Abstract
Analytic logic is obsessed with kee** reasoning free from error, as is seen most clearly in the developments of the philosophy of mathematics between Frege’s work in the late nineteenth century and Gödel’s second theorem: a sustained attempt at founding arithmetic on the bedrock of a logic that incorporated naïve set theory, the collapse of that attempt with the discovery of the paradoxes, Hilbert’s program of vindicating all mathematical theories by proving their consistency, and a final resolution that no such proof was possible, even for the most elementary mathematical theory. Mathematics was then going to be practiced in the context of an ineliminable risk of error—and that was understood as a limiting, negative result.
In dialectical logic, error has no currency, because everything anyone believes, at any stage of the total narrative, is dialectically justified. There is still plenty of room, however, for relative error: for something that is false with respect to a truer, later phase of the narrative. And, in this sense, dialectical logic embraces error: the starting point of every dialectical argument is a rough, or immediate, position that through the course of the argument evolves into greater and greater truth.
In oceanic logic, truth inevitably becomes error: the ordinary truth that a man is uncontroversially bald, or white, when subjected to the relentless attacks of sorites, turns out not to be true after all—the man is just as much non-bald or nonwhite. This is seen most clearly in a class of paradoxes that are not often lumped together with the sorites but in fact express the same attitude: Zeno’s paradoxes about space. There, too, uncontroversial truths about reaching a certain destination, or overcoming a contender in a race, are challenged and called in question. Ultimately, oceanic logic invites us to reject the arrogance of truth and learn to live in the territory of error.
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Notes
- 1.
“As I reflect on these matters more attentively, it occurs to me first of all that it is no cause for surprise if I do not understand the reasons for some of God’s actions; and there is no call to doubt his existence if I happen to find that there are other instances where I do not grasp why or how certain things were made by him. For since I now know that my own nature is very weak and limited, whereas the nature of God is immense, incomprehensible and infinite, I also know without more ado that he is capable of countless things whose causes are beyond my knowledge” (pp. 38–39). “I cannot … deny that there may in some way be more perfection in the universe as a whole because some of its parts are not immune from error, while others are immune, than there would be if all the parts were exactly alike” (pp. 42–43).
- 2.
“No one shall be able to drive us from the paradise that Cantor created for us” (Hilbert 1967, p. 376).
- 3.
“In the present situation … [the] problem of consistency is perfectly amenable to treatment. As we can immediately recognize, it reduces to the question of seeing that ‘1 ≠ 1’ cannot be obtained as an end formula from our axioms by the rules in force, hence that ‘1 ≠ 1’ is not a provable formula” (ibid., p. 383).
- 4.
“[T]here is no sin that cannot be forgiven, except for the sin against the Holy Spirit, the denial of spirit itself; for spirit alone is the power that can itself sublate everything” (1984/1988 III, p. 235). “The sins of him who lies against the Holy Spirit cannot be forgiven. That lie is the refusal to be a universal, to be holy, that is to make Christ become divided, separated” (1995 I, p. 74; translation modified).
- 5.
In (2005), originally published in 1949, after he reports witnessing events that, as he puts it, “followed vertiginously, as if all the Furies, long kept repressed, having torn down the gates, had rushed into the world, igniting wars and revolutions, offering spectacles of horror that surpassed those of the barbaric ages, because they had available, in great plenty, the technical means provided by civilization,” Croce offers as consolation the “pedagogical effect” these events had in exploding the “sense of security” people had acquired at the end of the XIX century, and in recalling everyone to the duty of “defending one’s own faith and spending for it one’s own forces” (pp. 301–302; my translation). A large part of what makes such complacent reconstructions possible, of course, is that they are reconstructions: that, as we have seen, they situate themselves at the end of a given development and draw what is invariably a positive moral from it. As Kierkegaard illustrated in vivid detail, things look quite differently when you are situated inside that very development, with no clue as to how it will turn out; then, self-serving after-the-fact wisdom gives way to anxiety and turmoil. The personal episode I described in Chap. 5 brings out a similar dilemma.
- 6.
Of the major works published during his lifetime, the Phenomenology of Spirit contains a 45-page preface and an introduction; the Science of Logic contains two prefaces (for the two editions), an introduction, and a section of uncertain editorial role, since it nominally belongs to Book I but is titled With What Must the Science Begin? (some 60 pages altogether); the Encyclopedia contains three prefaces (for the three editions), an introduction, and (again!) a section that nominally belongs to the First Part but is in fact a Preliminary Conception of it and summarizes the entire previous history of philosophy, followed by a More Precise Conception and Division of the Logic, for a total of over 130 pages; the Elements of the Philosophy of Right have a preface and an introduction that together come to over 50 pages.
- 7.
See Aristotle’s Poetics 1450b.
- 8.
“Knowledge is belief about things that are universal and necessary, and there are principles of everything that is demonstrated and of all knowledge (for knowledge involves reasoning). This being so, the first principle of what is known cannot be an object of knowledge, of art, or of practical wisdom; for that which can be known can be demonstrated, and art and practical wisdom deal with things that can be otherwise. Nor are these first principles the objects of wisdom, for it is a mark of the wise man to have demonstration about some things. If, then, the states by which we have truth and are never deceived about things that cannot—or can—be otherwise are knowledge, practical wisdom, philosophic wisdom, and comprehension [noûs], and it cannot be any of the three (i.e. practical wisdom, scientific knowledge, or philosophic wisdom), the remaining alternative is that it is comprehension that grasps the first principles” (Nicomachean Ethics 1140b–1141a).
- 9.
“Knowledge (epistéme), then, is a state of capacity to demonstrate and has the other limiting characteristics which we specify in the Analytics [where the theory of syllogisms is spelled out]” (ibid. 1139b). “[T]he wise man must not only know what follows from the first principles, but must also possess truth about the first principles. Therefore wisdom [sophía] must be comprehension combined with knowledge” (ibid. 1141a).
- 10.
“[D]ialectic is the only field of enquiry which sets out methodically to grasp the reality of any and every thing. All the other areas of expertise, on the other hand, are either concerned with fulfilling people’s beliefs and desires, or are directed toward generation and manufacture or looking after things while they’re being generated and manufactured. Even any that are left—geometry and so on, which we were saying do grasp reality to some extent—are evidently dreaming about reality. There’s no chance of their having a conscious glimpse of reality as long as they refuse to disturb the things they take for granted and remain incapable of explaining them. For if your starting-point is unknown, and your end-point and intermediate stages are woven together out of unknown material, there may be coherence, but knowledge is completely out of the question” (Republic 533b–c).
- 11.
“All men by nature desire to know. An indication of this is the delight we take in our senses; for even apart from their usefulness, they are loved for themselves; and above all others the sense of sight. For not only with a view to action, but even when we are not going to do anything, we prefer sight to almost everything else. The reason is that this, most of all the senses, makes us know and brings to light many differences between things” (Metaphysics 980b).
- 12.
“[I]t’s out of the question for a legal expert to be brought up, from childhood onwards, in the company of minds which are in a bad condition, and for his mind to have thoroughly explored the whole arena of immorality until it has become immoral itself and can quickly use itself as a criterion by which to assess the immorality of others’ actions…. Instead, his mind must, while young, have no experience of bad characters, and must not be contaminated by them, if it’s to become truly good at assessing the morality of actions in a reliable manner…. A good legal expert must have been slow to learn the nature of immorality, because he’s been observing something which is not an inherent quality in his own mind, but an alien quality in other people’s minds. He must have trained himself over many years to discern its badness by making use of information, not his own experience” (409a–c).
- 13.
This is not just a metaphor (is anything?): choosing immunization requires adopting a certain logic of “health,” and so does refusing vaccines. Similar remarks apply to, say, displaying your valuables versus kee** them under lock and key. (Venice has been compared to a theater and Genoa to a safe.)
- 14.
Without entering here into complex and controversial interpretative issues, I can at least point out that the most general understanding of Parmenides’ only work, the (fragmentary) poem On Nature (for which see Kirk et al. 1983), is germane to oceanic logic as I am presenting it here: it contrasts a world of reality that is whole, uniform, timeless, necessary, and continuous with a world of opinions and appearances that induces separation and change.
- 15.
The reason why these paradoxes are typically not lumped together with the sorites, I believe, is that superficially they seem to be going in opposite directions: the latter make us merge together what appeared radically divided; the former make us irreparably divided from what appeared to be in our grasp. But this superficial impression can be reversed (thus hel** us appreciate the deep similarity between them): you thought that you could go smoothly from a non-heap to a heap and yet, one grain of sand at a time, you will never get there; you thought there was a radical distinction between what you can and what you cannot reach, and yet there isn’t—the reachable is also unreachable. In fact, as will surface shortly in the text, both sorts of oddities originate from the same source: there is an irreducible heterogeneity between the continuity of the world and the discreteness of the measuring—or conceptual—devices we superimpose on it. A heap is not made of grains of sand, and a distance is not made of individual, separate points.
- 16.
“In its ordinary use the term ‘empty’ usually refers not to a place or space in which there is absolutely nothing at all, but simply to a place in which there is none of the things that we think ought to be there. Thus a pitcher made to hold water is called ‘empty’ when it is simply full of air; a fishpond is called ‘empty,’ despite all the water in it, if it contains no fish; and a merchant ship is called ‘empty’ if it loaded only with sand ballast. And similarly a space is called ‘empty’ if it contains nothing perceivable by the senses, despite the fact that it is full of created, self-subsistent matter; for normally the only things we give any thought to are those which are detected by our senses” (Principles of Philosophy, p. 230). “I have an appointment with Pierre at four o’clock…. But now Pierre is not there…. So that what is offered to intuition is a flickering of nothingness; it is the nothingness of the ground, the nihilation of which summons and demands the appearance of the figure, and it is the figure—the nothingness which slips as a nothing to the surface of the ground” (Sartre 1992, pp. 40–42).
- 17.
“[I]n principle the chair can never touch the wall, even if the space between them should be equal to zero. If the chair could touch the wall, this would presuppose that the wall is the sort of thing ‘for’ which a chair would be encounterable” (1962, p. 81).
- 18.
In Chap. 4 I showed that sorites are reversible: by reversing the one proving that every man is white, you could prove that every man is black. The same is true of Zeno’s paradoxes (which is another point of resemblance between them). Assuming that I am touching something, would moving away from it by an imperceptible, infinitesimal degree amount to no longer touching it?
- 19.
Mathematical physics will tell you that Achilles’ movement can be divided in such a way that one component of it is on the exact same line as the tortoise’s. But that, as will become clear below, is one more case of superimposing a measuring/conceptual scheme onto a world that is foreign to it.
- 20.
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Bencivenga, E. (2017). Truth. In: Theories of the Logos. Historical-Analytical Studies on Nature, Mind and Action, vol 4. Springer, Cham. https://doi.org/10.1007/978-3-319-63396-1_6
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