Ignorance: (On the Wider Implications of Deficient Knowledge)

Chapter 4 Cognitive Finitude

57 Problems of Detail 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. And once continuous quantities are introduced, the range of inferentially available statements becomes uncountable. “The length of the table exceeds x inches.” Once known, this straightaway opens the door to uncountable knowable consequences. And 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 knowledge beyond the reach 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 4 Cognitive Finitude rescher ign text.indd 57 12/19/08 9:45:43 AM 58 Cognitive Finitude our cognitive finitude becomes a crucial consideration, and the difference between finite and infinite knowers becomes 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 π always continue to agree at some future point with that of √2 for 100 decimal places?”) Finite knowers cannot manage this sort of thing. Finite knowers can, of course, know universal truths. After all, we must acknowledge the prospect of inductive knowledge of general laws, and we will have it that a knower can unproblematically know—for example—that “All dogs eat meat.”1 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—something they could not do without knowing who they individually are. So the issue now before us is that of the question of general truths that can be known only by assessing the situation of an intractable manifold of individual cases. Quasi-Quantities It is instructive to survey some situations in which the ways of the world impel ignorance upon us. One instance is afforded by quasiquantities that cannot be pinned down exactly. What is at issue here rescher ign text.indd 58 12/19/08 9:45:44 AM Cognitive Finitude 59 is a quantity that resists being specified precisely by a particular number . It admits of being located within a certain numerical range while nevertheless not letting itself be made precise. It is, in sum, a quantity X such that for any real number n the claim X = n is untenable, and not just false but even meaningless or silly, even though n1 < X < n2 may well be true. So when X is a quasi-quantity in the aforementioned sense, then there will be some values of u so small that X specifying X to within a limit of size u is no longer meaningful. With a quasi-quantity we cannot be exact beyond a certain point (a point about which itself we cannot be absolutely exact). The weight of a person is just like that. Just how much air is in his lungs at the moment? Does that hair or that flake of skin just now detaching from him count or not count? What about that drop of perspiration just dropping off his nose? We can meaningfully specify weight to within the nearest X for X down to perhaps a milligram, but we cannot push the matter far below that. Much the same goes for a person’s age, seeing that the moment of someone’s birth can be pinned down only so far. And of course the current length of Britain’s coastline can only be managed to within rough limits, what with tides and waves and such. An interesting question arises in this connection. Are there any situations in life (or, indeed, even in science) where extreme precision actually matters, where the fact that X is a quasi-quantity rather than a precise quantity makes a difference to anything—and “being as precise as one can reasonably be expected to be in matters of this kind” is not good enough? There will indeed be, specifically in those situations that physicists characterize as chaotic. In these instances, there will be quantities that operate in the context of functional relationships, where a difference in input, however minute, will always make a substantive difference in output. Hence the outcome depends on matters that rescher ign text.indd 59 12/19/08 9:45:44 AM 60 Cognitive Finitude ultimately are wholly below and beyond the threshold of our vision. The precision needed to go from speculation to calculation is simply beyond our reach in such cases. As far as we are concerned, the matter will be a thing of pure chance. We cannot effectively come to know the details of it. In drastic matters, quasi-quantities will impel us into ignorance Surd Features There are two sorts of properties of objects: namely generic properties that an object shares with all others of a particular natural kind to which it belongs, and idiosyncratic properties that characterize it uniquely without in anyway inhering in a natural kind to which it belongs. An object’s possession of such idiosyncratic properties cannot be derived from laws that are given its constitution in terms of general kinds. They cannot be determined by descriptions but require individual, specific inspection. Thus, if each individual of an infinite set has an idiosyncratic property, then no finite intelligence can ever know this array of fact in detail. (In the end this is bound to be the case with the positive integers, for example.) One cannot, of course, provide concrete examples of facts that are unknowable to finite knowers, seeing that a claim to factuality automatically carries a claim to knowledge in its wake. However, while we cannot know specifically which is such a fact, one can certainly substantiate the claim generally that there are such things. Let us consider this situation more closely. A feature F of an object/item x is surd if Fx cannot be deduced from the body of knowledge consisting of the following: • The identifying (discussion-introducing) descriptive characterization of the item x at issue rescher ign text.indd 60 12/19/08 9:45:44 AM Cognitive Finitude 61 • A specification of the various natural kinds (Ki) to which said item x belongs, together with • A specification of the various kind-correlative laws— all given by generalizations having the structure: “Everything of kind K has the property F” A second feature of an item is, in sum, one that cannot be established from general principles after that item has been duly identified . Such a feature is idiosyncratic to that item in conditions taken to all others of its kind with respect to each and every one of its several kinds to which it belongs. For example, it is not the case that “being a prime” is a surd property of 5. For the nondivisibility of 5 by any lesser integer (save 1) can be deduced from 5’s defining specification, together with the general laws of arithmetic that govern integers at large (a natural kind of which 5 belongs). By contrast, “being the number of books on that table” is a surd property of 5, seeing that there is no way of deriving it from the general principles at issue with the characterizing specification of 5, together with the laws that govern its correlative natural kinds. Accordingly, the specifically surd feature of objects/items are those facts about them that are not inferentially accessible from a knowledge of their nature—and thereby not explicable through recourse to general principles. As far as the relevant corpus of general principles is concerned, the feature is anomalous, contingent, and is by its very nature not law-rationalizable.2 Its possession by an object has to be determined by inspection: it cannot be established by inference from that object’s specifying features via general principles. Given this understanding of the matter, let us suppose that some mind feature or other stands in common by each and every member of some infinite or open-ended set of terms. Since this fact cannot rescher ign text.indd 61 12/19/08 9:45:44 AM 62 Cognitive Finitude (ex hypothesi) be established on general principles, it will have to be revised through case-by-case inspection, which—in these circumstances —is unpreventable with any finite knower. For example, it must be granted that as long as these astronomical objects persist there will, on any given day, be a number of meteors (with mass greater than 1 kg) closer to the earth than to the moon. And now consider the contention that this number is invariably less than five thousand. It may well be that this contention is true. But since the number of relevant days is—potentially—infinitely large and since the matter is not one that can be settled for each of these several days on the basis of general automated principles, the fact now hypothetically at issue (namely, that the number in question is never greater than five thousand) is something that finite intelligence cannot possibly come to know. And this illustrates the fact that finite knowers can never ascertain the surd/contingent general features of an infinite or indefinitely large collection . For our knowledge of the universal features of infinite groups is limited to the reach of lawful generalizations alone. Determination of surd generality would require an item-by-item check, which is by hypothesis impracticable for us with infinite or indefinite collections. Accordingly, where such large groups are concerned, secure general knowledge is confined to the region of nomic fact. Though there will doubtless be universal facts that are surd in character in our complex world, they remain, for us, in the realm of supposition and conjecture . For finite knowers, firm knowledge of surd universality is unrealizable . Such examples illustrate the general phenomenon that finite knowers can never decisively establish a surd/contingent general feature of an infinitely or indefinitely large collection. For 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, because determination of such kind-pervasive rescher ign text.indd 62 12/19/08 9:45:44 AM Cognitive Finitude 63 surdity would require an item-by-item check, which is by hypothesis impracticable for us. This situation clearly manifests yet another sort of inevitable ignorance in finite intelligences. The Principle of Epistemic Disparity Let us begin here with a somewhat extreme case. A knower is unrestrictedly omniscient. Whenever there is something to be known, this knower knows it. In other words, whenever p is a true matter of fact, the knower knows that it is so. Thus, x is omniscient in this sense if we have: (∀p)(p → Kxp) Such a knower knows everything that is knowable. This knower’s knowledge is literally unlimited: something is a truth if, and only if, our omniscient being x knows it. By contrast, a knower is restrictedly omniscient iff this knower knows everything that is known. That is, whenever anyone knows something, this knower knows it as well: Thus x is omniscient in this weaker sense if we have: (∀p)(∃yKyp → Kxp) The difference between the two modes of omniscience can come into operation only when there are unknowns—that is, truths which nobody knows at all. For when we have p & ~(∃y)Kxp the antecedent at issue with unrestricted omniscience is satisfied, while that at issue with restricted omniscience is not. How do I know that I am not omniscient? Certainly not because I can specify particular facts that I do not know. Rather, it is because there are questions I cannot answer—and because I realize full well at the level of generality that there are truth-determinate propositions whose truth-status I cannot decide, perfectly meaningful proprescher ign text.indd 63 12/19/08 9:45:44 AM 64 Cognitive Finitude ositions about authentic matters of fact that I know neither to be true nor false, even though I do know that they have to be one or the other. (That George Washington wondered if Martha was suitably dressed for the occasion of his first inauguration would seem to be a good example.) But what of others? There can be no doubt that ignorance exacts its price in incomprehension . And here it helps to consider the matter in a somewhat theological light. The world we live in is a manifold that is not of our making but of Reality’s—or of God’s, if you will. And what is now at issue might be called Isaiah’s Principle on the basis of the following verse: For My thoughts are not your thoughts, neither are your ways My way, says the Lord. For as the heavens are higher than the Earth, so are My ways higher than your ways, and My thoughts than your thoughts. (Isa. 55:8–9) A fundamental law of epistemology is at work here—to wit, the principle that a mind of lesser power is for this very reason unable to understand adequately the workings of a mind of greater power. To be sure, the weaker mind can doubtless realize that the stronger can solve problems it itself cannot. But it cannot understand how it is able to do so. An intellect that can only just manage to do well at tic-tac-toe cannot possibly comprehend the ways of one that is expert at chess. The knowledge of limited knowers is inevitably limited in point of detail. The lesser mind can know that the more powerful has certain things but cannot tell the what and the how of it. To the lesser mind, the performances of a more powerful one are bound to seem like magic. Consider in this light the vast disparity of computational power between a mathematical tyro like most of us and a mathematical prodigy like Ramanujan. Not only cannot our tyro manage to answer the number-theoretic question that such a genius resolves in rescher ign text.indd 64 12/19/08 9:45:44 AM Cognitive Finitude 65 the blink of an eye, but the tyro cannot even begin to understand the processes and procedures that the Indian genius employs. As far as the tyro is concerned, it is all sheer wizardry. No doubt once an answer is given, he can check its correctness. But actually finding the answer is something that lesser intellect cannot manage—the how of the business lies beyond its grasp. And, for much the same sort of reason, a mind of lesser power cannot discover what the questionresolving limits of a mind of greater power are. It can never say with warranted assurance where the limits of question-resolving power lie. (In some instances it may be able to say what’s in and what’s out, but never map the dividing boundary.) The Old Testament is strikingly explicit on these matters: “Who has understood the mind of the Lord, or instructed him as his counselor ? Whom did the Lord consult to enlighten him, and who taught him the right way? Who was it that taught him knowledge or showed him the path of understanding?” (Isa. 40:13–14). And Christian theo­­ logians proceed along the same line, as per the teachings of St. Thomas Aquinas: “The knowledge that is natural to us has its source in our senses and therefore extends just as far as it can be led by sensible things. But our understanding cannot reach to an apprehension of God’s essence from these.”3 It is not simply that a more powerful mind will know quantitatively more facts than a less powerful one, but that its conceptual machinery is ampler in encompassing ideas and issues that are quantitatively inaccessible in lying altogether outside the conceptual horizon of its less powerful compeers. Now insofar as the relation of a lesser toward a higher intelligence is substantially analogous to the relation between an earlier state of science and a later state, some instructive lessons emerge. It is not that Aristotle could not have comprehended quantum theory—he was a very smart fellow and could certainly have learned. But he could not have formulated quantum theory within his own conceptual framework, his own familiar terms of reference. The very ideas rescher ign text.indd 65 12/19/08 9:45:44 AM 66 Cognitive Finitude at issue lay outside of the conceptual horizon of Aristotle’s science, and, like present-day students, he would have had to master them from the ground up. Just this sort of thing is at issue with the relation of a less powerful intelligence to a more powerful one. It has been said insightfully that from the vantage point of a less developed technology , another, substantially advanced technology is indistinguishable from magic. And exactly the same holds for a more advanced conceptual (rather than physical) technology. It is instructive to contemplate in this light the hopeless difficulties that nowadays confront the popularization of physics—of trying to characterize the implications of quantum theory and relativity theory for cosmology into the subscientific language of everyday life. A classic obiter dictum of Niels Bohr is relevant: “We must be clear that, when it comes to atoms, language can be used only as in poetry.” If the thought that is conceptualized in the operations of physical reality is, so to speak, reflective of the work of a mind more powerful than ours, then an adequate apprehension of nature will prove beyond our grasp, so that here too an unavoidable ignorance falls to our lot. rescher ign text.indd 66 12/19/08 9:45:44 AM ...