Saturday


Recently in The Space Age turns 60! I wrote, “We are still in the very early stages of the Space Age; the inflection point of this developmental sequence has not yet arrived, so we are today still in the same shallow end of the exponential growth curve that was initiated sixty years ago.” What do I mean by an inflection point, and what is (or what would be) the inflection point for spacefaring civilization?

In a curve, an inflection point (according to Wolfram Mathworld) is, “…a point on a curve at which the sign of the curvature (i.e., the concavity) changes.” In this technical sense, then, I have misused “inflection point,” but it has become commonplace to speak of the inflection point of an exponential (or sigmoid) curve as the point at which the transition occurs from the long, shallow part of the curve, only incrementally growing over time, to the exponential growth part of the curve. In this sense, the inflection point is the transition from slow (sometimes very slow), incremental development to rapid, exponential development.

We have some good examples of inflection points from human history. The industrial revolution is a nearly perfect example of an inflection point. Human beings have been developing technologies since long before civilization. Pre-human ancestors were using stone tools more than two million years ago. However, technological development began to accelerate with the industrial revolution, and continues to develop at an expanding and increasing rate. Technological growth — both in terms of technological complexity and large-scale industrial application — has been exponential since the industrial revolution. Is something like this possible with spacefaring?

In Late-Adopter Spacefaring Civilization: the Preemption that Didn’t Happen and Stagnant Supercivilizations and Interstellar Travel I discussed one of my favorite themes, namely, that spacefaring civilization might have experienced its inflection point in the wake of the Apollo program, which latter demonstrated what was possible when significant resources are expended on a difficult goal. More recently, on The Unseen Podcast Episode We, Martians? I said that if we had gone to Mars as NASA once planned, building immediately following Apollo, it would have been a different mission than any mission to Mars undertaken at the present time. It would have been, in short, a mission much like the Apollo mission, meaning a transient presence on Mars sufficient to plant the flag of the sponsoring nation-state and to collect some samples to bring back to Earth. Paul Carr called this a “Flags and Footprints” mission, which is a good way to phrase this, and I subsequently heard this from others, so apparently it’s a thing.

These counterfactuals did not occur, so that they represent a permanently lost opportunity for human civilization. The door has closed on this particular shape for human history, but the door remains open for different shapes for human history if spacefaring technologies are eventually adopted, and when they are adopted (if they are adopted), will decisively and definitively alter the shape of human history — or the history of any intelligent species able to build spacefaring technologies. To consider this a little more carefully I am going to delineate three generic scenarios for the breakout to spacefaring civilization that might be experienced by a civilization that develops spacefaring technology. These three scenarios are as follows:

● Early Inflection Point when spacefaring is pursued with exponential frequency immediately upon the technology being available.

● Middling Inflection Point when spacefaring is pursued with exponential frequency only after it has been available for a substantial period of time, but within the longue durée in which the technology became available.

● Late Inflection Point when spacefaring is pursued with exponential frequency after the technology has been available throughout a longue durée period of history.

No great store need be placed on the time frames I have implied above; sufficient to our purposes is that spacefaring may become routine immediately upon, sometime after, or long after the technology is available. Each of these spacefaring inflection points can be taken separately, since each represents a different civilization as defined by the relationship between the civilizations of planetary endemism and spacefaring civilization. Moreover, we can justify the significance of the position of the spacefaring inflection point in the overall history of civilization by reference to the infinitistic possibilities available to a spacefaring civilization

Early Inflection Point

On several occasions I have written about the possibility of a spacefaring civilization emerging immediately upon the technology of the Space Race being available, specifically in Late-Adopter Spacefaring Civilization: the Preemption that Didn’t Happen. In this post I suggested that industrial-technological civilization as it has been known from the industrial revolution up to the advent of the Space Age might have been suddenly “preempted” by the emergence of a new kind of civilization — a spacefaring civilization — that changed the conditions of human life as radically as the industrial revolution changed the conditions of human life. This is what did, in fact, happen with the industrial revolution: as soon as the technology to drive machinery by fossil fuels became available, it was rapidly exploited, and western societies passed through a series of rapid social changes driven by industrialization.

While an early inflection point did not occur on Earth with the initial availability of spacefaring technology, we must consider the possibility that this is could occur with any civilization that passes the spacefaring technology threshold. I explored some of these possibilities in my Centauri Dreams post, Stagnant Supercivilizations and Interstellar Travel. In so far as an early spacefaring breakout would encourage a focus on spacefaring technologies (the relative neglect of other technologies being an opportunity cost of this alternative focus), the developmental trajectory of such a civilization might involve continual and rapid development of spacefaring technologies even while other technologies (say, for example, computing technologies) remain relatively undeveloped. Thus the technological profile of a given civilization is going to reflect the existential opportunities it has pursued, and when it pursues them.

We may also observe that, along with early-adoption spacefaring scenarios that did not occur with human civilization, it is also the case that a variety of counterfactual existential risk scenarios also did not occur. What I mean by this is that, once nuclear weapons were invented (shortly before the advent of the Space Age), human beings immediately realized that this gave us the power to destroy our own civilization. A number of novels were written and films were made in which human beings or human civilization went extinct shortly after the technology was available for this. These scenarios did not occur, just as the scenarios of early spacefaring adoption did not occur.

Middling Inflection Point

It has become a commonplace to speak of the recent development of space industries as “NewSpace.” If the technologies of NewSpace come to maturity in the coming decades and results in the following decades in a spacefaring breakout and the establishment of a truly spacefaring civilization, this would constitute an instance of a mediocre spacefaring inflection point. Given that the Space Age is now sixty years old, a few more decades of development would mean that spacefaring technologies will have been available for a century before they come to be fully exploited for a spacefaring breakout and a spacefaring civilization. In other words, the spacefaring inflection point did not occur immediately after spacefaring technology was available, but it also did not have to wait for an entirely new epoch of human history to come to pass for the spacefaring breakout to occur. (In terms of human civilization, we might identify a period of 100-300 years from advent to breakout as a mediocre spacefaring inflection point.)

As implied above, the current nominal spacefaring capacity of our civilization today is consistent with a middling spacefaring inflection point, if spacfaring expands rapidly in the wake of the maturity of NewSpace industries and technologies. Among these technologies we may count reusable spacecraft (Sierra Nevada’s Dream Chaser), including the booster stages of multi-stage rockets (SpaceX and Blue Origin), hybrid rocket engines (Reaction Engines LTD), and ion and plasma rockets (Ad Astra’s VASIMR), inter alia. These are the actual technologies of spacefaring; many industries that seek to exploit space for commercial and industrial uses are focused on technologies to be employed in space, but which are not necessarily technologies of spacefaring that will result in a spacefaring breakout.

Late Inflection Point

Say that the NewSpace technologies noted above come to maturity, but they prove to be impractical, or too expensive, or simply uninteresting to the better part of humanity. If this opportunity arises and then is passed over without a spacefaring breakout, like the initial existential opportunity presented by spacefaring technologies, the middling spacefaring inflection point will pass and humanity will remain with its nominal spacefaring capacity but no spacefaring breakout and no spacefaring civilization. In this case, if there is to be an eventual spacefaring breakout for human civilization, it will be a late spacefaring inflection point, and human civilization will change considerably in the period of time that passes between the initial availability of spacefaring technology and its eventual exploitation for a spacefaring breakout.

Just as in the meantime from initial availability of spacefaring technology to the present day, computer technology exponentially improved, a late spacefaring inflection point would mean that many technologies would emerge and come to maturity and industrial exploitation even as spacefaring technologies are neglected and experience little development (perhaps as an opportunity cost of the development of alternative technologies). Thus a late-adopter spacefaring civilization may develop a variety of fusion technologies, alternative energy technologies, genetic engineering technologies, quantum computing, human-machine interface technologies (or xenomorph-machine interface, as the case may be), artificial consciousness, and so on. Once a civilization possesses something akin to technological maturity on its homeworld, its historical experience will be radically different from the historical experience of a species that pursues an early spacefaring inflection point.

I can imagine a civilization that becomes so advanced that spacefaring technologies become cheap and easily available simply because the technological infrastructure of the civilization is so advanced. Thus even if there is no large-scale social interest in spacefaring, small groups of interested individuals can have spacefaring technologies for the asking, and these individuals and small groups will leave the planet one or two at a time, a dozen at a time, and so on. The homeworld civilization would be unaffected by this small scale spacefaring diaspora, since the technological and financial investment will have become so marginal as to be negligible, but these individuals and groups will take with them an advanced technology that will allow them to survive and prosper even at this small scale.

The worlds these small groups pioneer will grow slowly, but they will grow, regardless of whether the homeworld notices. Under these conditions, an ongoing nominal spacefaring capacity could develop over longer scales of time into a spacefaring capacity that is no longer nominal, though we would never be able to say exactly when this changeover occurred; this would be an evolutionary rather than a revolutionary transition. However, once these other worlds began to grow in population, eventually these populations would exceed the population of Earth, and at this point we could say with confidence that the late spacefaring inflection point had been reached, without spacefaring per se ever becoming a great civilizational-scale undertaking.

The Null Case

In addition to these three scenarios, there is also the null case, i.e., spacefaring technology is initially developed, but it is not further pursued, so that it is either forgotten or regarded with disinterest. A civilization that develops spacefaring technology and then either fails to pursue the development, or loses the capacity due to other factors (such as civilizational collapse), never achieves a spacefaring breakout and never becomes a spacefaring civilization. As I make a distinction between the nominal spacefaring capacity we now possess, and a spacefaring civilization proper, our contemporary civilization remains consistent with the null case scenario unless or until it experiences a spacefaring breakout.

The null case is the trajectory of a civilization toward permanent stagnation. Even if many technologies are developed and come to maturity and industrial exploitation, nothing essential will have changed in the human relationship to the cosmos (or the relation of any intelligent species that develops spacefaring technology but which does not exploit these technology for a spacefaring breakout). Spacefaring technologies, if exploited for a spacefaring breakout that results in a spacefaring civilization, would change the relationship of a species to the cosmos, as the species in question then has the opportunity to develop separately from its homeworld, and is therefore no longer tightly-coupled to the natural history of its homeworld. Without a spacefaring breakout, an intelligent species remains tightly-coupled to the natural history of its homeworld, and necessarily goes extinct when its homeworld biosphere is rendered uninhabitable.

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Addendum added Wednesday 25 October 2017: Further to the above discussion of early spacefaring inflection points, I happened upon Space That Never Was is one artist’s vision of a never-ending space race: Where else might we have gone? by Andrew Liptak, which led me to the work of Mac Rebisz, Space That Never Was, who writes of his artistic vision, “Imagine a world where Space Race has not ended. Where space agencies were funded a lot better than military. Where private space companies emerged and accelerated development of space industry. Where people never stopped dreaming big and aiming high.” Rebisz’s images might be understood as illustrations of early-adopter spacefaring civilization.

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Accelerationism

16 June 2017

Friday


Salvador Dali, ‘Geopoliticus Child Watching the Birth of the New Man’

In the Salvador Dali painting “Geopoliticus Child Watching the Birth of the New Man” (1943) we see a prophetic figure (sometimes identified as the old world) indicating to the Geopoliticus Child the emergence of a new order, represented by the New Man. Here the Earth is an egg, from which new life emerges, and the Geopoliticus Child, already itself new life, watches from safety the struggle of the New Man to be born. If one could place oneself in this archetypal context (perhaps, as a thought experiment, inhabiting the person of the Geopoliticus Child), there are at least three possibilities as to how one might respond:

one might passively observe the birth of a New Man while taking no action
one might actively seek to facilitate the birth of the New Man
one might actively seek to prevent the New Man from being born

The second of these possibilities represents what I will here term “accelerationism,” which is the conscious and purposeful effort to expedite an historical process so that the process in question will be more rapidly brought to its end or fulfillment.

The terms “accelerationism” and “accelerationist” are sometimes employed to discuss accelerating technological change, especially exponentially accelerating technological change (which is sometimes called “exponentialism”). That is not how I will use the term in this context. In the present discussion, I will use “accelerationism” to refer to the view that certain events or processes could or should “speed up” the collapse of existing political institutions, which can be understood as a good thing if one believes that the ground must be cleared in order to frame new institutions de novo.

Accelerationism in the sense of accelerating the collapse of a decaying and doomed social order is a species of contemporary apocalypticism. I have touched on apocalypticism in several posts, most recently in Vernacular Declensionism focusing on contemporary “preppers” (who were formerly called “survivalists”). There is both a vernacular apocalypticism (such as I wrote about in my “vernacular declensionism” post), which appears to be independent of political orientation, and a high-culture apocalypticism expressed in academic and scholarly terms. It has been my intention for some years to write more generally about apocalypticism, since it has become so widespread, and is rarely challenged on principle. This is a project that still remains in the offing.

It is of some interest to me that contemporary apocalypticism has become prevalent on both the left and the right, including being prevalent among the emerging political permutations that go beyond traditional left and right, and these are the social justice ideologues as the transfiguration of the left, and the alt-right and neo-reaction as the transfiguration of the right. (The most famous neoreactionary is Curtis Yarvin, blogging as Mencius Moldbug; the neoreactionary whose work I follow is Youtube vlogger Reactionary Expat, who has touched on accelerationism in some of his posts.) As I noted in my post on Vernacular Declensionism, this form of apocalypticism has mostly represented the political right, and the idea of the collapse of modern civilization easily plays into the narrative of a return to traditional forms of society. Obviously, a traditionalism predicated upon the destruction of existing social institutions is a radical form of traditionalism, but if the intention is to restore traditionalism by eliminating modernity, sooner rather than later (in virtue of accelerationism), then I guess this still counts as some form of traditionalism.

In recent years, the left has joined in vernacular apocalypticism with gusto, especially with scenarios of environmental apocalypse, to which a growing literature of popular fiction is devoted. However, there is little sign of accelerationism on the left; the hints I have glimpsed of accelerationism have been almost exclusively concerned with hastening the demise of corrupt modern society. There is, however, an important exception: anarchism. This will be discussed below. But, more importantly, accelerationism is apocalypticism with a purpose, and not apocalypticism for its own sake.

Accelerationism is not apocalypticism simpliciter, but rather it is a tactical apocalypticism, i.e., an apocalypticism only for the sake of that which will follow after the apocalypse; in other words, the means of social denudation will be justified by the end of the social order that replaces the existing social order of the present. What social order will replace the existing social order that is to be accelerated in its trajectory of self-destruction? Here there is a clear bifurcation of the visions of the future held by left and right.

It is possible that the surviving vestiges of the past will hamper the emergence of a truly new order to supplant the old order, and this could be an argument for a complete and total extirpation of the old order so that a new order can arise in its place. I am not advocating this argument, but I can see how the argument could be made. Many twentieth century communist regimes attempted to follow this line of reasoning, attempting to utterly obliterate traces of the pre-communist past (the entire Cultural Revolution in China could be framed in these terms). These efforts could be understood as an example of leftist accelerationism, attempting to more rapidly bring into being the communist utopia of a classless society.

Anarchic utopians have long held that the realization of a better social order is just around the corner if only we will take the radically appropriate action of extirpating traditional institutions that have held us back from realizing our human potential. This is an idea that goes back at least to Rousseau (for purposes of Enlightenment thought), and probably is much older. I will not, at present, attempt to elucidate a more thorough history of this idea. While utopians who project a peaceful anarchic society in the near future tend to identify with the political left, we cannot fully assimilate them to the traditional left, in the same way that we cannot fully assimilate social justice ideologues to the traditional left. I cannot, however, think of any anarchists on the right, as the right tends to believe in human fallibility (original sin), and so are distrustful of human nature released into the wild, as it were. The Rousseauvian dream is, for the right, a Hobbesian nightmare. And so we usually find the radical right looking not to anarchy, but to a reaffirmation of order, and of the symbols of order. The apocalypticism of the right thus plays into accelerationism; the two go together as tactic and strategy.

Implicit in the accelerationist view is that there are historical changes occurring anyway, albeit gradual and incremental change, and while this change must be accepted, it is nevertheless amenable to being managed. The accelerationist, then, understands that history transcends itself when an old order is replaced by a new order, so that the accelerationist may be characterized as facilitating historical transcendence, and that, moreover, the historical process must be brought to its fulfillment. In true Hegelian form, we cannot skip a step in the historical process, but not skipping a step in historical evolution does not preclude the possibility of accelerating a step so as to reduce the amount of time spent in a suboptimal form of civilization and therefore to maximize the amount of time spent in a preferred mode of civilization.

Accelerationism on the right, which I believe to be the more common form of accelerationism, understands the preferred mode of civilization to be a society dominated by traditional institutions. How are traditional institutions to be brought into being in the wake of accelerated apocalypticism? This, I think, is the nub of the problem, as the traditionalist favoring accelerationism as a means to realizing a traditional society must either hope for new traditionalist institutions to emerge, or for the reconstitution of defunct institutions. Both of these horns of the dilemma are a problem.

Part of Burke’s criticism of the French revolution was the folly of attempting to craft de novo institutions on the basis of abstract and theoretical propositions about human beings and human society, especially in the light of existing institutions that apparently are adequate to their institutional role, and which are, in some sense, the preserved wisdom of our ancestors. (The attempt to frame new institutions de novo was the source of Goya’s famous etching, “The sleep of reason produces monsters,” which was a symbolic response to the terror that followed the superficial rationalism of the French revolution; more simply, we can call this an instance of the law of unintended consequences.) Burke wrote before an evolutionary understanding of human beings and human society had been formulated, but in the light of evolutionary psychology and the slow evolution of human society we could easily reframe Burke’s critique so that any nebulous invocation of the wisdom of ancestors can be replaced by traditional institutions being the cumulative result of natural selection. This is far more satisfying from a scientific point of view.

The argument can be made that if an episode of social denudation stripped away existing social institutions, surviving human societies would revert to a model of social organization that is naturally emergent from the kind of beings that we are, that is to say, a social order predicated upon our particular cognitive endowments and cognitive biases (as well as that which I have called less than cognitive biases, which might be called “breaking human”). The traditionalist assumes, or would assume, that these naturally emergent institutions would be traditionalist institutions. In this view there is a hint of a venerable pre-modern idea, that truth lies at the source of things, so that if only we can return to the source of being, the source of our being, we will find the authentic truth that has been hidden from us by the overgrowth of thousands of years of extraneous developments that have led us far from our origins. This view stands in stark contrast to the idea that truth is a distant goal to which we aspire, and which we always approximate more closely, but which we never fully possess.

If, instead of seeking to frame traditionalist institutions de novo (which may be a contradictory idea anyway), the accelerationist seeks the reconstitution of defunct traditional institutions, I am skeptical that this effort would fare any better. There have been many times when regimes have attempted to turn back the clock on developments that did not seem to favor their vision of how things ought to be, but I cannot think of any of these attempts that were successful. Old or traditional institutions transplanted into new circumstances will neither function as these traditional institutions functioned, nor will they remain true to the tradition from which they are drawn. The same logic is to be found in arguments over the historically informed performance (HIP) movement in music: can we ever truly make our instruments and performances sound like those of the past, or must our contemporaneous recreations always be performed with modern instruments in a modern setting? This is an interesting debate, and many books of musicology have been devoted to the HIP controversy. Perhaps the discussion of the accelerationist reconstitution of defunct traditionalist institutions could learn something from this discussion.

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Plate 43 of Goya’s Los Caprichos series of etchings: ‘The Sleep of Reason Produces Monsters.’

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Monday


In The Industrial-Technological Thesis I characterized industrial-technological civilization as involving an escalating cycle of science, technology, and engineering, each generation of which feeds into the next so that science makes new technologies possible, new technologies are engineered into new industries, and new industries create the instruments for further scientific research. I further argued in Civilization, War, and Industrial Technology that the only property more pervasively inherent in industrial-technological civilization than escalating feedback is war — since escalating feedback is characteristic only of The Industrial-Technological Thesis, whereas war typifies all civilization. Thus technological growth and war are both structurally inherent in The Industrial-Technological Thesis, so much so that to entertain the idea of civilization without either is probably folly.

Now I realize that in recounting the escalating spiral of science, technology and engineering, that I was recounting only the “creative” side of the “creative destruction” of industrialized capitalism, and that the creative destruction of capitalism as it is played out in industrial-technological civilization also has a destructive side that is expressed in a way entirely consonant with the distinctive character of industrial-technological civilization. Each phase in the cycle of science, technology, and engineering fails in a distinctive (and in a distinctively interesting) way.

The counter-cyclical trend to that of the exponentially escalating spiral of science, technology, and engineering is the exponentially deescalating downward trend of science in model crisis, stalled technology, and catastrophic failures of engineering. Science falters when model drift gives way to model crisis and normal science begins to give way to revolutionary science. Human beings, being what they are, have invested science with the “truth” once reserved for matter theological; but science has no “truths” — there is only the scientific method, which remains the same even while the knowledge that this method yields is always subject to change. Technology falters when its exponential growth tapers off and its attains a mature plateau, after which time it changes little and becomes a stalled technology. Engineering falters when industries experience the inevitable industrial accidents, intrinsic to the very fabric of industrialized society, or even experience the catastrophic failures to which complex systems are vulnerable.

Industrial accidents are intrinsic to industrialized society, and cannot be wished away.

I hadn’t previously thought of these disruptions to industrial-technological civilization together, but now that I see them whole I see that I have already written separately about all the phases of failure that so closely parallel the successes of industrialization. Mostly, I think, these disruptions have taken place separately, and have therefore only proved to be temporary disruptions in the rapidly-resuming cycle of technological growth. However, once we see the possible failures as a systemic, counter-cyclical trend that destroys old knowledge, old technology, and old industries in order to make room for the new, we can easily see the possibility of an escalating disruption in which scientific model crisis would limit knowledge, limited knowledge would lead to long term stalled technologies, and stalled technologies would lead to escalating industrial accidents and complex catastrophic failures.

None of this, of course, is in the least bit surprising. Ever since the industrialized warfare of the twentieth century we have been discussing the possibility that industrial-technological civilization will more or less inevitably destroy itself. Civilization, when it was suddenly and unexpectedly preempted by industrialization, has opened Pandora’s box, and the evils that fly free cannot be shut back inside.

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Sunday


L. E. J. Brouwer: philosopher of mathematics, mystic, and pessimistic social theorist

A message to the foundations of mathematics (FOM) listserv by Frank Waaldijk alerted me to the fact that today, 14 October 2012, is the one hundredth anniversary of Brouwer’s inaugural address at the University of Amsterdam, “Intuitionism and Formalism.” (I have discussed Frank Waaldijk earlier in P or Not-P and What is the Relationship Between Constructive and Non-Constructive Mathematics?)

I have called this post “One Hundred Years of Intuitionism and Formalism” but I should have called it “One Hundred Years of Intuitionism” since, of the three active contenders as theories for the foundations of mathematics a hundred years ago, only intuitionism is still with us in anything like its original form. The other contenders — formalism and logicism — are still with us, but in forms so different that they no longer resemble any kind of programmatic approach to the foundations of mathematics. In fact, it could be said that logicism was gradually transformed into technical foundational research, primarily logical in character, without any particular programmatic content, while formalism, following in a line of descent from Hilbert, has also been incrementally transformed into mainstream foundational research, but primarily mathematical in character, and also without any particular programmatic or even philosophical content.

The very idea of “foundations” has come to be questioned in the past hundred years — though, as I commented a few days ago in The Genealogy of the Technium, the early philosophical foundationalist programs continue to influence my own thinking — and we have seen that intuitionism has been able to make the transition from a foundationalist-inspired doctrine to doctrine that might be called mathematical “best practices.” In contemporary philosophy of mathematics, one of the most influential schools of thought for the past couple of decades or more has been to focus not on theories of mathematics, but rather on mathematical practices. Sometimes this is called “neo-empiricism.”

Intuitionism, I think, has benefited from the shift from the theoretical to the practical in the philosophy of mathematics, since intuitionism was always about making a distinction between the acceptable and the unacceptable in logical principles, mathematical reasoning, proof procedures, and all those activities that are part of the mathematician’s daily bread and butter. This shift has also made it possible for intuitionism to distance itself from its foundationalist roots at a time when foundationalism is on the ropes.

Brouwer is due some honor for his prescience in formulating intuitionism a hundred years ago — and intuitionism came almost fully formed out of the mind of Brouwer as syllogistic logic came almost fully formed out of the mind of Aristotle — so I would like to celebrate Brouwer on this, the one hundredth anniversary of his inaugural address at the University of Amsterdam, in which he formulated so many of the central principles of intuitionism.

Brouwer was prescient in another sense as well. He ended his inaugural address with a quote from Poincaré that is well known in the foundationalist community, since it has been quoted in many works since:

“Les hommes ne s’entendent pas, parce qu’ils ne parlent pas la même langue et qu’il y a des langues qui ne s’apprennent pas.”

This might be (very imperfectly) translated into English as follows:

“Men do not understand each other because they do not speak the same language and there are languages ​​that cannot be learned.”

What Poincaré called men not understanding each other Kuhn would later and more famously call incommensurability. And while we have always known that men do not understand each other, it had been widely believed before Brouwer that at least mathematicians understood each other because they spoke the same universal language of mathematics. Brouwer said that his exposition revealed, “the fundamental issue, which divides the mathematical world.” A hundred years later the mathematical world is still divided.

For those who have not studied the foundations and philosophy of mathematics, it may come as a surprise that the past century, which has been so productive of research in advanced mathematics — arguably going beyond all the cumulative research in mathematics up to that time — has also been a century of conflict during which the idea of mathematics as true, certain, and necessary — ideas that had been central to a core Platonic tradition of Western thought — have all been questioned and largely abandoned. It has been a raucous century for mathematics, but also a fruitful one. A clever mathematician with a good literary imagination could write a mathematical analogue of Mandeville’s Fable of the Bees in which it is precisely the polyglot disorder of the hive that made it thrive.

That core Platonic tradition of Western thought is now, even as I write these lines, dissipating just as the illusions of the philosopher, freed from the cave of shadows, dissipate in the light of the sun above.

Brouwer, like every revolutionary (and we recall that it was Weyl, who was sympathetic to Brouwer, who characterized Brouwer’s work as a revolution in mathematics), wanted to do away with an old, corrupt tradition and to replace it with something new and pure and edifying. But in the affairs of men, a revolution is rarely complete, and it is, far more often, the occasion of schism than conversion.

Many were converted by Brouwer; many are still being converted today. As I wrote above, intuitionism remains a force to be reckoned with in contemporary mathematical thought in a way that logicism and formalism cannot claim to be such a force. But the conversions and subsequent defections left a substantial portion of the mathematical community unconverted and faithful to the old ways. The tension and the conflict between the old ways and the new ways has been a source of creative inspiration.

Precisely that moment in history when the very nature of mathematics was called into question became the same moment in history when mathematics joined technology in exponential growth.

Mars is the true muse of men.

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Mars, God of War and Muse of Men.

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