Romances

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Romances

A romance is a love affair between two people. But this doesn’t specify what kind of love affair: it could be sexual love, but it need not be. It could be an intellectual romance; then it connotes shared passions and interests of an intellectual nature. The idea is of a cooperative, committed, passionate relationship between two people. I want to extend this idea to include what I will call athletic romance: two people sharing their common interest in athletic activity. This too should be cooperative, committed, and passionate—like a sexual romance. It is also bodily and may involve touching. It should be intense and consuming. It should form a higher unity consisting of the two persons—a kind of team or companionship. It may involve problems of infidelity, insecurity, jealousy, illness, separation, time, and rivalry. It is intimate and emotional, involving a type of love. It may not last forever. Learning will be a big part of it, as will respect. I see it as having a therapeutic component, as producing feelings of well-being and transcendence of the ordinary. It has a religious dimension. Overcoming obstacles is part of it. An athletic romance must be romantic.

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Mathematical Philosophy

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Mathematical Philosophy

In 1919 Bertrand Russell published a book titled Introduction to Mathematical Philosophy, written while he was in jail; but the title is somewhat misleading, because the book isn’t about doing philosophy mathematically but about the philosophy of mathematics. Still, Russell did do a fair amount of mathematical philosophy: the theory of descriptions, existence as a propositional function, mathematical constructions of the external world, etc. So did Frege, Wittgenstein, Carnap, Quine, Tarski, Montague, Kripke, Davidson, and others. It was a movement, a meta-philosophy. We might call it formal philosophy or symbolic logic philosophy or technical philosophy or scientific philosophy—as opposed to informal philosophy or phenomenological philosophy or commonsense philosophy or ordinary language philosophy or humanistic philosophy. Its model is mathematical science not vernacular literature—arithmetic not art. It aligns itself with the mathematical member of the two cultures. I think we can add that analytic philosophy is by nature mathematical in that it employs conceptual analysis, which is a breakdown of the whole into its parts, isolated and enumerated. Concepts are composites of added parts. Such analysis results in equations—e.g., knowledge = true justified belief; it seeks conceptual equations. It also employs rigorous deduction like mathematics, hence the emphasis on logic: syllogisms, axioms and theorems, formalized arguments with premises and conclusions, logical variables. Thus, we have today formal epistemology, formal politics, formal ethics, formal semantics. The subject is steeped in formal methods and conceits. Twentieth century philosophy was awash in mathematics, for good or ill. The dispute between analytical philosophy and so-called continental philosophy might well be seen as a disagreement about the role of mathematical methods in philosophy—what do you prefer, aesthetics or arithmetic?

But I think this story has history skewed: philosophy has been mathematical for much longer—indeed, it has never really been non-mathematical. I think philosophy is, and always at heart has been, a mathematical science, more or less successfully. That is what it aspires to be, though we must be liberal about the meaning of “mathematical” and “science”. It is about abstract and abstruse questions, aiming for precision and exactitude, obsessed with number, methodologically a priori, in love with logic, proudly technical and symbolic. It is a kind of geometry or geography of concepts or facts or realities. One of its most fundamental questions is numerical: is the world One or Many—and if many, how many? Hence, monism, dualism, and pluralism. Is the world finite or infinite, continuous or discrete, qualitative or quantitative? What kind of mathematics best captures its nature? Philosophy is all about distinctions, divisions, differences, relations. Identity is always a central concept. For example, Plato’s problem of the one universal and the many particulars: this is a quintessentially philosophical problem, and it is overtly numerical. We have the abstract form and the concrete particulars instantiating it: the form is a unity, but the particulars are a plurality. These are essentially different kinds of entity, yet they are intimately connected: many things are white, say. How can the One be Many, and how can Many be One? What is the relationship between them? It seems puzzling. More generally, is the whole world a unity or is it a plurality? How many types of entity are there? Is a person two things in one (mind plus body)? Are there values and facts or just facts? Is appearance the same as reality? How many kinds of good are there?  Are pleasures all of one kind? How many types of truth are there? Is necessity unitary? Are all words names? Polytheism or monism?  And so on. We want to know how many; we want to count our ontological commitments. We want to mathematically analyze the world. There are odd numbers and even numbers—are there universals and particulars, minds and bodies, values and facts? We look for unity but accept multiplicity. In arithmetic it all comes down to addition (subtraction is just the opposite of addition)—is that the way it is with the world generally? Metaphysics is a kind of mathematics of reality in its broadest outlines. We are applying mathematical conceptions, albeit elementary ones (but are any mathematical conceptions truly elementary?). This is not all we are doing, to be sure, but it is a large part of it; we are employing a diverse range of knowledge in conjunction with our mathematical knowledge.

So, the mathematical paradigm did not suddenly emerge in the twentieth century; it was alive and well for two thousand years or more. But it is so natural to us that we don’t notice it; we mathematize instinctively. Philosophy could hardly exist without it—or any science. It’s how we think scientifically, rigorously, systematically. The only question is how much we insist on it or extend it. I am not myself enamored of a sharp distinction between science and literature (I think literature is a type of science, mainly psychological). I think the human mind is essentially mathematical, here as elsewhere. It’s just a matter of admitting it, making it explicit as part of one’s meta-philosophy. We don’t have to mathematize philosophy because it is already mathematical. Even Sartre’s existentialism is mathematical, because it insists on the duality of the in-itself and the for-itself: Sartre is an ontological dualist—two things not one. The philosophy of language as it now exists is clearly a branch of mathematics—what with model theory and computational linguistics. Mathematical thinking is so entrenched in philosophy that we risk accusations of triviality just by drawing attention to it. It is an interesting question in the pre-history of ideas whether philosophy evolved in the human mind as an offshoot of an antecedent mathematical faculty. We are natural accountants, calculators, and bean counters; we brought these talents to bear on the most general questions of the cosmos. Mathematicians often have philosophical predilections. Mathematics and philosophy belong to the same intellectual natural kind.[1]

[1] Pretty much all science is mathematical to one degree or another; it would be strange if philosophical science were the exception. Inside each philosopher is a little mathematician struggling to strut his stuff.

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God and Matter

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God and Matter

Berkeley was convinced that matter and God are incompatible. If matter exists, God doesn’t; and if God exists, matter doesn’t.[1] I want to spell this out a bit, because Berkeley is onto something important here. His basic point is that if you postulate matter, you open the way for skepticism, mystery, and atheism; but this is theologically unsound, so matter must be rejected. Suppose God wants to create a world in which a certain moral drama is played out, leading to reward for the virtuous and punishment for the vicious (heaven and hell). One way to do that is to create immaterial spirits and present them with moral choices; to do this you need to create an experiential world in which things happen (or seem to). Moral choice certainly doesn’t depend on the existence of a material world; it just requires agents capable of moral decisions. The agents can be disembodied beings to whom things appear a certain way. To create such a world God can use his own mind to conjure up ideas of objects and then plant these ideas in the minds of his immaterial agents. Reality is thus constituted by God’s mind, which finite spirits perceive (God arranges things this way). In other words, God creates an idealist ontology. Then he sets to work to confront these finite spirits with objects as so constituted. He wants his creatures to know these objects exist and that he exists, and they can know these things given their mental nature. The objects are right there in the mind, and it is evident that God is behind their existence because he put them there; now he just waits to see how virtuously they handle the world thus constituted. God could easily have made the world this way, given his omnipotence; and there would be no reason to doubt his purposes and existence. He made an intelligible world in which we finite beings play our ordained part—no worries about the metaphysics or epistemology of the situation. And yet, according to materialist metaphysics, he did none of these sensible things; instead, he created a world of mysterious unintelligible matter that naturally leads to a general skepticism that takes in God himself. He creates a world in which the moral purpose is obscured by the wacky metaphysics and problematic epistemology. Absurd! Away with matter!

For consider: this supposed matter stuff is completely alien in its nature to the nature of God its creator; you cannot infer God from this stuff. It could in principle exist without God because there is nothing Godly in it. Moreover, we don’t know what it really is, unlike our own ideas and ideas in general: people say matter is extension, but extension of what? It is a complete mystery, an epistemological black hole. How can it even have causal powers, as spirits, finite and infinite, have causal powers? It is just an inert soulless enigma. Why would God create a world unknowable to the creatures that populate it? Wouldn’t that lead them to a generally skeptical cast of mind that includes God’s own existence—and this would undermine their commitment to the moral project that is the point of the whole exercise. How indeed could God create such a thing from his own nature, since he is immaterial himself—and why? It makes no sense given his overall intentions. What would be the point of creating matter that coexists with God in the absence of human souls, as the materialists suppose? First matter, then minds: but matter itself has no moral status without minds. And how could matter create minds? The whole idea makes God redundant: once you have got matter, nature can be left alone (allegedly) to produce life, mind, and morality. Matter usurps the role of God in the management of his creation. But God wants to keep his hand on the wheel, his foot on the gas; and this is no trouble given his omnipotence. Why would God create a rival to his own dominion—a kind of second material God (not that the idea makes any sense when you think about it). If God wants to alter the course of history, he must intervene in the actions of matter, producing what mortals call miracles. Better to create a world in which everything already obeys his will—no need to puzzle the mortals by having two conflicting sources of causation in the world. There are no miracles in God’s preferred world, only his continual beneficent presence. And there is an even more sinister implication of the matter theory: people will derive their concept of existence from the supposed material substance, and this will make God’s existence seem unintelligible to them. For God does not exist in that way—by being extended in space, or solid in substance, or perceptible by the senses. This will certainly give rise to atheism. Why put such an obstacle in the way of sound religious belief? To exist is to be a spiritual substance not this inert unintelligible something that only promotes skepticism and atheism. Religion (the Christian kind) cannot therefore coexist with materialist metaphysics; the two are in deep tension with each other. So, God doesn’t create a world made according to this plan; he creates an idealist world. If the external world of ordinary objects were material, we would be rightly skeptical of its existence; but that would be to be skeptical about God’s own creation and would inevitably lead to skepticism about God himself (his own creation, meant for us, becomes inaccessible and senseless). Instead, we should accept the idealist theory, in which objects are ideas in our mind and God’s mind—perfectly accessible intelligible items. It’s common sense: go for what you know not what you don’t know—mind, not matter. Materialism about the external world is clearly an anti-religious doctrine; it is inherently skepticism-producing and atheism-encouraging. If you try to combine materialism with religion, you get an insidious form of cognitive dissonance, which can only undermine religious belief. This, at any rate, is Berkeley’s credo.

I think he is right: the two are incompatible. God has no time for matter. But if so, the real existence of matter as a non-mental substance would undermine God. It’s one or the other; there’s no halfway house. A materialist account of creation, such as that produced by the mechanists of the seventeenth century, is incompatible with traditional Christian theology: rocks must be in the mind if Christianity is to be believed (and likeminded religions). We can prove metaphysical idealism from the claims of religion: if God exists, then this desk must be an idea in his mind, not a lump of matter. If matter were the root of all evil, then it would be clearly incompatible with a good God creating it—surely a beneficent God would not create such a world (though the devil would)! But even if it isn’t, it still violates sound theology. So, contraposing, if we remain wedded to matter, we cannot also believe in God. If you are looking for a disproof of God’s existence, you need look no further than the existence of matter. The bishop of Coyne was right about that. If God had accidentally created matter, he would be highly displeased with his creation and scrap it and start again.[2]

  [1] See my “Matter and God”.

[2] It was the devout Descartes who laid the groundwork for atheism, contrary to his intentions. He wasn’t mistaken (Berkeley would say) in postulating an immaterial mind; his mistake was to postulate a material body. For that is not consistent with a belief in God as the creator of the universe. Nor does it help to follow Hobbes and postulate a material God, since this still leaves us with a theology inconsistent with materialism about anything. There is a further question here about whether an ontology of immaterial substances is also incompatible with traditional theology, given how little we know about them (how do we even know they exist?); I have not discussed this question here, but the question is real. We might be forced into an even more extreme idealism of free-floating ideas.

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The Selfless Gene

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The Selfless Gene

The logic of selfish gene theory is impeccable, but its usual formulation leaves something to be desired in point of strict and literal accuracy. I propose to remedy that. No harm will be done to the theory thereby; we will merely clarify its exact import. First, some terminology: by “gene” I will mean a particular piece of matter that exists on an individual strand of DNA—a specific localized chemical lump. I will not mean the type of gene of which this is a particular instance: this is a general form that can be instantiated in many individual instances; it might be viewed as the information contained in the chemical instance or the general type of which there are many instances (my genes, your genes, etc.—construed as local particular lumps of matter in individual bodies). We might call these gene tokens as opposed to gene types. This distinction is vital to the point I want to make, because we might mean the token gene is selfish or the type gene is selfish. Is the gene token (that particular lump) selfish?

To answer this let’s consider an ordinary case of selfishness: a person who cares only for himself and no one else—he always comes first. Such a person might coexist with copies of himself—replicas, twins. Perhaps they have been created by cloning him; they are exactly like him, but they are not him (that individual). Let’s suppose he doesn’t give a damn about these copies; he cares only for himself. He would happily let them die if it suited his own selfish purposes. Now contrast this selfish person with another person who also has many copies, but he does care about them; he goes out of his way to help them out, share with them, even sacrifice himself for them. This person is selfless not selfish; he is an altruist not an egoist. But suppose he is indifferent to anyone who is not a copy of himself; he is selectively selfless. It would be wrong to call him selfish, but his selflessness is sharply limited to his doubles, duplicates, twins. His altruism is confined to a certain select group of individuals: these he promotes and supports, feeds and protects—though anyone else gets the cold shoulder. It is this individual who best represents the concept of the selfish gene not the outright selfish person. The gene token seeks the welfare and survival of its copies not itself alone; it will go when the body it is in disintegrates, and it knows it. But its replicas will live on—and that is its mission in life. It may even sacrifice itself for the sake of its copies—as when a mother gives food to its babies even at a considerable cost to herself. It produces as many copies as possible, so that its type survives, possibly immortally; but it itself will perish with the bodily vehicle in which it exists. The copies will exist even when the individual gene that made them has gone. This gene is not selfish in any reasonable sense—though we might say the gene type is selfish because it sees to it that this type will survive into the future.

We really have two questions: are gene tokens selfish and are gene types selfish. No to the former, and yes to the latter. It depends on what we are referring to. I rather think people generally mean the former, in which case they misspeak, because the tokens are selfless not selfish. The token is selfless in the way the individual animal is selfless: the mother is concerned to protect her offspring copies even when this goes against her own individual interests (she might die as they suckle at her teats). It would be strange to say she was selfish because her type is trying to protect that type; she is protecting her copies not herself, as opposed to being a type that is protecting itself. So, the so-called selfish gene is not really selfish but selfless—altruistic, other-interested. Our genes (individual tokens) are actually generous and other-directed, but only within the class of recipients limited to type-identity with them. They are like a person only altruistic within his tribe or family, only more strictly. The Selfish Gene could have been called The Selfless Gene and said essentially all the same things, reserving for a footnote the observation that gene types can be said to be selfish in an extended sense of “selfish” (we don’t say person types are selfish but only person tokens). Still less is the information inside a person itself selfish. It is really a lot simpler simply to say that genes (those bits of matter) are selfless, adding that their sole focus of altruistic interest is their own copies. They don’t survive bodily destruction, though their replicas do. They are not physically passed on, though physical entities just like them survive in future bodies. It is certainly not true that my token set of genes is literally made up of the same material entities that existed in my ancestors; what survives are just the old types, or the information they contain. As I say, this doesn’t detract from the correctness of selfish gene theory, though it does recommend changing the rhetoric slightly. And I think the book would have attracted a lot less misunderstanding if it had been called The Unselfish but Selective Gene.[1]

[1] For philosophers steeped in the type-token distinction in application to the mind-body problem this paper will be pretty straightforward, but that distinction is not always apparent to people less prone to logic-chopping. I have tried to make it as clear as possible, based on past experience. The logical point is simply that “x is a replica of y” doesn’t entail “x is (numerically) identical to y”—in fact the opposite is true.

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Mathematical Knowledge

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Mathematical Knowledge

We can all agree that human beings have mathematical knowledge. We understand mathematical concepts. But how extensive is mathematical knowledge—what subjects are mathematical? The OED defines mathematics as “the branch of science concerned with number, quantity, and space, either as abstract concepts or as applied to physics, engineering, and other subjects”. But the question is which other subjects—psychology, sociology, geology, geography, history, ethics? It is tempting to divide all learning (academic and non-academic) into two groups, mathematical and non-mathematical; but is this right? One subject you study in math class, the other subjects in other classes (music, literature, Spanish). I am going to argue that this distinction is artificial, misguided, and simple-minded; mathematics is everywhere. All (or nearly all) knowledge is mathematical to one degree or another. In fact, mathematical knowledge is cognitively fundamental. Elsewhere[1] I have argued that logic and ethics are (partly) mathematical; now I will extend that thesis more broadly. It turns out that this is not that difficult to do once we have shed certain prejudices and curricular conventions. I will quickly survey the whole field of knowledge to establish this claim; then I will turn to theoretical conclusions.

Psychology has a mathematical side; mathematics is applied in it. Psychophysics, statistical methods, computational models, laws of memory (recency and frequency laws), theories of learning, IQ tests, and so on. You can’t be a psychologist and be mathematically illiterate. Sociology is much the same. Geology is concerned with questions of depth, hardness, age, weight, and molecular composition. Geography deals with land masses, distance, size, height, and so on. Biology involves morphology, genetic structure, population dynamics, cellular forms, and energy consumption. History records dates and times, movements of populations, rates of social change, distribution of political power. Economics states laws of supply and demand, discusses money, tracks financial markets. Poetry is concerned with meter. Literature may include word counts. Music is mathematical. Space and time are mathematically described and they condition everything. Both in class and in the street the knowing mind is working mathematically. The case of language is of special interest, because it has taken recent linguistics to recognize the mathematical character of linguistic competence: a natural language is a combinatorial capacity to operate on discrete symbols to produce infinitely many sentences; Chomsky’s Merge operation is set-theoretic and recursive. Grammar is basically mathematical (“computational”). There isn’t much that doesn’t have a mathematical side, whether salient or submerged. It isn’t just math proper along with physics and engineering; mathematics seeps into everything (astronomy, botany, cookery, art). Mathematical knowledge is ubiquitous.

This has a bearing on certain philosophical questions. First, mathematical and non-mathematical knowledge are not discrete states of mind but interwoven. The a posteriori is infected by the a priori—what we call applied math. Second, the problem of mathematical knowledge is not limited to pure math, or school math; it applies to all knowledge, give or take a bit. The farmer has this problem as much as the professional mathematician. Third, Quine’s indispensability thesis applies not just to physics but to every branch of science and beyond. Fourth, philosophy of mind (including cognitive science) must take account of the prevalence of mathematics in the human (and animal) mind. The mind is not just computational but also numerical. The brain too—neuroscientists must reckon with the mathematical brain. The mind-brain not only processes information; it calculates, does sums. This doesn’t fit classic empiricism in which perceiving is the fundamental operation of mind, unsullied by such rational faculties as mathematics. If mathematics in innately known, then all knowledge has elements of the innate in it. Epistemology also must acknowledge that the foundations of so-called empirical knowledge cannot be innocent of non-sensory contents. Math is foundational too. Empirical knowledge is a mixture of the purely sensory and the abstractly mathematical. Such knowledge is a fusion of the qualitative and the quantitative. Even colors need numbers (they have extent and intensity).

We are edging towards the doctrine that all knowledge is mathematical, necessarily so. This is a strong doctrine, but it has its appeal. Consider intentionality: all mental states are directed at some object or other—some discrete, distinguishable, countable object. Typically, we have multiple intentional objects before our minds simultaneously, each comparable with the others—larger, smaller, heavier, lighter, louder, quieter, closer, further away. These are all quantifiable relations. The visual field alone is replete with mathematical content (mainly geometry). Some of this mathematical complexity is imposed by the mind, which is mathematically pre-established. Thus, where there is intentionality there is mathematical content (quantity, number); and all knowledge presupposes intentionality. The idea of a spatiotemporal world consisting of discrete particulars builds in mathematical structure, and it is omnipresent. Plus, the very notion of plurality is implicitly mathematical: it is the idea of a set. Not for nothing did Frege liken predication to the function-argument structure in mathematics; it involves the idea of one thing being a member of a set of things. Set-theoretic thinking is basic to human cognition (perhaps all cognition). We are beginning to seem like natural-born mathematicians, at home with abstractions. Born to be mathematical (and wild). Baby, I was born this way (and in other ways too).

This prompts two further thoughts: is it the same with other animals, and does it also apply to perception. I think yes in both cases, though human knowledge no doubt has additional properties. Animals size each other up, travel great distances, compute the time from the position of the sun, estimate the reproductive fitness of potential mates, and so on. In these activities they demonstrate basic mathematical skills, make quantitative comparisons, count objects (up to a point).  They are not mathematical ignoramuses. I suspect bees are pretty math-savvy. Survival depends on getting your numbers right. Even worms need to estimate how deeply to bury themselves (see Darwin on worms). Number is part of Life on Earth, part of our biological heritage, embedded in our genes (the mathematical gene). The lioness must count the number of her cubs and estimate how much food they need. In the case of perception, it is eminently plausible to suppose that the eyes perform calculations to gauge distance and produce impressions of depth. People write books called things like “The Intelligent Eye”; someone could write one called “The Mathematical Eye”. The senses are calculating prodigies in the production of perceptual constancies, computing size (say) from perceived distance. So, mathematics is biologically primitive and no doubt ancient; number sense may first have evolved with the most primitive of organisms—but then went on to perform spectacular feats of mathematical cognition, culminating in Isaac Newton and the great mathematicians of history. Mathematics (the faculty) evolves. It has a natural history.

Philosophers are fond of the question of what is ontologically basic and universal. Some say sense-data, others material objects in space and time. You don’t have to be Plato or Pythagoras to see the universality of number: everything real is countable and hence subject to arithmetic. Kant located material objects in space and time, and these are quantifiable things; mathematics is part of the spatiotemporal manifold. And sense impressions can also be counted and measured (e.g., for vividness). Yet people seem reluctant to make numbers ontologically basic (the opposite in fact), probably because they are not “concrete”. In our conceptual scheme mathematics occupies a unique place as a kind of universal language or framework or background. It needs more respect; more room at the epistemological table. In any “descriptive metaphysics” it should be accorded a central place, because it is all-pervasive, an aspect of the lens through which we perceive reality. Sense, reference, and number: for reference depends on singling out one thing from other things, and that involves the deployment of concepts of identity, difference, and plurality—the very basis of number science. We refer to what we can count. If you imagine a mind transitioning from a blurred world to a differentiated world, thus achieving individuated reference, then you will see that mathematical concepts come into play when reference comes on the scene. We might speak of the “mathematical theory” of reference—the embedding of reference in a matrix of mathematical ideas. A space of points, a plurality of particulars. Referential competence presupposes mathematical competence (as linguistic competence in general does). Not school arithmetic, to be sure, but its cognitive underpinnings (quantifiers, sortal concepts, etc.).

It strikes me as not surprising that mathematics was the first science to develop: it is the most salient characteristic of the scientific mind, and the most rigorous. A species bad at arithmetic will not last long. It’s a pity, then, that in its modern academic incarnation it is so forbidding and inaccessible as a field of study (it should be easy!). We are born to be mathematical, as we are born to be linguistic, but the knowledge is not easily articulated or systematized. It is more of a practical skill than conscious theoretical knowledge. Yet everything mathematical seems to be in us from the start (as Plato taught). Of course, this knowledge is philosophically puzzling, even mysterious, but there it is, bubbling beneath the surface. Moreover, it is good knowledge to have (as Plato also taught); it isn’t epistemic junk (a type of gossip). You can be proud of your mathematical knowledge. Fortunately, the human mind has evolved to be in tune with mathematics. Evolution has produced something both useful and admirable. If God existed, he would be a mathematician. In the beginning was the number.[2]

[1] See my “Mathematical Ethics” and “Mathematical Logic”.

[2] Not the deed, not the sense-datum, not the material object, not the process, not the thought—but the number. For all the above are numerable. We have the finite and the infinite, the natural, the negative, the prime, the real, the imaginary, the number zero. You can’t get away from them. You can’t eliminate them from your conceptual scheme. Counting is part of human nature.

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Mathematical Logic

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Mathematical Logic

It has always been felt that logic and mathematics have a lot in common. This led to the idea that mathematics reduces to logic. I am going to argue that the converse is true: logic is basically mathematical; logic “reduces” (expands) to mathematics. It’s more complex than it looks in the textbooks. The first and vital point to get clear about is that logic is not concerned with language—words, symbols. It is concerned with thought—logical thought. If we imagine logical thought before language existed, logic is about that. There is no guarantee that the grammatical forms of language will correctly represent the structure of thought; indeed, that is unlikely to be true, given the prime purpose of spoken language, i.e., efficient communication. Logical thought (reasoning) takes place within the individual and has whatever complexity it needs. Logical concepts are not elements of speech. So, you need to open your mind to the possibility that logic de re is quite unlike the formulas you are familiar with from “symbolic logic” class (an oxymoron in my book). I am talking about conceptual logic—logic in the head. Be prepared, then, for some startling suggestions. Logic needs to be re-written from the ground up.[1]

My thesis then is that what we know as propositional and predicate calculus are species of arithmetic. This is easier to see for predicate logic because it is quantificational—it is about quantity, number. It is about “all” and “some”, or the concepts that underlie these words. These are part of a whole system of quantitative concepts corresponding to “a few”, “several”, “many”, “most”, “more”, “nearly all”, “innumerably many”, “infinitely many”, “one”, “two”, “twenty- seven”, and so on. These are all number words, words for counting, adding, etc. They answer the question “How many?”. It is to be noted that this quantificational logic is second-order: it concerns the number of things falling under a concept, as in “The concept dog has many things falling under it”. The thought is captured by that verbal formulation: we are thinking about concepts and assigning numbers to them. So, we are not doing the same thing in thought as we are when we ascribe a property to an object, as expressed by “That dog is brown”. This has long been recognized by philosophical logicians (or logical psychologists). The logical form of a quantificational thought is given by a property and a number: F (n)—e.g.,Dog (many). In predicate logic (quantification theory) we go up a level and think about concepts and their multiplicity; we don’t think about objects and their properties. Logic is meta.

But how does so-called propositional (or sentential) logic fit this description? In this logic we encounter the ideas of a sentence connective, a truth function, an operator on propositions. Natural language is taken very seriously as a guide to logical structure. There is scarcely a linguistic turn; the subject matter is already assumed to be linguistic (not that people are very clear what this means). How do these logical words and concepts assume a mathematical shape? The answer is not far to seek: “and” means “both”, to put it simply. The conjunction “p and q” means “Both p and q are true”; not one of them, or none, but both. This is a quantity word: the mind is working quantitatively when it thinks conjunctively—numerically, arithmetically. If both are true, then not just one is true—truth here requires a number of separate truths. If the conjunction is long, it will require that the corresponding number of propositions be true. The word “and” works like the word “plus”: it adds to the number of conjuncts. So, conjunctions in thought are mathematically conceived; you can’t form them unless you have some basic mathematical competence. Conjunction is often compared to universal quantification and the comparison is apt; but “all” is a number word, and so conjunction is a mathematical concept too, perhaps the simplest one. Disjunction works similarly: “or” means “at least one of the disjoined propositions is true”. It doesn’t have to be all disjuncts, so long as it is some; one will do. All this we grasp when we use the word “or”. We are thinking quantificationally. Hence “or” is often compared to “some”, aptly enough. Propositional logic is thus a species of quantificational logic, as complex and mathematical as it is. Moreover, it refers to propositions, so it is second-order, meta. But what about “not”? That little word has always been a thorn in the side of philosophers of logic (negative facts anyone?); it seems like a thing apart, curious in its connotation. What is this “not-ness”? It doesn’t yield easily to quantificational analysis—where is the “some” and “all”? What does “Not-p” mean? What do I mean when say that snow is not black? I suggest that I mean something like this: “There is no fact corresponding to the proposition that snow is black”, or “None of the facts in the world is the (putative) fact that snow is black”. It is a negative quantifier like “no one”. We could paraphrase our target sentence as “Nothing fits the description ‘black snow’”; or again, “There is no object such that it is both snow and black”. Thus, “not” is a simple word with a complex meaning; it takes a bit of brains to comprehend (animals are probably not up to it). You see why I said logical thought may not be properly reflected in ordinary language? The negative thought is a convoluted thought raising conceptual puzzles. In it the number zero presumably has its beginnings. It goes along with the word “false”: we can equally say “It is false that snow is black”, which is also second-order and meta. To think negatively is to entertain a counterfactual state of affairs and repudiate it as actual. In a way it lies at the very heart of logic, because logic is about necessity and possibility. The concept of possibility is itself tacitly quantificational and hence mathematical: “In some possible world, p”. We can naturally say “In hundreds of possible worlds, p” or “In no possible world, p”. Modal logic is likewise mathematical in character. Logic with negation is also logic with quantification. Similarly for tense logic in which we have quantification over times. All of these are mathematical, though primitively so (actually not so primitive). The logical mind is at bottom a mathematical mind. We would do well to render it more perspicuously in mathematical terms, thereby revealing its affinity with other types of mathematics. It is a kind of pre-school arithmetic.

I end with identity. Logic is sometimes said to include identity, and not without reason because identity is implicit in it. How does identity fit the mathematical analysis? As follows: to say that Hesperus is identical to Phosphorus is to say that Hesperus and Phosphorus are one. There are not two objects here but one object. We thought there were two (or more) and now we learn there is just the one. We counted wrong, as it turns out: if you put Hesperus and Phosphorus together in the same room, you find only one object there; there is one less object in the world than you supposed. In this thought we find the roots of the idea that a perceived plurality might reduce to an objective unity. Identity forces us to think numerically. It is an essentially mathematical concept. Accordingly, even elementary logic (with or without identity) is shot through with mathematical notions, though not particularly advanced ones; we might as well declare it a branch of mathematics. But remember it is not the symbolism that makes it mathematical but the underlying thought content. Mathematical logic is not a branch of logic but its very nature.[2]

[1] Compare my paper “Mathematical Ethics”.

[2] There is an irony in this in that the formulas of symbolic logic were supposed to lay bare the structure of logical thought, but actually it is the structure of logical thought that reveals the real content of those formulas. The thought is mathematical upon closer analysis, and this constitutes the meaning of the symbolic system. The hidden logical form of the formulas is given by the mathematical analysis of the underlying thought. Logic as we have it is a kind of baby math.

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The Bald Ape

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The Bald Ape

Desmond Morris christened us “the naked ape” in his 1967 book The Naked Ape.[1] His reason was that we are hairless compared to other apes; that is what marks us out from them. But this is wrong, because we are neither hairless nor naked. We are not hairless because we have a lot of hair—all over as well as in the obvious places. Our body hair is just finer and less visible than that of our ape relatives. You might say we are less hairy than them, but we are not hairless; by that standard non-human apes would be naked compared to polar bears. Further, being hairless is not the same as being bald: the palms of the hand are hairless, but it would be odd to say they are bald. To be bald is to have no hair where hair normally exists. A snake is not bald, or an elephant, or an ant, or a stone. We are hairy apes with some hairless body parts, but we are not bald (unless contingently on the head). We are fairly hairy animals prone to going bald on top. Are we naked in the ordinary sense? Of course not: we wear clothes. Our ape relatives are naked because unclothed; we are the non-naked ape. So, should we be designated “the clothed ape”? Well, that is not quite right either: some of us are clothed some of the time, and some of us are never clothed; and we existed as a species before we invented clothes. And isn’t it correct to say birds are clothed in feathers and polar bears in fur? The concept of clothes is not defined clearly enough to apply only to us, let alone all of us (are shoes clothes?). No one is ever fully clothed (except perhaps astronauts and Eskimos). We cannot then be defined by our outer covering or lack thereof: we are not defined by being hairless or naked or bald or clothed. Whether we can be singled out from other apes by any other distinctive characteristic remains a moot point—perhaps the ability to misdescribe ourselves.[2]

[1] This book made a big splash when it appeared and the idea of the naked ape captured the popular imagination. I first read the book in 1969 and re-read it recently. It stands up well.

[2] It is very tempting to want to find an observable trait that sets us apart from animal creation, but as good Darwinians we should eschew such an ambition. There is continuity not discontinuity. Beware the discrete mind! Is there an unobservable trait? It isn’t easy to find any such trait: some people cite language, but this trait is shared by other communicating species. If we say “language as humans have language”, we face the retort “That’s like saying we have eyes as humans have eyes”—a trivial tautology. It also implies that before language evolved humans were all of a piece with other animals. Kinship is the rule not transcendence.

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A Puzzle about Pornography

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A Puzzle about Pornography

Sex and food are clearly powerful forces in the psychic life of human beings. The question may be raised as to which is more powerful (is there a power imbalance?). But there is a curious asymmetry between them: we like to watch pornography but not its culinary equivalent. We like sex porn but food porn not so much. You never hear about people with a food porn addiction. Why is this? People like to watch other people having sex, but not other people eating. People don’t watch food porn videos when they feel hungry—why not? Why do people like watching sex porn when they feel horny? Isn’t it just as pointless as watching people eat when you feel hungry? Isn’t it irrational? The case is like imagination: people regularly have sexual fantasies, but it is not normal to have recurrent eating fantasies. Sexual fantasies are felt as gratifying, but not food fantasies. Teenagers don’t fantasize about chowing down all day. And people like pornography even when sexually active; but they don’t go in for food porn when well-fed. What need is satisfied by sexual images that is not satisfied by gustatory images? Why are there no magazines specializing in pictures of delicious meals? Why is a nude a lot more stimulating than a still life of a plate of food?

A variety of answers suggest themselves. Is it because pornography breaks a taboo but food images don’t? Is it the thrill of violating a taboo? It is true that we see naked food all the time but not naked bodies, and that it is regarded as taboo to want to see the latter. What if we inverted the two? Suppose people were general nudists but food is kept under wraps: you never see people eating food, but you constantly see them naked and fornicating. Would we then revel in food porn? Hardly. Pictures of food don’t have the power of pictures of naked bodies: food going into the mouth is not like the penis going into the vagina (etc.). The two types of “penetration” are not equivalent. Also, it would be breaking a taboo to watch film of people excreting, but hardly anyone would be enthusiastic about that. Taboo-breaking is neither necessary nor sufficient. What about the idea that sex is (partly) visual but eating isn’t? This sounds on the right track: the visual plays a role in actual sex, but not so much in eating. We are stimulated by the look of a naked body but not by the look of a ham sandwich. But is it true that vision plays no significant part in the pleasure of eating? That sounds wrong; it plays some part. And isn’t sex in the dark or blind sex pretty exciting too? Yet we don’t thrill to a purely auditory representation of a sex act. It isn’t that the sense of sight is absent from the food case but essential in the sex case. However, what is true is that sex is typically private and eating is not; and vision is what reveals the private most forcefully, as in voyeurism. Is it that sex porn satisfies our curiosity about this private matter, while food porn would do nothing analogous because eating is public? If we invert the two, the asymmetry seems to fade away: if all sex were public all the time, so that we were inundated with the stuff, we might lose our interest in porn; but if eating were always done privately, away from prying eyes, we might find some frisson in seeing images of it. It’s a question of privacy and curiosity. Yet we don’t have a comparable interest in toilet activity and it is also private; and I doubt there would be much of a market for food porn once the initial novelty wore off (how many times can you watch a man eating a hamburger?). The visual element seems part of the appeal of pornography, and lack of appeal of the food counterpart, but we have not yet identified the crucial difference. It begins to seem quite puzzling.

The puzzle deepens when we recall that food and sex are not just analogous but also connected. People seem to enjoy coupling the two (while coupling) and food-sex pornography is apparently quite popular. Yet no one is interested in pure food porn (except the odd gastronomic weirdo): people don’t want to salivate to the image of a roast chicken or a serving of sushi, no matter how much they may enjoy eating these things. Watching people eat is just not arousing or fascinating or exciting. It is the sex part of food-sex porn that gets people’s juices flowing not the food part. At this point we might be tempted to wax metaphysical: pornography is all about the human condition, death, embodiment, the soul. I don’t think this is completely wrong: thoughts of our animal nature, our talent for creative ideas, our strange juxtaposition of the godly and the goofy, our understanding of interpersonal relations, our grasp of the phenomenon of common knowledge, the mystery of reproduction, civilization and its discontents—all these may come into play. But they omit the primordial character of the depicted sex act—what really sets it apart viscerally. And here I want to mention an obvious fact: the genitals are on full display in all their anatomical glory. And the genitals are of enormous interest to us, because they are the organs of reproduction; the genes have programmed us to be obsessed with the genitals. Their future depends on them. The genitals are objects of endless fascination because of their biological centrality. If we had no interest in them, reproduction would grind to a halt (under normal conditions). I would bet that if you showed a video of the rear end of a female ape in heat to a male ape you would get a reaction. Copulation is genital, trivially. So, pornography is popular because of an ancient biological imperative: genital awareness, genital competence. Sexual selection is part of this: the genitals need to be seen to be present and healthy. Nothing like this holds in the case of food—the female is not going to reproduce with food! Genital know-how is what pornography efficiently enables, and that is useful knowledge to have. Genital curiosity is built into the genes and pornography satisfies this curiosity. A taste for porn is therefore innate and unavoidable. One might even conjecture that pornography works as a learning tool for the human species (think of the wall porn in old Pompei). If we combine this with the other suggestions mooted above, we can see why sex porn has a special interest not shared by food porn. We have no particular interest in plumbing the depths of the mouth and stomach, but the genitals are another ball of wax. We need to keep track of them in sexual encounters, hence the visual element, and their general hiddenness and privacy deprives us of ready access to their anatomy. Pornography enables us to remedy these difficulties. We can simulate in our minds the act depicted, which can then be applied in real life; it is a skill that has to be learned. That is the deep evolutionary meaning of pornography. Puzzle resolved.[1]

[1] I have relegated to a footnote one theory strangely popular these days, namely that the appeal of porn resides in its depiction of women as abused and degraded. This is not plausible as a general theory of the appeal of porn, though some porn no doubt trades on this deplorable proclivity. Obviously, much porn has nothing to do with it. Porn comes in many varieties, though the common factor is the genitals, one way or another. The genes program interest in the genitals and pornography caters to this instinctive interest.

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