by
William MacCartney
submitted to the
Department of Philosophy at Princeton University
in partial fulfillment of the requirements for the degree of
Bachelor of Arts
30 April 1990
The generalizations of empirical science are grounded not upon demonstrative proof, but upon induction.[1] A zoologist may remark, for instance, that among the mammals (i.e., animals which suckle their young) he has observed, all had hair on their bodies. He may then conjecture that the same is true of mammals in general. If he observes a great number of instances of his conjecture, and no counter-instances, he will be confident that it is true in all cases. He will, however, have to admit two things: first, that because he has not observed all mammals, his information is insufficient to guarantee that all mammals are hairy; second, that, even if all mammals in this world should be hairy, still some mammals might not have been hairy if the world had been built differently.
The generalizations of mathematics, on the other hand, are usually grounded upon demonstrative proof. Mathematicians believe that every prime greater than 2 is odd, not because they have observed that 3, 5, 7, 11, 13, 17, etc. are all odd, but because they can prove that it is so. (Reductio ad absurdum: Suppose x is an even prime greater than 2. Since x is even, 2 is a factor of x. Since x > 2, x has a factor distinct from itself and from 1. Thus x is not prime.) Since mathematicians regard proven mathematical theorems as universally and necessarily valid, they will admit neither that a never-before-examined prime might turn out to be even, nor that a prime greater than 2 might have been even if the world had been built differently.
Between the familiar domains of empirical science and demonstrative mathematics, there exists a little-noticed shadowland which might be called 'pseudo-empirical mathematics'. Like demonstrative mathematics, pseudo-empirical mathematics is concerned with the properties of mathematical objects, but like empirical science, it proceeds not by proof but by observation and induction.[2] For instance, in 1742 Christian Goldbach wrote a letter to Euler in which he observed that, among all the even numbers he had examined, all greater than 4 were the sum of two primes. For example:
6 8 10 12 14 etc. |
= = = = = |
3 + 3 3 + 5 3 + 7 5 + 7 3 + 11 |
= = |
5 + 5 7 + 7 |
Since he observed a great number of instances of this rule, and no counter-instances, he advanced the conjecture known today as the 'Goldbach Conjecture': "All evens greater than 4 are the sum of two odd primes." We can recast this in a simpler form if we define the predicate 'gold' as follows:
Now we can write the Goldbach Conjecture as:
Mathematicians, who prefer theorems to conjectures, have tried again and again to find a proof of the Goldbach Conjecture, but for two hundred years they have met only with failure. Recently, high-speed computers have allowed researchers to verify the Goldbach Conjecture for millions of cases, most recently in 1980, when Light, Forres, Hammond and Roe verified it out to one hundred million.[3] But because mathematicians have no proof that the conjecture is true, they must admit that a counterexample might be discovered. On the other hand, no mathematician is likely to admit that, even if the Goldbach Conjecture is true in this world, still it might have been false if the world had been built differently.
Nowadays, the value of pseudo-empirical mathematics is barely acknowledged. The triumphs of formalism in securing the foundations of mathematics have encouraged the notion that formalism is mathematics. But, as any practicing mathematician can attest, mathematical advances are not the result of a blind, mechanical extraction of consequences from axioms. Mathematical progress, like scientific progress, depends upon intuition, educated guessing, and revision. Still, the formalist dogma has convinced most mathematicians that a guess does not really count as a piece of mathematics until it is baptized as a theorem. Thus the predominant view is that pseudo-empirical mathematics is not mathematics at all.
It has not always been so. Some of the greatest figures of the history of mathematics have had enormous respect for pseudo-empirical mathematics. Consider, for instance, Gauss, who regarded pseudo-empirical induction as an important and reliable path to mathematical truth: "In the Theory of Numbers it happens rather frequently that, by some unexpected luck, the most elegant new truths spring up by induction."[4] Laplace was even more enthusiastic; he considered pseudo-empirical induction to be, not merely one path among many, but the primary route to mathematical knowledge: "Even in the mathematical sciences, our principal instruments to discover the truth are induction and analogy."[5] Among all mathematicians, Euler was perhaps the greatest proponent of the value of pseudo-empirical induction. He wrote:
The properties of the numbers known today have been mostly discovered by observation, and discovered long before their truth has been confirmed by rigid demonstrations. There are even many properties of the numbers with which we are well acquainted, but which we are not yet able to prove; only observations have led us to their knowledge. Hence we see that in the theory of numbers, which is still very imperfect, we can place our highest hopes in observations.[6]
As a result of their observations, Euler and others advanced countless conjectures about the properties of numbers. Some of these conjectures were later proven to be universally valid, but hundreds remain unproven to this day. Many, however, have been verified for thousands or millions of cases with the aid of high-speed computers. Here follows a brief survey of some prominent examples:
The Odd Perfect Number (OPN) Conjecture: No odd number is perfect. (A perfect number is equal to the sum of all its positive divisors other than itself). In 1973, Peter Hagis proved that any counter-example to the OPN Conjecture must be greater than 10^{50}.[7]
Euler's 8n + 3 Conjecture: For all n > 1, every integer of the form 8n + 3 is the sum of a square and the double of a prime. Euler himself verified the conjecture for all n < 200, and D. H. Lehmer has since verified it for all n < 1000.[8]
Fermat's Last Theorem: For all a, b, c, and for all n > 2, a^{n} + b^{n} ≠ c^{n}. This misnamed conjecture has some truly remarkable inductive support. It has been shown that in any counterexample to Fermat's Last Theorem, n is not less than 125,000, and a is not less than 10^{1,800,000}.[9] [*]
The Twin Prime Conjecture: There are an infinite number of twin primes. (A twin prime differs from another prime by 2.) More than one hundred thousand twin primes are known.[10]
The Collatz Conjecture: Let f be the function on the positive integers defined by:
f(n) = { | 3n + 1 | if n is odd, | |
n/2 | if n is even. |
Then for any positive integer n, there is a k such that f^{(k)}(n), the k^{th} iterate of f on n, equals 1. The Collatz Conjecture has been verified for all n less than three trillion.[11]
Massive amounts of numerical evidence are bound to increase our confidence in unproven conjectures, but we should never forget that pseudo-empirical induction, like empirical induction, is fallible. This has sometimes been demonstrated in rather embarrassing ways. Pierre Fermat, for example, having observed that 2^{2n} + 1 is prime for n = 1, 2, 3, and 4, conjectured that the same is true for all n. He did not prove his conjecture, but he felt so sure of it that he challenged the mathematical community to prove it. But Euler found that the very next case, 2^{32} + 1, is divisible by 641.[12] Euler was embarrassed in his turn when, having proved the n = 3 case of Fermat's Last Theorem, he advanced the stronger conjecture that if n > 2 then fewer than n n^{th} powers cannot sum to an n^{th} power. His conjecture was disproven in 1966 when Lander and Parkin discovered that 27^{5} + 84^{5} + 110^{5} + 133^{5} = 144^{5}.[13]
A more recent example concerns the prime-counting functions π_{1}(x) and π_{2}(x), where π_{i}(x) is the number of primes p ≤ x such that 3 divides p − i. In 1853, P. L. Chebyshev observed that π_{2}(x) − π_{1}(x) > 0 for all small values of x, and conjectured that the same might be true for all values of x.[14] His conjecture has since been verified for all x less than a quarter of a trillion,[15] but in 1914 J. E. Littlewood had proved that π_{2}(x) - π_{1}(x) changes sign infinitely often.[16] In 1976 Bays and Hudson finally determined that the first sign-change occurs at 608,981,813,029.[17] A similar example is connected with the Prime Number Theorem, which was first conjectured by Legendre in 1798, although it was not proven until 1896.[18] The Prime Number Theorem states that π(x), the number of primes less than or equal to x, converges to li(x) as x goes to infinity, where li(x) is defined as the integral from 2 to x of 1/ln t dt. Gauss and other early researchers were struck by the fact that, for all small values of x, π(x) − li(x) is negative. It has since been shown that π(x) − li(x) is negative for all x less than one trillion,[19] and R. S. Lehman has shown that π(x) − li(x) is probably negative for all x < 10^{20}.[20] But once again it was Littlewood who proved that π(x) − li(x) changes sign infinitely often.[21] No one knows where the first sign-change occurs; but Lehman has shown that it definitely occurs below 1.65 × 10^{1165}.[22] H. M. Edwards commented, "This example shows the danger of basing conjectures on numerical evidence, even such seemingly overwhelming evidence as Lehmer's computation of π(x) up to ten million."[23] Edwards, in fact, is an unapologetic skeptic when it comes to pseudo-empirical induction. In the introduction to his book on Fermat's Last Theorem, Edwards cites the staggering inductive evidence for the conjecture, but immediately warns, "There seems to me to be no reason at all to assume that Fermat's Last Theorem is true."[24] Thus Edwards does not restrict himself to reminding us that the predictions of pseudo-empirical induction are merely probable; he questions whether they deserve any credence at all.
Edwards's skepticism, while it may seem exaggerated, is bound to plant doubts in our own minds. If pseudo-empirical induction, like empirical induction, is fallible, then why should the generalizations of pseudo-empirical mathematics deserve our confidence? If the prime-counting conjectures were proven false despite a fantastic amount of inductive support, then why should we suppose that the Goldbach Conjecture will not suffer the same fate? Or to put it more directly: how can we justify pseudo-empirical induction?
But is this question worthy of our consideration? Is it any different from the question of justifying empirical induction—a question which, after two and half centuries of debate, can fairly be called a dead horse? The answer depends on how we interpret the propositions of pseudo-empirical mathematics (e.g., evidence statements like, '0 is gold', '2 is gold', etc., and predictions like, '10^{1000} is gold'). The answer is no, if we interpret these propositions as contingent statements about the outcomes of real-world calculations. For on this interpretation, pseudo-empirical induction becomes simply a special branch of empirical induction.
But mathematicians do not normally regard propositions like '0 is gold' as contingent. They normally regard mathematical propositions, whether true or false, as necessarily true or false. They consider mathematical truths to be independent of the way the world happens to be built. This position has come under criticism from various philosophers,[25] but if we agree to interpret the propositions of pseudo-empirical mathematics as necessarily true or necessarily false, then we will discover a significant and intriguing difference between pseudo-empirical induction and empirical induction. (Thus one of the starting-points of this paper must be an agreement to assume a Platonistic view of mathematics.) We hinted at this difference in §1 when we observed that although mathematicians must admit that a counterexample to the Goldbach Conjecture might be discovered, they will not admit that even if the Goldbach Conjecture is true in this world, still it might have been false if the world had been built differently. This is another way of saying that although the conjecture could be false as far as we know, it could not be false in another possible world if it is true in this one. Thus we are distinguishing between two senses of 'could', or (to put it another way) between two concepts of contingency, which we might title epistemological contingency and metaphysical contingency. A proposition is epistemologically contingent just in case we do not know whether it is true or false; it is metaphysically contingent just in case it is true in some possible worlds and false in others. Thus most generalizations of empirical science are both epistemologically and metaphysically contingent,[26] while most generalizations of demonstrative mathematics are neither epistemologically nor metaphysically contingent. The generalizations of pseudo-empirical mathematics, however, have the virtue of splitting the two concepts of contingency. For these generalizations are epistemologically contingent but metaphysically necessary.
To see why this separation might be interesting, we should go back to Hume, who questioned the foundations of empirical induction, first in A Treatise of Human Nature (1740), and then more concisely in An Enquiry Concerning Human Understanding (1748). Hume's discussion of induction reflects our distinction between epistemological and metaphysical contingency. Kripke observes that "Hume questions two...nexuses, related to each other: the causal nexus whereby a past event necessitates a future one, and the inductive inferential nexus from past to future."[27] The first of these may be characterized as metaphysical; the second as epistemological.
Hume addresses the epistemological nexus in §4 of the Enquiry, "Sceptical Doubts concerning the Operations of the Understanding." In this section he is concerned with the question, "What is the foundation of all our reasonings and conclusions concerning [the] relation [of cause and effect]?"[28] And he does not find the foundation to be very secure. He writes:
You say that the one proposition [concerning a prediction about the future] is an inference from the other [concerning the evidence of the past]. But you must confess that the inference is not intuitive; neither is it demonstrative: Of what nature is it then? To say it is experimental, is begging the question. For all inferences from experience suppose, as their foundation, that the future will resemble the past... If there be any suspicion, that the course of nature may change, and that the past may be no rule for the future, all experience becomes useless, and can give rise to no inference or conclusion.[29]
Thus Hume would question the validity of the inference from "All previously examined mammals have been hairy," to, "All mammals are hairy." For all we know, he would say, the generalization might turn out to be false.
Hume turns his attention to the metaphysical nexus in §7 of the Enquiry, "Of the Idea of Necessary Connexion." Here he is not concerned with the connection between one proposition and another, but with the connection between one event and another:
All events seem entirely loose and separate. They seem conjoined, but never connected. And as we can have no idea of any thing, which never appeared to our outward sense or inward sentiment, the necessary conclusion seems to be, that we have no idea of connexion or power at all, and that these words are absolutely without any meaning.[30]
Hume's tone in this passage warns us that he is aware of a method of avoiding this conclusion. But when we discover what this method is, it turns out not to be particularly reassuring. It is merely that the idea of necessary connection might be derived from observing one event repeatedly conjoined with another. But since
this idea arises from a number of similar instances, and not from any single instance; it must arise from that circumstance, in which the number of instances differ from every individual instance. But this customary connexion or transition of the imagination is the only circumstance, in which they differ.[31]
Thus the idea of necessary connection is not derived from anything in the exterior world, but only from the operations of our own mind. Hume seems to be arguing, therefore, that there is no necessary connection between matters of fact.[32] That all mammals are hairy, is entirely accidental. Even if it should be true of all mammals in this world, it might not have been true if the world had been built differently.
Unfortunately not all commentators on the problem of induction seem to have been as aware as Hume of the dual nature of the problem. Many seem to ignore the difficulties concerning the epistemological nexus and to regard the problem as solved if the metaphysical nexus can be secured. Kant, for instance, attempts to refute Humean skepticism by arguing, first, that empirical reality is not independent of us but is merely the sum of our experiences; second, that nothing can be an object of experience unless it is capable of being brought under the synthetic unity of apperception by the understanding; third, that the understanding does so by placing objects of experience under the categories; fourth, that cause and effect is one such category; and finally, that empirical reality must therefore conform to the law of cause and effect.[33] But this argument, even if successful, answers only the metaphysical issue. At best, the argument establishes only that nature is governed by causal laws. It does not explain how we can discover what those laws are. (It cannot explain, for instance, why his past observations should lead the zoologist to infer that all mammals are hairy. Why should he not infer instead that all mammals are hairy if born before 2000, and scaly if born thereafter? For this too is a law.) But if the epistemological issue cannot be resolved, then the resolution of the metaphysical one would seem to provide scant comfort.
And herein lies the advantage of considering the problem of pseudo-empirical induction: one cannot confound the metaphysical and epistemological aspects of the problem. For the metaphysical issue is, so to speak, already resolved. If the Goldbach conjecture is true, it is no accident. That 0, 2, 4, 6... are gold is necessarily bound to the goldness of an unexamined number like 10^{1000}. Thus the spotlight is cast fully on the epistemological issue: how can I justify the inference from any finite amount of mathematical evidence to a completely general conjecture?
The question how we can justify pseudo-empirical induction is so nebulous that it is difficult even to see what sort an answer the question demands. What we need, then, is a more precise formulation of the question. Since our question is so similar to the question of justifying empirical induction, a quick survey of the sorts of answers which have been proposed for this related problem may help us better to understand the nature of our own problem.
Most attempts to answer Hume's inductive skepticism argue, following Kant, that nature must conform to some necessary and universal principle of regularity. But, as we have seen, this sort of argument can answer only Hume's doubts concerning the causal nexus; it does not seem to address doubts about the inductive nexus at all. And since it is only with this last nexus that our problem is concerned, it seems unlikely that this argument could help us. (That mathematics is orderly is beyond doubt; the question is how to discover what this order is.)
On the other hand, we might hope the principle of the uniformity of nature could secure the inductive nexus if it, in combination with the evidence statements of empirical science, entailed the predictions of empirical science. In other words, we might hope to use the principle of the uniformity of nature to transform inductive inferences into deductive ones. But whether or not such a project could ever succeed in the empirical sphere, it does not seem that it could in the pseudo-empirical sphere. For how shall the premise itself be supported? If the premise is worded in vague terms like
then it might plausibly claim to be self-evident. But we cannot infer immediately from (1) and one hundred million evidence statements that the Goldbach Conjecture is true.[34] On the other hand, it doesn't look as if it will help to word the premise more precisely. For if the premise were given any form precise enough to allow us immediately to infer (with the aid of the evidence statements) the truth of the Goldbach Conjecture, such as
then the premise itself would demand justification. Now perhaps (2) could be demonstratively proved, but even this would not help us. For if we had a proof of (2), we would have a proof of the Goldbach Conjecture itself. But in the first place, if our confidence in the truth of the Goldbach Conjecture had no justification short of proof, then, since we have no proof of it, it would be entirely unjustified. And I do not think we would want to admit that. In the second place, if we did have a proof of the Goldbach Conjecture, the question of justifying the pseudo-empirical inductive inference to the conjecture would become moot. Who would care whether it were justified or not?
In his book Fact, Fiction and Forecast, Nelson Goodman takes a very different approach to the problem of justifying the inferences of empirical induction. He suggests that we consider how we justify a deduction. "Plainly," he writes, "by showing that it conforms to the general rules of deductive inference." These rules, in turn,
are justified by their conformity with accepted deductive practice. Their validity depends upon accordance with the particular deductive inferences we actually make and sanction.... This looks flagrantly circular.... But this circle is a virtuous one. The point is that rules and particular inferences alike are justified by being brought into agreement with each other.... In the agreement achieved lies the only justification needed for either.[35]
Thus a deductive inference is justified by showing that it conforms with our accepted deductive practice, and the task of justifying deductive inferences is nothing but the task of defining the difference between valid and invalid deductions. Since no other path seems to be open, I suggest that the same is true of the task of justifying pseudo-empirical induction, which we have set ourselves in this paper—namely, that it is just the task of defining the difference between valid and invalid inferences of pseudo-empirical induction.
Note that by 'valid' and 'invalid' we do not mean 'true' and 'false'. For an infallible method of distinguishing true conjectures from false ones would constitute a proof of the Goldbach Conjecture, the OPN Conjecture, Fermat's Last Theorem, etc. (that is, provided that these are true). And as we noted above, we do not want to admit that only a proof of a conjecture can justify the pseudo-empirical inductive inference to that conjecture. By 'valid' we mean simply 'intuitively acceptable'.
The problem of defining valid pseudo-empirical induction has many aspects. One question which must be answered is how much evidence, and what sort of evidence, is required to validate a conjecture. But we should recognize that some conjectures will not be validated by any amount of evidence. An example is:
Even if this conjecture should be verified for the first million digits of π, we would not accept it as a valid conjecture. That '7777' did not occur in the first million digits we would regard as merely an accident, albeit a rather surprising one. We would hardly conclude that the sequence never occurs. For we do not regard (3) as lawlike. Thus another question which must be answered is, how may we define the difference between lawlike and unlawlike mathematical conjectures? It is to this question that this paper will devote its primary attention.
In formulating the problem of pseudo-empirical induction, we have borrowed a great deal from Goodman's formulation of the problem of empirical induction. Like Goodman, we do not think the problem is "to find some way of distinguishing antecedently between true and false predictions".[36] Rather, we see the problem as "a problem of defining the difference between valid and invalid predictions."[37] Moreover we, like Goodman, are primarily concerned with the task of defining the difference between lawlike and unlawlike generalizations. Since Goodman's problem is so similar to our own, a glance at his approach to his problem may shed some light on our own problem.
Goodman was not the first to attempt to define the difference between valid and invalid hypotheses. Hempel and Carnap, among others, had already made significant contributions in this area. But their theories tried to determine the validity of a hypothesis solely on the basis of the amount and nature of the evidence for or against it. What Goodman pointed out was that some hypotheses, because they are not lawlike, cannot be validated by any amount of evidence. Thus any satisfactory definition of valid induction must exclude unlawlike hypotheses. And this is not simply a matter of ruling out a few aberrant cases. For Goodman showed that every well-supported hypothesis has an equally well-supported rival which is not lawlike.
To show this Goodman introduced the predicate 'grue', which applies to "all things examined before [the year 2000] just in case they are green but to other things just in case they are blue."[38] Now it is a consequence of the definition of 'grue', that all examined emeralds which have been green have also been grue. Thus the evidence for the hypothesis that
is no better and no worse than the evidence for the hypothesis that
Now (4) entails that emeralds first examined after the year 2000 will be green, while (5) entails that they will be blue. Naturally we feel confident that (4) is, and (5) is not, a valid projection. But the evidence alone cannot explain the difference in validity. Goodman, therefore, is concerned to explain what beyond the evidence could justify us in projecting (4) rather than (5). And the stakes are high, for every predicate will have a grue-like counterpart, and every hypothesis will have a grue-like rival. If we cannot justify our confidence that emeralds first examined after 2000 will be green, then we cannot justify any predictions about the future at all.
The same sorts of worries can be raised about the conjectures of pseudo-empirical mathematics. On the model of the predicate 'grue', let us introduce the predicate 'gase', which will apply to an even number just in case it is gold and is not equal to some colossal even number g (let's say, 10^{1000}), or is base and is equal to g (where a number is base just in case it is not gold). Since g has (we may safely assume) never been examined for goldness, all examined evens which have been gold have also been gase. Thus the evidence for the Goldbach Conjecture will be no better and no worse than the evidence for the Gasebach Conjecture, which claims that all evens are gase. Although we are confident that the Goldbach Conjecture is, and the Gasebach Conjecture is not, a valid conjecture, the evidence alone cannot explain this difference. Since every mathematical conjecture will have a gase-like rival, we must either find a way to justify our preference for the Goldbach Conjecture over the Gasebach Conjecture, or admit that we cannot justify any mathematical conjectures at all.
Now in the empirical example there is nothing we can do ahead of time to prove that emeralds first examined after 2000 will be green. We simply have to wait and see. But the same is not true of the mathematical example. If g really is gold, then we could (in principle) prove it so any time we wanted to, just by doing a few calculations. And proving that g is gold would certainly constitute one way of justifying our preference for the Goldbach Conjecture over the Gasebach Conjecture. So it seems that there is a justification available in the mathematical example which is not available in the empirical example. But just as (4) is challenged, not only by the hypothesis that emeralds first examined after 2000 will be blue, but also by analogous hypotheses for 2001, 2002, 2003, etc., so the Goldbach Conjecture will be challenged, not only by the conjecture that all numbers except g are gold, but also by analogous conjectures for g + 2, g + 4, g + 6, etc. In fact, for every set S of even numbers, we may define the predicate 'gase_{S}' as follows:
It is natural to hope that the solution Goodman provided to his own problem will offer a way out of our difficulties as well. What follows, then, is a quick summary of Goodman's Theory of Projectibility, which he introduced to solve the 'grue' paradox.
Although Goodman is primarily concerned with defining the difference between lawlike and accidental hypotheses, he frames his task as the task of defining 'projectible', which he says applies to a hypothesis if "the actual evidence supports and makes it credible."[40] Thus lawlikeness seems to be implied by projectibility. In contrast to Carnap and Hempel, Goodman intends to avail himself, not only of the evidence for or against the hypothesis, but also of the record of actual past projections.[41]
Goodman first eliminates as unprojectible all hypotheses which either have no positive cases (i.e., are unsupported), or have negative cases (i.e., are violated), or have no undetermined cases (i.e., are exhausted). The next step is to rule out all hypotheses which, though supported, unviolated, and unexhausted, are not lawlike. To accomplish this, Goodman introduces the concept of entrenchment. A predicate 'P' is said to be better entrenched than a predicate 'Q' if 'P' is the veteran of earlier and more past projections than 'Q'. (A predicate may also derive entrenchment through being co-extensive with some well-entrenched predicate, or through having a well-entrenched parent predicate.[42]) Likewise a hypothesis H is said to be better entrenched than a hypothesis H' if one of the predicates of H is much better entrenched than its counterpart in H', and the other is no worse entrenched than its counterpart in H'. H is said to override H' if the two conflict[43] and H is the better entrenched and is not itself overridden. We may now define:
Thus in any group of mutually conflicting hypotheses, either one will be (presumptively) projectible and the rest unprojectible, or some will be nonprojectible and the rest (if any) unprojectible. In the latter case, conflicts among nonprojectibles may be resolved through weighing the comparative projectibility inherited by each. Though a nonprojectible hypothesis has no presumptive projectibility, its final index of projectibility may be increased if it has a projectible positive overhypothesis,[44] or decreased if it has a projectible negative overhypothesis.[45] The effect of an overhypothesis on comparative projectibility is determined not only by its projectibility, but also by the amount of its supporting evidence and by its specificity.[46]
Every Gase_{S}bach Conjecture (including the Goldbach Conjecture) conflicts with every other, and all are unexhausted. Many, however, are violated or unsupported. But even if we eliminate every violated or unsupported Gase_{S}bach Conjecture,[47] we are left with an infinity of supported, unviolated, unexhausted, mutually conflicting conjectures. If we are to use Goodman's theory to justify our preference for the Goldbach Conjecture, we must show either (1) that the Goldbach Conjecture is presumptively projectible (in which case every other Gase_{S}bach Conjecture would be unprojectible), or (2) that the Goldbach Conjecture is nonprojectible, but it has a higher comparative projectibility than any other (unviolated) Gase_{S}bach Conjecture.
To prove that the Goldbach Conjecture is presumptively projectible requires that we prove 'gold' to be better entrenched than any other 'gase_{S}'. Intuitively, it seems that, if Goodman's theory can be applied to pseudo-empirical mathematics at all, then 'gold' must be well-entrenched. In the first place, we feel confident that the Goldbach Conjecture is projectible, and if Goodman is right, then this can only be so if 'gold' is well-entrenched. In the second place, 'gold' (like 'green') seems like a "normal" predicate, while its rivals (like 'grue') have an air of artificiality. But when we attempt to account for this intuited entrenchment within the framework of Goodman's theory, we run into difficulties. As we noted in §6, Goodman lists three sources from which a predicate may derive its entrenchment. First, it may have earned its entrenchment directly through repeated projection. Second, it may be co-extensive with another predicate which is well-entrenched. Third, it may inherit entrenchment from a well-entrenched parent. Unfortunately none of these sources seem able to account for the intuited projectibility of 'gold'.
First, 'gold' would not seem to have a very impressive history of projection, since it was only invented for the purposes of this paper. Since 'gold' is a coined abbreviation, the second possible source of entrenchment (a well-entrenched co-extensive predicate) would seem to be more promising. Indeed it was just such coinages which impelled Goodman to provide this source of entrenchment. We know 'gold' is co-extensive with 'six less than the sum of two odd primes', but how can we explain whence the entrenchment of this predicate should derive? For it would not seem to be a veteran of many actual projections. Indeed, in a mathematical context it is not even clear what we should mean by 'actual projection'. Should we call a mathematical generalization actually projected when it appears in a proof? Goodman's definition of actual projection (see note 41) would seem to exclude this possibility, but if we count only unproved and overtly adopted conjectures as actually projected, then precious few (relatively speaking) mathematical predicates would seem to have any entrenchment at all. Certainly an odd predicate like 'six less than the sum of two odd primes' has rarely, if ever, been projected outside this paper.
The reader will rightly object that our difficulties are only the result of our decision to recast the Goldbach Conjecture in the form of a conjecture about all even numbers. 'Six less than the sum of two odd primes' may never have been projected outside this paper, but the predicates of the original form of the conjecture, 'even greater than 4' and 'the sum of two odd primes', have each been projected thousands of times in the long history of the Goldbach Conjecture. True enough, but can a predicate really gain entrenchment merely through the repeated projection of one hypothesis in which in occurs? Isn't it necessary that the predicate occur in many different actually projected hypotheses? Now perhaps 'even greater than 4' and 'six less than the sum of two odd primes' have occurred in other mathematical projections (although I am unaware of any examples) and thus can claim to be entrenched. But even if the entrenchment of 'gold' could be thus explained, it would be easy enough to produce many other apparently projectible conjectures whose predicates have never occurred anywhere else.
Is there any other predicate with which we know 'gold' to be co-extensive? Well, come to think of it, what is the extension of 'gold'? We believe 'gold' is co-extensive with 'even' (a mathematical predicate which would seem to have as good a claim to independent entrenchment as any other), but we are not sure. That belief, in fact, is just what the Goldbach Conjecture expresses. Thus only if we assume that the conjecture is true can we borrow from 'even' the entrenchment required to make the conjecture projectible. But of course if we assume that the conjecture is true, the question of its projectibility becomes moot. Nor can we hope that 'gold' can be shown to be co-extensive with some other well-entrenched predicate previously unsuspected to be equivalent it. For Goodman admits that only known equivalences can yield entrenchment.[48]
Nor does the possibility of inherited entrenchment seem likely to solve our difficulties. A parent predicate of 'gold' must apply to the extension of 'gold', but we do not know what this extension is. Of course, a predicate like 'set of even numbers' is a parent of 'gold', and if it has any independent claim to entrenchment, 'gold' will inherit a certain amount of entrenchment. But every 'gase_{S}' will inherit an equal amount. No inherited entrenchment can render the Goldbach Conjecture projectible unless it derives from a predicate of which 'gold' is the unique child among its rivals. We believe 'set of all even numbers' is such a predicate (and it may well be entrenched), but we are not sure. And as before, the assumption that the belief is true renders moot the question which the assumption was intended to settle. Nor does there appear to be any other way finding a suitable parent for 'gold' without knowing its extension.
If Goodman's theory has so far failed to explain the apparent projectibility of the Goldbach Conjecture, we might begin to imagine that it is only because the Conjecture is not presumptively projectible but merely comparatively projectible. That is, 'gold' may not surpass some or all of its rivals in entrenchment, but there may be some overhypothesis which lends projectibility to the Goldbach Conjecture. But in searching for an appropriate overhypothesis we will run into the same difficulties we faced in the last paragraph. Clearly we need an overhypothesis which is not also an overhypothesis of some other Gase_{S}bach Conjecture. But we cannot locate such an overhypothesis unless we have a predicate of which 'gold' is the unique child among its rivals.[49]
So it seems that Goodman's theoretical framework is unable to explain the apparent projectibility of the Goldbach Conjecture. But this failure shows neither that the Goldbach Conjecture is not projectible nor that the framework is a failure, for the framework was never intended to cover the comparatively unusual case of pseudo-empirical induction. Nor does this failure mean that Goodman's theory can have nothing to say about pseudo-empirical induction. For it may be that in paying such close attention to the fine points of the theory of projectibility, we have missed the forest for the trees. Even if pseudo-empirical induction does not conform to the letter of Goodman's theory, it may yet conform to the spirit.
What, then, is the spirit of Goodman's theory? He first states his underlying thesis when he writes that inductive inferences, like deductive inferences, "are justified if they conform to valid canons of induction; and the canons are valid if they accurately codify accepted inductive practice."[50] Thus what justifies inductive inferences—or, to put it another way, what separates valid inferences from invalid ones—is not conformity to as-yet-undetermined truths (e.g., concerning the color of emeralds first examined after 2000), but merely conformity to the inferences we actually make and accept. Goodman later states his thesis more succinctly: "the roots of inductive validity are to be found in our use of language."[51] "Somewhat like Kant," he writes elsewhere, "we are saying that inductive validity depends not only upon what is presented but also upon how it is organized; but the organization we point to is effected by the use of language and is not attributed to anything inevitable or immutable in the nature of human cognition."[52] Since Goodman regards the problem of justifying induction as the problem of separating valid inferences from invalid ones, this is just another way of saying that "the line between valid and invalid predictions...is drawn upon the basis of how the world is and has been described and anticipated in words."[53]
Now perhaps the same is true of pseudo-empirical induction. Perhaps the line between valid and invalid mathematical conjectures is drawn upon the basis of how the mathematical world is and has been described and anticipated in words. The last section was an attempt to specify the precise connection between the past use of mathematical language and the validity of current conjectures. But even if that attempt failed, we may still hold that there is a connection. That is, we may still believe that our intuition, that 'gold' is in some sense a more natural predicate than its rivals, derives somehow from our past use of mathematical language. And in fact we seem to have good reason to do so. For the Gase_{S}bach Conjectures show that no amount of computational evidence can distinguish valid conjectures from invalid ones. But we do not want to admit that the only thing which could distinguish the Goldbach Conjecture as valid would be its as-yet-undemonstrated truth. For we like to believe that we are justified in believing it even though we cannot prove it. And if computational evidence and demonstrative proof are both excluded, it is difficult to see what besides linguistic history could separate valid projections from invalid ones.
But how seriously can we entertain the idea that different ways of talking about mathematics may lead to different prejudices about what conjectures are most justified? We might be willing to admit, with Goodman, that this sort of thing happens with empirical hypotheses. But we may be less comfortable when we cross into mathematics. For this view suggests that it is merely an accident of our linguistic history that we find the Goldbach Conjecture a better projection than every other Gase_{S}bach Conjecture. Moreover, it suggests that, if our linguistic behavior were to change, then 'gase_{S}' might eventually begin to seem more natural than 'gold', and the corresponding Gase_{S}bach Conjecture might become more natural, and indeed more justified, than the Goldbach Conjecture.
But isn't this an abhorrent conclusion? Isn't there really a fact of the matter about which conjecture is true, even if we don't know for sure what it is? And wouldn't the Goldbach Conjecture still be the right one, even if our linguistic behavior were to change?
It might seem that a similar worry could be raised in the case of empirical induction. Goodman's theory says (more or less) that our justification for projecting 'green' rather than 'grue' consists merely in the fact that we actually do project 'green'. But, as Sydney Shoemaker has pointed out, this answer is unlikely to "afford relief to anyone who is troubled by skeptical doubts about induction."[54] For such a person will surely wonder, "Why are we justified in projecting the predicates which we actually do project?" And this worrier might find it very disturbing that Goodman's theory seems to imply that if enough people began regularly to project 'grue', rather 'green', then 'grue' would eventually become more firmly entrenched, and projections involving 'grue' would be more justified. For aren't projections involving 'green' still the right ones, no matter how people talk?
Now Shoemaker remarks that this does not constitute a criticism of Goodman, since Goodman's account was not intended to afford such relief, and since skepticism is of little interest to him. But in fact Goodman answers just this objection. For someone who wonders why we are justified in projecting the predicates which we actually do project must have slept through Goodman's discussion of justification.[55] And someone who insists that projections involving 'green' are the right ones, no matter how people talk, must have missed Goodman's discussion of that very issue, which concludes, "The reason why only the right predicates happen so luckily to have become well entrenched is just that the well entrenched predicates have thereby become the right ones."[56]
But when we cross over into pseudo-empirical induction, Goodman's reassurances may fail to reassure. For if (as we agreed in §3) we adopt a Platonistic view of mathematics, it does seem as if some Gase_{S}bach Conjecture must be true, and thus the right one, no matter how people talk. How, then, can we apply to pseudo-empirical induction Goodman's justification, which, although it seems to be the only justification possible, also seems to imply that if our linguistic behavior were to change, then some Gase_{S}bach Conjecture might become more justified than the Goldbach Conjecture?
In his article "On Projecting the Unprojectible", Shoemaker attempts to relieve analogous worries about empirical induction. He argues that no one really could begin to project 'grue' rather than 'green'. Even if someone began to make projections involving the word 'grue', he says, nothing would count as good evidence that the person had a coherent projective policy which was truly (and not merely verbally) different from our own. If this is so, then the danger of 'grue' outstripping 'green' in entrenchment is illusory.
Shoemaker's argument may seem to offer a way out of our own worries. For if we can construct a parallel to his argument, replacing 'green' with 'gold' and 'grue' with 'gase',[57] then we may be able to show that no one really could begin to project 'gase' rather than 'gold'. Because of the considerable differences between the empirical and mathematical contexts, our parallel cannot be completely faithful. At times we will have to invent entirely new arguments to prove (the analogs of) Shoemaker's claims. But our argument will be just like his in that it will attempt to show that, in any situation in which we had good reason to believe that someone had adopted a 'gase'-projecting policy which was truly (and not merely verbally) different from our own, that policy would be incoherent. And, like Shoemaker, we will proceed by attempting to tell a story in which one character has a coherent 'gase'-projecting policy which is truly different from the projective policy of another character, and, at each successive failure, revising the first character's policy, until we have exhausted the possibilities.
At this point someone may interject, "Well, look, the question whether 'gase' projection is coherent just is the question whether the Gasebach Conjecture is true. For if the Gasebach Conjecture is true, then of course 'gase' projection is coherent, but if it isn't, then there will be some hidden inconsistency in any policy of 'gase' projection." In a sense, this is perfectly true, but it is rather beside the point. What we want to try to do is to prove 'gase' projection to be inconsistent without having to prove the Gasebach Conjecture—or, to be more precise, without having to prove every Gase_{S}bach Conjecture and the Goldbach Conjecture to be true. For, as we noted in §4, we do not want to admit that our confidence in the truth of the Goldbach Conjecture has no justification short of proof.
Imagine two mathematicians, Alpha and Beta. Neither Alpha nor Beta has ever tested g for goldness. The sensible Alpha, however, has a projective policy just like our own, and thus feels quite confident that the Goldbach conjecture is true (although he cannot prove it). Thus he predicts that when g is tested, it will be found to be gold. Beta, on the other hand, insists that the Gasebach conjecture is true (although he is equally unable to prove his contention). He therefore predicts that g will turn out to be base.
So far we have said nothing about the rest of Beta's projective policy. In determining the rest of his policy, Beta must make choices about how far his policy will diverge from Alpha's. First, he must decide whether to project 'gase' systematically (i.e., wherever Alpha projects 'gold') or selectively. If he chooses the former, he must decide whether to 'gase-ify' his projective vocabulary systematically (i.e., to replace all Alpha's predicates with predicates related to them as 'gase' is to 'gold') or selectively. Paralleling Shoemaker, we will attempt to argue that: (1) no policy of selective 'gase' projection is coherent; (2a) if Beta adopts any policy of systematic 'gase' projection with selective gase-ification, then the best explanation of his behavior is that he just means by 'gase' what Alpha means by 'gold' and thus that his projective policy is not truly different from Alpha's; (2b) any such policy is incoherent anyway;[58] and (3) if Beta adopts the only coherent policy available—a policy of systematic 'gase' projection and systematic gase-ification—we still cannot suppose that he means by 'gase' what Alpha means by 'gase'.
Before we begin, we must provide ourselves with a stock of raw material for our examples. Just as, in §5, we recast the Goldbach Conjecture in the form of a conjecture about all even numbers through the definition
so it will now be convenient likewise to recast the OPN Conjecture and Euler's 8n + 3 Conjecture. To this end, let us adopt the following definitions:
These definitions may be forgotten; what is important is that the three conjectures may now be re-written as follows:
It should be noted that each of these three predicates has the feature that there exists a test, guaranteed to terminate in a finite number of steps, which will determine, for any x, whether that predicate applies to x.
Suppose Beta takes the simplest option and adopts a projective policy which differs from Alpha's only in that he projects the Gasebach Conjecture (and its obvious consequences) rather than the Goldbach conjecture (and its obvious consequences).[59] Shoemaker's demonstration of the incoherence of this sort of policy begins, "If Mr. B follows this policy he will project the generalization (G1) 'Things visually indistinguishable from green things are themselves green.'"[60] Now how shall we parallel 'visually indistinguishable from green things' in our example? An obvious suggestion is 'goldtests like some gold even', where a 'goldtest' of x is a procedure (guaranteed to terminate) which checks all pairs of primes less than x + 6. If some such pair sums to x + 6, then result of the test is positive; otherwise it is negative.[61] If we adopt this suggestion, our parallel to (G1) will be,
Unfortunately, this suggestion will not work. Unlike (G1), (6) cannot be projected, for it is not epistemologically contingent. Since it is obviously true for all cases, there can be no question of supporting it inductively. A true parallel to (G1) must be epistemologically contingent, and yet must be well-supported and unviolated. These requirements suggest that we may be able to create a parallel by constructing a new predicate which involves another of the unproven conjectures mentioned in §2.
Let 'opengold' mean 'if open, then gold' (that is, 'either gold or not open'). Since the tests for openness and goldness are guaranteed to terminate, there also exists a terminating opengoldtest. So we may parallel (G1) by:
Note that (7), like the rest of the unproven conjectures, is well-supported, unviolated, and epistemologically contingent (in fact, it an obvious consequence of the OPN conjecture). So both Alpha and Beta will project it. Now our parallel to Shoemaker's argument can proceed smoothly:
Because it is well-supported, unviolated, and epistemologically contingent, both Alpha and Beta will also project
But, taken together, (7) and (8) imply that if g is open, then it is gold. (For if g is open, then it is alike in openness to 0, and thus—by (8)—will opengoldtest like 0. But since 0 is a gold even, we may conclude—by (7)—that g is gold.) Now, part of our story is that Beta's projective policy is as similar as possible to Alpha's. So Beta, like Alpha, will project the OPN Conjecture, and hence must admit that g is open. But the inevitable conclusion, that g is gold, conflicts with Beta's Gasebach Conjecture. Thus his projective policy is incoherent.[62]
Obviously we cannot solve the problem by selectively widening the use of 'gase' in Beta's projective canon, for the same argument can be made in a thousand ways. For example, it will do no good to replace (7) by
for we need only repeat the argument with
(And, in any case, we can only deny (7) if we deny the OPN Conjecture.)
We might also try to solve the problem through selective (i.e., limited) gase-ification of Beta's projective vocabulary. So, for instance, we might try changing (8) to
where 'opengasetest' has the obvious definition. But we cannot replace 'opengoldtest' by 'opengasetest' in both (7) and (8), or else the conclusion again follows that if g is open, then g is gold. But if (7) rests unchanged, we can construct the following argument: just as we constructed the opengoldtest to be a reliable, but not obviously certain, test of goldness, so we must now imagine a reliable, but not obviously certain, test for the holding of the relation 'opengoldtests alike'. Perhaps an (if x and y both oily, then x and y opengoldtest alike)-test would do. Let us call this test the 'trickytest'. Now both Alpha and Beta will project
But these, together with (7), once again imply that if g is open, then g is gold. And, as Shoemaker points out, it does not seem likely that we can help matters by supposing Beta to project, in some cases, a predicate gase-like with respect to 'trickytests alike'.
There might seem to be another possible solution (one which does not arise in Shoemaker's argument): Beta may be able to retain both (7) and (8) if he abandons the OPN Conjecture and adopts instead
where
For then Beta will not admit g to be open, and thus will be able to accept the consequence of (7) and (8), that if g is open, it must be gold.
Of course, since the same argument can be repeated with (10) and (11), Beta must also abandon Euler's 8n + 3 Conjecture, and adopt instead
where
And so on for all other conjectures.
But is this policy coherent? Can Beta maintain that all evens are gold, open, oily, etc. except g—or, to be more precise, except the members of S, which are none of these? That is, can Beta maintain that each of the conjectures has exactly the same set of counterexamples? It seems highly implausible.
In fact, we can prove that it cannot be so. First, recall that when we recast Euler's 8n + 3 Conjecture as a conjecture about all evens, our definition of 'oily' depended upon a mapping of the numbers 0, 2, 4, 6, 8... directly onto the numbers 19, 27, 35, 43, 51... through the function f(x) = 4x + 19. But of course this is not the only way to map the one set onto the other in such a way that there is a one-to-one correspondence between the two. I could also map 0, 2, 4, 6, 8... onto 27, 19, 35, 43, 51.... Or I could map 0, 2, 4, 6, 8... onto 35, 27, 19, 43, 51.... In general, we may define a family of functions f_{i} given by:
f_{i}(x) = | { |
4i + 19 19 4x + 19 |
if x = 0, if x = i, otherwise |
Now we may define the family of predicates oily_{i} given by:
Now for any i, f_{i} will map i onto 19, and since 19 is the sum of a square and the double of a prime, i will be oily_{i}. So now we can argue from
that, for any i ∈ S, since i is oily_{i}, it must be gold, in contradiction with the Gasebach conjecture.
Since no policy of projecting 'gase' selectively is coherent, we must consider the possibility of projecting 'gase' systematically. Here the simplest option will be for Beta to adopt a policy which differs from Alpha's only in that he projects 'gase' wherever Alpha projects 'gold'. Shoemaker argues that if Beta adopts such a policy, then the best explanation of his behavior will be to suppose that he means by 'gase' what Alpha means by 'gold', so that he does not really have a different projective policy at all. We may construct an analog to Shoemaker's argument as follows:
Alpha, as we may well imagine, is desperate to prove conclusively that g is gold. Like us, he regards Beta's projective policy as perverse, but he can find no philosophical argument for the unreasonableness of Beta's policy. Moreover he knows that g is too large for any computer ever to test. He concludes that his only recourse is prayer. He begins to beg God to tell him whether g is gold or base. Finally his prayers are answered—sort of. One day while he and Beta are out for a stroll, the heavens open and an angel appears before them, bearing two black boxes. The angel identifies the boxes as an opengoldtester and an oilygoldtester.
The mathematicians have unshakable faith that the boxes perform as advertised, and with complete accuracy. While they might have preferred that God had simply given them a goldtester, they decide not to look a gift horse in the mouth. After all, if the OPN Conjecture is right, then an opengoldtester is as good as a goldtester, and likewise for the oilygoldtester. Since both conjectures are very well supported, both tests are reliable (though not clearly infallible) indicators of goldness.
Breathless with excitement, Alpha enters g into the opengoldtester. A few moments later, a green light comes on. g opengoldtests positive! Alpha is pleased, although not very surprised. "Look," he says, "g opengoldtests positive, and you have to admit that the opengoldtest is a reliable indicator of goldness. So g is gold! Ha ha, guess you feel pretty stupid, huh?"
"Not really," Beta says calmly.
"Still not convinced?" Alpha asks. "Let's try the other test." He enters g into the oilygoldtester, and soon a green light comes on. "That proves it," says Alpha. "Just admit it: I was right and you were wrong."
"On the contrary," Beta says. "g is gase—and these results only confirm my predictions. You have forgotten that I always project 'gase' where you project 'gold'. One thing that was true of all small gase evens was that they opengoldtested positive. So of course I expected this one to opengoldtest positive. In the same way, I expected g to oilygoldtest positive. Your conviction that testing g would vindicate one of us and frustrate the other has always amused me. I realized all along that, by the nature of our projective policies, we would be vindicated or frustrated together."
But if Beta calls 'gase' whatever Alpha calls 'gold', and moreover bases these ascriptions on the same criteria as Alpha (viz., the opengoldtest and the oilygoldtest), then surely the most plausible hypothesis for Alpha to adopt is that Beta means by 'gase' what Alpha means by 'gold'—no matter what he says he means. If Beta should claim that he, like Alpha, uses 'x is gase' to mean 'x ∉ S and x is gold, or x ∈ S and x is base', then Alpha should infer that Beta just uses 'gold' to mean 'gase' and uses 'base' to mean 'bold'.[63]
If this explanation is correct, then we have failed to produce an example where Beta has a different projective policy than Alpha. The apparent conflicts between Beta's projections and Alpha's are merely verbal conflicts. Beta may say, "I project the Gasebach Conjecture," but what he really means is, "I project the Goldbach Conjecture." The example fails because it does not satisfy what we might call the 'same-ascriptions requirement'. Unless Beta ascribes 'gold' and 'gase' to the same evens to which Alpha ascribes them, we cannot suppose that he means by 'gase' what Alpha means by 'gase', and thus we cannot attribute to Beta a projective policy different from Alpha's.
Not only does this story fail to satisfy the same-ascriptions requirement, but the projective policy it attributes to Beta is outright incoherent, in two ways. The first of these incoherences does not arise in Shoemaker's original empirical example. It is this: since g open- and oilygoldtested positive, if Beta maintains that g is base, then he must deny that g is either open or oily. Or, to be more precise, Beta must hold that the three conjectures have exactly the same set of counterexamples, namely S. Not only is this highly implausible, but a small change in our story (the angel gives Alpha and Beta a full complement of oily_{i}goldtesters) renders it outright inconsistent, as we showed at the end of the last section.
Shoemaker does acknowledge the second incoherence, and in fact bases the further development of his argument upon it. It is this: Just as green and blue are colors, let us call gold and base 'lusters'. Now if Beta's projective policy differs from Alpha's only in that he replaces 'gold' by 'gase', then he, like Alpha, should project
But he cannot do so coherently. For if he does, he will have to predict that g will be alike to 0 in luster. But he also predicts that g will be gase, and so base. So he will have to predict that a gold even (viz., 0) will be alike in luster to a base even (viz., g). And this is inconsistent.
In order to avoid this difficulty, we might have Beta project 'schmuster' wherever Alpha projects 'luster', where two evens are alike in schmuster just in case both are gase or both are bold. Then Beta would project
And this solution might, at first glance, seem not only to eliminate the second incoherence but also to bring about the satisfaction of the same-ascriptions requirement. For now it seems that if Alpha finds g to be gold, Beta will agree with him. If g is alike in luster to a gold even, then it would be alike in schmuster to a (hypothetical) non-g base (and hence bold) even. So it seems that Beta should call g bold, and hence gold, thereby agreeing with Alpha.
But this line of reasoning (Shoemaker claims) is mistaken, for there's no reason to think that Beta will find g alike in schmuster to a non-g base even. Just the opposite is true. For since Alpha projects
Beta should project
Now since g opengoldtests like 0, Beta will call it alike in schmuster to 0, and hence call it gase. Once again the same-ascriptions requirement has been violated. Given Beta's ascriptions and the criteria on which they are based, Alpha should assume that Beta's 'gase' just means 'gold' and Beta's 'schmuster' just means 'luster'.
We might be tempted to resort to further gase-ification of Beta's projective vocabulary in order to satisfy the same-ascriptions requirement. We might, for instance, like to replace 'opengoldtest' by 'opengasetest' in Beta's projections. But this move will land Beta in difficulties with the 'trickytest' presented in §10. And once again, it does not seem likely that it will help matters to introduce a predicate which is gase-like with respect to 'trickytests alike'. (And, in any case, Beta will still be forced to claim that all the conjectures have the same set of counter-examples.)
At this point we might begin to wonder about the source of our difficulties. Shoemaker argues that, in his empirical example, the source of the difficulties is that
what properties we take observed items to have, and so what generalizations we take them to be positive instances of, is determined in part by what generalizations we already accept. For the generalizations we already accept determine what we count as evidence concerning what properties things have. Let us call generalizations that play this role "evidential generalizations." ...If in our thought experiments we tamper with inductive policies...we have got to expect this to affect their judgments about...objects. And this interferes with the satisfaction of the [same-ascriptions requirement].[64]
Whether these claims are also true of pseudo-empirical induction is a topic to which we shall return.
We seem to have shown that piecemeal gase-ification of Beta's projective vocabulary cannot (as it first seemed) bring about the satisfaction of the same-ascriptions requirement. But perhaps a massive dose of gase-ification, introduced all at once rather than in stages, could succeed where piecemeal gase-ification failed. Let us, then, examine the consequences of stipulating that the same-ascriptions requirement be met and gase-ifying as necessary to preserve this stipulation.
If it is true that "what generalizations we accept affects what properties we ascribe to observed items," then, as Shoemaker writes,
It might seem to follow...that if someone systematically projected predicates that are "grue"-like relative to predicates we project, he would necessarily differ from us in some of the evidential generalizations he accepts and hence in some of judgments about items first observed after t... If this were so, it would follow immediately that the [same-ascriptions requirement] cannot be satisfied. But this line of thought is mistaken. It assumes that if two people differ about the projectibility of a generalization, they necessarily differ about the acceptability, and the truth, of the generalization. And this is not so.[65]
Shoemaker gives as an example "All emerires are grue," which, since it is not supported by the fact that all so far examined emerires have been grue, is not projectible, but nevertheless is acceptable. He continues,
Hence, even if Mr. B's projective vocabulary were entirely gruified relative to Mr. A's, it still might be the case that a number of the generalizations Mr. A projects translate into generalizations Mr. B accepts, and [vice-versa]. And [this] might be enough to bring about the [satisfaction of the same-ascriptions requirement].[66]
Like Shoemaker, we will alter our example somewhat in order to avoid certain complications. First we observe that there may be more than one pair of odd primes which sum to a certain even. 10, for instance, is both 3 + 7 and 5 + 5. So let us divide the golds into two groups as follows:
Thus 4, 8, 10, 12, 14... are fine, while 0, 2, 6, 16, 18... are crude. Let's suppose that Alpha projects 'fine' and 'crude', but that Beta perversely projects 'fude' and 'crine', where
Before we go on we must refer to Shoemaker's original argument. Shoemaker asks us to consider the predicates 'emerald-like' and 'ruby-like', where
'emerald-like' stands for a cluster of properties that Mr. A takes as showing something to be an emerald, and [likewise for 'rubylike']. Mr. A of course projects these terms, and accepts the generalizations "If something is emerald-like, it is an emerald" and "If something is ruby-like, it is a ruby" (where these are to be understood as synthetic propositions).[67]
Now what properties shall we take as showing an even to be fine-like or crude-like? We cannot simply use the definitions of these predicates, for we want the propositions
to be, if not synthetic, then at least epistemologically contingent. So let's use our old trick of bringing in the OPN conjecture:
in which case he will call fine-like members of S crude, while Alpha calls them fine, thereby violating the same-ascriptions requirement. So instead let's have him project 'fude-like' and 'crine-like', where
Now Beta will project
If Beta bases his ascriptions of 'fude' and 'crine' on these generalizations, then before any member of Sis tested, Beta will apply 'fude' to just those evens to which Alpha applies 'fine', so Alpha might think 'fude' was just another word for 'fine'. But when members of Sare examined, Alpha finds that Beta stops applying 'fude' to fines and starts applying it to crudes. Since the same-ascriptions requirement is satisfied, it may seem that Alpha will be justified in supposing that Beta's 'fude' is his own 'fude'.
Shoemaker argues that, in his empirical example, such a supposition would not be justified. He writes:
Now from Mr. A's point of view, Mr. B would be [guilty of] irrelevance if he takes something first examined after t to be an emeruby on the grounds that it is emeruby-like and that prior to t things that were emeruby-like were emerubies. For this would amount, given the definitions, to inferring that after t things that are ruby-like are rubies on the grounds that prior to t things that were emerald-like were emeralds. ...Now if someone applies a term on the basis of irrelevant grounds, it is not to be expected that he will apply it to things it correctly applies to. And if it does nevertheless get applied to the thing it correctly applies to, this will be a coincidence; in such a case the application of the term to examined objects is not explained in the ordinary way by the speaker's meaning by the term what he does. So from Mr. A's point of view the fact that Mr. B applies the terms 'emeruby' and 'ruby' to the things to which he applies them is not explained by, and so does not provide evidence for, the hypothesis that Mr. B means by these terms what Mr. A means by them.[68]
So Shoemaker is arguing that even in this final resort to systematic gruification, Mr. A still has no reason to believe that Mr. B has a projective policy which is truly different from his own.
Can the same argument be made in our example? Does Beta apply the term 'fude' on irrelevant grounds? Beta does believe that
and that
If (33) constituted Beta's grounds for believing that (32), then Beta would be inferring that
on the grounds that
And (35) might appear to be irrelevant grounds for believing that (34). But, in the first place, Beta has another ground for believing that (34)—namely, that it is entailed by the well-supported OPN conjecture. And in the second place, (35) does constitute grounds for believing that (34). For (35) is equivalent to
and since 0, 2, 6, 16, 18... are not fine, (36) asserts the existence of many positive instances of the OPN conjecture, and provides no counter-instances. Therefore (36), if true, constitutes strong inductive support for the OPN conjecture, which in turn entails (34).
The argument of §12, then, seems to have failed. What was a fairly convincing argument in an empirical context could not be paralleled in a mathematical context. So it looks as if there is a coherent projective policy which projects 'fude' instead 'fine', and which does not force Beta either to misapply 'fude' or to apply it on irrelevant grounds. Does this mean, then, that the worries which motivated our discussion of Shoemaker were well-founded? Have we shown that one mathematical prediction is as good as another?
I think the answer is no. While Alpha and Beta may, in a technical sense, have different projective policies (they project different predicates, and there is good reason to believe that they mean the same things by these predicates), we have not shown that they make different predictions. Of the generalizations we have considered, all are generalizations that both would accept.
We may now be tempted to jump to the opposite conclusion: that our initial worries were groundless, and that any policy of mathematical projections which makes predictions different from our own must be incoherent. For the policy considered in §12 does not make predictions different from our own, and we have shown that the policies considered in §10 and §11 are incoherent.
But this conclusion would be equally mistaken. Although the story we told in §12 did not include any predictions which conflict with our own, it does not follow that no policy of systematic gase-ification can include such predictions. In the first place, even Shoemaker's argument does not demand that all of Alpha's projections translate into generalizations Beta accepts, and vice-versa, but only that "a number of them" do. In the second place, Shoemaker's belief that any generalizations at all need to be acceptable to both is based on his claim that "what generalizations we accept affects what properties we ascribe to observed items." But even if we understand 'generalizations' as including epistemologically contingent mathematical generalizations (though Shoemaker surely intended to exclude all metaphysically necessary propositions), this claim does not seem to be true of mathematics. We normally ascribe 'gold', not on the basis of an opengoldtest, but on the basis of a goldtest. And so our ascriptions normally depend, not on whether we accept
but simply on the epistemologically necessary truth that evens which goldtest like some gold even are gold.
In fact, we cannot fail to notice that throughout our attempt to parallel Shoemaker's argument, we have had to jump through hoops in order to provide Alpha and Beta with epistemologically contingent generalizations to project. In §10 we had to introduce the rather bizarre concept of 'opengoldtesting'. In §11 we were forced to give our heroes an opengoldtester and an oilygoldtester, instead of simply giving them a goldtester. And in §12 we introduced the opencrudetest and the openfinetest.
These quirks do not damage the arguments of §10 and §11—any projective policy which does not gase-ify systematically really is incoherent. But because it is not normally true that our mathematical ascriptions are affected by what epistemologically contingent generalizations we accept, there may be a projective policy which gase-ifies systematically and yet neither violates the same-ascriptions requirement, nor preserves the acceptability to both Alpha and Beta of any appreciable number of generalizations. Thus, for all we have shown, there may be a coherent policy open to Beta which makes projections which definitely conflict with Alpha's.
Of course, we haven't shown that such a policy is possible, either. So the question remains unsettled. Must we then begin again? Should we once again stipulate that the same-ascription requirement be satisfied, gase-ify as needed to preserve this stipulation, and then (as in §10) resort to more and more sophisticated mathematical arguments in attempt to show that the policy is incoherent? But we may begin to suspect that a successful argument will be equivalent to a proof of the Goldbach Conjecture. And we must not forget that the object of our inquiry was to see if we could show 'gase' projection to be incoherent without proving the Goldbach conjecture. And the answer seems to be that we cannot. At any rate, none of the arguments Shoemaker used will suffice.
Following Shoemaker, we have tried to argue that, if we should ever come upon a stranger who has a coherent policy of projecting the word 'gase', we should translate his word 'gase' as our word 'gold'. We have tried to argue, in fact, that even if he says that his 'gase' means 'x ∉ S and x is gold, or x ∈ S and x is base', then we should translate his 'gold' by our 'gase', and his 'base' by our 'bold'. And even if he should balk, and insist that by 'x is gold' he means 'x + 6 is the sum of two odd primes', then we should translate his function 'sum' by our function 'grum', where
grum(p, q) = | { |
p + q 0 |
if (p + q − 6) ∉ S, if (p + q − 6) ∈ S. |
And so on. But the argument did not succeed. For all we have shown, someone might have a coherent policy of 'gase' projection and yet might mean by 'gase' exactly what we mean by 'gase'.
But what if we turn the question around? Can we show that, if we should come upon a stranger who makes projections involving the word 'gold', we should translate his word 'gold' as our word 'gold'? For if we cannot establish whether a 'gase' projector really means 'gold' or 'gase', then we might begin to worry that we would be equally unable to establish what a 'gold' projector really means. After all, what's the difference between a strange 'gold' projector and a strange 'gase' projector? Of what importance is it that one happens to favor a predicate which sounds very much like a predicate we also favor?
Now, if in actuality we should ever begin to doubt that someone who projected 'gold' really means 'gold', and to suspect that he really means 'gase', we will simply ask him what he means by 'gold'. And if he answers that by 'x is gold' he simply means 'x + 6 is the sum of two odd primes', then we will probably be quite reassured. But if, in the case of the 'gase' projector, we could doubt that he meant by 'sum' what we mean by what we mean by 'sum', and suspect that instead he means 'grum', then why should we not also suspect the 'gold' projector? Are we prejudiced in his favor just by the sound of his words?
Our function 'grum' will undoubtedly remind the reader of the function 'quus' which Kripke introduced in On Rules and Private Language, his interpretation of Wittgenstein's Philosophical Investigations. In fact it would be surprising if the reader had struggled this far without noticing the striking resemblance between 'gase' and 'quus'. Both are mathematical terms related in a grue-like way to "normal" mathematical terms. And both present skeptical challenges to certain of our natural preconceptions. We might begin to wonder if the connection between them runs deeper.
If we recast Kripke's problem in terms of 'gold' and 'gase', it runs like this: Obviously there must be some even number which we have never before checked for goldness. Let g be such a number. Now suppose we check g and discover that there are, in fact, two odd primes which sum to g + 6. We therefore ascribe to g the predicate 'gold'. But now a skeptic comes along and suggests that we have misinterpreted our own past usage of the predicate 'gold'. He suggests that, when we have used 'gold' in the past, we meant not gold[69] but gase (that is, gase_{{g}}); and likewise that when we have used 'gase' in the past, we meant not gase but gold. Thus he holds that, since we have determined g to be gold, we can only accord with our previous linguistic intentions by calling g 'gase', and hence 'base'.
Although this suggestion is undoubtedly bizarre and implausible, it is not logically impossible, as we can see if we consider the reverse situation: Assume the common sense hypothesis that by 'gold' and 'gase' we did mean gold and gase. It would nevertheless be possible that, due to temporary delirium, we should misinterpret all our past uses of 'gase' as meaning gold. If we then determine g to be gold (but not gase), we will call g 'gase', mistakenly believing that we are according with our earlier usage. Now the skeptic is proposing that we have made a mistake precisely of this kind, but with gold and gase reversed. He is suggesting that we should call g gase, because it is gold, and because we used to mean gold by 'gase'.
Obviously we have been making projections involving the word 'gold', in the past as well as the present. But if Kripke is right, then it is possible that what we mean by that word has changed. Thus he thinks it is possible that our projective policy should have changed, and not merely on a verbal level. This is exactly the sort of possibility that Shoemaker tries to rule out. Both Kripke and Shoemaker think words and meanings can slip past each other. But whereas as Kripke would argue that the meaning of a projective policy can change even if its verbal form remains constant, Shoemaker argues that, even if the verbal form should change, the idea of a change in the meaning of our projective policy is incoherent. For his story about Mr. A and Mr. B was meant to show us something about ourselves. He writes,
Now what I hope to have shown is that nothing that could happen would be good evidence that someone else had adopted and was following such an alternative policy. But if this is so, is it intelligible for us to contemplate adopting such a policy in place of the one we now have? It seems to me that it is not.[70]
But what we hope to have shown is that no Shoemaker-like argument can demonstrate the incoherence of a shift in the meaning of our projective policy. That is, no Shoemaker-like argument can rule out the possibility that we may begin to project gase instead of our usual gold. And if this is a possibility, then so is its opposite: that we should begin to project gold instead of our usual gase. Thus the skeptic's claim, that we always have projected gase, is not logically impossible.
But that the skeptic's claim is not logically impossible does not show that it is true. Surely we can produce some fact which proves that we meant gold by 'gold'? Well, if Kripke and Wittgenstein are right, then we cannot. Certainly the record of our past ascriptions of 'gold' is not enough. Since all the numbers to which we have ascribed 'gold' have been both gold and gase, this evidence is just as compatible with the hypothesis that we meant gase by 'gold' as with the hypothesis that we meant gold by 'gold'.
Nor will it help to observe that in applying the predicate 'gold', we followed an explicitly formulated rule, namely: 'x is gold iff x + 6 is the sum of two odd primes'. For just as the skeptic suggests that we have misinterpreted our own past use of 'gold', he will now suggest that we have misinterpreted our own past use of 'sum'. When, in the past, we used 'sum', what we really meant was 'grum'. And of course it will not help to claim that we had a rule for computing the 'sum' function. For the skeptic will simply suggest that we have misinterpreted this rule too. Whenever we supply a rule for interpreting a rule, the skeptic can challenge our interpretation of that rule. And, as Wittgenstein points out, "Explanations come to an end somewhere."[71] (We see here how far Shoemaker opened the door to a Kripkean/Wittgensteinian attack.)
Kripke considers several other candidates for the alleged fact that we meant gold by 'gold' (including dispositions to apply 'gold' in certain ways, irreducible introspectible qualia, and others), but we need not tarry over these here. If we are willing to accept Kripke's (to my mind, quite convincing) arguments, then we come to the conclusion that there is no fact which constitutes our having meant gold by 'gold'.
But why should we concern ourselves with skeptical doubts about what we meant by 'gold' in the past? How does it bear on our primary question, which is the problem of justifying our preference for the Goldbach Conjecture over its equally well-supported rivals?
The answer is, that it bears quite heavily thereon, if we accept a Goodman-like justification of our preference. And although we had difficulties in §7 in applying the details of Goodman's theory to pseudo-empirical induction, we also observed, in §8, that some sort of Goodman-like answer seems to be our only hope. Thus even if we cannot explain precisely how our past use of mathematical language renders the Goldbach Conjecture valid, we must admit that there is some connection. Although we suggested in §7 that 'gold' is not itself entrenched, our discussion will proceed more smoothly if we suppose that 'gold' is entrenched, and that its entrenchment explains the projectibility of the Goldbach Conjecture. Of course, it will not be the word itself which is entrenched, but something which stands behind the word, something which remains constant through differences of tongue, use of coined abbreviations, etc. Since Goodman does not feel comfortable with the notion of 'meaning' or 'intension',[72] what stands behind the word is, for him, its extension. Goodman writes, "not the word itself but the class it selects is what becomes entrenched, and to speak of the entrenchment of a predicate is to speak elliptically of the entrenchment of the extension of that predicate."[73] Thus on Goodman's theory, even if we have projected the word 'gase' in the past, the Goldbach Conjecture will still be projectible, provided that we have used the word 'gase' to denote the class of gold numbers. But the converse is also true: if we have used the word 'gold' to denote the class of gase numbers, then it is not the Goldbach Conjecture, but the Gasebach Conjecture, which is truly projectible.
Now Kripke's skeptic frames his challenge as a doubt about what we meant by a word (its intension), not as a doubt about what we used the word to denote (its extension). But it is clear that if the former can be doubted, so can the latter. For what fact can we produce to prove that we used 'gold' to denote the class of gold numbers, and not the class of gase numbers? Or, what seems to be equivalent, what fact could we produce to show that we ascribe 'gold' to all (and only) gold numbers? All the numbers to which we have actually ascribed the predicate 'gold' have been both gold and gase, so that is no proof. If we could produce a fact which proved that we used 'gold' to mean gold, then that might be enough also to show that we used 'gold' to denote the class of gold numbers. But Kripke and Wittgenstein argue that there is no such fact. Thus the skeptic's challenge points an arrow at the very heart of Goodman's theory. For if his suggestion, that by 'gold' we really meant gase, cannot be answered, then apparently neither can the suggestion, that we used 'gold' to denote the class of gase things, and thus that it is not the Goldbach Conjecture but the Gasebach Conjecture which is really projectible. And that conclusion we cannot accept.
But can we not easily answer the skeptic? Can we not simply say: well, look, why need it be worrisome that we cannot produce a fact which proves the hypothesis that we ascribe 'gold' to all (and only) gold numbers? This hypothesis, like any other, is fallible, but that does not mean that we should reject it. On the contrary, we have every reason to adopt it, for it is incontestably far more natural and reasonable than any other hypothesis. The answer is: certainly it is, just as the hypothesis that all emeralds are green (while lacking any factual proof) is far more natural and reasonable than any other hypothesis. But the question is, why is it more reasonable? We began with a stock of evidence:
And so on, perhaps all the way to one hundred million. On the basis of this evidence, we adopted the hypothesis that
But why this hypothesis? Why not the equally well-supported hypothesis
Thus, it seems that, in order to explain why the Goldbach Conjecture is valid and the Gasebach Conjecture invalid, we must assume that (37) is valid and (38) is invalid. But how do we explain this? Will we argue that the consequent of (37) ('gold') is better entrenched than the consequent of (38) ('gase')? But this can only be if we have used 'gold' to denote the class of gold things rather than the class of gase things—which is exactly what (37) claims. Thus it seems that Goodman's theory can only show (37) to be valid by assuming that (37) is valid. We are caught in a vicious circle.
Even if we could avoid a vicious circle, we would not have solved our problems. If Goodman's theory can only justify a hypothesis by assuming a second hypothesis (namely, one about what we have used some predicate to denote), then the skeptic will always be able to frustrate Goodman by proposing that an equally well-supported grue-like rival to the second hypothesis. Goodman may be able to use his theory to justify the second hypothesis if he assumes a third hypothesis, but the skeptic can challenge this third one too. However far Goodman goes, the skeptic can go a step further. And justifications must come to an end somewhere.
What Goodman's 'grue' showed was that the facts about a particular topic (say, the color of emeralds) can, by themselves, never suffice to distinguish any one generalization from those facts as valid. Goodman's remedy for this shortcoming was to bring in "facts" from outside the topic, namely, "facts" about what classes we have meant by the predicates we have projected in the past. But what Wittgenstein and Kripke have shown is that there are no such facts. The real facts concern merely what we have said; what we meant by what we said seems to be merely a hypothesis like any other.
Now we said above that the facts about a particular topic can, by themselves, never suffice to distinguish any one generalization from those facts as valid. But what if our linguistic behavior is the topic under consideration? Worse yet, what if the state of the entire world is the topic under consideration? If the complete facts about the entire history of the entire world are not enough to distinguish any one extrapolation from those facts as valid, then what facts can be brought in from outside? Goodman's solution, if it works at all, works only on the local scale, not on the global scale. But what we hope to have shown is that the solution cannot work on the local scale if it does not work on the global scale. It can distinguish certain local hypothesis as valid only by presupposing that other, outside hypotheses have already been validated. But justifications must come to an end somewhere.
We may be able to make more sense of this conclusion if we can fit it into a historical background. We seem to have a primitive notion that what justifies our preference for certain predictions is a fact in the world, namely, a necessary causal connection between past and future events. What Hume showed is that this alleged fact is merely a phantom of our own imagination. But Hume did not leave us to despair. He provided a skeptical solution to his skeptical doubts. He argued that we need no fact to justify our inductive inferences. Conformity with the way we actually make such inferences is all the justification we need—or at least that is Goodman's interpretation of Hume.[74] For Goodman, our preference for certain projections is justified only by the facts about our past projection of certain predicates—or, more accurately, our past projection of the extensions of those predicates. That is, Goodman's justification of our preference for certain projections depends, not only on the facts about what we said, but also on the "facts" about what we meant by what we said.
But what Wittgenstein tries to show is that the "facts" about what we meant are just as much phantoms of our imagination as the "facts" about necessary connection. If this is true, then we would seem to have no justification at all for our projective preferences. But Wittgenstein also does not abandon us to utter despair. Like Hume, he proposes a skeptical solution to his skeptical doubts (a parallel which Kripke explicitly draws). In opposition to "the author of the Tractatus," he argues that we need no fact to justify attributions of meaning. Conformity with the way we actually make such attributions is all the justification we need.
The claim, that by 'gold' we meant gold, is simply one way of describing of the facts about how we have used 'gold'. (Just as to say that '0', '1', '2', etc. signify numbers is simply one way of describing of the facts about how we use these symbols. See the Investigations, §10.) Thus the claim, that by 'gold' we meant gold, is not a hypothesis, and does not need to be justified by other hypotheses about what we meant by other words. It is merely one way of describing the facts, and it is justified, if our language game, as we currently play it, sanctions this manner of describing the facts.
Assuming that we have habitually projected the word 'gold' in the past, a Goodman-like explanation says (loosely speaking, and without much regard for the intension/extension distinction) that the Goldbach Conjecture is justified just in case we have meant gold by the word 'gold' (and, conversely, the Gasebach Conjecture is justified just in case we have meant gase). Now Goodman seems to imagine that what we meant by 'gold' is a fact about the past, and thus that the facts about the past, in conjunction with the mathematical evidence, are sufficient to distinguish the Goldbach Conjecture as valid. What I hope to have shown is that the Goldbach Conjecture and the Gasebach Conjecture are equally compatible, not only with the mathematical evidence, but with the facts about our linguistic history. Nothing in the past suffices to distinguish the Goldbach Conjecture as valid. If the Goldbach Conjecture is more justified than the Gasebach Conjecture, it is only because our language game, as we currently play it, sanctions the Goldbach Conjecture. Which is just another way of saying that the Goldbach Conjecture is justified only because we choose to regard it so.
Thus Goodman was right to say that "the roots of inductive validity are to be found in our use of language." But he was wrong to suppose that anything in our past use of language forces us to regard one conjecture as more justified than another. In §8 we speculated, following Goodman, that the line between valid and invalid mathematical conjectures may be drawn upon the basis of how the mathematical world is and has been described and anticipated in words. We would have been much closer to the truth if we had left out the words "and has been." How we presently choose to project is all the justification we need for our present projections. If this seems like no justification at all, so be it. What Wittgenstein wrote concerning arithmetical calculations might just as easily be applied to pseudo-empirical induction: "The danger here, I believe, is one of giving a justification of our procedure where there is no such thing as a justification and we ought simply to have said: that's how we do it."[75]
Our conclusion—that we can give no (to use Wittgenstein's term) superlative justification of the inferences of pseudo-empirical induction, and that these inferences are valid only because we choose so to regard them—is not surprising in light of Wittgenstein's philosophy of mathematics. For Wittgenstein holds that even the proven generalizations of mathematics are acceptable only because we accept them. This radical view of mathematics is aptly summarized by Michael Dummett:
Even if [the] rules [of inference] had been explicitly formulated at the start, and we had given our assent to them, our doing so would not in itself constitute recognition of each transition as a correct application of the rules. ...Hence at each step we are free to choose to accept or reject the proof; there is nothing in our formulation of the axioms and of the rules of inference, and nothing in our minds when we accepted these before the proof was given, which of itself shows whether we shall accept the proof or not; and hence there is nothing which forces us to accept the proof. If we accept the proof, we confer necessity on the theorem proved; we "put it in the archives" and will count nothing as telling against it. In doing this we are making a new decision, and not merely making explicit a decision we had already made implicitly.[76]
Thus just as the facts about my past use of mathematical language fail to determine which conjectures I must accept, so, on Wittegenstein's view, do they fail to determine which theorems I must accept.
In pointing out the parallels between our conclusion and Wittgenstein's philosophy of mathematics, I am not trying to suggest that that the foregoing arguments compel us to accept Wittgenstein's radically intuitionist philosophy of mathematics. On the contrary, there is nothing in our conclusion which is inconsistent with the Platonist view of mathematics which we agreed, in §3, to assume. Let the mathematical world be perfectly and eternally ordered; let the proofs of mathematics be ever so apodeictic; still we can only justify our pseudo-empirical inductive inferences by saying, "That's how we do it."
Indeed, in retrospect we can see the clear advantage of having taken pseudo-empirical induction for our topic. In the course of arguing against the possibility of a superlative justification of pseudo-empirical induction, we have managed also to side-swipe the idea of a superlative justification of empirical induction. And the reader might well wonder why we did not simply attack Goodman head-on from the beginning. But if we had taken empirical induction for our topic, the conclusion, that no superlative justifcation for it can be given, might be thought to be wholly or partly due to some failure on the part of nature to connect her events according to a law of causality. By taking for our topic pseudo-empirical induction, whose field is the perfectly orderly world of mathematics, we hope to have shown that no such thing is in question. It is not the fault of Nature that no superlative justification of empirical induction can be given. Let the course of Nature be ever so regular; still we can only justify our predictions by saying, "That's how we do it."
[1] One exception is the class of generalizations which Goodman calls exhausted, for example, "All emeralds examined today were green."
[2] The sort of induction on which pseudo-empirical mathematics is grounded should not be confused with the method of mathematical proof also known as 'induction'.
[3] Wang Yuan, Goldbach Conjecture (Singapore: World Scientific, 1984), p. 1.
[4] Carl Friedrich Gauss, Werke (Göttingen: Königlichen Gesellschaft der Wissenschaften, 1876), II, 401.
[5] Pierre de Laplace, Oeuvres complètes de Laplace (Paris: Gauthier-Villars, 1886), VII, p. V.
[6] Leonhard Euler, "Specimen de usu observationum in mathesi pura," in Opera Omnia (Zürich: Teuloneri, 1915) ser. 1, II, 459.
[7] Peter Hagis, "A Lower Bound for the Set of Odd Perfect Numbers," Mathematics of Computation, 27 (1973), p. 951.
[8] George Polya, Patterns of Plausible Inference (Princeton: Princeton University Press, 1954), p. 4.
[9] Stan Wagon, "Fermat's Last Theorem," The Mathematical Intelligencer, 8 (1986), p. 59.
[10] Daniel Shanks, Solved and Unsolved Problems in Number Theory (Washington D. C.: Spartan Books, 1962), I, 30.
[11] Stan Wagon, "The Collatz Problem," The Mathematical Intelligencer, 7 (1985), p. 73.
[12] George Polya, Induction and Analogy in Mathematics (Princeton: Princeton University Press, 1954), p. 9.
[13] Wagon, "Fermat's Last Theorem," p. 59.
[14] P. L. Chebyshev, "Lettre de M. le Professeur Tchébychev à M. Fuss sur un nouveau théorème rélatif aux nombres premiers contenus dans les formes 4n ± 1 et 4n ± 3," Bulletin of Classical Physics of the Academy of Imperial Science, 11 (1853), 208.
[15] C. Bays and R. Hudson, "Details of the first region of integers x with π_{3,2}(x) < π_{3,1}(x)," Mathematics of Computation, 32 (1978), 571.
[16] J. E. Littlewood, "Sur la distribution des nombres premiers," Comptes Rendus de l'Académie de Science, 158 (1914), 1869.
[17] Bays and Hudson, p. 571.
[18] Hadamard and de la Vallée Poussin finally accomplished the proof. See Shanks, pp. 15-16.
[19] Wagon, "The Collatz Problem," p. 72.
[20] R. S. Lehman, "On the difference π(x) − li(x)," Acta Arithmetica, 11 (1966), 409.
[21] Littlewood, p. 1869.
[22] Lehman, p. 409.
[23] As quoted in Wagon, "The Collatz Problem," p. 72.
[24] H. M. Edwards, Fermat's Last Theorem, A Genetic Introduction to Algebraic Number Theory (New York: Springer-Verlag, 1977), p. vi.
[25] See Willard Van Orman Quine, "Two Dogmas of Empiricism," in From a Logical Point of View (Cambridge, Massachusetts: Harvard University Press, 1953), pp. 20-46; reprinted from The Philosophical Review, 60 (1951), 20-43. See also Ludwig Wittgenstein, Remarks on the Foundations of Mathematics (Cambridge, Massachusetts: MIT Press, 1978), esp. III, 66.
[26] Some exceptions are mentioned in note 1.
[27] On Rules and Private Language (Cambridge, Massachusetts: Harvard University Press, 1982), p. 62.
[28] (Indianapolis: Hackett, 1977), p. 20.
[29] Hume, p. 24.
[30] Hume, p. 49.
[31] Hume, p. 52.
[32] Alternatively, one could interpret Hume as arguing, not that there is no necessary connection between matters of fact, but merely that we can can know of no necessary connection between matters of fact. But since this interpretation would seem to assimilate Hume's doubts concerning necessary connection to his doubts concerning the operations of the understanding (which he seems to feel he has treated adequately in §4 and §5), the former interpretation seems more plausible.
[33] Kant's arguments for the second, third, and fourth points are mostly found in the Transcendental Deduction of the Categories, but a short (if not particularly perspicuous) summary of his main argument can be found in the Analytic of Principles, where he writes, "Experience...rests on the synthetic unity of appearances, that is, on a synthesis according to [the categories]. Apart from such a synthesis it would not be knowledge... Experience depends, therefore, upon a priori principles of its form, that is, upon universal rules of unity in the synthesis of appearances. Their objective reality, as necessary conditions of experience, and indeed of its very possibility, can always be shown in experience." (Immanuel Kant, Critique of Pure Reason, tr. Norman Kemp Smith, (New York: St. Martin's Press, 1965), p. 193 (A156-57, B195-96).
[34] I say "immediately infer" because, if the Goldbach Conjecture is true, then, as a necessary truth, it could be inferred from anything at all. What I am claiming here is merely that we could not know that (1) entailed the Goldbach Conjecture without having an independent proof of the conjecture.
[35] Nelson Goodman, Fact, Fiction, and Forecast, 4th ed. (Cambridge, Massachusetts: Harvard University Press, 1979), pp. 63-64.
[36] Goodman, p. 62.
[37] Goodman, p. 65.
[38] Goodman, p. 74.
[39] It might be thought that we could solve the problem by ruling out all 'non-qualitative' mathematical predicates, of which 'gase_{S}' is supposedly an example, since its definition mentions a particular set of numbers. But for arguments to the contrary, see Goodman, pp. 78-80. Here we will merely observe that, although we must mention a particular set of numbers in order to define 'gase_{S}' in terms of 'gold', the converse is also true. For 'gold' may be defined by:
[40] Goodman, p. 104.
[41] Goodman explains what he means by 'actual projection' as follows: "Actual projection involves the overt, explicit formulation and adoption of the hypothesis—the actual prediction of the outcome of the examination of further cases." (p. 88)
[42] The predicate 'P' is a parent of the predicate 'Q' iff 'P' applies to the extension of 'Q'.
[43] Goodman does not, as we might expect, use 'conflicts with' to mean 'is incompatible with'. He writes, "Two hypotheses conflict if neither follows from the other (and the fact that both are supported, unviolated, and unexhausted) and they ascribe to something two different predicates such that only one actually applies." (p. 99)
[44] H is a positive overhypothesis of H' iff the antecedent of H is a parent predicate of the antecedent of H', and the consequent of H is a parent predicate of the consequent of H'.
[45] H is a negative overhypothesis of H' iff the antecedent of H is a parent predicate of the antecedent of H', and the consequent of H is complementary to a parent predicate of the consequent of H'.
[46] Goodman also adds that "it is no news that the projectibility of hypotheses is affected by certain related hypotheses, or that the effect of correlative information is the greater the more of it there is and the more closely it is allied to the hypothesis in question." (p. 116)
[47] In practice we need only eliminate the violated conjectures, since no conjecture is unviolated and unsupported.
[48] "To say that coextensive predicates are equally entrenched is not to say that we must know what predicates are coextensive before making any inductive choices but to say how whatever judgments of coextensivity we do make are relevant to our inductive choices." Nelson Goodman, Problems and Projects (Indianapolis: Bobbs-Merrill, 1972) p. 360.
[49] It might be suggested that the Goldbach Conjecture could derive projectibility from one of the 'related hypotheses' mentioned in note 46. It is difficult to know precisely what sort of related hypotheses Goodman has in mind, but if we consider some likely candidates, it seems that we will again run into the difficulty, that any hypothesis which we know to be related to the Goldbach Conjecture in the right way will also be so related to its rivals.
[50] Goodman, p. 64.
[51] Goodman, p. 120.
[52] Goodman, pp. 96-97.
[53] Goodman, p. 121.
[54] Sydney Shoemaker, "On Projecting the Unprojectible," The Philosophical Review, 84 (1975), p. 180.
[55] Goodman, pp. 62-65.
[56] Goodman, p. 98.
[57] Rather than constantly tripping over the messy subscript of 'gase_{S}', we will frequently revert to our original term, 'gase', which of course is just 'gase_{{g}}'.
[58] It's a bit mystifying why, if Shoemaker thinks that such a policy is incoherent, he devotes a large portion of his paper to showing that it's no different from our own. (Strictly speaking, this would imply that our own policy is also incoherent. But surely Shoemaker did not intend us to read him so strictly.) But since the argument of (2a) introduces some concepts which will be important for the argument of (3), we will go ahead and include it.
[59] Beta must project the obvious consequences of the Gasebach Conjecture along with the conjecture itself if his policy is not to be obviously incoherent; however, we do not want to insist that he project all consequences of the Gasebach Conjecture, for this group will include all truths of mathematics, and probably all falsehoods as well. Of course, the line between obvious and non-obvious is not clearly drawn. We might sharpen the distinction by saying: Q is a non-obvious consequence of P just in case it is epistemologically contingent whether P implies Q.
[60] Shoemaker, p. 195.
[61] A proposition like 'x goldtests positive' should be understood as a metaphysically necessary assertion about mathematics, not as a metaphysically contingent report of a calculation.
[62] Our argument for this point is actually much stronger than its analog in Shoemaker's argument. A closer parallel to Shoemaker would argue, not that it is inconsistent for Beta to deny that g is open, but merely that it is unreasonable, since it would seem to be within the power of any sufficiently large computer to prove g open. But of course this assumes what remains to be shown, that the OPN Conjecture is true.
[63] 'Bold', naturally, is related to 'bleen' the way 'gase' is related to 'grue'. It is defined by:
[64] Shoemaker, p. 190.
[65] Shoemaker, p. 208.
[66] Shoemaker, pp. 208-9.
[67] Shoemaker, p. 209.
[68] Shoemaker, p. 211.
[69] I will use italics to indicate that I am referring to the meaning of the italicized expression (that is, its intension, rather than its extension or the expression itself). See Kripke's remarks on this typographical issue in his note 8, pp. 9-10.
[70] Shoemaker, p. 213.
[71] Philosophical Investigations (New York: Macmillan, 1953), §1.
[72] He mentions "obscurities in the notion of 'the meaning' of a predicate," Fact, Fiction, and Forecast, p. 79.
[73] Goodman, p. 95.
[74] "For in dealing with the question how normally accepted inductive judgments are made, [Hume] was in fact dealing with the question of inductive validity. The validity of a prediction consisted for him in its arising from habit, and thus in its exemplifying some past regularity." Goodman, Fact, Fiction, and Forecast, pp. 64-65.
[75] Wittgenstein, Remarks on the Foundations of Mathematics, III, 74.
[76] Dummett, pp. 425-27.
[*] In 1994, after this paper was written, a deductive proof of Fermat's Last Theorem was finally found by Andrew Wiles of Princeton University.
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