Relations as objects of knowledge. Ontological philosophy explains relations as aspects of the world that exist because of the essential nature of space and how space contains bits of matter at any moment, and correspondence to them explains, as we have seen, how mathematical propositions are true. That means that mathematics is prior to empirical science in the sense of being ontologically necessary. However, necessity in the sense of being certain is what has traditionally been thought to make mathematics different from empirical science. Certainty is what is relevant about mathematics when the project is justifying belief in certain propositions by how they are related to what is known in other ways. Thus, epistemological philosophy approaches mathematical objects as objects of knowledge, rather than as aspects of the world, and it is not obvious that what mathematics is about are the most basic relations that hold in the world.
The problem of mathematical knowledge. When the certainty of mathematics is taken for granted, the problem of mathematical knowledge is to explain how such certainty is possible, that is, why it is more certain than what is known by ordinary experience of what happens in the world.
It is somewhat misleading to think of the certainty of mathematics only as a problem, for in the beginning, that is what inspired belief in epistemological philosophy. In ancient Greece, mathematics was taken as an example to show the possibility of philosophy as a superior kind of knowledge of the world, one that revealed necessary truths. In the
Meno, for example, Plato describes Socrates as asking a slave boy a series of questions about some lines he draws in the sand which lead the boy to recognize the truth of a special case of Pythagoras’ theorem (that the square built on the diagonal of a square is twice the area of the first square). That put the slave boy in a position to defend what he knew rationally, and Plato used that story to illustrate how knowledge is different from true belief.
Beliefs about whose truth one can be certain are what philosophy pursues out of its love of wisdom, according to Plato. Above the entrance to Plato’s Academy, the first university, it was written that no one should enter who does not know mathematics.
It is hard to overstate how important mathematics has been to the credibility of philosophy’s claim to provide a kind of knowledge of that is superior to our ordinary ways of knowing what happens in the world through experience. But given its role in epistemological philosophy, the issue about how the certainty of mathematical knowledge is possible becomes the issue of how realism is possible.
Theories of mathematical knowledge. To set the stage for considering the received explanations of the certainty of mathematics, let us consider briefly what ontological philosophy implies about the knowledge of mathematics. We will then take up the epistemological theories.
Ontological theories of mathematical knowledge. We have explained why mathematics is true by showing how its propositions correspond to relations as basic aspects of a spatiomaterial world. Geometry corresponds to the structure that space has as (part of) its essential nature as a substance, and that explains why the propositions of geometry hold of bits of matter in space as well as points. Arithmetic holds of the particular substances postulated by spatiomaterialism, because they all have spatial relations to one another, making it possible to pick out particular substances and to group them together in sets. But that does not explain how it is possible for rational beings like us to know that these propositions are true — and to know that they are true in a way that makes them certain in comparison to empirical science and other ordinary ways of knowing about the world.
The short answer is that mathematics is not certain, but merely prior to empirical science. Mathematical propositions are among the necessary truths proved by ontological philosophy. They are ontologically necessary, because they are entailed by the best ontological explanation of the natural world, namely, spatiomaterialism. That is the foundation that ontological philosophy uses to prove that propositions are necessarily true about the world, and mathematical propositions are among them, because they correspond to basic aspects of any spatiomaterial world. But to prove that propositions are ontologically necessary is not necessarily to prove that they are certain, that is, epistemologically necessary. Since spatiomaterialism itself is an empirical truth, the justification of what follows from it is ultimately empirical and, thus, falsifiable by experience. It is nevertheless prior to empirical science, because ontological explanations are prior to efficient-cause explanations. What follows from spatiomaterialism could be false, because spatiomaterialism could be false. But if what follow from it is false, we must give up our otherwise empirically well-founded belief about the basic nature of existence and deny that the world is constituted by its two, opposite kinds of basic substances.
Nevertheless, mathematics seems to be certain. It was not without reason that traditional philosophy took an epistemological approach to necessary truths. And the long answer to the question about why beings like us believe that mathematics is true and believe that it is more certain than science has to do with the nature of reason. Reason is a cognitive capacity that evolves in certain animals, and as we shall see (in Change: Evolutionary stage 9 and following), reason has an ontologically necessary nature which involves two forms of imagination. But it will be easier to explain the received, epistemological philosophies of mathematics if we anticipate that explanation with a brief account of them here.
Animals become rational as they evolve the use of language, and in a world of space and matter in time, it is plausible to suppose that those animals already have a spatial imagination by which they can understand the structure of space.
By "spatial imagination", I mean a brain mechanism (a system of representation in what will be called the "animal behavior guidance system") that uses spatial images of objects and temporal sequences of them to represent objects, their spatial relations to one another in three dimensions, and how their spatial relations change as a result of motion or being manipulated. At its core, it is a memory mechanism that records the locations of objects by lining up images of them in the order they would appear as a result of locomotion in each direction in space, and since "covert locomotion," that is, motor commands for moving the body that are not actually executed, can call up those images in sequence, it serves as a form of imagination that gives animals a nonlinguistic way of thinking about the basic geometrical structure of space and the effects of motion on their relations. (Spatial imagination is this brain mechanism that makes it possible for computers to generate what is called "virtual reality.")
Furthermore, in animals that can manipulate objects, such as primates, spatial imagination also includes an ability to think about the geometrical structures of objects and how they interact when being manipulated. Acts of imagination call up spatial images of objects in sequences that represent the effects of manipulating them in various ways.
Spatial imagination gives even nonlinguistic animals an intuitive way of understanding the structure of space, that is, that spatial relations among objects. And as suggested in the last chapter, since such brain activity involves a form of matter whose intrinsic nature registers what is happening throughout the forebrain, spatial imagination is what makes it appear that sensory qualia are located in phenomenal space. That is, its structure is what gives rise to complex phenomenal properties and what we are calling the unity of consciousness. In this context, however, it phenomenal appearance explains the faculty of intuition on which epistemological philosophy typically bases its theory about the nature of reason.
It is not surprising that such a cognitive faculty evolves in a spatiomaterial world, given that animals acquire food by ingesting other objects in space, for it gives them more power over objects in space. Indeed, we shall see that its evolution is inevitable in worlds where evolution can occur at all. But this nonlinguistic understanding of the spatial and temporal aspect of the world is inherited by animals in which language evolves, and in such animals, spatial imagination comes under the control of verbal behavior.
In order to understand a sentence about objects in space, users of language must construct its meaning in imagination. Spatial imagination makes it possible to connect words to particular material objects in space, and thus, learning the meanings of words involves the development of "abstract images," which correspond to properties and relations, or the aspects of objects in space that are called "abstract objects." (As we shall see, they develop in the brain as states that represent many different particular objects of certain kinds indifferently.)
Furthermore, learning to combine such words grammatically involves the development of complex representations, in which properties are related to the objects that have them and states of affairs are represented. Thus, language is a second system of representation. The capacity of language to represent basic aspects of a spatiomaterial world derives therefore, from the spatial imagination of the (mammalian) animal system of representation. (This is the role of what I will later call "natural sentences.")
However, rational imagination, as I will call it, depends on another kind of linguistic representation, in addition to the linguistic representations based on spatial imagination (or "natural sentences') and the representations of spatial imagination itself. The use of language, as we shall see, eventually makes the animals in which it evolves reflective. (This further stage in the evolution of language introduces what I will call "psychological sentences.")
The ability to use more complex sentences enables language-using animals to represent to themselves the (brain) states (such as perceptions, memories, beliefs, desires, and intentions) that occur in the process of perceiving and thinking about the natural world and to think about the roles that such states play in causing behavior and beliefs. Thus, these animals can reflect on the causes of their beliefs and behavior.
But in reflective animals, such reflective (brain) states can themselves be causes of the conclusions they draw about how to behave or what to do, and thus, they have earned a special name. They are called "reasons." In other words, reasons are basically just causes of conclusions that are represented as causes as an essential part of the process of causing such conclusions. Considering how language depends on spatial imagination to connect words to objects in the world, the control that language has over spatial imagination transforms the animal faculty of imagination into rational imagination, a capacity to think about the possible reasons for certain conclusions.
These three elements — the animal's spatial imagination, how it connects linguistic representations to the world, and how language eventually enables the animals in whom it evolves to reflect on the reasons for their beliefs and intentions — are essential to reason, and they explain why it seems that mathematical truths are certain. Spatial imagination is an intuitive way of understanding the structure of space, and thus, if spatial relations among substances are the basic subject matter of mathematics, it is an intuitive understanding of mathematical propositions.
There is, of course, a longer story to be told about how reflection on its operations evolves into explicit knowledge of geometry and arithmetic. But for now, let us simply notice that, as rational beings with such knowledge reflect on the causes of their beliefs, a difference between mathematics and empirical science will inevitably appear. Though it is possible to know the propositions of geometry and arithmetic by perception, in the same way as other facts about nature, it is eventually noticed that they have reasons for believing mathematical propositions that do not depend on perceiving what actually happens in the world. They seem forced to believe, for example, that a straight line is the shortest distance between two points and that two plus three is five by their very understanding of those propositions. Those beliefs seem especially compelling, because those facts about the world are built into the structure of their spatial imagination.
Thus, when epistemological philosophers reflect on how they know that mathematical propositions are true, the first hypothesis is that geometrical objects and numbers are objects of a special kind which are revealed only to rational intuition (or what ontological philosophy explains as the subjective, phenomenal appearance of rational imagination). That is basically Platonism about mathematics.
Given how theorems about geometrical figures and numbers can be derived from axioms, however, another possible hypothesis is that mathematical propositions are a result of logic or reasoning. That leads to forms of anti-realism about mathematics.
In either case, however, there there seem to be reasons for believing mathematical propositions that are sufficient, but which do not depend on perceiving what happens in the natural world. That explains the apparent certainty of mathematics. It can be known in a way that does not seem to be vulnerable to what is learned about the world through perception.
These epistemological theories do not lead to errors in mathematics, because what seems certain in this way actually holds universally in a spatiomaterial world. That is, what spatial imagination corresponds to is the basic aspect of the world in which rational beings find themselves.
It is interesting to notice that, since that basic aspect of the world is its spatial structure, or the aspect of the world that, more than anything else, makes the world whole, mathematics is a way of knowing about the wholeness of the world. And since it is known by subjects who are part of that world, mathematics is the part's knowledge of the basic nature of the whole of which it is part.
This ontological explanation of the apparent certainty of mathematical knowledge is the foundation for its critique of epistemological philosophy of mathematics.
Epistemological theories of mathematical knowledge. The approach of epistemological philosophy is just opposite to ontological philosophy. Instead of starting with ontology and showing that mathematical truths are ontologically necessary, epistemological philosophy starts by reflecting on how we know about the truth of mathematical propositions and tries to show that they are necessary in the sense of being certain, or epistemologically necessary. The basic form of success in epistemological philosophy of mathematics is realism about entities beyond what is known by ordinary experience of the natural world, and as we have seen, the fate of epistemological philosophy is sealed, because its realism involves metaphysical dualism. The problems of metaphysical dualism eventually leads to anti-realism.
As the example of Socrates and the slave boy in the Meno suggests, mathematical knowledge was the original inspiration for philosophy’s claim to have a superior way of knowing about the world. It was the first way philosophy ever claimed to prove there are necessary truths. Since epistemological philosophy began with Plato's use of the certainty of mathematics to illustrate the success of realism, realism in the philosophy of mathematics is now called "Platonism." Given the fate of epistemological philosophy, Platonism eventually leads to anti-realism. But in the case of mathematics, even most anti-realists affirm the certainty of its propositions. There is, however, a form of anti-realism that denies the certainty of mathematics by assimilating it to empirical science, that is, by denying that there is any basic difference between mathematics and science.
A brief account of the history of epistemological philosophy of mathematics follows, and having seen how ontological philosophy can explain why mathematics appears to be certain to those who reflect on how they know it, I will use the ontological theory of reason to show not only what is true and false in the traditional theories of mathematics, but also how the philosophical problems caused by the epistemological approach are solved by ontological philosophy.
Realism: Platonism about mathematics. For philosophers who argue from how we know to what can be known, success comes from showing that we have knowledge of the real existence of entities of some kind beyond a kind of knowledge that is taken for granted, that is, knowledge of reality beyond appearance. In the philosophy of mathematics, realism is called "Platonism," after its founder. But Platonism takes different forms in the ancient and modern worlds.
Ancient Platonism about mathematics. Plato’s explanation of what the slave boy learned from Socrates is that beings like us have a faculty of reason that makes us aware of objects that are fundamentally different from the objects of perception. That is how all genuine knowledge (as opposed to mere belief) was explained by Plato, and it is the model for Platonism in mathematics. Numbers and geometrical objects are part of a reality that Platonists believe lies beyond appearances in natural world.
According to Plato, the Forms in the realm of Being are different from natural objects, which are known by perception (that is, empirical knowledge), because the Forms do not change and never appear differently from what they really are. What enables us to know about them is rational intuition, which Plato repeatedly contrasted to perception, as knowledge to mere belief. But it is the difference in the natures of the objects being cognized that was supposed explain the certainty and necessity of mathematical truths.
Rational intuition of mathematical objects does involve appearances, according to ontological philosophy, for there is a faculty of rational imagination in the brain and its activity has an appearance to the subject by way of phenomenal properties (by generating bits of matter whose intrinsic natures register brain activity). But that is not the appearance of objects that are outside space and time, and the belief that the objects being grasped are Platonic Forms involves an insuperable problem, namely, Platonic ontological dualism.
Mathematical propositions hold of objects found in the natural world, and in order for Platonists to explain how our knowledge of such propositions is certain, they need a way to explain why truths about abstract entities in a realm beyond nature reveal something about objects that exists in nature. The main problem with platonic realism, as Plato himself recognized, is that there is no way to explain how objects outside space and time can have any effect on objects in the natural world.
Ontological philosophy avoids the problems of Platonic realism by taking mathematical objects to be aspects of the natural world, rather than abstract entities that exist in a transcendent realm. But it can also explain why they appear to be abstract entities. In both cases, abstract entities are reifications of concepts based on spatial imagination.
Though geometrical structures are always concrete parts of space, they seem to be universal, because space exists everywhere with the same three-dimensional structure. Since reflective subjects with spatial imagination recognize such geometrical structures in many different particulars, it is not surprising that they think of them as universals or abstract entities.
Likewise, though the material objects they count are concrete particular substances existing independently of one another, their spatial relations are what makes it possible for them to be grouped together, and since that makes the results of arithmetic operations the same everywhere, numbers seem to abstract entities.
Modern realism (Platonism) about mathematics. With the rise of modern philosophy, the problems with Platonism about mathematics were transformed, but not solved. Plato was a naive realist about both perception and reason. He believed that the objects of both forms of intuition (that is, perception and rational imagination) exist independently of the subject, but are nevertheless immediately present to the subject. The modern period began with the recognition that perception is mediated by appearances (or "ideas’) that are part of the mind, and that meant that rational intuition is likewise just another kind of appearance in the mind (what Descartes called clear and distinct ideas). That eliminated the problems caused by Plato's attempt to explain the relationship between the objects of perception and the objects of reason as the relationship between Forms in a realm of Being and visible objects in the realm of Becoming. But modern philosophers were still Platonists, in effect, because they believed that what makes knowledge of mathematics certain, in contrast to empirical knowledge, is that it is about abstract objects that exist independently of both the subject and the natural world. But instead of existing in a realm of Being, mathematical objects were taken to exist as ideas in the mind of God. In short, as an offspring of the marriage of Platonism and Christianity in the medieval period, the modern era had inherited a rationalistic theology in which God played the role of the realm of Being.
Modern philosophy still had to explain, however, why mathematics is true of the natural world. Indeed, the question was even more pressing, because the new science discovered laws of nature that are highly mathematical. Those laws described precise quantitative relationships among properties, such as distance, mass, time, and velocity, and since relations among different quantities of the same property are arithmetical, those physical descriptions required the truth of mathematical propositions.
Modern philosophy had, however, a ready solution, at least, until doubts about theistic supernaturalism late in the eighteenth century, because the objects of mathematics were assumed to be ideas in God’s mind. God created the natural world according to a rational plan, and since God had used mathematics to create a world governed by natural laws, the discovery of those laws was basically seeing into God’s mind. In The Assayer (1610), for example, Galileo described nature as a book that God had written in the language of mathematics. And Descartes used God to prove that our clear and distinct ideas about geometry corresponded to extension, the essential nature of the bodily substance. In other words, it was possible for rational beings to recognize the truth of mathematical propositions, because rationality comes from their being created in God’s image.
Anti-realism about mathematics. The problems of supernaturalism eventually made Platonism in either its ancient or modern form untenable. There is simply no way to prove the existence of entities existing beyond the natural world. But anti-realism about mathematics takes two fundamentally different forms, because mathematics still seems to be certain, even if realism is doubtful.
One form continues to accept the certainty of mathematics and tries to explain how there can be such self-evident truths without having to prove the existence of entities beyond what is given to ordinary experience of the world.
The other form takes the denial of the existence of platonic entities beyond the natural world to mean that mathematics must be about the natural world, and by assimilating mathematics to empirical science, denies that mathematics has the kind of certainty that is taken for granted by realists and other anti-realists.
Anti-realism that affirms the certainty of mathematics. The nineteenth century was a transitional period in the history of mathematics. Not only did the rise of naturalism made Platonism less attractive, but developments in mathematics itself also made it less plausible that mathematics describes the essential nature of a reality beyond the subject, natural or supernatural.
Though Euclidean geometry had once made it seem obvious that one kind of rational certainty somehow reveals the inherent nature of the natural world, the discovery of non-Euclidean geometry cast doubt on that assumption. The certainty of geometry seemed to depend more on the deducibility of theorems from axioms. And recognition that the arguments (about infinity) on which the calculus had been based were logically faulty focused mathematicians on the project of making mathematical proofs more rigorous. Though physics undoubtedly required mathematics for its spectacularly successful descriptions of regularities in the natural world, it was, by the beginning of the twentieth century, plausible to hold that the certainty of mathematics does not come from knowing a special kind of object that exists independently of the subject. Instead, it seemed possible to explain its special certainty as deriving from the nature of the rational subjects themselves.
Various theories of the certainty of mathematical truth have been proposed in the twentieth century, and disputes among them tend to be technical. But a rough sketch of two opposite approaches and their problems will put us in a position to see why naturalists now seem to have little choice but to treat mathematics as a species of empirical, scientific truth. Both of the following views give up the belief in independently existing, abstract entities, and both explain its certainty by holding that it is a kind of truth that is discovered within the mind. And both are just what would be expected of epistemological philosophers, given how ontological philosophy explains the ability of rational beings to know the truth of mathematical propositions. One takes account of the role of spatial imagination and attempts to reduce all of mathematics to objects of rational intuition, and the other takes account of the role of language in expressing those intuitions and attempts to reduce mathematics to logic or the structure of language.
Intuitionism. Intuitionism derives historically from Kant, and it reflects the assumptions of modern philosophy. Kant argued that mathematics is a priori knowledge about the natural world because it describes the structure of the forms of intuition (space and time) in which nature itself is presented in experience. Proofs of mathematical propositions involve the construction of mathematical objects in imagination, and thus, they must conform to the mind’s pure forms of intuition, space and time. But that means that mathematical truth hold necessarily and universally in experience of the natural world, because the two forms of intuition are also conditions of possible experience.
Kant was describing the process by which rational beings do actually come to accept the certainty of mathematical propositions, according to our ontological explanation of reason. The role that Kant ascribed to space and time as forms of intuition in understanding mathematics is explained by spatial imagination, and that accounts for knowledge of geometrical and arithmetical propositions.
The use of a language to control and to reflect on the structure of spatial imagination gives one a nonlinguistic understanding of what is meant by such concepts as "point", "line", "plane", and "sphere," and thus, one can "see" that there is a shortest distance between two points and that a line and a point not on it determines a plane. One can also recognize the truth of simple propositions, such as that exactly three lines intersecting at a point can be mutually perpendicular, that three planes can be mutually perpendicular, and that any closed plane figure with just three internal angles has three sides.
Ontological philosophy confirms Kant's theory of arithmetical knowledge in a similar way. Spatial imagination enable reflective subjects to think about the operations of singling objects out, combining them as groups, adding and subtracting members, and the like, and thus, they can recognize the truth of arithmetic axioms and construct theorems of arithmetic in imagination. That makes it seem that such truths about the world can be known prior to discovering their truth by perception, because what makes arithmetic true is the way in which space makes different bits of matter parts of the same world and that aspect of the world is represented in spatial imagination.
It also seems, in a similar way, that they can know the truth of theorems in other mathematical systems constructed from arithmetic and geometry, such as calculus, prior to discovering their truth by perception.
Though Kant did not develop his constructivist approach to mathematical propositions in much detail, intuitionism was taken up by many mathematicians in the twentieth century (including Henri Poincaré, 1854-1912) and given a detailed defense by L. E. J. Brouwer (1881-1966).
In the end, however, intuitionism was not acceptable to most mathematicians, because the requirement that all mathematical objects be constructed in imagination required giving up too much of mathematics. (Brouwer rejected the axiom of choice, actually infinite sets, Cantor’s transfinite numbers, and any arguments for the existence of mathematical objects based on the law of excluded middle.)
Even if it were possible for intuitionists to construct all of mathematics, however, this way of explaining the certainty of mathematics implies that its truth comes from the structure of thought. Kant believed that nature is just the phenomenal world, which is "in the mind," so to speak, and though he never doubted there is a noumenal world (or things in themselves) beyond the phenomenal world, he denied that mathematical truths hold of it (or them). That may be plausible to Kantians, but it is not plausible to naturalists. Naturalists believe that what exists independently of the subject is a world of material objects with spatial relations that change over time, and intuitionism does not imply that mathematics is true of that world.
Logicism and formalism. The other way of explaining the certainty of mathematics in epistemological philosophy without accepting Platonism tries, in effect, to reduce mathematics to language. Historically, it has taken two main forms, logicism and formalism.
Logicism holds that all of mathematics is derivable from logic. Its philosophical roots are in Leibniz, but it was developed more rigorously by Gottlob Frege (1848-1925) and Bertrand Russell (1872-1970) around the turn of the twentieth century.
It turned out, however, that the laws of logic needed to generate number theory involved several axioms that hardly seemed to be laws of logic at all. They included, for example, the axiom of reducibility (which holds that propositions about higher types, or sets of sets, could be reduced to propositions about first order members of sets), the axiom of infinity (which affirms the existence of infinite sets), and the axiom of choice (which says that from any set of non-empty, non-overlapping sets, it is possible to form a set of one member from each). If one were to insist that these are laws of logic and that they are known by rational intuition of some kind, logicism would be a kind of Platonism in which the laws of logic, rather than the mathematical objects themselves, have an independent existence as abstract objects outside of space and time. But to many, these axioms seemed more doubtful than the propositions about numbers that were derived from them.
"Formalism" is the name of the project pursed by David Hilbert (1862-1943) in order avoid the problems of logicism. He did not believe that mathematics could be reduced to the laws of logic. He held that each branch of mathematics requires its own axioms and rules of inference. But he believed that logicism was on the right track in taking logical entailments to be what explains mathematical truth. Thus, Hilbert set out to prove the certainty of mathematics by reconstructing each branch of mathematics as a formal system with its own axioms, rules of inference, and theorems. But these statements were to be stripped of any meaning outside the formal system and treated as meaningless symbols, mere marks on paper, which were written down in sequence according to strict rules. Each formal system would include all the propositions in some branch of mathematics, and the rigor of these symbolic manipulations was supposed to prove the certainty of its theorems.
The formalists’ explanation of mathematical certainty, however, required systems constructed in this way to be free of contradictions, and so Hilbert saw the main challenge as demonstrating their consistency. For this purpose, he developed a special formal system for describing formal systems, a "metamathematics," which was supposed to be beyond reproach. In the end, however, it was not possible to demonstrate the consistency of arithmetic, or even of set theory, as Gödel showed. (This is the origin of the puzzles encountered by set theory that were solved in the ontological explanation of the truth of arithmetic in Relations: Solutions to puzzles.)
What logicists and formalists are getting at can be understood from our sketch of the nature of rational subjects. They are also epistemological philosophers reflecting on how we know the truth of mathematical propositions. But by contrast to intuitionists, they abstract from spatial imagination and focus on the use of language. They identify brain states by the linguistic representations they involve, and they use logical relations to keep track of the causal roles that brain states play in drawing conclusions about what to believe. By focusing exclusively on the formal relations among brain states identified in that way, whole systems of mathematical proofs can be reconstructed as formal deductive systems. The logical structure of language represents the elements in such reasoning completely enough that there are formal tests of the validity of those inferences, making it seem that their truth can explained by their deducibility from certain axioms and definitions.
Though the validity of deductive relationships in formal systems does afford a certain concept of certainty, it does not explain how mathematics is true. Even logicists complained that formalism cannot account for the truth of the axioms or the usefulness of the definitions that are assumed. But neither do the axioms used in geometry and arithmetic follow from the laws of logic. Indeed, any consistent set of sentences could be used as axioms and definitions, because as far as formal logic is concerned, deductive systems are just rule-governed ways of transforming assumptions as inscriptions that preserve their truth. Formalism has no explanation of why the axioms used in mathematics should be singled out as true. Nor does it explain why they, or the theorems derived from them, should be useful in describing the natural world.
Anti-realism that denies the certainty of mathematics. Applicability to the natural world is, however, as crucial to the nature of mathematics as its apparent certainty, and since neither the intuitionists nor logicists/formalists were able to explain its certainty in a way that would also explain its applicability to nature, naturalists could not help being attracted to the view that mathematical objects are somehow part of the natural world. That would be, like Platonism, a kind of realism about mathematical objects. But since our way of knowing about the natural world is perception, it would be more like scientific realism, for there would be no basic difference between mathematics and empirical science. And it would have to deny the certainty of mathematics.
The view that mathematics is a form of empirical knowledge was first defended by John Stuart Mill in the nineteenth century, but it was renewed in 1983 by Philip Kitcher. Kitcher rejected what he called "apriorism", the belief that the certainty of mathematical knowledge comes from its being epistemologically prior to experience of nature, and proceeded to explain mathematics as a species of scientific knowledge. Kitcher bases knowledge of mathematics on perception, by thinking of arithmetic operations as "idealizations" of publicly observable manipulations of natural objects.
The price of explaining how mathematics is true about nature seems to be giving up the belief that it has a certainty that is basically different from natural science. Kitcher explains the appearance of certainty by the extremely general character of the regularities described by mathematical hypotheses. But since there is no essential difference between mathematics and scientific hypotheses, he agrees that they are confirmed in basically the same way.
Ontological philosophy agrees with Kitcher in rejecting apriorism. It also takes mathematics to be a form of empirical knowledge in the end. But the end does not come so quickly as it does for Kitcher, because ontological philosophy recognizes two levels of explanations (ontological-cause explanations and efficient-cause explanations) and, accordingly, two levels of empirical truths (empirical ontology and empirical science). In other words, instead of taking mathematics to be knowledge of very general regularities about what happens in the world, it sees mathematics as knowledge about the most basic (or categorical) features of what exists in the world, namely, how space makes the world whole. That means that mathematics is still prior to empirical science in a philosophically relevant way. But the priority is ontological rather than epistemological.
Furthermore, when this explanation of mathematical truth as ontologically necessary is combined with what ontological philosophy holds about the nature of reason, there is even a sense in which mathematics is epistemologically more certain than empirical science. As we have seen, it holds that mathematical knowledge is not merely a correspondence of linguistic representations to the world, but also involves a correspondence of representations in the brain’s spatial imagination to the world. Thus, unlike Kitcher’s theory, it can explain the role that constructions in imagination play in proving mathematical truths according to intuitionists as well as the role of formal deductive relationships among sentences that logicists and formalists take to be basic.
But as we have seen, what explains the truth of both geometry and arithmetic ontologically are the spatial relations that particular substances have in a spatiomaterial world. Thus, since the rational subject is part of the world, mathematical knowledge involves a relationship between subject and object that is a correspondence between the structure of spatial imagination in a part of the world and the basic structure of the whole world of which he is part. It is within that basic correspondence that rational being discover what happens in the world by perception, and thus, if this is a spatiomaterial world, mathematics is not only ontologically necessary, but epistemologically certain.
 For an accessible discussion of the problem of certainty in the philosophy of mathematics, see Kline (1980). A somewhat more technical, but still readable discussion of issues about infinity is Lavine (1994).
 Covert manipulation also makes it possible to combine images of effects of motion from locomotion imagination into a single geometrical structure in manipulative imagination to think about all the relations of the objects in some territory at once, like a map.
 In a much discussed paper, Benacerraf (1973) argues against Platonism on the ground that it is not compatible with a causal theory of mathematical knowledge. Bigelow (1988) nevertheless takes universals to be the objects of mathematics, and he avoids this problem with abstract entities by following Armstrong (1983) and assuming, in effect, that universals are just spatial relations of bits of matter, which are always instantiated (that is, as so-called "tropes").
 This is similar to the "skeptical fictionalist" view of mathematical truth defended by naturalists like Field (1980) and Papineau (1993, pp. 193-197). They think that they must deny the existence of mathematical objects because such objects are abstract. But that is because they do not recognize the ontological role that space plays in making arithmetic true. They are implicitly materialists (taking space to be just spatial relations) who are nominalists about the concepts used in science, and though they recognize that mathematics can facilitate complex inferences about the natural world, they believe that all those inferences could, in principle, be made without referring to numbers or geometrical figures as abstract entities. Thus, they take such numbers and geometrical figures to be useful fictions and are skeptical about their existence. But that makes it just as puzzling why mathematics holds of the natural world as Platonism does. However, if space as a substance containing all the bits of matter is the ontological cause of geometrical figures and the groupings of material objects called numbers, there is an alternative, non-fictionalist defense of geometry and arithmetic truth. Though there is every reason to be skeptical about the existence of mathematical objects that are abstract entities, that is no reason to believe that numbers are just useful fictions. They could describe something concrete -- a very general, ontological effect of the structure of space on the bits of matter it contains.
 Kitcher's approach is endorsed by other philosophers of mathematics, such as J. Bigelow (1988, p. 3). Bigelow holds that mathematics is about universals, but he follows Armstrong's (1983) "a posteriori realism" in taking universals to be physical, and thus, in the terms used here, he is a materialist. Indeed, the subtitle of his book is "A Physicalist's Philosophy of Mathematics".
 The traditional theory about mathematical truth that comes closest to Kitcher's is abstractionism, the view originally defended by Aristotle that mathematical objects are abstractions from perceived objects. See Körner (1960, pp. 18ff). Kitcher has a more sophisticated theory about how mathematics is derived from perception than Aristotle. According to his "evolutionary theory of mathematical knowledge" (p. 92), the abstractions come from idealizing the operations of arithmetic, though Kitcher insists that this is compatible with saying that "arithmetic describes the structure of reality" (p. 109).
 The assumption that the use of language makes spatio-temporal imagination a cognitive capacity of reflective subjects enables spatiomaterialists to answer objections that Kitcher (1983, pp. 50ff) raises to intuitionism (or "constructivism'). Contrary to Kitcher, it is possible to distinguish essential properties from those that are accidental, because imagination is not just "pictures in the mind", but images that reflective subjects construct and manipulate within its structure. When a geometrical figure, such as a triangle, is defined, it is constructed in imagination, and thus, assuming that language-using subjects can reflect on what they are doing, they can see the effects of varying triangles in all possible ways on their inferences about them. Second, although Kitcher is right to insist that infinite sequences of operations, such the division of a line, cannot be carried out in practice, subjects who can reflect on what they do in imagination and its effects can come to see that what will happen each time is limited in a certain way and, thus, infer what would, and would not, happen if the operations were taken to infinity. Finally, the problems about exactness that may arise with Kitcher's "mental pictures" do not arise with spatio-temporal imagination. For example, imagined straight lines cannot be crooked, for they are constructed according to the understanding of the structure of space that is built into the structure of spatio-temporal imagination, that is, as the path of the shortest distance between two points. In short, since ontological philosophers postulate space as a substance containing all the matter in the world, they need only recognize the basic role that a spatio-temporal imagination would play in the reflective subject's knowledge of the world to explain how reflective subjects have a priori knowledge of mathematical truth, because its structure corresponds to the structure of space. Indeed, without the capacity to see what is given in perception against the background of what imagination tells us is possible in three dimensional space, it is hard to see how we could perceive a line as straight, a set of three lines as a triangle, or anything as a mathematical object.