Anti-Realist Theories

There are a range of anti-realist solutions to the challenge of Platonism. These Anti-Realist Theories are united in denying that there are any such things as mathematical objects. Mathematical objects simply do not exist. There are no such things.

Theological Significance
These anti-realist solutions immediately remove the challenge posed by the existence of abstract objects to God’s being the sole ultimate reality because on anti-realism there just are not any such objects, and therefore God is the only uncreated, self-existent, necessary, eternal being.

In order for Platonism to be a defeater so to speak of God’s unique self-existence, the person would have to prove that Platonism is true and that all of these anti-realists alternatives are false. There is no realistic prospect of doing that.

Anti-Realist Solutions
These are just some of the anti-realisms that are on offer today. There is a real potpourri of alternatives. There are others that are not even on this list. As seen, there are a great number of options available today which would not commit somebody to the reality of uncreated objects of any sort. Platonism is only one view – a tiny view – in the whole range of views about the reality of these objects. These other views, many of them, are very plausible.

Free Logic
Free Logic is a type of logic that has only been developed since about the 1970s. It is a very recent development in the study of logic.

According to Free Logic, terms to refer to things can be used even though those things do not exist. For example, a hole in a shoe can be referred. The shoe exists, but it is not as though in addition to the shoe there is something else, namely, the hole in the shoe. The hole is not a thing. It is not an object that exists. What there simply is is a shoe that is shaped in a certain way, but the hole is not something different. Or if a person says, “There is a lack of compassion in the world.” The person is not committing himself or herself to things called “lacks.” There are not things out there in the world – objects - “lacks” when a person says there is a lack of compassion in the world. Or if a person says “Wednesday is the day of the faculty meeting.” The person is saying something true but he or she is not committing himself/herself to the reality of Wednesdays. The person is not saying that Wednesdays are objects that actually exist. Or if a person says, “The pilot’s quick thinking averted the terrible accident on the Hudson river.” The accident never happened but the person can still refer to it in a true sentence.

So Free Logic is a logic that enables one to talk about and refer to things even though those things do not exist. What the Free Logician can say is that mathematics sentences, like 2+2=4, are true even though the terms in those sentences do not actually refer to anything. There is no such thing as 2+2 or 4 anymore than there is such a thing as the hole in a shirt or a lack or a Wednesday. That is the alternative of Free Logic.

Figuralism
Figuralism is a different form of anti-realism. Figuralism points out that much of our language, a great deal of ordinary language, is figurative in nature. It is metaphorical in nature. If somebody says, “It is raining cats and dogs outside,” the person said something that is true but it does not mean there are animals falling from the sky. This is a figure of speech for saying that it is raining hard outside. So it would be inept to take that statement literally. It is figuratively true that it is raining cats and dogs outside. Or if somebody is angry, another person might say, “She has a bee in her bonnet.” That is true, but not in a literal way. That is a figure of speech.

Similarly, the Figuralist will say that mathematical discourse is very plausibly interpreted as a sort of metaphorical or figurative discourse. It is not meant to be taken literally as referring to things like numbers. These are what one philosopher calls existential metaphors. They are figurative ways of speaking of things but there really are not such things in a literal sense. That would be Figuralism.

Neutralism
Neutralism is yet a third form of anti-realism. Neutralism agrees with Free Logic that terms to refer to things that do not exist can be used. When things are referred, the statements are just neutral with respect to whether those things exist. So if somebody says, “The weather in Atlanta today is balmy” that person is not committing himself or herself to an object called “the weather” as though “the weather” is something that exists. Or if somebody says, “The view of the Jezreel Valley from atop Mount Carmel was gorgeous” that person is not committing himself/herself to an object “the view” of the Jezreel Valley. It is not as though there is an object that is in the world called “the view of the Jezreel Valley.” Or if somebody says, “The price of the tickets was ten dollars” that person is not committing himself or herself to the reality of objects called “prices.” In many, many different ways terms in ordinary language are used to talk about things without committing to the reality of those things.

Certainly sometimes people do mean to speak in a metaphysically committing way. If somebody says, “This table is made out of wood laminate,” there that person is pretty clearly committing himself or herself to the reality of the table. What will tip people off to whether or not a person is thinking that there is a real object will usually be personal effects – rhetorical statements maybe emphasizing like “it really does exist” or the context. But the Neutralist will agree with the Free Logician that people often use terms to talk about things without thinking there are objects that correspond to those. So the person would agree with respect to mathematical objects that, when statements like 3 x 3 = 9 are made, those terms are just neutral as to whether or not someone is committed to the reality of mathematical objects.

The Neutralist goes farther, however, than the Free Logician because the Free Logician thinks that if someone says “there is” something then that person is committing himself or herself to the reality of that thing. The Neutralist would say that even expressions like “there is” are ontologically neutral. Someone can say, for example, “There are deep differences between Republicans and Democrats” without thinking that he or she committed himself or herself to objects in the world called “differences” and that some are “deep.” The expression “there is” and “there are” in English are very light in their ontological commitments. It will be, again, personal factors such as context, inflection of somebody’s voice, saying “there really is an abstract object” that will tip someone off as to whether or not the person meant to make an ontological commitment. So the Neutralist goes even further than the Free Logician. The Neutralist will say that really there is not anything in language that in virtue of its meaning commits somebody to saying there really are those objects that either correspond to the terms that person uses or are the things that person says “there is” or “there are.”

Neutralism is a view gives a very plausible account of ordinary language. When applied to mathematical discourse, it allows a person to affirm the truth of mathematics but to simply say it is neutral in terms of its commitments to objects.

Fictionalism
Fictionalism is a quite different form of anti-realism. The Fictionalist, like the Platonist, agrees that if somebody uses terms to refer to something, or if somebody says “there is” or “there are” something, then that person is committed to the reality of the things that he or she says “there are” or that he or she refers to. So the Fictionalist accepts those criteria for ontological commitment that the Neutralist and the Free Logician reject. But why is the Fictionalist then not a Platonist? Because Fictionalists think that those statements referring to or saying that “there is” or “there are” certain things are false. They are fictional. They are not true. So the Fictionalist will take the radical line that it is not true that 2+2=4. It is not true that 3 is greater than 1. It is not true that there is a prime number between 2 and 4. If somebody says that is crazy – those seem to be obvious truths, even necessary truths – the Fictionalist will remind that person that on his or her view to say 2+2=4 is to make a radical metaphysical statement that there is an abstract object named “2+2” and there is an abstract object named “4” and that those two objects are the same object. And that is not at all obvious. So the Fictionalist will say if a person accepts these criteria for how people make ontological commitments then it is far from obvious that statements of elementary arithmetic are true. That actually turn out to be radical metaphysical assertions that we have no reason to think are true.

The Fictionalist will say these statements are true within the story of mathematics; within standard arithmetic they are true. In this sense they are like statements of fiction. It is not true that Sherlock Holmes lives at 221B Baker Street. There is no such place. There was no such person. But in the Arthur Conan Doyle stories, it is true that Sherlock Holmes lived at 221B Baker Street. So the Fictionalist will say statements of mathematics are true in the story of mathematics in the same way that some of these statements are true in the Conan Doyle stories of Sherlock Holmes. But other statements will be false. It is not true, for example, that Sherlock Holmes met Henri Fauveau in those stories. That would not be true in the story of Sherlock Holmes. Neither is it true in the story of standard mathematics that 2+3=4. That is Fictionalism.

Pretense Theory
Pretense Theory is another anti-realism that takes inspiration from theories of fiction. They work largely off of the brilliant pioneering work of a philosopher at the University of Michigan named Kendall Walton. Walton’s work on fiction holds that fiction is an extension of children’s games of make-believe. Walton notes that children invest enormous amounts of time and energy in games of make-believe. He says it would be very surprising if, when people reach adulthood, they just give this up all of a sudden and no longer make-believe. Walton says, in fact, we do not give it up. This is what fiction and drama and film and literature and art is all about. These are, in effect, adult games of make-believe. He says what is crucial to fiction is not that the statements are false. A novel about the future like George Orwell’s 1984 could turn out to be true. It could all turn out to be true, but it is still fiction. Or the story of Hamlet might be true on some other planet somewhere in another galaxy in the universe for all we know, but Hamlet is still fiction. It is not the falsity of the story that makes something fictional. Rather, in Walton’s analysis, what makes something fictional is that it is prescribed to be imagined as true. We are to imagine that there is a Danish prince named Hamlet and that he did such-and-such. Or we are to imagine that there was a detective living in London who had a colleague named John Watson who did tremendous exploits. Then within this fictional world of imagination the story can be unfolded. So what is essential to fiction, in Walton’s view, is this act of making believe or imagining something to be true. The statements are prescribed to be imagined as true. They may or may not actually be true. But in either case what is essential to fiction is the prescription to be imagined as true.

Apply this to mathematics. In mathematics, people are, in a sense, prescribed to imagine the axioms to be true. A person is prescribed to imagine the elementary arithmetic axioms to be true. Then it is possible to derive all of the theorems. Or someone imagines the axioms of set theory to be true. Then the mathematician can derive all of his or her theorems. So the whole thing is a sort of species of make-believe. Far from being a crazy view of mathematics, this is a view of mathematics that many mathematicians themselves actually entertain. They would say that the mathematical axioms are postulates which a person postulates and then the person derives his or her deductions from them. But somebody is quite free as a mathematician to adopt a different set of postulates, a different set of axioms, and to explore that. So there is a wide variety of set theories that are on offer today. There is not simply a single set theory in mathematics. There is a range of set theories. These have different ontological commitments. Some commit people to sets. Some commit people to a different sort of objects called classes which are different from sets. So Pretense Theory will say that because a person is merely pretending or imagining these things to be true, he or she is not committing himself or herself to the reality of these objects anymore than committing himself or herself to the reality of Sherlock Holmes in imagining that this fictional world is the case.

Neo-Meinongianism
Neo-Meinongianism is one of the wildest anti-realisms. This stems from an Austrian philosopher Alexius Meinong who lived at the end of the 19th and into the early 20th century. Meinong was concerned to develop a theory of objects. He called his philosophy “Object Theory” – in German, Gegenstande Theorie. What Meinong maintained is that there are objects that do not exist. He says although it may sound paradoxical, there are things of which it is true that there are no such things. Unicorns. Centaurs. Fairies. The accident that was prevented. Holes. There are things which do not exist, Meinong would say. He develops this whole theory about these objects. On this view the Neo-Meinongian (that is to say, the modern follower of Meinong of which there are several in the world today) would say that mathematical objects are objects that do not exist. That would be one form of anti-realism.

Modal Structuralisn
This is a view that has been defended by Geoffrey Hellman, who is a philosopher of mathematics. On this view, mathematics is about structures. The idea here is that numbers are positions in structures. So the third position in a structure would be 3, the fourth position would be 4, and so forth. There are not objects called numbers. They are just positions in a sort of abstract structure. Some Structuralists in mathematics think that structures exist. This would be a kind of abstract object. A Platonist would say there are not numbers but there are structures. But the Modal Structuralist will say there could have been such structures. What he would say is that if there were numbers then 2+2 would equal 4[4]. Or if there were numbers the square root of 9 would be 3. So you can see he reinterprets mathematics in such a way that it does not commit someone to the reality of the objects that appear in those statements. These are simply paraphrased in such a way as to say that it is possible that there are such things, and if there were then they would have these properties.

So Modal Structuralism is is a kind of Paraphrastic strategy. That is to say somebody offers paraphrases of mathematical sentences that do not involve that person in commitments to these abstract objects.