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Probability Theory Pioneer Mark Kac on the Duality of the Creative Life, the Singular Enchantment of Mathematics, and the Two Types of Geniuses

“Creative people live in two worlds. One is the ordinary world which they share with others and in which they are not in any special way set apart from their fellow men. The other is private and it is in this world that the creative acts take place.”

Probability Theory Pioneer Mark Kac on the Duality of the Creative Life, the Singular Enchantment of Mathematics, and the Two Types of Geniuses

The great Polish-American mathematician Mark Kac (August 3, 1914–October 26, 1984) possessed one of the most dazzling minds of the twentieth century. In pioneering probability theory, he paved the way for a radical new conception of truth and ushered in the first generation of scientists trained to think probabilistically — a more accurate assessment of knowledge, making room for uncertainty, be it scientific or otherwise. This probabilistic mode of judgment is all the more necessary today as the growing complexity of the world is swirling us into exponentially increasing uncertainty, which we attempt to tame through artificial absolutism.

Mathematics literally saved Kac’s life. His student work earned him a post-doctoral fellowship to study abroad, so he left Poland for Johns Hopkins University in December of 1938. World War II broke out months later. His entire family, along with millions of other Jews, was killed by the Nazis.

Kac went on to lead a long and creatively fertile life — one he considered, despite this unfathomable share of misfortune, a tremendously fortunate one. “I must pay tribute to that powerful but capricious lady, Chance, who chose to bestow her beneficence on my personal life even though I spent much of my mathematical life trying to prove that she does not really exist,” he wrote with his characteristic mix of wit and wisdom in Enigmas of Chance: An Autobiography (public library) — a small, wonderful 1976 book I discovered via a passing mention in an interview with the trailblazing astronomer Vera Rubin. (Here is further proof of my longstanding conviction that literature is the original internet — such citations, allusions, and cross-references between books are the wondrous “hyperlinks” connecting human knowledge throughout our “common record.”)

Mark Kac
Mark Kac

Kac writes:

Creative people live in two worlds. One is the ordinary world which they share with others and in which they are not in any special way set apart from their fellow men. The other is private and it is in this world that the creative acts take place. It is a world with its own passions, elations and despairs, and it is here that, if one is as great as Einstein, one may even hear the voice of God. The two worlds are intimately and intricately connected. Jealousy, the desire for recognition and competitiveness, for example, are part of the ordinary world but they are among the forces which propel into the second. Similarly, dreams and triumphs in the second have a way of merging with less than lofty thoughts of rewards in the first.

With an eye to the particular challenge that autobiography presents to the creative person, he adds:

To create a coherent and truthful picture of life in the two disparate and yet interrelated worlds is a nearly impossible task.

In discussing his great heroes and influences, Kac delineates another dichotomy in creative culture — the bifurcation of brilliance by degree and by kind:

In science, as well as in other fields of human endeavor, there are two kinds of geniuses: the “ordinary” and the “magicians.” An ordinary genius is a [person] that you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what he has done, we feel certain that we, too, could have done it. It is different with the magicians. They are, to use mathematical jargon, in the orthogonal complement of where we are and the working of their minds is for all intents and purposes incomprehensible. Even after we understand what they have done, the process by which they have done it is completely dark. They seldom, if ever, have students because they cannot be emulated and it must be terribly frustrating for a brilliant young mind to cope with the mysterious ways in which the magician’s mind works.

He points to the nuclear physicist Hans Bethe as an example of an “ordinary genius” and to Richard Feynman as a “magician.” (Kac’s distinction appears in James Gleick’s superb biography of Feynman and there is a high probability that it inspired the title of BBC’s documentary about the legendary physicist, No Ordinary Genius.) I would add Alan Turing to the “magicians” category, and of course Albert Einstein.

Illustration by Vladimir Radunsky for On a Beam of Light: A Story of Albert Einstein by Jennifer Berne

In the postscript, Kac considers what lends mathematics its enduring enchantment — what renders people besotted with it:

Mathematics is an ancient discipline. For as long as we can reliably reach into the past, we can find its development intimately connected with the development of the whole of our civilization. For as long as we have a record of man’s curiosity and his quest for understanding, we find mathematics cultivated and cherished, practiced and taught. Throughout the ages it has stood as an ultimate in rational thought and as a monument to man’s desire to probe the workings of his own mind.

[…]

The urge to understand and to create mathematics has always been remarkable, considering that those who have devoted their lives to the service of this aloof and elusive mistress could expect neither great material rewards nor widespread fame.

A champion of nuance, Kac challenges Bertrand Russell’s famous assertion that “mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.” Kac finds it to be a “dull and rather miserable picture of mathematics” and yet “not wholly wrong,” “just hopelessly incomplete and one-sided.” Instead, he considers what grants mathematics its richness as a mode of illuminating reality:

I am reminded of something Balthazaar van der Pol, a great Dutch scientist and engineer who was also a fine musician, remarked to me about the music of Bach. “It is great,” he said, “because it is inevitable and yet surprising.” I have often thought about this lovely epigram in connection with mathematics… The inevitability is, in many cases, provided by logic alone, but the element of surprise must come from an insight outside the rigid confines of logic.

It warrants noting that the altogether marvelous Enigmas of Chance is part of a series of scientists’ autobiographies funded by the Alfred P. Sloan Foundation, which has probably done more for science and its social life than any other entity in the past half-century. Complement it with this beautiful love letter to mathematics by the pioneering 19th-century English mathematician James Joseph Sylvester, the illustrated life of the eccentric mathematical genius Paul Erdos, and the great English mathematician John Horton Conway on tinkering, thinkering, and the art of being a professional nonunderstander.

BP

Philosopher Martha Nussbaum on Anger, Forgiveness, the Emotional Machinery of Trust, and the Only Fruitful Response to Betrayal in Intimate Relationships

“All too often, anger becomes an alluring substitute for grieving, promising agency and control when one’s real situation does not offer control… Anger is often well-grounded, but it is too easy for it to hijack the necessary mourning process.”

Philosopher Martha Nussbaum on Anger, Forgiveness, the Emotional Machinery of Trust, and the Only Fruitful Response to Betrayal in Intimate Relationships

“We’ve got to be as clear-headed about human beings as possible, because we are still each other’s only hope,” James Baldwin told Margaret Mead in their terrific forgotten conversation about forgiveness and the difference between guilt and responsibility. “To forgive is to assume a larger identity than the person who was first hurt,” philosopher David Whyte echoed half a century later in contemplating anger, forgiveness, and what maturity really means. And yet the dance of anger and forgiveness, performed to the uncontrollable rhythm of trust, is perhaps the most difficult in human life, as well as one of the oldest.

The moral choreography of that dance is what philosopher Martha Nussbaum explores in Anger and Forgiveness: Resentment, Generosity, Justice (public library).

Martha Nussbaum

Nussbaum, who has previously examined the intelligence of the emotions and whom I consider the most incisive philosopher of our time, argues that despite anger’s long cultural history of being seen as morally justifiable and as a useful signal that wrongdoing has taken place, it is a normatively faulty response that masks deeper, more difficult emotions and stands in the way of resolving them. Consequently, forgiveness — which Nussbaum defines as “a change of heart on the part of the victim, who gives up anger and resentment in response to the offender’s confession and contrition” — is also warped into a transactional proposition wherein the wrongdoer must earn, through confession and apology, the wronged person’s morally superior grace.

Nussbaum outlines the core characteristics and paradoxes of anger:

Anger is an unusually complex emotion, since it involves both pain and pleasure [because] the prospect of retribution is pleasant… Anger also involves a double reference—to a person or people and to an act… The focus of anger is an act imputed to the target, which is taken to be a wrongful damage.

Injuries may be the focus in grief as well. But whereas grief focuses on the loss or damage itself, and lacks a target (unless it is the lost person, as in “I am grieving for so-and-so”), anger starts with the act that inflicted the damage, seeing it as intentionally inflicted by the target — and then, as a result, one becomes angry, and one’s anger is aimed at the target. Anger, then, requires causal thinking, and some grasp of right and wrong.

[…]

Notoriously, however, people sometimes get angry when they are frustrated by inanimate objects, which presumably cannot act wrongfully… In 1988, the Journal of the American Medical Association published an article on “vending machine rage”: fifteen injuries, three of them fatal, as a result of angry men kicking or rocking machines that had taken their money without dispensing the drink. (The fatal injuries were caused by machines falling over on the men and crushing them.)

Beneath this tragicomic response lies a combination of personal insecurity, vulnerability, and what Nussbaum calls status-injury (or what Aristotle called down-ranking) — the perception that the wrongdoer has lowered the social status of the wronged — conspiring to produce a state of exasperating helplessness. Anger, Nussbaum argues, is how we seek to create an illusion of control where we feel none.

Art by JooHee Yoon from The Tiger Who Would Be King, James Thurber’s parable of the destructiveness of status-seeking

She writes:

Anger is not always, but very often, about status-injury. And status-injury has a narcissistic flavor: rather than focusing on the wrongfulness of the act as such, a focus that might lead to concern for wrongful acts of the same type more generally, the status-angry person focuses obsessively on herself and her standing vis-à-vis others.

[…]

We are prone to anger to the extent that we feel insecure or lacking control with respect to the aspect of our goals that has been assailed — and to the extent that we expect or desire control. Anger aims at restoring lost control and often achieves at least an illusion of it. To the extent that a culture encourages people to feel vulnerable to affront and down-ranking in a wide variety of situations, it encourages the roots of status-focused anger.

Nowhere is anger more acute, nor more damaging, than in intimate relationships, where the stakes are impossibly high. Because they are so central to our flourishing and because our personal investment in them is at its deepest, the potential for betrayal there is enormous and therefore enormously vulnerable-making. Crucially, Nussbaum argues, intimate relationships involve trust, which is predicated on inevitable vulnerability. She considers what trust actually means:

Trust … is different from mere reliance. One may rely on an alarm clock, and to that extent be disappointed if it fails to do its job, but one does not feel deeply vulnerable, or profoundly invaded by the failure. Similarly, one may rely on a dishonest colleague to continue lying and cheating, but this is reason, precisely, not to trust that person; instead, one will try to protect oneself from damage. Trust, by contrast, involves opening oneself to the possibility of betrayal, hence to a very deep form of harm. It means relaxing the self-protective strategies with which we usually go through life, attaching great importance to actions by the other over which one has little control. It means, then, living with a certain degree of helplessness.

Is trust a matter of belief or emotion? Both, in complexly related ways. Trusting someone, one believes that she will keep her commitments, and at the same time one appraises those commitments as very important for one’s own flourishing. But that latter appraisal is a key constituent part of a number of emotions, including hope, fear, and, if things go wrong, deep grief and loss. Trust is probably not identical to those emotions, but under normal circumstances of life it often proves sufficient for them. One also typically has other related emotions toward a person whom one trusts, such as love and concern. Although one typically does not decide to trust in a deliberate way, the willingness to be in someone else’s hands is a kind of choice, since one can certainly live without that type of dependency… Living with trust involves profound vulnerability and some helplessness, which may easily be deflected into anger.

One of William Blake’s illustrations for Milton’s Paradise Lost

Another element that makes intimate relationships a special case is how deeply we experience their breakdown. Nussbaum writes:

The damage involved in the breakdown of an intimate relationship … is internal and goes to the heart of who one is… Beyond a certain point there is really no place to go, except into your own heart — and what you find there is likely to be pretty unpleasant. So there is something lonely and isolating about these harms; they involve a profound helplessness. Once again, this helplessness can easily be deflected into anger, which gives the illusion of agency and control.

She points to one more singular feature of intimate relationships and their breakdown — the simultaneous and often confusing coexistence of positive and negative emotions toward the person whom we once loved and whose painful betrayal has now spun us into anger. (This might well share psychological underpinnings with the paradox of why frustration is essential for satisfaction while falling in love.) Nussbaum considers the complexities of this duality:

We typically form intimate relationships with people we like. We choose our spouses, and even though parents do not choose their children or children their parents, there is typically, in cases that are not really awful, a symbiosis that produces liking on both sides, though adolescence certainly obscures this. Most other people in the world, by contrast, are not people with whom one would choose to live. It’s pretty easy to find them irritating, or off-putting, or even disgusting. How many people who sit next to one by chance on an airplane are people with whom one would be happy living in the same house for an extended period of time? But a spouse, a lover, a child — these people are welcomed, and there usually remains something nice about them that is not utterly removed by whatever it is they have done. The target of anger is the person, but its focus is the act, and the person is more than the act, however difficult it is to remember this. This nice something could become another knife to twist in the wound of betrayal (to the extent that a person is appealing, it’s harder to say good riddance), but on the other hand it could also be a basis for constructive thought about the future — in a restored relationship or some new connection yet to be invented.

Nussbaum, who has written brilliantly about the nuanced relationship between agency and victimhood, turns a skeptical eye toward the common cultural mythology of anger as a response indicative of self-respect. (I am reminded of Joan Didion’s unforgettable assertion that self-respect springs from “the willingness to accept responsibility for one’s own life.” How, then, can anger — the artificial and ill-fated attempt to take responsibility for another’s life — be a wellspring of self-respect?) Instead, Nussbaum points to a Transition — a mental pivot in which one turns from anger to more constructive, forward-oriented considerations of what can be done to increase welfare rather than to inflict harm in vengeance — as the proper self-caring response to a breach of trust. She writes:

Anger is such a large and corrosive problem that much of the literature focuses on how to manage it so that it does not destroy one’s entire life. And it is especially here that there’s a widespread feeling that, bad though anger is, people (and women especially) owe it to their self-respect to own, nourish, and publicly proclaim their anger.

Anger, Nussbaum suggests, is a mask for the profound grief we don’t want to or simply can’t let ourselves feel when confronted with an intimate betrayal:

Such breakdowns typically, and rightly, involve deep grief, and grief needs to be dealt with. Grief is amply warranted: intimate relationships are very important parts of a flourishing life. (Here the Stoics are wrong.) But grief, and the helplessness it typically brings with it, are usually not well addressed by allowing anger to take the center of the stage. All too often, anger becomes an alluring substitute for grieving, promising agency and control when one’s real situation does not offer control… The way to deal with grief is just what one might expect: mourning and, eventually, constructive forward-looking action to repair and pursue one’s life. Anger is often well-grounded, but it is too easy for it to hijack the necessary mourning process. So a Transition from anger to mourning — and, eventually, to thoughts of the future — is to be strongly preferred to anger nourished and cultivated.

Nussbaum considers the particularly charged betrayals of intimate spousal relationships:

Because the couple pursues jointly some of the most important life goals of each, these goals themselves become shared goals and are shaped by the partnership. The vulnerability involved in such a relationship therefore goes very deep… Even though it would still be possible, and, I believe, highly desirable, to preserve a core sense of oneself as a person who could continue no matter what, this is often difficult to achieve, and it is always difficult to strike a balance between this healthy self-preservation and a kind of self-withholding that is incompatible with deep love.

In ongoing relationships, Nussbaum argues, there are bound to be many strains and possible breaking points, since the very premise of a long-term intimate relationship is that two different people with different goals (however similar their values may be) must somehow reconcile the autonomy of their individual personhood with the cohesion of their shared life. She examines these elemental dynamics:

It’s clear that there will be more strains when people are inflexible and intolerant, seeing every divergence from what they want as a threat… Anger will also be more common when one or more of the parties feels a lot of insecurity, because so many things can seem threatening, including, indeed, the sheer independent existence of the other person. (Proust makes the point that for a deeply insecure person, the other person’s very independent will is a source of torment and, often, rage.) A good deal of marital anger is really about this desire for control — and since such projects are doomed, that sort of anger is likely to be especially hard to eradicate. Intimacy is scary, and it makes people helpless, since deep hurt can be inflicted by the independent choices of someone else; so, as with other forms of helplessness, people respond by seeking control through anger. People never dispel their own insecurity by controlling someone else or making that person suffer, but many people try — and try again. Furthermore, people are adept rationalizers, so insecure people seeking control are good at coming up with a rational account of what the other person has done wrong…

One of Maurice Sendak’s illustrations for Melville’s Pierre

Nussbaum argues that while anger is an understandable response when spousal trust is breached, it is ultimately more destructive than constructive for the person feeling it, for it prevents her or him from processing the deeper emotions and healing the wounds from which they ooze. She examines the innermost machinery of betrayal:

What’s the real problem? It is one of deep loss. Two selves have become so intertwined that the “abandoned” one has no idea of how to have fun, how to invite friends to dinner, how to make jokes, how to choose clothes even, if not for and toward the other one. So it’s like learning to walk all over again, and that is particularly true of women without strong independent careers and social networks, since those who do have careers have many parts of their lives that have not been blasted by the betrayal, friends of their own who are not attached to the spouse, and lots of useful work to do. Children have all of their adolescence to learn, gradually, how to live apart from their parents, and they expect to do so all along. A betrayed spouse often has no preparation for separateness, and no skill at leading a separate life.

[…]

It is easy, in that situation, to think that the best future is one involving some type of payback, since that future, unlike the future of self-creation, is easy to imagine. It’s still intertwined with the other person. It is like not breaking up. You can go on being part of a couple, and keeping that person at the center of your thoughts.

One of William Blake’s illustrations for Dante’s Divine Comedy

But this default response of anger, Nussbaum cautions, does nothing to address our actual problems — in fact, it obstructs their solution:

[Anger] diverts one’s thoughts from the real problem to something in the past that cannot be changed. It makes one think that progress will have been made if the betrayer suffers, when, in reality, this does nothing to solve the real problem. It eats up the personality and makes the person quite unpleasant to be with. It impedes useful introspection. It becomes its own project, displacing or forestalling other useful projects. And importantly, it almost always makes the relationship with the other person worse. There was something likable about the person, and even if marriage is no longer possible or desirable, some other form of connection might still be, and might contribute to happiness. Or it might not. But the whole question cannot be considered if angry thoughts and wishes fill up the mental landscape. Far from being required in order to shore up one’s own self-respect, anger actually impedes the assertion of self-respect in worthwhile actions and a meaningful life.

The only reasonable requirement, Nussbaum argues, is an acknowledgement of wrongdoing on behalf of the wrongdoer:

Being heard and acknowledged is a reasonable wish on the part of the wronged party, and asking for truth and understanding is not the same thing as asking for payback. Indeed, it often helps the Transition. However, often the extraction of acknowledgment shades over unpleasantly into payback and even humiliation, and this temptation should be avoided.

Nussbaum concludes:

Intimate relationships are perilous because of the exposure and lack of control they involve. Being seriously wronged is a constant possibility, and anger, therefore, a constant and profoundly human temptation. If vulnerability is a necessary consequence of giving love its proper value, then grief is often right and valuable. It does not follow, however, that anger is so.

Anger and Forgiveness, based on Nussbaum’s 2014 Locke Lectures in Philosophy at Oxford University, is a tremendous read in its totality and goes on to explore such facets of this perennial subject as payback, mercy, shame, our ideas about strength and weakness, what everyday justice means in the political realm, and how false social values warp our interior lives. Complement it with psychologist David DeSteno on the psychology of trust in work and love and Maria Konnikova on what con artists reveal about the psychology of deception, then revisit Nussbaum on human dignity and how to live with our fragility.

BP

Nobel-Winning Physicist Frank Wilczek on Complementarity as the Quantum of Life and Why Reality Is Woven of Opposing Truths

“You can recognize a deep truth by the feature that its opposite is also a deep truth.”

Nobel-Winning Physicist Frank Wilczek on Complementarity as the Quantum of Life and Why Reality Is Woven of Opposing Truths

“The truth is, we know so little about life, we don’t really know what the good news is and what the bad news is,” Kurt Vonnegut lamented in his terrific lecture on storytelling. But one of life’s greatest confusions stems from our tendency to divide the world into such polarities in the first place — something Susan Sontag considered an immensely limiting impulse. When confronted with the world’s complexity, we default into navigating it by creating artificial binaries, perceiving contradiction where they might in fact only be complementarity. Cheryl Strayed captured this perfectly: “Two things can be true at once — even opposing truths.” Then there is, always, Whitman: “Do I contradict myself? / Very well then I contradict myself, / (I am large, I contain multitudes.)”

In A Beautiful Question: Finding Nature’s Deep Design (public library), Nobel-winning physicist Frank Wilczek considers this paradoxical notion of complementarity not only as raw material for the philosophical and the poetic but as one of the four cornerstones of modern physics, alongside relativity, symmetry, and invariance.

Art by Salvador Dalí for a special edition of Alice in Wonderland

Complementarity — the idea that two different ways of regarding reality can both be true, but not at the same time, so in order to describe reality we must choose between the two because the internal validity and coherence of one would interfere with that of the other — is a centerpiece of quantum theory. Wilczek points to one familiar example — the fact that light is neither inherently a particle nor inherently a wave, but can be either depending on how we measure it.

True knowledge, Wilczek intimates, progresses not toward simplifying our answers but toward improving our questioning mechanisms to better address complexity — Newton fancied the idea that light was a particle but was also curious about alternatives; a century and a half later, Maxwell ushered in electromagnetism and rendered wave theory victorious; when quantum mechanics came into bloom three generations later, scientists pointed to the photon, an elementary particle, as the ultimate quantum of light. Wilczek writes:

Particle and wave offer complementary perspectives on the reality of light. Newton’s practice of keeping many alternatives in play, while refusing to put forward any one Hypothesis exclusively, anticipates modern complementarity.

Newton at work by William Blake (1795-1805)
Newton at work by William Blake (1795-1805)

But between Newton and modernity stood the Romantic era, in which artists rebelled against scientific reductionism and what they perceived to be its assault on complementarity. Wilczek points to one particularly resplendent example involving Newton himself:

William Blake protested against reductionism’s blinkered vision. In this depiction of Isaac Newton at work, Blake’s conflicted feelings for his subject are on display. His Newton is a figure of extraordinary concentration and purpose, not to mention superhuman anatomy. On the other hand, he is shown looking down, lost in abstractions having literally turned his back on the strange, colorful landscape. Yet Blake admitted (as did Keats) that mathematical order governs the world. In Blake’s complex mythology Urizen, depicted here, is a dualistic Father figure, who both brings life and constrains it. One can hardly fail to notice a certain resemblance to the preceding drawing. Is Newton Urizen’s interpreter, or his incarnation?

The Ancient of Days, William Blake's depiction of Urizen (1794)
The Ancient of Days, William Blake’s depiction of Urizen (1794)

Indeed, although rooted in physics, complementarity’s central proposition extends into the metaphysical — a dimension that goes all the way back to quantum theory pioneer Niels Bohr, who originated the complementarity principle. Wilczek writes:

[Bohr] was fond of a concept he called “deep truth.” It exemplifies Ludwig Wittgenstein’s proposal that all of philosophy can, and probably should, be conveyed in the form of jokes. According to Bohr, ordinary propositions are exhausted by their literal meaning, and ordinarily the opposite of a truth is a falsehood. Deep propositions, however, have meaning that goes beneath their surface. You can recognize a deep truth by the feature that its opposite is also a deep truth.

Bohr was so enchanted by complementarity and its manifestations beyond science that he became fascinated with the unified duality of yin-yang in the Eastern philosophy — so fascinated that he placed the yin-yang symbol in the middle of the coat of arms he designed for himself, under the banner Contraria sunt complementa [Opposites Are Complementary].

Niels Bohr's coat of arms
Niels Bohr’s coat of arms

Wilczek writes:

From his immersion in the quantum world, where contradiction and truth are near neighbors, Niels Bohr drew the lesson of complementarity: No one perspective exhausts reality, and different perspectives may be valuable, yet mutually exclusive. The yin-yang sign is an appropriate symbol for complementarity, and was adopted as such by Niels Bohr. Its two aspects are equal, but different; each contains, and is contained within, the other. Perhaps not coincidentally, Niels Bohr was very happily married. Once recognized, complementarity is a wisdom we rediscover, and confirm, both in the physical world and beyond.

(Although Wilczek’s remark about marriage is a facetious wink, the poet Mary Oliver has written beautifully about the vitalizing role of complementarity in love.)

Wilczek synthesizes the larger truth to which complementarity speaks:

To address different questions, we must process information in different ways. In important examples, those methods of processing prove to be mutually incompatible. Thus no one approach, however clever, can provide answers to all possible questions. To do full justice to reality, we must engage it from different perspectives. That is the philosophical principle of complementarity. It is a lesson in humility that quantum theory forces to our attention… Complementarity is both a feature of physical reality and a lesson in wisdom.

Complement the wholly magnificent A Beautiful Question with Simone Weil on how quantum theory changed science and society and Alice in Quantumland — an allegory of quantum mechanics inspired by the Lewis Carroll classic — then treat yourself to Wilczek’s enchanting On Being conversation with Krista Tippett:

You have to view the world in different ways to do it justice, and the different ways can each be very rich, can each be internally consistent, can each have its own language and rules. But they may be mutually incompatible — and to do full justice to reality, you have to take both of them into account.

BP

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