Textuality, reality, and the limits of knowledge *.
HOW MUCH CAN A PERSON KNOW? A Leibnizian Perspective on Human Finitude. How much can someone possibly know? What could reasonably be viewed as an upper limit of an individual's knowledge--supposing that factually informative knowledge rather than performative how-to knowledge or subliminally tacit knowledge is to be at issue?
In pursuing this question, let us suppose someone with perfect recall who devotes a long lifespan to the acquisition of information. For seventy years this individual spends 365 days per annum reading for twelve hours a day at the rate of sixty pages an hour (with 400 words per page). That yields a lifetime reading quota of some 7.4 x [10.sup.9] words. Optimistically supposing that, on average, a truth regarding some matter of fact or other takes only some seven words to state, this means a lifetime access to some [10.sup.9] truths, that is, around a billion of them. No doubt most of us are a great deal less well informed than this. But is seems pretty well acceptable as an upper limit to the information that a human individual could probably not reach and certainly not exceed.
After all, with an average of 400 pages per book, the previously indicated lifetime reading quota would come to some 46,000 books. The world's largest libraries, the Library of Congress for example, nowadays have somewhere around 20 million books (book-length assemblages of pamphlets included), and it would take a very Hercules of reading to make his way through even one-quarter one of one percent of so vast a collection (= 50,000), which is roughly what our aforementioned reading prodigy manages. If mastery of Library of Congress-encompassed material is to be the measure, then few of us would be able to hold our heads up very high. (1) This means that while a given individual can read any book (so that there are no inherently unreadable books), the individual cannot possibly read every book (so that for anyone of us there are bound to be very many unread books indeed).
All this, of course, still only addresses the question of how much knowledge a given person--one particular individual--can manage to acquire. There yet remains the question of how much is in principle knowable--that is, can be known. Here it is instructive to begin with the perspective of the great seventeenth-century polymath G. W. Leibniz.
Leibniz took his inspiration from The Sand Reckoner of Archimedes, who in this study sought to establish the astronomically large number of sand grains that could be contained within the universe defined by the sphere of the fixed stars of Aristotelian cosmology--a number Archimedes effectively estimated at [10.sup.50]. Thus even as Archimedes addressed the issue of the scope of the physical universe, so Leibniz sought to address the issue of the scope of the universe of thought. (2)
Leibniz pursued this project very much in the spirit of the preceding observations. He wrote:
All items of human knowledge can be expressed by the letters of the alphabet ... so that it follows that one can calculate the number of truths of which humans are capable and thus compute the size of a work that would contain all possible human knowledge, and which would contain all that could ever be known, written, or invented, and more besides. For it would contain not only the truths, but also all the falsehoods that men can assert, and meaningless expressions as well. (3)
Thus if one could set an upper limit to the volume of printed matter accessible to inquiring humans, then one could map out by combinatorial means the whole manifold of accessible verbal material--true, false, or gibberish--in just the manner that Leibniz contemplated. This is exactly what he proceeded to do in a fascinating 1693 tract, De l'horizon de la doctrine humaine. (4)
Any alphabet devisable by man will have only a limited number of letters (Leibniz here supposes the Latin alphabet of twenty-four). So even if we allow a word to become very long indeed (Leibniz overgenerously supposes thirty-two letters (5)), there will be only a limited number of words that can possibly be formed (namely 24 exp 32). And so, if we suppose a maximum to the number of words that a single run-on, just barely intelligible sentence can contain (say 100), then there will be a limit to the number of potential "statements" that can possibly be made, namely 100 exp (24 exp 32). (6) This number is huge indeed--far bigger than Archimedes' sand grains. Nevertheless, it is still finite, limited. Moreover, with an array of basic symbols different from those of the Latin alphabet, the situation is changed in detail but not in structure. (This remains the case even if one adds the symbols at work in mathematics, where Descartes's translation of geometrically pictorial propositions into algebraically articulated format stood before Leibniz's mind, to say nothing of his own project of a universal language and a calculus ratiocinator. (7))
The crux of Leibniz's discussion is that any propositionalizable contention can in principle be spelled out in print. There is only so much, so finitely much, that can be stated in sentences of intelligible length--and so also that can explicitly be thought of beings who conduct their thinking in language. Moreover, since this encompasses fiction as well, our knowledge of possibility is also finite, and fiction is for us just as much language-limited as is the domain of truth.
The Leibnizian Perspective. These considerations mean that as long as people transact their thinking in language--broadly understood to encompass diverse symbolic devices--the thoughts they can have--and afortiori the things they possibly can know--will be limited in number. The moment one sets a realistic limit to the length of practicably meaningful sentences, one has to realize that the volume of the sayable is finite--vast thought it will be.
Moving further along these lines, let it be that the cognitive (in contrast to the affective) thought-life of people consists of the language-framed propositions that they consider. Let us suppose that people can consider textualized propositions at about the same speed at which they can read--optimistically, say, some sixty pages per hour where each page consists of twenty sentences. Assuming a thought-span of sixteen waking hours on average, it will then transpire that in the course of a year a person can entertain a number of propositional thoughts equal to 365 x 16 x 60 x 20 [congruent to] 7 x [10.sup.6].
So subject to the hypotheses at issue, this is how much material one would need to replicate in print the stream of consciousness thought-life of a person for an entire year. Once again, this number of seven million, though not small, is nevertheless limited. These limits will again finitize the combinatorial possibilities. There is only so much thinking that a person can manage. In the context of a finite species, these limits of language mean that there are only so many thoughts to go around--so many manageable sentences to be formulated. Once again we are in the grip of finitude.
Now as Leibniz saw it, matters can be carried much further. For the finitude at issue here has highly significant implications. Consider an analogy. Only a finite number of hairs will fit on a person's head--say 1,000. So when there are enough individuals in a group (say 1001 of them) then two of them must have exactly the same number of hairs on their heads. And so also with thoughts. If only there are enough thinking intelligences in the cons of cosmic history while the number of thoughts--and thus also thought-days and thought-lives--are finite, then there will inevitably be several people in a sufficiently large linguistic community whose thoughts are precisely the same throughout their lives.
It also becomes a real prospect that language supposes limits on our grasp of people and their doings. Suppose that the detailed biography of a person is a minute-by-minute account of his doings, allocating (say) ten printed lines to each minute, and so roughly 15,000 lines per day to make up a hefty volume of 300 fifty-line pages. So if a paradigmatic individual lives 100 years we will need 365 x 100 or roughly 36,500 such substantial tomes to provide a comprehensive blow-by-blow account of his life.
But, as we have seen, the number of such tomes, though vast, is limited. In consequence, there are only so many detailed biographies to go around, so that it transpires that the number of detailed biographies that is available is also finite. This, of course, means that: If the duration of the species were long enough--or if the vastness of space provided sufficiently many thinkers--then there would have to be some people with exactly the same detailed biography. Given enough agents, eventual repetitions in point of action became inevitable.
Now moving on from biographies (or diaries) to public annals, Leibniz thought to encounter much the same general situation once again. Thus suppose that (as Leibniz has it) the world's population is one hundred million (that is [l0.sup.8]) and that each generation lives (on average) for fifty years, then in the 6,000 years during which civilized man may be supposed to have existed, there have existed some 1.2 x [10.sup.10] people--or some [10.sup.10] of them if we assume smaller generations in earlier times. (8)
Recall now the above-mentioned idea of 36,500 hefty tomes needed to characterize in detail the life of an individual. It then follows that we would need some 36.5 x [10.sup.13] of them for a complete history of the species. To be sure, we thus obtain an astronomically vast number of possible overall annals for mankind as a whole. But though vast, this number will nevertheless be finite. Moreover, if the history of the race is sufficiently long, then some part of its extensive history will have to repeat itself in full with a parfaite repetition mot pour mot since there are only so many possible accounts of a given day (or week or year). For once again there are only a finite number of possibilities to go around and somewhere along the line total repetitions it will transpire that life stories will occasionally recur in toto (ut homines novi eadem ad sensum penitus tota vita agerent, quae alii jam egerunt (9)).
As Leibniz thus saw it, the finitude of language and its users carries in its wake the finitude of possible diaries, biographies, histories--you name it, including even possible thought-lives in the sense of propositionalized streams of consciousness as well. Even as Einstein with his general relativity (initially) saw himself as finitizing the size of the physical universe, so Leibniz's treatise saw the size of mankind's cognitive universe as a manifold of limited horizons--boundless but finite.
It was accordingly a key aspect of Leibniz's thought that the human understanding cannot keep up with reality. For Leibniz, the propositional thought of finite creatures is linguistic and thereby finite and limited. But he also held that reality--as captured in the thought of God, if you will--is infinitely detailed. Only God's thought can encompass it, not ours. Reality's infinite detail thus carries both costs and its benefits in its wake. Its cost is the unavoidability of imperfect comprehension by finite intelligences. Its benefit is the prospect of endless variability and averted repetition. The result is a cognitively insuperable gap between epistemology and metaphysics. Everything that humans can say or think by linguistic means can be comprehended in one vast but finite universal library.
But what do these Leibnizian ruminations mean in the larger scheme of things?
Statements Are Enumerable. Let us now shift the theme of discussion from the issue of what a given person can know to the more massive issue of what can be known by people at large.
The preceding deliberations have unfolded on the basis of the emphatically contingent supposition that there are certain limits to human capabilities--and, in particular, to the length of the words and sentences with which our discourse can effectively operate. But let us now also waive this (otherwise surely realistic) restriction and break through the limits of finitude in the interests of getting a grip on the general principles of the matter.
Even if one construes the idea of an "alphabet" sufficiently broadly to include not only letters but symbols of various sorts, it still holds that everything stateable in a language can be spelled out in print through the combinatorial combination of some sequential register of symbols. (10) With a "language" construed as calling for development in the usual recursive manner, it transpires that the statements of a language can be enumerated in a vast and indeed infinite but nevertheless ultimately countable listing. (11) Thus, since the world's languages, even if not finite in number, are nevertheless at most enumerable, it follows that the set of all statements--including every linguistically formidable proposition--will be enumerably infinite.
As a matter of principle, then, we obtain the following contention:
Thesis 1: The Enumerability of Statements. Statements (linguistically formulated propositions) are enumerable and thus (at most) denumerably infinite.
Our linguistic resources for describing concrete states of affairs are thus subject to quantitative limitation. Insofar as our thought about things proceeds by recursively developed linguistic means, it is inherently limited in its reach within the confines of countability. And so, the upshot is that the limits of textuality impose quantitative limitations upon propositionalized thought--albeit not limits of finitude.
Truth vs. Facts. It serves the interests of clarity to introduce a distinction at this stage, that between truths and facts. Truths are linguistically formulated facts, correct statements which, as such, must be formulated in language (broadly understood to include symbol systems of various sorts). A "truth" is something that has to be framed in linguistic/symbolic terms--the representation of a fact through its statement in some language--so that any correct statement formulates a truth.
A "fact," on the other hand, is not a linguistic item at all, but an actual aspect of the world's state of affairs. A fact is thus a feature of reality. (12) Facts correspond to potential truths whose actualization as such waits upon their appropriate linguistic embodiment. Truths are statements and thereby language-bound, but facts outrun linguistic limits. Once stated, a fact yields a truth, but with facts at large there need in principle be no linguistic route to get from here to there.
Truths Though Infinite in Number Are Denumerable. Being inherently linguistic in character, truths are indissolubly bound to textuality, seeing that any language-framed declaration can be generated recursively from a sequential string of symbols--that is, that all spoken language can in principle be reduced to writing. Since they correspond to statements, it follows that truths cannot be more than countably infinite. On this basis we have it that:
Thesis 2: The Denumerability of Truth. While the manifold of the truth cannot be finitely inventoried, nevertheless, truths are no more than denumerably infinite in number.
The Inexhaustibility of Fact. Facts, however, are another matter altogether. It is a key facet of our epistemic stance toward the real world that its furnishings possess a complexity and diversity of detail so elaborate that there is always more to be said than we have so far managed. Every part of reality has features beyond the range of our current cognitive reach--at any juncture whatsoever.
Moreover, any adequate account of inquiry must recognize that the process of information-acquisition at issue in science is a process of conceptual innovation. Caesar did not know--and in the then extant state of the cognitive art could not have known--that his sword contained tungsten and carbon. There will always be facts about a thing that we do not know because we cannot even express of them in the prevailing conceptual order of things. To grasp such a fact means taking a perspective of consideration that as yet we simply do not have, because the state of knowledge (or purported knowledge) has not reached a point at which such a consideration is feasible. The ongoing progress of scientific inquiry always leaves various facts about the things of this world wholly outside the conceptual realm of the inquirers of any particular period. Accordingly, the facts about any actual physical object are theoretically inexhaustible. After all, any such thing has dispositions that run off into infinity.
Its susceptibility to further elaborate detail--and to changes of mind regarding this further detail--is built into our very conception of a "real thing." And so the range of fact about anything real is effectively inexhaustible. There is, as best we can tell, no limit to the world's ever-increasing complexity that comes to view with our ever-increasing grasp of its detail. The realm of fact and reality is endlessly variegated and complex. And so we also arrive at:
Thesis 3: The Inexhaustibility of Fact. Facts are infinite in number. The domain of fact is inexhaustible: there is no limit to facts about the real.
In this regard, however, real things differ in an interesting and important way from fictive ones. For a key about fictional particulars is that they are of finite cognitive depth. In discoursing about them we shall ultimately run out of steam as regards their nongeneric features. A point will always be reached when one cannot say anything further that is characteristically new about them, when one cannot present nongeneric information that is not inferentially implicit in what has already been said. (13) New generic information can, of course, always be forthcoming through the progress of science. When we learn more about coal-in-general then we know more about the coal in Sherlock Holmes's grate. But the finiteness of their cognitive depth means that the presentation of ampliatively novel nongeneric information must by the very nature of the case come to a stop when fictional things are at issue.
With real things, on the other hand, there is no reason of principle why the elaboration of information that is nongenerically idiosyncratic need ever end. On the contrary, we have every reason to presume these things to be cognitively inexhaustible. The prospect of discovery is open-ended here. A precommitment to description-transcending features--no matter how far description is pushed--is essential to our conception of a real thing. Something whose character was exhaustible by linguistic characterization would thereby be marked as fictional rather than real. (14)
And so we have it that facts regarding reality are infinite in number. But just how infinite?
Facts Are Transdenumerable. While statements in general--and therefore true statements in particular--can be enumerated, so that truths are denumerable in number, there is no reason to suppose that the same will be true of facts. On the contrary, there is every reason to think that, reality being what it is, there will be an uncountably large manifold of facts.
The reality of it is that facts, unlike truths, cannot be enumerated. For no listing of fact-presenting truths--not even one of infinite length--can possible manage to constitute a complete register of facts. Any attempt to register-fact-as-a-whole will founder: the list is bound to be incomplete because there are facts about the list-as-a-whole that no single entry can encompass.
We thus arrive at the next principal thesis of these deliberations:
Thesis 4: The Transdenumerability of Facts. The manifold of fact is transdenulnerably infinite.
The idea of a complete listing of all the facts is manifestly impracticable. Consider the following statement: "The list F of stated facts fails to have this statement on it." But now suppose this statement to be on the list. Then it clearly does not state a fact, so that the list is after all not a list of the facts (contrary to hypothesis). And so it must be left off the list. But then in consequence that list will not be complete since the statement is true. Thus, facts can never be listed in toto because there will always be further facts--facts about the entire list itself--that a supposedly complete list could not manage to register.
This conclusion can be rendered more graphic by the following considerations. Suppose that the list F: [f.sub.1], [f.sub.2], [f.sub.3], ... were to constitute a complete enumeration of all facts. Now consider the statement
(Z) the list F takes the form, [f.sub.1], [f.sub.2], [f.sub.3], ...
By hypothesis, this statement will present a fact. So if F is indeed a complete listing of all facts, then there will be an integer k such that Z = [f.sub.k]. Accordingly, Z itself will occupy the kth place on the F listing, so that
[f.sub.k] = the list L takes the form [f.sub.1], [f.sub.2], [f.sub.3], ...[f.sub.k], ...
But this would require [f.sub.k] to be an expanded version of itself, which is absurd. With the kth position of the F listing already occupied by [f.sub.k], we cannot also squeeze that complex [f.sub.k]-involving thesis into it.
The point here is that any supposedly complete listing of facts [f.sub.1], [f.sub.2], [f.sub.3] ... will itself exhibit, as a whole, certain features that none of its individual members can encompass. Once those individual entries are fixed and the series is defined, there will be further facts about that series-as-a-whole that its members themselves cannot articulate.
Moreover, the point at issue can also be made via an analogue of the diagonal argument that is standardly used to show that no list of real numbers can manage to include all of them, thereby establishing the transdenumerability of the reals. Let us begin by imagining a supposedly complete inventory of independent facts, using logic to streamline the purported fast history into a condition of greater informative tidiness through the elimination of inferential redundancies so that every item adds some information to what has gone before. The argument for the transdenumerability of fact can now be developed as follows. Let us suppose (for the sake of reductio ad absurdum argumentation) that the inventory [f.sub.1], [f.sub.2], [f.sub.3], ... represents our (nonredundant but yet purportedly complete) listing of facts. Then, by the supposition of factuality we have ([for all]i)[f.sub.1]. Further, by the supposition of completeness we have it that
([for all]p)(p [right arrow] ([there exists]i)[([f.sub.1][right arrow]p])
Moreover, by the aforementioned supposition of nonredundancy, each member of the sequence adds something quite new to what has gone before.
([for all]i)([for all]j)[i <j [right arrow] ~ [([f.sub.1] & [f.sub.2] & ... & [f.sub.1]) [right arrow] ?[f.sub.1])]
Consider now the following course of reasoning.
(1) ([for all]i)[f.sub.1], by "factuality"
(2) ([for all]j)[f.sub.j] [right arrow] ([there exists]i)([f.sub.1] [right arrow] ([for all]j)[f.sub.j]), from (1) by "completeness" via the substitution of ([for all]j)[f.sub.j] for p
(3) [there exists]i]([f.sub.1] [right arrow] ([for all]j)[f.sub.1]),
But (3) contradicts nonredundancy. This reductio ad absurdum of our hypothesis indicates that the facts about any sufficiently complex object will necessarily be too numerous for complete enumeration. In such circumstances, no purportedly comprehensive listing of truths can actually manage to encompass all facts.
More Facts than Truths. The long and short of it is that the domain of reality-characterizing fact inevitably transcends the limits of our capacity to express it, and afortiori those of our capacity to canvas it completely. In the description of concrete particulars we are caught up in an inexhaustible derail: There are always bound to be more descriptive facts about things than we are able to capture explicitly with our linguistic machinery. Given that concrete reality is--so we must suppose--endlessly complex, detailed, and diversified in its make-up, the limitedness of our recursively constituted linguistic resources means that our characterizations of the real will always fall short.
Thesis 5: There are quantitatively more facts than truths. Facts, then, are too numerous for enumerabilty.
Cognition, being bound to language, is digital and sequentially linear. Reality, by contrast, is analogue and replete with feedback loops and coordinated in nonsequentially systemic interrelations. It should thus not be seen as all that surprising that the two cannot be brought into smooth alignment. But, of course, the aspect of the situation that is paramount for present purposes is the resultant discrepancy of numbers.
The basic reason why the domain of fact is ampler than that of truth is that language cannot capture the entirety of fact. We live in a world that is not digital but analog, and so the manifold of its states of affairs is simply too rich to be fully comprehended by our linguistically digital means. The domain of fact inevitably transcends the limits of linguistically formulable truth. Truth is to fact what film is to reality--a merely discreptized approximation.
When facts and language play their game of musical chairs, some facts are bound to be left in the lurch when the music of language stops. The discrepancy manifests itself in the difference between "any" and "every." Any candidate can possibly be accommodated. (We have ([for all]x)[??]([there exists]y)Syx.) But it is not possible to accommodate every candidate. (We do not have [??]([for all]x)([there exists]y)Syx.) The same situation of a quantitative discrepancy between fact and language-formidable knowledge imposes the cognitive limits and limitations that have been highlighted in the present discussion. For the limits of knowledge are in the final analysis quantitative. The crux of the problem is a discrepancy of numbers. They have roots in the musical chairs perplex--in the fact that propositionalized knowledge is language bound, and that the realm of fact is too vast to be encapsulated in the confines of language.
Twentieth-century philosophers of otherwise the most radically different orientation have agreed on preeminentizing the role of language. "The limits of my language set the limits of my world" ("Die Grenzen meiner Spache bedeuten die Grenzen meiner Welt") says the Wittgenstein of the Tractatus at 5.6. "There is nothing outside text" ("Il n'y a pas de hors de texte") say the devotees of French constructionism. But already centuries earlier Leibniz had taken the measure of this sort of textualism. He looked at it up close and saw that it could not be more wrong.
There is a fundamental qualitative discrepancy between the realms of the real and that which is text-characterizable. Realty bursts the confines of textualization. (15) And that this occurs must be accepted despite the inherent and unavoidable impossibility of indicating just where it does so. For, of course, we cannot possibly adduce any concrete example of an unstateable fact.
Limits of Knowledge. The cognitive beings that concern us here are language-dependent finite intelligences, and these are by their very nature imperfect knowers. For the information at their disposal by way of propositional knowledge that something or other is the case will--unlike how-to knowledge--have to be verbally formulated--unlike performative how-to knowledge. Language-encompassed textuality will, as we have seen, be outdistanced by the facts themselves. But what is one to make of the numerical disparity between facts and truths, between what is knowable in theory and what we finite intelligences can actually manage to know? Just what does this portend?
It means that our knowledge of fact is incomplete--and inevitably so!--because we finite intelligences lack the means for its comprehensive characterization. Reality in all its blooming, buzzing complexity is too rich for faithful representation by the recursive and enumerable resources of our language. We do and must recognize the limitations of our cognition, acknowledging that we cannot justifiably equate reality with what can explicitly be known by us through the resources of language. What transpires here for the circumstantial situation of our sort of mind also obtains for any other sort of finite mind as well. Any physically realizable sort of cognizing being can articulate--and thus can know--only a part or aspect of the real.
The situation as regards the knowing of facts is akin to that of the counting of integers--specifically in the following respects: (16)
(1) The manifold of integers is inexhaustible. We can never come to grips with all of them as particular individuals. Nevertheless--
(2) Further progress is always possible: we can always go beyond whatever point we have so far managed to reach. In principle we can always go beyond what has been attained. Nevertheless--
(3) Moving forward gets ever more cumbersome. In moving onward we must be ever more prolix and make use of ever more elaborate symbol complexes. Greater demands in time, effort, and resources are inevitable here. Accordingly--
(4) In actual practice there will be only so much that we can effectively manage to do. The possibilities that obtain in principle can never be fully realized in practice. However--
(5) Such limitations nowise hamper the prospects of establishing various correct generalizations about the manifold of integers in its abstract entirety.
A parallel situation characterizes the cognitive condition of all finite intelligences whose cognitive operations have to proceed by a symbolic process that functions by language. Inductive inquiry, like counting, never achieves completeness. There is always more to be done: In both cases alike we can always do better by doing more. But we can never do it all.
Does this state of affairs mean that those unknown facts are unknowable? The answer is neither "yes" nor "no." As already foreshadowed above, it all depends upon exactly how one is to construe this possibilistic matter of "knowability." Using Kxf to abbreviate "the individual x knows the fact f," there will clearly be two rather different ways in which the existence of an inherently unknowable fact can be claimed, namely,
([there exists]f)[??] ([for all]x) ~ Kxf or equivalently ~ ([for all]f]) [??] ([there exists]x)Kxf
[??] ([there exists]f) ([for all]x) ~ Kxf or equivalently ~ [??]([for all]f])([there exists]x)Kxf
The first of these logically entails the second. This second is in the circumstances inevitable, there being more facts than finite humans ever will or can know. However, the first, stronger contention is clearly false. For as long as the nonexistence of an omniscient God is not a necessary circumstance, there can be no fact that is of necessity unknown.
The difference in the quantifier placement in the preceding two formulas is crucial when one contemplates the prospect of unlimited knowability--of the idea that all facts are knowable. Insofar as the issue is problematic, the idea of unknowable facts will have to pivot on the acceptability of the first thesis.
But of course even though there are--or may well be--unknowable facts (in the indicated sense of this term), they can never be identified as such, seeing that to identify a fact as such, namely as a fact, is effectively to claim knowledge of it. It is, accordingly, in principle impossible for us ever to give an example of one.
Cognitive Finitude. First the good news. Generalizations can of course refer to everything. Bishop Butler's "Everything is what it is and not another thing" holds with unrestricted universality. Once continuous quantities are introduced, the range of inferentially available statements becomes uncountable. "The length of the table exceeds x inches." This clearly opens the door to uncountably many alternatives. However--and this is the crucial point--only a countable number of statements can ever be specifically made. (17)
Fortunately, a case-by-case determination is not generally needed to validate generalizations. We can establish claims about groups larger than we can ever hope to inventory. Recourse to arbitrary instances, the process of indirect proof by reductio ad absurdum, and induction (mathematical and scientific) all afford procedures for achieving generality beyond the range of an exhaustive case-by-case check.
But will this always be so? Or, are there also general truths whose determination would require the exhaustive surveying of all specific instances of a totality too large for our range of vision?
At this point our cognitive finitude becomes a crucial consideration. The difference between a finite and an infinite knower is of fundamental importance and requires closer elucidation. For an "infinite knower" need not and should not be construed as an omniscient knower--one from whom nothing knowable is concealed (and so who knows, for example, who will be elected U. S. President in the year 2200). Rather, what is at issue is a knower who can manage to know in individualized detail an infinite number of independent facts. Such a knower might, for example, be able to answer such a question as: "Will the decimal expansion of [pi] always continue to agree at some future point with that of [square root of 2] for 100 decimal places?" (Of course, the circumstance that an infinite knower can know some infinite set of independent facts does not mean that he can know every such set.)
Finite knowers can, of course, know universal truths. After all, we must acknowledge the prospect of inductive knowledge of general laws. We will have it that a knower can unproblematically know, for example, that "All dogs eat meat." (18) But what finite knowers cannot manage is to know this sort of thing in detail rather than at the level of generality. They cannot know specifically of each and every u in that potentially infinite range that Fu obtains--that is, while they can know collectively that all individuals have F, they cannot know distributively of every individual that it has F. Finite knowers can certainly know (via the U. S. Constitution) that every President is over the age of thirty-five. But of course one has this knowledge without knowing of every President (including those one never heard of, let alone the yet unborn) that each individual one of them is over the age of thirty-five--something one cannot do without knowing who they individually are.
Surd Facts and Unknowability. One cannot, of course, provide concrete examples of specific facts that are unknowable for finite knowers, seeing that a claim to factuality automatically carries a claim to knowledge in its wake. However, while we cannot know what is such a fact, we can certainly establish that there are such things.
Given any collection of items, there are two importantly different kinds of general properties: those that all members of the collection do have in common, and those that all member of the collection must have in common. The latter are the necessity-geared general features of the collection, the former its contingency-geared features. Thus that all prime numbers greater than 2 are odd is a necessity-geared feature of these primes. Or consider the set of all post-Washington U. S. presidents. That all of them are native born and that all of them are over thirty-five years of age is a necessity-geared feature of the collection in view of our Constitution's stipulations. However, that all were the favored candidates of a political party will (if indeed true) be a contingently geared feature of the collection that is nowise necessitated by its constituting characterization.
Now the crucial consideration for present purposes is that the necessary features of a collection must inhere in (and be derivable from) the generalities that govern the collection at issue as a matter of principle. But its contingent features will be "surd" in that they cannot be established on the basis of general principles. When and if they actually hold, this can only be ascertained through a case-by-case check of the entire membership of the collection. This means that finite knowers can never decisively establish a surd/contingent general feature of an infinite collection. Whenever a generality holds for a collection on a merely contingent basis, this is something that we finite intelligences can never determine with categorical assurance. The determination of such kind-pervasively surd would thus require an item-by-item check, which is ex hypothesi impracticable for us.
Whenever a situation of this kind actually obtains--which for aught we know to the contrary is often the case--then we can never manage to ascertain all the facts regarding an unsurveyable totality! Actual knowledge of the matter is beyond our reach here. The best and most that we can ever do here is to employ inductive or plausible or probabilistic reasoning in a way that leaves the issue beclouded with a shadow of doubt.
Consider an illustration. The New York Times is an English-language newspaper. As such, it is a necessary feature of the Times that throughout the history of its publication, mostly English words appear on its front page. This circumstance is inherent in the general principle (the "laws") of the matter. With these general principles in hand we can settle the issue of front-page vocabulary. With such law-constrained facts--let us call them "nomic"--we certainly do not need to carry out a case-by-case check through every issue; and knowledge reacts on the general principles of the situation. A nomic property of something is a necessary feature for its kind: one that everything of its type not only does but must exhibit as a member of that particular "natural kind."
However, it must also be presumed to be a fact that as long as the paper exists, every issue of The New York Times will be such that the word "the" occurs more than ten times on its front page. This is almost certainly a fact. But to determine that it is actually so, a case-by-case check becomes unavoidable. Such a fact--one whose determination cannot be settled by general principles (laws) but whose ascertainment requires a case-by-case check--is generally characterized as surd. Such a property of something is contingent: it cannot be accounted for on the basis of the general principles at issue.
Consider now a set of objects of a certain sort S that is infinite or interminably open-ended (lions, say, or sunrises at Acapulco). Let P be a surd/contingent property of some S-item X that, while in principle applicable to S-members, is nevertheless unique to X--that is, is such that no other S-member actually has P. Now note that this uniqueness could only be determined on a case-by-case check across the whole range of S. That X is unique within S in point of P-possession is (by hypothesis) a truth that no finite intelligence could ascertain, seeing that an item-by-item canvas of an infinite/indefinite range is beyond its capacity. Such truths illustrate the prospect of truths beyond the cognitive grasp of finite knowers.
Of course, "unknowably true" is a vagrant predicate--one that has no determinate address in that it admits of no identifiable instance. Instantiating this sort of thing can only be done at the level of schematic generality and not that of concrete instantiation. But we can convince ourselves--for good reason--that there indeed are such things even though it is in principle impracticable to provide examples of them.
Lessons. Overall, however, the situation is not as bleak as it may seem. For even though the thought and knowledge of finite beings is destined to be ever finite, it nevertheless has no fixed and determinate limits. Return to our analogy. As is counting integers, there is a limit beyond which we never will get. But there is no limit beyond which we never can get. For the circumstance that there is always room for linguistic variation--for new symbols, new combinations, new ideas, new truths and new knowledge--creates a potential for pushing our though ever further. While the thought of finite beings is destined ever to be finite, it nevertheless has no fixed and determinable limits.
The line of thought operative in these deliberations was already mooted by Kant:
[I]n natural philosophy, human reason admits of limits ["excluding limits," Schranken], but not of boundaries ["terminating limits," Grenzen], namely, it admits that something indeed lies without it, at which it can never arrive, but not that it will at any point find completion in its internal progress.... [T]he possibility of new discoveries is infinite: and the same is the case with the discovery of new properties of nature, of new powers and laws by continued experience and its rational combination. (19)
Here Kant was right--even on the Leibnizian principles considered at the outset of this discussion. The cognitive range of finite beings is indeed limited. But it is also boundless because it is not limited in a way that blocks the prospect of cognitive access to ever new and continually different facts, thereby affording an ever ampler and ever more adequate account of reality. (20)
Some writers analogize the cognitive exploration of the realm of fact to the geographic exploration of the earth. But this analogy is profoundly misleading. For the earth has a finite and measurable surface, and so even when some part of it is unexplored terra incognita its magnitude and limits can be assessed in advance. Nothing of the kind obtains in the cognitive domain. The ratio and relationship of known truth to knowable fact is subject to no fixed and determinable proportion. Geographic exploration can expect eventual completeness, cognitive exploration cannot. There are no boundaries--no determinate limits--to the manifold of discoverable fact.
Further Implications. It is worthwhile to note that the numerical discrepancy between truths and facts that textuality imposes recurs time and again in other contexts, and in particular as between
* names and entities
* statements and possibilities
* descriptions and objects
* novels and plots
* instructions and actions
* explanations and phenomena
The same quantitative discrepancy between the verbal and the ontological occurs throughout. In each the former is a verbalized indicator for the later. There are just not enough of the former to go around, so that there is a recurrence of the musical chairs situation touched upon above: any one of these "players" can find a seating accommodation in language, but not every one. The problem is that of deficient accommodation for an oversize group of candidates. We have ([for all]x)[??]([there exists]y)Sxy but emphatically not [??]([for all]x)([[there exists]y)Sxy.
In particular, consider names. Of course everything is capable of being named. Nothing is name-resistant. We could (as someone has quipped) simply name everything Charlie. The real question is if everything could have a unique name characteristic of itself alone, an identifying name.
Now everything that is in fact identified could be named via the specification: "the item identified in such and such a way." Or at least this would work if the identification process answered to some verbalized formula or other. Now supposing this to be the case, the real question becomes: Are there enough verbal/textual identifiers to go around? Can everything that has an identity be individuated by verbalized formulas?
The answer is "no." Select any language you please--take your pick. As long as it is produced recursively--like any other human language-it will only have countably many expressions (words, sentences, texts). But we know full well that the number of objects is transdenumerable: uncountably infinite. (Think of the real numbers for example.) So there just are not enough names for everything. In musical chairs not everybody gets to be seated. In reality not everything gets to be named.
Of course, things will stand differently if we radically revise the concept of language. Thus, if we are prepared to countenance a thing-language (rather than a word language), then we could adopt the rule that everything names itself. Then, of course, everything is namable. But this sort of thing is clearly cheating.
The question, "Is everything nameable?", bears the format: "Can everything of type 1 be uniquely associated with something of type 2?" Only if we are definite about what type 1 and type 2 are will a question of this format be well defined and meaningful. As long as that type at issue is construed so broadly as to include the real numbers, and "naming" calls for specification within a recursively articulated language, a negative answer to our question becomes unavoidable.
And so, while nothing is name-resistant and everything is namable in the sense of being able, in principle, to bear a name, the possibility of realizing this prospect across the board--with everything whatsoever bearing a name--is precluded by the general principles of the situation. Things and names also engage in a game of musical chairs in a way that renders it unavoidable that some of the former must lose out. (21)
University of Pittsburgh
Correspondence to: Department of Philosophy, University of Pittsburgh, 1012 Cathedral of Learning, Pittsburgh, PA 15260.
* Presidential Address at the 57th annual meeting of the Metaphysical Society of America, March 2005, at the University of Pittsburgh.
(1) To be sure, there lurks in the background have the question of whether having mere information is to count a having knowledge. With regard to this quantitative issue it has been argued that authentic knowledge does not increase propositionally with the amount of information as such, but only proportionally with its logarithm. See chapter 13 of the author's Epistemology (Albany: State University of New York Press, 2003). This would suggest that the actual knowledge within the Library of Congress's many volumes might be encompassed in some far more modest collection. But this sort of complication can be put aside as irrelevant to the course of reflection that will be unfolded.
(2) On Archimedes' estimate see T. C. Heath, The Works of Archimedes (Cambridge: Cambridge University Press, 1897).
(3) See G. W. Leibniz, De l'horizon de la doctrine humaine, ed. Michael Fichant (Paris: Vrin, 1991). The quotation is from a partial translation of Leibniz's text in Philip Beeley, "Leibniz on the Limits of Human Knowledge," The Leibniz Review 13 (December 2003): 93-7 (see p. 95).
(4) It is well known that Leibniz invented entire branches of science, among the differential and integral calculatus, the calculus of variations, topology (analysis situs), symbolic logic, and computers. But there is also one branch in which subsequent developments have not arrived even now, namely epistemetrics, the measurement of knowledge. For while intelligence measurement (IQ assessment) is a well developed field, knowledge measurement is not--notwithstanding the proliferation of quiz shows comparing different people's knowledge of various fields. The actual measurement of how much people do know--and no less importantly, or how much they can know--is still a substantially unrealized domain of investigation.
(5) The longest word I have seen in actual use is the 34-letter absurdity "supercalifragilisticexpialidocious" from the musical "Mary Poppins."
(6) G. W. Leibniz, De l'horizon, 11. This of course long antedates the (possibly apocryphal) story about the Huxley-Wilbefforce debate which has Huxley arguing that sensible meaning could result from chance process because a team of monkeys typing at random would eventually produce the works of Shakespeare--or (on some accounts) all the books in the British Library, including not only Shakespeare's works but the Bible as well. (The story--which goes back, at least, to Sir Arthur Eddington's The Nature of the Physical World [London: McMillan, 1929], 72-3) is doubtless fictitious since the Huxley-Wilberforce debate of 1860 antedated the emergence of the typewriter.) However, the basic idea goes back at least to Cicero: "If a countless number of the twenty-one letters of the alphabet ... were mixed together it is possible that when cast on the ground they should make up the Annals of Ennius, able to be read in good order" (De natura deorum, bk. 2, chap. 27). The story launched an immense discussion on the contemporary scene, as is readily attested by a Google or Yahoo search for "typing monkeys." It has also had significant literary repercussions, as exemplified by Jorge Luis Borges's well-known story of "The Library of Babel," which contains all possible books.
(7) Louis Couturat, La logique de Leibniz (Paris: Alcan, 1901), is still the best overall account of this Leibnizian project.
(8) Leibniz, De l'horizon, 112.
(9) Ibid., 54.
(10) Compare Philip Hugly and Charles Sayward, "Can a Language Have Indenumerably Many Expressions?" History and Philosophy of Logic 4 (1983): 112-26.
(11) This supposes an upper limit to the length of intelligible statements. Even if this restriction were waived, the number of statements will still be no more than countably finite.
(12) Our position thus takes no issue with P. F. Strawson's precept that "facts are what statements (when true) state." See his "Truth," Proceedings of the Aristotelian Society, supplementary vol. 24 (1950): 129-56 (see p. 136). Difficulty would ensue with Strawson's thesis only if an "only" were added.
(13) To deny inferentially implicit information the title of authentic novelty is not, of course, to say that it cannot surprise us in view of the limitations of our own deductive powers.
(14) This also explains why the dispute over mathematical realism (Platonism) has little bearing on the issue of physical realism. Mathematical entities are akin to fictional entities in this--that we can only say about them what we can extract by deductive means from what we have explicitly put into their defining characterization. These abstract entities do not have nongeneric properties since each is a "lowest species" unto itself.
(15) The circumstance that not every actual fact can be articulated in a (true) statement shows a fortiori that not every possible situation can be characterized linguistically. If the domain of fact outruns the bounds of language articulation, then the manifold of possibility must certainly do so as well. We must accordingly acknowledge that not everything is sayable!
(16) We here take "counting" to be a matter of indicating integers by name--for example, as "thirteen" or "13"--rather than descriptively, as per "the first prime after eleven."
(17) To make a statement is effectively to name it, and the inherently limited resources of language will only put countably many names at our disposal. On this issue see the Appendix below.
(18) To be sure, the prospect of inductively secured knowledge of laws is a philosophically controversial issue. But this is not the place to pursue it. For the author's view of the matter see his Induction (Oxford: Blackwell, 1980).
(19) Prolegomena to Any Future Metaphysics, sec. 57. Compare the following passage from Charles Sanders Peirce: "For my part, I cannot admit the proposition of Kant--that there are certain impassable bound to human knowledge.... The history of science affords illustrations enough of the folly of saying that this, that, or the other can never be found out. Auguste Comte said that it was clearly impossible for man ever to learn anything of the chemical constitution of the fixed stars, but before his book had reached its readers the discovery which he had announced as impossible had been made. Legendre said of a certain proposition in the theory of numbers that, while it appeared to be true, it was most likely beyond the powers of the human mind to prove it; yet the next writer on the subject gave six independent demonstrations of the theorem"; Collected Papers, 2d ed. (Cambridge, Mass.: Harvard University Press, 1931-58), vol. 6, sec. 6.556.
(20) This discussion has profited from the constructive comments of several Pittsburgh colleagues, including Jason Dickinson, Mickey Perloff, and Laura Ruetsche.
(21) This Appendix has benefited from exchanges with C. Anthony Anderson.
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|Publication:||The Review of Metaphysics|
|Date:||Dec 1, 2005|
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