Ontology, Understanding, and the A Priori

© Blackwell Publishing Ltd. 2003, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA. Ratio (new series) XVI 2 June 2003 0034–0006

ONTOLOGY, UNDERSTANDING, AND THE A PRIORI Ernest Sosa Abstract How might one explain the reliability of one’s a priori beliefs? What if anything is implied about the ontology of a certain realm of knowledge by the possibility of explaining one’s reliability about that realm? Very little, or so it is argued here.1

If one can know a fact only if that fact causes one’s belief in it, then the facts of mathematics lie unknowably beyond the causal realm of spacetime. According to Gödel, we do gain access to that realm through direct insight, and he may even have supposed that people themselves exist beyond spacetime. But naturalists who defend a priori warrant can appeal to no such perception. According to Alvin Goldman, for example, warrant is provided rather by intramental causal processes of belief formation that reliably enough yield truth rather than error. Among such processes are processes of calculation or reasoning that lead to beliefs in domains of necessary truths and falsehoods; it is such beliefs that enjoy a priori warrant.2 Some intramental processes lead reliably to beliefs about the physical objects in a subject’s environment. Taking our experience at face value3 is normally such a process on the surface of the earth. Processes of reasoning or calculation also yield true beliefs reliably enough to make them warranted. Again, such processes for Goldman comprise no quasi-perceptual relations to the objects or facts that they enable us to know. So it is not by including such relations to abstracta that his processes of a priori warrant cause a problem; but they do cause a problem anyway. We can know and understand how it is that taking experience at face value is a reliable gateway to the shapes and colors of visible 1 This paper was presented at the Symposium, Current Issues in Ontology, held at the University of North Carolina at Greensboro, March 31–April 2, 2000, directed by Joshua Hoffman and Gary Rosenkrantz. 2 Alvin Goldman, “A Priori Warrant and Naturalistic Epistemology,” Philosophical Perspectives, 13, Epistemology (1999): 1–28. 3 I.e., believing that p on the basis of experiencing as if p.

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surfaces. But this explanation will apparently involve postulating perception of surfaces in appropriate conditions.4 By contrast, abstracta are imperceptible; so how could our intramental processes of reasoning or calculation put us reliably in touch with how it is with such abstracta? True, ethereal perception does not constitute the intramental processes. But it is still a mystery how these processes could be reliable about mind-transcendent facts without perception or some other causal mechanism to connect the two.

*************** George Bealer proposes an explanation of a priori warrant that apparently requires no perception of abstracta.5 On his view there is a tight relationship among understanding, appropriate intuition-driven theorizing, and the truth, a relationship explained in two stages: (a) A proposition P is “settled with a priori stability” by a subject S IFF while understanding that proposition fully enough S believes it as a result of good enough intuition-based theorizing, and necessarily no better such theorizing would make S change his mind so long as he continued to understand P fully enough. (b) Propositions of a certain sort (which we will not stop to specify) that a subject understands fully enough are true if and only if the subject could possibly settle with a priori stability that they are true. So intuitions are said to have this kind of connection with understanding. Certain sorts of truths may be reached on the basis of intuitions if we theorize appropriately on this basis so as to reach such truths understood well enough. Compare our reliability about the colors and shapes of facing surfaces. How do we account for it? Here is one way. 4 Goldman accordingly needs more argument for his claim (p. 7) that, given his view, “. . . a priori warrant does not require the sort of trans-mental, perception-like process that Benacerraf was discussing.” 5 George Bealer, “A Theory of the A Priori,” Philosophical Perspectives, 13, Epistemology (1999): 29–57. This is part of a long, connected series of important papers on the epistemology of the a priori and of philosophy more specifically.

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Through a theory of perception. Thus we might explain how we are related to facing surfaces by means of light, eyes, rods and cones, optic nerves, etc. Increasing scientific sophistication would thus deepen our insight into the causal mechanisms that enable us to know shapes and colors. Alternatively, we might offer an acount by way of a theory of understanding. How do we grasp properties of shape or color, how do we gain the required concepts? Our grasp of such properties is presumably bound up both (a) with appropriate belief formation as to when they are exemplified before us, and (b) with the truth about such exemplification. Thus, our grasp of a color or shape seems to involve our ability to tell when it is present in our visual field, provided the cognitive and perceptual conditions are good enough. Sensory experiences of color or shape may well bear such a relation to subjects’ understanding or grasp of them. Experiences may thus be a gateway to truths about environing colors and shapes, so long as we theorize appropriately from them to such truths understood fully enough. In two ways then might we explain our reliability on shapes and colors: first, through a theory of perception; second, through a theory of understanding. Let us grant that a theory of understanding does somehow “account” for our reliability, and might even serve as an “explanation.” Is this enough for the platonist? No, absent a theory of perception to accompany the theory of understanding, we are left wondering how it is that we gain so much as a grasp of observable properties. According to the sketched theory of understanding, one important requirement is that we be sensitive enough to the presence or absence of such properties in appropriate cognitive and perceptual conditions. Unaided, the theory of understanding does not explain the required sensitivity. It is the theory of perception that fills this gap. Bealer’s way of binding together understanding, intuition, and the truth about the a priori, includes no account of how we attain our understanding of the a priori. But we apparently need some explanation of our supposed sensitivity to the truth via intuitions. What indeed makes our states of intuition so systematically correlated with the facts in the realm of abstracta? This explanation © Blackwell Publishing Ltd. 2003

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is unlikely to feature anything like light bouncing off surfaces and getting into our eyes thus affecting our retinas, etc. No correlate of light, nor any organ like our eyes, can give substance to any such explanation. And we are left wondering just how we manage so systematically to believe in accordance with the truth, avoiding so well the no less numerous adjoining falsehoods.6 *************** “Plenitudinous platonism” has recently been proposed as a solution to our problem by Mark Balaguer.7 Here is the key argument: 1. Every consistent mathematical sentence is true under some interpretation. 2. Therefore, so long as we are able to restrict our mathematical beliefs to consistent mathematical sentences, we shall be right in our mathematical beliefs. 3. But we are able to restrict our mathematical beliefs to consistent mathematical sentences. 4. So we can be sure that we are generally right in our mathematical beliefs. A mathematical belief is not just a belief that a certain mathematical sentence is true, however, much less that it is true under some interpretation.8 Consider, accordingly, the retreat to the concept of truth under the intended interpretation. It is hard to 6 I will eventually sketch a proposed solution below, upon canvassing other alternatives in the literature. My proposal has points of contact with the Bealer account, but differs in crucial respects. My “intuitions” are inclinations to believe, which seem more closely tied to understanding than are “intuitions” viewed as sui generis propositional attitudes (which presumably might become detached from beliefs and even inclinations to believe, at least might possibly become so detached). Moreover, I would connect understanding, intuitions, and the truth, not only via what it is metaphysically possible for us to attain, but via our abilities in the here and now. And other differences will also emerge. 7 Mark Balaguer, Platonism and Anti-Platonism in Mathematics (Oxford University Press, 1998). Although I will object to its answers for our questions here, there is much to admire in this very substantial book. 8 Perhaps “restriction” of our beliefs to consistent sentences does not involve our believing that any such sentence is true. It may involve rather restricting our believing-true to consistent sentences (and, moreover, so restricting our believing-true-under-theintended-interpretation). Otherwise, if we opt for “believing that sentence x is true,” rather than “believing-true sentence x,” we are open to devastating objections. Thus consider the knowledge that the sentence ‘2 + 2 = 4’ (or any other such sentence, conditional or not) is true under the intended interpretation. This would seem to be a knowledge about us and about what interpretation we intend. But arithmetical knowledge is distinct from any such sociological or psychological knowledge. In believing that 2 + 2 = 4,

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see how this could help. What now takes the place of even the first premise? Try this: 1¢. Every consistent mathematical sentence is true under the intended interpretation. When you make a mistake in calculation, however, you believetrue a mathematical sentence under-a-certain-intendedinterpretation, but that sentence, although consistent, is not then true. Perhaps by “consistent sentence (or theory)” we should mean “consistent sentence (theory) as interpreted.” But then still: isn’t the interpreted sentence (theory) “2 + 2 = 5” consistent enough, though false? Unless one is a logicist (which I assume no-one is any longer, not even about arithmetic, and certainly not about the whole of mathematics) then the interpreted sentence “2 + 2 = 5” is just false, not inconsistent. Alternatively a mathematical belief may be said to be invariably conditional in the final analysis: in mathematics we always believetrue a conditional sentence whose antecedent conjoins the axioms of a certain mathematical theory along with various definitions, and whose consequent is the immediately attended to sentence. But this would make our belief in the axioms trivial, and in any case seems plain false. Could you right now formulate a sentence stating the Peano axioms and correlated definitions, etc., so as to come up with the required conditional sentence? If not, would that preclude your appropriately believing that 2 + 2 = 4? Finally, in being aware of the consistency of a sentence is one not aware of something concerning an abstract object (the sentence)? So how can we reliably tell when these abstract objects are consistent? This seems just a special case of the original problem. We cannot retreat here to beliefs about contingent, spatiotemporal sentence tokens, moreover, on pain of the following consequence: that if the token involved had not existed, then it would not have been true that 2 + 2 = 4. When we believe that 2 + 2 = 4, we are not just believing something about tokens. moreover, I do not just believe that ‘2 + 2 = 4’ is true (either true under some interpretation, or under the intended interpretation, or true simpliciter). One can believe that 2 + 2 = 4 without ever having heard of the sentence ‘2 + 2 = 4’, without understanding it, and in any case while having no beliefs about it.

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*************** A theory about the perceptual instruments used in a certain field – the microscopes, telescopes, and even eyeglasses – can help us understand how we know facts in that field. The same goes for direct observations aided by our most basic instruments, our sensory organs. But none of that illuminates how we can know by introspection facts of our own mental life, or how we can know such contingencies as the fact that one now thinks or even the fact that one now exists. In these cases the explanation of reliability stays largely within the realm of the a priori in the fashion of Descartes (in a broad sense of the a priori as that open to epistemically justified belief independent of perceptual observation). It is because we see how such beliefs are and must be infallible that we understand how perfectly reliable they are. No doubt here derives from ignorance of causal mechanisms. Moreover, such explanation of reliability is extensible far beyond cogito propositions. For example, we may similarly explain how we are so reliable regarding propositions expressed by “This exists” affirmed about whatever one picks out thus in thought, and how we are so reliable regarding propositions expressed by “This is thus” where the thought latches on to an image, say, and attributes to it a color or shape intrinsic to that image, as one that then characterizes the image.9 So, perhaps here again we might appreciate a priori how reliable such thoughts must be, as reliable as the cogito thoughts distinguished by Descartes. Some traditional paradigms of reliability are comprehensible, then, through reflection about the content-determining conditions of our thought. Might we possibly extend such reflection, driven by similar curiosity, so as to help explain how reliable we are in other realms, including that of logic, propositional or quantificational? Results in proof theory, or in metatheory more generally, might thus explain why it is that our thoughts in the relevant fields are likely to be right, or even bound to be right, if 9 I assume here that the conditions of contextual reference through the use of “thus” enable its use, aided by some mechanism of attention, to pick out a color or shape as present in that very image. Of course the requirement of such a mechanism of attention would seem to import a need for some presupposed nondemonstrative concepts to give content to the attention, which means that such demonstrative reference could not be conceptually fundamental.

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we follow certain methods. What rules out the possibility of such general understanding of our own reliability on the a priori, precisely by means of properly directed a priori theorizing? Compare our potential explanation of how perceptual observation is reliable as a source of true perceptual beliefs, and how scientific theorizing is reliable as a source of scientific beliefs. It is hard to see how these explanations could fail to derive precisely from perceptual observation and scientific theorizing. It is true that if we move beyond theorems known by proof, to items known directly, with no benefit of proof, then the circle tightens threateningly. Our reliable intuitive knowledge of modality and other matters open to a priori access seems subject to limits not knowable except through such intuition. But this gives sufficient reason neither to reject such intuition nor to judge its ontological status, not without first comparing our knowledge of the cogito, a knowledge defensible independently of any specific ontological commitments. Our cogito knowledge is defensible via appeal to its infallible reliability, in the fashion of Descartes, without this requiring any such commitments. And the same seems true for our introspective knowledge more generally. Unless one’s critical attack is to extend beyond philosophy and beyond platonism in mathematics, therefore, one will need to show what distinctive features of modal and mathematical intuition entail that it must be shown reliable, perhaps through a causal or judgment-dependence account, in some way that would require specific ontological commitments. Why should this be so, if our knowledge of the cogito and of our own headaches requires no such commitments? Your belief at a given moment as to whether you exist at that very moment can enter into no causal relationship with the fact believed so as to enable a causal explanation of the reliability of that belief. There is here no more possibility of a causal relation between our beliefs and the relevant facts than there is between mathematical beliefs and the mathematical facts believed. Nor is it plausible that your existence is at that moment determined by what you believe or might best believe on the matter. It is plainly false that you exist simply because your best opinion as to whether you exist would be affirmative. For the cogito the explanation of infallible reliability proceeds therefore in some way that skirts both causal tracking and construction or judgmentdependence. © Blackwell Publishing Ltd. 2003

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Could we reasonably expect to acquire the required epistemic perspective on the reliability of our mathematical beliefs through a priori reflection of the sort featured in Descartes’s philosophy of the cogito? On a superficial view, even Descartes seems to complete only half the job. He seems able to explain our infallible reliability in cogito thoughts, but not thereby to explain how we so effectively avoid any contrary thoughts. It is only when we see the cogito as not only infallible but also indubitable (upon consideration) that we grasp the fuller Cartesian account. Once we see that you cannot concurrently assent to a proposition and also to its very negation, it becomes clear how for Descartes we are really reliable about whether the cogito is true, and not just reliable in thinking that it is true. And that stronger reliability is captured more fully by the fact that at any given time we would believe it upon consideration, infallibly so, and never its negation. Why is this so? Thinking that we think and exist is safe even from the evil demon, but that is not quite enough; it is required also that we explain why it is that we opt always for the positive and never for the negative on whether we think and exist. And what could possibly explain this if not the fact that we would not so much as understand the question if we opted for the negative. After all, you lack so much as the ability to pose the question whether you now think and exist unless you always answer yes, never no. Analogously, we need to account not only for the reliability of our correct arithmetical beliefs, but also for how we so reliably avoid incompatible and, of course, false arithmetical beliefs. A theory of understanding seems again an attractive recourse, one that might explain how the very understanding of certain concepts – such as our concepts of existence, and of thinking, and simple numerical concepts – requires some complement and preponderance of correct beliefs involving them. This might potentially help explain both our tendency to be right in cogito thoughts and arithmetical beliefs, and also our avoiding error on these subjects. Whether that project can succeed or not, anyhow, the fact remains that its success would give us an a priori component for our desired epistemic perspective, a component that in the respect of being substantially a priori would match the Cartesian epistemic reflections traditionally accorded the highest explanatory efficacy in epistemology. And, again, these components of our epistemic perspective would require no specific ontological © Blackwell Publishing Ltd. 2003

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commitments, nor any a posteriori causal story of the sort pertinent to perceptual knowledge.10 ************ Accounts of a priori justification often postulate some basic sui generis source of such justification. It might be called intuition, or insight, or apparent insight. However labelled, such a source is proposed as fundamental, although further inferences can then be drawn from its immediate deliverances, as in some proofs of theorems from given axioms. And it is often argued that such intuition or insight works by means of a distinctive intellectual appearance that grounds the a priori as does sensory appearance the a posteriori. Must such appearances be infallible? Recent opinion preponderantly answers in the negative. Intuition is said to serve as fundamental source of a priori justification despite its fallibility – an answer supported by familiar paradoxes. It is plausible to consider simple arithmetical beliefs as again based on no first-level evidence or inference or calculation (unlike the solution of a complex arithmetical problem). Such simple beliefs seem justified by an intellectual appearance of truth derived from the bare understanding of the proposition in question. And, in simple arithmetic, this understanding seems largely based on certain drills. The rote learning of the tables must be viewed not on the model of testimony, however, but rather as a way to acquire not only belief but also understanding. For many simple propositions, the tables coordinately provide both belief and understanding. Phenomenologically such beliefs would seem to derive simply from one’s understanding of their contents. Admittedly, the drills that foster understanding of arithmetic concepts might be flawed in particular cases, limiting the value of the resulting appearances as sources of justification. Already the paradoxes suggest that an appearance can be false and no less robust, powerful, and entrenched for all that, and still apparently a source of justification. Here again we see intellectual appearance that is robust, powerful, and entrenched, but now perhaps only a limited and imperfect source of justification. 10 Further discussion of how empirical considerations might bear on a priori knowledge may be found in Albert Casullo’s “A Priori Knowledge Appraised,” in Albert Casullo, ed., A Priori Knowledge (Aldershot: Dartmouth Publishing Company, 1999).

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Simple arithmetical beliefs are the case in point, but only if the drills can be reliable enough to instill understanding without being reliable enough to yield the degree of epistemic justification requisite for knowledge. For example, perhaps the manual used has enough nearby errors to make it doubtful that the subject is epistemically justified in believing that 9 ¥ 8 = 72, even while remaining overall a good enough manual to give understanding of arithmetic in general and of that proposition in particular. Nevertheless, the conditions that attach contents to the relevant thoughts and sayings come with a guarantee that the contents attached will be true at least preponderantly, and this seems sufficient for some at least minimal degree of epistemic justification. Moreover, the degree of justification would seem to depend on how crucial it is to believe the particular proposition involved (or how crucial it is to believe a large enough preponderance of the members of some limited family to which that proposition belongs) if one is to grasp the relevant properties and concepts. It is this element in particular that would be shared with cogito thoughts. In both sets of cases, the conditions determinative of content determine also that the contents must be true, in every case or at least in most cases. And it is this understanding-based appearance that is characteristic of a priori intuition or insight, and that secures reliability, for both the cogito and arithmetic. Neither for the cogito nor for arithmetic does our explanation imply any conclusion about the allowable pertinent ontology (well, other than the thinnest of conclusions, namely that one does exist; what is left wide open is the ontological nature of this self). The reliability of the cogito is compatible with nearly any ontology of the self. And the like may then be true of arithmetic. How, it may be asked, could the tables impose a reference to platonic items outside space and time? Isn’t a platonic world of numbers ruled out? Well, how does one secure reference to oneself through the cogito despite the absence of any relevant relation, whether causal or constructive (or of judgment-dependence), between one’s existence and one’s thought that at that very moment one does exist? Here the connection is secured through a fact known a priori: namely, that when someone has a thought of that sort the thought refers to the thinker. What precludes the platonist’s saying that when we have thoughts of the arithmeticaltables sort, we can know a priori that such thoughts are bound to © Blackwell Publishing Ltd. 2003

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be right? This would still leave wide open the ontology of arithmetic, just as the cogito and Descartes’s explanation of its reliability leave open the ontology of the self. All we need for now is our ability to see a priori that thoughts and sayings of certain sorts are bound to be right (at least preponderantly).

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