Waves and Stones
Waves and Stones
On the Ultimate Nature of Reality
Graham Harman
LANE an imprint of
ALLEN
ALLEN LANE
UK | USA | Canada | Ireland | Australia India | New Zealand | South Africa
Allen Lane is part of the Penguin Random House group of companies whose addresses can be found at global.penguinrandomhouse.com.
Penguin Random House UK One Embassy Gardens, 8 Viaduct Gardens, London SW 11 7BW penguin.co.uk
First published in Great Britain by Allen Lane 2025 001
Copyright © Graham Harman, 2025
Penguin Random House values and supports copyright. Copyright fuels creativity, encourages diverse voices, promotes freedom of expression and supports a vibrant culture. Thank you for purchasing an authorized edition of this book and for respecting intellectual property laws by not reproducing, scanning or distributing any part of it by any means without permission. You are supporting authors and enabling Penguin Random House to continue to publish books for everyone. No part of this book may be used or reproduced in any manner for the purpose of training artificial intelligence technologies or systems. In accordance with Article 4(3) of the DSM Directive 2019/790, Penguin Random House expressly reserves this work from the text and data mining exception.
The moral right of the author has been asserted
Set in 12/14.75 pt Dante MT Std Typeset by Six Red Marbles UK, Thetford, Norfolk Printed and bound in Great Britain by Clays Ltd, Elcograf S.p.A.
The authorized representative in the EEA is Penguin Random House Ireland, Morrison Chambers, 32 Nassau Street, Dublin D 02 YH 68
A CIP catalogue record for this book is available from the British Library
ISBN : 978– 0– 241– 39286– 7
Penguin Random House is committed to a sustainable future for our business, our readers and our planet. This book is made from Forest Stewardship Council® certified paper.
‘For if it does not belong to the philosopher, then who will be the investigator of whether Socrates and Socrates seated are the same?’
Aristotle
Prologue: The Continuous and the Discrete
Nikos Kazantzakis (1883–1957) is a major figure of modern Greek letters who is most famous for his novels The Last Temptation of Christ and Zorba the Greek, both of them turned into Hollywood films.1 Yet he also took the bold step of writing an epic poem, a genre largely absent from recent literature. Entitled The Odyssey: A Modern Sequel, it contains a passage where the hero, Odysseus, recalls three moments in his life that towered above all others:
Sweet, very sweet had been his dread on that first night when in the dark he’d laid his hand on a maid’s body; how like a hawk he’d shrieked, how all the world had sighed when in his arms he’d held a son for the first time! And then that third dread shriek when on a distant plain he’d held on high his foe’s slain head for the first time!2
His first sexual experience, the first time holding a child of his own, and his first time killing someone in battle: it’s easy to see why these three moments might stand out as turning points in a human life. Although hoisting the head of one’s enemy on a spike is now biographically rare, we can all remember especially dramatic moments in our own lives. Kazantzakis is poetically capturing a vision of the human lifespan as characterized by discrete, monumental instants during which we experience irreversible change.
A different intuition can be found in a passage from Japanese author Yukio Mishima (1925–70), an intense, nationalistic figure who died via ritual suicide at a military base outside Tokyo. His novel Runaway Horses offers an account of human life that takes the opposite tack: ‘How oddly situated a man is apt to find himself at age thirty-eight! His youth belongs to the distant past. Yet the
Prologue: The Continuous and the Discrete period of memory beginning with the end of youth and extending to the present has left him not a single vivid impression.’3 Here Mishima imagines nearly two decades of life as a continuous span during which no particular experience changes us significantly. This uneventful continuum generates the illusion that in our late thirties, and perhaps beyond, we are still close to the now-distant era of youth.
Which author’s standpoint seems more accurate? Do our lives consist of a small number of dramatic turning points, or is there nothing but a series of gradual changes from infancy to old age? The same type of question often arises in daily life. In the case of the weather, temperature is the sort of thing we treat as continuous, meaning that changes from day to day are usually experienced as differences of degree rather than radical differences in kind. Yet the freezing and boiling of water happen at specific points on the thermometer: 0 degrees Celsius for freezing and 100 degrees Celsius for boiling, assuming standard atmospheric pressure at sea level. In American politics there is the constitutional requirement to hold a presidential election in early November every fourth year, and at no other time. This seems to punctuate public life into predictable, discrete segments. But some elections come to be seen as transformational, while others seem in retrospect to have led to nothing but continued business as usual. Within this system elections are usually called transformational if they involve a substantial realignment of previous political coalitions. The Democratic politician Franklin D. Roosevelt was first elected President in 1932, and was eventually elected a total of four times, which is no longer even legal. He led the country out of the Great Depression and through most of World War II with a series of liberal policies that came to typify an entire era. But later, in 1980, a large number of labour union members permanently abandoned their allegiance to the Democratic Party to vote for the Republican candidate Ronald Reagan: ‘Reagan Democrats’, as they are called. In the wake of the surprising 2024 election, there has been early talk of ‘Trump Democrats’ as well. In any case, although American politics schedules discrete
Prologue: The Continuous and the Discrete four-year administrations in advance, some are largely continuations of the existing spirit of the age while others mark relatively abrupt shifts in national policy and mood. But are transformational elections real, or are they merely arbitrary points along a gradually shifting cultural timeline?
Far from being abstract or arcane, these two ways of looking at the world – continuous and discrete – have given rise to some of the most bitter controversies of our time. To list just one: are there two fully distinct genders existing by nature, or just numerous gradations between the male and female poles?4 The same opposition between the continuous and the discrete pulls modern physics in two directions: Einstein’s General relativity treats gravity as something varying continuously, while quantum theory treats the other three fundamental forces of nature (see p. 242) as working by way of discontinuous jumps. Incredibly enough, these two honoured theories remain in contradiction, although both have been experimentally confirmed to a high degree of accuracy. This book will show that the issue both predates and exceeds contemporary physics. Since the days of ancient philosophy, an equal number of complications have arisen whether we think of the world as made of tiny discrete units or as a continuum where everything blends together into a single whole. At the close of his recent work Hysteresis, the Italian philosopher Maurizio Ferraris (b. 1956) shares an insight drawn from the writings of the German polymath G. W. Leibniz (1646–1716): ‘There are two mazes in which human thought gets lost. The first is that of predestination, the second is that of continuity and movement.’5 Ferraris notes that both problems boil down to the interplay of the continuous and the discrete: freedom requires a discontinuous gap between causation and human decision, while movement is impossible if space consists of an infinite number of actual discrete points.
But in fact, every area of life is entangled with the paradoxical relation between the continuous and the discrete. Most other issues do not cross disciplinary boundaries in this way. While astronomers and physicists puzzle over the crucial unsolved problems of dark
Prologue: The Continuous and the Discrete matter and dark energy, these topics mean very little to researchers in electrical engineering or sociology. The same holds for the question of whether humans first reached the Americas via a land bridge with Asia: an issue pivotal for anthropology and perhaps linguistics, but of little relevance to criminal justice or Shakespeare studies. But in cases where certain topics arise again and again in every place we look, we seem to be dealing with a basic feature of reality. These are the moments when philosophy, sometimes dismissed as idle speculation with little practical value, is called to the scene like Sherlock Holmes to the site of a murder. So it is with the Case of the Continuous and the Discrete: the question of whether reality is made up primarily of sudden jumps or is laid out instead along a gentle gradient, with no clear divisions between the various different things in the world.
But before entering further into this topic, I should introduce myself and the standpoint from which I speak. This will help clarify why the duel of waves and stones – the continuous and the discrete – is so decisive for illuminating the nature of reality. Otherwise we will continue to fall into the same paradoxes as ancient philosophy, which the ancients often grasped more firmly than we do today.
Skipping Stones: The Riddle of Thixis
Object-Oriented Ontology (OOO)
My colleagues and I have been developing Object-Oriented Ontology (OOO ) since the 1990s.1 This began when I noticed an important glitch in the mainstream interpretation of the influential German philosopher Martin Heidegger (1889–1976), a committed Nazi who nonetheless had some extremely suggestive ideas about reality.2 Indeed, I am just one of countless commentators who have been fascinated by the analysis of tools found early in his major work Being and Time. 3 Although Heidegger was referring to such obviously useful items as hammers, screwdrivers and railway platforms, it seemed to me that his analysis held good for any object at all. His basic insight is that most of the time we do not consciously perceive the objects in our midst; instead, we silently rely on them unless something goes wrong, such as a hammer breaking apart in our hands or a train not arriving on time. This was an important insight at a time when phenomenology, the descriptive philosophy of consciousness that Heidegger initially defended, gave excessive priority to our direct mental awareness of things. Whereas phenomenology was a rationalism committed to explaining all human experience clearly, Heidegger’s mutant version of the doctrine emphasized those aspects of reality that tend to remain hidden: the mysteries of poetic language, the pre-rational impact of human moods, or our dependence on historical structures far predating our birth.
On the whole, I always found Heidegger’s tool-analysis convincing. What bothered me was an additional claim, to which I was first
Skipping Stones: The Riddle of Thixis alerted by my academic advisor Alphonso Lingis (1933–2025). Namely, Heidegger also thought that all tools ultimately blend together in a single system, one that gains its meaning from my human purposes: the house I am trying to build, the money I am trying to earn. But to treat tools as belonging to a single holistic system shaped by each person is to ignore two major complicating factors. First, tools cannot fit snugly into a system, since we know that they sometimes break or otherwise go haywire; this means that the system never fully deploys any tool as a whole. The act of building a house makes use of the hammer, but takes no account of a crack in its handle that eventually causes it to shatter, bringing the construction project to a halt.
Second, and more controversially, it is not just human beings who reduce objects to specific purposes while forgetting their plenitude of hidden qualities. Instead, objects do this to each other as well. Recent philosophy has little or nothing to say about object–object interactions when no human observer is on the scene, and simply assumes that natural science should have a monopoly on this topic. This is one reason why even the early version of OOO faced a good deal of hostility from other philosophers. To this day, in fact, OOO remains more influential in other disciplines, which are professionally less committed to the specific anthropocentric bias of modern philosophy.
The phrase ‘object-oriented’ was borrowed from computer science, where it refers to a type of programming language. Originally a computer program functioned as a unified whole on which each of its parts was dependent. By contrast, object-oriented programming languages are based on independent modules that perform autonomous tasks and can be moved between different programs with relative ease, given their autonomy from the system as a whole. As an analogy from outside the world of computers, consider a coffee machine. The complex internal functions of the machine are hidden from the user, who deals instead with an interface offering a small number of brewing options. Furthermore, the coffee machine can be moved if necessary from our kitchen to a café or restaurant, assuming that we own one of these businesses.
In other words, the user’s coffee-brewing experience is independent of the machine’s internal workings (as long as it does not break) and also independent of the context in which it is placed. This is what an object-oriented program is like, and also what an object in OOO is like. As we will see, objects have a degree of independence from both their internal components and their external effects.
As for the ‘object’ part of OOO, it’s important to note that OOO objects are not limited to mid-sized physical things, but refer to anything that can be considered as a unit irreducible to its internal workings or external context. As simplistic as this might sound, we will see that a surprising amount of talk about objects involves reducing them either upward or downward. Although in everyday English ‘reduce’ refers to making something smaller, in philosophy it indicates that one thing is declared to be merely derivative of another. Note that this is not always a bad thing. If we successfully reduce heat to an effect of atomic motion, or reduce belief in witches to social paranoia and the desire to confiscate the belongings of widows, we have already learned quite a lot. Yet something is always left out when we reduce an object: namely, the flexibility to understand heat or belief in witches on their own terms before explaining them away in terms of something else. In philosophy, for instance, we have seen that Heidegger tends to reduce objects to a wider system that embraces them all. It is a powerful idea, but one that risks losing any robust sense of the independence and individuality of things.
As for the ‘ontology’ part of OOO, this term stems from ancient Greek, though it was first coined in 1613 by the little-known German philosopher Rudolf Göckel (1547–1628). Ontology can be defined roughly as ‘the study of being’, and any subtler explanation of the term would only lead to pedantic remarks inconsistent with the unpedantic spirit of this book. Generally speaking, ontology is the most fundamental branch of philosophy. It is concerned with the basic structure of reality as opposed to more specific pursuits such as the philosophies of law, art or language.
For more than a century, professional philosophy has been
Skipping Stones: The Riddle of Thixis polarized between two opposed but partly interlocking traditions. Analytic philosophy, which dominates the elite universities of the Anglo-American world, has a culture that values precisely focused technical research articles in the manner of the natural sciences. Continental philosophy has largely French and German roots, and generally works in a more literary vein. What is most valued here are major books by a relatively small number of superstar thinkers, with the result that others tend mostly to write commentaries on the works of the superstars: Hegel or Heidegger, Hannah Arendt or Walter Benjamin, Judith Butler or Slavoj Žižek. Both traditions have typical strengths and weaknesses. As I see it, one of the strengths of the continental approach is its greater appeal to those who work in disciplines other than philosophy, such as art, architecture, anthropology, psychology or organization studies. This is because continental work, at its best, is less narrowly aimed at a readership of philosophy professors.
But although I come from the continental tradition myself, I am frustrated by some of its most prominent trends. The one most relevant to the present book is that present-day continental thought is overly enamoured of the notion that the world is made up of continuities, turbulent fluxes and flows, and gentle gradations rather than abrupt cut-offs between one thing and the next. As novel and brilliantly counterintuitive as this flux-based model might seem, it is already rather old, and faces defects of the sort that quickly doom any theory. It leads to the idea that reality is what I have called a ‘Blend-o-Rama’, or what my friend Timothy Morton once colourfully termed an ‘everything-is-everything-else Deleuzean Hinduism’, referring to the French philosopher Gilles Deleuze (1925–95), a noted advocate of the continuous approach to reality.4 The price paid by such a theory, and in a different sense by Heidegger’s tool-analysis, is an excessive focus on the whole and a weakened ability to account for the status of individual entities. If we say that so-called objects are just transient patterns in a molten world undergoing constant change, in the manner of a hippy’s groovy lava lamp, we still need to explain why it seems that
there are enduring individual things in the world: hammers, ducks, stars in the sky. Inevitably, the answer from the flux-lovers will be that the biases of human thought and perception deceive us into thinking that objects exist in their own right, when they are really just arbitrary portions of a single vibrating whole. According to this line of thought, the biases of human cognition have the power to misconstrue the nature of reality radically, seeing distinct individuals where there are really just resonant flows. But if that is the case, it means that humans are already being treated as something different enough from the rest of the cosmos that we can make mistakes about it. The contradiction should be obvious: if the whole of reality is in constant flux, then humans too should be part of that flux; hence, we should not be different enough from the rest to be able to misinterpret it in the first place. Some version of this contradiction has haunted every theory of the oneness of reality since the preSocratics in ancient Greece.
To summarize, an object-oriented standpoint is important as a counterweight to the view that the world does not contain any genuine discrete entities. If we treat the cosmos as a giant ball of throbbing flux, we strip all agency from individual things. If we quickly assume that everything is in such dynamic pulsation that nothing has enduring identity at all, we fail to grasp the way an object lies concealed behind the many faces it shows us at different times. And if everything is a system, we lose all sense of the resilience, rebelliousness and counterstrike capacity of individual things, including human beings themselves. If the universe is just a raging river, we are not liberated, but left empty-handed.
Two Kinds of Objects
As mentioned, OOO ’s sense of ‘object’ is not limited to midsized everyday things like horses and trees. Instead, it includes all specific entities of any status or origin: whether they be physical, non-physical, natural, artificial, mathematical, theological, simple,
Skipping Stones: The Riddle of Thixis compound, technological, delusional, contradictory or anything else. Readers familiar with philosophy might wonder how this differs from the theory of objects of the Austrian philosopher Alexius Meinong (1853–1920). While there is certainly some overlap, Meinong was interested in objects as possible objects of thought, and largely ignored the interaction of objects with each other; in this way his theory still reflected the chief bias of modern philosophy, which places human thought at the centre of everything.5 In any case, OOO contends that objects can be of any size, but that they come in two and only two basic kinds. These are real objects (which exist apart from their relations) and sensual objects (which exist only for other objects that encounter them).
As for the case of real objects, I will assume that the Convention Centre across the street from my apartment exists when I am not looking at it, even if this cannot be ‘proven’ by any means other than pointing. It is certainly possible that I am a delusional psychotic who hallucinates buildings that aren’t there, but for the most part we can and do assume that human perception has some sort of relation to objects that exist outside it; that topic lies beyond the scope of this book. At any rate, I am convinced that the Convention Centre is a real object even when I’m not looking at it. Others are equally convinced, as seen from the fact that it is scheduled to host the water polo competition during the 2028 Summer Olympics. I find that I have no problem thinking of the building as something that will still exist in that year, even if by then I am dead and buried, or still alive but nowhere in the vicinity, or fallen into a post-accident coma for the duration of the Olympic Games.
But alongside the real Convention Centre there is a sensual version of the building, where ‘sensual’ refers not to the senses, but to the pleasures of immediate contact: as with a smooth fabric or refreshing liquid. The sensual does not just mean what we know through the five senses: any form of human cognition, including the most abstract forms of logical thought, encounter sensual objects rather than real ones. In the example now at hand, the sensual Convention Centre is entirely dependent on my attention. If I
Waves and Stones fall asleep or drift into daydream, the sensual building disappears, even though the real one does not. The building might look very different to a cat, mosquito or seagull; it might seem frighteningly large to a young child. To repeat, real objects are what they are no matter what is happening around them. But sensual objects exist only insofar as they are encountered by something else, whether that something be me, my wife, a building inspector, a raindrop, an Olympic athlete or a dog. In principle, all objects can be both real and sensual at the same time, although there are cases (such as hallucination) where sensual objects do not correspond at all to anything beyond our experience. Most importantly, no one’s and nothing’s encounter with an object (sensual) will ever be interchangeable with the object in its own right (real). OOO is especially interested in the relations between real and sensual objects.
To repeat, there is no way to gain direct access to real objects. We cannot appeal to some direct intuition of reality, whether through mystical experience, mathematical exactitude, the rigours of logical notation or even a mood such as anxiety (as Heidegger does). Our experience deals solely with sensual objects, and the question of how these correspond to real ones is always a fragile and complicated matter, consisting of indirect links between mediated pieces of knowledge. But sensual objects are also not just bundles of qualities. The world we encounter is made up of bona fide units, with the same tree or bicycle persisting despite our seeing it from constantly different angles and in different lighting conditions and moods.
Earlier I noted that continental philosophy has become excessively devoted to a model of reality featuring nothing but continuities (waves) while excluding all genuine discrete entities (stones). The cosmos is treated as a continuous field of energy that sometimes gives rise to local illusions of individual things. This tendency increased with the rise of the aforementioned Deleuze, who from the mid-1990s (also the time of his death) began to replace Jacques Derrida as the standard avant-garde continental author. On a personal level I am grateful for the irreverent sense of humour that Deleuze brought to our discipline, which I hope is here to stay.
Skipping Stones: The Riddle of Thixis
Even so, the almost pornographic level of flux and flow in continental thought today (to which Deleuze is a major contributor) is a grotesque misportrayal of reality. Despite its trendy counterintuitive power, it cannot do justice to both sides of reality simultaneously. The central idea of OOO, instead, is that the world consists of discrete objects that also possess continuous qualities. And furthermore, these discrete objects also require a continuum where they can interact. Stated differently, not only does reality consist of both waves and stones, but to some extent every individual is half-wave, half-stone, like a hybrid creature from a summer blockbuster film.
Undermining, Overmining and Duomining
Let’s turn now to a different aspect of OOO : its strong distrust for the idea that explicit knowledge is the only form of cognition worth having. By explicit knowledge I mean the sort that can be adequately expressed either in clear prose language (‘the cat is on the mat’) or in equations (e=mc2). After years of considering the matter, I have concluded that there are just two basic forms of knowledge, which I describe with the technical names ‘undermining’ and ‘overmining’. If someone asks you what something is, you can either (a) tell them what it’s made of, or (b) tell them what it does. That’s it. The point of emphasizing this limitation is to remind the reader that knowledge is just one part of a vast cognitive landscape.
To see how the two types of knowledge work, imagine that a doctor has just prescribed you an unfamiliar medication. Let’s invent one called ‘Cardiomoxin’, which obviously sounds like it has something to do with the heart. Imagine now that a close friend sees the bottle of Cardiomoxin in your cabinet and asks you what it is. In the unlikely event that you and your friend are both chemically literate, you might tell them something like this: ‘Cardiomoxin features a complex, multi-ring structure with a trifluoromethylsubstituted phenyl group attached to an imidazole ring.’ If this description is over their head (or yours), you can instead give the
personal backstory of how the doctor came to prescribe it to you, what medical incident prompted them to do so or something along those lines. In this case you are undermining Cardiomoxin (in OOO ’s technical sense of the term) by reducing it to its physical and autobiographical underpinnings.6
You might also tell your friend the history of how this medicine was discovered, if you happen to know it. This too would be a form of undermining, since it turns our attention away from the object at hand to a discussion of how it was physically and historically produced, or of how its composition became relevant to you personally. Instead of telling your friend directly about the medicine, you are telling them about its subcomponents or about various aspects of its history. In this respect undermining resembles what magicians call ‘misdirection’, as when they ask you to look closely at the clown gesticulating in the aisle while a rabbit is secretly smuggled into a hat on stage. Undermining is the first of the two forms of knowledge.
The second form of knowledge is called overmining . This happens when we reduce an object not downward to its pieces, but upward to its actions, effects, symptoms or visible traits. In the case of our mythical heart medicine, you might tell a knowledgeable friend something like this: ‘Cardiomoxin functions as a selective inhibitor of the cardiac sodium channel and exhibits potent antiarrhythmic properties. It stabilizes the inactive state of the sodium channel, reducing the likelihood of aberrant cardiac electrical activity and restoring normal rhythm in cases of ventricular tachycardia and atrial fibrillation.’ If this too is over your friend’s head (as it would be for most of us) you might simply tell them that Cardiomoxin makes you feel more energetic, or safer, or that it helps you sleep better at night. All of these answers are informative, but all are versions of overmining. If you live in the United States, the land of sky-high drug prices, you might also complain about your $275 out-of-pocket co-payment for Cardiomoxin after insurance. You could add as well that the drug generated $5 billion in sales last year for Pierrot Biotech, the fictitious company that
Skipping Stones: The Riddle of Thixis synthesized it, and that this led to the creation of over 700 wellpaying laboratory jobs.
These are all examples of overmining. Rather than speaking about Cardiomoxin itself, you have changed the subject to the various things that Cardiomoxin does. This is an especially popular way of thinking today. The French philosopher Bruno Latour (1947–2022) treats objects as actors, meaning that they are equal to the sum total of everything they do in the world. Among his key inspirations in saying so were the American pragmatist philosophers of the early twentieth century, such as Charles Sanders Peirce (1839–1914), William James (1842–1910) and John Dewey (1859–1952). Peirce (pronounced ‘purse’, not ‘pierce’) claimed that the way to make our ideas clear is to rephrase any philosophical dispute as a question about what practical difference it would make if one answer were true rather than another.7 If there is no such difference, then we have an empty dispute over words rather than a true intellectual difficulty. For instance, a pragmatist might say that it’s pointless to argue about whether a world exists outside the mind, since the answer will not affect any of our actions. The more recent Latour made a number of similar statements during his lifetime.
Many people have found the pragmatist principle liberating, since it permits the disposal of seemingly trivial questions inherited from the past. But I happen to think it runs too big a risk. We don’t always immediately know the practical consequences of any philosophical debate, and for this reason we should not be too quick to dismiss questions that have no obvious stakes. For example, the seemingly stale old dispute over the existence of a world outside the mind could become crucially important once virtual reality provides a greater portion of human entertainment. Should we be allowed to assault, torture or murder humans (including children) who exist ‘only in virtual reality’ rather than outside the mind? The legal stakes of this question may prove to be enormous.
Returning to the topic of undermining and overmining, you might now ask what is wrong with undermining an object. Nothing is really wrong with it, since we learn something important when
we read about the ‘complex, multi-ring structure’ of our new medication. Yet when we analyse a thing by breaking it down to its basic elements, we forget that every genuine object is more than the sum of its parts, as the old phrase goes. As a rule, if an object is not just an aggregate of pieces, it will have properties not found in those pieces: an aeroplane can fly, but none of its parts alone is able to fly. To undermine an object is to forget that the object as a whole is just as real as its parts. This idea is so important in philosophy that it has its own technical name. When an object has a surplus of qualities or features exceeding those found in its parts, we call this ‘emergence’.8
You might also ask: what is wrong with overmining an object? Again, nothing is inherently wrong with talking about an object this way. We learn a lot from overmining, such as the fact that Cardiomoxin ‘stabilizes the inactive state of the sodium channel’, if we happen to know what that means. But while undermining misses out on emergence, overmining misses out instead on what I call submergence. This means that an object always has unexpressed qualities hidden beneath its currently visible effects. If a factory performs 100 tests on a new model of car, we still don’t gain a complete picture of the car. It may be capable of other successes and failures that were not ascertained in any of the tests: maybe it handles especially well on a rare type of bridge surface, or perhaps it tends to explode if struck by lightning. The reason that vehicle recalls happen is precisely because no one can foresee all of their possible mechanical troubles.
Overmining also fails in counterfactual historical scenarios, in which we ask how a given human or non-human object might have behaved in a completely different context. Consider the following prompt: ‘Imagine that Napoleon had been a French general during World War II . Given his past military strategies, how might he have tried to prevent the Nazi invasion of France in 1940?’ While this question could be dismissed as unverifiable speculation, I think that’s too harsh a judgement. The reason is that Napoleon isn’t just the sum total of things that actually happened in his military career from 1779 to 1815, during which he engaged in sixty battles,
Skipping Stones: The Riddle of Thixis winning fifty-three of them while losing only seven. For we can safely assume that the winning battles did not draw on the whole of Napoleon’s skill set, and that the losing battles were not sufficiently numerous to exploit all his weaknesses as a commander. Nothing forbids us from speculating about how a World War II version of Napoleon might have fought, despite the later availability of tanks and fighter planes that he never lived to see. In doing so, historians force themselves to reflect more carefully on aspects of Napoleon that were never fully detected amidst the limitations of his era. When I asked ChatGPT to answer this very question, it came up with some feasible answers, such as the likelihood that Napoleon might have persuaded the British military to join him in a pre-emptive campaign in the Rhineland, or that he might have heavily fortified the Ardennes region in northern France.9
The point of showing that an object cannot be exhaustively undermined or overmined is to see that it is primarily discrete rather than continuous, in the sense that every object is something distinct from both its components and its actions. This is not to say that there’s no relation between the three levels; of course there is. If a molecule of gold contained no gold atoms, it would not be a molecule of gold; if it were gold but somehow did not have the typical colour of that metal, no purchaser would value it as gold. But within certain limits, an object can endure as the same even when its pieces or outward effects change. Looking downward, any chunk of gold will lose atoms from time to time, though we don’t stop calling it the same piece of gold for that reason. And looking upward, the gold can be incorporated into different pieces of jewellery without losing its identity. Objects have parts, and objects are usually parts of larger objects in turn. Yet each layer of this process has a certain degree of autonomy, an independence from its smaller components and the larger contexts in which it is placed. This means that while the various objects in the world are somehow capable of connection, they are connected somewhat loosely. Every object has a degree of resistance to what happens to it, a certain deafness to the noise that surrounds it.
The point I am making is that whether we reduce an object downward or upward, similar difficulties arise. Douglas Hofstadter’s widely read bestseller Gödel, Escher, Bach pits ‘reductionism’ against what he calls ‘holism’, the idea that everything is connected.10 But while he’s right to say that reductionism is an extreme way of reducing things downward to their tiniest pieces, I would add that holism is an equally extreme way of reducing things upward to a widest context that governs them all. The problem with holism is that it would be purely arbitrary to say that random objects such as my toothbrush or shoes have systematic relations with the entire cosmos, even though important philosophers such as Alfred North Whitehead (1861–1947) have suggested as much.11 Essentially, holism is too easy on itself: it simply asserts with cavalier abandon that everything is connected. Yet it fails to explain the mechanics of how limited connections between distinct things are possible without the world being dissolved into a single bubbling pudding. Returning now to the main point, there are only two basic forms of knowledge: undermining and overmining. I do not believe there are any others. But it’s also worth noting that they do not always appear on their own. Instead, it frequently happens that undermining and overmining are performed simultaneously in a gesture we can call ‘duomoning’.12 For instance, we might answer our friend’s question about Cardiomoxin by explaining what it’s made from and what it does. But as comprehensive as this sounds, duomining still does not tell us everything about an object. What makes this combined tour de force of knowledge fall short is that knowledge is not the only form of human cognition. Consider the realm of art, which is not primarily meant to produce knowledge. Imagine that an artist paints a portrait of your bottle of Cardiomoxin: Damien Hirst (b. 1965) comes to mind as a likely suspect. You would never expect this painting to give us knowledge of either the components or the effects of the drug. Yes, Hirst could always undermine the painting by telling us about the specific canvas and pigments he used, or where he got the idea for such a painting, but that would hardly count as an understanding of the artwork itself. He could also try to
Skipping Stones: The Riddle of Thixis overmine the painting by going around the room and asking everyone how the painting makes them feel, but this too would fall short of true aesthetic appreciation.
The fact is that the two forms of knowledge (undermining, overmining) as well as their combination (duomining) give us nothing more than a rough approximation of what an object is. Objects are one thing and knowledge quite another. The painting of Cardiomoxin, like artworks more generally, points to this gap between object and knowledge by alluding to something situated midway between the drug’s components and its effects: namely, Cardiomoxin itself. Every object exists at a halfway point between its parts and its effects, irreducible to either of the two forms of knowledge.
I have mentioned art as one form of cognitive activity that helps exhibit a thing in a form that surpasses knowledge. But perhaps an even better example is philosophy itself. In ancient Greek the word is philosophia, which means ‘love of wisdom’, not wisdom pure and simple. As Socrates explains in the Platonic dialogues, humans find themselves midway between the gods and animals, partly but not fully understanding any topic under discussion. Although Socrates is famous for always insisting that people should give definitions of the terms they use, he ought to be more famous for the fact that he is never satisfied with any definition. The reason, as I see it, is that nothing is ever fully describable by a definition, or indeed by any sort of prose statement or mathematical equation. Analytic philosophy has often insisted that philosophy, just like the sciences, should aim at becoming clearer and more exact. I answer that this is a misguided goal, since reality itself is neither clear nor exact; hence knowledge is not always the best way of doing justice to reality. This is a point to which we will return later, when considering the difficulty of translating between the continuous and the discrete. When I walk on the beach in southern California, where I live, I often pick up stones of unusual beauty or intriguing shape. Typically a stone is a durable unit, hardened by millennia of geological pressure. It does not crumble in my hands, and there is never any doubt about its boundaries. This makes stones a perfect metaphor
Waves and Stones
for individual objects. However complex the process that forms an individual stone, to some extent that process is no longer relevant once it is a finished product in my hand; at this late stage it is very clear where the stone ends and the surrounding world begins. By contrast, the waves at the beach alert us to a different face of reality. A wave is less a material object than a pattern of movement that is tied to no particular set of objects; it moves successively through different parts of the ocean before crashing at last into the sand. Although a wave is always finite – it does not directly affect the entire ocean, much less the universe as a whole – we cannot say that it consists of any definite number of parts. Even if we analyse it into smaller and smaller segments, we will never reach a smallest unit of the wave. How different this is from the stone, which cannot be cut at all without destroying it. From this contrast we gain a sense of the two-faced character of reality that is the central topic of this book.
Defining the Terms
Discrete things exist in the world. When we open our eyes in the morning we do not encounter the world as a single lump where everything is melted together. There are hammers and dolphins alongside planets, laws, fictional characters and mathematical objects. It is true that these discrete things are not completely isolated; they interact. But they do not interact with everything, and no interaction is automatic or total. The aforementioned Latour includes an amusing passage to this effect in his 1992 book Aramis, which traces the sad demise of an automated Metro system once slated to open in Paris. Here the young engineer narrator is speaking of his mentor, a professor called Norbert who speaks in the final line:
the violent blow he struck with his fist on the desk had no visible influence on the chapter of Aristotle’s Metaphysics that was filed
Skipping Stones: The Riddle of Thixis under the letter A at the top of his bookshelf. ‘You see: not everything comes together, not everything is connected.’13
Even when objects do interact and transform each other, the resulting changes are finite and not always reciprocal. Living in Egypt for sixteen years changed me in lasting ways, but when I left Cairo in 2016, I was still recognizable to friends and family as the person who first arrived there in the year 2000. Moreover, whatever reciprocal influence I may have had on the venerable city of Cairo was surely minuscule when contrasted with its life-changing effect on me.
Continua (the plural of ‘continuum’) also exist. A continuum is something that can be cut up into as many pieces as you please, as small as you wish, but which is still effectively one thing rather than a sum total of tiny parts. As an example, consider the number line you probably remember from your student days; we will return to it more than once. Please note that most mathematicians would not agree that the number line is just an idea. For most of them mathematical objects are every bit as real as material physical things, if not more so. In any case, the number line is usually drawn with zero in the middle, negative numbers extending to the left, and positive numbers to the right. The reason the number line is continuous is because there are no holes in it. Imagine that someone wrongly claimed to notice a gap on the line between the numbers 2 and 3. This person could easily be refuted by adding 2.5 between them. If that person complained further that there are still holes on the line, we could add as many new numbers as needed to keep filling up the supposed gaps. This process would never end, since even if we have a rather tiny gap between numbers (say, between 2.49978361 and 2.49978362) we can still add infinitely many more numbers into any gap we find. That is what makes the number line a continuum. Space and time are also continua as far as we know, though in advanced physics the jury is still out on this question. When I teach a philosophy seminar at the Southern California Institute of Architecture (SCI -Arc), it lasts in principle for two hours and fifty minutes. Just as with the number line, we can mentally split the length of my
Waves and Stones
seminar into three parts, seven parts, 4,000 parts, a billion parts, or as many as we like, to the verge of infinity. That’s because time is a continuum and consists of no definite number of smallest instants. But in another sense, the full length of the seminar is just a single unified stretch of time. The reason is that if the class were truly made up of infinitely many moments, I could never pass through them all and reach the end of the lecture. That is what it means to say that even though a continuum can be carved up indefinitely, it always remains one thing.
The same holds for space, another important continuum. When I fly from Los Angeles to Istanbul during summer vacation, the air route can be divided into four, twenty-five, or 7,986 parts at my whim, and I can always increase the number of divisions as far as I want. But if there were really infinitely many points in space, as with the infinitely many numbers on the number line, it would be impossible for me to fly from California to anywhere else. More than that, it would be impossible even to take off from the runway at LAX Airport, since there would be infinitely many points to cross before the plane could even begin to fly. This type of thinking will soon lead us to consider Zeno’s Paradoxes, which ranked among the most prominent intellectual puzzles in ancient Greece and are still discussed today.
It’s also important to distinguish between a continuum and the related but looser notion of a spectrum. A good example of a spectrum is the range of political positions, running from Left to Right, represented by the parties of democratic nations. Take present-day Germany, for instance. We can put the Green Party towards the left of the spectrum, with the Social Democrats closer to the centre. At right of centre you’ll find the classically liberal Free Democrats, followed by the Christian Democrats further to the right, and finally the radical anti-immigrant Alternative für Deutschland party. Why do we call this a spectrum and not a continuum? Because the German political system consists of a finite number of voters and parties, so that we cannot always find another party between any two given parties. Yes, someone could always try to launch a new
Skipping Stones: The Riddle of Thixis party midway between the Greens and the Social Democrats, but there is not endless room to create an infinite number of subtly different parties in that section of the spectrum. Even if we imagine the German population growing at a completely out-of-control rate, eventually reaching billions of residents, there will always be a finite number of parties and hence there will always be gaps between them. This makes the German political spectrum (or any other) very different from the continuum between the numbers 4 and 5, which truly contains an infinite number of numbers. As an analogy, while our continuum can be understood as a wave, a spectrum is more like a finite series of stones ranging from largest to smallest, or brightest to darkest in colour.
Radical Stances
This book claims that the most common mistake about the continuous and the discrete is the assumption that one of them must be more fundamental than the other. In other words, when it comes to waves and stones, there is disagreement between one set of theories that treats reality as a continuous, pulsating flux, and another that asserts the absolute individuality and mutual separation of specific things. We will soon meet with examples of both extreme positions. At present, there is no question that the model of unremitting change has the sexier reputation of the two, whether in philosophy departments, popular science books or creative fields such as literature and the visual arts. The widespread desire to embrace ceaseless turbulence is captured in the following passage from the eminent microbiologist Carl R. Woese (1928–2012):
Imagine a child playing in a woodland stream, poking a stick into an eddy in the flowing current, thereby disrupting it. But the eddy quickly reforms. The child disperses it again. Again it reforms, and the fascinating game goes on. There you have it! Organisms are resilient patterns in a turbulent flow– patterns in an energy flow.14
Waves and Stones
The prolific American philosopher Thomas Nail (b. 1979) writes in a similar spirit, presenting this river-like model of reality as the irrevocable verdict of science and history alike. In fact, he links his grand theory of motion both to state-of-the-art physics and to the increasing concern with migration and refugee issues in contemporary politics.15 As Nail puts it:
the flux, turbulence, and movement of energy are more primary than the relative or metastable fixity of classical bodies . . . The old paradigm of a static cosmos, linear causality, fundamental particles, and classical space-time no longer fits the twenty-first-century reality of cosmic acceleration, turbulence, and continuously vibrating fields.16
Rein Raud (b. 1961), one of Estonia’s most prominent intellectuals, has made similar arguments in his fine book Being in Flux 17
It’s easy to see why the idea of replacing stagnant things with dynamic processes is thrilling to so many people. If the everyday world of solid physical objects, monotonous daily errands and unjust political and economic conditions seems dull and familiar, how revolutionary it feels to think that everything is in constant motion! From this daring standpoint, all apparently stable forms or patterns are nothing but derivative products of an energetic turbulence in the heart of things. But how can this type of theory account for the fact that individual things at least seem to exist? As we will see, Parmenides and his disciple Zeno said that our senses are simply deluded when they display many different things; the world is really an unarticulated primal whole. Other philosophers have conceded that individual things do exist, but only because human thought arbitrarily cuts a single world-lump into pieces that meet its own practical needs.
If we insist that reality is just a continuum in constant motion, we avoid addressing the most important problem of all: how to account for the fact that the continuous and the discrete interact. Individual things (which are discrete) engage in relations in space and time
Skipping Stones: The Riddle of Thixis (which are continuous). An object (which is discrete) possesses qualities (which can be continuously varied, say, from brighter to darker red). A photon (a discrete quantum of light) moves through space warped by gravity (a continuous effect). To view the world as nothing but a constantly raging firehose of change is to pretend that there are no discrete entities at all, when in fact there are.18
In the history of philosophy there have also been extreme positions on the other side of the debate: theories claiming that everything is so discrete that individuals are completely cut off from mutual contact. Such positions are rare today, since they were linked with a style of theologically based philosophy that has long been out of fashion. The name for this way of thinking is occasionalism; it flourished in early medieval Islam and again in early modern Europe. The occasionalist idea of how radically discrete beings influence each other is to say that God is not just the only creator, but the only causal agent at all. In this way all causal power is removed from individual things. When objects collide, they only seem to collide; God alone makes contact with the various things in the world. The problem with this extreme form of discreteness is not that it draws on religion, which has been widely disdained by secular intellectuals since the Enlightenment. Instead, the true problem is that if we say God alone is capable of causal influence when nothing else in the universe is capable, this does nothing to clarify the dynamics of interrelation between things, even if such a theory quickly satisfies whatever degree of piety we might have. For instance, what features of God can we deduce from his purported ability to touch an individual thing when nothing else is able to do so? What enables a homely terrestrial object to make contact with the divine? Such questions are usually ignored in favour of vague appeals to God’s power and goodness. But as missionaries know very well, if philosophical arguments are to be effective they must also address the unconverted.
In the modern era a more widely accepted view is that causation is not monopolized by God, but by the human mind. This idea was made popular by the eighteenth-century philosophers David Hume
Waves and Stones and Immanuel Kant. Both argued in different ways that we can only grasp causation as a feature of human experience and cannot assume it exists in the world itself. While this position has a better chance of convincing secular-minded intellectuals, it is not much of an improvement on theological occasionalism. The same difficulty remains: the causal monopoly granted by occasionalism to God is transferred to the human mind, in a manner similar to sweeping dust from one part of the floor to another. We still have no idea how discrete entities affect each other in continuous time and space. This is no small problem, since hasty appeals to God or the structure of human experience prevent us from considering the interaction of objects among themselves.
But there has long existed an alternative to the radical stances of pure continuity and pure discreteness, in the writings of philosophers who recognize that we must account for both. One example is the ancient master Aristotle, who devoted one of his great books (the Physics ) to the continuous character of nature, and another (the Metaphysics ) to the discrete character of individual things. Another example is the more recent French thinker Henri Bergson, who, despite being known as a remorseless thinker of flux and flow, was aware that we must also address the apparent endurance of individual things. Since neither the continuous nor the discontinuous can single-handedly account for the whole of reality, we need a model of the cosmos that recognizes both while clarifying how they coexist and interact. To put it plainly, both the discrete and the continuous exist: often simultaneously, and even within the same object, insofar as an individual thing is not identical with the continuous qualities it also possesses. A continuous number line would mean nothing without the existence of specific numbers, and a discrete individual horse cannot run except through continuous space and time. Likewise, a stone could not disrupt a wave if it simply melted into a wave-like continuous universe. To understand the paradoxical interaction between the continuous and the discrete requires an exploration of the very fabric of reality, and could change our deepest sense of how the world works.
Skipping Stones: The Riddle of Thixis
This book will pose the following basic question: what sort of relation do the continuous and the discrete have to each other? In one sense the two have highly distinct properties, which means they are both discrete; in another sense they do interact, which means they are also continuous. That is our ultimate problem: how can the continuous and the discrete be both continuous and discrete with respect to each other without leading to a mess of contradictions? Let’s break it down. When it comes to the continuous and the discrete, three important questions appear at centre stage:
1. Given that entities are discrete, how can they touch at all?
2. Given that each continuum is one, how can it contain infinitely many parts?
3. Given that the continuous and the discrete are not the same thing, they are discrete with respect to each other. That being the case, how can they also become continuous with each other? If this were not possible, they would not be able to interact.
It’s easy to imagine someone making the following complaint about our first question: ‘Objects can obviously touch and interact. We see this happen constantly, so why waste time asking about it?’ I have heard this objection frequently. But we can see how misguided it is when we apply it to other cases. For example, imagine someone saying this: ‘Obviously children inherit features from both of their parents – so why waste time finding genetic explanations?’ Or this: ‘Obviously human personality is shaped by both heredity and environment, so why waste time figuring out the exact mechanisms of each?’ Or this: ‘We already know that earthquakes happen. So why waste time figuring out how?’ The fact that something is known to happen is not an argument against studying how it occurs. Quite the contrary: the existence of a thing or phenomenon is an argument in favour of such study; to acknowledge that something exists is an opening for conversation rather than an end. Stated differently, our goal is not to ‘prove’ that objects affect each other (no such proof is needed) but to determine the mechanisms that bring
it about. If we proceed in the opposite direction and assume that everything is fundamentally related to everything else – remember our holists – then we lose sight of the fact that while some objects have tremendous mutual influence, others sit side by side for years without affecting each other at all. For instance, the genetic writings of Gregor Mendel lay mostly unread for decades before Darwinians turned Mendel into a powerful ally in the combined doctrine known as neo-Darwinism. Another example: quite often a book sits on my shelf for years or even decades before I rediscover it exactly when needed.
Our second question is focused on the continuum. As we will see when discussing Zeno’s Paradoxes in the coming chapter, a continuum cannot actually consist of infinitely many points, since this would lead to impossible results. For one thing, we would not be able to travel from one place to the next, given that any change of place would need to pass through infinitely many points to do so. For another, we would not be able to experience an hour or even a second of time: after all, we would first need to pass through infinitely many moments before completing the full hour or second. Nonetheless, it’s impossible for a continuum only to be a unified whole, since we do not move through any continuum instantly, but only cover part of it during any interval of time. Although a walking trip from central Manhattan to central Brooklyn can be treated as a single journey, the route this entails is not a simple whole, since throughout the journey we are always at some specific place along the route.
The third question is the one that really gets to the heart of the matter. How can something continuous interact with something discrete, given that the two play by such different rules? What we’ll see is that two continua can only make contact through something discrete, and that two discrete objects can only make contact in a continuum. Everyday life teaches that the discrete and the continuous do interact, since discrete objects exist in continuous space and time; conversely, continua can be cut up into discrete slices. As stated earlier, what we need is less a proof that interactions between
Skipping Stones: The Riddle of Thixis things actually happen (few would deny it) and more an explanation of how these interactions occur, or how and why discrete objects and continua meet and affect one another. This is not a false problem summoned from thin air; instead, such questions haunt our conception of the ultimate nature of reality.
Thixis
At its core, this book is an examination of why contact between two things is more difficult than we imagine. I’d like to propose a technical term for this problem, based on one of the many ancient Greek words for ‘touch.’ Thixis refers to the contact between surfaces. This makes it an appropriate term for OOO, which holds that contact between two objects is always a matter of surfaces. Later we will consider the notion of thixis in greater detail. Remembering that OOO recognizes two different kinds of objects (the real and the sensual) it will also turn out that objects can only touch objects of the opposite kind. Neither two real objects nor two sensual ones can make direct contact; they can only touch their opposite kind, just as the poles of a magnet repel each other if they are both north or both south. But this insight soon gives way to the deeper truth that discrete objects can only make contact with something continuous, and that two continua can only touch through something discrete. Let’s coin the word heterothixis as a technical term for this principle. In due course we will return to this point.
Our philosophical journey into the paradoxical coexistence of the continuous and the discrete begins with Aristotle in ancient Greece, and will take us all the way to modern physics and mathematics. But to clarify this problem does not mean to dissolve it once and for all. Problems in philosophy, unlike those in mathematics, engineering or medicine, can neither be eliminated nor definitively solved. Once a new philosophical problem is detected, it tends to return intermittently in new variations across the centuries. The philosopher’s job is not to solve it, but to propose new options for dealing with it;
philosophy is driven by the love of wisdom, not some knowledge safely in our possession once and for all. For instance, Kant never ‘solved’ the famous mind-body problem that emerged from René Descartes’s philosophy, but in some sense made it more extreme than ever, with his new split between visible phenomena and nevervisible noumena, or ‘things-in-themselves’.
There is a tendency to think of all the branches of knowledge as modernizing in a single direction, like improved cancer treatments or the increasing speed of computers. But this incremental picture of progressive knowledge does not work well for everything. Politics does not get ‘better and better’ over time; in any nation it shifts between high points of human flourishing and low points of stagnation and civil conflict. We would not say that painting is ‘better’ today than it was during the Florentine Renaissance, even if certain pigments can now be produced more reliably, and even if the system for training vast numbers of painters is bureaucratically better organized than in fifteenth-century Italy. The same holds for philosophy, which generates new insights in every century, but does not produce irreversible forward progress. We cannot say that the philosopher Baruch Spinoza (1632–77) ‘proved that God is nature’ in the same way that the mathematician Carl Friedrich Gauss (1777–1855) proved the fundamental theorem of algebra. Mathematicians are bound by Gauss’s result; philosophers are not bound by Spinoza’s.
Stated differently, ‘proof’ is not really the medium in which philosophy is conducted; indeed, every attempted proof is met by an army of counterproofs extending over long stretches of time. This is not what happens in astronomy or chemistry, not to mention such technical fields as industrial engineering or the design of superior submarines. Yes, attempts are made in academic journals to ‘refute’ various statements by Plato or Aristotle on a weekly basis. Yet most of these attempts vanish forever into the mist, while the great ancient thinkers repeatedly return from the dead to guide our thinking. This is not because they show us perennial truths, as conservatives like to believe, but because classic works make basic decisions on fundamental issues and thus inspire us to make our
Skipping Stones: The Riddle of Thixis
own. They do not whittle every stick to the point where only dust remains, as academic discourse too often does.
In closing, let’s return to the beach and its medley of waves and stones. If I skip a stone through an oncoming wave, they have clearly interacted. I have made a cut in the continuous wave by puncturing it at one discrete point instead of another. But in using the stone to make this cut, I see that the stone is not just a self-contained discrete unit, but belongs to a continuous space and time just like the wave itself. Without this shared citizenship in a continuum, nothing could interact. By gaining a better understanding of this simple image from the beach, we will learn something important about the ultimate nature of reality. At the close of this book we will learn the strange lesson that the continuum where objects meet lies on the interior of another, larger object.
The philosophers Aristotle (Chapter 2) and Henri Bergson (Chapter 4) are not normally viewed as allies, but we will see that they are joined in recognizing that neither the continuous nor the discrete is alone in the world; both principles play significant roles. By considering the Arab historian Ibn Khaldun’s theory of generations and the medieval death match between nomadic hordes and civilized urban centres (Chapter 3) we will gain a sense of how the continuous and the discrete function in human history. Turning to the philosopher of science Thomas Kuhn (Chapter 5), we will examine the distinction he draws between the gradual progress of mainstream cumulative ‘normal science’ and the sudden jumps that he links with scientific revolutions. In evolutionary theory we will encounter the strife between Charles Darwin’s gradualist model of species change and Niles Eldredge and Stephen Jay Gould’s idea of ‘punctuated equilibrium’ (Chapter 6), a debate that provides one of the most famous intellectual battlegrounds between the continuous and the discrete. Zeroing in on a debate between architects that unfolded from 1988 onwards (Chapter 7), we will examine the key role played by the continuous and the discrete in aesthetics. We will then return to Gould’s notion of ‘Nonoverlapping Magisteria’ (Chapter 8), which conceives of human knowledge as made up of