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Induction on Identity


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#21 Stephen Boydstun

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Posted 08 December 2010 - 12:10 PM

—Nicolaus – Reasoning to Substance

 

As one would expect, Nicolaus discounted inferences to material causes along with inferences to material substance. Of causes in general, Nicolaus would ask whether the effect is identical to the cause. If they are identical, then they are not truly cause and effect. If they are not identical, then existence of the effect can be denied and existence of the cause affirmed without contradiction. We may believe there are necessary connections between repeatedly conjoined things or events, but if they are truly distinct, then there is no logical necessity in their conjunction. (Al-Ghazali, a philosopher-theologian in the Islamic tradition, came to the same point in the eleventh century; Weinberg 1965, 135, 89–90.) Moreover, we have no direct perception of any necessity between repeatedly conjoined things or events; we do not perceive any agency or patience in things or events observed to be conjoined (Weinberg 1969, 109–11).

 

This last contention would seem to be an overstatement since we plainly do perceive agency and patience in our own interactions with the world. Perhaps Nicolaus was thinking here only of physical occurrences to which one is not a party, but only an observer.

 

The contention of Nicolaus, and of Ockham, that an effect cannot be identical to its purported cause would also seem to be an overstatement, though only a mild one. There are cases in which it seems sensible to say that an effect is virtually identical to part of its cause. The image of an object in a mirror is virtually identical with the optical object before the mirror. There are other cases in which it seems sensible to say that a thing bears in itself part of the identity of its cause. Examples would be die-castings, fingerprints, and DNA sequences. Another example would be the causation of thunder by lightning. There are patterns in the lightning that persist in and can be perceived in the complex waveform of the ensuing thunder. A signature of the lightning is borne by the thunder it generates.

 

There is one sort of case in which it seems sensible to say that an effect is truly identical with its cause (cf. Nozick 1981, 118–21). We say, for instance, that our experience of the impenetrability of solid objects is caused, in part, by the repulsions of electrons bound to separate objects. In some sense, what we experience is just those electronic repulsions, yet we say the electronic repulsions caused the experience of impenetrability. Should we say that impenetrability is a form in which we can experience electronic repulsions? “Whatever is in something is in it according to the mode of that in which it is” (Aquinas 1975, 1.49.3). Very well, but it still seems sensible to say that the electronic repulsions are part of the cause of our form of experiencing them, that is, our experience of impenetrability (see further, Peikoff 1991, 41–48; Kelley 1986, 36–37, 83–95, 105–11; and Heil 1983, 63–69, 81–86).[9]

 

Lastly, in every case of alteration, it seems we may sensibly say that a necessary part of the cause of the altered thing is that same thing prior to its alteration. An altered thing inherits from its former self. What it was is partial cause of what it becomes.

 

More fundamentally, Nicolaus was wrong to hold that objective necessity obtains only across transformations that map a thing, or some parts of it, identically. There are necessary connections also between different things; at least there are prima facie necessary connections between different things. Could it be that where there is an ineradicable necessity, there is either a simple identity transformation, as Nicolaus proposed, or there are underlying identity transformations? In the latter case, where an antecedent and a consequent are underlain by an identity transformation, it might well be that denial of the consequent and affirmation of the antecedent does not itself compose a contradiction. Could it be, though, that identity is a more basic principle that non-contradiction, that identity is the deeper principle of reality, that contradictions are local conflicts with the principle of identity, that identity is not only a local but a global condition?

 

By the time one were four years old (earlier, by Macnamara 1986, 147–49), one had come to rely on underlying identities of the objects of perception. In identifying kinds of objects, especially natural objects, one had begun to rely not only on properties presently apparent in perception but to ask about presently unapparent properties. One had begun to seek the kind of thing a thing is and to use knowledge of kind, of categorical identity, to override obvious characteristics on their misleading occasions (Markman 1989, 95–135; Keil 1989, 195–215, 249–53).

 

“Children were taught a property of one object and a different property of a second one. They were shown a third object that looked much like one of the first two but was given the same category label as the other. For example, children saw a tropical fish and were told that it was a fish and that it breathes underwater. They saw a dolphin and were told it was a dolphin and that it pops out of the water to breathe. They then had to decide how a second fish, a shark that looks like a dolphin, breathes. Children relied on the shared category to promote inductions even in this stringent case where perceptual similarity would lead to a different conclusion. Moreover, [studies showed that] children’s inferences are based on common category membership and not just on identity of labels.” (Gelman and Markman 1986, 203)

 

Inference to the existence of the atomic chemical elements of matter is a very important case of reliance on underlying categorical identities. It is an inference to existents never directly perceived at all, just the sort of inference Nicolaus attacks when he attacks inference to material substance. Occasionally one sees in modern science texts or periodicals what appear to be photographs of individual atoms arrayed in some material. These are really not straight photographs. We have not perceived atoms so directly as in straight photographic perception. Objects in straight photographs are susceptible of being seen also directly or through simple optical magnification. Our “perception” of atoms is even now not direct but inferential (Wichramasinghe 1989).[10]

 

Well before man had such persuasive images of atoms, their existence could not be rationally doubted. By 1908 evidence of their existence was overwhelming. Evidence of their existence is fortified with every passing year in our glorious age, though our usual focus now is on exploring their properties rather than confirming their existence. We can take their existence for granted.

 

Inference to the existence of atoms is a case of induction in the genre of what William Whewell (1794–1866) termed the consilience-induction (Butts 1982, 153–55). By 1900 atoms and molecules were evidenced by Dalton’s law of multiple proportions, Gay-Lussac’s law pertaining to the volume of gases, Avagadro’s law (which made possible the determination of molecular weights), and the kinetic theory of gases (which could approximately predict molar heat capacities). After 1908, when Jean Baptiste Perrin published his results on the sedimentation distribution of (visible) particles suspended in a still liquid and his measurement of Avogadro’s constant, the existence of atoms could not be reasonably doubted (Wehr and Richards 1967,* 4–26; Nye 1972).[11] These lines of induction, and many others, converged in favor of the atomic hypothesis. The evidence was and is several and joint. In conjunction the strength of the evidence magnifies.

 

Consilience-inductions pervade the history of science. Newton’s gravitational law was such an induction, analytically, if not genetically. It unified phenomena formerly disparate, most conspicuously, planetary orbits and projectile trajectories on earth. Newton’s law of gravity and his concomitant theory of space have been displaced by Einstein’s general relativity. It seems to me that this can never happen to the atomic hypothesis (nor to the hypothesis that the earth is rotating for that matter). The atoms are here to stay. Our best theory of many of their properties, the theory of quantum mechanics, might someday be superseded, but atoms will remain theoretical entities for us and our descendents. And serious minds with some scientific education will continue to regard them not only as theoretical entities, but as concretely real things (Friedman 1983, 238–50). Neither atoms nor our atomic lasers (Newton-Smith 1981, 196) are going to go away however we may perfect or supersede our theories of matter in the future (see also Putnam 1988, 9–14, 22–26, 30–37).

 

I pound the atoms constituting these pages to make a philosophic point[12]: There is such a thing as non-deductive demonstration. There is such a thing as demonstrative ampliative induction.

 

Now I have contradicted myself. I said earlier* that ampliative inductive inferences, unlike deductive inferences, were not absolutely conclusive. In a significant sense, that is false. Usually it is true but not always.

 

My contention that there are some demonstrative, absolutely conclusive inductions would seem to bring me into opposition with the consensus among authorities on the subject: “Induction is not a demonstrative form of inference like deduction” (Braithwaite 1953, 257). “The distinction between valid deduction and non-demonstrative inference is completely exhaustive. Take any inference whatsoever. It must be deductive or non-demonstrative” (Salmon 1966, 20). I think perhaps we pass too hastily from (1) the thesis that if premises in a valid deduction are true, the conclusion cannot (cannot ever) be false and (2) the thesis that that is not so for inductive inference to (3) the result that inductions cannot be absolutely conclusive, that is, to the result that true premises in an inductive argument can never ensure truth of the conclusion. But as we know, by our knowledge that atoms exist, this last is not strictly true. It is sometimes the case that the truth of the premises in an inductive argument ensure the truth of the conclusion, but unlike any case of valid deduction, we are not informed of this by the principle of non-contradiction. This circumstance is what one might expect of indeed identity is the broader and deeper principle of reality and the mainstay of induction.

 

We should also underscore an error made by Ockham. He held that we could never demonstrate existence, only attributes. He was mistaken. Atoms exist. This we now know, proof positive. “Thus we understand truth by considering a thing of which we posses truth” (Aquinas 1947, Pt. 1, Q. 84, Art. 7, Reply Obj. 3; see also Goodman 1983, 64).

 

Continued below—

Hume – Experience of Cause and Effect

Hume – Reasoning to Cause or Effect
Hume – Necessity
Hume – Uniformity
Existence is Identity

 

References

 

Aquinas, T. 1947 [c. 1265-73]. Summa Theologica. New York: Beniziger Brothers.

——. 1975 [1259–68]. Summa Contra Gentiles. A. C. Pegis, translator. Notre Dame, IN: Notre Dame University Press.

 

Braithwaite, R. B. 1953. Scientific Explanation. Cambridge: Cambridge University Press.

 

Butts, R. E., editor, 1989 [1968]. William Whewell: Theory of Scientific Method. Indianapolis: Hackett Publishing.

 

Friedman, M. 1983. Foundations of Space-Time Theories. Princeton: Princeton University Press.

 

Gelman, S. A. and E. M. Markman 1986. Categories and Induction in Young Children. Cognition 23:186–209.

 

Godman, N. 1983 [1954]. Fact, Fiction, and Forecast. 4th ed. Cambridge, MA: Harvard University Press.

 

Heil, J. 1983. Perception and Cognition. Berkeley: University of California Press.

 

Keil, F. C. 1989. Concepts, Kinds, and Cognitive Development. Cambridge, MA: MIT Press.

 

Kelley, D. 1986. The Evidence of the Senses. Baton Rouge: Louisiana State University Press.

 

Macnamara, J. 1986. A Border Dispute: The Role of Logic in Psychology. Cambridge, MA: MIT Press.

 

Markman, E. M. 1989. Categorization and Naming in Children. Cambridge, MA: MIT Press.

 

Newton-Smith, W. H. 1981. The Rationality of Science. Boston: Routledge & Kegan Paul.

 

Nozick, R. 1981. Philosophical Explanations. Cambridge, MA: Belknap Press of Harvard University Press.

 

Nye, M. J. 1972. Molecular Reality: A Perspective on the Scientific Work of Jean Perrin. Canton, MA: Watson Publishing.

 

Peikoff, L. 1991. Objectivism: The Philosophy of Ayn Rand. New York: Dutton.

 

Putnam, H. 1988. Representation and Reality. Cambridge, MA: MIT Press.

 

Salmon, W. C. 1966. The Foundations of Scientific Inference. Pittsburgh: University of Pittsburgh Press.

 

Wehr, M. R. and J. A. Richards 1967 [1959]. Physics of the Atom. 2nd ed. Reading, MA: Addison-Wesley Publishing.

 

Weinberg, J. R. 1965. Abstraction, Relation, and Induction. Madison: University of Wisconsin Press.

——. 1969 [1948]. Nicolaus of Autrecourt. New York: Greenwood Press.

 

Wickramasinghe, H. K. 1989. Scanned-Probe Microscopes. Sci. Amer. (Oct.):98–105.

 

~~~~~~~~~~~~~~~~

This essay of 1991 had no notes. I will now add a few endnotes to indicate changes or emendations to the positions I took in this essay nineteen years ago. I will also add some hyperlinks within the text.

~~~~~~~~~~~~~~~~

 

Notes

9. See also a, b.

 

10. See also a, b, cd.

 

11. David Harriman (2008) places the point at which the atomic theory was inductively proven sometime after Maxwell’s kinetic theory of gases (1866) and not later than the confirmation of Mendeleev’s prediction of gallium (1875). His is not a claim about when all knowledgeable scientists accepted the atomic theory, but a claim about when all the elements of a rational proof of the theory were at hand. I hope later in this thread to look into whether the additional evidence and theory to 1908, my point (in the 1991 essay) of definitive proof for the atomic theory, fits naturally and entirely within the criteria Dr. Harriman has proposed for inductive proof of a theory. Meanwhile, note that criteria for rational induction purportedly sufficient to establish scientific theory in chemistry go back to Jakob Friedrich Fries (1801, 1822).

 

12. Written in 1991, when typewriter keys and printing press struck paper.



#22 Stephen Boydstun

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Posted 19 December 2010 - 01:41 PM

—Hume – Experience of Cause and Effect

We possess only fragments of the writings of Nicolaus of Autrecourt. Ecclasiastical authorities censured his ideas. Nicolaus was ordered to burn his writings publicly and to recant condemned propositions. This he did in Paris in 1347. We possess the writings of David Hume. Of special interest to us are Book I (Of the Understanding) of A Treatise of Human Nature (T), which appeared in 1738–40, and Enquiries Concerning Human Understanding (E), which appeared in 1748.*

“All the objects of human reason or enquiry may naturally be divided into two kinds, to wit, Relations of Ideas, and Matters of Fact. Of the first kind are the sciences of Geometry, Algebra, and Arithmetic; and in short, every affirmation which is either intuitively or demonstrably certain. . . . Propositions of this kind are discoverable by mere operations of thought, without dependence on what is anywhere existent in the universe” (E 20).[13] Mathematical, demonstrative science is a sort of rationalist’s paradise. “Every proposition, which is not true, is there confused and unintelligible” (E 132).

“Matters of fact . . . are not ascertained in the same manner” (E 21). When it comes to fact or existence, “whatever is may not be” (E 132). Why does Hume think that? Because “no negation of a fact can involve a contradiction” (E 132). Why does Hume think that? Because “whatever is intelligible, and can be distinctly conceived, implies no contradiction” (E 30) and “the non-existence of any being, without exception, is as clear and distinct an idea as its existence. The proposition, which affirms it not to be, however false, is no less conceivable and intelligible, than that which affirms it to be” (E 132).

Hume’s reasoning here is very rationalist and very weak. That one can conceive of the contrary of some matter does not always mean that that contrary is possible nor that it could not eventually be shown to be self-contradictory. One can conceive of the possibility that there is some odd counting number n such that (n2 – 1) is not evenly divisible by 4, but in truth there is no such number n. One’s ability to conceive of the “possibility” of things truly impossible could be due to a liberality or poverty in one’s concepts or due merely to a present unconcern for truth (Kneale 1949, 79–80; Quine 1980, 20–46; Peikoff 1990, 88–121; Rasmussen 1983; Armstrong 1989, 7–13, 45–51, 54–76).

Let us follow Hume’s enquiry into “the nature of that evidence which assures us of any real existence and matters of fact, beyond the present testimony of our senses, or the records of our memory” (E 21). Hume observes:

All reasonings concerning matter of fact seem to be founded on the relation of Cause and Effect. By means of that relation alone we can go beyond the evidence of our memory and senses. If you were to ask a man, why he believes any matter of fact, which is absent; . . . he would give you a reason; and this reason would be some other fact . . . . A man finding a watch or any other machine in a desert island, would conclude that there had once been men in that island. All our reasonings concerning fact are of the same nature. And here it is constantly supposed that there is a connexion between the present fact and that which is inferred from it. Were there nothing to bind them together, the inference would be entirely precarious. (E 22; see also T I.3.2)

Hume contends that the connection we suppose to bind distinct factual states of affairs is the causal connection. That is the connection on which we rely when inferring one state of affairs from another. If we conceive causality in a very general way—along the lines of Ayn Rand’s conception—as the law of identity applied to action and to becoming, then we can say that Hume is roughly right in this contention. Hume will soon argue, however, that though we suppose these causal connections, we have no rational basis for believing in them.

In saying that distinct states of affairs are bound together by the causal connection, Hume is only roughly right. Where the distinct states of affairs under consideration are just a single, selfsame state of affairs at different times, the more salient connection is temporal, rather than causal. That is, when we are considering simple identity through time, it is more natural to speak of identity through time than of identity applied to action or becoming.[14]

Hume insisted that time is atomic. It comes in indivisible minimal, finite units. In his arguments for this conclusion, Hume reasons not so much about time as about extension and about quantity in general. He clearly intends these reasonings to be carried over to time as well (T I.2.2). In the imagination, Hume observes, we can subdivide an extension only so far; we cannot subdivide an extension into an infinite number of parts. “’Tis the same with the impressions of the senses as with imagination” (T I.2.1). There are thresholds of perception. As an object recedes, it does not become infinitely smaller yet visible; it becomes finitely small and then vanishes. As the impressions of our senses are not infinitely small and as extension is not infinitely divisible in imagination, physical space, and time, are themselves not infinitely divisible. “Wherever ideas are adequate representations of objects, the relations, contradictions and agreements of the ideas are all applicable to the objects . . . . Our ideas are adequate representations of the most minute parts of extension” (T I.2.2). However small a part might be, we can imagine it, and if we can imagine it, it will be finite (see also Fogelin 1988).

I think Hume resorted to such rationalism out of desperation. He was desperate to establish that time was not continuous. If time were continuous, his skeptical fun concerning the existence of enduring objects and the reality of causality would be hard getting started.[15] As we shall see, Hume’s programme would be in big trouble at the start anyway, though time were discrete.

“From the succession of ideas and impressions we form the idea of time. . . . Time cannot make its appearance to the mind, either alone, or attended with a steady unchangeable object, but is always disover’d by some perceivable succession of changeable objects” (T I.2.3; see also I.4.2). This is too simple. Were we perceiving a constant scene, we would still sense the passage of time, for we are living beings, and there is activity in us. Indeed, we are a living activity (Whitehead 1967, 143–44; Minsky 1988, 288). We cannot dwell on this defect of Hume’s account of experience here.

Focus instead on the following: “Wherever we have no successive perception, we have no notion of time, even tho’ there be a real succession in the objects” (T I.2.3). This proposition is inimical to the skeptical side of Hume’s project. He really cannot afford to be admitting that there are objective instants shorter than we might sense. If there are such, then when we sense things in an instant, we have really sensed them over an objective succession. (Hume cannot simply renounce objective grounds of succession if he intends to leave memory intact; see Bennett 1990, 222–29, and Harper 1984, 125–32.) As it turns out, this is just the way it woks. An instant of consciousness is of the order of milliseconds (10-3 s) or longer (Macar 1985; Edelman 1989, 127–33). Yet we know that as we observe an object for an instant the atomic constituents of the object are making insensible (and unimaginable) transitions down at 10-18 seconds, and some nuclear transitions are occurring in the object down at 10-23 seconds. In an instant of observation, there is an ocean of objective time. In an instant, we observe an object’s existence through time, but to us it is all at once. Hume does not want to impugn present testimony of the senses (nor memory); he needs present sense impressions (and memory) as the only valid source(s) for factual knowledge; he hopes to show that against present sense, inferential factual knowledge pales into pretension. Hume does want to impugn the selfsame identity of any object we observe through perceptible time. That project can be of no avail if in fact we already observe objects selfsame across objective time in every instant of observation (see Stroud 1988, 96–117).

We return to Hume’s charges against causal connections. First Hume emphasizes, like Locke and Ockham before him, that causes and effects are not discoverable by a priori reasoning but only by experience. That proposition seems true, at least if our conception of what it means is not too narrow. We should leave open the possibility that the human infant may be organically, genetically predestined to learn certain elementary facts about the world on the basis of the slightest personal experience (Minsky 1988, 115). If this be true, we would want to include pertinent learning (evolution) of the biological species in addition to individual learning per se when asserting that causal relations are gathered from experience only. Such a circumstance should worry Hume somewhat (but it does not; E 45). Some principles of demonstrative reasoning, the principle of non-contradiction in particular, as well as some principles of causal reasoning (which Hume takes to be apart from demonstrative reasoning) might be acquired not so much by experience of the individual as by organic development of the brain in infancy and antecedent evolution of the human brain. Then it becomes more difficult than ever to maintain that there really are any propositions “discoverable by mere operations of thought, without dependence on what is anywhere existent in the universe,” and the alternative “discoverable by a priori reasoning or discoverable by experience” becomes less momentous. We should also leave open the possibility that causes and their effects as such are in some cases, to some extent, perceived directly (Leslie and Keeble 1987; Heil 1983, 36–41).[16]

As every student of philosophy knows, Hume wants to establish more than the wholly empirical origins of our knowledge of causal relations. He wants to establish the further, extreme thesis that no object or event logically implies anything about its causes or its effects.

For the effect is totally different from the cause and consequently can never be discovered in it. . . . When, I see, for instance, a Billiard-ball moving in a straight line towards another; even suppose motion in the second should by accident be suggested to me, as the result of their contact or impulse; may I not conceive, that a hundred different events might as well follow from that cause? May not both these balls remain at absolute rest? May not the first ball return in a straight line, or leap off from the second in any line or direction? All these suppositions are consistent and conceivable. Why then should we give the preference to one, which is no more consistent or conceivable than the rest? (E 25)[17]

Hume’s answer is the same as Nicolaus’: habit. We are simply accustomed to seeing the second billiard ball recoil in particular ways (E 35–39). On the principles of habit, the child, like the animal, can come to anticipate that fire will burn (E 82–85). Now no one will deny that we retain our animal nature; we have cognitive resources in common with other animals. What will concern us here, though, is what further intellectual resources we have.

Hume asserts that effects are totally different from their causes, and this is somewhat misleading. The second billiard ball is not the first ball; in this sense, they are totally different. But the second billiard ball is very much like the first, and the motion of the second much like the motion of the first. There are symmetries. We earlier adduced cases of causality in which effects bore more conclusive signatures of their causes on their faces, but even in Hume’s example, the effect resembles the cause (cf. APo 2.16–17).

Hume concludes his famous billiards passage with the claim that all our a priori reasonings can never show any foundation for our expectations of what the second billiard ball will do (E 25). The sense of “a priori” here is unclear. If by “a priori reasonings” he means simply reasoning we might engage in just now while the first ball is on its way (very slowly, I hope) to the second, then I contest his claim.

Hume mentions some equally “consistent and conceivable” responses of the second ball to impact by the first. These responses are suspiciously tame; each could be readily engineered in reality. Imagine, instead, the following: upon being struck by the first ball, the diameter of the second is reduced, or the second disappears, or becomes a pulley, a loaf of bread, a live rabbit, a melody, or the letter B. These fanciful outcomes are as “conceivable” as Hume’s, but they are hardly as “consistent.” Why not?

One never observes simply two billiard balls colliding. There is always more than that present. There are accompaniments, usually taken for granted and not brought into focal awareness. We know (here, imagine) not only that before us there is one billiard ball rolling toward another, but that they reside on the table, that the table rests on the floor, that beneath the building is the earth (the same earth one was upon the day before and the day before that), that one has a body, that it has weight, that one is breathing, that the song playing from the jukebox was popular during WWII.

One billiard ball is rolling toward another. Upon collision something will happen. One knows that already. If the target ball has been screwed to the table from underneath, things would be just as they are now? Not quite. The target ball was rolling freely a few moments ago and has been resting quietly since then. We would have noticed someone securing it to the table (contrast with Nozick 1981, 222–23, and Goodman 1983, 72–81; against the latter, see Hesse 1974, 75–88). If the target ball is going to become a rabbit or the letter B, things have to be radically different than they are if the target ball is just going to continue being the same old ball. One knows that already (cf. Armstrong 1991, 46–49; Harper 1984, 119–22; Brittan 1978, 197–205).

It is not only the two billiard balls that are subject to the law of identity; it is not only their behavior that must be “consistent.” Everything else that there is or has been or will be, everything else along with the two billiard balls, every thing, individually and altogether, is subject to the law of identity. From identity, manifold identity, comes consilience-induction. “And all things are ordered together somehow, but not all alike—both fishes and fowls and plants; and the world is not such that one thing has nothing to do with another but they are all connected” (Metaph. 1075a15–17).

Continued below—
Hume – Reasoning to Cause or Effect
Hume – Necessity
Hume – Uniformity
Existence is Identity

References

Aristotle 1984 [c. 348–322 B.C.]. The Complete Works of Aristotle. J. Barnes, editor. Princeton. Princeton University Press.

Armstrong, D. M. 1989. A Combinatorial Theory of Possibility. Cambridge: Cambridge University Press.
——. 1991 [1983]. What Is a Law of Nature? Cambridge: Cambridge University Press.

Bennett, J. 1990 [1966]. Kant’s Analytic. Cambridge: Cambridge University Press.

Brittan, G. G. 1978. Kant’s Theory of Science. Princeton: Princeton University Press.

Edelman, G. M. 1989. The Remembered Present: A Biological Theory of Consciousness. New York: Basic Books.

Fogelin, R. 1988. Hume and Berkeley on the Proofs of Infinite Divisibility. Philosophical Review 97(Jan):47–69.

Goodman, N. 1983 [1954]. Fact, Fiction, and Forecast. 4th ed. Cambridge, MA: Harvard University Press.

Harper, W. L. 1984. Kant’s Empirical Realism and the Distinction between Subjective and Objective Succession. In Kant on Causality, Freedom, and Objectivity. W. L. Harper and R. Meerbote, editors. Minneapolis: University of Minnesota Press.

Heil, J. 1983. Perception and Cognition. Berkeley: University of California Press.

Hesse, M. 1974. The Structure of Scientific Inference. Berkeley: University of California Press.

Hume, D. 1975 [1893, 1748]. Enquiries Concerning Human Understanding. 3rd ed., L A. Selby-Bigge, editor. Oxford: Clarendon Press.
——. 1978 [1888, 1740]. A Treatise of Human Nature. 2nd ed., L. .Selby-Bigge, editor. Oxford: Clarendon Press.

Kneale, W. C. 1949. Probability and Induction. Oxford: Oxford University Press.

Leslie, A. M. and S. Keeble 1987. Do Six-Month-Old Infants Perceive Causality? In Thought without Language. Oxford: Clarendon Press.

Macar, F. 1985. Time Psychophysics and Related Models. In Time, Mind, and Behavior. J. A. Michon and J. L. Jackson, editors. Berlin: Springer-Verlag.

Minsky, M. 1988 [1985]. The Society of Mind. New York: Simon & Schuster.

Nozick, R. 1981. Philosophical Explanations. Cambridge, MA: Belknap Press of Harvard University Press.

Peikoff, L. 1990 [1967]. The Analytic-Synthetic Dichotomy. In Introduction to Objectivist Epistemology. Expanded 2nd ed., H. Binswanger and L. Peikoff, editors. New York: Penguin Books USA, Meridian.

Quine, W. 1980 [1953]. Two Dogmas of Empiricism. Reprinted in From a Logical Point of View. 2nd ed. Cambridge, MA: Harvard University Press.

Rasmussen, D. B. 1983. Logical Necessity: An Aristotelian Essentialist Critique. The Thomist 47:513–40.

Stroud, B. 1988 [1977]. Hume. London: Routledge.

Whitehead, A. N. 1967 [1925]. Science and the Modern World. New York: The Free Press.

~~~~~~~~~~~~~~~~
This essay of 1991 had no notes. I will now add a few endnotes to indicate changes or emendations to the positions I took in this essay nineteen years ago. I will also add some hyperlinks within the text.
~~~~~~~~~~~~~~~~

Notes
13. Correct affirmations discoverable by mere operations of thought are a priori, but in Hume’s account, they are not also analytic. That does not mean they are therefore synthetic. Hume’s category of things known with certainty by intuition or demonstration is a category of things known neither analytically nor synthetically (Allison 2008, 63–64, 76–83). Hume’s partition is not the analytic-synthetic partition, whose necessary and sufficient criterion of analyticity is self-contradiction under denial, every proposition not analytic being classed as synthetic (as with Leibniz’ fundamental division between truths of reason and truths of fact).

Known by mere operations of thought, in Hume’s view: the relation resemblance, the relation contrariety between existence and non-existence, the relation degree in a quality, and the relation proportion in quantity and number. Known only with information from sensory experience are the standing of things in: the relation identity (self-sameness across time), relations spatio-temporal, and relations causal (T I.1.5, I.3.5).

These seven relations are called philosophical relations by Hume. The first four guide thought, reflective imagination, with certainty; the last three guide with probability. Notice that association of ideas is not among these normative relations. Association is not a philosophical relation. It is a natural mental relation influenced by resemblance, contiguity, and cause and effect (T I.1.4; E 19). The sight of a wound leads one to think of the pain (E 19). The relation of cause and effect is both philosophical and mentally natural (T I.1.5). It would seem Hume must also regard resemblance and contiguity as not only philosophical relations, but mentally natural ones. How else can they be sources of the purely natural relation of association?

Hume tries to use the natural relation association to invalidate causality as a philosophical and rational epistemological guide (as conceived by his predecessors). In the Enquiry, Hume omits the list of philosophical relations, as well as the distinction between philosophical and natural relations.

I would say, contrary to Hume, that none of his philosophical relations are known apart from sensory experience of the natural world, however directly or reflectively they are connected to that experience. Furthermore, habit from association is not an adequate substitute for reason animated by what Hume had called the philosophical relation of cause and effect.

14. In the Treatise, Hume distinguishes between causation and simple identity through time (I.1.5). The latter is left out of account in the Enquiry.

In his 1908 book Identity and Reality, Emile Meyerson writes: "The principle of causality is none other than the principle of identity applied to the existence of objects in time" (page 43 in the 1930 English translation*). That leaves simple identity through time as a form of the causal relation, or at least it makes numerical identity through time a more salient part of the causal relation than in Rand’s formula: “The law of causality is the law of identity applied to action. All actions are caused by entities. The nature of an action is caused and determined by the nature of the entities that act; a thing cannot act in contradiction to its nature” (1957, 1037). (See further: a, b.)

Part of Rand’s law of identity is that one thing cannot simply turn into another without there being a distinctive reason. (Mere wishing, magical words, or mystical words are ruled out as not real reasons.) The birth of the infant mind “is the day when he grasps that the streak that keeps flickering past him is his mother and the whirl beyond her is a curtain, that the two are solid entities and neither can turn into the other, that they are what they are, that they exist” (1957, 1041).

15. It is not only for the sake of arguing the invalidity of the philosophical relation cause and effect that Hume maintains the atomicity and perceptual correspondence of time. The atomicity of time and its perfect concordance with moments of perception is an integral part of Hume’s way of judging the validity of empirical ideas: solely by their traceability to the mental particulars that are simple sense impressions (T I.1.1).

16. See also Gopnik and Schulz 2007.

17. I have exhibited the relationship between (i) the principle that every action-bearing entity bears certain kinds of action and not other kinds of action and (ii) the principle of non-contradiction here. Investigation of what are the kinds of action a certain entity bears will have its own distinctive relationships to the principle of non-contradiction, and constantly the investigation will presuppose (i) in its specially intimate relation with (ii). Not all imaginings of what actions a billiard ball might bear are equally consistent in the logical sense.

#23 Stephen Boydstun

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Posted 25 December 2010 - 01:53 PM

—Hume – Reasoning to Cause or Effect

Any and all things are situated. Billiard balls are in situations. The situations are in situations too. The weather is in a situation no less than a billiard ball. The range of situations in which a thing might be is a concomitant of a thing’s identity. The range of situations in which a thing might be is learned from one’s total experience.[18] The task of sorting out what is what and what can become what in what situations is undertaken not only by scientists and philosophers (Armstrong 1991, 137–39), but by everyone, child and adult (Keil 1989, 159–215).

Hume would want to emphasize that learning from experience is a matter of forming habits. He tried to squeeze far too much into that mode of learning. (On Hume’s attempt to explain generalization in terms of habit, see Stroud 1988, 38–41. Habituation accounts of conceptual knowledge have not fared well in recent years; see Rips 1989 and Jackendoff 1987, 143–48.) In his earlier work, the Treatise, Hume seems to be cognizant of complications. He allows that we sometimes reason about causes and their effects not only according to custom arising from similar past conjunctions, but from a principle of identity. He allows that in cases “more rare and unusual,” we may assist elementary custom by conscious reflection on past experience and arrive at “custom [belief] in an oblique and artificial manner.”

I explain myself. ’Tis certain, that not only in philosophy, but even in common life, we attain the knowledge of a particular cause merely by one experiment, provided it be made with judgment, and after a careful removal of all foreign and superfluous circumstances. Now as after one experiment of this kind, the mind, upon the appearance either of the cause or the effect, can draw an inference concerning the existence of its correlative; and as a habit can never be acquir’d merely by one instance; it may be thought, that belief cannot in this case be esteem’d the effect of custom. But this difficulty will vanish, if we consider, that tho’ we are suppos’d to have had only one experiment of a particular effect, yet we have many millions to convince us of this principle; that like objects, plac’d in like circumstances, will always produce like effects; and as this principle has establish’d itself by a sufficient custom, it bestows an evidence and firmness on any opinion, to which it can be apply’d. (T I.3.8)

We might want to qualify and refine the general principle to which Hume appeals, the principle that “like objects, placed in like circumstances, will always produce like effects,” but it is at any rate clear that Hume is here squirming out of his official position that knowledge of fact is simply a matter of custom or habit.[19] Also in the Treatise, when writing on our knowledge of the continued and independent existence of bodies (T I.4.2) and when writing on our understanding of why things, e.g., clocks, may sometimes behave one way and sometimes another (T I.3.7), Hume senses the inadequacies of his habituation account of inductive inference and tries to make accommodations. In his later work, the Enquiries, he deals with these complications either by not bringing them up or by offering only meager “hints “ of their solution (E 47).

Hume’s commonsense principle that same causes yield same effects was endorsed also by Aristotle: “It is a law of nature that the same cause, provided it remain in the same condition, always produces the same effect” (GC 336a27-28). Ockham endorsed the principle in a form close to Hume’s: “Causes of the same kinds are effective of effects of the same kinds” (Weinberg 1965, 142). Ockham took this principle to be necessary and self-evident. As the principle is formulated by Ockham or Hume, it is subject to two interpretations. One, a broad one, I shall endorse in a moment. The other—and this is what both Ockham and Hume (E 64) most likely meant—is just the principle as stated without ambiguity by Aristotle. I think we should be wary of Aristotle’s principle. Hereafter, I shall refer to it as the narrow mode of causality. Although it obtains throughout vast regions of our experience, throughout much of existence, it evidently does not obtain for physical processes in quantum regimes nor in classical chaotic regimes. I suggest we reformulate the principle more broadly, thus: “Identical existents, in given circumstances, will always produce results not wholly identical to results produced by different existents in those same circumstances.” Application of the law of identity to action or becoming would seem to require only this much (contrary to Peikoff 1991, 14–15).[20]

It is not always the case that identical things placed in the same circumstances yield a single (repeated) result. Some existents yield single distinctive results; others yield distributions of distinctive results. Only if one allows for the latter possibility in one’s construction of the principle that “same causes yield same effects” is it universally true. Within the realm of classical mechanics, which covers billiards, Hume is correct in saying that same causes yield same (single) effects.

Out of all the conditions that obtain in a situation, we typically take only one or a few as cause of some distinctive result, only a select portion of the ways in which the law of identity applies to an action or a becoming (Minsky 1988, 129; van Fraassen 1991, 318–27; Gasper 1991, 293–95). We try to discover among antecedent conditions ones that will make a certain result under a wide range of variations in the remaining variable conditions. In a general commonsense way, everyone knows that characteristic of causes. Then they know, though perhaps only dimly, the principles of induction enshrined by John Stuart Mill: the principle of agreement, the principle of difference, and the principle of concomitant variation (Mill 1973, 3.8; Copi 1961, 363-407; Kelley 1988, 276–87). Aristotle knew something of these principles (Phys. 199b15–20; Top. 146a2–12); more so did Robert Grossette, Albert the Great, Duns Scotus, Ockham, and Francis Bacon (Weinberg 1965, 136–45, 153). That these are effective techniques for arriving at causes is knowable from everyday experience and all the more from modern scientific practice.[21] To rely on these techniques is to affirm the principle of identity operative. In reasoning to causes, we follow most conspicuously not habit but identity.

In a primary sense, causes make things happen.[22] We witness one billiard ball setting another in motion. We would usually say that what happened was that one billiard ball struck the other and set it in motion. According to Hume, we have gotten the idea that the first ball made the second move by custom of having seen like conjunctions of billiard balls in the past. Then why could we not as well say that the second billiard ball’s acceleration is what causes the first ball to strike it rather than what we usually say? (In some circumstances, young children do take effects to precede causes; Bullock, Gelman, Baillargéon 1982, 216.) Perhaps we are more often interested in prediction than postdiction; animals are surely more oriented to the future than the past. Nevertheless, when we look only backwards, why can we not believe that the acceleration of the second caused the first to strike it? I suggest it is because we have direct experience of causal powers, contrary to the fabulous sayings of Hume (E 50), and we know where they lie in this situation (Boyd 1991, 355–66). The idea of causality is more than the (tightly fastened) idea of regular conjunction in regular temporal order even for the simple billiards case.

Hume acknowledges that we think of causes as having some sort of necessary connection with their effects. This sense of necessity he supposes to be a psychological compulsion arising from our past observations of repeated conjunctions between causes and their effects. “We immediately feel a determination of the mind to pass from one object to its usual attendant. . . . Upon the whole, necessity is something, that exists in the mind, not the objects. . . . Necessity is nothing but that determination of the thought to pass from causes to effects and from effects to causes, according to their experienced union” (T I.3.14). As Hume would have it, even the necessities we find in the interactions of our bodies with other objects are merely projections from necessities we find in our mental operations (T I.4.4). This is absurd. (This error has more subtle relatives, e.g., the thesis of Aquinas that necessity in affirmation of existence is gotten from necessity in intelligibility; see Hoenen 1952, 165–67, 179–81.)

By about three years of age, we have grasped the general commonsense adult principles of causality; we understand that occurrences have causes and that these causes come down to specific mechanisms. The striking of one billiard ball by another is perceptually salient. It is a sharp event. We naturally would seek a trigger of such an event, and in this case, what we would seek is readily apparent. All are agreed that the second ball’s being set in motion had a cause, namely, the first ball’s striking it. This is a causal explanation at the level of common sense. Even at this level, though, not all causal relations are so obvious. That the snowman melted because it became too warm or that the living room became filled with smoke because the damper was not open are a little less obvious. Identification of these causal relations requires, in addition to specific experience, a little more causal reasoning. Such reasoning is a major mode of induction.

It is not from more repetitions of experience, but from a more extensive network of experience and from greater skill in reasoning on causal principles, that we become capable of composing commonsense causal explanations (Bullock, Gelman, Baillergéon 1982).[23] Contrary to the implications of Hume’s model, repetition of experience is not the main influence in our identification of causes. Hume recognized the importance, for humans, of causal reasoning, but he managed to not notice that this is at variance with his insistence on the preeminent role of habit (E 84n1).[24]

When one comes to formulate ideas about inanimate motion more generally, things become much less obvious, and customary experience can put one in the wrong frame of mind. It was very difficult for man to get straight which motions needed to be explained, which motions have causes in the primary sense. Today, students of physics learn the answer when they are taught the law of inertia, the law that a body will continue at constant speed and in a straight line (or will remain at rest) unless acted upon by a force. At a more advanced level, students learn that and how the inertia principle has been recast in more general forms: in the Lagrangian mechanics, as an extremal principle which tells how a body will move when or when not subjected to external forces; and in general relativity, as the principle that free bodies travel along geodesics of spacetime whether curved or flat.

It is only the elementary form of the law of inertia that concerns us here. Aristotle and his followers held to principles contrary the inertia principle. Terrestrial bodies, when moving, naturally tend to move towards certain places of repose (Phys. 199b14–19, 208b9–15, 215a1–21, 230a19–231a17). The heavenly spheres, upon which ride the heavenly bodies, naturally and always move in circular ways. Here let us confine attention to terrestrial bodies whose natural direction of motion is towards the earth: the earthly bodies, not water, air, nor fire. Aristotle’s camp took the free fall of earthly bodies as natural and as standing in no need of special explanation; no external force is being applied to keep such bodies falling to the earth. Any other motion of an earthly body, any motion that is not free fall, needs special external explanation. Moreover, any motion at all requires some explanation. “Everything that is in motion must be moved by something. For if it has not the source of its motion in itself it is evident that it is moved by something other than itself, for there must be something else which moves it” (Phys. 241b34–36).

What could be more sensible? In our life experiences here on the surface of the earth, we have countless confirmations of Aristotle’s thesis every day. To get an object moving requires effort, to keep it moving requires effort, and the object will sooner or later return to rest. The strings of the harp will return to silence. For Galileo and his followers to propose that motion, provided it be uniform, required no explanation, no efficient cause, but that non-uniform motion, including coming to rest, did, they had to put on new thinking caps. (Although it does not affect my argument, I should note for historical accuracy that it was not until Descartes that uniform motion was surely taken to be along a straight line; Galileo took it to be along a circle of constant radius, with center at the center of the earth.) These men could not leave the answer to habitual experience (nor to tradition). Formulating physical principles simply according to the most usual observations would not have led men to the law of inertia. Until this law and its conceptual vantage were discovered, the scientific revolution could not happen (Butterfield 1965, 14–28, 67–85; McClosky 1983).[25]

In common sense and in most scientific reasoning, we make the tried and true presumption that occurrences have causes. Hume discussed this principle in the venerated form “whatever begins to exist, must have a cause of existence.” He contended that this principle is not a necessary truth. He may have actually doubted the truth of the principle (T 78–82). We should distinguish two interpretations of the principle. In one we take cause as material cause, and in the other, we take cause as efficient cause. As to material cause, the principle seems to have held up perfectly in the two and a half centuries of science since Hume. It holds for all elementary particles; every particle gets made from some others. As to efficient cause, the principle holds always for cause in the broad mode; each type of elementary particle has its distinctive ways of coming about. (It seems to me that Kant’s defense of the principle as pertaining to efficient causality, in his celebrated Second Analogy, succeeds for the broad mode, but not for the narrow; Kant 1965 A190-211 B233–56; Brittan 1978, 170–71, 181–82.) Again as to efficient cause, the principle evidently does not hold in the narrow mode for elementary particles. There is no narrow cause of a particle decay, so there can be no narrow cause of the decay products. Remember, too, the proverb of particle physicists: “Seek not reasons for decay, but seek the barriers to decay.” At the level of elementary particles, we seek reasons for stability (Frauenfelder and Henley 1974, 83–87; Sachs 1987, 100–103, 175-77; Weinberg 1981).[26]

We have observed that Mill’s methods of induction—agreement, difference, and concomitant variation—are essential to the growth of scientific knowledge and that they are clearly wedded to a fundamental law of all existents, the law of identity. We have observed also that Hume’s proposed basis for induction—habit—makes no suggestion as to why those techniques should be effective, why they should expose new regularities, orders, and unities of nature. Hume’s account also leaves utterly opaque that great engine of scientific discovery, the hypothetico-deductive method (Copi 1961, 433–51; Hempel 1966, 10–28; Kelley 1988, 344–65).[27]

From the statement of a hypothesis (say, Newton’s law of gravitation), together with some established truths (e.g., the orbit of Uranus) and plausible presumptions, we deduce consequences (a planet beyond Uranus). On the character of confirmation, see Mackie 1981; Newton-Smith 1981, 183-97, 226–32; and Armstrong 1991, 41–46. Both the hypothesis and its consequences are claims about the world. Hypotheses are arrived at by some mix of induction and imagination. We really do not know all that much about how hypotheses are formed (Reilly 1970, 36–38; Hempel 1966, 15–18; Drake 1980; Cohen 1981). We really do not know that much either about how one selects relevant established truths. I assume that not all are selected ahead of the grasping of consequences, but my argument does not depend on this. We do know how the consequences are constructed, or at any rate, how they can be reconstructed (Minsky 1988, 186–89). The consequences are deductions from the hypothesis in combination with select established truths. Now consider the case of a hypothesis that has been subsequently well confirmed by observation of predicted, deduced consequences. On Hume’s model of induction and causality, is this not a minor miracle? Why, on Hume’s view, should deduction yield consequences about what is in the world?

On the view that identity is a deep and general law of reality, the success of the hypothetico-deductive method is intelligible. Validity of our deductions assures us in some measure that we are not in opposition to the universal law of identity. Presumably, that is why validity is desirable. From the vantage of Rand’s principle of identity, we have some idea of why the hypothetico-deductive method works. It is marvelous but not miraculous.

In Hume’s view, as in Ockham’s, the existence of one object cannot be inferred from the existence of another (E 132). Hume contends that, excepting mathematical objects, nothing can be demonstrated of any objects of reasoning (E 131). “All belief of matter of fact or real existence is derived merely from some object, present to the memory or senses, and a customary conjunction between that and some other object” (E 38). Yet in reasoning from a hypothesis to an observable, but as yet unobserved consequence, we can hardly be relying merely on custom. Apparently, Hume never squarely confronted the hypothetico-deductive method. He speaks of “hypothetical arguments, or reasoning upon supposition,” but in these he says there is no “belief of a real existence” (T I.3.4; E 37). Insofar as Hume does begin to consider scientific reasoning, he turns from custom to identity. He may boldly proclaim that “all inferences from experience, therefore, are effects of custom, not of reasoning” (E 36), but in a note, he remarks that scientific knowledge cannot be established purely by experience, but requires “some process of thought, and some reflection on what we have observed, in order to distinguish its circumstances and trace its consequences” (E 36n1; see also E 84n1).

Continued below—
Hume – Necessity
Hume – Uniformity
Existence is Identity

References

Aristotle 1984 [c. 348–322 B.C.]. The Complete Works of Aristotle. J. Barnes, editor. Princeton. Princeton University Press.

Armstrong, D. M. 1991 [1983]. What Is a Law of Nature? Cambridge: Cambridge University Press.

Brittan, G. G. 1978. Kant’s Theory of Science. Princeton: Princeton University Press.

Boyd, R. 1991 [1985]. Observations, Explanatory Power, and Simplicity: Toward a Non-Humean Account. In Boyd, Gasper, and Trout 1991.

Boyd, R., Gasper, P., and J. D. Trout, editors, 1991. The Philosophy of Science. Cambridge, MA: MIT Press.

Bullock, M., Gelman, R., and R. Baillargéon 1982. The Development of Causal Reasoning. In The Developmental Psychology of Time. W. J. Friedman, editor. New York: Academic Press.

Butterfield, H. 1965 [1957]. The Origins of Modern Science. Revised ed. New York: Free Press.

Cohen, I. B. 1981. Newton’s Discovery of Gravity. Sci. Amer. (Mar):150–56.

Copi, I. M. 1961 [1953]. Introduction to Logic. 2nd. Ed. New York: Macmillan.

Drake, S. 1980. Newton’s Apple and Galileo’s Dialogue. Sci. Amer. (Aug):150–56.

van Fraassen, B. C. 1991 [1977]. The Pragmatics of Explanation. In Boyd, Gasper, and Trout 1991.

Frauenfelder, H., and E. M. Henley 1974. Subatomic Physics. Englewood Cliffs, NJ: Printice-Hall.

Gasper, P. 1991. Causation and Explanation. In Boyd, Gasper, and Trout 1991.

Hempel, C. G. 1966. Philosophy of Natural Science. Englewood Cliffs, NJ: Prentice-Hall.

Hoenen, P. 1952. Reality and Judgment according to St. Thomas. H. F. Tiblier, translator. Chicago: Henry Regnery.

Hume, D. 1975 [1893, 1748]. Enquiries Concerning Human Understanding. 3rd ed., L A. Selby-Bigge, editor. Oxford: Clarendon Press.
——. 1978 [1888, 1740]. A Treatise of Human Nature. 2nd ed., L. .Selby-Bigge, editor. Oxford: Clarendon Press.

Jackendoff, R. 1987. Consciousness and the Computational Mind. Cambridge, MA: MIT Press.

Kant, I. 1965 [A-1781 B-1787]. Critique of Pure Reason. N. Kemp Smith, translator. New York: St. Martin’s Press.

Keil, F. C. 1989. Concepts, Kinds, and Cognitive Development. Cambridge, MA: MIT Press.

Kelley, D. 1988. The Art of Reasoning. New York: W. W. Norton.

Mackie, J. L. 1981 [1963]. The Paradox of Confirmation. Reprinted in The Philosophy of Science. P. H. Nidditch, editor. New York: Oxford University Press.

McClosky, M. 1983. Intuitive Physics. Sci. Amer. (Apr):122–30.

Mill, J. S. 1973 [1843]. A System of Logic Ratiocinative and Inductive. Toronto: University of Toronto Press, Routledge & Kegan Paul.

Minsky, M. 1988 [1985]. The Society of Mind. New York: Simon & Schuster.

Newton-Smith, W. H. 1981. The Rationality of Science. Boston: Routledge & Kegan Paul.

Peikoff, L. 1991. Objectivism: The Philosophy of Ayn Rand. New York: Dutton.

Reilly, F. E. 1970. Charles Peirce’s Theory of Scientific Method. New York: Fordham University Press.

Rips, L. J. 1989. Similarity, Typicality, and Categorization. In Similarity and Analogical Reasoning. S. Vosniadour and A. Ortony, editors. Cambridge: Cambridge University Press.

Sachs, R. G. 1987. The Physics of Time Reversal. Chicago: University of Chicago Press.

Stroud, B. 1988 [1977]. Hume. London: Routledge.

Weinberg, J. R. 1965. Abstraction, Relation, and Induction. Madison: University of Wisconsin Press.

Weinberg, S. 1981. The Decay of the Proton. Sci. Amer. (Jun):64–75.

~~~~~~~~~~~~~~~~
This essay of 1991 had no notes. I will now add a few endnotes to indicate changes or emendations to the positions I took in this essay nineteen years ago. I will also add some hyperlinks within the text.
~~~~~~~~~~~~~~~~

Notes
18. Learned here means established in one’s conceptual framework, as in Harriman 2010, 31–34. Compare those pages with Critique of Pure Reason, Bxiii–xiv. On Kant’s connection of induction to taxonomic systematization of nature in the organization of experience, see Allison 2008, 140–51. On the connection of induction to conceptualization, in Francis Bacon and in William Whewell, see McCaskey 2004. On abstractive induction, see here.

19. Allison points out that in warranting induction from experience of a single case, Hume is not only relying on his general principle that “like objects, placed in like circumstances, will always produce like effects.” Hume is also relying on his principle that every event has some cause (Allison 2008, 156). On this latter principle, see third paragraph from last in the present section (and Allison 2008, 93–111, 137–38).

Allison observes also that those two conditions are required, in Hume’s account, not only for induction based on a single case, but for virtually all inference from something observed to something unobserved (e.g., T I.3.13). Hume’s text on inductive inference from the single case (T I.3.8), which I quoted in the third paragraph, shows acutely the power of judgment as acting independently of custom concerning the case at hand. Making the right judgment concerning causally relevant factors in this single-case base for inference in subsequent cases is not helped by appeal to the general rule that like objects placed in like circumstances produce like effects. As a matter of fact, Hume’s two general rules are also not helpful in making the required particular judgment where the observed cases are multiple or numerous. Sorting objects and actions into classes according to usual, manifest similarities does not help either (Allison 2008, 159–60; see also here and Harriman 2010, 9, 31–34).

Hume helps himself, in the quoted passage and in others, to the human power of judgment. That is a power Hume is not entitled to invoke, given his model of human cognition. Barry Stroud writes that Hume’s theory of ideas

obstructs proper understanding of the role, or function, or point of various ideas in our thought about the world because in representing “having” an idea as a matter of a certain object’s simply being “in” the mind, it leaves out, or places in a secondary position, the notion of judgment, the putting forth of something that is true or false. For Hume, ideas exist in the mind and have their identity completely independently of any contribution they might make to judgments or statements that have a truth-value. He sees judging as just a special case of an object’s being present to the mind. . . . He does not see that without an account of how ideas combine to make a judgment or a complete thought he can never explain the different roles or functions various fundamental ideas perform in the multifarious judgments we make, or in what might be called the “propositional” thoughts we have. Consequently, he does not arrive at even the beginnings of a realistic description of what “having” the idea of causality actually consists in. (1988, 232)

20. There were two errors in this paragraph. Firstly, unlike quantum regimes, classical chaotic regimes hold no exceptions to Aristotle’s principle (a, b). Secondly, my broader formula intended to be the minimum implied by the law of identity did not quite reach the minimum, which would be: “For some given circumstance or other, identical existents will produce results not wholly identical to results produced by different existents in those same circumstances.”* As is seen in the link, that picayune revision to the broad formula is occasioned by a consideration that applies to all physical regimes, including the classical regular regime.

The gravamen of the broad formula was captured perfectly well in my 1991 statement: “Identical existents, in given circumstances, will always produce results not wholly identical to results produced by different existents in those same circumstances.” In contrast Leonard Peikoff had maintained earlier that year that Rand’s law of identity entails the following: “In any given set of circumstances, there is only one action possible to an entity, the action expressive of its identity” (1991, 14). Dr. Peikoff’s formula can be read as not in contradiction with mine if his phrase only one action possible is taken to mean only one kind and range of action possible. But that is not the plain reading of his text. In his 1976 lectures The Philosophy of Objectivism (Lecture 2), also, he had maintained that Rand’s law of identity applied to action entailed that only a single action was physically possible to a thing in a given circumstance. Rand gave notice that those lectures were an accurate representation of her views, so I expect she shared the erroneous view expressed by Peikoff concerning uniquely determined outcome. (That there is a unique outcome in all cases is not in dispute; the issue is whether in all cases only that unique outcome was physically possible; see my 1997 reply to Rafael Eilon, 159–62.)

So I expect Rand meant “uniquely determined” in her 1973 formula for the law of physical causality: “All the countless forms, motions, combinations, and dissolutions of elements within the universe—from a floating speck of dust to the formation of a galaxy to the emergence of life—are caused and determined by the identities of the elements involved” (MvMM, 25). In any case, the error is easily corrected without major revision to her metaphysics or to its counters to Hume’s account of causation.

21. Cf. Harriman 2010, 67–71, on the first two methods, the third glossed only as a species of the second. Kelley 1988 discusses all three methods, the third on pages 283–85.

22. An ambitious attempt at developing this elementary thesis into a full-blown account of causation and explanation in science is made in Woodward 2003.

23. On the origins and elaboration of causal understanding in development, see also Chapter 6 of Carey 2009 along with Gopnik and Schulz 2007.

24. It was misleading for me to say that Hume “managed to not notice” that the importance of causal reasoning is at variance with his insistence on the preeminent role of habit in causal attributions. What he managed to “not notice” was the ineffectiveness of his gestures at reconciling that variance. That is, the variance remains.

25. Cf. Harriman 2010, 14–15, 44–46, 49–53. See further, Miller 2006.

26. More on the stability of matter: Lipkin 1995 and Lieb 2009.

27. Contrast my representation of the hypothetico-deductive method in science with its representation by Harriman (2010, 145–46). I have not supposed that the method entails that hypotheses are mere guesswork, which is not the way the method has been employed by any research in physical science with which I am familiar. However much later philosophers of science took hypotheses to be guesswork, that was not the view of William Whewell (Snyder 2006).

Rand’s theoretical philosophy and my own understanding of the methods of science are consonant with the following superbly informed view of Ernan McMullin 1992. (See also.)

Let us restrict the term abduction to the process whereby initially plausible and testable causal hypotheses are formulated. This is inference only in the loosest sense, but the extensive discussions of the logic of discovery in the 1970’s showed how far, indeed, it differs from mere guessing. The testing of such hypotheses is of the most varied sort. It does, of course, involve deduction in a central way, as consequences are drawn and tried out. Some of these may be singular, others may be lawlike and hence involve induction. But we shall not restrict induction to the testing of causal hypotheses, as Peirce came to do. (89–90)

[Our concern] is with the process of theoretical explanation generally, the process by which our world has been so vastly expanded. This is the kind of inference that makes science into the powerful instrument of discovery it has become. . . . As a process of inference, it is not rule-governed as deduction is, nor regulated by technique as induction is. Its criteria, like coherence, empirical adequacy, fertility, are of a more oblique sort. They leave room for disagreement, sometimes long-lasting disagreement. Yet they also allow controversies to be adjudicated and eventually resolved.

It is a complex, continuing, sort of inference, involving deduction, induction, and abduction. Abduction is generally prompted by an earlier induction (here we disagree with Peirce). The regularity revealed by the induction may or may not be surprising. Deductions are made in order that consequences may be tested, novel results obtained, consistency affirmed. The process as a whole is the inference by means of which we transcend the limits of the observed, even the instrumentally observed.

Let us agree to call the entire process retroduction. We are “led backwards” from effect to cause, and arrive at an affirmation, not simply a conjecture. Retroduction in this sense is more than abduction. It is not simply the initial plausible guess. It is a continuing process that begins with the first regularity to be explained or anomaly to be explained away. It includes the initial abduction and the implicit estimate of plausibility this requires. It includes the drawing of consequences, and the evaluation of the match between those and the observed data, old or acquired in light of the hypothesis. Tentative in the first abduction, gradually strengthening if consequences are verified, if anomalies are successfully overcome, if hitherto disparate domains are unified, retroduction is the inference that in the strongest sense “makes science.” (92–93)


#24 Stephen Boydstun

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Posted 13 January 2011 - 03:57 PM

—Hume – Necessity

Hume could find no necessary connection between distinct objects or events. “One event follows another, but we never observe any tie between them. They seem conjoined, but never connected” (E 58). Why is Hume seeking ties and connections between events? Because, for one thing, between some events, he and we have direct experience of such. “Draw a bucket of water from the well, Mr. Hume. Between your arms and the rising bucket, there is a tie, the rope. Place your walking cane on top of either foot and bear down. Notice the connection between your exertion and the pressure on your foot.”[28]

It might be reasonably protested that what Hume really seeks and fails to find is any logical necessity linking distinct events rather than any physical necessity linking distinct events. Hume was surely seeking logical necessitation; as between distinct things, he says repeatedly, one may be denied and the other affirmed without contradiction; there is no logical connection between distinct things. But he was just as surely seeking (and avoiding?) physical necessitation; he complains of being unable to observe any tie between distinct events and speaks of what it is possible for a billiard ball to do. He swings freely back and forth between physical and logical necessity.[29]

Hume’s vacillation is perfectly understandable, I think, because logical necessities (or possibilities) and physical necessities (or possibilities) are in fact not worlds apart. They are not even as far apart as next-door neighbors. Here are six reasons for thinking that logical necessity and physical necessity are intimately related:

(1) Yesterday a grey squirrel visited my balcony. I witnessed it from the library. It is true that yesterday that grey squirrel made that visit to my balcony. It is a physical fact. Tomorrow it will be true that two days past that grey squirrel made that visit. That is a physical truth and a logical truth. Everything that has been will have been, and will have been just as it was, for one day more tomorrow. That is so necessarily, both physically and logically.

(2) The principle of non-contradiction, with its necessity, emerges in development. Physical constraints found in experience may well inform that emergence. For the most part, during the first year of life, infants do not have language. The very young child evidently acquires a working principle of non-contradiction spontaneously, before acquiring language. It seems that in order to begin to acquire language, the child must have the notions truth and falsity and a working principle of non-contradiction. Without these he cannot grasp the notion proposition (Macnamara 1986, 33–37, 105–9, 114–17).

I mentioned in an earlier section* experiments by Renée Baillergéon (1986; 1987) which indicate that by four months, infants apprehend that one solid object cannot pass through another. I suggested that this might be the most primitive form in which one grasps the principle of non-contradiction (see further, Leslie 1989, 194–200). During the last half of the first year, apparently on account of maturation of the frontal cortex, the normal human infant becomes capable of inhibiting prepotent responses: Until about nine months, the infant will reach impulsively for all objects, but by the end of the first year, he can be more selective. He can give a NO command to some otherwise interfering habitual responses in order to execute the reach required for a desired object (Diamond 1989; Edelman 1989, 44–50, 57–63, 120–27, 159–62). This would seem to be an important step along the way to attaining a working principle of non-contradiction. It is plausible that the ability to represent negations and employ a principle of non-contradiction arises from the preverbal child’s perceptual and motor experience of the physical world (Dretske 1988, 62–79, 95–107; see also Peikoff 1985).

(3) In formal reasoning, we use physical props (Rumelhart 1989, 306–11).

(4) There are no purely logical necessities; every logical necessity has some physical bases, namely, neuronal processes. In his seminal study of the relationship of logic and psychology, John Macnamara points out that although logicians study ideal reasonings, “inferences carried out by logicians in conformity with the rules of the ideal logic are still actual” (1986, 7).

(5) The difference between logical necessity and physical necessity would seem to be a graded difference. Geometrical necessities holding in the physical world would seem to lie between the poles of logical and physical necessity. Under physically realized geometrical necessities, I mean to include not only spacetime structures, from relativity, but patterns of spacetime occupation, from quantum mechanics.

(6) There are physical necessities nearly as pervasive as logical necessities: All physical things are in spacetime; at all times, there is a population of physical things, and they have some composition (especially important facts for induction; see Williams 1963, 153–56, 138–39, 160–61); any physical thing has mass-energy; any population has entropy. There are, moreover, three physical quantities that are constants, virtually come what may: mass-energy, electrical charge, and angular momentum (Misner, Thorne, and Wheeler 1973, 875–92; Wald 1984, 312–24).

Concerning physical connections, manifest or obscure, between distinct physical events, Hume would surely ask what necessity there is that tomorrow things will work as they have in the past. Everyone agrees that there is no deductive necessity that the drawing of water from the well tomorrow will go as it has in the past. There is no physical necessity either. Water drawing is physically contingent on a great many things; contrast with the mass-energy of the universe, which is evidently not physically contingent on anything, that is, which is evidently ineradicably necessary. We know there is no necessity that water drawing will succeed tomorrow, and that is not all we know about the subject. We know that if our attempt to draw water tomorrow fails, there are reasons for that, reasons growing out of the way things have been. In addition to this general principle, we have some ideas, from our past experience and consequent general understanding of the world, of what could and could not account for a water-drawing failure.

The general principle that the ways things will be grow out of the ways things have been is not a brand new philosophic insight. Immanuel Kant stressed that an effect “not only succeeds upon the cause,” but “is posited through it and arises out of it” (1965 A91 B124). The contemporary philosopher Robert Nozick employs the same idea: “To say that something is a continuer of x is not merely to say its properties are qualitatively the same as x’s, or resemble them. Rather it is to say they grow out of x’s properties, are causally produced by them, are to be explained by x’s earlier having had its properties, and so forth” (1981, 35).

By our general principle, that the ways things will be grow out of the ways things have been, we assert the law of identity in application to action and becoming, and we assert the reality and unity of material substance in all things physical. Our principle actually applies to nominal variations and constancies as well as to physical ones. A term used in an identical sense throughout a discourse is a case of growing out of—in this case, simply remaining the same—previous character. Notice also that conclusions grow out of premises. Of course all nominal items, and their characters, have physical substrata consisting of brain activities. So all propagations of nominal items through time are supported, though not necessarily isomorphically, by physical propagations, Because our primary concern in this essay is with ampliative induction, our focus is on the physical, but our principle that the ways things will be grow out of the ways things have been would seem to pertain to everything. Hereafter I shall refer to this general principle as the principle of substantive propagation.

Even were Hume to agree that our general principle and our particular principles of water drawing are formulated truly and in accord with past experience, he would emphasize that they cannot be established purely deductively (T 86–87). Therefore, belief in these principles cannot be rational. Our belief is instinctual but not rational. Virtually all contemporary philosophers dispute Hume on this point. Hume’s apparent presumption that only the deductively demonstrable is rational is wrong. The conception of rationality in Hume’s era was very restricted. Today inductive reasoning is widely regarded as complementary to deductive reasoning and every bit as rational.[30]

Continued below—
Hume – Uniformity
Existence is Identity

References

Baillargéon, R. 1986. Representing the Existence and the Location of Hidden Objects: Object Permanence in 6- and 8-Month-Old Infants. Cognition 23:21–41.
——. 1987. Object Permanence in 3.5- and 4.5-Month-Old Infants. Developmental Psychology 23:655–64.

Diamond, A. 1989. Differences Between Adult and Infant Cognition: Is the Crucial Variable Presence or Absence of Language? In Weiskrantz 1989.

Dretske, F. 1988. Explaining Behavior: Reasons in a World of Causes. Cambridge, MA: MIT Press.

Edelman, G. M. 1989. The Remembered Present: A Biological Theory of Consciousness. New York: Basic Books.

Hume, D. 1975 [1893, 1748]. Enquiries Concerning Human Understanding. 3rd ed., L A. Selby-Bigge, editor. Oxford: Clarendon Press.
——. 1978 [1888, 1740]. A Treatise of Human Nature. 2nd ed., L. .Selby-Bigge, editor. Oxford: Clarendon Press.

Kant, I. 1965 [A–1781 B–1787]. Critique of Pure Reason. N. Kemp Smith, translator. New York: St. Martin’s Press.

Leslie, A. M. 1989. The Necessity of Illusion: Perception and Thought in Infancy. In Weiskrantz 1989.

Macnamara, J. 1986. A Border Dispute: The Role of Logic in Psychology. Cambridge, MA: MIT Press.

Misner, C. W., Thorne, K. S., and J. A. Wheeler 1973. Gravitation. San Francisco: W. H. Freeman.

Nozick, R. 1981. Philosophical Explanations. Cambridge, MA: Belknap Press of Harvard University Press.

Peikoff, L. 1985. Aristotle’s “Intuitive Induction.” The New Scholasticism 59(2):185–99.

Rumelhart, D. E. 1989. Toward a Microstructural Account of Human Reasoning. In Similarity and Analogical Reasoning. S. Vosniadou and A. Ortony, editors. Cambridge: Cambridge University Press.

Weiskrantz, L. editor, 1989. Thought without Language. Oxford: Clarendon Press.

Wald, R. M. 1984. General Relativity. Chicago: University of Chicago Press.

Williams, D. 1963 [1947]. The Ground of Induction. New York: Russell & Russell.

~~~~~~~~~~~~~~~~
This essay of 1991 had no notes. I will now add a few endnotes to indicate changes or emendations to the positions I took in this essay nineteen years ago. I will also add some hyperlinks within the text.
~~~~~~~~~~~~~~~~

Notes
28. Hume wrote the Treatise during the two years he stayed in La Flèche, France (age 25–26). In the Introduction to that work, he places himself in the empirical tradition of Bacon and Locke. While traveling from La Flèche back to Paris, having just completed the Treatise, Hume wrote a letter to a friend in England urging the following preparatory reading for study of his own book: Malebranche’s Search after Truth, Berkeley’s Principles of Human Knowledge,* portions of Bayle’s Dictionary,* and Descartes’ Meditations* (Mossner 1969, 12–13). Like Berkeley before him, Hume was significantly influenced by Nicolas Malebranche (Luce 1934). Neither Malebranche nor Hume would be turned around by my examples of the rope and the cane.

The psalmist writes of God: “He spake, and it was done” (33:9). For Malebranche the mind of God is the model of effective understanding and causation. “God contains within Himself in an intelligible fashion the perfection of all the beings He has created or can create, and . . . through these intelligible perfections He knows the essence of all things, as through His volitions He knows their existence. Now, these perfections are also the human mind’s immediate object” (ES 10).

Malebranche was an Augustinian. “The Lord Jesus . . . has intimated to us that the human soul and rational mind which is in man, not in the beast, is invigorated, enlightened, and made happy in no other way than by the very substance of God” (Augustine tractate 23.5; S III.ii.6). Malebranche was a Cartesian:

 

The mind’s pure ideas are clear and distinct. (S V.10)
Everything one clearly conceives is precisely as one conceives it. (S IV.11.2)
We should reason only about things of which we have clear ideas . . . . The state of the question we propose to resolve must be distinctly conceived. (S VI.ii.1)
Real ideas produce real science, but general or logical ideas never produce anything but a science that is vague, superficial, and sterile. We must, then, carefully consider distinct, particular ideas of things in order to discover the properties they contain, and study nature in this way, rather than losing ourselves in chimeras that exist only in certain philosophers’ minds. (S II.ii.8; further, S I.16, III.i.3, VI.ii.6)

Malebranche innovated philosophy extending beyond Augustine and Descartes. (S – The Search after Truth* [1674–75]; ES – Elucidations of the Search after Truth [1678])

The human mind has the idea of the infinite and has it even before the idea of the finite. “For we conceive of infinite being simply because we conceive of being, without thinking whether it is finite or infinite. In order for us to conceive of a finite being, something must necessarily be eliminated from this general notion of being, which consequently must come first” (S III.ii.6). (Like Descartes, Malebranche does not keep firmly distinct the indefinite and the infinite.)

We conceive an infinitely perfect being, and in this idea is included clearly that such a being exists necessarily. The same could not be said of the concept of an infinitely perfect body. For a body is particular and finite being, not infinite being. Being has its existence of itself. It cannot not be. Everything that exists comes from being. Bodies exist by imperfect participation in being. Being would exist were there no bodies (S IV.11.2). God is infinite perfect being. God exists.

One cannot see a body in itself or of itself. One sees a body only through certain perfections of it in God, perfections that represent the body (S VI.11.3). We experience our sensations without anything resembling them outside us; they are modifications of our minds (S III.ii.5). Contrary to what empiricists claim, we are not able to produce ideas of things after the likeness of objects that seem to make sensory impressions on us (S III.ii.2).

Perception consists of sensation and pure idea (S III.ii.6). The presence of the object of its idea is necessary in perception (S III.ii.1), but from this one should not conclude that the object is the true cause of its idea in us (S III.ii.6).

Ideas differ from one another and have real properties. They are real beings. They are spiritual beings, radically different from bodies they may represent. Ideas cannot be made from matter. Men come to think the mind can form its own ideas because men “never fail to judge that a thing is the cause of a given effect when the two are conjoined, given that the true cause of the effect is unknown to them. This is why everyone concludes that a moving ball which strikes another is the true and principal cause of the motion it communicates to the other, and that the soul’s will is the true and principal cause of movement in the arms, and other such prejudices” (S III.ii.3).

One should conclude, rather, “the collision of the two balls is the occasion for the Author of all motion in matter to carry out the decree of His will, which is the universal cause of all things” (S III.ii.3). God wills “that the latter ball should acquire as much motion in the same direction as the former loses, for the motor force of bodies can only be the will of Him who preserves them” (S III.ii.4). (Malebranche, by the way, was skilled in billiards.)

Bodies cannot move themselves. We could conclude that our minds move them were we to see the “necessary connection” between the idea of our finite minds and motions of bodies. We see none. Only in the idea of the infinitely perfect being, the all-powerful being, is there necessary connection between its will and a body’s motion (S VI.ii.3).

That is the verdict of our conceiving things clearly. It is not the verdict of the senses. The latter see a ball communicate its motion to another. We may say a ball is the natural cause of the motion it communicates. “But natural causes are not true causes; they are only occasional causes that act only through the force and efficacy of the will of God” (S VI.ii.3). A true cause “is one such that the mind perceives a necessary connection between it and its effect” (S VI.ii.3).

To my examples of perceiving causal connections—the rope between straining arm and rising bucket, the cane conveying force from hand to foot—Malebranche will say that such connections are natural causal connections, but lack conceptual, logical necessity. Only in an infinitely powerful and absolutely creative being do power and thought coincide. The orderliness we find in everyday experience and in science are from the perpetual wisdom and creative power of that being (S III.ii.6).

Hume helps himself amply to Malebranche’s arguments against logically necessary connections between causes and their effects in nature (T I.3.14; S VI.ii.3). However, as to Malbranche’s doctrine

 

that the connexion betwixt the idea of an infinitely powerful being, and that of any effect, which he wills, is necessary and unavoidable; I answer, that we have no idea of a being endow’d with any power, much less one endow’d with infinite power. But if we will change expressions, we can only define power by connexion; and then in saying, that the idea of an infinitely powerful being is connected with that of every effect, which he wills, we really do no more than assert, that a being, whose volition is connected with every effect, is connected with every effect; which is an identical proposition, and gives us no insight into the nature of this power or connexion (T I.4.5; see also T I.3.14 and E 55–57).

In moving or straining one’s arm one has no direct experience of causal power in one’s will, according to Hume (E 51–52), following Malebranche (S VI.ii.3; ES 15, 17 [25–26]; see further, Nadler 2000, 121–25, and 2011). For the logical necessity of the causal links to be known the links must be known. But we do not know how we move our arm. We do not know how the mind interacts with matter, specifically how the mind interacts with nerves and muscles. This argument is weak in Hume, as it is weak in Malebranche. Early modern philosophers sleepwalk back and forth between what they know in first-person experience and what they know from third-person scientific physiology. They appeal to the latter to invalidate the former, which is fallacious (cf. Kelley 1986, 36–43). Hume would commit this error again in contending that one does not sense extension and solidity of the cane directly, rather, one senses only sensory impressions (T I.4.4).*

Hume rejected the traditional distinction of types of cause: efficient, formal, material, exemplary, and final. If any such cause is not essentially an efficient cause, the type he had labored so hard to reduce to constant conjunctions, then it is no cause at all. Furthermore, “we must reject the distinction betwixt cause and occasion, when suppos’d to signify any thing essentially different from each other. If constant conjunction be imply’d in what we call occasion, ’tis a real cause” (T I.3.14).

29. Hume formulates two definitions of cause. He maintains that ultimately they are the same. In the first, causal necessity is physical. The second definition is that of which the first is a projection; its necessity is psychological.

In the first, cause is any resembling class of objects (say, flame) that are followed in time by another resembling class of objects (heat) with which they are contiguous. A cause is “an object precedent and contiguous to another, and where all the objects resembling the former are plac’d in like relations of precedency and contiguity of those objects, that resemble the latter” (T I. 3.14; cf. E 60). Necessity between the conjoined classes would have to include the necessity in what we call kinematics in our physics and engineering, the possible motions in a situation, leaving aside forces and energetics. Hume would take kinematical characterization to stand to experience and reason as geometry stands to them (T I.3.1–2; E 20, 27; see further, Allison 2008, 83–87).

That occasions of billiard balls in motion or rest are followed by occasions of billiard balls in motion or rest is not enough for the implication of causation under the scope intended by Hume’s first definition. Dynamics and other physical restrictions must contract the possibilities left open by kinematics. Hume does not allow that there is anything more to our concepts force and energy fundamentally than there is to our concept cause in general (T I.3.14; E 49, 54). Hume’s concept of force in mechanics would be the Newtonian concept: that external influence which produces an acceleration of a body and whose measure is the time rate of change of the momentum of that body. Hume’s concept of energy in mechanics would also be focused on external influence on a body, specifically communication of motion from one body to another.

 

In reality, there is no part of matter, that does ever, by its sensible qualities, discover [uncover, display] any power or energy, or give us ground to imagine that it could produce any thing, or be followed by any other object, which we could denominate its effect. Solidity, extension, motion; these qualities are all complete in themselves, and never point out any other event which may result from them. The scenes of the universe are continually shifting, and one object follows another in an uninterrupted succession; but the power or force, which actuates the whole machine, is entirely concealed from us, and never discovers itself in any of the sensible qualities of body. (E 50)

We experience that heat is a constant attendant of flame, but to know a putative power or energy of the flame to produce heat is not possible to us. Were we to know such a power, “we could foresee the effect, even without experience; and might, at first, pronounce with certainty concerning it, by mere dint of thought and reasoning” (E 50). That is straight out of Malebranche. Godlike knowledge is the model for human knowledge.

Causal power is something we project onto occasions of cause, cause as under Hume’s first definition, on account of an unavoidable psychological compulsion named in his second definition of cause: “An object precedent and contiguous to another, and so united with it in the imagination, that the idea of the one determines the mind to form the idea of the other, and the impression of the one to form a more lively idea of the other” (T I.3.14; E 60)

30. For example, at a deep level, consider Hintikka’s distinction between the definitory and strategic rules of logic (cf. those types of rule in chess) and his base for both in interrogative logic. See Inquiry as Inquiry: A Logic of Scientific Discovery (1999); also Suppes, 743–45, and Sintonen, chapter 25, in The Philosophy of Jaakko Hintikka (2006). See also Haugeland’s “Truth and Rule-Following” in Having Thought (1998).



#25 Stephen Boydstun

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Posted 18 February 2011 - 07:28 AM

—Hume – Uniformity

Hume thought that if we did have a valid rational basis for induction, it would be the principle “that instances, of which we have had no experience, must resemble those, of which we have had experience, and that the course of nature continues always uniformly the same” (T I.3.6). Taken superficially, this principle, the ‘uniformity of nature’ principle is false and uninteresting (Mill 1973, 3.2.2). Wells do not always work, let alone work “uniformly,” and the atomic chemical elements did not always exist. I think the principle that Hume was really trying to get at was an identity principle, something like the principle of substantive propagation, something like, “whether things continue the same or change, there will be reasons, growing out of the way things were at earlier times, for the way things are at later times.”

Hume says it is possible the uniformity principle is false since we can conceive of it being false. The principle is false under the superficial reading, quite apart from Hume’s rationalism. Under the identity reading, the principle stands up to all of our experience, provided we construe “growing out of” in the broad causal mode. The principle stands up to much of our experience even where “growing out of” is construed in the narrow mode.[31]

“Very well,” a modern Humean might say, “how do we know that this identity principle, the substantive propagation principle, with the broad mode of cause, will be true in the future?” Well, how do we know there will be a future? Because we know that our being here just now does not make now cosmically special. “Then all collapsed, and the great shroud of the sea rolled on as it rolled five thousand years ago” (Melville). At this stage of modern science [1991], we reasonably ask whether the universe will eventually contract to a singularity, whether such singularity would be final, whether time would stop. But the end of time, if there be such, is very far off and is not predicated on any fantasies one might indulge concerning the world ending tomorrow (Dicus et al. 1983; Gott et al. 1976).

Immediately after setting out the uniformity principle, Hume reformulates it so as to make it atemporal (T I.3.6). This is close to the assumption we make in attempting to select representative samples from a population. Of course our aim is really only that the samples resemble closely enough the remaining population, only in ways pertinent of our inquiry, and it is often not the case that we have no experience of the population not selected in the sample. We often have some experience of the whole population, and this experience should inform our choice of sampling procedures and our statistical analysis. In these circumstances, Hume’s atemporal form of the uniformity principle affords little skeptical fun. Even where we cannot range freely over a population to take samples for closer examination, we may have reasons to expect the inaccessible population to follow the suite in hand (Mill 1973, 3. 3.3). There are pretty strong reasons of consilience to suppose that atomic chemical elements will continue to be found through electromagnetic radiation spectra, in regions of the universe today [1991] beyond our reach and that the redshifts will continue to increase (cf. Davis 1972, 37–45).

Hume can have his most skeptical fun when he casts the uniformity principle as explicitly about predicting how things will be in the future. “If there be any suspicion that the course of nature may change, and that the past may be no rule for the future, experience becomes useless, and can give rise to no inference or conclusion” (E 32).

As I have already remarked, there is no reason to suppose that the present time is cosmically special. Things as they are over a present time span are a sample of things as they are over other such spans. Time itself does not make a physical difference. Things in a recent time span are presumptively more uniform with things in nearby time spans than in farther time spans because the ways things will be grow out of the ways thing have been. Notice the obvious boundary condition: as time spans approach zero, uniformity tends to perfect identity. This boundary condition seems appropriate for fermions, nuclei, atoms, molecules, and the objects they compose; bosons are another story.

Though time itself makes no physical difference, time is physical. Under the supposition that time is homogeneous, we can derive from Hamilton’s principle (the fundamental dynamical principle in the Hamiltonian formulation of classical mechanics, a formulation under which the transition from classical to quantum mechanics can be smoothly accomplished) the principle of the conservation of energy (Marion 1965, 214–51). The conservation of mass-energy is a very robust principle, experimentally and theoretically. Were the universe eventually to come to a final singularity, the conservation of mass-energy would obtain all along the way. The conservation of mass-energy can be derived also as a consequence of Einstein’s field equation and one of the geometrical identities known as Bianchi identities. (Perhaps the reader demonstrated some trigonometric identities in high school; then the reader has demonstrated some geometric identities. Readers with vector calculus should think of Stokes’ theorem.) It is Einstein’s field equation that warrants our contemplation of the possibility of the universe coming to a final singularity. On the left of that equation, we have geometric structure of spacetime; on the right, we have matter (stress-energy tensor), the source of the geometric structure. Bianchi identities on the left correspspond to conservation of mass-energy on the right (Misner, Thorne, and Wheeler 1973, 364–81; Wald 1984, 5973, 285–95, 450–56).

The modern Humean would continue shamelessly: “Yes, Einstein’s field equations, Hamilton’s principle, and the conservation of energy may have all held up until now, but what about tomorrow? We concede that since the beginning of time there has been nothing cosmically special about present times per se, but what about present times per se in the future? And again, what about this principle of substantive propagation, this principle that the ways things will be grow out of the ways things have been? That may have been true yesterday, but what about tomorrow?”

I shall give a negative argument, then a positive argument. If the ground of induction is the law of identity and no principle is entirely independent of this law, then my arguments should be ultimately circular—well, perhaps spiral, perhaps widening for the positive argument and narrowing for the negative. At any rate of curvature, we may count our circularity on this issue benign, even virtuous. This is not meant to shield my arguments from other types of rational criticism.

The negative argument is this: The principle that the way things will be grow out of the ways things have been is not a bald analytic statement. It is not a stupid tautology. One could go ahead and say “perhaps it is not always the case that the way things will be grow out of the ways things have been” without having misunderstood the simple meanings of the terms of the principle. Suppose the principle were false. Would the principle of non-contradiction then be true tomorrow? If the principle of non-contradiction is true tomorrow without having been an identity in time with itself today, then it is a radical, Humean contingent truth. Then, in the Humean mentality, it could be false day after tomorrow. Then was it true today? Suppose the principle of non-contradiction is true for any tomorrow, but only contingently so. Better yet, suppose we just return to reality. There we find all the necessity worth having. There we find the principle of non-contradiction and our principle of substantive propagation always true.

Immanuel Kant has a very strange explanation for that. I will here resist taking up his account.

The positive argument is this: The principle that the ways things will be grow out of the ways things have been comprehends all of its specific occasions. An oak grows from an acorn, a pair of gamma photons is born from the annihilation of an electron with a positron, mammals evolved from reptiles, the view became clear because I washed the window—these are specific occasions of the principle. Why not just stick to the specific occasions of the principle? Why try to summarize them all in a single principle? What work does our substantive propagation principle do? It helps us think and learn.

Our substantive propagation principle is an example of what Marvin Minsky calls a uniframe (1988, 121–23). That is a description constructed to apply to several different things at once. “We know only a very few—and, therefore, very precious—schemes whose unifying power cross many realms” (126). Our principle that the ways things will be grow out of the ways things have been is one of those schemes (218–19). It enables us to move fluidly among multiple descriptions of a particular occasion of change (393–95), in addition to facilitating assimilation of any and all occasions.

Substantive propagation is not the grandest of uniframes. Highest honor belongs to substantive predication (Minsky 1988, 267; Jackendoff 1987, 148–58; Edelman 1989, 147–48).

Our substantive propagation principle is metaphorical, or analogical (cf. Metaph. 1048a35–b9), but “every thought is to some degree a metaphor” (Minsky 1988, 299; see also Rasmussen 1982, 425–26). Understanding requires that new thoughts be grown from old thoughts (Rumelhart 1989; Brown 1989, 369–85, 390–98, 404–7; Vosniadou 1989, 422–28, 432–33; Collins and Burstein 1989).

"Once scientists like Volta and Ampere discovered how to represent electricity in terms of the pressures and flows of fluids, they could transport much of what they already knew about fluids to the domain of electricity. Good metaphors are useful because they transport uniframes intact, from one world into another. Such cross-realm correspondences can enable us to transport entire families of problems into other realms, in which we can apply to them some already well-developed skills" (Minsky 1988, 299).

Transference of our substantive propagation principle from one domain of change to another does not convey so much detail of form as the fluid-electricity analogy conveys. The form conveyed by our principle must be approximately as general as it is to comprehend all the forms of change there are. I say approximately since a detailed consideration of the various specific principles of change might well yield a more refined, tighter statement of our principle.

David Hume is very pleased to have received these insights into how it is useful to suppose always that the ways things will be grow out of the ways things have been. He really would like to see our positive justification for thinking this principle always true.

Our principle provides a unifying explanation of our successes in finding more specific unifying explanations (cf. Nozick 1981, 268–80, and Rescher 1973, 323–31). Our principle has the connective character of justified true belief; it is a belief which tracks truth (Nozick 1981, 169–78). Had the principle been false in our experience, we could have noticed it; some people think they have noticed it; they are mistaken. Were the principle to be false in some corner in the future, we could notice that (assuming that what could be grows out of what could have been in the past, i.e., assuming our principle true in a looser way). The principle grew out of experience; it is a self-subsuming principle; it is an instance of itself. New experience has grown from the principle (Mill 1973, 3.3.1). Our principle has bearings of a fundamental justified justification (Nozick 1981, 137–40, 641–42).

Continued
Existence is Identity

References

Aristotle 1984 [c. 348–322 B.C.]. The Complete Works of Aristotle. J. Barnes, editor. Princeton. Princeton University Press.

Brown, A. L. 1989. Analogical Learning and Transfer: What Develops? In Vosniadou and Ortony 1989.

Collins, A. and M. Burstein 1989. A Framework for a Theory of Comparison and Mapping. In Vosniadou and Ortony 1989.

Davis, W. H. 1972. Peirce’s Epistemology. The Hague: Martinus Nijoff.

Dicus, D. A., Letaw, J. R., Teplitz, D. C., and V. L. Teplitz 1983. The Future of the Universe. Sci. Amer. (Mar):90–101.

Edelman, G. M. 1989. The Remembered Present: A Biological Theory of Consciousness. New York: Basic Books.

Gott, III, J. R. , Gunn, J. E., Schramm, D. N., and B. M. Tinsley 1976. Will the Universe Expand Forever? Sci. Amer. (Mar):62–79.

Hume, D. 1975 [1893, 1748]. Enquiries Concerning Human Understanding. 3rd ed., L A. Selby-Bigge, editor. Oxford: Clarendon Press.
——. 1978 [1888, 1740]. A Treatise of Human Nature. 2nd ed., L. .Selby-Bigge, editor. Oxford: Clarendon Press.

Jackendoff, R. 1987. Consciousness and the Computational Mind. Cambridge, MA: MIT Press.

Marion, J. B. 1965. Classical Dynamics of Particles and Systems. New York: Academic Press.

Mill, J. S. 1973 [1843]. A System of Logic Ratiocinative and Inductive. Toronto: University of Toronto Press, Routledge & Kegan Paul.

Minsky, M. 1988 [1985]. The Society of Mind. New York: Simon & Schuster.

Misner, C. W., Thorne, K. S., and J. A. Wheeler 1973. Gravitation. San Francisco: W. H. Freeman.

Nozick, R. 1981. Philosophical Explanations. Cambridge, MA: Belknap Press of Harvard University Press.

Rasmussen, D. B. 1982. Necessary Truth, the Game Analogy, and the Meaning-is-Use Thesis. The Thomist 46(3):423–40.

Rescher, N. 1973. The Coherence Theory of Truth. Oxford: Clarendon Press.

Rumelhart, D. E. 1989. Toward a Microstructural Account of Human Reasoning. In Vosniadou and Ortony 1989.

Vosniadou, S., and A. Ortony, editors, 1989. Similarity and Analogical Reasoning. Cambridge: Cambridge University Press.

Wald, R. M. 1984. General Relativity. Chicago: University of Chicago Press.

~~~~~~~~~~~~~~~~
This essay of 1991 had no notes. I have now added some endnotes to indicate changes or emendations to the positions I took in this essay nineteen years ago. I have also added some hyperlinks within the text.
~~~~~~~~~~~~~~~~

Notes
31. Broad causal mode: Identical existents, in given circumstances, will always produce results not wholly identical to results produced by different existents in those same circumstances. Narrow causal mode: Identical existents, in given circumstances, will always produce identically same results. See the fifth and sixth paragraphs of “Hume – Reasoning to Cause or Effect.”*

Is there a cognitive concept of the negative of the principle of uniformity transformed into my principle of substantive propagation “Whether things continue the same or change, there will be reasons, growing out of the way things were at earlier times, for the way things are at later times” with “growing out of” understood as requiring only the broad mode of causality? No. Without this principle, no concepts are cognitive. Nothing is known to be same as self and different from other. Possibility is not known to be different from impossibility. Concepts and propositions are not known to be different from their negations; premises are not known to be different from conclusions, truth different from falsity, knowing different from not knowing. Against the “principle of uniformity” pared down to my principle of substantive propagation, Hume’s indirect proof of the indemonstrability of the principle cannot get started (T I.3.6).

Kant would assimilate the preceding counter to Hume’s case against the “principle of uniformity” under his Kant’s own counter by: thinking of my principle of substantive propagation as a condition of the possibility of the use of the understanding, which then is also a condition on the possibility of experience. Such a principle is not a reflection of the human propensity to project past regularities into the future. No, Kant would see such a principle as “a principle of reason (or reflective judgment) that licenses rather than causally determines such a projection” (Allison 2008, 155; see further 133–60). In that Kantian result, I concur, although in my Randian view, the principle of substantive propagation is a matter of human consciousness as identification, which is prescriptive on account of the base circumstance that existence is identity.

The justification of the principle of substantive propagation given in “Induction on Identity” could be profitably compared with my defense of the axiom “Every action-bearing entity bears certain kinds of action and not others.”* The considerations in the second paragraph of the present Note strike me as sufficient to establish the principle of substantive propagation as an axiom rather than designating it a postulate; the principle cannot be denied without self-contradiction. If it were shown, further, that in the case of concrete entities “growing out of” in the principle of substantive propagation logically entails causal determination, then the postulate (Ia) of “Exclusions of Non-Contradiction: Actions,” the postulate which includes the claim “Every concrete entity is capable of acting and being acted upon,” could be shown to be a metaphysical axiom, rather than a postulate. Because kinematics does not entail dynamics (or statics or kinetics), I do not expect it can be shown that the “growing out of” in the principle of substantive propagation logically entails causal determination. Substantive propagation is consistent, of course, with universal causal determination, but it does not entail it. That is, the principle of substantive propagation does not entail the true and important principle that “All the countless forms, motions, combinations and dissolutions of elements within the universe . . . are caused and determined by the identities of the elements involved” (MvMM).

That important principle is Rand’s statement of the law of causality in her metaphysics. Harriman (2010) writes that the essence of the law of causality is that “an entity of a certain kind necessarily acts in a certain way under a given set of circumstances” (21). Does that formulation of the essence of the causal law coincide with causality in the broad mode or with causality in the narrow mode? Harriman’s statement is slightly ambiguous between the two, though it leans towards the latter mode. Indeed, in further elaboration, he states that future actions can be inferred from past actions because the past actions were effects of causes, and because “if the same cause is operative tomorrow, it will result in the same effect” (21). As I argued in the 1991 text above, application of the law of identity to action and becoming entails only the conception of causality in the broad mode, not the narrow, and taking the latter to apply to all existents is an error. See the preceding section “Hume – Necessity” and its Note 20.

We should notice that were the narrow mode true in application to any and all existents, its epistemological status would be postulate, not axiom. For, to my knowledge, no one has demonstrated the principle of narrow-mode causality to be axiomatic in the required way, such as I have done for certain other propositions belonging to the family “Existence is Identity.”

Mr. Harriman appeals to a brother of my principle of substantive propagation in maintaining that the “justification for inferring the future from the actions of the past is the fact that the past actions occurred . . . for a reason, a reason inherent in the nature of the acting entities themselves” (21). Harriman erroneously supposes this principle entails universal causality in the narrow mode. That is, for all the “forms, motions, combinations and dissolutions of elements within the universe” (Rand’s fine phrase), “if the same cause is operative tomorrow, it will result in the same effect” (21).

With regard to generalizations about kinds of joined actions, such as push of a ball and its rolling, Harriman rightly says they are made true by “some form of causal relationship between the two” (21). C. S. Peirce wrote: “General principles are really operative in nature. This is the doctrine of scholastic realism” (1903, 193). Peirce famously was a proponent of scholastic realism in theory of universal concepts. Particularly, his realism was close to the realism of Duns Scotus, as informed by and as informing modern scientific practice. As applied to generalizations, Peirce saw realist concepts at work in the following way. Take any two occasions of releasing a stone from the hand and watching it fall. However much the two occasions are alike, between them there is any number—an infinity, denumerable or higher—of like possible occasions of its release allowing a stone to fall. A real relation of mediation unites the particular occasions, actual and possible, of the generalization “released stones fall.” Behind that uniformity of nature there must be not mere chance, like a run of straight sixes, but “some active general principle” (ibid.) Every sane person must accept that last statement, where it is understood that the principle does not merely accidentally coincide with moments in which one makes predictions based on it (ibid. See also Peirce 1901, chapter 18, and 1902, chapter 15).

Rand’s theory of concepts is not what has traditionally been called realist. Rather, hers is an objectivist theory. Concepts are “produced by man’s consciousness in accordance with the facts of reality . . . [they are] products of a cognitive method of classification whose processes must be performed by man, but whose content is dictated by reality” (ITOE 54). That cognitive method of classification requires the ability to regard items as (at least) substitution units along a real dimension(s) shared by the items. Regarding things as units, whether as substitution units or also as measure-value units along shared dimensions,* is a “method of identification or classification according to the attributes which a consciousness observes in reality. . . . Units do not exist qua units, what exists are things, but units are things viewed by a consciousness in certain existing relationships” (ITOE 6–7).

Rand’s objectivist theory of conceptual identification, set in her metaphysics, as supplemented by the principle of substantive propagation, is a strong competitor to Peirce’s realist way of tying inductive generalization to universal concepts.

By four months of age, an infant expects objects to fall if not supported (¶8). This is an example of what Mill called eduction, inference from particular past cases to the next particular like case, rather than inductive inference from particulars to general (¶9). Animals also have the limited power that is eduction. Harriman writes that animals “cannot project from their percepts what future to expect” (28). While that is a slight overstatement, it is surely correct with respect to all the expectations we have from induction, which requires conceptual generalization.

Harriman inclines to think that higher animals have direct experience of causation (cf. Enright 1991, §II). Like us, they “perceive that various actions they take make certain things happen. But they cannot go on to infer any generalizations from these perceptions” (28). The important thing is that Harriman rightly affirms that the human animal perceives some causal relations directly. (See “Hume – Experience of Cause and Effect” above* and Yale.) From those percepts, general causal principles (from “Pushed balls roll” to “Applied torque causes onset of rolling”) are formed after the general pattern of how universal concepts are formed from percepts. Harriman’s book is an attempt to spell out more specifically the abstraction process from elementary causal principles such as “pushed balls roll” to general scientific principles—the tremendous abstraction process that is ampliative induction—illustrated by episodes in the history of science (join with Note 27).

David Hume was dead set against the idea that we have any direct perception of causal power operating in the world (Note 29). “We never have any impression, that contains any power or efficacy” (T I.3.14). And Hume was dead set against alleged human powers of abstraction (T I.1.7). Moreover, “a general idea being impossible without an individual; where the latter is impossible, ’tis certain the former can never exist” (T I.3.14). “We never therefore have any idea of power” (ibid.).



#26 Stephen Boydstun

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Posted 15 April 2011 - 05:31 AM

—Existence is Identity

Then existents are identities. There are two broad types of identity. There is what I shall term particular identity and what I shall term specific identity. The identities of existents would seem always to be of both types. Always look for both. Existence is identity, particular and specific.

Particular identity is the it-ness, or that-ness, of a thing. Sometimes this is called material identity; more often it is called simply identity. This is the identity we trace by indexicals or nouns, proper or common. It is the target of reference. It is just the existent(s) itself as against all others and through all time of its existence. We have spoken of simple identity through time, and this falls under particular identity. The relation of a thing with itself at another time is a relation of particular identity.

So too, a thing may in certain respects stand in a relation of particular identity to another thing. Here we have the identity of logical and mathematical equivalence as well as the identity in physical symmetries and functional relations. Here we have the identity of one electron with another (having the same spin); apart from their separate world lines in spacetime, they are identical. This is rather stronger than saying two electrons are identical in their essential aspects, and presumably this is why individuation of identical elementary particles is so tricky (Teller 1983; van Fraassen 1984).[32] Again, a thing under one description stands in a relation of particular identity to itself under another description. Here we have the identity of token and type; the identity of Spot and a dog, as when we say “Spot is a dog.” The is here is first and foremost the is of particular identity; Spot is one of the members in the extension of the concept dog. Particular identity is the identity of correspondence. (Macnamara 1986, 49–62, 69–76, 83–94, 123–36, 14–47; Nozick 1981, 29–37, 99–104).

Mention of Spot being a dog brings us to specific identity. The intension of the concept dog reflects Spot’s specific identity, his what-ness. Classifications, attributions, and descriptions, definite and indefinite, capture specific identity. Here we have conditions and their satisfactions. Specific identity is the identity of contrasts and character (Macnamara 1986, 124–28, 144–45).[33]

Consciousness is specific identification, and consciousness is particular identification. The form of consciousness of special interest to us here is the propositional form. In propositional consciousness, one notices both types of identifications immediately.

The compositions between subject and predicate in a true propositional consciousness capture objective identity relations. There are at least three such relations affirmed simultaneously in every true proposition. Consider the proposition “Spot is brown and white.” The copula is executes a triple coordination. It wires the subject, the proper name Spot, and the predicate, brown and white coloration, to the same existent, namely, the dog who is Spot. This is a relation of particular identity. Secondly, it specifies Spot in respect of his coloration. This is a relation of specific identity. Thirdly, it affirms that this particular identity and this specific identity obtain in reality. This last is a relation of particular identity between the proposition and reality. For the third identity to obtain, for the proposition to be fully true, the proposition’s first particular identity and its specific identity must obtain in reality (cf. Hoenen 1952, 62–84; see also Shaffer 1969).

I suggest unsurely how these identities may change when we are speaking clearly make-believe, as in the proposition “A unicorn is a quadruped.” The first particular identity coordinates the predicate with a merely posited subject. The subject is not posited as real, but as unreal. There is no direct, concretely real subject. (Perhaps horses are the subject in a roundabout way.) The subject has become only a topic. The first particular identity no longer wires a predicate and subject to a real thing; it jumpers a predicate directly to a subject. Propositions in make-believe are more analytical. Attention shifts to terms, to stipulative definitions and descriptions. The predicate still specifies. The third identity, the particular identity between the proposition and reality, does not obtain.

Our view of earnest predication is much like the Aquinas-Aristotle view. The underlying metaphysics is different. Aristotle’s schemes of substance-accident, matter-form, and potency-act are largely discarded in favor of the modern scientific scheme of material substance and a rich metaphysical principle of identity. Yet our view of predication is much like the view of Aquinas-Aristotle, for we too hold that we cognize true propositions about the physical world and that the structures of these propositions coincide with structures in reality. In our view, the structures are identity structures.

Our view, like the Aquinas-Aristotle view, is squarely opposed to the view of Kant. We maintain that, through-and-through, reality is predicative. This is an aspect of Rand’s dictum “Existence is identity.” In Kant’s view, predicative structure is imposed on the world by the mind (Kant 1965, A50–136, A148–62; B74–175, B188–202; Patton 1967; Bennett 1990, 86–125). In our view, the mind conforms itself to predicative structure in the world.[34]

It might be thought that Nicolaus could have reached our identity theory of predication; he came after Aquinas, and identity and non-contradiction held a large sway with him. Admittedly, he was hindered somewhat by his concern to keep the world safe for fideism, but he had also more honest intellectual handicaps. His concept of identity was too meager. It could reach only as far as the reach of non-contradiction. It was really little more, if anything more, than simple numerical identity. Perhaps man could not begin to see the true richness and fecundity of identity until he had witnessed the birth and growth of modern science.

It might be wondered to what extent Rand would have concurred with the senses of identity I have set out or with the role of identity in induction or in predication I have framed. I do not know. I began with Rand’s thought. I have aimed at truth.

Our present work is not quite done. I promised* to treat mathematical induction. The time has come for making good that promise. The general form of mathematical induction was stated earlier, but I will restate it here for convenience. A statement P(n) to be proved for integers n>t, where t is a fixed positive integer, is true, provided:

(i) P(t) is true, and
(ii) If P(t), P(t+1), . . . P(n) are true, then P(n+1) is true.

First verify (i), that P(n) is true for n=t. Then try to show that assuming the antecedent of (ii) true implies that the consequent of (ii) is also true.

For illustration of the technique, let us prove that for all odd counting numbers n, n2 – 1 is divisible by 4. That is, (n2 – 1) = 4m, where m is some whole number. We take t = 1 since we want to prove the property for all odd counting numbers. For n = t = 1, we see that (12 – 1) = 0 = 4 x 0. So n2 – 1 is divisible by 4 for n = t = 1. Assume n2 – 1 is divisible by 4 for all the odd numbers 1 through n, with n odd. The next odd number after n is n + 2. We see that (n+ 2)2 – 1 = n2 + 4n +4 – 1 = (n2 –1) + 4(n + 1). But we have assumed, for all odd numbers 1 through n, that (n2 – 1) = 4m. So (n + 2)2 – 1 = 4 (m + n +1), that is, (n + 2)2 – 1 is divisible by 4.

In the course of proof by mathematical induction, we come always to a stage at which we insert n into the thesis to be proved generally and, by a series of elementary arithmetic operations, arrive at the thesis with n +1 in it. We did not insert n + 1 into the thesis; n + 1 in the thesis is just what resulted from our insertion of n into the thesis. From this perspective, the thesis seems to function as a counter. Such a thesis resembles a time-invariant structure, an identity in time.

I remarked earlier that mathematical induction is absolutely conclusive. I remarked also that despite the inductive character of (ii), mathematical induction is overall a deduction. I am a little unsure now whether it really comes to deduction. In saying that mathematical induction is absolutely conclusive, we mean, here anyway, to say specifically that if (i) and (ii) are true for a thesis, in no case is the thesis false. That this is so, we support as follows: Suppose there is a thesis P of which (i) and (ii) are true though P is in some case(s) false. Then there is some integer N>t for which the thesis is false. Let N1 be the smallest such integer. The thesis is true for t, t + 1, . . . , N1– 1, but it is not true for N1. Since the thesis is true for N1 –1, it is by induction true for (N1 – 1) + 1 = N1. This contradicts the supposition that P is in some case false. We have confirmation that mathematical induction is overall a deductive form of argument, and at the same time, we have confirmation that it is absolutely conclusive.[35]

Consider again the visit of that grey squirrel to my balcony.* I said that it was both logically and physically true that a day hence it will have been true for one more day past that that squirrel made that visit. The logical force would seem to be just the logical force of mathematical, or recursive, induction. So long as we are physically assured that nothing can affect the past, then what has been in the past is a pure, untouchable identity through all future time. This would seem not to be a case of ampliative induction, not a case of things growing from the ways things were, at least not once the utter impossibility of affecting the past is granted.

Should we conjecture then that identities through physical time, both substantive propagations and the inert endurance of the past, are the physical and logical bases of mathematical induction? It is tempting. Time probably is infinitely divisible. It is, moreover, probably a continuum, of the same order as the continuum of the real numbers. But there is the rub. We already know certainly that there are the infinite sets of integer and rational numbers, and we already know certainly that there is the higher-order infinite set of real numbers. We already know, quite apart from the possible, even probable, infinite divisibility and continuity of physical quantities such as time, length, or angle.

Suppose these quantities and their usual cohorts, such as velocity, mass, and momentum, are ultimately nowhere continua, unlike the real number system. Let us be even more pessimistic and suppose that not only are these physical quantities not continua, they are not even infinitely divisible. They do not admit even of rational fractioning on-and-on. Then we should still be grateful to Blaise Pascal 1654) for having discovered mathematical induction Surely known physical numerosities are so vast that we could still put mathematical induction to good use. I should mention that the technique of mathematical induction can be put to uses not purely mathematical. Augustin-Louis Cauchy (1826) used it to prove that the parallelogram rule for addition of two forces applied to a point holds for any angle between the forces. Cauchy’s proof does not demonstrate that physical angles about a point are a continuum, only that the parallelogram rule will hold for any fineness of angle up to and including a continuum (Benvenuto 1990, 116–42; Moigno 1966, 5–18). The mathematicians are magnanimous in the tools they provide the physicists.

Perhaps identity across a physical time or across a physical extension or around a physical point are, after all, the intuitions plied in the technique of mathematical induction, but the proper intensity—denumerable finitude, denumerable infinitude, or continuum—of the process is not set by those intuitions. For the appropriate intensity, in physical applications, we turn to physics. True, physics can today only give upper bounds to any hypothetical discreteness or discontinuity of space or time, but we should not sell future physics short on this issue. It is quite possible that physics will someday dispose the issue completely. Remember the power of consilience-inductions. Remember also the mathematical prowess of the modern relativists (e.g., Friedman 1983, 165–76, 309–20).

We might rest semi-content here over mathematical induction, but we have been passing by an issue that must not be left unstated, unbroached. All the integers, all the rationals, and all the reals exist. They probably exist physically all around us, but they certainly exist mathematically. What is mathematical existence? How is it grounded in the real world?

It is sometimes said that mathematical entities, qua mathematical, have a purely intentional existence (Hoenen 1952, 176–77; Shaffer 1969, 140–41). In this way they are like unicorns. They are make-believe. There may or may not be any real extension falling under the concept of a certain mathematical entity, but when thinking mathematically, we are thinking only in the “as-if” mode. This much seems true, and if we remain cognizant of the real extensions falling under or falling nearly under the more elementary mathematical concepts (Jetton 1991), then we have a start on the reasons for “the unreasonable effectiveness of mathematics in the natural sciences” (Eugene Wigner’s memorable phrase).

The intentional existence of mathematical entities differs profoundly, however, from unicorns in a fable. Mathematics is the study of an objective realm waiting to be discovered. We must discover whether “there is a real number such that its cube minus seven is zero” or whether “there are functions that are everywhere continuous and nowhere differentiable.” Mathematicians may posit a new mathematical entity, or structure, but they may then pull out of it more than they put into it. Roger Penrose makes the point well:

 

Later we find many other magical properties that these complex numbers possess, properties that we had no inkling about at first. These properties are just there. They were not put the by Cardano, nor by Bombelli, nor Wallis, nor Cotes, nor Euler, nor Wessel, nor Gauss, despite the undoubted farsightedness of these, and other great mathematicians; such magic was inherent in the very structure that they gradually uncovered. When Cardano introduced his complex numbers, he could have no inkling of the many magical properties which were to follow—properties which go under various names, such as the Cauchy integral formula, the Riemann mapping theorem, the Lewy extension property. These, and many other remarkable facts, are properties of the very numbers, with no additional modifications whatever, that Cardano had first encountered about 1539. (1989, 96)

In mathematical discovery, deduction rules. The deductive demonstration is required, and it disposes of the case. Because of this, it is sometimes said that mathematical necessity is of the “if-then” sort (Armstrong 1989, 119–24). Yes, but that—the principle of non-contradiction—is not the only source of necessity in mathematics. The realms of mathematics have their own distinctive, objective characters. In mathematics, too, existence is identity. Nothing but identity, it seems to me.

It is sometimes said that the physical sciences are of actual existence, whereas mathematics is of possible existence (ibid., 124–26). That is true, but what is the sense of possible here? It is not merely the non-contradictory. It might be reasonably said that it may as well be merely the non-contradictory, provided we confine attention to the right sort of intentional objects. What are the right sort? I suggest that mathematics is the study of predicative structure, with focus on quantification and mapping.[36]

Now we rest. Some will say things were just getting good and they are unable to rest. These may continue into the light; I recommend Chihara 1990 and Geroch 1985.

References

Armstrong, D.M. 1989. A Combinatorial Theory of Possibility. Cambridge.

Bennett, J. 1990 [1966]. Kant’s Analytic. Cambridge.

Benvenuto, E. 1991. An Introduction to the History of Structural Mechanics: Statics and Resistance of Solids. Springer-Verlag.

Chihara, C. S. 1990. Constructibility and Mathematical Existence. Oxford.

Friedman, M. 1983. Foundations of Space-Time Theories. Princeton.

Geroch, R.1985. Mathematical Physics. Chicago.

Hoenen, P. 1952. Reality and Judgment According to St. Thomas. H. F. Tiblier, translator. Henry Regnery.

Jetton, M. 1991. Philosophy of Mathematics. Objectivity 1(2):1–23.

Kant, I. 1965 [A–1781 B–1787]. Critique of Pure Reason. N. Kemp Smith, translator. St. Martin’s.

Macnamara, J. 1986. A Border Dispute: The Role of Logic in Psychology. MIT.

Moigno, M. 1966 [1868]. Leçons de Méchanique Analytique: Statique. Paris: GauthierVillars. Johnson Reprint.

Nozick, R. 1981. Philosophical Explanations. Harvard.

Paton, H.J. 1967 [1931]. The Key to Kant’s Deduction of the Categories. Reprinted in Kant: Disputed Questions. M. S. Gram, editor. Quadrangle Books.

Penrose, R. 1989. The Emperor’s New Mind. Penguin.

Shaffer, J. 1969 [1962]. Existence, Predication, and the Ontological Argument. Reprinted in The First Critique. T. Penelhum and J. J. MacIntosh, editors. Wadsworth.

Teller, P. 1983. Quantum Physics, the Identity of Indiscernibles, and Some Unanswered Questions. Philosophy of Science 50:309–19.

Van Fraassen, B. C. 1984. The Problem of Indistinguishable Particles. In Science and Reality. J. T. Cushing C. F. Delaney, and G. M. Gutting, editors. Notre Dame.

~~~~~~~~~~~~~~~~
This essay of 1991 had no notes. I have now added some endnotes to indicate changes or emendations to the positions I took in this essay nineteen years ago. I have also added some hyperlinks within the text.
~~~~~~~~~~~~~~~~

Notes
32. More recently: Teller 1997; French and Krause 2006.

33. Today I make the distinction simply this way. Particular identity answers to that and which, to where and when, and to how much. Specific identity answers to what, to character such as kind, form, capability, or susceptibility. As stated in the 1991 text, every existent consists of both a particular and a specific identity.

A distinction between particular and specific identity may be implicit in Rand’s writings addressing identity, as when she writes of the child’s identity stage of “awareness of specific, particular things” (ITOE 6; see also a, b).

It is convenient to divide the how much particular identity into two sorts. There is on the one hand what I call the item-measure particularity of the existent. That would be its possibilities for being placed in sequences, for being counted, and for being placed in frequency distributions, discrete or continuous. On the other hand, there is what I call the trait-measure particularity of the existent. I am using trait as a short cover for attributes, actions, and dimensioned relations (including classical and quantum states). The relations implicated in item-measure particularity are not relations along dimensions; they are purely numerical relations.

Item-measure coincides with the type of measurement now called absolute in the measurement literature. Trait-measure includes the types the experts call ordinal, hyper-ordinal, interval, and ratio. Trait-measure includes also all the multidimensional forms of measurement, such as the use of topological vector spaces in physics or the use of trigonometry in making drapery. Measurement restricted to the sense of trait-measures is what Rand presumes when she writes that “entities (and their actions) are measured by their attributes (length, weight, velocity, etc.)” (ITOE 7). I should note that where and when are trait-measures, though I spotlight them apart from all other trait-measures.

Sameness “applies in the most strict sense to what is numerically one” (Topics 151b28–29). A belly is one and the same as a tummy. A two-footed terrestrial mammal is one and the same as a man. That man assisting in the childbirth is one and the same as the philosopher Socrates.

Leibniz would add that a triangle is one and the same as a trilateral (NEU 363 [181]), and Frege shows us a less obvious geometric example of particular identity (1879, 21). All of these identity statements may be represented by the schema X=Y, where the relational sign (=) is the logical identity sign, at present restricted to the that and which and item-measure portions of particular identity.

We learn that the morning star is one and the same celestial object as the evening star, namely the planet Venus. The fact that Venus is Venus (A=A) is the bottom explanation for the fact that the morning star is the evening star (X=Y). But for the particular identity of Venus to support the truth that the morning star is one and the same celestial object as the evening star, we are presuming not only that Venus is numerically one with itself. We are further presuming, with reason, that Venus is in a determinate sequence of locations across particular times and not in any other sequence. Venus has a determinate world line (or avenue) in spacetime and not any other world line.

It is only a fuller particular identity of Venus, answering not only to that and which and item-measure, but to where and when, that allows the particular identity of Venus to adequately support the truth that the morning star is one and the same celestial object as the evening star. The morning star, evening star, and Venus are all in spacetime in (the same) excluding ways. Venus rising from one’s eastern horizon excludes Venus simultaneously rising from one’s horizon due north, and so forth.

I take it as a postulate that all concrete particulars (short of the universe itself), whether physical or mental, have delimited loci in spacetime. This is how I divide concrete particulars such as stones, throws, chills, clouds, rainbows, and thoughts from, on the other side of the division, abstract particulars such as sets, mathematical groups, and operator actions on Hilbert spaces. Concrete particulars have particular histories in the space and time had by the physical universe. In thinking about abstract particulars as such, we suspend consideration of any potential exemplification they may have in actual space and time. (See further, here.)

34. My triple-identity view of predication needs to be cast in appropriate measurement terms. That journey would include the transition from logical quantification in predicate logic to numerically definite quantification (which is still purely logical quantification) to sets to arithmetic (see Quine 1982). For the transition from logic and set theory to measurement theory, one-dimensional and multidimensional, I recommend the three volumes of Foundations of Measurement by Krantz, Luce, Suppes, and Tversky (2006).


35. On logic and mathematical induction, see Machover 1996 and Srivastava 2008.

36. In philosophy of mathematics, I incline to structuralism and to empiricism in the broad sense of Philip Kitcher. For the major varieties of structuralism in contemporary philosophy of mathematics, see Hellman 1989; Shapiro 1997; Resnik 1997; Chihara 2004. The empirical theory of Philip Kitcher is advanced in The Nature of Mathematical Knowledge (1984). This view and the way in which it is an empiricist view is examined in the Objectivity essay “Mathematic Empiric” by Daniel Ust (1993).



#27 Stephen Boydstun

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Posted 15 November 2012 - 09:14 AM

A Material Dissolution of the Problem of Induction
John D. Norton (2010)

A Material Theory of Induction
John D. Norton (2003)

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Induction on Identity
Varieties of Induction
Ockham – Contingency
Nicolaus – Experience of Substance
Nicolaus – Reasoning to Substance
Hume – Experience of Cause and Effect
Hume – Reasoning to Cause or Effect
Hume – Necessity
Hume – Uniformity
Existence is Identity





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