Ordinary problems, both practical and theoretical, tend to be highly circumscribed. If we want to prove the hypotenuse-leg theorem or get from Boston to Charlottesville, a variety of factors limit our options. In the one case, we restrict our thinking to the resources, the axioms, and inference rules of Euclidean geometry; in the other, to available air routes, railroad lines, and roads, and to current schedules. We summarily exclude many conceivable alternatives. In attempting to prove the theorem, for example, induction is not an option. If millions of cases confirm the hypothesis and no exceptions have been found, we have evidence that the hypothesis is true. But no amount of empirical evidence constitutes a mathematical proof. Nor will resort to theology help. One could argue: God made the theorem true. Whatever God makes true is true. Hence the theorem is true. This is a valid deduction. If one believes the premises, one has reason to believe the conclusion, but again, not a mathematical reason. Indeed, not even every mathematical maneuver is acceptable. One easy way to prove the theorem is this: Make it an axiom, then derive it from itself, since p entails p. Clearly this won’t do, even though Euclidean geometry is re-axiomatizable, and different Euclidean propositions provide equally serviceable axioms. Having good reasons isn’t enough; having good deductive reasons isn’t enough; even having good deductive reasons within a Euclidean system isn’t enough. We require a derivation from some standard or antecedently specified axiomatization of Euclidean geometry.
The practical problem of getting from Boston to Charlottesville is less regimented, but still subject to a variety of tacit constraints. We don’t entertain such options as beaming from place to place à la Star Trek or tunnelling through the Earth. We don’t consider going by way of Paris or chartering a plane. We standardly exclude options that require damming rivers, constructing highways, or creating air routes. Except perhaps for beaming, all of these are possible. We reject them out of hand because they are impractical. We automatically exclude alternatives that are inefficient, technologically unfeasible, or exorbitant. Typically, they don’t even leap to mind. This suggests that the problem we set out to solve has more constraints than its statement indicates. The problem isn’t how to get from Boston to Charlottesville simpliciter, but how to get from Boston to Charlottesville given available transportation routes and constraints on one’s time and money. If we tried to be fully explicit, a good deal more would have to be said. If we switched contexts, the options would change. The Department of Transportation may have reason to seriously entertain alternatives that the ordinary traveler does not—building new roads or revising air routes, for example. Problems turn out to be like icebergs. However large they loom on the horizon, most of their magnitude lies beneath the surface, out of sight.
Everyday problems are far more complex than their ordinary statement suggests. They are circumscribed by vast networks of presuppositions, some so deeply tacit that we don’t even recognize that we are making them. But far from making problems harder to solve, I suggest, this makes them easier. My point is not psychological. I’m not saying that the constraints provide cues like “It would be a good idea to check a map,” or “Remember the Pythagorean Theorem.” Rather, I believe, the constraints figure in the demarcation of the problem and the specification of what constitutes an adequate solution. They shape the ends in view. What is wanted is not just to prove the theorem somehow, but to prove the theorem using some standard—or even some particular—axiomatization of Euclidean geometry, not to get to Charlottesville somehow, but to get there relatively inexpensively, directly, and efficiently using currently available modes of transportation. Typically, tacit constraints circumscribe a problem enough to allow for a solution, or at least for an understanding of what a solution requires.
If this is right, it helps explain why philosophical problems, as standardly framed, seem so intractable. In philosophical contexts, we waive presuppositions that ordinarily supply the needed constraints and fail to supply replacements to take up the slack. Rather than settling for a proof of the hypotenuse-leg theorem that takes the rules and axioms of Euclidean geometry for granted, philosophers of mathematics ask what underwrites those rules and axioms. They then endeavor to justify the axioms, validate the rules, explicate the concept of proof, determine the role of proof in mathematics, and spell out the relation of proof to truth. Instead of taking the problem of going from Boston to Charlottesville as well defined, metaphysicians back off until they confront Zeno’s paradoxes: What is motion? How is it possible to move from one place to another if you have to traverse half the distance first? And so forth. The moves are familiar and, I would argue, valuable. When we uncover tacit presuppositions and subject them to critical scrutiny, we can discover whether they are well founded and whether they admit of viable alternatives. But philosophy is relentless. In Kant’s words, it seeks after the unconditioned.11xImmanuel Kant, Critique of Pure Reason, trans. and ed. Paul Guyer and AllenWood (Cambridge: Cambridge University Press, 1997), 112. Ideally, we want to disclose, explicate, and vindicate all the considerations that underlie our convictions and actions. We seek to justify the justifiers and explicate the explicators, hoping to reach the unconditioned, where no further justification or explication can or need be given. Then and only then will conclusions be transparently correct.