Naturalism and Objectivity in Mathematics†

 

Feng Ye*

 

Nominalism in the philosophy of mathematics denies the objective existence of abstract mathematical entities, but intuitively it is obvious that mathematics is objective in some sense. Nominalism must then explain in what sense and on what aspects mathematics is objective, and it must explain what the basis for objectivity in mathematics is, if not the objective existence of abstract mathematical entities. I offer a nominalistic answer for these questions, based on a truly naturalistic approach to the philosophy of mathematics.

 

1. Introduction

Nominalism (or anti-realism) in the philosophy of mathematics denies the objective existence of abstract mathematical entities. (See Shapiro [2005] for surveys on the contemporary philosophy of mathematics.) However, obviously, mathematical practices are objective in many aspects. For instance, even if mathematical entities do not really exist, our mathematical knowledge should still have objective content in some proper sense. We are not making assertions out of our wishes in doing mathematics. One could wish that the Goldbach Conjecture is true, but we know that there is something objective and independent of our wishes there. A natural attempt to explain such objectivity by nominalists is to claim that correctness in following logical rules in a mathematical proof is an objective matter. However, admitting objective correctness in rule following appears to commit to rules as abstract entities and commit to objective truths about such abstract entities. In particular, remember that this physical universe could be finite and discrete. If correctness in rule following means correctness in following rules literally up to infinity, it has to commit to something not belonging to this physical universe. This is certainly the case when rules are understood as mathematical functions that can operate on infinitely many instances of arguments. Nominalists must account for objectivity in rule following without committing to rules as abstract entities.

Another obvious aspect of objectivity in mathematics is about the relationships between the mathematical and the physical. First of all, even if mathematical entities do not exist, intuitively, there must be some genuine relationships between the mathematical and the physical, in particular, in the contexts of mathematical applications. For instance, when physicists use Riemann spaces to model space-time, there must be some genuine relationship between space-time and something else that are on the mathematical side, no matter what they are and even if they are not Riemann spaces as abstract entities. Realists characterize this relationship as an ‘approximate isomorphism’ between space-time and Riemann spaces as abstract entities. (It is approximate, because differentiable Riemann manifolds model only large scale space-time structures, with micro-scale details ignored.) Nominalists must certainly account for that relationship in other ways, without assuming that Riemann spaces literally exist.

Then, the point here is that this relationship must be objective, not what scientists wishfully assert. It is not the case that scientists simply pretend that the relationship exists. For example, Hoffman’s (2004) recent exposition of fictionalism seems to imply that scientists pretend that Riemann spaces exist and are (approximately) isomorphic to space-time structures, just like kids pretend that a sofa is a mountain in playing games. For working scientists, the (approximate) structural isomorphism between Riemann spaces and real space-time structures is objective and is the objective reason for our successes in modeling space-time structures by Riemann spaces. In general, if scientists indulge in wishfully pretending things even a little bit, they will not be successful in their work. Realists certainly believe that such objective relationships between the mathematical and the physical and such objective reasons for the applicability of mathematics can only be accounted for by accepting the objective existence of abstract mathematical entities. To meet realists’ challenge, nominalists must explain what this objective relationship between the mathematical and the physical consists in and how it is the objective reason for the applicability of Riemann spaces.

In the previous papers (Ye [2007a], [2007b], [2007c]), I introduced a research project for exploring a truly naturalistic and completely scientific account for human mathematical practices. Its philosophical basis is naturalism, which means viewing human minds as functions of brains, parts of the physical world, and the results of evolution (Ye [2007b]), and viewing human mathematical practices as human brains’ cognitive activities in interacting with environments. Then, a truly naturalistic and completely scientific study of human mathematical practices will describe the cognitive functions of human mathematical mental representations in brains and their relations with things in environments (Ye [2007c]). It is also nominalistic, because it talks only about brains, neurons, mental representations supposedly realized as neural circuitries in brains, and other physical entities that brains and bodies interactive with in environments. It does not speculate about the alleged abstract entities that a brain ‘commits to’ (Ye [2007b]). In particular, it does not assume infinity in any sense, because all human brain cognitive activities are finite. They are natural phenomena within this finite universe.

This paper belongs to that research project. This paper aims at offering an account for objectivity in mathematical practices, under that naturalistic and nominalistic framework. I will start with identifying various aspects of objectivity in mathematical practices from the naturalistic point of view. Then, I will discuss the objective relationships between the mathematical and the physical, the objective basis for successfully communicating mathematical concepts among humans, objectivity in rule following, and the objective constraints for brains’ mathematical activities.

Such a naturalistic and scientific account for aspects of human mathematical practices as brains’ cognitive activities must rely on some basic assumptions on human cognitive architecture and the cognitive functions of mathematical concepts and thoughts in brains. These are given in Ye [2007c]. In the following, I will introduce relevant terminologies and briefly recapitulate those assumptions.

First, we assume a Representational Theory of Mind, which means that mental representations are the building blocks of human cognitive architecture. Mental representations include concepts and thoughts, which are expressed by nominal phrases and declarative sentences in languages respectively, and are supposedly realized as neural circuitries in brains. Some concepts (e.g. DOG) are meant to represent physical entities or their properties directly. They are called ‘realistic concepts’. This representation relation between a realistic concept and physical entities is assumed to be a natural relation between two types of physical entities, the concept as some neural circuitries in a brain and other physical objects. That is, the semantic representation relation is supposed to be naturalized. Some other concepts are not meant to represent physical entities directly. They are called ‘abstract concepts’. Mathematical concepts belong to this category. Similarly, some thoughts in brains are meant to represent physical states of affairs directly. These are realistic thoughts. With the semantic representation relation between realistic concepts and physical entities being naturalized, truth is then a naturalized correspondence relation between realistic thoughts in brains and other physical states of affairs. Note that this applies to realistic thoughts only. Abstract thoughts in brains, including mathematical thoughts, are not meant to represent physical states of affairs directly.

Abstract thoughts play other cognitive functional roles in brains. For instance, an abstract thought ‘about Riemann spaces’ that mathematicians or physicists entertain in their brains can be translated into realistic thoughts about space-time, with, for example, the abstract concepts ‘point’ and ‘line’ in the Riemann space theory translated into the realistic concepts ‘very small space-time region’ and ‘trace of a photon’ representing real physical entities. A mathematical thought, or a collection of mathematical thoughts in a mathematical theory, allows multiple ways of translating into realistic thoughts directly representing physical states of affairs. We call them ‘schemes of translations’. A collection of abstract mathematical thoughts, plus a scheme of translation, provides a way for representing our scientific knowledge about real things in the universe. This is one of the major cognitive functional roles of abstract mathematical thoughts in brains.      

A naturalistic and scientific study of human mathematical practices and mathematical applications then describes the cognitive functional roles of abstract mathematical concepts and thoughts in brains and describes their indirect relations with physical entities through translation schemes in mathematical applications. As a philosophical theory, it may ignore all psychological details and focus on aspects and issues that interest philosophers and logicians. Issues related to objectivity are among these.

 

2. Aspects of Objectivity in Mathematics 

Naturalism implies that there is no Transcendental Cognitive Subject alienated to nature. All that exist are brains as the results of evolution, on a par with other physical entities in the universe. (See Ye [2007b] for more discussions on naturalism.) Therefore, from this naturalistic point of view, we do not ask if something is subjective/objective per se, or subjective/objective relative to a unique, peculiar, Transcendental Subject not belonging to nature. We only ask if something in nature is subjective/objective relative to the mental processes in a brain (or in some brains) in nature.

Then, there are several senses in which we can say that something is objective relative to a subject’s brain:

(1) The existence and properties of an external physical object out of a brain are objective relative to the mental processes in the brain. In contrast, the existence and structures of mental representations in the brain are not.

(2) Whether or not a realistic thought in a brain truthfully represents external physical states of affairs is partially objective, since it is a naturalized correspondence relation with an objective component. The existence of such a naturalized correspondence relation for a realistic thought in a brain is objective, although the existence of the thought is not objective relative to the brain.

(3) Given a scheme of translation, whether or not a mathematical thought is translated into a true realistic thought directly representing physical states of affairs is also objective, since truth of the latter is an objective matter in the sense (2) above. However, we must remember that this is relative to a given scheme of translation, and there could be many schemes of translations for a mathematical thought (or a collection of mathematical thoughts in a mathematical theory), and different abstract thoughts can be translated into the same realistic thought by different translation schemes.

(4) Two concepts as mental particulars in two persons’ brains can have similar internal structures and functions. In that case, we may say that they are ‘the same concept’. Objectivity here means that different people can share ‘the same concept’ in some sense. This also includes the cases where the concept is a rule and then it becomes the issue of how two people can follow ‘the same rule’ and the issue of objectivity of correctness in following a rule.

(5) A sense of objectivity can also come from the awareness that we are constrained in creating and manipulating mathematical concepts in brains, given some purpose for doing so. For instance, if the purpose is to be able to truthfully represent space-time given some translation scheme or just for any possible translation schemes, then one is highly constrained in creating and manipulating geometrical concepts, because of the objectivity in (3) above. These constraints also include the fact that, to serve our purpose, we have to follow some logical inference rules in manipulating mathematical thoughts in brains.

 Finally, (6) our innate cognitive architectures may constrain how we can possibly create and manipulate our mental representations. For instance, we are not able to create a perceptual image in which something is both round and square, and we are not able to imagine directly and clearly microscopic particles with quantum effects. Objectivity here means that we are innately constrained in some way in our fundamental capabilities in creating and manipulating mental representations, or in imagining things.

These are several aspects of objectivity. Mathematics is not objective in the sense (1) and (2), since there are no external mathematical entities, and mathematical thoughts are not mean to represent external states of affairs directly. Realists’ concern that anti-realism may annihilate objectivity in mathematics is about objectivity in the sense (1) and (2). However, there is objectivity in the sense (3) for mathematical practices, in particular, in mathematical applications. Similarly, there is objectivity in the senses (4), (5), and (6).

Objectivity in (6) actually concerns with the apriority and necessity of logic and arithmetic, which will be discussed in a separate paper (Ye [2007e]). In this paper, I will focus on objectivity in the senses (3), (4) and (5) above. I will discuss what the true naturalistic basis for the kind of objectivity is and why objectivity there does not imply the objective existence of abstract mathematical entities.

 

 

 

3. The Objective Relations between the Mathematical and the Physical

From the naturalistic point of view, saying that the approximate isomorphism between Riemann spaces and space-time is objective actually means that, given the translation scheme chosen by physicists, whether or not mathematical thoughts in the Riemann space theory are translated into true realistic thoughts about space-time is an objective matter. This is objectivity in the sense (3) above. The true naturalistic objective basis for the relationship between the mathematical and the physical and for the applicability of the Riemann space theory is the objective fact that after those abstract thoughts in the Riemann space theory are translated into realistic thoughts about space-time according to the translation scheme, they become true realistic thoughts about space-time (based on the naturalized correspondence relation). This objective relation is some natural regularity in a special type of natural phenomena, namely, natural phenomena involving neural processes in brains and the naturalized correspondence relation between brains and external physical entities.

Noting the presence of translation schemes helps to clear up many misconceptions about objectivity related to explaining the applicability of mathematics, in particular, misconceptions that lead to realism about abstract entities or structures. First, the translations preserve the structures of those mathematical thoughts, not what those thoughts ‘represent’ if any. In other words, the structures of abstract thoughts in the Riemann space theory determine if the results of translations are true realistic thoughts about space-time, and whether or not those abstract thoughts directly represent any states of affairs of abstract mathematical entities is irrelevant. If the structures are right, the results of translations will be true of real physical things, no matter if those abstract thoughts directly represent anything; and if the structures are not right, the translations will result in false realistic thoughts, even if those abstract thoughts directly represent something. In the same manner, a fiction together with an appropriate translation scheme can encode our true knowledge about real things. If terms in a fictional discourse are translated into terms directly representing real things in the real world, then the structures of the thoughts in the fictional discourse determine if the results of translations are literally true of those real things in the real world. Terms in the fictional discourse do not have to refer to anything.

Secondly, note that the translations are between thoughts in scientists’ brains in the real world. Therefore, what are really relevant are the structures of those abstract thoughts in brains, not the alleged abstract structures described by those thoughts. One may speculate about the alleged abstract structures described by those abstract thoughts, but what really happens in the real world are some mental processes in scientists’ brains that result in some realistic thoughts corresponding to some physical states of affairs based on the naturalized correspondence relation. Similarly, the so-called possible structures described by those abstract thoughts are not relevant either.

Therefore, the applicability of a collection of abstract thoughts does endow value to those thoughts, but it has nothing to do with the literal truth of those thoughts and has nothing to do with the alleged abstract entities represented by those thoughts. Applicability raises a collection of abstract thoughts from a game for entertainment only, to a useful collection of representational means for representing genuine knowledge about real things in the universe, together with a translation scheme. The presence of translation schemes makes insisting on the literal truth of those abstract thoughts scientifically irrelevant, because the alleged abstract entities or structures represented by those abstract thoughts do not enter into our naturalistic description of the brains’ cognitive activities in applying mathematics. What are really relevant are the structures o those abstract thoughts.

Note that the existence of physical entities represented by realistic thoughts is relevant in our naturalistic explanation of human cognitive activities, because physical entities are connected with realistic thoughts in brains representing them by natural connections, which are in the end based on causal and other physical connections. True realistic thoughts in brains instruct brains to control bodies to grab some represented physical entities as foods, for instance. Therefore, the literal existence of physical entities represented is certainly relevant in describing human cognitive activities.

On the other side, note that two different collections of abstract thoughts may represent the same set of physical states of affairs when combined with different translation schemes. It means that brains have some latitude of freedom in choosing which collection of abstract thoughts together with which translation scheme in representing physical states of affairs. The story about the choices between ‘2’, ‘{{Æ}}’, or ‘{Æ, {Æ}}’ is well-known (Benacerraf [1965]). One translation scheme may translate ‘2’ into ‘2-inches’, representing a physics property, and another may translate ‘{{Æ}}’ into ‘2-inches’, and still another may translate ‘{Æ, {Æ}}’ into ‘2-inches’. They will result in the same set of realistic thoughts if the structures of the original abstract thoughts are appropriate. This latitude of freedom entices some people to claim that mathematics is a free creation of minds. If one stays within doing mathematics, that is, creating and manipulating abstract concepts and thoughts without paying attention to their possible connections with physical entities out of brains, this is true (except for the objective constraints in the sense (5) and (6) above). However, fixing a translation scheme, it becomes an objective matter if those abstract thoughts are translated into true realistic thoughts about physical states of affairs.

Our brains have ingenious inventions to allow them to represent physical states of affairs in abstract and very flexible ways. When one takes the subjective stance and tries to project abstract concepts and thoughts in one’s own mind onto the external, then one is naturally enticed to believe that those abstract concepts and thoughts directly represent the alleged abstract entities. However, as objective naturalistic observers looking into a human subject’s brain, we see only the cognitive functional roles of those abstract concepts and thoughts in the brain and the objective relationships between those abstract thoughts and other physical entities out of the brain. If details are filled in for that scientific description, then there is simply no more to explain. Insisting that those abstract concepts correspond to the alleged abstract entities will only unnecessarily add an insurmountable task for us naturalists, namely, the task of offering a naturalistic characterization of the correspondence relation between brains and abstract entities. There is no obvious way to do this (Ye [2007b]) and it does not add any real scientific value to our scientific description of human brains’ cognitive activities.

 

4. The Objective Basis for Successful Communications

A well-known Fregean criticism on psychologism claims that there must be objective, public concepts as abstract entities independent of each individual’s mind so that we can grasp ‘the same concept’ and can communicate successfully. There are two issues here.

First, it is intuitively valid to claim that two people can have ‘the same concept’ in some sense. The genuine task from the naturalistic point of view is to characterize an equivalence relation among concept instances as mental particulars existing at various moments in all brains, so that equivalent instances are intuitively the instances of ‘the same concept’. This is the issue of concept individuation in a naturalistic theory of concept. I discussed it in Ye [2007d] and will not repeat the details here. I only want to mention that the Fregean faith in a public concept as an abstract entity is not just an illusion. It is scientifically and technically misleading for a scientific description of the social level phenomena regarding concepts in human brains and their mutual influences in communications.

For instance, the fact of matter is that what are associated with the word ‘carburetor’ in all brains vary greatly, from an expert to someone who knows only that ‘carburetor’ means a part in the old cars, and people can influence each other in communications regarding what are associated with the word in their brains. Then, from the scientific point of view, we might want to model the phenomena by a weighted set of entities, each representing a concept expressed by the word ‘carburetor’ in an individual person’s brain, with the weights to model the fact that some people (e.g. the experts) have bigger social wide influence in communications with the word. Moreover, we might want to add some dynamic structure, such as some game theoretical structure, to simulate the fact that people can influence each other regarding what are associated with the word in their brains in their social communications. If the alleged unique, public Fregean concept expressed by the word ‘carburetor’ is used to model the real phenomena, it is obviously scientifically too naïve and too simplistic. Otherwise, it is scientifically irrelevant. As long as the more complex model accurately simulates what are in people’s brains and how they influence each other in communications, the is no more to say from the scientific point of view. The Fregean idea again comes from taking the subjective stance and trying to project what are in one’s own mind onto the external. It is not a scientific observation.

Second, there is the issue of explaining the objective basis for the successes of communications. The straightforward answer is that the true objective basis for the successful communications between humans regarding their concepts is the fact that humans share the same innate cognitive architecture as a result of evolution. This, together with social trainings regarding language uses, makes successful communications possible. Sharing the same innate cognitive architecture is obviously more fundamental here, because social trainings must rely on it to be possible.

For instance, consider how humans learn to recognize the tokens of the letter ‘a’ in various fonts, including hand-written formats, or learn to recognize the tokens of a word. By some innate neural structures and some training experiences, humans memorize the perceptual visual images of some of these tokens and make them mutually connected in memory as the perceptual memories ‘of the same letter or word’. Then, in seeing a new token, brains match the newly received visual image with what are stored in memory. Recognizing a token as a token of the letter ‘a’, or as belonging to the letter type ‘a’, only means that brain neurons process the new visual image and match it with some images in the collection of mutually connected visual images ‘of the same letter’ in memory. From the scientific point of view, a research into how humans can consistently recognize letters and words (and how some animals cannot do that even with trainings), we should look into the innate structures of human brains (and how they differ from the innate structures of some animals’ brains), and describe the natural regularities of human brains (or the lack of some natural regularities among other animals’ brains) that enable human brains to work in the way they do. The true objective basis for the fact that human brains have the ability to recognize words consistently is some innate human brain structures, and the true objective reason for the fact that some animals’ brains do not have that ability is also the lack of some innate brain structures. Assuming that there is an abstract entity as a letter type (vs. tokens) is apparently irrelevant for a scientific explanation of these natural phenomena.

Similarly, consider how humans communicate their concept PEGASUS to others. It seems that one creates a perceptual mental image associated with the word by merging one’s perceptual images of birds and horses in one’s memory, based on one’s understanding of a description associated with the word ‘Pegasus’. The true objective basis for successful communications here is the fact that human brains have similar abilities in creating and manipulating mental representations, including doing so in responding to external linguistic stimuli (i.e. hearing a description of Pegasus).

Perhaps, the most basic concepts must directly represent natural classes or natural properties of external things. Then, as a result of evolution, human brains have some innate structures that assure that humans have consistent innate tendencies in recognizing some patterns, so that communications among humans are in the end possible. For instance, innate brain structures that allow human brains to consistently recognize the visual patterns of the tokens of a letter or the aural patterns of the tokens of a phoneme should belong to these, as well as innate brain structures that allow human brains to consistently recognize familiar physical objects. However, besides similarity among human brains, the external things as objective anchors for interpersonal communications here are concrete external physical things, which work as objective anchors through their causal or other natural connections with human brains. They are not the alleged public concepts as abstract entities. Then, when humans communicate their imaginations ‘about Pegasus’, the objective anchors are still concrete external physical things, including word tokens in the description of Pegasus and other concrete things such as birds and horses as the objective anchors for communicating concepts BIRD and HORSE expressed by those word tokens. Assuming a Fregean concept of ‘Pegasus’ as a public, abstract entity is again irrelevant for a scientific explanation of how human brains actually communicate their imaginations successfully.

The same is true for communicating concepts in mathematics. A vague concept, a not clearly defined or half-defined concept, and even a concept that causes paradoxes can be communicated by one mathematician’s brain to another’s, but a clearly defined concept may not be communicable to a brain that is not ‘mathematically mature enough’. I am not sure if vague concepts, half-defined concepts and paradox-causing concepts all exist as abstract entities according to conceptual realism, but obviously, if we look at things from the scientific point of view and see human communication activities as natural phenomena involving brains and their physical communications, then the really scientifically relevant assumptions for explaining the phenomena are assumptions about brains, their structures and natural regularities, and as the results, their consistent tendency in processing communication signals. Assumptions about the alleged public concepts as abstract entities will not have any position in a scientific explanation.

Actually, assuming a Fregean concept as a public, abstract entity only adds troubles, namely, the trouble of offering a naturalistic and scientific description of how human brains recognize or ‘grasp’ that Fregean concept ‘out of brains’. Apparently, it comes from taking the subjective stance and trying to project what are in one’s own mind onto the external, not from an objective and naturalistic observer’s point of view.

 

5. Objectivity in Rule Following

Now, consider the issue of following a rule or two people’s following the same rule. While sharing ‘the same concept’ is only about similarities among people’s individual concepts in their brains, following a rule involves another dimension, unique projectibility to new things. There are many issues related to rule following. Here we are only interested in two questions. What is the true objective, naturalistic basis for people’s following the same rule? Does admitting objective correctness in rule following commit us to rules as abstract entities?

Consider the well-known example of PLUS and QUUS by Kripke [1982]. For a computer, observing a finite number of input-output correlations generated by the computer will not uniquely determine which function is computed by the computer either. It is actually determined by which internal program is controlling the operations, and which function a given program computes is in turn determined by the internal structure of the program and by the mechanical and electric-magnetic regularities of CPU and other hardware components. Note that, as naturalists, we assume that there are such objective natural regularities among physical things. Our question here is whether or not admitting the objective meaning of a rule, or objective correctness in rule following, will commit us to a rule as an abstract entity, or instead normativity in following a rule can be naturalized, that is, reduced to natural regularities among physical things.

For computers, errors in following a rule can happen in two senses. First, there can be hardware malfunctioning, which means the breaking of mechanical and electric-magnetic regularities and is a purely statistical abnormality. Second, a program can be an error relative to a specification. A specification can also be seen as a program (e.g. an inefficient program doing exhaustive search and verification). Then, errors in this second sense only mean that one program fails to output the same result as another program, where for both programs, what they compute are still determined by the internal structures of the programs and by the mechanical and electric-magnetic regularities of hardware components.

Similarly, the internal structure of neurons realizing a rule concept (as a mental particular) that is controlling a person’s motor actions in doing calculations determines which function is being calculated by the person, and this is determined by human physiological and neurological regularities of those neurons and their controls on bodies among biologically normal humans. Errors in following a rule can happen in the sense that a person’s neurons and bodies malfunction in the statistical sense, relative to what they function in most biologically normal situations. Errors can also happen in another sense, that is, one’s internal rule concept expressed by a linguistic term (e.g. ‘addition’) computes a different function (i.e. quus) from those computed by the rule concepts expressed by the same linguistic term in other people’s brains (i.e. plus). In that case, for both that person and other people, what they compute is still determined by the structures of concepts that actually exist in their brains and by the physiological and neurological regularities by which those concepts (as neural structures) control the bodies’ counting motor actions. Error in this second sense is what we call a deviation from the social norm. However, the social norm here is nothing mysterious, and nor does it assume the objective existence of any abstract entities. It supervenes on what are in all people’s brains and it is explained on the basis of physiological and neurological regularities. It is naturalized.

These explain which rule, ‘plus’ or ‘quus’, one is actually following, as an objective fact of matter. As for how one can know what are inside others’ brains in order to know if others are following the same rule as one does, it is a separate issue. We can usually make very reliable guesses about what are in others’ brains based on observing their very limited external behaviors. That is not surprising, since evolution guarantees that humans have this ability of correctly guessing others’ behavior patterns (controlled by their brains) so that they can form a society. Reliability is guaranteed by the fact that humans have the same innate brain structure and have very similar developmental environments. The result is that they have very similar natural propensities in developing some kind of inner states because of some kind of external stimuli, and they have very similar natural propensities in associating what kind of external behaviors with a particular kind of inner states. Then, by observing some very scant external behaviors on a human subject, we can reliable guess what extremely complex inner states are in the subject’s brain and what sophisticated following up behaviors will occur. 

In particular, it seems that evolution guarantees that humans have the same natural tendency to develop similar basic level concepts representing the basic level regularities in nature. For instance, concepts GREEN and BLUE are perhaps innate and are directly realized by some special types of neural circuitries connected with some special types of retina cells, directly controlled by human genes. These are basic level concepts. On the other hand, the concept GRUE has to be expressed by a linguistic expression, memorized as a linguistic expression, and learned through linguistic communications. Humans have the same natural tendency to form concepts GREEN and BLUE, not GRUE, in observing external things. Similarly, humans have the same natural tendency to create PLUS, not QUUS, in their minds in learning how to count in their normal learning experiences. This common natural tendency is the objective basis for their sharing the same basic level concepts and rules and for guessing which rules others are following. It means that, in ordinary situations, after observing a human subject who is adding numbers for a few instances, we can quite reliably decide that the rule concept that is in the subject’s brain and is controlling the subject’s action is PLUS not QUUS. We can be wrong, but the assertion has a high degree of reliability as a scientific assertion. 

Note that basic level concepts and rules are usually lexical concepts, expressed by single words without explicit linguistic definitions in the natural languages. More complex concepts and rules are usually expressed by complex linguistic expressions and are supervened on basic level ones. Sharing the same concept GRUE or QUUS means memorizing the same linguistic expressions and sharing the same basic level component concepts expressed by each linguistic item in the expression. To know if two people have the same complex rules expressed by a name, we examine if they have the same (or equivalent) linguistic expressions associated with the name in their brains.

Moreover, note that regularities that guarantee humans having the same concepts or following the same rule are regularities among concrete things in the universe, among real humans and their real physiological-neurological regularities. They literally do not extend to infinity. For instance, if one asks if some gigantic computer in the universe will generate some big output, the literally correct answer might actually be that such a gigantic computer in the universe will not be able to sustain its own gravity and will collapse into a black hole immediately. The answer should be similar if one asks if a human can follow a rule up to infinity. On the other hand, when we talk about Turing machines with unlimited physical memory, for instance, we are actually imagining things. We use expressions such as ‘forever’, ‘for every item in the sequence there is another item such that …’ and so on in manipulating and communicating our imaginations ‘about infinity’. The mental representations in such imagination activities are what really exist, together with the fact that within some finite extend we can translate our imaginations into literally true realistic thoughts about real things in this universe by some appropriate translation schemes. However, with our knowledge in modern physics, we are aware that real things far away from us (e.g. quarks, superstrings, or the universe) can be quite beyond our imaginations. Science never literally generalizes its assertions to infinity. Therefore, in the realistic sense, we do not have to admit that objectivity in rule following generalizes to infinity, and we do not have to conceive of a rule as a mathematical function that applies to infinitely many instances.

In summary, which rule a brain is following is determined by the neural structure that is inside the brain and is controlling the behavior and by the neural-physiological regularity of the neural structure’ controlling bodies among biologically normal humans. Whether or not two people are following the same rule is objectively determined by these natural entities in brains and natural regularities. As for why people tend to have the same basic rules in their brains, we can refer to the fact that people’s brains have the same innate structure and similar developmental environments. Then, to know if two people have the same complex rule in their brains, we examine their linguistic expressions for expressing the rules. Finally, to say that a person is following a rule expressed by a name correctly is to say that the person has the same rule in his or her brain expressed by the name as other people in the community have. None of these commits to a rule as an abstract entity. In particular, the regularity that determines how a neural structure will control bodies in the future is a regularity of real things in the universe and it does not extend to infinity. Therefore, none of these commits to an objective infinity.

We can also see that assuming that a rule is an abstract entity does not really help in all these. One still has to describe in what sense we can say that a brain is ‘grasping’ this but not that rule. It is conceivable that one still has to refer to neural structures in brains and neural-physiological regularities in explaining this. That is, neural-physiological regularities guarantee that what exists in a brain implements this but not that rule. Then, calling the situation ‘grasping the rule as an abstract entity’ does not add any real scientific value to the explanation. What really matter are still the relevant neural structures and neural-physiological regularities that are controlling bodies.

 

6. Objective Constraints on Imagination Activities

Creating and manipulating abstract mathematical concepts and thoughts in brains is like imagining things. In particular, I have mentioned above that we are imagining things when we talk ‘about infinite Turing machines’. There are indeed objective constraints on how we can imagine things or how we should imagine things given our purpose. This is objectivity in the sense (5) above.

For instance, given our purposes and given our innate imagination capability, there may be a single most natural way to imagination simple infinity (a simply infinite sequence such as the sequence of natural numbers) as a straightforward generalization of some regular finite pattern. Mathematical induction is one of the rules that we follow in imagining simple infinity. Imagining otherwise would look ad hoc. This may also be the reason why people tend to think that the universal laws in arithmetic, such as the universal commutative law of addition, are objective.

However, being constrained in imagining something does not imply that ‘the thing one imagines’ must be real. For naturalists, what are real are not so closely related to what brains can entertain. Real things are things in this universe, of which brains and bodies are its parts. Brains cannot clearly imagine quantum particles, but quantum particles really exist in the universe. Brains can vaguely imagine an infinite universe, or imagine time before the Big Bang, but some mental representations created in imagining these in brains may not correspond to real things out of brains. Even if brains are uniquely constrained in its way to imagine infinity, it does not automatically make mental representations created in imagining infinity correspond to anything external. Besides, apparently, constraints do not uniquely constrain how we can imagine infinity. For instance, different ways of extending the set theory ZFC may all look natural.

Some philosophers resort to assumptions about infinity of the physical universe to save objective literal truth values for universal arithmetic statements (Field [1998]). This seems to put the priority upside-down for a nominalist and naturalist. If the universe is indeed finite and discrete, then, in this universe, no rules can really be followed for arbitrarily large steps, and no operations can really be repeated infinitely, and there are literally no unlimitedly large physical quantities. Objectivity ‘involving infinity’ should only be objectivity in our constrained manner of imagining infinity. A nominalist’s job should then be explaining how such imaginations ‘about infinity’ are applicable to a finite and discrete world. The motivation to save objective literal truth values for universal arithmetic statements, such as the universal commutative law of addition, can only be a motivation to defend Platonist truths beyond and above this concrete world.

There is a more subtle type of attempts to project inner things onto the external because of such objectivity. When one is aware that one is highly constrained in creating concepts (as mental particulars) for some purpose, one may believe that there is a unique set of concepts that is best for that purpose. This belief might be correct. It actually says that given human brain architectures, creating so and so concepts (which may include our concepts ‘involving infinity’) is the best way for humans to cope with this (finite) world. This is in itself a scientific assertion about human brains and it is perfectly meaningful scientifically.

However, one must also be aware that what is the best for some purpose given human brain architectures may change when external circumstances change. For instance, imagining infinity as we do it now may be the best way for humans to model physics phenomena so far, but with more and more powerful computers available, it could happen that another kind of language that directly refers to computer simulations becomes equally efficient or even more efficient for humans for that purpose. That may require a different kind of imaginations. Then, when one claims that classical mathematics consists of objective conceptual truths, one is trying to elevate the classical mathematical conceptual scheme into some sort of uniquely correct conceptual scheme. The consequence could be some dogmatism that actually hinders humans from exploring new imaginations, or new conceptual schemes and new cognitive strategies, when the external circumstances have changed.

 

Reference

Benacerraf, P. [1965]: ‘What numbers could not be’. Philosophical Review 74, pp. 47–73.

Field, H. [1998]: ‘Which Undecidable Mathematical Sentences Have Determinate Truth Values?’ in H. G. Dales and G. Oliveri (ed.), Truth in Mathematics, Oxford University Press.

Hoffman, S. [2004]: ‘Kitcher, Ideal Agents, and Fictionalism’, Philosophia Mathematica (3) 12, 3-17.

Kripke, S. [1982]: Wittgenstein on Rules and Private Language. Cambridge: Harvard University Press.

Shapiro (ed.) [2005]: The Oxford Handbook of Philosophy of Mathematics and Logic, Oxford: Oxford University Press.

Ye, F. [2007a]: ‘What Anti-realism in Philosophy of Mathematics Must Offer’, available online at http://www.phil.pku.edu.cn/cllc/people/fengye/index_en.html.

Ye, F. [2007b]: ‘Naturalism and Abstract Entities’, ibid.

Ye, F. [2007c]: ‘On What Really Exist in Mathematics’, ibid.

Ye, F. [2007d]: ‘On Some Puzzles about Concepts’, ibid.

Ye, F. [2007e]: ‘Naturalism and the Apriority of Logic and Arithmetic’, ibid.