What Anti-realism in Philosophy of Mathematics Must Offer

 

FENG YE

This article proposes a new approach to anti-realism in the philosophy of mathematics. The realism versus anti-realism debate in the philosophy of mathematics comes from some conflicting intuitions regarding mathematics. The basic intuition favoring realism (or anti-anti-realism) is (Burgess 2004; Rosen and Burgess 2005; Colyvan 1999, 2002; Baker 2001, 2005):¹ as long as mathematicians and scientists attempt to refer to mathematical entities and assert mathematical theorems in their best theories, we already have our best reason to believe that mathematical entities exist and mathematical theorems are true, because we should respect scientists' understanding of their own theories and take their words literally, and there are no other stronger or more superior standards for justifying existence and truth. Note that this does not require mathematical entities to be strictly indispensable in the sciences. Burgess and Rosen emphasize respecting working mathematicians' literal understanding of their own assertions, against various attempts by anti- realists to paraphrase away references to mathematical entities in the sciences; Colyvan and Baker further emphasize that various pragmatic values of mathematics in the sciences make mathematical entities indispensable for our best scientific theories, even if they are not strictly indispensable. This position implies that it is the duty of philosophers to resolve any philosophical problems due to accepting the existence of abstract entities, e.g., the epistemological problem for abstract entities (Benacerraf 1973), and it is not the right of any philosopher to claim, based on any philosophical principle (or actually philosophical prejudice), that mathematical entities do not exist and mathematicians and scientists are systematically wrong.

However, some philosophers are not convinced by these. They hold the opposite intuition that mathematical entities are unlike robustly real physical objects, and that scientists seem to treat mathematical entities differently. Therefore, for instance, some anti-realistic philosophers contend that scientific confirmation cannot reach mathematics (Maddy 1997, 2005a,b; Sober 1993; Leng 2002); some argue that we are entitled to “take back” the assertions on the existence of mathematical entities in the sciences (Melia 2000), or to treat mathematical entities as fictions (Field 1980; Hoffman 2004), or to interpret mathematical assertions non-literally (Yablo 2001, 2002; Chihara 2005; Hellman 2005). The debate sometimes boils down to the question: Who has the burden of proof (Rosen and Burgess 2005)? Realists presume that realism is the prima facie position, because scientists apparently hold the realists' view, and they claim that anti-realists must prove that scientists are systematically wrong. On the other side, anti-realists presume that the epistemological difficulty for abstract entities makes it clear that anti-realism is the prima facie position, and they claim that realists must prove the existence of abstract entities and conclusively refute anti-realism, and that realists must also resolve the epistemological problem for abstract entities.

Now, a philosophical theory should not be like a legal self-defense, where one can presume one's own innocence and wait for the other side to prove that one is guilty. In particular, anti-realists should not just argue that realists have not conclusively proved that we have to be committed to abstract entities. We expect a philosophy of mathematics to be able to resolve puzzles due to genuinely conflicting intuitions. For that, one has to analyze intuitions from both sides impartially and carefully. In particular, anti-realists have to pay attention to realists' points carefully and sympathetically. They should positively account for all the intuitive observations that realists cite to support realism. They should not just try to find faults in realists' well-constructed arguments aiming to refute anti-realism and prove that mathematical entities really exist (such as the indispensability argument).

Therefore, in this article, I will try to explore the strongest challenges to anti-realism by interpreting the intuitions that appear to support realism sympathetically. A fundamental idea underlying these challenges is that, after denying that abstract mathematical entities exist, anti-realists should explain what then really exists in mathematics and should provide a literally truthful account of every aspect of mathematical practices by referring to what really exists. Anti-realists should especially account for the features of mathematical practices that are taken by realists as evidence supporting realism, such as the objectivity of mathematics, the apparent apriority and necessity of mathematics, and the applicability of mathematics. In other words, wherever realists have (or appear to have) an explanation of some feature of mathematical practices by assuming the existence of abstract mathematical entities, anti- realists should not simply reject drawing the realistic conclusion about the existence of abstract entities without providing another equally “realistic” (i.e., literally truthful) explanation of that feature of mathematical practices. Only then can anti-realists really resolve the genuine puzzles about mathematics and can they hope to convince realists or working mathematicians and scientists.

Moreover, I will argue that, for anti-realists to be coherent, they must not assume the objectivity of infinity in any format (not even the objectivity of potential infinity), and they must work under the assumption that there are only strictly finitely many concrete objects in total. That is, anti-realists must show that assumptions implying infinitely many entities are in principle dispensable in mathematical applications, which means showing that some sort of strict finitism is in principle sufficient for the applications.

These may be the strongest challenges to anti-realism and the strictest constraints on anti-realism. No current anti-realistic philosophies can meet all these challenges and constraints. I will argue that this is why they cannot convince realists, and why they are subject to serious objections from realists, if one reads realists' intuitions sympathetically.

I must make it very clear that I am not trying to refute anti-realism here, although some of my comments on current anti-realistic philosophies may sound like attempts to refute anti-realism. I am only trying to say that current anti-realistic philosophies have not offered (and some of them have not even tried to offer) the most essential thing that an anti- realistic philosophy of mathematics must offer. For instance, sometimes they simply label the phenomena to be explained by names such as “empirical adequacy” (Hoffman 2004) or “nominalistic adequacy” (Melia 2000), without really offering (or trying to offer) a “realistic” explanation for the phenomena and resolving the genuine puzzles (i.e., explaining why mathematics is empirically or nominalistically adequate if not true). Sometimes, they do try to offer an explanation (e.g., Field 1980), but in those explanations, they are committed to something that is equally suspicious from the nominalistic point of view (i.e., infinity and continuity of space-time, etc.). I will elaborate on these issues in due course. In commenting on these anti-realistic accounts, I am only raising issues to motivate a new anti-realistic philosophy of mathematics that is perhaps better. I am not trying to refute the basic tenets of anti-realism.

Then, at the end of the article, I will introduce an on-going research project for a new and positive anti-realistic account for mathematical practices, to meet all the challenges and requirements for anti-realism raised in this article. The philosophical bases of this approach are naturalism and physicalism. It pursues a completely scientific study of human mathematical practices, viewing human mathematical practices as human brains' (versus non-physical minds') cognitive activities. It tries to answer philosophical questions about mathematics on a truly naturalistic and completely scientific basis. These will include questions regarding the meanings of mathematical language, the objectivity of mathematics, the apriority and necessity of mathematics, and the applicability of mathematics.

Challenges for Anti-realism

Challenge 1: Anti-realism must explain what really exists in mathematics (if not mathematical entities), and must show how these real things can account for the meanings of mathematical statements and mathematicians' knowledge, intuitions, and experiences.

Mathematical statements are certainly meaningful for mathematicians (in a broad sense) and mathematicians certainly have knowledge, intuition, and experiences regarding mathematics. After denying that mathematical entities exist, anti-realists must then say what really exists in mathematics and must provide a literally truthful account for meaning, knowledge, and intuition, and so on in human mathematical practices by referring to these real things. For instance, mathematical statements as concrete syntactical entities (realized as ink marks on papers, for instance) are certainly real things involved in human mathematical practices, but it seems that there must be something else in order for those statements to be meaningful. Recall that formalism is sometimes understood as the claim that a mathematical theory is a formal system of meaningless symbols. However, ordinary mathematical statements are certainly meaningful for working mathematicians. An anti-realist philosopher's job should be to describe, very realistically (i.e., using literally truthful assertions), how a mathematical sentence can be meaningful without assuming an abstract reality independent of our minds.

Verificationism is therefore in a better position to claim that the meanings of symbols consist in their uses and that some of our knowledge is knowledge about the uses of language, not knowledge about external entities. However, it is still unclear if this is sufficient to account for meaning, knowledge, and intuition in mathematical practices. For instance, it is unclear if this can account for mathematicians' intuitions “about mathematical structures,” or their apparent knowledge that “a Riemann space is approximately isomorphic to real space-time.” These intuitions and knowledge do not seem to consist of intuitions or knowledge about how to manipulate symbols. At least much more work needs to be done by verificationists to show that verificationism can indeed account for meaning, knowledge, and intuition in mathematical practices. Moreover, very importantly but unfortunately, instead of exploring what the meanings of the language of classical mathematics consist of and what scientists' actual knowledge consists of when they use the language of classical mathematics, some verificationists (e.g., Dum- mett 1973) declare scientists' successful uses of the language of classical mathematics as illegitimate uses and suggest that only intuitionistic uses, which are almost never practiced by scientists, are legitimate.

Fictionalism (e.g., Field 1980) accepts the literal interpretation of the meanings of mathematical statements. That is, according to fiction- alists, mathematical statements purport to refer to abstract mathematical entities. However, fictionalism claims that existential mathematical statements are thus “literally false” (and universal statements may be “vacuously true”) since mathematical entities do not exist. This is also misleading. If this “literal meaning” is what working scientists assign to mathematical statements and it accounts for scientists' knowledge, experiences, and intuitions regarding mathematics and its applications, then, as realists like Burgess (2004) contend, we should respect scientists and admit that mathematical entities exist; if this “literal meaning” is not what working scientists mean and it cannot account for scientists' knowledge, experiences, and intuitions regarding mathematics and its applications, then the true meanings of mathematical statements should be more properly construed in other ways, and calling mathematical theorems “literally false” is unhelpful and confusing. It hints that scientists are just making false assertions about nothing pointlessly, and that scientists have neither mathematical knowledge nor valid mathematical intuitions. In other words, if mathematical entities do not exist and mathematical theorems are false (or vacuously true), then what does mathematicians' grasp of the meanings of mathematical statements consist of, and what are mathematicians' knowledge and intuitions about? Fictionalism actually leaves the true meaning, knowledge, and intuition, and so on in mathematical practices unaccounted for. However, I am not saying that fictionalism cannot account for meaning, knowledge, and intuition in mathematical practices. Rather, I am suggesting that if they do try, then they may realize that they need to adopt the new approach that I will introduce at the end of this article, and then fictionalism as a distinctive philosophical position will be unnecessary.

Some fictionalists might think that accounting for meaning, knowledge, and intuition is unimportant because what makes mathematics applicable has nothing to do with what mathematicians grasp as meaning or what they have as knowledge and intuitions. However, this amounts to saying that working mathematicians and scientists come across the right mathematical theory for a type of applications by chance, or that scientists just arbitrarily choose statements that may be false and apply them in the sciences. This is incredible. More importantly, this clearly puts some philosophical principle (i.e., the nominalistic intuition) over scientists' judgments. For scientists, there are scientifically valid reasons why, for instance, the Riemann space theory is applicable for describing real space-time, for instance, the reason that Riemann spaces are (approximately) isomorphic to space-time. Such reasons are based on scientists' understanding of the meanings of statements in the Riemann space theory, based on their geometrical intuitions “about Riemann spaces,” and based on their knowledge “about Riemann spaces.” The realists' claim is that scientists' understanding, knowledge, and intuitions imply the literal existence of Riemann spaces. Anti-realists must account for scientists' understanding of meaning, their knowledge, and intuitions positively (without assuming that Riemann spaces really exist, of course). Otherwise, anti-realists will not be able to meet the realists' challenge.

Moreover, accounts for meaning, knowledge, and intuition, and so on must be realistic accounts, or accounts consisting of literally true assertions. For instance, anti-realists should not say, “scientists use Riemann spaces as models for simulating real space-time,” because this assertion is “literally false” for anti-realists, since Riemann spaces do not exist. It amounts to saying that scientists use nothing for simulating real space-time. Similarly, one should not say, “mathematicians have geometrical intuition about Riemann spaces as fictional entities,” because fictional entities do not exist and one cannot have intuitions about nothing. I will not venture into the metaphysics of fictional entities here. From the nominalistic and naturalistic point of view, the real problem in these “literally false” explanations seems to be that they have not really explained what it means to talk about “fictional things,” or to have knowledge or intuitions “about fictional things.” Until we have a literally truthful account without referring to alleged fictional things, realists can always come back and claim that the indispensability of alleged fictional entities in the sciences shows that they are not mere fictions.

Finally, an anti-realistic account for meaning, knowledge, and so on should focus on the current practices of classical mathematics, and should respect mathematicians' understanding of classical mathematics. It should not invent new mathematics or paraphrase mathematical statements into something unrecognizable by mathematicians. Because, the real issue that is at stake is: Do our actual mathematical practices and working scientists' understanding of them imply realism? For example, if figuralism (Yablo 2001, 2002) is applied to statements about Riemann spaces, it will imply that statements about Riemann spaces have some real content that is not about Riemann spaces and is actually not about any particular things, because, according to figuralism, mathematical theorems are logical truths, and logical truths are true about everything. Mathematicians could not have understood that alleged real content of the statements in the Riemann space theory. No geometrical intuition conveyed by the original statements is in that alleged real content. That alleged real content could not be what mathematicians really mean. Again, the realists' challenge is that working mathematicians' actual understanding, knowledge, and intuitions imply the literal existence of mathematical entities such as Riemann spaces (Burgess 2004).

Similarly, mathematicians are obviously not talking about ideal agents, possible concrete inscriptions on papers, etc., (cf. Hoffman 2004; Chi- hara 2005). Approaches to anti-realism by paraphrasing mathematical statements into statements about ideal agents, possible inscriptions on papers, and so on, have not directly met the realists' challenge. (Besides, ideal agents do not exist. Therefore, the claim that mathematicians are referring to ideal agents is again literally false.)

On the other hand, it seems that the realistic reading of mathematical statements is not the only one assumed by working mathematicians and scientists, as some realists seem to imply. Many authors (e.g., Leng 2005) have pointed out that mathematicians and scientists do not have a unanimous view regarding the nature of mathematics. In particular, physicists sometimes like to call mathematics a language or formalism, and they sometimes speak as if mathematics is just manipulating symbols. Therefore, there are genuine puzzles due to genuinely conflicting intuitions about the nature of mathematics. This is essentially different from the issue of the existence of atoms, about which perhaps no working scientists have any doubt today. A philosopher's task, then, is to solve those puzzles for scientists.

The real point is that anti-realists must not start from some obscure philosophical principles alien to scientists, such as the metaphysical intuition of nominalism, or even Ockam's razor principle. For a naturalist, these principles must be less certain than what working scientists unanimously hold. Instead, anti-realists must speak in the language of science, and must provide an account that is acceptable by working scientists, according to their scientific methodologies and standards. If anti- realists can provide a literally truthful scientific account of scientists' understandings, knowledge, and intuitions regarding mathematics and its applications, without assuming that mathematical entities literally exist (or exist in any mysterious sense that only some philosophers appear to be able to understand), then they may be able to clarify the mystery surrounding the nature of mathematics for working scientists, and they may be able to convince working scientists.²

On this view, philosophers do not even have to be so modest as to claim that philosophical analyses never do any good to the sciences, or that philosophers should never suggest anything to working mathematicians (cf. Burgess 2004; Leng 2005). Mach's analysis of relativity of space was alleged to have influenced Einstein. Philosophical analyses certainly cannot substitute for constructive scientific work. What philosophical analyses can offer is to dispel away dogmatic faiths without real empirical supports or illusions due to our thinking habits, which, under some circumstances, may actually hinder new scientific explorations. The realistic faith about classical mathematics could be such a dogmatic faith, if it indeed comes solely from our thinking habits, and if it is not really justified by the sciences and is not compatible with our overall scientific worldview.³ Rejecting that realistic faith will suggest taking a more liberal view on possible mathematical practices and will encourage exploring new ways of doing mathematics.

Challenge 2: Anti-realism must account for the genuine relationships between some (alleged) mathematical entities (or structures) and some physical things.

Scientists choose Riemann spaces to model space-time structures for good reasons. Even if Riemann spaces do not literally exist, it is still a matter of fact that in some sense, Riemann spaces and real physical space- time structures are structurally similar. Structural similarity appears to be a genuine relationship between some mathematical structures and some physical things. There are also other types of relationships between the mathematical and the physical. For instance, a function may approximately represent the states of a physics system in some way, and a stochastic process may approximately simulate some real random events. Such relationships all seem to be genuine and are the objective reasons why mathematical theories are applicable in the sciences. On the other hand, nothingness certainly could not structurally resemble any real things and could not be related to any real thing in any meaningful way. Therefore, if mathematical entities do not exist, what resembles those real physical things in these genuine relationships between the mathematical and the physical? Anti-realists should not simply deny such relationships, which will again leave working scientists' reasons for the applicability of mathematics unaccountable. Anti-realists must explain what really exists on the mathematical side, and then show that such genuine relationships between the mathematical and the physical are realistically (literally truthfully) accountable based on what really exists on the mathematical side. Anti-realists must also show that our knowledge of such relationships is explainable.

Moreover, anti-realists must show how the content of a specific mathematical theory is relevant to the existence of such relationships. For example, the content of the definition of Riemann spaces is certainly relevant to the fact that Riemann spaces resemble real space-time structures, and the content of the theory of finite groups is relevant to the fact that a finite group does not in any way resemble real space-time structures. In other words, it is not enough to say generally that pragmatic consequences decide which mathematical theory is useful to model reality in a particular area. Scientists do not randomly pick some literally false statements about nothing and then try to apply them in an arbitrary area in the sciences. They choose (or discover, or define, or invent, or imagine) Riemann spaces to model large-scale space-time structures, because they really discern some genuine relationship between these two, in particular, based on their understanding of statements “about Riemann spaces” and their knowledge “about Riemann spaces.” Anti-realists have to admit scientists' actual intuitions and judgments regarding the relationship between Riemann spaces and physical space-time, and have to explain them realistically and scientifically, by referring to what really exists in mathematical practices (without assuming that Riemann spaces really exist).

There are anti-realistic approaches that resort to some general concepts that apply to mathematics as a whole, such as nominalistic (or empirical) adequacy, to account for the usefulness of mathematics (Melia 2000; Hoffman 2004). These concepts may be of some interest, but they say nothing about such genuine relationships between the mathematical and the physical, and nothing about scientists' reasons for the applicability of a particular mathematical theory to a particular type of natural phenomena based on such relationships. “Nominalistic (or empirical) adequacy” is a name for the observed results in scientists' mathematical practices. It is not an explanation of why those results obtain. It does not explain, for instance, what is special about the Riemann space theory that makes it applicable in modeling space-time, and why scientists did not use the theory of finite groups, or anything else, to model space- time. (Riemann spaces and finite groups do not exist anyway.) Actually, if Riemann spaces did exist and were literally (though approximately) isomorphic to physical space-time, there would be an explanation of their nominalistic or empirical adequacy. The realists' claim is exactly that this justifies the existence of Riemann spaces. Anti-realists cannot meet this challenge without providing an equally literally truthful account for scientists' valid judgments without assuming that Riemann spaces literally exist.

Recall that in answering the same question for empirical adequacy, van Fraassen reminds us that any explanation must stop somewhere anyway, and then he claims that it stops at explaining the empirical adequacy of postulating unobservable (physical) entities in the sciences. Similarly, some anti-realists appear to be claiming that there is no more explanation for the applicability of a mathematical theory for a type of natural phenomena (e.g., the applicability of Riemann space theory to space-time), and that scientists' explanation for it (i.e., by assuming the literal existence of Riemann spaces and a real isomorphism between Riemann spaces and space-time) is just wrong. For instance, Hoffman (2004) takes her fictionalist view to be a completion of van Fraassen's views. I will not try to contest such claims here, but I would like to point out that anti-realists will certainly be in a better position to meet the realist challenge if anti-realists can offer a literally truthful explanation of such genuine relationships between the mathematical and the physical, as well as an explanation of the applicability of mathematics based on such relationships.

Challenge 3:Anti-realism must identify and account for various aspects of objectivity in mathematical practices and applications.

Even if mathematical entities do not exist, our mathematical knowledge should still have objective content. We are not making assertions out of our wishes in doing mathematics. One could wish that Goldbach's conjecture is true, but we know that there is something objective and independent of our wishes there. A natural attempt to explain such objectivity from anti-realists' perspective is to claim that correctness in following the logical rules in a mathematical proof is an objective matter. Then, the challenge for anti-realists is: admitting such objective correctness in rule following appears to be committed to rules as abstract entities and be committed to objective truths about abstract entities, in particular, when rules are understood as mathematical functions that can operate on infinitely many instances of arguments.

Another aspect of objectivity in mathematics has to do with the relationships between the mathematical and the physical. Hoffman's (2004) recent exposition of fictionalism appears 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 when playing games. However, the (approximate) structural isomorphism between Riemann spaces and real space-time structures seems to be objective, not wishful pretending, and it seems to be the objective reason for our successes in modeling space-time structures by Riemann spaces. If scientists were indulging in wishfully pretending things as kids do in games, scientists would not be successful in their work. Frege's claim that applications raise mathematics from a game to science is well known. Realists' charge against anti-realism is just that such strong objectivity in the sciences, which is not in kids' games, shows that mathematics is not merely a make-believe. Until anti-realists can clearly explain what this objective relation between the mathematical and the physical consists of and how it is the objective reason for the applicability of Riemann spaces (without assuming that Riemann spaces exist), they have not yet met the realists' challenge.

Anti-realists who completely deny any objective realistic truths (or seek to account for mathematics and our scientific knowledge, in general, only as socio-cultural constructions or conventions) may not care about this challenge. Criticizing them is usually the realists' job. I propose this as a challenge to anti-realism in mathematics, because I take anti-realism in mathematics as a clarification of and a defense for common-sense realism and scientific realism. It tries to solve puzzles due to alleged mathematical truths about infinity and abstract objects, which appear to be “out of this universe.” For this, anti-realism in mathematics must distance itself from views that deny objectivity altogether.

Challenge 4: Anti-realism must explain the apparent obviousness, universality, apriority, and necessity of simple arithmetic and set theoretical theorems, and they must also provide a consistent account of logic.

We have a strong intuition that “5 + 7 = 12” expresses some obvious, universal, necessary, and a priori truth. It does not help to say that “5 + 7 = 12” is “literally false,” as some anti-realists seem to be saying, which only adds more puzzles. “5 + 7 = 12” is certainly meaningful to everyone. It has content. Kids do learn something when they learn “5 + 7 = 12.” There must be some truth in it even if numbers do not “literally exist” and even if “5 + 7 = 12” is “literally false” in whatever sense. Anti-realism must explain what the content of “5 + 7 = 12” is and why it is obviously true in some proper sense. It must also answer questions regarding the universality, apriority, and necessity of“5 + 7 = 12,” and give reasonable explanations as to why we strongly believe that “5 + 7 = 12” is a priori, necessary, and universal. Moreover, arithmetic, set theory, and logic are tightly entangled. Some simple theorems in arithmetic and set theory, such as “5 + 7 = 12” or “A U B = B U A,” appear to be logical truths in disguise. The common wisdom is that logical truths are universal, a priori, and necessary truths. The universality, apriority, and necessity of arithmetic are obviously closely related to the same characteristics of logic. Anti-realism must provide an account of logic consistent with the anti-realistic ontology and epistemol- ogy, and consistent with anti-realists' accounts of arithmetic and simple set theory.

One attempt to explicate the truth of “5 + 7 = 12,” adopted by figuralism (Yablo 2002), claims that the real content of the statement is expressed by the following logical truth in the first-order logic

This may be fine; however, for arithmetic statements with quantifiers, Yablo's suggestion is that they are logical truths expressed by infinitely long sentences, namely, infinite conjunctions and disjunctions of logical truths in the above format in the first-order logic. Now, we do not speak infinitely long sentences. Infinitely long sentences are actually mathematical constructions and are thus abstract entities. If infinitely long sentences do not really exist as abstract entities, it is unclear whatYablo has said about the alleged real content of quantified arithmetic statements. Nothingness surely cannot express any meaningful content. If infinitely long sentences simply do not exist, there is nothing there to express that alleged content. Therefore, one naturally suspects that what really express the alleged content are actually still the original quantified statements about numbers.

It seems that what Yablo actually has there is another mathematical theory “about infinitely long sentences” as mathematical entities, which could be defined by using set theory with the axiom of infinity, and Yablo has a “true” predicate for those infinitely long sentences, recursively defined in set theory. Therefore, Yablo actually translates real, concrete quantified statements “about numbers,” into real, concrete statements in this mathematical theory “about infinitely long sentences” (as abstract entities), and then claims that those infinitely long sentences (as abstract entities) are “true,” that is, they satisfy that recursively defined “true” predicate in the mathematical theory “about infinitely long sentences.” In the end, the alleged real content is still the content of statements that appears to refer to abstract entities (i.e., infinitely long sentences). This perhaps shows that one has to deal with the content of statements that appears to refer to alleged abstract entities (or fictional entities) directly. There is no magic to get “real content” out of these statements.

Challenge 5: Anti-realism must be able to account for mathematics under the assumption that there are only finitely many concrete objects in total, but it must also account for our apparent valid intuitions “about infinity.”

Philosophers favoring anti-realism or nominalism may still hold different views regarding infinity. For example, both Field (1998) and Yablo (2002) claim that arithmetic theorems involving infinity are objectively true (when those theorems are properly stated). Field refers to a “cosmological assumption” on infinity of the universe in defending that arithmetic statements implying infinity have objective truth values. However, as of today, physics has not provided a definite account regarding whether the universe is infinite. More importantly, in almost all areas of the sciences so far, the applicability of mathematics is independent of the conjectures of physics about whether space-time is infinite or finite, or discrete or continuous. We apply infinite mathematics also in economics, which is certainly about finite and discrete things. If an account for mathematics and its applications depends on a specific assumption from physics about space-time, it must have missed something essential about the nature of our mathematical knowledge, and must have missed the true reasons for the applicability of mathematics.

Now, if anti-realists accept the objectivity of infinity, but claim that it does not mean that this physical universe is infinite, then they must explain where infinity is if this physical universe is indeed finite and discrete. These anti-realists will also face a similar epistemological problem as realists face, namely, explaining how knowledge about objective infinity is possible, given that we are finite beings with finite experiences. They might try to exploit some concept of scientific confirmation. Then, it is likely to be again some sort of holistic confirmation based on the pragmatic values of assuming infinity in the sciences, and it is likely that realists can simply take over this strategy to explain our knowledge about any abstract objects, as Quine does. After all, as long as the door is open to accept one objective truth that appears to be about things essentially beyond this concrete universe, it should not be too difficult to go ahead and accept others for similar reasons.

Similarly, some philosophers resort to an objective “mathematical possibility,” according to which infinity is “mathematically possible.” Now, if the universe is indeed finite and there are only finitely many concrete objects in total, what justifies this claim about the mathematical possibility of infinity? It seems that it will be again some sort of holistic justification based on their usefulness in the sciences. Then, the door to Quinean realism is opened again.

Therefore, rejecting an objective infinity is the only way to be a coherent nominalist. This assertion was already made long ago by Goodman and Quine (1947) and it seems to be ignored by some contemporary philosophers. This assertion implies that for any statement containing quantifiers intended to range over infinite domains, anti-realists must either offer an anti-realistic interpretation of the statement, or admit that the statement can be vacuously false or vacuously true. These include statements about the consistency of formal systems, statements about Turing machines, and statements expressing simple universally quantified arithmetic laws.

We do seem to have a strong intuition that there are objective truths involving infinity, for instance, the commutative law of addition, or the commutative law of addition expressed as an assertion about a Turing machine implementing the addition function, or the consistency of some very simple formal systems, etc. Therefore, anti-realists must show that the literal truth of such assertions involving infinity is neither presumed in the sciences or implied by the sciences, and nor is it needed in the anti-realistic account for mathematical practices and mathematical applications. In other words, similar to the cases for meaning, knowledge, and relationships between the mathematical and the physical, anti-realists must account for our various intuitions “about infinity” without assuming the objectivity of infinity.

Challenge 6: Anti-realism must explain the applicability of mathematics, and for that purpose, one has to show that the apparent references to infinity and abstract mathematical entities in mathematical applications are in principle dispensable.

Field (1980) tried to explain how a bunch of “false statements about nothing” could be useful by using the notion of conservativeness. His strategy depends on the assumption that one can nominalize scientific theories by eliminating references to abstract mathematical entities, and thus prove that classical mathematics is conservative over a nominalistic version of science. There were other similar programs for nominalizing mathematics in the 1980s and 1990s (see Burgess and Rosen 1997). However, all these programs assume infinity in one way or another. Therefore, as I have argued above, they are not really nominalistic.

On the other hand, if a nominalization program is successful, it does help anti-realism. There are objections to this claim. First, nominalized scientific theories must be much more complex than the theories stated in classical mathematics. One may argue that simplicity justifies realism about classical mathematics. However, consider this example. We can simulate the population growth on the Earth by a differentiable function and a differential equation. This is fairly simple. Now, if we have a gigantic computer (or just a gigantic brain) that can simulate all people on the Earth (on aspects related to reproduction), then we will have a literally more accurate description of the population growth on the Earth and it will give us more accurate predictions on future population growths. This literally more accurate description will refer to finite and discrete entities only and it will not refer to infinity or abstract mathematical entities such as real numbers, differentiable functions, etc. It will be a nominalis- tic description. On comparing these two descriptions of the population growth on the Earth, we see that classical mathematics can offer a simpler description of discrete and finite things only because it helps us to ignore some details and build a simpler but literally less accurate model. We will have to eliminate infinity, if we want a more accurate description of the phenomena. It is doubtful that such simplicity can justify realism about infinity and abstract entities in classical mathematics. Note that this is not a peculiar example. Considering the fact that our physics has not yet reached things below the Planck scale (about 10~^3S m, 10~^4S s etc.), all current applications of continuity and infinity in the sciences (at least except for physics at the Planck scale) gloss over microscopic details. The kind of simplicity that classical mathematics brings in these applications does not seem to confirm the existence of infinity. These applications are similar to the cases where one imagines some non-existent fictional things as simpler substitutes for real but more complex things.

Melia (2000) claims that mathematics brings a simpler theory, not a simpler world, and he argues that such simplicity does not justify mathematics. However, if we agree with realists to put the mathematical world on a par with the physical world, this combined world does become significantly simpler if we postulate infinite mathematical entities, and if we ignore the fact that there is a price for it, that is, its part that describes the physical world becomes literally less accurate. Ignoring this price, realists seem to be at a good position to claim that positing mathematical entities does bring the same kind of simplicity as positing atoms does. Therefore, a mere distinction between simplicity of the world and simplicity of a theory cannot convince realists, for whom the world includes mathematical entities as well (or, they claim that you should at least have an open mind to allow your world to include abstract mathematical entities).

Colyvan (1999, 2002) and Baker (2001, 2005) imply another objection to the thesis that successful nominalizations of scientific theories will help anti-realism. They claim that mathematics is not only used to build models to represent phenomena and deduce known conclusions. They cite examples to show that mathematics has unification powers, predicting and discovering powers, the ability to apply to new and unexpected areas of natural phenomena, and genuine explanatory powers. They claim that these pragmatic values support realism, by which they seem to imply that those nominalized theories will not have these pragmatic values. However, a nominalized physics theory, if that can be worked out at all, will have the same pragmatic values as the classical theory except for simplicity, because they express the same physics laws, and even their mathematical formats can also be very similar (because the nominalized version typically parodies the classical version, e.g., Field 1980; Ye 2011).

From the logical point of view, the real puzzle about applicability is due to the gap between infinity in mathematics and finitude of the real world (from the Planck scale to the cosmological scale). In order to resolve this puzzle, we must clarify, for instance, the logic of using infinite and continuous models to approximate and simulate finite, discrete things. This puzzle is independent of any philosophical view about the ontological and epistemological status of those alleged models as abstract entities. What anti-realists can hope is that a complete logical clarification of the puzzle may turn out to favor anti-realism since it is likely that a complete logical clarification will eventually show that infinity and apparent references to abstract entities all in principle can be eliminated in mathematical applications, which will then show exactly how mathematics helps to derive literally true nominalistic conclusions about finite concrete things from literally true nominalistic premises aboutfi- nite concrete things in plain logic. Moreover, it may show that the literal existence of infinity and abstract entities is irrelevant for explaining applicability, because what really explains applicability is the fact that mathematical proofs used in applications allow eliminating apparent references to infinity and abstract entities, so that the proofs can preserve literal truths about finite concrete things. That is, a real logical explanation of this puzzle of the applicability of infinity to finite things may in the end have to imply that infinity is in principle dispensable.

With doubts about dispensability, some recent anti-realists try to look for some easy arguments to show that even if abstract mathematical entities are indispensable, we still do not have to be committed to them. For instance, Leng (2002) claims that mathematical entities are used to build models to represent physical things, and she claims that science confirms the existence of those physical things only, but not the existence of those models.

Now, computers are used to build models to simulate other things. However, computers really exist. Indeed, thefailure of a computer modeling should not be taken as evidence that the computer does not exist or that our assertions about data and programs in the computer are false, but successes of computer modeling do require that computers literally exist and do confirm that our assertions about data and programs in computers are literally true. Similarly, realists claim that while the failure of mathematical modeling does not imply that the mathematical model does not exist or that our assertions about the model are false, but successful modeling does require that the model literally exists and does confirm that our assertions about the model are literally true. For instance, if one makes an error in doing calculations about a mathematical model, one will not be successful in using the model in applications. Anti-realists may insist that models are fictional entities. However, this cannot convince realists. They claim that scientific applications raise mathematics from a game or fiction to science, and they charge that anti-realists are intellectually dishonest and are placing their principle of nominalism above scientists' judgments.

Besides, remember that according to anti-realists, the claim “scientists are using the fictional model X to simulate Y” is literally false, because X does not exist. Therefore, these anti-realists are making literally false assertions in their philosophical papers, according to themselves. It clearly implies that these anti-realists have not yet really explained how mathematics is applied, or how mathematical models work. They have not provided a literally true explanation of how “fictional models” are applied to derive literal truths about real things. Since fictional things do not exist, in a literally true explanation, one should not refer to “fictional models” again. That is, one has to show that they are in principle dispensable.

Melia (2000) is another instance. Melia claims that classical mathematics is not conservative over nominalistic theories because some assertions about concrete things are not expressible in a nominalistic language. Melia cites some examples to show how assertions about concrete things have to be expressed by referring to abstract entities. Then, Melia proposes the so-called “weaseling strategy,” which suggests that we can take back what we asserted previously about the existence of abstract entities in applying mathematics to the sciences. It means that classical mathematics is nominalistically adequate in the following sense: the consequences about concrete things obtained by applying mathematics are true of concrete things.

Now, since all scientifically reliable assertions about concrete things in this universe are accurate only up to some finite precisions (i.e., above the Planck scales 10~^3S m, 10~^4S s, etc.), Melia's examples regarding how assertions about concrete things have to be expressed by referring to abstract entities are beside the point for a real nominalist, because these examples all assume infinity. We may need apparent references to infinity and abstract mathematical entities to give a simple description about finite concrete things. An example will be using a differentiable function to represent population growth on the Earth. However, apparent references to infinity and abstract mathematical entities may not be strictly indispensable.

On the other side, sometimes we do take back what we asserted at first in everyday life and in the sciences. However, if we do so and are then confronted with the accusation that we have to take back our conclusions as well, we usually have to show that we do not really have to commit to that which was asserted previously. That is, we have to show that what was asserted previously can in principle be eliminated. That is the case, for instance, when we refer to a rigid body in elementary mechanics. We believe that such references to fictional entities can in principle be eliminated. Now, what can one offer to justify the nominalistic adequacy of one's “weaseling practice” if one admits that abstract entities are strictly indispensable? It seems that the only available strategy is again some sort of holistic confirmation based on the pragmatic values of positing abstract entities. Then, why does not this lead to Quinean realism? Again, we need an explanation of nominalistic adequacy, and it has to be a literally truthful and nominalistic explanation, which means that it should not refer to abstract entities again. If we can explain the applicability of infinite mathematics to finite real things in the universe by showing that apparent references to infinity and abstract entities can in principle be eliminated, then we will have such an explanation of nominalistic adequacy. Otherwise, anti-realists either have to leave nominalistic adequacy unexplained or have to open a door for the Quinean holistic justification for abstract entities.

These are the challenges for anti-realists. In summation, anti-realists must provide a positive account for the practices of classical mathematics including its applications. Especially, anti-realists must account for those aspects that are taken as evidence supporting realism by realists. They should not simply label some phenomena by a name (i.e., “nominalistic adequacy”) without giving a real explanation, and should not be negative only (i.e., arguing that they do not have to be committed to something). They should not fall into resorting to some holistic confirmation to justify some obscure things, such as infinity, possibility of infinity, “weaseling strategy,” and so on. They must explicitly say what are real on the mathematical side in mathematical practices, and then refer to those real things to give a very realistic account for mathematical practices. Finally, they must do these without assuming that the universe is infinite or that there are infinitely many concrete objects in total. They must realize that the real puzzle of the applicability of mathematics is just the logical puzzle of how infinite mathematics is applied for describing finite real things, which constitute almost all mathematical applications in the current sciences.

Toward a Scientific Account for Mathematical Practices

These challenges and requirements seem to have cornered anti-realism. Is there still a chance for anti-realism in the philosophy of mathematics? The answer is yes. Actually, the analyses in the last section very naturally lead to a completely naturalistic and scientific account for human mathematical practices.

First, the analyses above show that what is missing from current anti- realists is a realistic and literally truthful account for aspects of mathematical practices, by referring to what really exists in mathematical practices. Therefore, what really exists and what is really happening in mathematical practices? Since the alleged “Riemann spaces” do not literally exist, if one asks, “What is that mathematician doing when she talks 'about Riemann spaces'?”, the only straightforward answer seems to be, “She is imagining something.” This idea is not new. As far as I can trace it, the earliest explicit exposition of it is Renyi (1967). Renyi explicitly suggested that mathematical entities are our imaginations. On the other side, perhaps all contemporary anti-realists more or less have this picture in mind.

Therefore, what we really need is a very realistic and literally truthful explanation of what it is to imagine something. The natural thought is to characterize imagining something as having relevant mental representations with the same or similar structures as mental representations of real external things, but without any corresponding external things to be directly represented. I emphasize “directly represented” here, because these mental representations are indirectly related to real external things in some way. In other words, our imaginations do not literally create “imaginary entities.” Only our hands can create things out of pre-existing materials. When we imagine, our minds create mental representations that reside in our brains.

Then, for mathematics, this means that while there are no mathematical entities, there are scientists' mental representations that they create and manipulate in doing and applying mathematics. These are what really exist on the mathematical side in mathematical applications (vs. other physical things as the subject matter of application on the other side), and they are the real things that are used as models for simulating real things. (Scientists use their brains to model other things much like they use computers to model other things.) Anti-realists' task will then be to describe the cognitive functions of these mathematical mental representations in human mathematical practices, and to describe the relationships between these mental representations and other real things in the physical world, to explain various aspects of mathematical practices, including the applicability of mathematics. An account for mathematical practices is thus a continuation and extension of cognitive science, dealing specifically with human mathematical cognitive activities. It is a completely scientific description of a class of natural phenomena. This is what is missing from the current anti-realistic philosophies. With that missing, one cannot answer, in realistic terms, what the meanings of mathematical statements are, or what the relationships between the mathematical and the physical consist of, or what exactly are used to model real things in mathematical applications, or how exactly those models work.

This picture of human mathematical practices is consistent with the following nominalistic but completely naturalistic ontological and epis- temological assumptions: (1) there is this physical universe, which may be finite and discrete (and we do not know yet), and only things in this universe really exist; (2) humans and their brains are parts of this natural world, and their knowledge (including mathematical knowledge) stored in their brains and realized as neural structures comes from their finite brains' interactions with finite concrete things in this universe, either individually or programmed into their genes as a result of evolution; and (3) there is no existence or truth beyond and above this concrete universe. These basic assumptions, which will be simply called naturalism here, seem to be clear and coherent. It is not Quinean naturalism. It is consistent with physicalism in contemporary philosophy of mind (e.g., Papineau 1993), and it is consistent with the common-sense realism, scientific realism, and the general scientific and naturalistic worldview, except for the fact that classical mathematics generates a puzzle.

The puzzle is that classical mathematics appears to include knowledge about things essentially out of this concrete universe. Quinean pragmatic mathematical realism actually implies that successful applications of classical mathematics in modern sciences force us to reject this naive naturalism and force us to accept a more sophisticated view of existence that puts abstract mathematical entities on a par with other concrete things in this universe. I take anti-realism in philosophy of mathematics as an effort to resolve the puzzle and defend this naive naturalistic worldview.

A research project following these ideas is underway. The project will account for various aspects of human mathematical practices by referring to the cognitive functions of mathematical mental representations in brains and their physical connections with physical entities outside brains. This article actually presents the motivation and sets the goals for the project. Another article (Ye 2010a) elaborates on the kind of naturalism this project relies on. It also argues that the Quinean indispensability argument is actually an argument from the point of view of a Transcendental Subject, and is therefore not a naturalistic argument. It should not disturb a true naturalist. In other words, the article argues that Quine is implicitly inconsistent with himself regarding the basic tenets of naturalism.

Then, a few other articles (Ye 2010b, online-a,-b,-c) and a monograph (Ye 2011) present the positive accounts for various aspects of mathematical practices. They constitute a part of the work done so far in the project. Ye (online-a) discusses some aspects of meaning, knowledge, and intuition in mathematical practices, as well as the relationships between the mathematical and the physical, by referring to the cognitive functions of mathematical concepts and thoughts as mental representations in brains, and by referring to their physical connections with physical entities outside brains.Ye (online-b) identifies various senses of objectivity from the naturalistic point of view and explains why admitting objectivity in mathematical practices does not imply the existence of abstract entities. Ye (online-c) discusses the apriority of logic and arithmetic from the naturalistic point of view. Finally, the monograph Ye (2011) first explains how the question of applicability of mathematics can be formulated as a scientific question and transformed into a logical question. Then, it develops a strategy for explaining the applicability of mathematics, in particular, the applicability of infinite mathematics to this finite physical world. The strategy involves showing first that the applications of classical mathematics are in principle reducible to the applications of strict finitism, a fragment of the quantifier-free primitive recursive arithmetic, and then showing that the applications of strict finitism can be interpreted as sound logical inferences from literally true premises about strictly finite, concrete physical objects, to literally true conclusions about them. These will constitute a logically plain explanation of why infinite mathematics can preserve literal truths about strictly finite things in mathematical applications, by showing that infinity can in principle be eliminated. Two short articles (Ye 2010b, online-d) give a summary of the monograph.

The project is still in progress. It is possible that even if it is successful, it still cannot convince some realists. In particular, since its basis is naturalism, it will not convince those who explicitly reject naturalism (e.g., Godel). However, this completely naturalistic and scientific description of human mathematical practices as human brains' cognitive activities will show that it is scientifically redundant and meaningless to assume that human mental representations created in human brains in mathematical practices “represent” or “correspond to” the alleged abstract entities. It offers a more coherent scientific and naturalistic picture of human mathematical cognitive activities. Then, this can perhaps convince those whose primary concern is about respecting science versus metaphysical (i.e., nominalistic) intuitions. Moreover, such research should have its own values, independent of any philosophical positions, since it is scientific research into human mathematical cognitive processes and into the exact logic in applying infinite mathematics to this finite physical world.

Finally, I will briefly compare this research project with other related approaches. Some cognitive scientists have studied the origin and psychological nature of mathematical concepts from the psychological point of view (e.g., Lakoff and ^nez 2000). However, they did not discuss issues that concern philosophers and logicians, such as objectivity in mathematics, the apriority of logic and arithmetic, and the applicability of mathematics and so on. This research project focuses on these philosophical and logical issues, not on the psychological aspects of mathematical practices.

This research also follows the spirit of philosophical naturalism (or physicalism) pursued by Papineau (1993) and is intended to be a substantial improvement on it. In particular, I suggest addressing philosophical issues on meaning and so on, by directly referring to mathematical mental representations from the point of view of cognitive science, while Papineau relies on Field's fictionalism (Field 1980), which is not a truly naturalistic and realistic scientific theory. The strategy for explaining the applicability of mathematics here is reminiscent of Field's notion of conservativeness, but Field assumed infinity in his nominaliza- tion program and did not really explain how infinite mathematics is applied to strictly finite physical things. The mathematical tool for explaining applicability here is strict finitism.

Acknowledgements

The research for this article is supported by the Chinese National Social Science Foundation (grant number 05BZX049). My research in philosophy of mathematics began during my graduate study at Princeton many years ago. I am deeply indebted to Princeton University and my advisors John Burgess and Paul Benacerraf for the scholarship assistance and all the help and encouragement they offered. An earlier version of this article was presented at the Shanghai Conference on Philosophy of Mathematics and the Beijing Conference on Analytic Philosophy, Philosophy of Science, and Logic, both in May 2005. I would like to thank the participants for their comments. Finally, I would like to thank Esther Rosa- rio, the language editor for Synthese, for the numerous corrections on my English presentations and for her many suggestions that help make the article clearer and more readable.

Notes

1. See Burgess (2004) for a characterization of the anti-anti-realism position. In this article, I will use the term “realism” in a broad sense. It will apply to both anti-anti-realism and other stronger realistic positions.

2. This is exactly what my truly naturalistic and scientific approach to philosophy of mathematics wants to do. See the last section for more details, including an explanation of what I mean by “true” naturalism.

3. In particular, the view that cognitive subjects can “refer to” or “be committed to” abstract entities appears incompatible with the physicalist view that cognitive subjects are just human brains, that is, some special kind of physical systems, which can have physical interactions with their physical environments only. See the last section of this article and other articles of mine on this subject.

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