Naturalism and Abstract Entities

 

Feng Ye*

 

I argue that the most popular versions of naturalism imply nominalism in philosophy of mathematics. In particular, there is a conflict in Quine’s philosophy between naturalism and realism in mathematics. My strategy is to explore the consequences of naturalism on the nature of human cognitive subjects and then argue that these consequences imply that assuming the existence of abstract entities is superfluous for a naturalistic account of human cognitive activities including mathematical practices. I will also argue that the indispensability of classical mathematics for scientific applications, methodological naturalism within mathematics, disquotational reference and truth, and holism cannot save abstract entities under naturalism. This argument differs from the Benacerrafian arguments against realism. It does not rely on any conception of reference or knowledge. It differs from current criticisms on the indispensability argument for realism as well. It does not deny indispensability or holism. This argument motivates a new, radically naturalistic and nominalistic approach to philosophy of mathematics.

 

1. Introduction

The epistemological difficulty for realism in philosophy of mathematics says that even if abstract mathematical entities exist, we cannot know them, because they do not exist in space-time and they are causally inert (Benacerraf 1973). The difficulty formulated by Benacerraf assumes a causal theory of knowledge, which may be too stringent, and therefore other Benacerrafian arguments against realism have also been proposed. However, some philosophers are apparently not convinced (Baker 2005, Colyvan 1999, 2002, Rosen and Burgess 2005). The problem seems to be that Benacerrafian arguments have to rely on assumptions about the limitation of our capacity to refer to or know something. Reference and knowledge are complex and controversial philosophical concepts. Therefore, sympathizers of realism can always question some of the assumptions about reference or knowledge made by Benacerrafian arguments. Besides, some philosophers seem to believe that holism, disquotational reference and truth, or methodological naturalism within mathematics can explain our epistemic access to mathematical entities.

This paper presents another argument against realism in mathematics. It does not rely on any specific assumption about the nature of reference or knowledge. Instead, it assumes naturalism and relies on a consequence of naturalism regarding the nature of cognitive subjects, that is, physicalism about cognitive subjects. I will first argue in Section 2 that if physicalism about cognitive subjects is true, then assuming the existence of abstract entities is superfluous for an account of human cognitive activities including mathematical practices and applications. Some immediate doubts about the argument will also be addressed there. Then, the rest of the paper will elaborate the argument and address other potential objections. I will explain which versions of naturalism support the argument, why people are unaware of the conflict between naturalism and realism in mathematics, and why methodological naturalism within mathematics, disquotational reference and truth, and holism cannot save abstract entities. I will also mention how this argument differs from common criticisms on the indispensability argument for realism.

 

2. From Physicalism about Cognitive Subjects to Nominalism

Consider the following thesis which I will call Physicalism about Cognitive Subjects:

Human cognitive subjects are brains as physical systems, and human cognitive processes are ultimately physical processes.

This is a consequence of physicalism as a general metaphysical position. Some influential philosophers including Quine hold both physicalism and realism in mathematics and this is still a popular position today. Therefore, an argument against realism assuming Physicalism about Cognitive Subjects has some value. Alternatives to Physicalism about Cognitive Subjects include views that take human cognitive subjects to be immaterial minds, souls, or transcendental egos in the Kantian philosophy. I will discuss different versions of naturalism and physicalism later. For now, let us assume Physicalism about Cognitive Subjects and let us understand it as reductive physicalism, which means that mental properties and processes are in principle reducible to physical properties and processes. Most philosophers agree that chemical, physiological, and neural properties and processes are in principle reducible to physical properties and processes. Therefore, the essence of this thesis is that mental properties and processes are in principle reducible to neural properties and processes. Moreover, I understand reduction in the ordinary scientific sense, for instance, in the sense that there are (a posteriori) scientific laws that connect a chemical property of water with the physical properties of neutrons, protons, and electrons. In particular, I do not mean analytic reduction, which implies that a chemistry predicate of water has the same meaning as a very complex physics description of neutrons, protons, and electrons. 

Now, consider a brain B that is doing mathematics and applying mathematics to physical things in a physics lab. Imagine that someday scientists can describe all the details of neural activities in B, as well as the details of other physical things in the lab. Perhaps real humans will never be able to do this, but we can imagine that an ideal human agent can do it. This will be a complete physicalistic description of everything in the physical world related to doing and applying mathematics by B in that scenario. B is using mathematical terms, but in this physicalistic description, we never need to say which abstract entities those mathematical terms as ink marks written by B or as neural circuitries in B (associated with those terms as their memories) refer to or semantically represent. Actually, we do not use any semantic notion in this complete physicalistic description. We just describe the structures of those neural circuitries and how they interact with other neural circuitries and other physical things in the lab.

One may think that this claim is trivial, since semantic notions such as reference are not physicalistic notions and they certainly do not appear in a physicalistic description of the cognitive activities of brains. One may further think that ‘referring to an abstract entity’ is a mental predicate, and that is why it does not appear in any physicalistic description. Yes, under physicalism this is trivial. However, firstly, if cognitive subjects were immaterial minds or transcendental egos, then some semantic notions would perhaps be among the irreducible fundamental notions in a description of the cognitive activities of the subjects. It is Physicalism about Cognitive Subjects that makes semantic notions or mental predicates superfluous. Secondly, many self-proclaimed physicalists seem to be unaware of this consequence of physicalism. They still use semantic notions as strictly irreducible fundamental notions in their philosophy, and the reason appears to be exactly that they are not serious about Physicalism about Cognitive Subjects. Unconsciously, they still take cognitive subjects to be immaterial minds or transcendental egos. My argumentation strategy in this paper is just to push these two points.

On the other hand, one may also contend that we have to use mathematical terms in describing neural activities and other physical things, and therefore we still have to refer to mathematical entities anyway. Similarly, one may contend that although we do not say which abstract entity a neural circuitry S in B ‘refers to’, we do say something like ‘the number of neurons in S = 100’, and therefore we do mention other relations between neurons and abstract entities. This is a misunderstanding. We are also brains like B, and our descriptions of B and other physical things in the lab are also physical things, such as ink marks written down by our hands or neural circuitries in our brains. They interact with other neural circuitries in our brains and other physical things in the environments in the same way as neural circuitries in B do. As physicalists, we similarly do not ask what our mathematical terms, including the term ‘100’ in the sentence ‘the number of neurons in S = 100’, ‘refer to’, just as we do not ask what the mathematical terms used by B, or the neural circuitry S inside B, ‘refer to’.

Similarly, one may contend that mathematicians accept the sentence ‘there exist prime numbers greater than 1000’, and therefore they are committed to the existence of numbers. However, accepting a sentence is a physical event of some neural activities in a brain reacting to the sentence as a physical thing. As physicalists, we do not ask what the sentence ‘means’ or if it ‘means that numbers exist’. A sentence is just a physical thing that is physically connected with other physical things, mediated by brains sometimes. Accepting the sentence ‘there exist tables in this room’ may cause the brain to control the eyes to look around for tables, but accepting the sentence ‘there exist prime numbers greater than 1000’ will immediately cause the brain to conduct other neural activities of doing mathematical inferences only.  (Of course, these neural activities may in turn cause the body to interact with the environment in some way.) Therefore, accepting the sentence ‘there exist prime numbers greater than 1000’ by a brain has nothing to do with alleged numbers as entities.

Then, as honest physicalists, we naturally hold nominalism in the following sense. We use mathematical terms, but they are physical things interacting with other physical things, and we do not ask what they ‘refer to’. We accept the sentence ‘there exist prime numbers greater than 1000’, but this again is a neural event and has nothing to do with the alleged ‘accepting the existence of numbers’. A critical point here is that we should treat ourselves as physical systems as well. We should not pretend that other people are merely brains as physical systems and their words are merely ink marks or neural circuitries that are physically connected with other physical things, but we ourselves are somehow exceptions and our words can somehow magically ‘refer to’ or ‘mean’ abstract entities. I will comeback to this point in Section 4.

Moreover, the indispensability of mathematics for science does not affect this conclusion. Some philosophers object to the indispensability argument for realism in mathematics by trying to show that the apparent references to abstract mathematical entities in the sciences are not strictly indispensable (e.g. Field 1980; Chihara 2005; Hellman 2005). Some supporters of the indispensability argument retort that pragmatic indispensability is enough to confirm the existence of mathematical entities (Baker 2005, Colyvan 1999, 2002). However, for physicalists, indispensability can only mean that some type of neural activities, namely, neural activities in doing and applying classical mathematics (vs. a nominalistic version of mathematics), is strictly (or pragmatically) indispensable for brains to do science. It has nothing to do with abstract entities.

One may suspect that the essence of this argument is that physicalism cannot accommodate semantic notions such as reference. Then, this is a challenge to physicalism in general, and as an argument from physicalism to nominalism, it says nothing new. This is a misunderstanding. This argument does not depend on whether physicalism can accommodate semantic notions. More specifically, there are three possible outcomes regarding semantic notions under physicalism: (1) There can be a physicalistic theory of semantic reference as a relation between brains and other physical things; (2) The notion of semantic reference cannot fit into physicalism but physicalism is still valid; (3) Physicalism is wrong. I do not consider the outcome (3), since the goal here is to argue that physicalism implies nominalism. The outcome (1) means that the word ‘table’ used by B (as an ink mark or neural circuitry) bears a special physical connection with tables in the lab and we can describe the details of this physical connection in physicalistic terms, and then we can define the semantic reference relation ‘refer to’ by this physicalistic description. However, mathematical terms do not refer to anything in this sense, because nothing in the environment bears the same kind of physical connection with a mathematical term. One may wonder if we can have a physicalistic theory of the reference relation between brains and abstract entities, but the argument above says exactly that this is superfluous for a physicalistic description of human cognitive activities. As for the outcome (2), it actually means that the alleged semantic reference relation is an illusion. That is, one has to hold eliminativism regarding semantic notions under physicalism. Then, some other physical connections between brains and physical things in the environments must be what are really relevant in the cognitive activities of brains. For instance, some philosophers suggest that a reliable indication relation between brains and things in the environments is such a connection (cf. Maddy 2007). In other words, physical things in the environments are physically connected with brains in many ways even if none of them is the semantic reference relation. However, if the semantic reference relation between brains and abstract entities is an illusion and our mathematical terms do not refer to anything, assuming the existence of abstract entities is completely gratuitous. Therefore, this argument says actually that no matter what stance on the issue of semantic notions you take, as long as you do hold physicalism, you should also hold nominalism in order to be consistent with yourself.

One may contend that what really exist are abstract mathematical concepts, but not abstract mathematical entities. However, to deserve the name ‘realism’, naturalistic realism has to hold that abstract mathematical concepts are independent of human brains in some way. Brains can indeed create concepts in a different sense, that is, concepts as neural circuitries. However, after we have a complete physicalistic description of the functions of these neural circuitries, it is again gratuitous to assume that there are abstract concepts independent of the brains and that brains can somehow ‘grasp’ those brain-independent abstract concepts.

One may insist that there must be a brain-independent concept or meaning as a public and abstract entity, so that two brains can mean the same thing by the same word and can communicate successfully. However, we usually do not think that a punctuation mark such as a comma expresses a brain-independent (or mind-independent) concept or meaning as a public and abstract entity, although we also agree that a comma ‘means the same’ for two brains, by which we actually mean that two brains process a comma in a similar manner in their language parsing and understanding processes. The same is true for other linguistic symbols, from punctuation marks to articles, prepositions, or nouns. A word means the same for two brains not because it mysteriously ‘expresses’ any brain-independent concept or meaning as a public and abstract entity, but because different brains, because of their similar innate architectures, language parsing and understanding functions, and memories, react to tokens of the word in similar ways.

Similarly, a brain recognizes some letter tokens to be sufficiently similar to ‘a’ and classifies them into the same category, which means associating them with the same piece of memory (the memory of ‘a’) in the brain. That is how the idea of a letter type comes about. The pattern recognition functions of brains determine which tokens are sufficiently similar to ‘a’ and are therefore tokens of the same letter type. Assuming that there exists the letter type ‘a’ as a public and abstract entity and using this assumption to explain why those tokens are tokens of ‘a’ is putting the carriage before the horse. Talks about types should be considered a simplified and figurative manner of speech. (See Ye (online-b) for more on objectivity under naturalism.)

In this paper, I will talk about linguistic terms and sentences freely, by which I will mean similar tokens recognized and classified by brains. Moreover, sometimes I will talk as if the terms and sentences used by a brain are themselves inside the brain. There are two ways to realize this. One is to assume that there are memories of those terms and sentences as neural circuitries in the brain. The other is to consider an ‘extended brain’ that contains a normal human brain plus some papers or books, printed with linguistic terms and sentences and used by the brain to assist its work. Then, we can take human cognitive activities to be activities inside this ‘extended brain’ and interactions between this ‘extended brain’ and its environments.   

The above is an argument from Physicalism about Cognitive Subjects to nominalism in philosophy of mathematics. I have considered some simple and immediate doubts and objections. The rest of this paper will address other potential doubts and objections.

 

3. Varieties of Naturalism and Abstract Entities

There are many versions of naturalism. First, there are methodological naturalism and ontological naturalism (Papineau 2007). Then, under ontological naturalism, there are physicalism and property dualism, and physicalism further includes reductive and non-reductive physicalism (Stoljar 2001). This section will discuss which versions of naturalism imply Physicalism about Cognitive Subjects and what will happen if cognitive subjects are a little more than physical systems, for instance, if cognitive subjects are brains with mental properties that are not reducible to physical properties even in principle. Since no philosophers seem to include substance dualism under naturalism, I will not discuss substance dualism here.   

First, methodological naturalism holds that scientific methods are the best methods for obtaining knowledge (e.g. Maddy 2007). Now, contemporary science describes humans as material beings and results of evolution. No mainstream working scientists today are studying immaterial minds, souls, or the so-called transcendental egos. Therefore, methodological naturalism implies that human cognitive subjects are material things and cognitive processes are ultimately material processes. It seems that all naturalists today accept methodological naturalism. However, some of them believe that the relationship between the mental and neural properties of a brain is still undecided by the sciences. Different versions of ontological naturalism then try to give different answers.

Reductive physicalism claims that mental properties are in principle reducible to neural properties. This directly implies Physicalism about Cognitive Subjects and the argument in the last section is based on this.

The most popular version of non-reductive physicalism is Davison’s anomalous monism. It claims that (1) a mental event token is identical with a physical event token, and (2) a mental property supervenes on physical states, that is, no two brains with exactly the same complete physical state can have different mental properties, and (3) we can recognize our own mental properties by introspection or guess the mental properties of other brains by their behaviours, but there is no strict law for deriving the mental properties of a brain from its complete physical state (Yalowitz 2005). The last point also implies that we cannot express the condition classifying the physical states instantiating a mental property in physical terms. Quine seems to endorse this version of physicalism (Quine 1995, p.87). Under anomalous monism, cognitive subjects are still physical things and cognitive processes are still physical processes ultimately. Therefore, abstract entities are irrelevant to human cognitive activities at the physical level. However, one may contend that a mental property of brains is ‘is referring to 9’. It is a relation between brains and an abstract entity at the mental level, and it supervenes on non-mental properties and relations of physical and abstract entities. Is this interpretation of anomalous monism coherent? Is it the best interpretation?

To answer these questions, first note that there is a nominalistic reading of anomalous monism. It claims that mental properties are not metaphysically different from physical properties. A mental property is instantiated by multiple complete physical states. It is a property of physical things alone, like other physical properties, and is never a relation with abstract entities. A brain can recognize that its own physical state at a moment instantiates a mental property, although it cannot characterize the class of physical states instantiating the mental property in physicalistic terms, according to the idea of anomalism. This may be simply due to the limited memory and computation capacity of a brain, or it may be because of some deeper reasons, but it does not affect the fact that mental properties are properties of physical things alone and not relations with abstract entities. A brain may have an intention to refer to alleged abstract entities, which is itself a mental state of the brain, but the mathematical terms used by the brain do not really refer to anything. The cognitive functions of mathematical terms and sentences for a brain consist in their syntactical inferential roles in the cognitive activities of the brain, not in bearing the semantic reference relation with alleged abstract entities. In doing pure mathematics, a brain plays mathematical terms and sentences in head, and in applying mathematics to physical things, a brain transforms mathematical terms and sentences into terms and sentences about physical things, which are then physically connected with physical things in the environments. There are certainly many details in such a nominalistic and physicalistic account of human mathematical practices. See Ye (online-a) for a research project a long that line. As for the fact that we name a mental property by ‘is referring to 9’, which appears to refer to an abstract entity, we have a similar explanation as the one given in the last section. That is, our mathematical terms do not refer to anything just as the named mental property is not a relation with an abstract entity.

With this nominalistic reading of anomalous monism available, one can still insist that another realistic reading is also coherent. For instance, one may claim that a brain is referring to the abstract entity 9 as long as it is using words in some ‘right manner’. Now, at the physical level, assuming the existence of the abstract entity 9 is still gratuitous, because ‘using words in so and so manner’ refers to some neural states. However, one may insist dogmatically that the reference relation exists between a brain and the abstract entity 9 at the mental level just in case the brain is in those neural states. This satisfies the supervenience requirement, because if another brain has the same neural state, it will be using words in the same manner and will refer to the abstract entity 9 as well. Therefore, this appears to be a coherent position. However, in this realistic picture, abstract entities and the reference relation between brains and abstract entities at the mental level are superfluous additions. Without them, the nominalistic picture is already self-sufficient. One may compare this with epiphenomenalism in philosophy of mind (as a version of substance dualism), which claims that there are mental entities but mental entities do not causally affect physical entities. Few philosophers today accept epiphenomenalism, and this realistic picture of abstract entities is actually worse. Epiphenomenalism identifies human cognitive subjects with mental entities. Therefore, assuming the existence of mental entities is necessary there. Since cognitive subjects are brains in anomalous monism, the dogmatic assumption about the existence of abstract entities and a reference relation between brains and abstract entities is completely superfluous.

I cannot argue that no other interpretation of anomalous monism can make abstract entities necessary. It is certainly up to the defenders of realism to work out such an interpretation. For now, this nominalistic anomalous monism seems more reasonable.

Besides reductive and non-reductive physicalism, ontological naturalism includes various kinds of property dualism. They admit that humans are material and there are no immaterial minds or souls, but they claim that brains have mental properties that do not supervene on physical states. That is, two brains can have exactly the same complete physical state but different mental properties. For instance, a zombie can have exactly the same molecular structure as a normal human but has no consciousness. Then, one may contend that ‘is referring to 9’ is a genuine relation between brains and an abstract entity at the mental level. Will this save abstract entities? I will only give two comments.

First, methodological naturalism suggests us to trust science. Depending on a stringent or lenient interpretation of this suggestion, property dualism may or may not contradict methodological dualism. Under a stringent interpretation, methodological naturalism suggests philosophers to trust only the theories in mainstream science. Property dualism does not seem to belong to these. That is why few philosophers support property dualism today. This does not rule out property dualism as a proposal for a new scientific research topic, but then defenders of realism in mathematics will be in a bad position if they want to rely on property dualism to explain the relevance of abstract entities to the cognitive activities of brains. Nominalistic physicalism will be more consistent with contemporary well-accepted scientific theories. I am not sure which contemporary defenders of realism in mathematics are willing to take property dualism as the basis for explaining how the cognitive activities of brains involve abstract entities. Note that Gödel’s view on minds is substance dualism. He seems to argue reversely from realism in mathematics to substance dualism about minds (Gödel 1951, 1972).

Second, even if one accepts some version of property dualism, it is still far away from clarifying the semantic reference relation between brains and abstract entities, which is required for defending realism. For instance, one version of property dualism is emergentism, which claims that new, irreducible properties can emerge from complex systems such as brains. From this conception of new, irreducible emergent mental properties, it is hard to imagine how they can be genuine relational properties between brains and abstract entities.

 

4. The Subjective Perspective and Abstract Entities

A natural doubt about this argument is that if there is such a conflict between physicalism and realism in mathematics, why are many philosophers unaware of it? It seems popular to characterize physicalism as the view that existent entities include physical entities plus some abstract entities such as numbers or sets. Realism also appears to be the default position among working mathematicians. Some philosophers argue that applying methodological naturalism to mathematical practices should result in realism (Burgess 2004, Rosen and Burgess 2005). Here I will explain why realism is intuitively appealing and why people are unaware of the conflict between physicalism and realism. I will also argue that nominalism does not violate methodological naturalism applied to mathematics.

The idea is that philosophers who hold both physicalism and realism in mathematics forget the fact that physicalism implies that they themselves are literally physical systems, and instead they pretend to be subjects in the following Egocentric View of Cognition: A cognitive subject is not itself a part of this physical world, but it stands against a world external to it. The external world contains objects, including physical objects and possibly abstract objects as well. The subject uses a language to ‘refer to’ or ‘posit’ objects in the external world. This language bridges a gap between the subject and the world external to the subject, allowing the subject to ‘access’ objects in the external world. The subject is not physical and does not have any direct physical connection with physical objects in the external world. All kinds of objects in the external world, physical or abstract, are ‘on a par’ and are equally ‘referred to’ by the subject using language. Methodological naturalism then says that scientific methods are the best methods for the subject to know things in the world external to the subject. Applying this to mathematics and considering mathematics a branch of science, one concludes that a part of the external world is the mathematical world consisting of mathematical entities.

In contrast, Physicalism about Cognitive Subjects is an objective and naturalistic picture of cognition, where cognitive subjects are just brains as physical things and parts of this physical world and they are in direct physical interactions with other physical things in the world. There is no gap between a subject and an alleged external world; there is only a distinction between physical things inside and outside a skull. Moreover, words and sentences are merely patterns of air vibrations or ink marks, which are recognized, processed, and memorized by the neural network in a brain. They do not have that mysterious function of bridging a gap between a subject and a world external to the subject. Brains are physical things and they interact directly with physical objects in the environments, through eyes, hands, and so on. This picture makes abstract mathematical entities gratuitous. Human mathematical practices are some kind of neural activities in brains, and when brains apply mathematics, those neural activities are physically connected with physical things in the environments, through other neural activities, and through eyes, hands, and so on. Similarly, scientific methods are a special type of neural activities developed in human brains in recent centuries, which have been very effective for brains to cope with their environments, and accepting methodological naturalism by a brain means conducting that type of neural activities in the brain. All these have nothing to do with alleged abstract entities. The idea of abstract entity comes from pretending to be a subject in that Egocentric View of Cognition, thinking from the subjective perspective, and projecting one’s ideas onto a world external to oneself. Only from that subjective perspective will one imagine that there is a mathematical world external to oneself, and then methodological naturalism allegedly requires one to accept its objective existence.

If a cognitive subject is an immaterial mind or soul assumed by substance dualism, or a transcendental ego in the Kantian philosophy, then the Egocentric View of Cognition can be the view from the perspective of the subject. Similarly, it is the view you assume if you think of yourself as a little human hiding inside your brain and trying to know a world external to you by utilizing your brain. However, these conceptions of a cognitive subject literally contradict physicalism.

One may contend that this Egocentric View of Cognition should be understood as a simplified picture that ignores the details inside a cognitive subject and treats a subject as an empty point of perspective only. Moreover, it ignores the details of cognitive processes and focuses on the language that a subject uses to describe the world, and it uses intuitive and intentional descriptions such as ‘referring to objects’ or ‘positing objects’ to describe the relation between the subject’s language and objects in the world. We ignore those details because they are too complex or opaque to us. We want to focus on the aspects of cognition that are simpler, more transparent, and treatable by logical analysis. Now, in science, a simplified model can indeed be very useful if it is sufficiently accurate, but we must always remember that it is a simplified model only. If this is how you see the Egocentric View of Cognition, then any conclusion you draw from it must be compatible with the full and more accurate picture of cognition, which should be Physicalism about Cognitive Subjects under physicalism. The argument in this paper is saying exactly that even if we just think a little more about the details of cognitive subjects and processes under physicalism, we can already see that abstract entities are superfluous in the full physicalistic picture of human cognitive activities.

Therefore, my suggestion is that people who are unaware of the conflict between physicalism and realism and feel that the existence of abstract entities is obvious unconsciously adopt the Egocentric View of Cognition and think from the subjective perspective (out of their egocentric instinct, perhaps). To see an example, let us examine Quine a little more closely. In From Stimulus to Science (Quine 1995), Quine first talks about neural receptors and their reactions to environmental stimuli. These are completely objective and physicalistic descriptions of human cognitive activities. Then, Quine talks about reification of physical objects by using pronouns or variables in language. This can be understood in two ways. First, it can be understood from the objective perspective, as a naturalistic description of the cognitive and linguistic developments of a human brain. It means that, at first, a brain reacts to environmental stimuli globally, recognizing similar stimulus patterns. The linguistic sign of this is holophrastic sentences pronounced by the brain, for instance, “cold!”, or “raven!” Then, in the second stage, the brain starts to focus its attention on a physical object and its properties and to track the identity of a physical object based on its space-time continuity. The linguistic sign of this is using a pronoun to refer to the same object that the brain speaks of before. However, Quine’s description of reification of physical objects can also be understood from the subjective perspective, as a description of when we as cognitive subjects start to ‘posit’ physical objects in a world external to us. That is, we are subjects facing a world external to us. At first, the external world looks amorphous to us. There is no separation of objects and properties, and there are no distinctions between objects. Then, we ‘posit’ physical objects in the external world and describe the external world (or a part of it) as physical objects with properties and relations. Here, we try to figure out what the external world looks like and what we do is using language to ‘posit’ objects and then draw verifiable conclusions about our own experiences. What objects we ‘posit’ is determined by the way we use pronouns and variables to describe the external world. This is perhaps how most people understand Quine, and this apparently assumes the Egocentric View of Cognition and thinks from the subjective perspective.

This subjective understanding of Quine’s description of reification of physical objects is harmless when only physical objects are concerned, because we can reinterpret it from the objective perspective, understanding ‘positing a physical object’ as a physical event involving physical interactions between an object and a neural circuitry in a brain tracking the object. That is, it can be compatible with the objective and physicalistic picture. However, after talking about positing physical objects, Quine starts to talk about positing abstract entities. Then, the situation is very different. From the objective perspective, according to Quine’s descriptions, this is what is happening in the physical world when a brain allegedly ‘posits abstract entities’: The brain uses pronouns and variables in a manner similar to the cases when the brain ‘posits physical objects’, but there are no physical objects in the environment bearing the same kind of connection with those pronouns and variables in the brain. Then, it is puzzling why such a physical event consisting of neural activities in a brain alone is described as ‘positing an abstract entity by the brain’. It seems that here one ignores the fact that the so-called ‘positing a physical object’ is actually a physical process of developing a neural circuitry in a brain tracking a physical object in the environment, and one takes it to be merely the result of using pronouns and variables in some manner. Then, when one uses pronouns and variables in a similar manner but not to ‘posit’ physical objects, one takes it to be positing abstract objects. In other words, ‘positing abstract objects’ makes sense from the subjective perspective with that Egocentric View of Cognition only. If we try to describe it from the objective perspective and in physicalistic terms, we see that it has nothing to do with alleged abstract entities. It is merely a physical event of neural activities alone.  

Traces of such subjective perspective with the Egocentric View of Cognition can be found in many writings by Quine and other philosophers. For instance, in discussing the indeterminacy of reference and ontology, Quine says (Quine 1992, p. 36),

A lesson of proxy functions is that our ontology, like our grammar, is part of our own conceptual contribution to our theory of the world. Man proposes; the world disposes, but only by holophrastic yes-or-no verdicts on the observation sentences that embody man’s predictions.  

Here, Quine seems to conceive of us as little humans hiding inside brains and trying to construct a picture of the external world using language. Then, some portions in the picture are our contributions and the picture touches the external world at the peripheral observation sentences only. He then concludes that the central part of the picture, namely, the mathematical sentences accepted, posits abstract entities. This is his well-known web of belief metaphor. There are debates among scholars regarding if Quine is really a realist, which I cannot discuss here. I will follow most philosophers of mathematics, treating Quine as a realist. However, the issue of exegesis aside, under true physicalism, it should be very clear that the sentences in a web of belief are themselves physical things in a brain (or in an ‘extended brain’ explained in Section 2). The peripheral observation sentences are physically connected with physical things in the environment directly, although the mathematical sentences at the centre are connected with physical things in the environment indirectly only, mediated by peripheral observation sentences. These sentences in a brain, plus another part of the brain that physically manipulates these sentences, are just the entire cognitive subject. They together as a physical system interact with physical things in the environment physically. This is the objective and physicalistic conception of web of belief. There is no little human hiding inside the brain and viewing the sentences in a web of belief as a picture of a world external to that little human. Then, it is very clear that it is pointless to imagine that those mathematical sentences at the centre ‘posit’ abstract entities in the external world.

Here we also see that holism has nothing to do with abstract entities. Holism is considered the trick by which Quine’s philosophy can bypass the epistemological difficulty of abstract objects. We cannot get in touch with abstract objects, but if a web of belief is accepted as a whole, then it appears that we already gain epistemic access to abstract mathematical objects as the posits of the mathematical part of the web of belief. Some critics of the indispensability argument reject confirmation holism (Sober 1993; Maddy 1997). However, let us examine more closely what holism means under physicalism. First, the meanings of peripheral observation sentences in a web of belief partially consist in their physical connections with things in the environments and partially consist in their inferential roles in the web of belief, and the meanings of mathematical sentences at the centre consist in their inferential roles in the web of belief only. This is what meaning holism means in this objective and physicalistic picture of web of belief. Second, a brain accepts the entire web of belief, not each sentence individually by verifying it individually. This is confirmation holism. Therefore, holism is a feature of the way a complex physical system such as a brain works in interacting with its environments. That is, its components work holistically. One cannot assign a definite function to a single component without considering other components. It has nothing to do with alleged abstract objects.

Similarly, speculations about Thin Realism (Maddy 2007) or weak Realism vs. full-blooded Realism (Balaguer 1998) seem to assume the stance of a subject that is not physical but faces an external world that may contain physical objects as well as other less robust entities. If cognitive subjects and processes are completely physical, thin objects are as superfluous as full-blooded abstract entities. Moreover, in criticizing the indispensability argument, some philosophers criticize Quine’s criterion of ontological commitment (e.g. Azzouni 1998). However, it seems that both the criterion and the criticisms on it are understood with the Egocentric View of Cognition in mind.  

One may suspect that I am assuming a God’s eye view here in talking about brains, neurons and so on. This is again a misunderstanding. It assumes that any description of the world has to be from a viewpoint outside the world. This is wrong. Our descriptions about other brains and their interactions with physical objects are also neural circuitries in our brains, and they are physically connected with the brains we describe in the same way as the neural circuitries in those brains are connected with other physical things they interact with. Physicalism is self-coherent and self-sufficient. We do not need to assume a viewpoint outside the physical world in describing the physical world. The real inconsistency is exactly between physicalism and the assumption that there is such a viewpoint from outside, no matter if it is from a subject facing an external world in the Egocentric View of Cognition or from God.

The issue of naturalism and subjectivity is a big philosophical issue, involving many areas in contemporary philosophy. I cannot hope to address it adequately in this paper. However, I hope it is already clear that if you do accept methodological naturalism and accept the scientific description of yourself as a material thing literally and seriously, then this objective and naturalistic description of yourself is in conflict with the idea that you posit abstract entities in a world external to you.

 

5. Disquotational Reference and Abstract Entities

Many philosophers accept the disquotational theory of reference and truth. Some even claim that this is the only theory of reference and truth available to naturalism. The theory holds that the following disquotation schema already summarizes everything that can be said about reference and truth:

（1）           ‘N’ refers to N; ‘P’ is true if and only if P.

Here, N can be any referential term, and P can be any declarative sentence. Then, one may contend that there is no specific difficulty in referring to abstract entities. ‘9’ just refers to 9, which is just what ‘refer to’ means. If this is correct, then it appears that we refer to mathematical entities in a trivial manner, as long as we use mathematical terms. Some philosophers claim that this is all that realism says. I will argue that there is an illusion here. I will focus on reference here, but the discussions apply to truth as well.

First, let us examine more closely what the schema (1) implies under physicalism. Instantiating (1) with names for physical objects, we have

（2）           ‘dog’ refers to dogs; ‘cat’ refers to cats; ‘apple’ refers to apples …

Under physicalism, (2) literally says that there exists a special connection between some word tokens (as air vibrations or ink marks) on the one side, and some other physical things on the other side. This connection is apparently mediated by human brains. It is the reference relation under physicalism. In some cases, the existence of historical physical events connecting word tokens, brains and other physical things is intuitively a necessary condition for the existence of this reference relation. For instance, the fact that English speaking people used to point to dogs and pronounce the sound ‘dog’ simultaneously is intuitively a necessary condition for the existence of the reference relation between the sound pattern ‘dog’ and dogs. In some cases, for instance, in the case of ‘electron’, the physical connections between word tokens, brains and the referents may be rather indirect. Moreover, similarity can extend referents to objects that never had and will never have any physical interaction with brains. For instance, ‘electron’ refers to electrons in the other part of the universe that never had and will never have any physical interaction with brains. That also appears to be how our words refer to future objects. Furthermore, these are necessary conditions for the existence of the reference relation in some cases only. It is extremely difficult if not impossible to give a complete necessary and sufficient condition for the existence of the reference relation in all possible scenarios. Therefore, the reference relation is very complex. It is nothing trivial, unlike what the disquotational theory hints. The schema (1) or the instances (2) are correct, but they merely state the phenomenon of reference. They do not constitute a theory about the phenomenon. (2) is similar to the following statement of the phenomenon of heredity:

（3）           Dogs beget dogs; cats beget cats; apple seeds grow apples …

This is also a correct statement of the phenomenon of heredity, but a theory of heredity should give the underlying mechanism of heredity. Similarly, a theory of reference should give the underlying conditions by which the sound ‘dog’ is connected with dogs. To say that (1) or (2) is a theory of reference is like to say that (3) is a theory of heredity.

 On the other side, the schema (1) can also be understood from the subjective perspective with the Egocentric View of Cognition in mind. Then, a motivation for the disquotational conception of reference and truth is the following. A subject can only access and describe things in the world external to the subject through a language and conceptual scheme. The subject cannot have a ‘God’s eye’ to see reference or truth as a relation between the language of the subject and things in the external world directly; the subject cannot ‘jump out of itself’ to describe that relation without using any language and conceptual scheme. For the subject, to refer to the referents of N in the external world is just to use N, and to assert that P is true of the external world is just to assert P. This disquotational conception of reference and truth avoids treating reference and truth as relations between subjects and the external world directly. The subject still stands behind the mask of language, using its language to speak of the external world without ‘jumping out of itself’. Most people seem to have this in mind when they accept the disquotational theory of reference and truth. However, this motivation for the disquotational conception of reference and truth is based on the Egocentric View of Cognition and it should not concern physicalists. Reference to physical objects in the objective and physicalistic picture of cognition is a physical connection between physical things in the physical world. It is not essentially different from any other relations between physical things studied by the sciences. There is no need to avoid addressing it directly, and we should not avoid it if we do want to study the cognitive activities of human brains.

Therefore, the so-called disquotational theory of reference fails to address the details in referring to physical objects by brains. It gives the illusion that reference is trivial, and then referring to abstract entities appears equally trivial. Moreover, one gets this illusion because one thinks from the subjective perspective with the Egocentric View of Cognition in mind, which literally contradicts physicalism. If you are a true physicalist, you will see that referring to physical objects by brains is a very complex natural phenomenon involving very complex physical systems, namely, brains. On the other side, the assertion that that brains ‘refer to abstract entities’ by using mathematical terms is gratuitous. Using mathematical terms is just some neural activities inside a brain, with possible effects on the interactions between the brain and its environment.

 

6. Conclusion

I argue that science is not merely about a world external to you. It also says what you are. Therefore, after accepting methodological naturalism and science, you should examine your previous pre-scientific ideas about yourself as a cognitive subject. You should carefully reject any pre-scientific conceptions or claims contradicting the scientific description of yourself as a material body interacting with the environment. Then, you can see that abstract entities are illusions. If this argument is correct, it suggests a more radically naturalistic but nominalistic approach to philosophy of mathematics. It will treat human mathematical practices as brain activities and conduct a completely scientific study of mathematical practices. In particular, the applicability of mathematics becomes a natural regularity in a class of natural phenomena, just like other natural regularities studied by the sciences (Ye forthcoming-b). A research project following this approach is underway. An article (Ye online-a) introduces the project. This paper helps to clarify the philosophical foundation of the research project, and another paper (Ye forthcoming-a) compares this approach with other anti-realistic approaches.

 

Acknowledgements

The research for this paper is supported by the Chinese National Social Science Foundation (grant number 05BZX049).

 

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