The Online Library of Liberty

A project of Liberty Fund, Inc.

• Introduction
• Textual Introduction
• Preface to the Third Edition[*]
• Chapter I: Introductory Remarks
• Chapter II: The Relativity of Human Knowledge
• Chapter III: The Doctrine of the Relativity of Human Knowledge, As Held By Sir William Hamilton
• Chapter IV: In What Respect Sir William Hamilton Really Differs From the Philosophers of the Absolute
• Chapter V: What Is Rejected As Knowledge By Sir William Hamilton, Brought Back Under the Name of Belief
• Chapter VI: The Philosophy of the Conditioned
• Chapter VII: The Philosophy of the Conditioned, As Applied By Mr. Mansel to the Limits of Religious Thought
• Chapter VIII: Of Consciousness, As Understood By Sir William Hamilton
• Chapter IX: Of the Interpretation of Consciousness
• Chapter X: Sir William Hamilton’s View of the Different Theories Respecting the Belief In an External World
• Chapter XI: The Psychological Theory of the Belief In an External World
• Chapter XII: The Psychological Theory of the Belief In Matter, How Far Applicable to Mind
• Chapter XIII: The Psychological Theory of the Primary Qualities of Matter
• Chapter XIV: How Sir William Hamilton and Mr. Mansel Dispose of the Law of Inseparable Association
• Chapter XV: Sir William Hamilton’s Doctrine of Unconscious Mental Modifications
• Chapter XVI: Sir William Hamilton’s Theory of Causation
• Chapter XVII: The Doctrine of Concepts, Or General Notions
• Chapter XVIII: Of Judgment
• Chapter XIX: Of Reasoning
• Chapter XX: On Sir William Hamilton’s Conception of Logic As a Science. Is Logic the Science of the Laws, Or Forms, of Thought?
• Chapter XXI: The Fundamental Laws of Thought According to Sir William Hamilton
• Chapter XXII: Of Sir William Hamilton’s Supposed Improvements In Formal Logic
• Chapter XXIII: Of Some Minor Peculiarities of Doctrine In Sir William Hamilton’s View of Formal Logic
• Chapter XXIV: Of Some Natural Prejudices Countenanced By Sir William Hamilton, and Some Fallacies Which He Considers Insoluble
• Chapter XXV: Sir William Hamilton’s Theory of Pleasure and Pain
• Chapter XXVI: On the Freedom of the Will
• Chapter XXVII: Sir William Hamilton’s Opinions On the Study of Mathematics
• Chapter XXVIII: Concluding Remarks
• Appendices
• Appendix A: Manuscript Fragments
• Appendix B: Textual Emendations
• Appendix C: Corrected References
• Appendix D: Bibliographic Index of Persons and Works Cited In the Examination, With Variants and Notes

CHAPTER XXIV

Of Some Natural Prejudices Countenanced by Sir William Hamilton, and Some Fallacies Which He Considers Insoluble

we have concluded our review of Sir W. Hamilton as a teacher of Logic; but there remain to be noticed a few points, not strictly belonging either to Logic or to Psychology, but rather to what is inappropriately termed the Philosophia Prima. It would be more properly called ultima, since it consists of the widest generalizations respecting the laws of Existence and Activity; generalizations which by an unfortunate, though at first inevitable mistake, men fancied that they could reach uno saltu, and therefore placed them at the beginning of science, though, if they were ever legitimate, they could only be so as its tardy and final result. Every physical science, up to the time of Bacon, consisted mainly of such first principles as these: The ways of Nature are perfect: Nature abhors a vacuum: Natura non habet saltum: Nothing can come out of nothing: Like can only be produced by like: Things always move towards their own place: Things can only be moved by something which is itself moving; and so forth. And the Baconian revolution was far indeed from expelling such doctrines from philosophy. On the contrary, the Cartesian movement, which went on for a full century simultaneously with the Baconian, threw up many more of these imaginary axioms concerning things in general, which took a deep root in Continental philosophy, found their way into English, and are by no means, even now, discredited as they deserve to be. Most of these were fully believed by the philosophers who maintained them, to be intuitively evident truths—revelations of Nature in the depths of human consciousness, and recognisable by the light of reason alone: while all the time they were merely bad generalizations of the vulgarest outward experience; rough interpretations of the appearances most familiar to sense, and which therefore had grown into the strongest associations in thought; never tested by the conditions of legitimate induction, not only because those conditions were still unknown, but because these wretched first attempts at generalization were deemed to have a higher than inductive origin, and were erected into general laws from which the order of the universe might be deduced, and to which every scientific theory for the explanation of phenomena must be required to conform. It is a material point in the estimation of a philosopher and of his doctrines, whether he has taken his side for or against this mode of philosophizing; whether he has countenanced any of these spurious axioms by his adhesion. Sir W. Hamilton cannot be acquitted of having done so, in more than one instance.

In treating of the problem of Causality, Sir W. Hamilton had occasion to argue, that we ought not to postulate a special mental law in order to explain the belief that everything must have a cause, since that belief is sufficiently accounted for by the “Law of the Conditioned,” which makes it impossible for us to conceive an absolute commencement of anything.^[*] I do not mean to return to the discussion of this theory of Causality; but let us ask ourselves why we are interdicted from assuming a special law, in order to account for that which is already sufficiently accounted for by a general one. The real ground of the prohibition is what our author terms the Law of Parcimony; a principle identical with the famous maxim of the Nominalists, known as Occam’s Razor—Entia non sunt multiplicanda præter necessitatem;^[†] understanding by Entia, not merely substances but also Powers. Sir W. Hamilton, instead of resting it on this logical injunction, grounds it on an ontological theory. His reason is, “Nature never works by more and more complex instruments than are necessary.”^* He cites, with approbation, the maxims of Aristotle, “that God and Nature never operate without effect (οὐδὲν μάτην, οὐδὲν έλλειπῶς, ποιοῦσι); they never operate superfluously (μηδὲν περίεργον—περιττῶς—ἀργῶς); but always through one rather than through a plurality of means (καθ’ ἔν, μᾶλλον ἢ κατὰ πολλὰ):”^† thus borrowing a general theory of the very kind which Bacon exploded, to support a rule which can stand perfectly well without it. Have we authority to declare that there is anything which God and Nature never do? Do we know all Nature’s combinations? Were we called into counsel in fixing its limits? By what canons of induction has this theory ever been tried? By what observations has it been verified? We know well that Nature, in many of its operations, works by means which are of a complexity so extreme, as to be an almost insuperable obstacle to our investigations. On what evidence do we presume to say that this complexity was necessary, and that the effect could not have been produced in a simpler manner? If we look into the meaning of words, of what kind is the necessity which is supposed to be binding on God and Nature—the pressure they are unable to escape from? Is there any necessity in Nature which Nature did not make? or if not, what did? What is this power superior to Nature and its author, and to which Nature is compelled to adapt itself?

There is one supposition under which this doctrine has an intelligible meaning—the hypothesis of the Two Principles. If the universe was moulded into its present form by a Being who did not make it wholly, and who was impeded by an obstacle which he could only partially overcome—whether that obstacle was a rival intelligence, or, as Plato thought, an inherent incapacity in Matter;^[*] it is on that supposition admissible, that the Demiourgos may have always worked by the simplest possible means; the simplest, namely, which were permitted by the opposition of the conflicting Power, or the intractableness of the material. This is, in fact, the doctrine of Leibnitz’s Théodicée; his famous theory that a world, made by God, must be the best of all possible worlds, that is, the best world which could be made under the conditions by which, as it would appear, Providence was restricted.^[†] This doctrine, commonly called Optimism, is really Manicheism, or, to call it by ^aits^a more proper name, Sabæism. The word “possible” assumes the existence of hindrances insurmountable by the divine power, and Leibnitz was only wrong in calling a power limited by obstacles by the name Omnipotence: for it is almost too obvious to be worth stating, that real Omnipotence could have effected its ends totally without means, or could have made any means sufficient. This Sabæan theory is the only one by which the assertion, that Nature always works by the simplest means, can be made consistent with known fact. Even so, it remains wholly unproved; and, were it proved, would be but a speculative truth of Theology, incapable of affording any practical guidance. We could never be justified in rejecting an hypothesis for being too complicated; it being beyond our power to set limits to the complication of the means that might possibly be necessary, to evade the obstacles which Ahriman or Matter may have perversely thrown in the Creator’s way.

The “Law of Parcimony” needs no such support; it rests on no assumption respecting the ways or proceedings of Nature. It is a purely logical precept; a case of the broad practical principle, not to believe anything of which there is no evidence. When we have no direct knowledge of the matter of fact, and no reason for believing it except that it would account for another matter of fact, all reason for admitting it is at an end when the fact requiring explanation can be explained from known causes. The assumption of a superfluous cause, is a belief without evidence; as if we were to suppose that a man who was killed by falling over a precipice, must have taken poison as well. The same principle which forbids the assumption of a superfluous fact, forbids that of a superfluous law. When Newton had shown that the same theorem would express the conditions of the planetary motions and the conditions of the fall of bodies to the earth, it would have been illogical to recognise two distinct laws of nature, one for heavenly and the other for earthly attraction; since both these laws, when stripped of the circumstances ascertained to be irrelevant to the effect, would have had to be expressed in the very same words. The reduction of each of the two generalizations to the expression of only those circumstances which influence the result, reduces both of them to the same proposition; and to decline to do so, would be to make an assumption of difference between the cases, for which none of the observations afforded the smallest ground. The rule of Parcimony, therefore, whether applied to facts or to theories, implies no theory concerning the propensities or proceedings of Nature. If Nature’s ways and inclinations were the reverse of what they are supposed to be, it would have been as illegitimate as it is now, to assume a fact of Nature without any evidence for it, or to consider the same property as two different properties, because found in two different kinds of objects.

In another place, Sir W. Hamilton says that the Law of Parcimony, which he terms “the most important maxim in regulation of philosophical procedure when it is necessary to resort to an hypothesis,” has “never, perhaps, been adequately expressed;” and he proposes the following expression for it: “Neither more nor more onerous causes are to be assumed, than are necessary to account for the phænomena.”^* This conception of some causes as “more onerous” to the general scheme of things than others, is a distinction greatly requiring what our author says it has never yet had—to be “articulately expressed.” He does not, however, articulate it in general terms, but only in its application to the particular question of Causality. From this we may collect,—1st. That a “positive power” is a more onerous hypothesis than a “negative impotence.” 2nd. That a special hypothesis, which serves to explain only one phænomenon, is more onerous than a general one which will explain many. 3rd. That the explanation of an effect by a cause of which the very existence is hypothetical, is more onerous than its hypothetical explanation by a cause otherwise known to exist. The last two of these three canons are but particular cases of the general rule, that we should not assume an hypothetical cause of a phænomenon which admits of being accounted for by a cause of which there is other evidence.^* The remaining canon, that we should prefer the hypothesis of an incapacity to that of a power, is, I apprehend, only valid when its infringement would be a violation of one of the other two rules.

The time-honoured, but gratuitous, assumption, respecting Nature, on which I have now commented, is not the only generality of the pre-Baconian type which Sir W. Hamilton has countenanced. He gives his sanction to the old doctrine that “a thing can act only where it is.” The dictum appears in this direct form in one of the very latest of his writings, the notes for an intended memoir of Professor Dugald Stewart.^† He has so much faith in it as to make it the foundation of two of his favourite theories. One is, that

Conformably to this, we have seen him not only maintaining, in opposition to Reid,^[*] that we do not see the sun—that we see only an image of it in our eye—but also, that we directly perceive Extension, whether by sight or touch, only in our own bodily organs: thus preferring the à priori axiom, that a thing can only act where it is, to the authority of those “natural beliefs” which he, in other cases, so strenuously asserts against impugners, and so often affirms that we ought either to accept as a whole, or never appeal to at all.

The other theory which our author maintains on the authority of the same dictum, is that the mind acts directly throughout the whole body, and not through the brain only.

Sir W. Hamilton should at least have shown how this hypothesis can be reconciled with the fact, that a slight pressure on the nerve at a place intermediate between the finger and the brain, takes away the mind’s power of feeling in the finger, while at any point above the ligature the feeling is the same as before. ^cIf he object that the mode in which the pressure impedes sensation need not be by interrupting the communication between the finger and the brain, but may be by disturbing the functions of the nerve itself, we may ask, why is this disturbance confined to the part of the nerve which is below the point of pressure, while above that point the functions remain unimpaired? Many other objections might be brought against Sir W. Hamilton’s theory, if my object were to discuss the physiological question; but my object is only^c to show the amount of evidence which Sir W. Hamilton will disregard, rather than admit that one thing can act directly upon another without immediate contact.^† What he would have thought of the application of his doctrine to the solar system, he has not told us ^d(the recent developments of the doctrine of the Unity of Force being posterior to his time)^d : but it commits him to the opinion, that gravitation acts through an intervening medium, which he must postulate, first as existing, and secondly, as possessed of inscrutable properties; in palpable repugnance to his own Law of Parcimony, and to all the canons grounded thereon. Descartes postulated his vortices in obedience to the same axiom.^[*]

What, however, is the worth of this doctrine, that things can only act upon one another by direct contact? Mr. Carlyle says, “a thing can only act where it is; with all my heart; only where is it?”^[*] In one sense of the word, a thing is wherever its action is: its power is there, though not its corporeal presence. But to say that a thing can only act where its power is, would be the idlest of mere identical propositions. And where is the warrant for asserting that a thing cannot act when it is not locally contiguous to the thing it acts upon? Shall we be told that such action is inconceivable? Even if it was, this, according to Sir W. Hamilton’s philosophy, is no evidence of impossibility. But that it is conceivable, is shown by every fairy tale, as well as by every religion. Then, again, what is the meaning of contiguity? According to the best physical knowledge we possess, things are never actually contiguous: what we term contact between particles, only means that they are in the degree of proximity at which their mutual repulsions are in equilibrium with their attractions. If so, instead of never, things always act on one another at some, though it may be a very small distance. The belief that a thing can only act where it is, is a common case of inseparable, though not ultimately indissoluble, association. It is an unconscious generalization, of the roughest possible description, from the most familiar cases of the mutual action of bodies, superficially considered. The temporary difficulty found in apprehending any action of body upon body unlike what people were accustomed to, created a Natural Prejudice, which was long a serious impediment to the reception of the Newtonian theory: but it was hoped that the final triumph of that theory had extinguished it; that all educated persons were now aware that action at a distance is intrinsically quite as credible as action in contact, and that there is no reason, apart from specific experience, to regard the one as in any respect less probable than the other. That Sir W. Hamilton should be an instance to the contrary, is an example of the obstinate vitality of these idola tribûs,^[†] and shows that we are never safe against the rejuvenescence of the most superannuated error, if in throwing it off we have not reformed the bad habit of thought, the wrong and unscientific tendency of the intellect, from which the error took its rise.^*

Though but remotely connected with the preceding considerations, yet as belonging in common with them to the subject of Fallacies, I will notice in this place the curious partiality which our author shows to a particular group of sophisms, the Eleatic arguments for the impossibility of motion. He ^fdeemed^f these arguments, though leading to a false conclusion, to be irrefutable; as Brown thought concerning Berkeley’s argument against the existence of matter—that as a mere play of reasoning it was unanswerable, while it was impossible for the human mind to admit the conclusion;^[*] forgetting that if this were so it would be a reductio ad absurdum of the reasoning faculty. There is no philosopher to whom, I imagine, Sir W. Hamilton would have less liked to be assimilated, than Brown; and he would probably have defended himself against the imputation, by saying that the Eleatic arguments do not prove motion to be impossible, but only to be inconceivable by us. Yet if a fact which we see and feel every minute of our lives, is not conceivable by us, what is? Our author does not enter at any length into the question, but expresses his opinion on several occasions incidentally. “It is,” he says, “on the inability of the mind to conceive either the ultimate indivisibility, or the endless divisibility of space and time, that the arguments of the Eleatic Zeno against the possibility of motion are founded; arguments which at least show, that motion, however certain as a fact, cannot be conceived possible, as it involves a contradiction.”^* We have been told in very emphatic terms by Sir W. Hamilton, that the Law of Contradiction is binding not on our conceptions merely, but on Things.^[†] If, then, motion involves a contradiction, how is it possible? and if it is possible, and a fact, as we know it to be, how can it involve a contradiction? The appearance of contradiction must necessarily be fallacious, even were we unable to point out the fallacy. Our author, apparently, has attempted to resolve it, and failed. He calls the argument “an exposition of the contradictions involved in our notion of motion,” and says that its “fallacy has not yet been detected.”^† And, again, “The Eleatic Zeno’s demonstration of the impossibility of motion is not more insoluble than could be framed a proof that the Present has no reality: for however certain we may be of both, we can positively think neither.”^* It must, one would suppose, be a great difficulty, which could appear insoluble to Sir W. Hamilton. The “demonstration,” at all events, cannot yet have been refuted, and superhuman ingenuity must be needed to refute it. Yet the fallacy in it has been pointed out again and again; and the contradictions which Sir W. Hamilton regards it as an exposure of, do not exist.

Zeno’s reasonings against motion, as handed down by Aristotle,^[*] consist of four arguments, which are stated and criticised with considerable prolixity by Bayle.^[†] Several of these are substantially the same argument in different forms, and if we examine the two most plausible of them it will suffice. The first is the ingenious fallacy of Achilles and the Tortoise. If Achilles starts a thousand yards behind the tortoise, and runs a hundred times as fast; still, while Achilles runs those thousand yards, the tortoise will have got on ten; while Achilles runs those ten, the tortoise will have run a tenth of a yard; and as this process may be continued to infinity, Achilles will never overtake the tortoise. In our author’s opinion, this argument is logically correct, and evolves a contradiction in our idea of motion. But it is neither logically correct, nor evolves a contradiction in anything. It assumes, of course, the infinite divisibility of space. But we have no need to entangle ourselves in the metaphysical discussion whether this assumption is warrantable. Let it be granted or not, the argument always remains ^gfallacious. The fallacy lies in the assertion that “this process may be continued to infinity.” Infinity is here ambiguous. The conclusion drawn is that the process may be continued for an infinite duration of time. But the premise is only true in the sense, that it may be continued for an infinite number of divisions of time. The argument confounds infinity and infinite divisibility. It^g assumes that to pass through an infinitely divisible space, requires an infinite time. But the infinite divisibility of space means the infinite divisibility of finite space: and it is only infinite space which cannot be passed over in less than infinite time. What the argument proves is, that to pass over the infinitely divisible space, requires an infinitely divisible time: but an infinitely divisible time may itself be finite; the smallest finite time is infinitely divisible; the argument, therefore, is consistent with the tortoise’s being overtaken in the smallest finite time. It is a sophism of the type Ignoratio Elenchi, or, as Archbishop Whately terms it, Irrelevant Conclusion;^[*] an argument which proves a different proposition from that which it pretends to prove, the difference of meaning being disguised by ^han ambiguity^h of language.

The other plausible form of Zeno’s argument is at first sight more favourable to Sir W. Hamilton’s theory, being a real attempt to prove that the fact of motion involves impossible conditions. The usual mode of stating it is this. If a body moves, it must move either in the place where it is, or in the place where it is not: but either of these is impossible: therefore it cannot move. First of all, this argument, even if we were unable to refute it, does not exhibit any contradiction in our “notion” of motion. We do not conceive a body as moving either in the place where it is, or in the place where it is not, but from the former to the latter: in other words, we conceive the body as in the one place and in the other at successive instants. Where is the “contradiction” between being in one place at this moment, and in another at the next? As for the fallacy itself, it is strange that when everybody sees the answer to it, a practised logician should have any difficulty in putting that answer into logical forms. It is not necessary that motion should be in a place. ⁱA bodyⁱ must be in a place; but motion is not ^ja body^j —it is a change: and that a change of place should be either in the old place or in the new, is a real contradiction in terms. To put the thing in another way; Place may be understood in two senses: it may either be a divisible, or an indivisible part of space. If it be a divisible part, as a room, or a street, it is true that in that sense, every motion is in a place, that is, within a limited portion of space: but in this meaning of the term the dilemma breaks down, for the body really moves in the place where it is; the room, the field, or the house. If, on the contrary, we are to understand by Place an indivisible minimum of space, the proposition that motion must be in a place is evidently false; for motion cannot be in that which has no parts; it can only be to or from it.

A parallel sophism might easily be invented, turning upon Time instead of Space. It might be said that sunset is impossible, since if it be possible, it must take place either while the sun is still up, or after it is down. The answer is obvious: it is just the change from one to the other which is sunset. And so it is the change from one position in space to another which is motion. The parallelism between the two cases was evidently seen by Sir W. Hamilton, and the sophism was too hard for him in both: and this is what he must have meant by saying that we cannot “positively think” the Present. That he should have missed the solution of the fallacy is strange enough: but, as a matter of fact, the assertion that we have no positive perception, on the one hand of Motion, on the other, of present time, deserves notice as one of the most curious deliverances of so earnest an asserter of “our natural beliefs.”

These paralogisms are only part of a long list of puzzles concerning infinity, which, though by no means hard to clear up, appear to our author insoluble. I append in a note the entire list.^* Many of them are resolved by the observations already made their difficulty being merely that of separating the two ideas of Infinite and Infinitely Divisible. To our author’s thinking, infinite divisibility and the Finite contradict one another. But even allowing (which, as was seen in a former chapter, I do not) that infinite divisibility is inconceivable, it does not therefore involve a contradiction. The remaining puzzles mostly result from inability to conceive that one infinity can be greater or less than another: a conception familiar to all mathematicians. Our author refuses to consider that a space or a time which is infinite in one direction and bounded in another, is necessarily less than a space or a time which is infinite in every direction. The space between two parallels, or between two diverging lines or surfaces, extends to infinity, but it is necessarily less than entire space, being a part of it. Not only is one infinity greater than another, but one infinity may be infinitely greater than another. Mathematicians habitually assume this, and reason from it; and the ^kresult^k always coming out true, the assumption is justified. But mathematicians, I must admit, seldom know exactly what they are about when they do this. As the results always prove right, they know empirically that the process cannot be wrong—that the premises must be true in a sense; but in what sense, it is beyond the ingenuity of most of them to understand. The doctrine long remained a part of that mathematical mysticism, so mercilessly shown up by Berkeley in his Analyst, and Defence of Freethinking in Mathematics.^[*] To clear it up required a philosophical mathematician—one who should be both a mathematician and a metaphysician: and it found one. To complete Sir W. Hamilton’s discomfiture, this philosophic mathematician is his old antagonist Mr. De Morgan, whom he described as too much of a mathematician to be anything of a philosopher.^* Mr. De Morgan, however, has proved himself, as far as this subject is concerned, a far better metaphysician than Sir W. Hamilton. He has let the light of reason into all the logical obscurities and paradoxes of the infinitesimal calculus. By merely following out, more thoroughly than had been done before, the rational conception of infinitesimal division, as synonymous with division into as many and as small parts as we choose, ^lwithout any limit,^l Mr. De Morgan, in his Algebra,^[*] has fully explained and justified the conception of successive orders of differentials, each of them infinitely less than the differential of the preceding, and infinitely greater than that of the succeeding order. Whoever is acquainted with this masterly specimen of analysis, will find his way through Sir W. Hamilton’s series of riddles respecting Infinity, without ever being at a loss for their solution. I shall therefore trouble the reader no further with them in this place.

[[*] ]See Lectures, Vol. II, pp. 376ff.

[[†] ]This formulation is mistakenly attributed to William of Ockham; it appears to have originated with John Ponce, in an annotation to Duns Scotus, Opera omnia, ed. Luke Wadding, John Ponce, et al., 12 vols. (Lyons: Durand, 1639), Vol. VII, p. 723.

[* ]Appendix [I(A)] to Discussions, p. 622.

[† ]Ibid., p. 629. [For the Greek passages, cf. Aristotle, On the Heavens (Greek and English), trans. W. K. C. Guthrie (London: Heinemann; Cambridge, Mass.: Harvard University Press, 1939), p. 30 (I, iv, 271a34-5); Parts of Animals, in Parts of Animals, Movement of Animals, Progression of Animals (Greek and English), trans. A. L. Peck (London: Heinemann; Cambridge, Mass.: Harvard University Press, 1937), p. 396 (IV, xi, 691b4); On the Heavens, pp. 206-8 (II, xii, 292a22-b25).]

[[*] ]See Timæus, in Timæus, Critias, Cleitophon, Menexenus, Epistles (Greek and English), trans. R. G. Bury (London: Heinemann; New York: Putnam’s Sons, 1929), pp. 108ff. (47eff.).

[[†] ]See Théodicée, pp. 115ff. (§§8ff.).

[a-a]651, 652 a

[* ]Appendix [I(A)] to Discussions, p. 628n.

[* ]This is what Newton meant by a vera causa, in his celebrated maxim, “Causas rerum naturalium non plures admitti debere quam quæ et veræ sint, et earum phænomenis explicandis sufficiant.” [Isaac Newton, Philosophiæ Naturalis Principia Mathematica, in Opera, ed. Samuel Horsley, 5 vols. (London: Nichols, 1779-85), Vol. III, p. 2.] It is singular that Sir W. Hamilton does not seem to have understood, that by veræ causæ Newton meant agencies the existence of which was otherwise authenticated: for he says, “In their plain meaning, the words et veræ sint are redundant; or what follows is redundant, and the whole rule a barren truism.” (Foot-note to Reid, p. 236n.) bBut in the Appendix [I(A)] to the Discussions (p. 631) Sir W. Hamilton puts the right interpretation on Newton’s maxim.b

[† ]Lectures, Vol. II, App. i, p. 522.

[‡ ]Ibid.

[[*] ]See Reid, Inquiry, pp. 182-6.

[* ]Lectures, Vol. II, pp. 127-8.

[c-c]651, 652 I shall not here enquire how much is positively proved by this experiment, or with what hypotheses it is inconsistent: my object is

[† ]In the Lectures, I mean: for, in the “Dissertations on Reid” the doctrine, that we feel in the toe, and not in a sensorium commune, is at least so far retracted, that the possibility of the opposite theory is explicitly acknowledged. ([Note D,] p. 861n.)

[d-d]+67, 72

[[*] ]See pp. 198, 375 above.

[[*] ]Thomas Carlyle, Sartor Resartus, 2nd ed. (Boston: Munroe, 1837), p. 59 (I, viii).

[[†] ]See Bacon, Novum Organum, in Works, Vol. I, pp. 163 and 169 (Bk. I, Aphs. 41 and 52).

[* ]In the course of his speculations our author comes across a fact which is positively irreconcileable with his axiom; the fact of repulsion. This brings him to a dead stand. He knows not whether to advance or recede. Repulsion, he says, “remains, as apparently an actio in distans, even when forced upon us as a fact, still inconceivable as a possibility.” He is soon afterwards obliged to confess that the same is true of attraction: “As attraction and repulsion seem equally actiones in distans, it is not more difficult to realize to ourselves the action of the one, than the action of the other.” (“Dissertations on Reid,” [Note D,] p. 852.) Action from eae distance being “a fact,” though inconceivable, this fact would seem to require of him the retractation of his axiom: yet he does not retract it. I need hardly remark that attraction and repulsion are not inconceivable; except indeed in another of the numerous senses of that equivocal word; that in which it is used when our author tells us that all ultimate facts are inconceivable, meaning only that they are inexplicable.

[f-f]651, 652 believed

[[*] ]See Brown, Lectures, Vol. II, p. 19.

[* ]Lectures, Vol. II, p. 373. To the same effect, Vol. IV, p. 71.

[[†] ]See ibid., Vol. III, p. 81.

[† ]Foot-note to Reid, p. 102n.

[* ]Appendix [I(A)] to Discussions, p. 606.

[[*] ]See Aristotle, The Physics (Greek and English), trans. Philip H. Wickstead and Francis M. Cornford, 2 vols. (London: Heinemann; Cambridge, Mass.: Harvard University Press, 1963), Vol. II, pp. 176-90 (VI, Chap. ix, 239b-240b).

[[†] ]See Pierre Bayle, Dictionnaire historique et critique, 2 vols. (Rotterdam: Reinier Leers, 1697), s.v. Zenon d’Elée, Vol. II, pp. 1267-9.

[g-g]651, 652 a fallacy. For it

[[*] ]See Elements of Logic, p. 187.

[h-h]651, 652 similarity

[i-i]651, 652 An object

[j-j]651, 652 an object

[* ]“Contradictions proving the Psychological Theory of the Conditioned.

1. Finite cannot comprehend, contain, the Infinite.—Yet an inch or minute, say, are finites, and are divisible ad infinitum, that is, their terminated division incogitable.

2. Infinite cannot be terminated or begun.—Yet eternity ab ante ends now; and eternity a post begins now. So apply to Space.

3. There cannot be two infinite maxima.—Yet eternity ab ante and a post are twoinfinite maxima of time.

4. Infinite maximum if cut in two, the halves cannot be each infinite, for nothing can be greater than infinite, and thus they could not be parts; nor finite, for thus two finite halves would make an infinite whole.

5. What contains infinite quantities (extensions, protensions, intensions) cannot be passed through,—come to an end. An inch, a minute, a degree contains these; ergo, &c. Take a minute. This contains an infinitude of protended quantities, which must follow one after another; but an infinite series of successive protensions can, ex termino, never be ended; ergo, &c.

6. An infinite maximum cannot but be all-inclusive. Time ab ante and a post infinite and exclusive of each other; ergo, &c.

7. An infinite number of quantities must make up either an infinite or a finite whole. I. The former.—But an inch, a minute, a degree, contain each an infinite number of quantities; therefore an inch, a minute, a degree, are each infinite wholes; which is absurd. II. The latter.—An infinite number of quantities would thus make up a finite quantity, which is equally absurd.

8. If we take a finite quantity (as an inch, a minute, a degree), it would appear equally that there are, and that there are not, an equal number of quantities between these and a greatest, and between these and a least.

9. An absolutely quickest motion is that which passes from one point to another in space in a minimum of time. But a quickest motion from one point to another, say a mile distance, and from one to another, say a million million of miles, is thought the same; which is absurd.

10. A wheel turned with quickest motion; if a spoke be prolonged, it will, therefore, be moved by a motion quicker than the quickest. The same may be shown using the rim and the nave.

11. Contradictory are Boscovich Points, which occupy space, and are unextended. Dynamism, therefore, inconceivable. E contra,

12. Atomism also inconceivable; for this supposes atoms,—minima extended but indivisible.

13. A quantity, say a foot, has an infinity of parts. Any part of this quantity, say an inch, has also an infinity. But one infinity is not larger than another. Therefore an inch is equal to a foot.

14. If two divaricating lines are produced ad infinitum from a point where they form an acute angle, like a pyramid, the base will be infinite, and, at the same time, not infinite; 1°. Because terminated by two points; and, 2°. Because shorter than the sides; 3°. Base could not be drawn, because sides infinitely long.

15. An atom, as existent, must be able to be turned round. But if turned round, it must have a right and left hand, &c., and these its signs [sides?] must change their place: therefore, be extended.” (Lectures, Vol. II, App. iii, pp. 527-9.) [Mill’s square brackets.]

[k-k]651, 652, 67 results

[[*] ]In Works, Vol. II, pp. 401-5, and Vol. III, pp. 1-62, respectively.

[* ]Appendix [II(B)] to Discussions, p. 707.

[l-l]+67, 72

[[*] ]Augustus De Morgan, The Elements of Algebra (London: Taylor, 1835).

[* ]This is what Newton meant by a vera causa, in his celebrated maxim, “Causas rerum naturalium non plures admitti debere quam quæ et veræ sint, et earum phænomenis explicandis sufficiant.” [Isaac Newton, Philosophiæ Naturalis Principia Mathematica, in Opera, ed. Samuel Horsley, 5 vols. (London: Nichols, 1779-85), Vol. III, p. 2.] It is singular that Sir W. Hamilton does not seem to have understood, that by veræ causæ Newton meant agencies the existence of which was otherwise authenticated: for he says, “In their plain meaning, the words et veræ sint are redundant; or what follows is redundant, and the whole rule a barren truism.” (Foot-note to Reid, p. 236n.) bBut in the Appendix [I(A)] to the Discussions (p. 631) Sir W. Hamilton puts the right interpretation on Newton’s maxim.b

[* ]In the course of his speculations our author comes across a fact which is positively irreconcileable with his axiom; the fact of repulsion. This brings him to a dead stand. He knows not whether to advance or recede. Repulsion, he says, “remains, as apparently an actio in distans, even when forced upon us as a fact, still inconceivable as a possibility.” He is soon afterwards obliged to confess that the same is true of attraction: “As attraction and repulsion seem equally actiones in distans, it is not more difficult to realize to ourselves the action of the one, than the action of the other.” (“Dissertations on Reid,” [Note D,] p. 852.) Action from eae distance being “a fact,” though inconceivable, this fact would seem to require of him the retractation of his axiom: yet he does not retract it. I need hardly remark that attraction and repulsion are not inconceivable; except indeed in another of the numerous senses of that equivocal word; that in which it is used when our author tells us that all ultimate facts are inconceivable, meaning only that they are inexplicable.

[bBut in the Appendix [I(A)] to the Discussions (p. 631) Sir W. Hamilton puts the right interpretation on Newton’s maxim.b]+72

[eae]+67, 72