Introduction to a Naturalistic Philosophy of Mathematics
Feng Ye Department of Philosophy, Peking University, China
This is a research project exploring a naturalistic and scientific account for various aspects of human mathematical practices, including the applicability of mathematics. Its philosophical basis is naturalism and it supports nominalism (vs. realism or Platonism) in the philosophy of mathematics. It views human mathematical practices as human brains’ cognitive activities and tries to refer to mathematical concepts and thoughts as mental representations in brains to account for aspects of mathematical practices, especially, aspects that interest philosophers. It tries to offer a logical explanation of the applicability of mathematics by showing 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. (This technical work grew out of the early work in my dissertation [1], [2].) It intends to show that we can reject all metaphysical speculations in accounting for human mathematical practices and turn all questions about mathematics, including the question about applicability, into ordinary scientific questions, in particular, into questions in cognitive sciences, whose answers are to be evaluated by ordinary scientific standards and methodologies.
The project is still in progress. This article introduces the basic ideas in the project and the work done so far.
1. The Philosophical Basis ¾ Naturalism
I take the central thesis of naturalism to be
Human minds are functions of human brains as parts of the physical world and results of evolution.
For the philosophy of mathematics, this means that human mathematical practices are human brains’ cognitive activities. A study of human mathematical practices should be similar to a cognitive scientific study of other human cognitive activities. It should be about how human brains represent and process relevant information in human mathematical practices, or about how brains work as physical systems. On the other side, this also means that one should not pretend to be an essentially non-physical Mind, not itself belonging to an External Reality but standing at the opposite of that External Reality, trying to speculate if there are the alleged abstract mathematical entities in that External Reality. There is no such an essentially non-physical Mind. There are only brains as physical things in this physical world and in physical connections with other physical things in human environments. A brain does not have any magical power to ‘grasp’ the alleged abstract entities. Speculating about the alleged abstract entities comes from taking the subjective stance and trying to project what are in one’s own brain onto the external. As a scientific researcher studying human mathematical practices, one should instead take the objective stance of a scientific observer and look into human brains to study how brains work as physical systems in mathematical practices. In doing these, we see concrete physical things only, things inside and outside the brains and their mutual connections. What really exist in human mathematical practices are human brain activities in doing mathematics and their connections with physical things outside the brains in applying mathematics, not the alleged abstract mathematical entities as the results of that projection.
In particular, as long as we do not project what are in our brains onto the external from that subjective stance, we see only finite concrete things in this universe. We do not see any infinity in this real universe. There are only brain activities in imagining infinity. We know that, beyond some finite scope, real things in this universe are quite unimaginable. For instance, consider ancient people’s idea of ‘cutting a rod into halves forever’. Since 2120»1036, after cutting a rod into halves for less than 120 times, we will reach the Planck scale (i.e. 10-35m), and the space-time may be discrete or not 4-dimensional there according to physicists. Literally, the operation ‘cutting a rod into halves’ will become meaningless after being repeated for less than 120 times. Only in our imaginations can we confidently say that we can repeat an operation forever, because we decide to imagine this ‘forever’ in some manner. What really exist are brain activities in imagining infinity in some manner, not the projected infinity. Whether or not there is infinity in this real universe is still unknown to scientists and it likely that this will be unknown to humans ‘forever’. (See Chapter 1 of the monograph [12] for more discussions on this point.)
This central thesis of naturalism is perhaps openly accepted by all self-professed naturalists, but many of them do not seem to take this thesis very seriously and literally. In discussing issues in the philosophy of mathematics, they are not talking about how a brain works as a physical system. They still implicitly assume the stance of an essentially non-physical Mind not belonging to nature but facing an External Reality at the opposite of the Mind, and they still speculate, from the point of view of that Mind, about if the Mind has to ‘commit to’ Abstract Objects in that Reality external to the Mind. That ‘ontological commitment’ relation is an intentionality relation between a Mind and Objects in that External Reality, not a physical connection between a brain and things in the brain’s environments. They never try to naturalize that intentionality relation (see Section 4 below), that is, to characterize that relation using ordinary scientific and naturalistic terms, without assuming an essentially non-physical Mind with a special mental power of intentionality not reducible to physical interactions between a brain and its environments even in principle.
The article [4] elaborates this point. It discusses the consequences of taking the central thesis of naturalism seriously, which many self-professed naturalists do not seem to be aware of. It points out that the ‘ontological commitment’ relation with the alleged abstract entities was never naturalized in the Indispensability Argument for realism in mathematics. It further argues that confirmation holism makes sense in a truly naturalistic context, for a brain, but it does not help in explaining ‘committing to abstract objects’. Therefore, a true naturalist does not have to worry about the Indispensability Argument, because that argument resorts to primitive notions that are not naturalistic. It is thus meaningless for a true naturalist.
2. Connection with Psychology or Cognitive Sciences
Taking this naturalistic stance means that a study of human mathematical practices is a continuation or extension of cognitive sciences, dealing specifically with human mathematical cognitive activities. Some people might object to introducing psychology in the philosophy of mathematics. However, the fact of matter is that if you do not deal with psychology, then you have to assume an essentially non-physical Mind and have to speculate about how such a Mind can ‘grasp’, or ‘commit to’, mathematical concepts or entities independent of the Mind. This is worse from the naturalistic point of view. Epistemology is about how something ‘knows’ something else. Either it is about how brains as physical systems work in the physical universe, or it is about how essentially non-physical Minds ‘grasp’, ‘commit to’, or ‘know’ something independent of the Minds. Note that the traditional focus on language in discussing the epistemological questions actually leaves the question of how human language hooks to things unanswered. Either brains associate words with things by receiving aural and visual signals, recognizing patterns in the signals (i.e. recognizing them as word tokens), and associating them with other physical things in the brains’ environments, through the brains’ controls on eyes, hands and so on to touch those things, or a Mind somehow ‘grasps’ the alleged ‘Mind-independent meanings of linguistic terms’ or somehow ‘refers to’ Mind-independent entities. That is, focusing on language cannot really bypass both psychology and speculations about essentially non-physical Minds. There is no magic. Focusing on language only leaves an equivocation there. Either you have to go with true naturalists and talk about how brains work in the natural world, or you have to go all the way with Gödel (and perhaps Husserl as well) and explicitly reject naturalism and openly admit essentially non-physical Minds and admit a transcendental stance and then seriously speculate about how a Mind ‘grasps’ mind-independent concepts. Therefore, if you are a naturalist, dealing with psychology is just unavoidable.
On the other hand, since our interests here are on the philosophical and logical aspects of human mathematical practices only, we can try to ignore psychological details as much as we can. Introducing psychology is only for clarifying the philosophical foundation, namely, naturalism, and to set a clear background for discussing philosophical issues in mathematical practices. When we start discussing specific philosophical and logical questions about human mathematical practices, we can reply on highly simplified cognitive models of the brain and abstract away all psychological details as much as possible, as long as that is reasonable and that will not affect the validity of our conclusions. What kind of simplification is reasonable and what may affect the validity of out conclusions will be judged by our current psychological knowledge, depending on the nature of the specific philosophical and logical questions concerned. Most importantly, after we settle on naturalism, this is a scientific question. It is to be judged by our scientific knowledge about how human brains work. It is not a metaphysical question in the traditional sense. Our analyses based on a simplified cognitive model of the brain will certainly be fallible, but they are fallible in the scientific sense, like any scientific theory, because we are pursuing a scientific description and explanation of some real phenomena in the universe, the phenomena of human mathematical practices.
In particular, this research project will assume the Representational Theory of Mind, which claims that the basic building blocks of human cognitive architecture are mental representations (realized as neural circuitries) in brains. More specifically, there are concepts and thoughts in brains as mental representations (realized as neural circuitries). Some concepts and thoughts directly represent physical entities or their states of affairs in human environments. (See Section 4 below on how this representation relation can become a subject matter of scientific researches.) However, some other concepts and thoughts do not directly represent physical entities or their states of affairs. In particular, mathematical concepts and thoughts belong to this second category. Mathematical concepts and thoughts have other cognitive functions in the cognitive processes in brains. For instance, they help to organize, represent and manipulate other concepts and thoughts that do directly represent physical entities and states of affairs in human environments, and therefore they also bear some indirect but more flexible and more abstract connections with physical things in human environments. A part of our task in this naturalistic account for human mathematical practices is to describe such cognitive functions of mathematical concepts and thoughts in human mathematical cognitive processes.
Moreover, in explaining the applicability of mathematics, we actually make a further simplification by assuming that concepts and thoughts in brains are like syntactical entities in a formal language and that the representation relation between concepts, thoughts and physical entities and states of affairs is like the satisfaction relation between a formal language and its semantic models. Then, the explanation of applicability is actually a logical explanation, without mentioning any psychological notions. (See Section 6 below and see the article [11] and Chapter 1 of the monograph [12].)
3. Why not satisfied with Current Anti-realistic Philosophies of Mathematics?
In the article [3], I summarize the tasks that a nominalistic account for human mathematical practices must accomplish:
(1) Explaining what the genuine human understanding, knowledge, intuition and experience in mathematical practices consist in, if they are not ‘grasping’ the alleged mind-independent mathematical concepts or abstract mathematical entities, and if they are not knowledge, intuition and experience ‘about’ the alleged abstract entities. (2) Explaining what the genuine relationships between the mathematical and the physical consist in, if mathematical entities do not exist. These genuine relationships include, for instance, the apparent (approximate) isomorphism between Riemann spaces and the physical space-time, which are the reasons for the applicability of mathematics. (3) Identifying various aspects of objectivity in mathematical practices and showing that admitting objectivity does not imply admitting the existence of abstract entities. These aspects of objectivity include the objective fact that people can have the same mathematical concept, compute the same arithmetic function, or follow the same rule, and they include that fact that there is objective correctness in following a rule. Moreover, they also include the fact that, for instance, the apparent (approximate) isomorphism between Riemann spaces and the physical space-time is something objective, not a wishful pretence. (4) Explaining the apparent obviousness (or apriority), universality and necessity of logic and arithmetic, without assuming any abstract entities (neither as the semantic models for characterizing logical validity, nor as numbers and semantic values for arithmetic terms). (5) Explaining the applicability of mathematics.
Moreover, in doing these, nominalists must work under the assumption that, possibly, there are only finitely many concrete objects in total in the universe. The range of real things within this universe to which our current sciences can have access is strictly finite, from the Planck scale up to the currently recognized cosmological scale. Any philosophical account assuming the reality of infinity in any sense must offer an explanation of the nature of that infinity and must offer a justification of it. It seems likely that any such justification will be the sort of Quinean holistic justification, based of the pragmatic values of ‘talking about infinity’ in human cognitive activities to cope with the strictly finite and discrete things in the universe, from the Planck scale up to the cosmological scale. We naturally suspect that if such a holistic justification were valid, it would also lead to justifying all abstract entities that we appear to ‘talk about’ in doing sciences. Therefore, a consistent nominalist should not assume the reality of infinity in any sense.
Then, we can see why current anti-realistic philosophies of mathematics are not satisfactory. First, they are not ‘realistic enough’. They have not yet offered literally true answers for the questions in (1) to (5) above. After denying that abstract mathematical entities exist, the first thing a nominalist should do is to answer what then really exist in human mathematical practices, and then he or she should offer literally true philosophical claims about human mathematical practices by referring to what really exist. In particular, one should not say that mathematical entities are ‘fictional entities’ and scientists use ‘fictional entities’ to model real things in mathematical applications. Since the alleged fictional entities do not really exist, the claim ‘scientists use fictional entities to model real things’ is a literally false claim. This only shows that one has not really explained what scientists are really doing. On the one side, this cannot stand against realists’ charge that nominalists are ‘intellectually dishonest’, or the claim that applications raise mathematics from a mere fiction to science, to objective truths in some sense (if not the same sense as truths about concrete objects). On the other side, apparently, some truths are slipping through nominalists’ fingers by saying merely that scientists are using fictional entities to model real things. We want strictly literal truths about what really happens in mathematical applications, that is, about how exactly mathematics is applied. For instance, a psychologist or sociologist studying the psychological and sociological functions of human fiction making activities will not refer to fictional characters in the manner as if they were real things with real properties. If mathematical entities are fictional and do not exist, we must refer to what really exist to offer our scientific and truthful account for human mathematical practices. Apparently, one has to refer to human brain activities in doing mathematics in a truly scientific account for human mathematical practices, because there is simply nothing else that can be the bearer of literal truths about mathematical practices (unless one is willing to speculate about a Transcendental Mind and the Mind’s ‘free creative activities’, as intuitionists do).
Note that it is not sufficient to say that mathematical theories exist. First, aren’t theories just abstract entities? Second, it needs a brain (or a Transcendental Mind) to interpret ink marks on papers into theories. Apparently, what really matter are brains’ cognitive activities. Certainly, we can make scientific simplifications and adopt a scientifically simplified model of the brain, that is, we can assume that theories are inside brains, or that mental representations are like syntactical entities in theories. However, it is still critical to be fully aware that we are talking about cognitive activities within brains and their physical connections with physical entities in human environments. This truly naturalistic background allows us to clarify much philosophical confusion and keep away from all metaphysical speculations, which are haunting the philosophical discussions about mathematics since the old time, including those in the Quinean self-proclaimed naturalistic tradition. It also offers a framework for pursuing literally true, scientific answers to the questions in (1) to (5) above.
On the other hand, current anti-realistic philosophies of mathematics are sometimes ‘too realistic’ about infinity. For instance, in explaining the applicability of mathematics, Hartry Field (in Science without Numbers) assumes the infinity of space-time in his nominalistic mathematics. This threatens the coherence of the position as nominalism. Moreover, the real puzzle of applicability of mathematics is the puzzle about how exactly ‘infinite mathematics’ is applied to this strictly finite and discrete physical world. (See Section 6 below.) Reducing the applications of classical mathematics to the applications of another ‘infinite mathematics’ cannot really resolve this puzzle.
More comments on the problems in current anti-realistic philosophies of mathematics are given in the article [3].
4. The Starting Point ¾ Naturalizing Semantics, Truth and Logical Validity
The first step in pursuing a truly naturalistic account for human mathematical practices is to naturalize the semantic representation relation between some mental representations in brains (realized as neural circuitries) and the physical things represented by those mental representations, that is, to characterize the semantic representation relation as a natural relation between physical entities, using ordinary scientific (non-semantic, non-intentional) terms, just like any other relations studied by sciences. This is called ‘naturalizing content’ or ‘naturalizing intentionality’ in the contemporary philosophy of mind. It first means naturalizing the semantic representation relation between some concepts in brains and the physical properties or categories of physical entities represented by those concepts. Then, this also means naturalizing the correspondence relation between some thoughts in brains and the physical states of affairs represented by those thoughts. It is thus a naturalized correspondence theory of truth, where the correspondence relation is a natural relation between natural entities, namely, thoughts in brains and other physical entities.
Note that these apply only to concepts and thoughts that can directly represent physical things and their states. In particular, naturalizing content does not apply to mathematical concepts and thoughts directly. Mathematical concepts and thoughts bear more indirect connections with physical things and their states.
With truth being naturalized, the logical validity of a logical inference rule is also naturalized. A logical inference rule is an inference process pattern in brains that produces thoughts in some format as conclusions from some other thoughts in some formats as premises. A logical inference rule is valid, if, as a natural regularity, in any inference instance with that pattern and with the premises and conclusion that can represent physical states of affairs, whenever the natural correspondence relation exists between the premises and some physical states, it exists between the conclusion and those physical states as well. Therefore, an assertion about the validity of a logical inference rule is an assertion about some natural regularity in a special type of natural processes. It says that whenever some natural (relational) property exists at the beginning of a special type of natural processes, it exists at the end of those processes as well. It is not unlike any other scientific assertions about regularities in nature.
Naturalizing semantics and logical validity serves several purposes. First, it shows that a completely scientific description of the brains’ cognitive interactions with physical things in human environments is possible and no scientifically unexplainable mental power of intentionality needs to be assumed. It makes possible a truly naturalistic study of logic, by allowing a naturalistic description of the cognitive functions of logical concepts in brains and by naturalizing the notion of logical validity. Moreover, it is also the basis for a naturalistic characterization of the indirect connections between mathematical concepts, thoughts in brains and physical things and their states out of brains. That is, mathematical concepts and thoughts in brains are connected with physical entities and their states of affairs through their effects on concepts and thoughts that do directly represent physical entities and their states through the naturalized representation relations. With that, we will be able to propose the question of applicability of mathematics as a completely scientific question about natural regularities in the natural phenomena of human mathematical practices. (See Section 6 below.)
On the other side, in contrast, naturalizing the semantic representation relation between some mental representations and physical entities and their states also reveals the fact that, in speculating about how a Mind ‘grasps’ or ‘commits to’ mind-independent abstract concepts or abstract entities, people actually implicitly assume the non-naturalistic picture of a Transcendental Mind with a non-naturalizable mental power of intentionality. There is no obvious way to naturalize the semantic representation relation between concepts in brains and the alleged abstract entities represented by those concepts, and actually no philosophers (in the Quinean tradition) ever tried to do that. (See the article [4] for more on this.)
Moreover, it also shows that naturalism is not the so-called psychologism, if that means the view that valid logical rules are the rules that are most frequently followed by human brains. A brain can frequently conduct logical inferences that are not valid according to the naturalistic characterization of validity above. Normativity in truth and logical validity comes from normativity in the semantic representation relation and is a naturalized normativity. In particular, the true basis for the semantic and logical normativity is not the alleged existence of abstract concepts independent of Transcendental Minds and the Minds’ alleged ability to ‘grasp’ those mind-independent concepts. It is some special natural relation between what are in brains and physical things in environments.
There have been several theories for naturalizing content proposed in the contemporary philosophy of mind, by authors such as Jerry Fodor, Fred Dretske, David Papineau, Ruth Millikan and so on. There are difficulties in those theories. The articles [5], [6] and [7] propose a new theory for naturalizing content. It seems to be able to resolve those difficulties. Naturalizing content is a big topic. This new theory is not a complete theory yet. In particular, a naturalistic theory of logical concepts is still to be explored.
5. Accounting for the Philosophical Aspects of Human Mathematical Practices
Another major part of this naturalistic study of mathematical practices is to account for aspects of human mathematical practices with some philosophical interest. These include answering questions (1) to (4) raised in Section 3 above. Work in this part is still in progress. In the following, I will report the work done so far.
The article [8] summarizes the basic assumptions about human cognitive architecture adopted by the research project and discusses understanding, knowledge, experience and intuition in mathematical practices. Genuine understanding, knowledge, experience and intuition in mathematical practices are explained by referring the cognitive functions of mathematical concepts and thoughts in brains. In particular, they are not about the alleged abstract entities or mind-independent mathematical concepts. For instance, having mathematical knowledge on a particular subject matter means having some relevant mathematical concepts and thoughts in the brain, having some relevant ability to manipulate those concepts and thoughts, that is, to do inferences on those concepts and thoughts, having some relevant knowledge about the expected outcomes of such inferences, and finally having relevant knowledge and capability in translating mathematical concepts and thoughts into concepts and thoughts that can directly represent physical entities, properties and their states of affairs in applying mathematics to real things.
In other words, a brain’s having such knowledge means the brain’s having some relevant brain capability for doing relevant inferences and having relevant representational knowledge about the outcomes of doing such inferences. They have nothing to do with the alleged abstract entities or abstract concepts ‘out of the brain’. They are related to physical things out of brains in some way, but that is through translations schemes in brains that translate mathematical concepts and thoughts in brains into those concepts and thoughts that can directly represent physical things and their states of affairs, through the naturalized representation relation, not by any magical mental power of intentionality.
The article [9] discusses aspects of objectivity in mathematics. The traditional notion of objectivity is understood from the point of view of a Transcendental Mind. Ideas, sense data and so on belong to the Mind and are subjective, and physical Objects or the alleged abstract Objects belong to the External Reality and are objective. Since there is no Transcendental Mind in the naturalistic picture of the world, being objective can only mean being objective relative to a brain (or relative to all human brains). The article discusses several notions of objectivity in the context of naturalism and argues that admitting objectivity does not imply admitting the existence of abstract entities or mind-independent (or brain-independent) concepts.
For instance, it argues that the true naturalistic objective basis for people’s sharing ‘the same concept’, successfully communicating the concepts (as mental particulars) in their brains, and following ‘the same rule’ and so on is the objective fact that human brains share the same fundamental architecture, innate capability and natural tendencies, and the fact that they obey the same biological, neural-physiological and psychological laws, and the fact that they have similar individual developmental environments. A truly scientific study of how exactly human brains communicate with each other will have to refer to such natural regularities among real things in the universe. Assuming the existence of a concept as a public, abstract entity common to all human brains is completely irrelevant from the scientific point of view. It comes from a subjective intention to project what are in one’s own brain onto the external.
Similarly, which rule a brain actually follows in controlling a body to perform some operations (i.e. PLUS or QUUS) depends on what neural structure is actually inside that brain and is controlling the body, and depends on the neural-physiological regularities in the neural structure’s controlling the body among biologically normal humans. Recall that no humans can literally perform arbitrarily many steps of operations in this strictly finite real universe. Therefore, objective regularities in rule following literally do not generalize up to infinity. It is literally false that a human can follow a rule (e.g. doing addition) up to arbitrarily large numerals. We only imagine that we can perform some operation arbitrarily many times, and what is truly objective here is the fact that we humans share the same innate capability and tendency in doing some type of imaginations, due to our same innate brain architecture. Therefore, admitting objectivity in rule following does not imply admitting the objectivity of infinity. As for how we can know if others are following the same rule as we do, again, we can usually make very reliable guesses about others’ innate behavior patterns based on observing their very limited external behaviors, because we humans share the same natural tendencies to develop some behavior patterns, but not some others, in our normal learning experiences. For instance, for the normal learning experiences for learning addition in schools, it is a scientifically reliable prediction that a child will develop an inner neural structure that will control the child’s body to perform PLUS, not QUUS. That is, the neural structure associated with the word ‘PLUS’ in that child’s brain will, in the future, control the child’s body to produce the same outputs, given the same inputs, as the neural structure associated with the word ‘PLUS’ in our brains will do. More details are in the article [9].
The article [10] discusses the apparent apriority, necessity and universality of logic and arithmetic from the naturalistic point of view. Once more, the traditional notion of apriority assumes a Transcendental Mind. A priori knowledge means knowledge that comes from the Mind’s form of sensibility or the Mind’s innate reason that is somehow in predetermined harmony with things external to the Mind. For naturalists, apriority can only mean apriority relative to a brain or relative to the human brain’s innate architecture. A brain grows out of a cell, controlled by genes, and it constantly receives external stimuli since its birth. Nothing is a priori in that traditional sense for a brain. However, we can still define a meaningful notion of apriority under this naturalistic background. It will be a scientific notion in cognitive sciences. It will be related to the different ways by which a brain gets, stores and processes its representations of things in its environments. For instance, as a result of evolution, a brain has some innate architecture, which may have determined, ‘a priori’, that some inner representations always correspond to external things in the brain’s environments. In other words, there can be some ‘predetermined harmony’ between a brain and its environments, as a result of evolutional selections that finally produce the brain. This will result in some ‘a priori’ knowledge for a brain. In particular, logic seems to be ‘a priori’ in this sense. However, the case for arithmetic is more subtle. More details are given in the article [10].
On the other hand, note that logic and arithmetic are not absolutely universal. As researches in quantum logic suggest, the classical logical constants may not be directly applicable for describing properties of quantum particles. However, human brains are evolved in interacting with deterministic, medium size physical objects in human immediate environments. As a consequence, human brains are evolved in such a way that humans can only clearly conceive of objects similar to those deterministic, medium size physical objects. (That is, we cannot clearly conceive of quantum particle with wave-particle duality.) The classical logic and arithmetic represent some most general features of these deterministic, medium size physical objects. Therefore, classical logic and arithmetic are universal and necessary in a limited sense, namely, in the sense that the classical logic and arithmetic are always true when applied to anything that are in human immediate environments and are observable by humans, or anything that are directly and clearly conceivable by human brains.
6. Explaining the Applicability of Mathematics
Explaining the applicability of mathematics is a challenge for any philosophy of mathematics. Realists claim to have an explanation, by assuming that the classical mathematics is literally true of a realm of mathematical entities and/or structures. However, this is an illusion. The physical world to which we apply mathematics is strictly finite, from the Planck scale up to the cosmological scale. No scientific theory is literally true of things in the universe below the Planck scale. Infinity and continuity in mathematical applications in sciences are always for glossing over details and providing simplified, literally false (but sufficiently accurate) models. In that sense, all applications of mathematics are essentially like the applications of mathematics in economics, population studies and so on, where the subject matter is more obviously finite and discrete. The real puzzle of applicability is then the puzzle of how exactly infinite and continuous models accurately simulate discrete, finite real things. This is a logical problem. It is due to the fact that the exact logic in using infinite mathematics to help deriving literal truths about strictly finite and discrete things in the universe is not very clear. We naturally expect that a logically completely clear explanation of applicability will in the end have to show that resorting to infinity in mathematical applications is in principle dispensable, for then we will have a logically plain demonstration of how, in mathematical applications, our conclusions about strictly finite real things in the universe logically follow our scientific hypotheses about those strictly finite real things in the universe. This is the basic idea for explaining applicability in this research project. The work is still in progress and the work done so far is reported in the monograph [12], of which the article [11] is a summary. In the following, I will very briefly introduce the basic ideas.
First of all, with semantics, truth and logical validity being naturalized (see Section 4 above), we can naturalize the problem of applicability of mathematics. In a mathematical application scenario, a brain first has some thoughts directly representing physical states of affairs as physical premises. These physical premises can bear the naturalized correspondence relation with some physical states of affairs. Then, the brain translates these thoughts into mathematical thoughts as mathematical premises. In some cases, the brain directly uses some mathematical thoughts to express physical premises about physical things, which means that the brain has a translation scheme for translating those mathematical thoughts into thoughts that can directly represent physical states of affairs, through the naturalized representation relation. Then, the brain draws a mathematical conclusion from those mathematical premises, together with other mathematical axioms, by some logical inferences. Finally, the brain translates the mathematical conclusion into a thought that can directly represent physical states of affairs, through the naturalized representation relation again, namely, which is the physical conclusion of the application. Applicability means that the whole brain process preserves the naturalized ‘true’ property from those physical premises to that physical conclusion. Therefore, an assertion about applicability, like an assertion about logical validity, is an assertion about some natural regularity in a type of natural processes. Applicability is thus a completely scientific issue. As a scientific problem, its answer should be a scientific answer. An explanation of applicability means a scientific explanation of how some natural property, i.e. the naturalized ‘true’ property, is preserved at the beginning and the end of a type of natural processes, and the difficulty in offering such an explanation is due to the fact that that naturalized ‘true’ property is not applicable to those intermediate mathematical thoughts in those processes.
Then, we can make a simplification by assuming that concepts and thoughts in the brain are syntactical entities in a formal language and we can ignore the details in the naturalized semantic representation relation and treat it as the satisfaction relation between a formal language and a semantic model. Note that the semantic model will be a strictly finite model, because it consists of strictly finite and discrete physical entities in the universe. Then, the question becomes a question about how truth in a strictly finite semantic model is preserved by employing ‘infinite mathematics’ in the intermediate steps in deriving a conclusion from the premises. This is a logical question.
Our strategy for explaining applicability is then as follows: First, we develop formal system called strict finitism. It is a fragment of the quantifier-free primitive recursive arithmetic, with the accepted functions restricted to elementary functions, the proper sub-class of primitive recursive functions, constructed from the successor function, addition, multiplication, and the power function, by composition, bounded primitive recursion, bounded minimalization, finite sum and finite product. Strict finitism can be interpreted as a literally true theory about strictly finite, concrete computational devices (computers or brains) in this universe. Terms in strict finitism can represent programs in finite, concrete computational devices and sentences in strict finitism are assertions about the outputs of such programs. (Remember that the ratio of the linear cosmological scale to the Planck scale is less than 10100. Therefore, primitive recursive functions beyond elementary functions have no chance to be really implemented by real finite computational devices in the universe.) Applying mathematics in strict finitism in sciences means using finite, concrete computational devices in the universe to simulate other physical entities in the universe. Mathematical axioms in strict finitism are literally true of those computational devices and some bridging hypotheses state how those programs simulate other physical entities. Then, an application of strict finitism consists of sound logical deductions from literally true premises about concrete, finite physical entities (computational devices and the physical entities simulated by those computational devices) to literally true conclusions about them. That is, the applicability of strict finitism is logically transparent. Here, truth and logical validity are the naturalized truth and logical validity. Therefore, this is a completely naturalistic explanation of applicability.
Then, to explain the applicability of the classical mathematics we show that the applications of classical mathematics are in principle reducible to the applications of strict finitism. We do this by trying to develop a sufficiently rich applied mathematics in strict finitism. The monograph [12] is devoted to that work. Developing ordinary mathematics in strict finitism is very similar to developing ordinary mathematic in Bishop’s constructive mathematics. The monograph [12] contains the development of the basics of calculus, metric spaces, complex analysis, Lebesgue integration, and (bounded and unbounded) linear operators on Hilbert spaces. It also shows that more can be developed in strict finitism in the same manner. If a sufficiently rich applied mathematics can be developed in strict finitism, it means that the language of strict finitism is sufficient for expressing scientific theories and conducting scientific calculations and proofs. This then in turn shows that the applications of classical mathematics in sciences are in principle reducible to the applications of strict finitism.
Note that reducing to the applications of strict finitism is not what really occurs in scientists’ brains in applying mathematics in sciences. Therefore, the type of scientific explanation of applicability is something like the following. You have a transition path in a state space of a physics system and you observe that a property (i.e. the naturalized ‘true’ property) is preserved at the beginning state and the end state of the transition path, but you also observe that the property is not preserved or is not applicable for the intermediate states in the transition path (i.e. those mathematical thoughts in the classical mathematics). To explain why the property is preserved at the beginning and the end, you demonstrate that the real transition path can be transformed into a virtual path in the state space (i.e. the path through strict finitism) from the same binning state to the same end state. It is a virtual path, because it does not really occur for the system, but it is a physically possible path. The virtual path is such that the property (the naturalized ‘true’) is obviously preserved in all the intermediate states in the path. This then explains why the property is preserved at the beginning and the end states of the original path.
This strategy relies on the following Conjecture of Finitism:
Strict finitism is in principle sufficient for formulating scientific theories and conducting proofs and calculations in sciences.
There are evidences supporting this conjecture. First, recall that infinity, and continuity and so on are used only for glossing over microscopic details in mathematical applications in sciences. They should not be strictly logically indispensable. For instance, if the continuity condition in a continuous model of fluids in fluid dynamics is in some sense ‘strictly logically indispensable’ for deriving a conclusion about the fluids, we have good reason to suspect that the conclusion is scientifically unreliable, because we know that the condition is only an approximation and should not be taken too literally. Experiences in developing mathematics within strict finitism show that a proof’s being reducible to a proof in strict finitism is naturally connected with the fact that the proof does not really ‘use up’ the infinity or continuity conditions in its premises.
Moreover, recall that the ratio of the linear cosmological scale to the Planck scale is less than 10100. The super-power function obtained by iterating the power function never appears in any natural contexts in scientific applications, not to mention those fast growing functions that are not provably recursive in strong axiomatic systems such as ZFC or its extensions. We suspect that the genuine complexity among real things in this universe is quite different from the type of complexity entertained by logicians. The logical power of those logically stronger and stronger axiomatic systems is not really relevant for resolving the complexity among strict finite real things in this universe.
Finally, the fact that some advanced applied mathematics can be developed within strict finitism certainly supports the conjecture as well. In particular, experiences in developing applied mathematics in strict finitism seem to show that we can always strengthen the assumptions in a classical theorem, in order to reveal its realistic computational content and to make it available to strict finitism. Then, when applying the theorem to real things in the universe, we expect that the strengthened assumptions are still valid for real things in the universe, because real things in the universe strictly finite. For instance, the Jordan Curve Theorem in its classical format is perhaps not available for strict finitism. However, remember that the physical space could be discrete microscopically. Therefore, if we can apply the theorem to the real physical space in some way, what is really relevant for the application is likely to be some finitistic and approximate version of the theorem, not its strictly classical format. That finitistic and approximate version is likely available to strict finitism.
These are certainly intuitive reasons only and they are not conclusive. More technical researches are required for the strategy of explaining applicability to succeed. The monograph [12] is an initial effort for that.
Acknowledgements
The research for this paper is supported by Chinese National Social Science Foundation (grant number 05BZX049). I would like to thank Princeton University and my advisors John P. Burgess and Paul Benacerraf for the graduate fellowship and all other helps they offered during my graduate studies at Princeton many years ago. Without them, my researches would not have started.
References
[1] Strict Constructivism and the Philosophy of Mathematics, PhD dissertation, Princeton University, 2000. [2] ‘Toward a constructive theory of unbounded linear operators on Hilbert spaces’, Journal of Symbolic Logic, 65(2000), no. 1. [3] ‘What Anti-realism in Philosophy of Mathematics Must Offer’, to appear [4] ‘Naturalism and Abstract Entities’, to appear [5] ‘A Structural Theory of Content Naturalization’, to appear [6] ‘On Some Puzzles about Concepts’, to appear [7] ‘Truth and Serving the Biological Purpose’, to appear [8] ‘On What Really Exist in Mathematics’, to appear [9] ‘Naturalism and Objectivity in Mathematics’, to appear [10] ‘Naturalism and the Apriority of Logic and Arithmetic’, to appear [11] ‘The Applicability of Mathematics as a Scientific and a Logical Problem’, to appear [12] Strict Finitism and the Logic of Mathematical Applications, draft
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