The first reads: what does rule-following consist of? It seems that rule-following practices must accommodate the pattern condition and the normative condition, but this is only an insight, not a well-developed idea. Furthermore, one needs to ask, how is rule-following possible, or what is the mechanism underlying this human ability? The third question pertains to the phenomenon I have deemed normative variety: rules serve to justify actions according to various criteria; in consequence, there are many types of rules – moral, linguistic, legal, mathematical, prudential, etc. Do they have something in common, or rather are different kinds of entities? Still, there is normative relativity: the same rule may justify action when given one normative criterion, but fail to do so when given another. How to account for this?

The following chapters are devoted to investigating these issues. In chapter 1, by analyzing Wittgenstein’s insightful remarks on rule-following, I try to understand better what is involved in the problem under consideration. I follow the argument developed in Wittgenstein’s Philosophical Investigations in order to reject some traditional conceptions of rules, as well as to formulate a philosophical hypothesis pertaining to rule-following, namely that one should distinguish two kinds of this activity: observing a rudimentary rule and being guided by an abstract rule. Chapter 2 attempts to depict the biological mechanisms behind rule-following. I posit that there are three interconnected such mechanisms that are responsible for our rule-following practices: the ability to imitate, the tendency to imitate and unconscious decision-making. In chapter 3, I ask what are language rules, and – more particularly – I try to show that the question concerning the normativity of meaning is ill-stated, being based on an out-dated, formal view of language. I further posit that recent findings in neuroscience, evolutionary theory, primatology and linguistics strongly suggest a different conception, one of embodied and embedded language. Chapter 4 is devoted to the question of what is the contemporary picture of human morality. I argue that it largely settles the long philosophical discussion pertaining to the role of emotions and reason in moral decision-making, as well as the fact that the resulting conception of morality fits within the philosophical view of rule-following developed in this study. Chapter 5, in turn, is centred on the problem of the neuroscientific foundations of mathematical thinking. I argue that a view of mathematics which is compatible with the findings of the contemporary neuroscience enables one to speak of mathematical rules, which emerge in the same way as other kinds of rules. I posit further that there are two facts about mathematics – the necessary relations between mathematical structures and the unreasonable effectiveness of mathematics in uncovering the laws of nature – that cannot be, at least at this stage, but probably never, explained with the use of the methods of neuroscience. This observation leads to the argument of chapter 6, where I propose an ontology of rules, based on a significantly modified version of Popper’s theory of three worlds.

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Footnotes

[1] It is possible to speak of normativity in many other ways. In particular, the prevalent idiom seems to be that a rule of behaviour is normative if it generates an objective reason for action, where reasons are understood as facts which justify undertaking some action or accepting some belief (cf. J. Hage, Reasoning with Rules. An Essay on Legal Reasoning and Its Underlying Logic, Kluwer Academic Publishers: Dordrecht 1997). Thus, one may say that rules are normative because they have the power of turning some facts into reasons. However, I have decided not to embrace this popular conceptual scheme, as I consider speaking of facts as constituting reasons a contingent feature of some languages (e.g. English), while in other languages (e.g. Polish) this mode of speaking is incorrect. At the same time, I believe that the theory I develop can relatively easily be translated into the idiom that takes ‘reason’ to be the central normative notion.

[2] Plato, Parmenides, Hackett Pub.: London 1996, p. 42.

[3] Cf. I. Kant, Critique of Pure Reason, transl. by P. Guyer and A.W. Wood, Cambridge University Press: Cambridge 1998.

[4] Ibidem, A713-A714.

[5] Ibidem, A183/B177.

[6] Ibidem, A140/B180.

[7] Ibidem, A105.

[8] For more details see B. Brożek, A. Olszewski, The Mathematics of the Transcendental Ego, “Copernicus Center Reports” 2011, no. 2, pp. 75–124.

[9] I. Kant, Critique of Pure Reason, op. cit., p.78.

[10] Cf. R. Holton, Principles and Particularisms, “Aristotelian Society Suppl. Volume” 2002, vol. 76, p. 193.

[11] For a detailed analysis of this problem cf. B. Brożek, Rationality and Discourse. Towards a Normative Model of Applying Law, Wolters Kluwer: Warszawa 2007.

[12] See in more detail B. Brożek, Normatywność prawa (The Normativity of Law), Wolters Kluwer: Warszawa 2012, chapter 1.

[13] Cf. ibidem.

[14] Th. Aquinas, Summa Theologica, transl. by Fathers of the English Dominican Province, online edition, http://www.newadvent.org/​summa/​index.html, 2008, 93.5.

[15] Ibidem, 90.1.

[16] M. Montaigne, The Complete Essays, Penguin: London 1991, pp. 653–654.

[17] Ibidem, p. 1216.

[18] Cf. B. Brożek, Normatywność prawa, op. cit., chapter 1.

[19] Cf. J.L. Mackie, Ethics: Inventing Right and Wrong, Pelican Books: New York 1977.