"General Relativity: It's Funny Cuz It's True."
-- Elliam Fakespeare
(That's the thesis of these two related posts.)
[It dawned on a couple days later to add to the picture the idea that my responsibility might not even obtain unless it were also coincidentally determined that the judge convicted me of homicide. Failing that, at the end of the whole ordeal, I would not be guilty––despite the facts of my intentions and actions!]
POSTULATE 1: An object A cannot be in more than one place P at one time T. If A were in more than one place P at T, it would eo ipso cease to be a single object A. That is just what we mean by A being an object as opposed to two objects B and C. This is a basic characteristic of what it means to be a physically delimited (and thus quantifiable) object. A's objective unity allows for compossible, internal margins, but not noncontiguous, external boundaries.
CONSEQUENT 1 from AXIOM 1 + POSTULATE 1: Nature as a whole cannot be in two places at once, if for no other reason that there is no larger “space” in which nature can exist at this or that location. A further reason why nature as a whole can no more be in one place P than A can at a single T, is that A's objective existence in a presumed whole-nature at T would require A to be as objectively existent in some other whole-nature* at T, which, again, violates POSTULATE 1 and the unity of corporeal objects.
OBJECTION 1: Perhaps nature is infinite (and eternal) and therefore at all places (and all times) at once.
REPLY 1: An infinite quantity is a contradiction in terms, since a quantity is only possible by being physically, or even just conceptually, distinct from and delimited by some other quantity.
CONSEQUENT 2 from REPLY 1: Nature as a whole is not infinite and eternal. If it is, then it is no longer a physical reality, subject to quantification-mensuration.
POSTULATE 2: An infinitely thin surface cannot exist in nature. In our normal experience, the thinner something is, the sharper it is and the more easily and deeply it can cut into something else. But if we extend our mind along an infinitely thinner and thinner blade, we can quickly see that something goes wrong, as it were, in natural terms. For, while any surface asymptotically on its way towards infinity with indeed cut magnificently into physical reality, once it so to speak “reaches infinite thinness,” it will no longer cut anything, since an infinitely thin cut into something is physically equivalent to no cut at all. An infinitely small gap between two objects is actually no gap at all. When infinity is applied to nature as a presumably physical reality, we see that, to put it mildly, funny things happen.
CONSEQUENT 3 from POSTULATE 2 + AXIOM 1: Therefore, again, infinitely great––or minor––measures do not pertain to nature and natural objects, nor to nature as a whole. As such, it is illegitimate to refer to nature as a whole as both physical and infinite-eternal.
Some thinking (judgment) is determinate in a way no physical process can be. Consequently, such thinking cannot be (wholly5) a physical process. If all thinking, all judgment, is determinate in that way, no physical process can be (the whole of) any judgment at all. Furthermore, "functions" among physical states cannot be determinate enough to be such judgments, either. Hence some judgments can be neither wholly physical processes nor wholly functions among physical processes.
We do not really add, either; we just simulate addition. Pure addition is just as much an idealization as E = mc^2. Of course, we can define such pure functions but cannot realize them; that is just a case of the many functions we can define which cannot be computed by any finite automation, or any other computer either. In a word, the fact that there is no pure addition and no pure conjunction or modus ponens is no odder than the fact that there are no perfect triangles.
1 After three centuries of amazingly successful science, we do not have a successful explanation of animal cognition, not even for a spider or a fish. Probably, we have been misconceiving the project in ways that makes science both less productive and less helpful.
2 Thinking here means "judgmental understanding"––what Aristotle thought to be the actuality of the intellect (De Anima, bk. 111, ch. 4, 429b, 30: "Mind is in a sense potentially whatever is thinkable, though actually it is nothing until it has thought"). There are many kinds of thinking; some thinkings are bodily doings, like my pouring a liquid. But it is only the processes of understanding that I am now trying to show cannot be wholly physical; understandings that involve feeling cannot be entirely nonphysical either, any more than my going for a walk can be a mere willing.
3 See Aristotle's argument (De Anima, bk. 111, ch. 4, 429a, 10–28; see also Aquinas's commentary in Aristotle's De Anima in the Version of William of Moerbeke and the Commentary of St. Thomas Aquinas, Kenelm Foster and Silvester Humphries, trans. (New Haven: Yale, 1959 repr.), sec. 684–6, pp. 406–7) that the understanding cannot have an organ as sight has the eye (and nowadays philosophers suppose thinking has the brain), because the limited physical states of an organ would fall short of the contrasting states of understanding that we know we can attain.
4 Philosophers should not recoil with distaste at such remarks about thought, because they attribute even odder features to propositions, e.g., being infinite in number, belonging to a tight logical network with formal features like "excluded middle," and being such that every one is determinately either logically related, by implication or exclusion, or logically independent of every other; in fact, in a system of material implication, no proposition is logically independent of any other.
5 But in part, yes, in the sense that my utterances are physical. Moreover, the thought may not even be possible apart from feeling or sense, just as a gesture is not possible without bodily movement. The target in this paper is theories that thoughts are "no more than" physical or functions determined physically; not that, for us, they are "at least physically realized."
6 I am not, of course, suggesting that a valid course of reasoning is not also a case of a variety of invalid forms, e.g., "P, therefore, C." But it must determinately be a case of some valid form.
7 Some conjunction tasks seem possible that are not: e.g., to conjoin all statements that can be expressed in English. That impossibility is not because of some fuzziness about the function "conjoin," but because the supposed totality is incoherent. You cannot add up all the even numbers, taken pairwise, just as you cannot conjoin all the sentences of English. See note 10.
8 We can even add certain nonterminating decimals, like .33333 and .66666 carrying from infinity to get 1. That is a form of understanding.
9 Equivalent but nonsynonymous functions would give the same arrays from inputs to outputs. Besides, a device that went to an address for the answer, and took it out in an envelope (encoded), which it did not open (decode) but handed to you (displayed for you to decode), could be made to produce the same array of outputs as addition. Yet it would not be adding. Besides, look at this function: 10 Z = X*X*X, 20 Print Z; 30 X = X + 1; 40 GOTO 10. That is a machine function for an endless loop to print the cube of every number beginning with zero. You can see that no matter what outputs the machine gives, it might have been doing something other than printing successive cubes, unless it produces all cubes––which cannot be done.
10 Postulating an infinity of cases will not suitably discriminate the functions that are the same for even numbers but differ for odd numbers after N. Postulating that "all" the cases are actual involves an incoherent totality, because the machine cannot both do all that it does and all that it might have done instead. Consequently, a pure function does not reduce to a pattern of inputs and outputs.
"All the additions" is as incoherent as "all the sets." So "what" addition is cannot be explained by "all the outcomes": rather, each and every outcome is determined by what addition is. It is impossible that all cases of addition be actual, even if infinities are performed because, even if we used up all the suitable numbers, the function itself would still be repeatable, say, for the same additions, but now done in a different order. The function cannot be exhausted by its cases, however many there are.
11 See "The New Riddle of Induction," in Fact, Fiction and Forecast, 2nd ed. (Indianapolis: Bobbs-Merrill, 1968), pp. 63-86.
12 Wittgenstein on Rules and Private Language (Cambridge: Harvard,
1982), p. 9. and passim.
l3 There is a complementary line of inquiry about immateriality. Christopher Cherniak argues (Minimal Rationality (Cambridge: MIT, 1986), p. 127) that because a physical object cannot be in an infinity of states, the mind treated as a brain computer is of limited understanding. That would be an understatement, were it true. Most of what actually happens would be unintelligible to us. An infinity of English sentences would be unintelligible, as would "most" truths of arithmetic.
For even if each of the finite number of electrochemical states the brain is capable of realizing actually happened, say, 10140 different thoughts, there would be an infinity of mathematical theorems we could not even understand because there would be no brain state or function among brain states to realize them.
The opposite seems to be true: there is nothing that is in principle unintelligible insofar as it has being, as Plato and Aristotle both thought. And we are able to be in an infinity of states of understanding, not successively but qualitatively. That is, we have the active ability to understand anything (accidents of presentation and of intelligence quotient being ignored for now). Thus, there is no arithmetical theorem we cannot understand, accidents ignored for now. Nor are there any well-formed utterances of any of the conjectured 10,000 human languages (most now lost) that we would not understand in the appropriate circumstances. But any one of those languages would require more than all the brain states. Brain states would have to be vehicles for varying content, perhaps media for thought and not the same thing.
Nothing is excluded because of its subject matter. Ours is not a successive infinite capacity (if we do not exist forever) but a selective infinite capacity. That is why the brain cannot even be the organ of thought, the way the eye is the organ of sight, as Aristotle, Avicenna, Averroes, Aquinas, and many others argued; otherwise, there would be something (that might be actually) that is unintelligible. Our corporeality imposes accidental limitations on understanding, the most important of which is that our contents of judgment have to be made by dematerialization (abstraction) and our intelligence cannot directly access immaterial being (e.g., angels or God). One consequence is the indeterminacy of contingent truth (see note 17).
How the dematerialization involved in our understanding something as shape (without consideration of which thing it is, or of its particular material composition) or our understanding something as in being (without consideration of its being material) could even come about is totally beyond the resources of any known experimental or formal science.
l4 See Nancy Cartwright, How the Laws of Physics Lie (New York: Oxford, 1983); and Ian Hacking, Representing and Intervening (New York: Cambridge, 1983).
l5 General natures (e.g., structural steel) do "have" abstract forms, but are not "pure functions." Two humans, proteins, or cells are the same, not by realizing the same abstract form, but by a structure "solid" with each individual (but not satisfactorily described without resort to atomic components) that does not differ, as to structure or components, from other individuals. There can be mathematical abstractions of those structures, many of which we can already formulate (cf. Scientific Tables (Basel: CIBA-GEIGY, 1970)).
l6 I think Kripke (op. cit., pp. 21, 65, and 71) interprets what he regards as indeterminacy as to whether I meant plus or quus as the basis for alleging an indeterminacy about what I do. ("There is no fact of the matter.") I say this gets the explanatory order backward and invites mistaken conclusions.
l7 All thought, as content, is immaterial in two other ways. (1) It lacks the transcendent determinacy of the physical. A true judgment, "someone is knocking on my door," requires for its physical compliant reality a situation with an infinity of features not contained (or logically implied) in the true judgment. Thus, an infinity of determinate but incompossible physical situations could make the same statement true. (2) Any physical-object truth requires its truth-making reality to overflow the thought infinitely in the detail of what obtains. So every compliant reality is infinitely more definite then anything contingently true we can say about it. It takes a lakeful of reality for one drop of truth.
A second argument: Products of physical processes are transcendently determinate. But no product of the understanding has an infinity of content, not contained therein logically. So no physical product can ever be such a content of the understanding.
Some thinking is as much physical as it is immaterial. My walking, as an action, is as much a mode of thought as it is a mode of movement; yet no movement, however complex, could ever make a thought.
Leibniz says in section 17 of the Monadology (in Philosophical Papers and Letters, Leroy Loemker, ed. and trans. 2nd ed. (Dordrecht: Reidel, 1969), p. 644) that, if perception were supposed to be produced by a machine, we could make the machine on large scale and walk around in it like a mill; we would never find a perception, only the movements of wheels, gears, and pulleys. Similar reasoning is given in Leibniz's Conversation of Philarete and Ariste (Loemker, p. 623). I thank Margaret Wilson for pointing these passages out to me.
A third argument: The present cases concern the definiteness of the form of the thinking. A third, parallel argument can be constructed from the definiteness of the content of thought, that thought is definite among incompossible contents in a way no physical process can ever be. Similar underdetermination arguments apply.
Machines do not process numbers (though we do); they process representations (signals). Since addition is a process applicable only to numbers, machines do not add. And so on for statements, musical themes, novels, plays, and arguments.