What Anti-realism in Philosophy of Mathematics Must Offer†

 

Feng Ye*

I will present the challenges to anti-realism in philosophy of mathematics, based on some sympathetic interpretations of the intuitions supporting realism. I will argue that current anti-realistic philosophies have not met these challenges, and that is why they cannot convince those realists (or anti-anti-realists) who emphasize respecting working scientists’ understanding of mathematics. This will motivate a new approach to an anti-realistic philosophy of mathematics that can meet the challenges and can perhaps convince those anti-anti-realists.

 

1. Introduction

The goal of this paper is to set the goals for a new approach to an anti-realistic philosophy of mathematics.

The realism vs. 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 & Burgess 2005; Colyvan 1999, 2002; Baker 2001, 2005): As long as mathematicians and scientists seriously attempt to refer to mathematical entities and seriously assert mathematical theorems in their best theories, they already justify that mathematical entities exist and mathematical theorems are true in some proper sense, because mathematicians’ and scientists’ methodologies and judgments should be respected, and there are no stronger or more superior standards for justifying existence and truth. Note that this does not require mathematical entities to be strictly indispensable in sciences. Burgess and Rosen emphasize respecting working mathematicians’ literal understanding of their own assertions, and Colyvan and Baker further emphasize various pragmatic values of mathematics in sciences besides being used as tools for stating theories and drawing logical consequences. This position implies that it is philosophers’ duty to answer any philosophical problems generated by abstract entities, e.g. the epistemological problem (Benacerraf 1973), and it is not philosophers’ right to claim, based on their philosophical principles (or prejudice), that mathematical entities do not exist and mathematicians and scientists are so wildly and systematically wrong.

However, some philosophers are not moved by these. They hold the opposite intuition that mathematical entities are unlike robustly real physical objects and that scientists seem to treat them differently in their theories. Therefore, some of them contend that scientific confirmation cannot reach mathematics (Maddy 1997, 2005a, 2005b; Sober 1993; Leng 2002); some argue that we are entitled to “take back” the assertions on the existence of mathematical entities in science (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 of who has the burden of proof (Rosen & Burgess 2005).

Now, a philosophical theory should not be like a legal self-defense, where one can presume one’s own innocence and wait the other side to prove that one is guilty. In particular, anti-realists should not just argue that realists or anti-anti-realists have not conclusively proved that we have to commit to abstract entities. A philosophical theory is to resolve genuine 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 reasons supporting realism. They should not just poke holes in the arguments refuting anti-realism.

Therefore, in this paper, I will try to explore the challenges for anti-realism by interpreting the intuitions supporting realism. The fundamental idea is that, after denying that abstract mathematical entities exist, anti-realists should explain what then really exist in mathematics and should provide a literally truthful account for every aspect of mathematical practices by referring to what really exist, especially, aspects that are taken by realists as evidences supporting realism. In other words, wherever realists have (or appear to have) an explanation of some aspect of mathematical practices by resorting to abstract mathematical entities, anti-realists should not simply reject drawing the realistic conclusion about abstract entities without providing an equally ‘realistic’ (i.e. literally truthful) explanation of the phenomenon as a substitute. In particular, they should not label the phenomenon by a name and walk away, without offering a literally truthful explanation by referring to what really exist. 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 objectivity of infinity in any format and they must work under the assumption that there are only finitely many concrete objects in total. This will imply that they must show that references to abstract entities and assumptions implying infinitely many entities are in principle dispensable in mathematical applications. That is, anti-realists must directly reject the Indispensability Thesis and must do so by showing that some sort of strict finitism is in principle sufficient for describing this finite physical world.

These could be the strongest challenges and the harshest requirements for an anti-realistic philosophy. No current anti-realistic philosophies can meet all these and I will argue that that is why they suffer from some serious objections from realists or anti-anti-realists, if one reads realism or anti-anti-realism sympathetically.

I must remind that I am not trying to refute anti-realism here, although some of my comments on current anti-realistic philosophies may sound like it. I am only trying to say here that current anti-realistic philosophies have not offered and sometimes have not even tried to offer the most essential thing that an anti-realistic account for mathematics must offer. Sometimes, they simply label the phenomena to be explained by a name (e.g. “empirical adequacy”, “nominalistic adequacy”) without really offering (or trying to offer) a ‘realistic’ explanation for the phenomena and resolving the genuine puzzles. Sometimes, they commit to something equally suspicious for a naturalist and nominalist (e.g. infinity, continuity of space-time, etc.) in the attempted explanations. Since I am not refuting anti-realism, in commenting on current anti-realistic philosophies, I am only raising issues to motivate a new anti-realistic approach that is perhaps better. I am not trying to refute their basic tenets.

At the end of the paper, I will briefly introduce a research project for a positive anti-realistic account for mathematical practices, to meet all these challenges and requirements. It will be a completely naturalistic and scientific study of human mathematical practices, viewing human mathematical practices as human brains’ cognitive activities.  

 

2. The Challenges for Anti-realism

Challenge 1: Anti-realism must explain what really exist on the mathematical side in mathematical practices (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 say what then really exist on the mathematical side and must provide a literally truthful account for meaning, knowledge, intuition and so on 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 on the mathematical side, but it seems that there must be something else so that those statements can be meaningful and mathematicians’ knowledge, intuition and so on can be accountable. Recall that formalism claims that a mathematical theory is a formal system of meaningless symbols. This is the typical wrong way of saying things by an anti-realist. A philosopher’s job is exactly to answer, very realistically (i.e. using literally truthful assertions), how a mathematical sentence can be meaningful although it does not describe an independent mathematical reality. Ordinary mathematics is not ‘meaningless’ for mathematicians.

Verificationism is therefore in a better position in claiming 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 enough. 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’. At least much more work needs to be done. Moreover, very importantly but unfortunately, instead of exploring what scientists’ actual knowledge consists in when they use the language of classical mathematics, some verificationists declare scientists’ successful uses of the language of classical mathematics as illegitimate uses (and suggest that only the intuitionistic uses, which were almost never practiced by scientists, are legitimate).

Fictionalism is also misleading in claiming that mathematical entities do not exist and therefore mathematical theorems are “literally false” (or “vacuously true”). If this “literal” meaning is what accounts for scientists’ knowledge, experiences and intuitions regarding mathematics and its applications, then, as realists or anti-anti-realists like Burgess (2004) contend, we should respect scientists and admit that mathematical entities exist; if it is not, then the meanings of mathematical statements should be more properly construed in other ways, and calling mathematical theorems “literally false” is unhelpful and confusing. Without answering what really exist after denying the existence of mathematical entities, fictionalism simply leaves meaning, knowledge, intuition and so on in mathematical practices unaccounted.

Some fictionalists might think that accounting for these is unimportant, because what makes mathematics applicable has nothing to do with what mathematicians understand and grasp as meaning, knowledge, and intuition and so on. However, this amounts to saying that working mathematicians and scientists just hit upon the right mathematical theory for a type of application by chance. That is incredible, and more importantly, that is actually based on the assumption that some philosophical principle (i.e. the nominalism intuition) can overrule scientists’ understandings and judgments. For scientists, there are scientifically valid reasons why, for instance, the Riemann space theory is applicable for describing real space-time. These reasons, for instance, the reason that Riemann spaces are (approximately) isomorphic to space-time, are related to their understanding of meanings of statements in the Riemann space theory, their geometrical intuitions ‘about Riemann spaces’, and their knowledge ‘about Riemann spaces’. Realists’ claim is exactly that scientists assert literal existence of Riemann spaces in their understandings. If anti-realists dodge accounting for scientists’ understanding of meanings and their knowledge and intuitions, they will not be able to meet realists’ this challenge.

Moreover, accounts for meaning, knowledge, and intuition and so on must themselves be realistic accounts, or accounts consisting of literally true assertions. For instance, anti-realists should not say, “scientists are using Riemann spaces as models for simulating real space-time”, because this assertion is “literally false” for them, since Riemann spaces do not exist. Similarly, one should not say, “mathematicians have geometrical intuition about Riemann spaces as fictional entities”, because fictional entities do not exist and one cannot literally have intuitions ‘about’ nothingness. I will not venture into the metaphysics of fictional entities here. From the nominalistic and naturalistic point of view, the real problem for these “literally false” explanations seems to be that they have not really explained what is to ‘use fictional things’, to talk ‘about fictional things’, or to have knowledge or intuition ‘about fictional things’. Until we have a completely realistic account (without again referring to ‘fictional things’) for these, realists or anti-anti-realists can always come back and claim that their indispensability in sciences shows exactly 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 at stake is: Do our actual mathematical practices and do working scientists’ understanding of them imply realism? For example, if figuralism (Yablo 2001, 2002) is applied to the statements about Riemann spaces, it will imply that the statements about Riemann spaces have some real content that is not about Riemann spaces and is actually not about any particular things, because they are logical truths. Mathematicians could not have understood that alleged real content. No geometrical intuition conveyed by the original statements is in that alleged real content. That could not be what mathematicians really mean. Again, this does not meet realists’ challenge that working mathematicians’ actual understanding, knowledge and intuitions imply literal existence of mathematical entities (Burgess 2004).

Similarly, mathematicians are obviously not talking about ideal agents, possible concrete inscriptions on papers, and so on (cf. Hoffman 2004, Chihara 2005). Approaches paraphrasing mathematical statements into statements about ideal agents, possible inscriptions on papers and so on have not directly met 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 like to talk 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 existence of atoms, about which perhaps no working scientists have any doubt today, no matter in their “scientific moments” or in their “philosophical moments”.

Then, a philosopher’s task is to clarify the 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 the so-called Ockam’s razor principle. For a naturalist, these principles must be less certain than what working scientists unanimously hold. Instead, they must speak in a language understandable by scientists, and must provide an account that is acceptable by working scientists, judged by their knowledge and academic standards. If they could provide a realistic and literally truthful scientific account for scientists’ understandings, knowledge and intuitions without assuming that mathematical entities literally exist or exist in any mysterious sense that only some philosophers appear to understand, then they might be able to convince working scientists.

On this respect, philosophers do not even have to be so modest as to claim that philosophical analyses never do anything good to sciences, or that philosophers should never suggest anything to working mathematicians. (cf. Burgess 2004 and Leng 2005) Mach’s analysis of relativity of space was alleged to have good influence on Einstein. Philosophical analyses certainly cannot substitute constructive scientific work. What they can offer is to dispel away some dogmatic faiths or some type of illusions due to our thinking habits, which, under some circumstances, may actually be hindering new scientific explorations. The realistic faith about classical mathematics could be such a dogmatic faith, if it indeed comes solely from our thinking habits, not really justified by sciences. Rejecting that faith can only mean taking a more liberal view on possible mathematical practices and encouraging more ways of trying mathematical practices.

 

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 some 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 those areas in sciences. On the other side, nothingness certainly could not structurally resemble any real things and could not be related to any real thing in any meaningful way. Anti-realists should not simply deny such relationships, which will again leave working scientists’ reasons for applicability of mathematics unaccountable. Anti-realists must explain what really exist 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 exist on the mathematical side, and that our knowledge of such relationships are also 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 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 understandings of Riemann spaces. Anti-realists have to admit scientists’ actual intuitions and judgments, and have to explain them realistically and scientifically, by referring to what really exist in mathematical practices (without assuming that Riemann spaces themselves 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 as a whole (Melia 2000; Hoffman 2004). These concepts may be of some interests, but they say nothing about such genuine relationships between the mathematical and the physical, and nothing about scientists’ reasons for applicability of a particular mathematical theory to a particular type of natural phenomena based such relationships. ‘Nominalistic (or empirical) adequacy’ is a name for the results observed. It is not an explanation of why those results obtain. It does not explain, for instance, what is special about Riemann spaces that makes Riemann spaces applicable in modeling space-time, and why scientists did not use the theory of finite groups, or anything else, to model space-time. (None of them exists anyway.) Actually, if Riemann spaces did exist and were literally (though approximately) isomorphic to the physical space-time, there would be an explanation of their nominalistic or empirical adequacy. Realists’ claim is just 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 (or exist in some mysterious sense).

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 things in his cases. However, many mathematical applications are for observable things. Some anti-realists appear to be claiming that there is just no more explanation for the applicability of a mathematical theory for a type of natural phenomena even at the observable level (e.g. Riemann spaces and space-time), and that scientists’ explanation for it is just wrong. These claims seem to have gone beyond what van Fraassen originally accepts. (Hoffman 2004 takes her fictionalist view to be a completion of van Fraassen’s views.)

 

 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 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 commit to rules as abstract entities and commit 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 is about 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 in playing games. However, the (approximate) structural isomorphism between Riemann spaces and real space-time structures seems to be objective and is the objective reason for our successes in modeling space-time structures by Riemann spaces. If scientists were even a little bit 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 sciences, which is not in kids’ games, shows that mathematics is not merely a bunch of make-beliefs. Until anti-realists can clearly explain what this objectivity consists in and how this objective relation between the mathematical and the physical is indeed the objective reason for applicability of Riemann spaces (without assuming that Riemann spaces exist), they have not met 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 social-cultural constructions or conventions) may not care about this challenge. Criticizing them is usually realists’ job. I propose this as a challenge here, because I take anti-realism in mathematics as a defense for common sense realism and scientific realism. It tries to clarify puzzles due to the 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 realism and objectivity altogether.

 

Challenge 4: Anti-realism must explain the apparent obviousness, universality, a priority and necessity of simple arithmetic and set theoretical theorems, and they must also provide a consistent account for 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, a priority and necessity of “5+7=12”, and give reasonable explanations as to why we strongly believe that “5+7=12” is so. Moreover, arithmetic, set theory and logic are tightly entangled. Some simple theorems in arithmetic and set theory, such as “5+7=12” or “AÈB=BÈA”, appear to be logical truths in disguise. The common wisdom is that logical truths are universal, a priori and necessary truths. The universality, a priority and necessity of arithmetic are obviously closely related to the same characteristics for logic. Anti-realism must provide an account for logic consistent with their general ontology and epistemology, consistent with their accounts for arithmetic and simple set theory, and consistent with their general accounts for mathematics.

One attempt to explicate the truth in “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  

$₅xP(x)Ù$₇xQ(x)ÙØ$x(P(x)ÙQ(x))®$₁₂x(P(x)ÚQ(x)).

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 what all these say 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 will suspect 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 appear to refer to abstract entities (i.e. infinitely long sentences). This perhaps shows that one has to deal with the content of statements that appear to refer to the alleged abstract entities (or fictional entities) directly. There is no magical ways 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 must also account for our apparently valid intuitions ‘about infinity’.

Philosophers favoring anti-realism or nominalism may still hold different views regarding infinity. For example, Field (1998) and Yablo (2002) both accept the objectivity of arithmetic truths involving infinity (when properly interpreted). Field refers to a so-called “cosmological assumption” on infinity of the universe in defending the objectivity of arithmetic statements involving infinity. However, as of today, physics has not given a definite opinion regarding if space-time is infinite and continuous or finite and discrete. More importantly, in almost all areas of sciences so far, the applicability of mathematics is independent of the physics conjectures about continuity or discreteness of real space-time. We apply infinite mathematics also in economics, which is certainly about finite and discrete things. If an account for mathematics and its applications depend on a specific physics assumption 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 infinity of this physical universe, then they must explain what that infinity means. They also face a similar epistemological problem as realists face, namely, explaining how the 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 sciences, and it is likely that realists can simply take over their explanations to explain our knowledge about just any abstract objects, as Quine already did. After all, as long as the door is open for accepting 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 the notion of an objective “mathematical possibility”, according to which infinity is “mathematically possible”. Now, if the universe is indeed finite and there are really only finitely many concrete objects in total, what justifies this mathematical possibility of infinity? It seems that it will be again some sort of holistic justification based on their usefulness in sciences. Then, the door to Quinean realism is opened.

Therefore, rejecting the objectivity of infinity is the only way to be a coherent nominalist. This assertion was already made long ago by Goodman and Quine (1947) and it appears to be ignored by some contemporary philosophers. This means that for any assertion containing quantifiers intended to range over infinite domains, either it should subject to a proper anti-realistic interpretation, or one has to be ready to admit that it can be vacuously false or vacuously true. These include assertions about the consistencies of formal systems, assertions about Turing machines, and assertions expressing simple universally quantified arithmetic laws and so on.

We do seem to have a strong intuition that there are objective truths involving infinity, for instance, the commutative law of addition, or the law expressed as an assertion about a Turing machine implementing the addition function, or the consistency of some very simple formal systems and so on. Therefore, anti-realists must show that literal truth of such assertions involving infinity is not presumed in sciences nor implied by sciences, and is not needed in the anti-realistic account for mathematical practices and mathematical applications either. On the other side, anti-realists must also answer what are truly objective in our mathematical knowledge that appears to involve infinity if it is not really from an objective infinity, and they must also provide a satisfactory explanation for why humans have a strong intuition on the objectivity of infinity. In other words, similar to the cases for meaning, knowledge, relationships between the mathematical and the physical and so on, anti-realists must account for our various apparently valid intuitions ‘about infinity’ without assuming the objectivity of infinity.

 

Challenge 6: Anti-realism must show that references to infinity and abstract mathematical entities in mathematical applications are in principle dispensable.

Field (1980) used to try to explain how a bunch of ‘false statements about nothing’ could be useful by the notion of conservativeness. This 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. A continuous mathematical model provides a simple way of modeling population growths on the Earth. If we had a gigantic computer (or just a gigantic brain) that could simulate all people on the Earth (on aspects related to reproduction), than we would have a literally more accurate description of population growths on the Earth and it may give us more accurate predictions on future population growths. This literally more accurate description will refer only to finite and discrete physics quantities, such as the number-of-people property, as well as their representations in the computer, and it will not refer to infinity or abstract mathematical entities such as real numbers, differentiable functions and so on. It will be a nominalistic theory. This is not a peculiar example. Considering the fact that everything below the Planck scales (about 10^-35m, 10^-45s etc.) is obscure to current physics, actually almost all current applications of continuity and infinity are for glossing over microscopic details. There is a chance this example essentially represents all applications of mathematics. (This is exactly what anti-realists must show in demonstrating that infinity and abstract entities are in principle dispensable in applications.)

Now, this nominalistic theory will be more complex than the continuous model of population growth. However, the kind of simplicity that infinity and abstract mathematical entities brings here is just to help to gloss over details in finite but very complex things so that we can have a human-tractable (and current-computer-tractable) but literally less accurate theory, and infinity (and therefore abstract entities) can in principle be eliminated to get a literally more accurate description of nature. It is doubtful that such simplicity can justify realism about infinity and abstract entities. References to infinity and abstract entities are perhaps practically indispensable for constructing a human-tractable theory here, but they may be indispensable only for constructing a simpler but literally less accurate theory, not for getting the real, more accurate literal truths.

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, namely, 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 move realists, for whom the world includes mathematical entities (or they claim that you should at least have an open mind to allow your world to include abstract entities).

Colyvan (1999, 2002) and Baker (2001, 2005) imply another objection to the thesis that the successful nominalizations of scientific theories will help anti-realism. They claim that mathematics is not just used to build models to represent phenomena or to deduce known conclusions. They cite examples to show that mathematics has unification powers, predicting and discovering powers, values in boldness for applying to new areas of phenomena, and genuine explanation 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, e.g. a discretisized and computerized version of a classical theory, actually has the same pragmatic values as the classical one except for simplicity, because they express the same physics regularity of nature and even state the physics regularity literally more accurately (although lengthier and more tediously). Moreover, it seems that all those examples by Colyvan and Baker can be accounted for with the picture that mathematical entities are used for modeling real things (Leng 2002). If a structurally richer model can simulate two types of phenomena unrelated to each other on apparent, that is the model’s unification power. Models are actually primarily used for predicting and discovering, not just for representing or stating logical inferences. For instance, computer simulations are used to predict future phenomena and discover unexpected consequences. Moreover, models are frequently migrated to simulate another type of phenomena unexpectedly. These are the pragmatic values of models corresponding to those pragmatic values mentioned by Colyvan.

In Baker’s latest example (Baker 2005), it is claimed that the arithmetic properties of prime numbers explain why the numbers of dormant years for some species of cicada are all prime numbers. This is considered an example of genuine explanation power of mathematics. However, obviously, what really explains those cicadas’ life-cycle patterns is the physical fact that the real physical quantities 13-years, 17-years and so on cannot be evenly divided into two or more smaller periods of years. It is not any properties of the alleged abstract entities 13 and 17 that explain the real phenomena. When we use a model to simulate some real phenomena, then what really explain the phenomena are the properties of real things that happen to be correctly represented by the model, not the properties of the model itself. For instance, if we successfully use a computer to predict the formation of a hurricane, what really explain the formation of the hurricane are the temperatures and motions of airs that are correctly represented by the computer, not the properties of the data and programs in that computer. The existence of the computer and truth of our assertions about data and programs in the computer are indeed necessary for the modeling to be successful, which is another issue and will be discussed below. The point here is that what really explain some phenomena in the universe are other real things and their properties in the universe, while the models or the alleged abstract entities 13 and 17 and so on are things that we appear to refer to in describing those real things. If anti-realists can explain how exactly mathematical models work without assuming that mathematical entities exist, then the genuine explanation of the phenomena in the universe is intact and the pragmatic values of the explanation (saving simplicity) are the same, and Colyvan and Baker’s objections can be dismissed.

Come back to the relevance of nominalization programs. From the logical (not philosophical) point of view, a real puzzle about applicability is due to the gap between infinity in mathematics and the finitude of the real world (from the Planck scale to the cosmological scale). To resolve the puzzle, we must clarify, for instance, the logic of using infinite and continuous models to approximate and simulate finite, discrete things. This puzzle is quite independent of any philosophical view about the ontological and epistemological status of those 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. Because, it is likely that a complete logical clarification will eventually show that infinity and apparent references to abstract entities can all in principle be eliminated in mathematical applications, which will then show exactly how mathematics helps to derive nominalistic truths about finite concrete things from nominalistic premises about finite concrete things (i.e. conservativeness) in plain logic. (It may also show how infinite mathematics greatly simplifies the derivation processes, as long as we are satisfied with less meticulously accurate conclusions.) Moreover, it may show that literal existence of infinity and abstract entities is irrelevant for explaining applicability, because what really explains applicability is on the contrary the fact that mathematical proofs used in applications allow eliminating infinity and abstract entities, so that the proofs can preserve truths about finite concrete things. That is, a real logical explanation of this puzzle of applicability of infinity to finite things may in the end have to imply dispensability.

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 commit to them. For instance, Leng (2002) claims that mathematical entities are used to build models to represent physical things, and claims that science confirms only the existence of those physical things, but cannot confirm the existence of the models themselves.

Now, computers are used to build models to simulate other things. However, computers really exist. Indeed, the failure of a computer modeling should not be taken as evidence that the computer does not exist or that our assertions about the 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 the data and programs in computers are literally true. Similarly, realists claim that while the failure of a mathematical modeling does not imply that the mathematical model does not exist or that our assertions about the model are false, but successes of modeling do require that the model literally exists and do 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. Of course, for anti-realists, models are fictional entities. However, this cannot convince realists, who claim that scientific applications raise mathematics from a game or fiction to science and who charge that anti-realists are intellectually dishonest or 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. So, these anti-realists are making literally false assertions in their philosophical papers according themselves. It clearly implies that these anti-realists have not 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. Some examples are cited to show how assertions about concrete things have to be expressed by referring to abstract entities. Then, Melia proposes the so-called “weaseling strategy”, namely, taking back what one asserted previously about the existence of abstract entities. It means that classical mathematics is nominalistic adequate simply in the sense that its consequences about concrete things 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 scales10^-35m, 10^-45s, etc.), Melia’s examples are beside the point for a real nominalist, because they all assume infinity. On the other side, sometimes we do take back what we asserted at first in everyday life. 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 what were asserted previously. That is, we have to show that what were asserted previously can in principle be eliminated. That is the case, for instance, when we refer to a rigid body in elementary mechanic. 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 pragmatic values. Then, why doesn’t this lead to Quinean realism?

These are the challenges for anti-realists. In summary, anti-realists must provide a positive account for the practices of classical mathematics including its applications, especially those aspects taken as evidences supporting realism by realists. They should not simply label some phenomena by a name (i.e. “nominalistic adequacy”) without giving real explanations, and should not be negative only (i.e. arguing that they do not have to commit to something). They should not again fall into resorting to some holistic confirmation to justify some obscure things. They must explicitly say what are real on the mathematical side and 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 and under the assumption that there are only finitely many concrete objects in total. They must realize that the real puzzle of applicability of mathematics is the logical puzzle of how exactly infinite mathematics is applied for describing finite real things.

 

3. Toward a Completely Scientific Account for Mathematical Practices

These challenges and requirements seem to have cornered anti-realism. Is there still a chance for anti-realism in 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 exist in mathematical practices. So, what really exist and what are 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 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 external things in some way. Our imagination activities do not literally create any ‘imaginary entities’. Only our hands can create things out of preexisting materials. All our minds do in imagination activities is creating mental representations residing 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 and what are really used as models in mathematical applications. (Scientists use their brains to model other things much like they use computers to model other things.) Anti-realists’ task will then be describing the functions of these mental representations in human mathematical practices and the relationships between them and other real things in the physical world, to explain aspects of mathematical practices, including the applicability of mathematics. An account for mathematical practices is thus a continuation and extension of cognitive sciences, dealing specifically with human mathematical cognitive activities. It is a completely scientific description of some natural phenomena. This is what is missing from the current anti-realistic philosophies. With that missing, one cannot answer, in realistic terms, what are the meanings of mathematical statements, or what the relationships between the mathematical and the physical consist in, 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 utterly naturalistic basic ontological and epistemological assumptions: (1) there is this physical universe, which could be finite and discrete or otherwise, 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 are no existences or truths beyond and above this concrete universe. These basic assumptions, which will be called nominalistic naturalism, seem to be clear and coherent. It is not Quinean naturalism. It is consistent with physicalism in the contemporary philosophy of mind (e.g. Papineau 1993), and consistent with the commonsense realism, scientific realism, and the general scientific naturalistic worldview, except for the fact that classical mathematics generates some puzzles.

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

A research project following these ideas is underway. The project will account for aspects of human mathematical practices by referring to the cognitive functions of mathematical mental representations in brains and their natural connections with physical entities out of brains. This paper actually presents the motivation and sets the goals for the project. Ye [2007a] will elaborate 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.

Then, papers Ye [2007b], [2007c], [2007d], [2007e] and the monograph [2007f] present the positive accounts for various aspects of mathematical practices. They include the work done so far in the project. Ye [2007b] discusses some aspects of meaning, knowledge, intuition and so on 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, as well as their connections with physical entities out of brains. Ye [2007d] 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 [2007c] discusses the apriority of logic and arithmetic from the naturalistic point of view. Finally, Ye [2007e] and [2007f] first explain how the question of applicability of mathematics can be formulated as a scientific question and transformed into a logical question. Then, they develop a strategy for explaining the applicability of mathematics, in particular, the applicability of infinite mathematics to this finite physical world. The strategy is first to show 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 to show 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 conclusions about them.

More work is still to be done in the research project. It is possible that this still cannot convince some realists. In particular, since its basis is naturalism, it will not convince those who loath naturalism (e.g. Gödel). 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 vs. metaphysical (i.e. nominalistic) intuitions. Moreover, such a research should have its own values, independent of any philosophical positions, as researches into human mathematical cognitive processes and researches into the exact logic in applying infinite mathematics to this finite physical world.

Finally, I will briefly compare this research project with other closely 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 Núñez 2000). However, they did not discuss many issues that concern philosophers and logicians, such as objectivity in mathematics, the apriority of logic, 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 over 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 is reminiscent of Field’s notion of conservativeness, but Field assumed infinity in his nominalization program and did not really explain how infinite mathematics is applied to finite physical things. The mathematical basis here is strict finitism.

 

References

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Burgess, J. P.: 2004, ‘Mathematics and Bleak House’, Philosophia Mathematica (3) 12, 18–36.

Chihara, C.: 2005, ‘Nominalism’, in Shapiro (2005), 483–514.

Colyvan, M.: 1999, ‘Confirmation Theory and Indispensability’, Philosophical Studies 96, 1–19.

Colyvan, M.: 2002, ‘Mathematics and Aesthetic Considerations in Science’, Mind 111, 69–74.

Field, H.: 1980, Science without Numbers, Basil Blackwell, Oxford.

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

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Lakoff, G. and R. Núñez: 2000, Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, Basic Books, New York.

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Leng, M.: 2005, ‘Revolutionary Fictionalism: A Call to Arms’, Philosophia Mathematica (III) 13, 277-293.

Maddy, P.: 1997, Naturalism in Mathematics. Clarendon Press, Oxford.

Maddy, P.: 2005a, ‘Three Forms of naturalism’, in Shapiro (2005), 437–59.

Maddy, P.: 2005b, ‘Mathematical Existence’, Bulletin of Symbolic Logic 11, 351–376.

Melia, J.: 2000, ‘Weaseling away the Indispensability Argument’, Mind 109, 455-479.

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Yablo, S.: 2001, ‘Go Figure: A Path through Fictionalism”, Midwest Studies in Philosophy 25(1), 72-102.

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Ye, F. [2007a]: ‘Naturalism and Abstract Entities’, available online at http://sites.google.com/site/fengye63/ .

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

Ye, F. [2007c]: ‘Naturalism and Objectivity in Mathematics’, ibid.

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

Ye, F. [2007e]: ‘Applicability of Mathematics as a Scientific and Logical Problem’, ibid.

Ye, F. [2007f]: Strict Finitism and the Logic of Mathematical Applications, ibid.