14 October 2012
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.
. . . . .
. . . . .
. . . . .
. . . . .
10 October 2012
Addendum on Civilization and the Technium
in regard to human, animal, and alien technology
One of the virtues of taking the trouble to formulate one’s ideas in an explicit form is that, once so stated, all kinds assumptions one was making become obvious as well as all kinds of problems that one didn’t see when the idea was just floating around in one’s consciousness, as a kind of intellectual jeu d’esprit, as it were.
Bertrand Russell wrote about this, or, at least, about a closely related experience in one of his well-known early essays, in which he discussed the importance not only making our formulations explicit, but of doing so by way of putting some distance between our thoughts and the kind of facile self-evidence that can distract us from the real business at hand:
“It is not easy for the lay mind to realise the importance of symbolism in discussing the foundations of mathematics, and the explanation may perhaps seem strangely paradoxical. The fact is that symbolism is useful because it makes things difficult. (This is not true of the advanced parts of mathematics, but only of the beginnings.) What we wish to know is, what can be deduced from what. Now, in the beginnings, everything is self-evident; and it is very hard to see whether one self-evident proposition follows from another or not. Obviousness is always the enemy to correctness. Hence we invent some new and difficult symbolism, in which nothing seems obvious. Then we set up certain rules for operating on the symbols, and the whole thing becomes mechanical. In this way we find out what must be taken as premiss and what can be demonstrated or defined. For instance, the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions. But without a symbolism it would have been very hard to find this out. It is so obvious that two and two are four, that we can hardly make ourselves sufficiently skeptical to doubt whether it can be proved. And the same holds in other cases where self-evident things are to be proved.”
Bertrand Russell, Mysticism and Logic, “Mathematics and the Metaphysicians”
Russell’s foundationalist program in the philosophical of mathematics closely followed the method that he outlined so lucidly in the passage above. Principia Mathematica makes the earliest stages of mathematics notoriously difficult, but does so in service to the foundationalist ideal of revealing hidden presuppositions and incorporating them into the theory in an explicit form.
Another way that Russell sought to overcome self-evidence is through the systematic pursuit of the highest degree of generality, which drives us to formulate concepts that are alien to common sense:
“It is a principle, in all formal reasoning, to generalize to the utmost, since we thereby secure that a given process of deduction shall have more widely applicable results…”
Bertrand Russell, An Introduction to Mathematical Philosophy, Chapter XVIII, “Mathematics and Logic”
These are two philosophical principles — the explication of ultimate simples (foundations) and the pursuit of generality — that I have very much taken to heart and attempted to put into practice in my own philosophical work. Russell’s foundationalist method shows us what can be deduced from what, and gives to these deductions the most widely applicable results. To these philosophical imperatives of Russell I have myself added another, parallel to his pursuit of generality, and that is the simultaneous pursuit of formality: it is (or ought to be) a principle in all theoretical reasoning to formalize to the utmost…
Russell also observed the imperative of formalization, though he himself did not systematically distinguish between generalization and formalization, and it is a tough problem; I’ve been working on it for about twenty years and haven’t yet arrived at definitive formulations. As far as provisional formulations go, generalization gives us the highly comprehensive conceptions like astrobiology and civilization and the technium that allow us to unify a vast body of knowledge that must be studied by inter-disciplinary means, while formalization gives us the distinctions we must carefully observe within our concepts, so that generalization does not simply give us the night in which all cows are black (to borrow a phrase that Hegel used to ridicule Schelling’s conception of the Absolute).
Foundationalism as a philosophical movement is very much out of fashion now, although the foundations of mathematics, pursued eo ipso, remains an active and highly technical branch of logico-mathematical research, and today looks a lot different from what it was when it was first formulated as a philosophical research program a hundred years ago by Frege, Peano, Russell, Whitehead, Wittgenstein, and others. Nevertheless, I continue to derive much philosophical clarification from the early philosophical stages of foundationalism, especially in regard to theories that have not (yet) been reduced to formal systems, as is the case with theories of history or theories of civilization.
I am still a long way from reducing my ideas about history or civilization to first principles, much less to symbolism, but I feel like I am making progress, and the discovery of assumptions and problems is a sure sign of progress; in this sense, my post on Civilization and the Technium marked a stage of progress in my thinking, because of the inadequacy of my formulations that it revealed.
In my Civilization and the Technium I compared the extent of civilization — a familiar idea that has not yet received anything like an adequate definition — with the extent of the technium — a recent and hence unfamiliar idea for which there is an explicit formulation, but since it is new its full scope remains untested and untried, and therefore it presents problems that the idea of civilization does not present. I formulated concepts of the technium parallel to formulations of astrobiology and astrocivilization, as follows:
● Eotechnium the origins of the technium, wherever and whenever it occurs, terrestrial or otherwise
● Esotechnium our terrestrial technium
● Exotechnium any extraterrestrial technium exclusive of the terrestrial technium
● Astrotechnium the totality of technology in the universe, our terrestrial and any extraterrestrial technium taken together in their cosmological context
I realize now that when I did this I was making slightly different assumptions for civilization and the technium. The intuitive basis of this was that I assumed, in regard to the technium, that the technium I was describing was all due to human activity (a clear case of anthropic bias), so that the distinction between the exotechnium and the exotechnium was the distinction between terrestrial human technology and extraterrestrial human technology.
When, on the other hand, I formulated the parallel concepts for civilization, I assumed that esocivilization was terrestrial human civilization and that exocivilization would be alien civilizations not derived from the human eocivilization source.
Another way to put this is that I assumed the validity of the terrestrial eotechnium thesis even while I also assumed that the terrestrial eocivilization thesis did not hold. Is that too much technical terminology? In other words, I assumed the uniqueness of the human technium but I did not assume the uniqueness of human industrial-technological civilization.
This points to a further articulation (and therefore a further formalization) of the concepts employed: one must keep the conception of eocivlization (the origins of civilization) clearly in mind, and distinguish between terrestrial civilization that expands into extraterrestrial space and therefore becomes exocivilization from its eocivilization source on the one hand, and on the other hand a xeno-eocivilization source that constitutes exocivilization by virtue of its xenomorphic origins. If one is going to distinguish between esocivilization and exocivilization, one must identify the eocivilization source, or all is for naught.
All of this holds, mutatis mutandis, for the eotechnium, esotechnium, exotechnium, and astrotechnium, although I ought to point that my formulations in Civilization and the Technium, and repeated above, were accurate because they were formulated in Russellian generality. It was in my following exposition that I failed to observe all the requisite distinctions. But there’s more. I’ve since realized that further distinctions can be made.
As I thought about the possibility of a xenotechnium, i.e., a technium produced by a sentient alien species, I realized that there is a xenotechnium right here on Earth (a terrestrial xenotechnium, or non-hominid technium), in the form of tool use and other forms of technology by non-human species. We are all familiar with famous examples like the chimpanzees who will strip the leaves off a branch and then use the branch to extract termites from a termite mound. Yesterday I alluded to the fact that otters use rocks to break open shells. There are many other examples. Apart from tool use, beaver damns and the nests of birds, while not constructed with tools, certainly represent a kind of technology.
If we take all instances of animal technology together they constitute a terrestrial non-human technium. If we take all instances of technology known to us, human and non-human together, we have a still more comprehensive conception of the technium that is more general that the concept of the human-specific technium and therefore less subject to anthropic bias (the latter concept due to Nick Bostrum, who also formulated existential risk). This latter, more comprehensive conception of the technium would seem to be favored by Russell’s imperative of generalization to the utmost, although we must continue to make the finer distinctions within the concept for the formalization of the conception of the technium to keep pace with its generalization.
There is a systematic relationship between terrestrial biology and the terrestrial technium, both hominid and non-hominid. Eobiology facilitates the emergence of a terrestrial eotechnium, of which all instances of technology, hominid and non-hominid alike, can be considered expressions. This is already explicit in Kevin Kelly’s book, What Technology Wants, as one of his arguments is that the emergence and growth of the technium is continuous with the emergence of growth of biological organization and complexity. He cites John Maynard Smith and Eors Szathmary as defining the following thresholds of biological organization (p. 46):
One replicating molecule -» Interacting population of replicating molecules
Replicating molecules -» Replicating molecules strung into chromosome
Chromosome of RNA enzymes -» DNA proteins
Cell without nucleus -» Cell with nucleus
Asexual reproduction (cloning) -» Sexual recombination
Single-cell organism -* Multicell organism
Solitary individual -» Colonies and superorganisms
Primate societies -» Language-based societies
He then suggests the following sequence of thresholds within the growth of the technium (p. 47):
Primate communication -» Language
Oral lore -> Writing/mathematical notation
Scripts -» Printing
Book knowledge -» Scientific method
Artisan production -» Mass production
Industrial culture -» Ubiquitous global communication
And then he connects the two sequences:
The trajectory of increasing order in the technium follows the same path that it does in life. Within both life and the technium, the thickening of interconnections at one level weaves the new level of organization above it. And it’s important to note that the major transitions in the technium begin at the level where the major transitions in biology left off: Primate societies give rise to language. The invention of language marks the last major transformation in the natural world and also the first transformation in the manufactured world. Words, ideas, and concepts are the most complex things social animals (like us) make, and also the simplest foundation for any type of technology. (p. 48)
Thus the genealogy of the technium is continuous with the genealogy of life.
Considering this in relation to the possibility of a xenotechnium, one would expect the same to be the case: I would expect a systematic relationship to hold between xenobiology and a xenotechnium, such that an alien eobiology would facilitate the emergence of an alien eotechnium. And, extending this naturalistic line of thought, that assumes similar patterns of development to hold for peer industrial-technological civilizations, I would further assume that a xenotechnium would not always coincide with the xenocivilization with which it is associated. If there is a “first contact” between terrestrial civilization and a xenocivilization, it is likely that it will be rather a contact between the expanding terrestrial technium (which is, technically, no longer terrestrial precisely because it is expanding extraterrestrially) and an expanding xenotechnium.
There remains much conceptual work to be done here, as the reader will have realized. I’ll continue to work on these formulations, keeping in mind the imperatives of generality and formality, and perhaps someday converging on a foundationalist account of biology, civilization, and the technium that is at once both fully comprehensive and fully articulated.
. . . . .
. . . . .
. . . . .
27 December 2011
Yesterday in The Philosophy of Fear I quoted Descartes from his Discourse on Method, from the section in which he introduces an implicit distinction between the theoretical principles he will use to guide his philosophical activities and the practical moral principles that he will employ in his life while he is going about his theoretical activity. Here is his exposition of his four theoretical principles:
● The first was never to accept anything for true which I did not clearly know to be such; that is to say, carefully to avoid precipitancy and prejudice, and to comprise nothing more in my judgement than what was presented to my mind so clearly and distinctly as to exclude all ground of doubt.
● The second, to divide each of the difficulties under examination into as many parts as possible, and as might be necessary for its adequate solution.
● The third, to conduct my thoughts in such order that, by commencing with objects the simplest and easiest to know, I might ascend by little and little, and, as it were, step by step, to the knowledge of the more complex; assigning in thought a certain order even to those objects which in their own nature do not stand in a relation of antecedence and sequence.
● And the last, in every case to make enumerations so complete, and reviews so general, that I might be assured that nothing was omitted.
Anyone who knows Descartes’ works will recognize that he has here stated, much more simply and compactly, the principles that he was working on in his unfinished manuscript Rules of the Direction of Mind. Here, by way of contrast, is a highly condensed version of Descartes’ practical and provisional moral principles:
● The first was to obey the laws and customs of my country, adhering firmly to the faith in which, by the grace of God, I had been educated from my childhood and regulating my conduct in every other matter according to the most moderate opinions, and the farthest removed from extremes, which should happen to be adopted in practice with general consent of the most judicious of those among whom I might be living.
● My second maxim was to be as firm and resolute in my actions as I was able, and not to adhere less steadfastly to the most doubtful opinions, when once adopted, than if they had been highly certain; imitating in this the example of travelers who, when they have lost their way in a forest, ought not to wander from side to side, far less remain in one place, but proceed constantly towards the same side in as straight a line as possible, without changing their direction for slight reasons, although perhaps it might be chance alone which at first determined the selection; for in this way, if they do not exactly reach the point they desire, they will come at least in the end to some place that will probably be preferable to the middle of a forest.
● My third maxim was to endeavor always to conquer myself rather than fortune, and change my desires rather than the order of the world, and in general, accustom myself to the persuasion that, except our own thoughts, there is nothing absolutely in our power; so that when we have done our best in things external to us, all wherein we fail of success is to be held, as regards us, absolutely impossible: and this single principle seemed to me sufficient to prevent me from desiring for the future anything which I could not obtain, and thus render me contented…
Descartes wrote a lot a extremely long run-on sentences, so that one must cut radically in order to quote him (except for his theoretical principles, above, which I have quoted entire), but I have tried to include enough above to give a genuine flavor of how he expressed himself. Although Descartes did not himself make this distinction between theoretical and practical principles explicit, although the distinction is explicitly embodied in his two sets of explicitly stated principles, he does provide a justification for the distinction:
“…as it is not enough, before commencing to rebuild the house in which we live, that it be pulled down, and materials and builders provided, or that we engage in the work ourselves, according to a plan which we have beforehand carefully drawn out, but as it is likewise necessary that we be furnished with some other house in which we may live commodiously during the operations, so that I might not remain irresolute in my actions, while my reason compelled me to suspend my judgement, and that I might not be prevented from living thenceforward in the greatest possible felicity, I formed a provisory code of morals, composed of three or four maxims, with which I am desirous to make you acquainted.”
After I quoted this in The Philosophy of Fear I realized that it constitutes a perfect antithesis to the conception of the rational reconstruction of knowledge embodied in the image of Neurath’s ship, which I have quoted several times.
Rational reconstruction was an idea that fascinated early twentieth century philosophers, especially the logical positivists, whose philosophical tradition would eventually mature and transform itself into mainstream analytical philosophy. It was logical positivism that gave us an enduring image of rational reconstruction, as related by Otto Neurath:
“There is no way of taking conclusively established pure protocol sentences as the starting point of the sciences. No tabula rasa exists. We are like sailors who must rebuild their ship on the open sea, never able to dismantle it in dry-dock and to reconstruct it there out of the best materials. Only the metaphysical elements can be allowed to vanish without trace.”
Quine then used this image in his Word and Object:
“We are like sailors who on the open sea must reconstruct their ship but are never able to start afresh from the bottom. Where a beam is taken away a new one must at once be put there, and for this the rest of the ship is used as support. In this way, by using the old beams and driftwood the ship can be shaped entirely anew, but only by gradual reconstruction.”
These two epistemic paradigms — what I will call Descartes’ house and Neurath’s ship — represent antithetical conceptions of the epistemological enterprise. Neurath’s ship is usually presented as an anti-foundationalist parable, which would suggest that Descartes’ house is a foundationalist parable. There are certain problems with this initial characterization. The logical positivists who invoked Neurath’s ship with approval were often foundationalists in the philosophy of mathematics while being anti-foundational in other areas.
There is a sense in which it is fair to call Descartes’ house a foundationalist parable: Descartes is suggesting a radical approach to the foundations of knowledge — utterly tearing down our knowledge in order to construct entirely anew on the same ground — and he attempted to put this into practice in his own philosophical work. He doubted everything that he could until he arrived at the fact that he could not doubt his own existence, and then on the basis of the certainty of his own existence he attempted to reconstruct the entire edifice of knowledge. The result was not radical, but actually rather conventional, but the method certainly was radical. It was also total.
Whether or not Neurath’s ship is anti-foundational, it is certainly incrementalist. If we were to attempt to rebuild a ship while at sea, we would need to proceed bit by bit, and very carefully. Nothing radical would be attempted, for to attempt anything radical would be to sink the ship. There is a sense in which we could identify this effort as essentially constructivist in spirit, though not exclusively constructivist: constructivism is certainly not the only motivation for Neurath’s ship, and many who invoked it employed non-constructive modes of reasoning.
Are Descartes’ house and Neurath’s ship mutually exclusive? Not necessarily. We do remodel houses while living in them, although when we do we need to keep some basic functions available during our residency. And we can demolish certain parts of a ship at sea; as long as the hull remains intact, we can engage in a radical reconstruction (as opposed to a rational reconstruction) of the masts and the rigging.
One ought not to push an image too far, for fear of verging on the ludicrous, but it can be observed that, while living in a house, we can tear down half of it to the ground and rebuild that half from scratch while living in the other half, and then repeat this process in the half we have been living in. In fact, I know people who have done this. There will, of course, be certain compromises that will have to be made in wedding the two halves together, so that the seam between the two has the incrementalist character of Neurath’s ship, while each half has the radical and total character of Descartes’ house.
It is difficult to imagine a parallel for the above scenario when it comes to Neurath’s ship. The hull of the ship can only be rebuilt incrementally, although almost everything else can be radically reconstructed. And it may well be that some parts of epistemology must be approached incrementally while other parts of epistemology may be radically reconstructed almost with impunity. This seems like an eminently reasonable conclusion. But it is no conclusion — at least not yet — because there is more to say.
What underlies the image of Descartes’ house and Neurath’s ship is in each case a distinct metaphor, and that metaphor is for Descartes the earth, the solid ground upon which we stand, while for Neurath it is the sea, to which we must go down in ships, and where we cannot stand but must swim or be carried. So, we have two epistemic metaphors — of what are they metaphors? Existence? Being? Human experience? Knowledge? If the house or the ship is knowledge, then the ground or the sea must be that upon which knowledge rests (or floats). This once again suggests a foundationalist approach, but points to very different foundations: a house stands on dirt and stones; a ship floats on water.
Does knowledge ultimately rest upon the things themselves — the world, existence, or being, as you prefer — or upon human experience of the world? Or is not knowledge a consequence of the tension between human experience and the world, so that both the world and human experience are necessary to knowledge?
Intuitively, and without initially putting much thought into this (although I will continue to think about this because it is an interesting idea), I would suggest that the metaphor of the earth implies that knowledge ultimately is founded on the things themselves, while the metaphor of the sea implies that knowledge ultimately is founded on the ever-changing tides of human experience.
Therefore, if knowledge requires both the world and human experience, either the metaphor of Descartes’ house or Neurath’s ship alone, in isolation from the other, is inadequate. We need something more, or something different, to illustrate our relation to knowledge and how it changes.
. . . . .
. . . . .
. . . . .
. . . . .