While the axioms of demonstrative sciences thus appeared [Pg 283]to be experimental truths, the definitions, as they are incorrectly called, in those sciences, were found by us to be generalizations7 from experience which are not even, accurately8 speaking, truths; being propositions in which, while we assert of some kind of object, some property or properties which observation shows to belong to it, we at the same time deny that it possesses any other properties, though in truth other properties do in every individual instance accompany, and in almost all instances modify, the property thus exclusively predicated. The denial, therefore, is a mere9 fiction, or supposition, made for the purpose of excluding the consideration of those modifying circumstances, when their influence is of too trifling10 amount to be worth considering, or adjourning11 it, when important, to a more convenient moment.
From these considerations it would appear that Deductive or Demonstrative Sciences are all, without exception, Inductive Sciences; that their evidence is that of experience; but that they are also, in virtue12 of the peculiar13 character of one indispensable portion of the general formul? according to which their inductions are made, Hypothetical Sciences. Their conclusions are only true on certain suppositions, which are, or ought to be, approximations to the truth, but are seldom, if ever, exactly true; and to this hypothetical character is to be ascribed the peculiar certainty, which is supposed to be inherent in demonstration14.
What we have now asserted, however, cannot be received as universally true of Deductive or Demonstrative Sciences, until verified by being applied15 to the most remarkable16 of all those sciences, that of Numbers; the theory of the Calculus17; Arithmetic and Algebra18. It is harder to believe of the doctrines20 of this science than of any other, either that they are not truths à priori, but experimental truths, or that their peculiar certainty is owing to their being not absolute but only conditional21 truths. This, therefore, is a case which merits examination apart; and the more so, because on this subject we have a double set of doctrines to contend with; that of the à priori philosophers on one side; and on the other, a theory the most opposite to theirs, which was at one time very generally [Pg 284]received, and is still far from being altogether exploded, among metaphysicians.
§ 2. This theory attempts to solve the difficulty apparently22 inherent in the case, by representing the propositions of the science of numbers as merely verbal, and its processes as simple transformations23 of language, substitutions of one expression for another. The proposition, Two and one are equal to three, according to these writers, is not a truth, is not the assertion of a really existing fact, but a definition of the word three; a statement that mankind have agreed to use the name three as a sign exactly equivalent to two and one; to call by the former name whatever is called by the other more clumsy phrase. According to this doctrine19, the longest process in algebra is but a succession of changes in terminology24, by which equivalent expressions are substituted one for another; a series of translations of the same fact, from one into another language; though how, after such a series of translations, the fact itself comes out changed (as when we demonstrate a new geometrical theorem by algebra,) they have not explained; and it is a difficulty which is fatal to their theory.
It must be acknowledged that there are peculiarities25 in the processes of arithmetic and algebra which render the theory in question very plausible26, and have not unnaturally27 made those sciences the stronghold of Nominalism. The doctrine that we can discover facts, detect the hidden processes of nature, by an artful manipulation of language, is so contrary to common sense, that a person must have made some advances in philosophy to believe it: men fly to so paradoxical a belief to avoid, as they think, some even greater difficulty, which the vulgar do not see. What has led many to believe that reasoning is a mere verbal process, is, that no other theory seemed reconcileable with the nature of the Science of Numbers. For we do not carry any ideas along with us when we use the symbols of arithmetic or of algebra. In a geometrical demonstration we have a mental diagram, if not one on paper; AB, AC, are present to our imagination as lines, intersecting [Pg 285]other lines, forming an angle with one another, and the like; but not so a and b. These may represent lines or any other magnitudes, but those magnitudes are never thought of; nothing is realized in our imagination but a and b. The ideas which, on the particular occasion, they happen to represent, are banished28 from the mind during every intermediate part of the process, between the beginning, when the premises30 are translated from things into signs, and the end, when the conclusion is translated back from signs into things. Nothing, then, being in the reasoner's mind but the symbols, what can seem more inadmissible than to contend that the reasoning process has to do with anything more? We seem to have come to one of Bacon's Prerogative31 Instances; an experimentum crucis on the nature of reasoning itself.
Nevertheless, it will appear on consideration, that this apparently so decisive instance is no instance at all; that there is in every step of an arithmetical or algebraical calculation a real induction4, a real inference of facts from facts; and that what disguises the induction is simply its comprehensive nature, and the consequent extreme generality of the language. All numbers must be numbers of something: there are no such things as numbers in the abstract. Ten must mean ten bodies, or ten sounds, or ten beatings of the pulse. But though numbers must be numbers of something, they may be numbers of anything. Propositions, therefore, concerning numbers, have the remarkable peculiarity32 that they are propositions concerning all things whatever; all objects, all existences of every kind, known to our experience. All things possess quantity; consist of parts which can be numbered; and in that character possess all the properties which are called properties of numbers. That half of four is two, must be true whatever the word four represents, whether four hours, four miles, or four pounds weight. We need only conceive a thing divided into four equal parts, (and all things may be conceived as so divided,) to be able to predicate of it every property of the number four, that is, every arithmetical proposition in which the number four stands on one side of the equation. Algebra extends the generalization still farther: every number represents that particular [Pg 286]number of all things without distinction, but every algebraical symbol does more, it represents all numbers without distinction. As soon as we conceive a thing divided into equal parts, without knowing into what number of parts, we may call it a or x, and apply to it, without danger of error, every algebraical formula in the books. The proposition, 2(a + b) = 2a + 2b, is a truth co-extensive with all nature. Since then algebraical truths are true of all things whatever, and not, like those of geometry, true of lines only or angles only, it is no wonder that the symbols should not excite in our minds ideas of any things in particular. When we demonstrate the forty-seventh proposition of Euclid, it is not necessary that the words should raise in us an image of all right-angled triangles, but only of some one right-angled triangle: so in algebra we need not, under the symbol a, picture to ourselves all things whatever, but only some one thing; why not, then, the letter itself? The mere written characters, a, b, x, y, z, serve as well for representatives of Things in general, as any more complex and apparently more concrete conception. That we are conscious of them however in their character of things, and not of mere signs, is evident from the fact that our whole process of reasoning is carried on by predicating of them the properties of things. In resolving an algebraic equation, by what rules do we proceed? By applying at each step to a, b, and x, the proposition that equals added to equals make equals; that equals taken from equals leave equals; and other propositions founded on these two. These are not properties of language, or of signs as such, but of magnitudes, which is as much as to say, of all things. The inferences, therefore, which are successively drawn33, are inferences concerning things, not symbols; though as any Things whatever will serve the turn, there is no necessity for keeping the idea of the Thing at all distinct, and consequently the process of thought may, in this case, be allowed without danger to do what all processes of thought, when they have been performed often, will do if permitted, namely, to become entirely34 mechanical. Hence the general language of algebra comes to be used familiarly without [Pg 287]exciting ideas, as all other general language is prone35 to do from mere habit, though in no other case than this can it be done with complete safety. But when we look back to see from whence the probative force of the process is derived36, we find that at every single step, unless we suppose ourselves to be thinking and talking of the things, and not the mere symbols, the evidence fails.
There is another circumstance, which, still more than that which we have now mentioned, gives plausibility37 to the notion that the propositions of arithmetic and algebra are merely verbal. That is, that when considered as propositions respecting Things, they all have the appearance of being identical propositions. The assertion, Two and one are equal to three, considered as an assertion respecting objects, as for instance "Two pebbles38 and one pebble39 are equal to three pebbles," does not affirm equality between two collections of pebbles, but absolute identity. It affirms that if we put one pebble to two pebbles, those very pebbles are three. The objects, therefore, being the very same, and the mere assertion that "objects are themselves" being insignificant40, it seems but natural to consider the proposition, Two and one are equal to three, as asserting mere identity of signification between the two names.
This, however, though it looks so plausible, will not bear examination. The expression "two pebbles and one pebble," and the expression, "three pebbles," stand indeed for the same aggregation41 of objects, but they by no means stand for the same physical fact. They are names of the same objects, but of those objects in two different states: though they denote the same things, their connotation is different. Three pebbles in two separate parcels, and three pebbles in one parcel, do not make the same impression on our senses; and the assertion that the very same pebbles may by an alteration42 of place and arrangement be made to produce either the one set of sensations or the other, though a very familiar proposition, is not an identical one. It is a truth known to us by early and constant experience: an inductive truth; and such truths are the foundation of the science of Number. The [Pg 288]fundamental truths of that science all rest on the evidence of sense; they are proved by showing to our eyes and our fingers that any given number of objects, ten balls for example, may by separation and re-arrangement exhibit to our senses all the different sets of numbers the sum of which is equal to ten. All the improved methods of teaching arithmetic to children proceed on a knowledge of this fact. All who wish to carry the child's mind along with them in learning arithmetic; all who wish to teach numbers, and not mere ciphers—now teach it through the evidence of the senses, in the manner we have described.
We may, if we please, call the proposition, "Three is two and one," a definition of the number three, and assert that arithmetic, as it has been asserted that geometry, is a science founded on definitions. But they are definitions in the geometrical sense, not the logical; asserting not the meaning of a term only, but along with it an observed matter of fact. The proposition, "A circle is a figure bounded by a line which has all its points equally distant from a point within it," is called the definition of a circle; but the proposition from which so many consequences follow, and which is really a first principle in geometry, is, that figures answering to this description exist. And thus we may call "Three is two and one" a definition of three; but the calculations which depend on that proposition do not follow from the definition itself, but from an arithmetical theorem presupposed in it, namely, that collections of objects exist, which while they impress the senses thus, ooo, may be separated into two parts, thus, o o o. This proposition being granted, we term all such parcels Threes, after which the enunciation43 of the above mentioned physical fact will serve also for a definition of the word Three.
The Science of Number is thus no exception to the conclusion we previously44 arrived at, that the processes even of deductive sciences are altogether inductive, and that their first principles are generalizations from experience. It remains45 to be examined whether this science resembles geometry in the further circumstance, that some of its inductions are not [Pg 289]exactly true; and that the peculiar certainty ascribed to it, on account of which its propositions are called Necessary Truths, is fictitious46 and hypothetical, being true in no other sense than that those propositions legitimately47 follow from the hypothesis of the truth of premises which are avowedly48 mere approximations to truth.
§ 3. The inductions of arithmetic are of two sorts: first, those which we have just expounded49, such as One and one are two, Two and one are three, &c., which may be called the definitions of the various numbers, in the improper50 or geometrical sense of the word Definition; and secondly51, the two following axioms: The sums of equals are equal, The differences of equals are equal. These two are sufficient; for the corresponding propositions respecting unequals may be proved from these, by a reductio ad absurdum.
These axioms, and likewise the so-called definitions, are, as has already been said, results of induction; true of all objects whatever, and, as it may seem, exactly true, without the hypothetical assumption of unqualified truth where an approximation to it is all that exists. The conclusions, therefore, it will naturally be inferred, are exactly true, and the science of number is an exception to other demonstrative sciences in this, that the categorical certainty which is predicable of its demonstrations52 is independent of all hypothesis.
On more accurate investigation53, however, it will be found that, even in this case, there is one hypothetical element in the ratiocination54. In all propositions concerning numbers, a condition is implied, without which none of them would be true; and that condition is an assumption which maybe false. The condition, is that 1 = 1; that all the numbers are numbers of the same or of equal units. Let this be doubtful, and not one of the propositions of arithmetic will hold true. How can we know that one pound and one pound make two pounds, if one of the pounds may be troy, and the other avoirdupois? They may not make two pounds of either, or of any weight. How can we know that a forty-horse power is always equal to itself, unless we assume that all horses are of equal strength? It is [Pg 290]certain that 1 is always equal in number to 1; and where the mere number of objects, or of the parts of an object, without supposing them to be equivalent in any other respect, is all that is material, the conclusions of arithmetic, so far as they go to that alone, are true without mixture of hypothesis. There are a few such cases; as, for instance, an inquiry55 into the amount of the population of any country. It is indifferent to that inquiry whether they are grown people or children, strong or weak, tall or short; the only thing we want to ascertain56 is their number. But whenever, from equality or inequality of number, equality or inequality in any other respect is to be inferred, arithmetic carried into such inquiries57 becomes as hypothetical a science as geometry. All units must be assumed to be equal in that other respect; and this is never accurately true, for one actual pound weight is not exactly equal to another, nor one measured mile's length to another; a nicer balance, or more accurate measuring instruments, would always detect some difference.
What is commonly called mathematical certainty, therefore, which comprises the twofold conception of unconditional58 truth and perfect accuracy, is not an attribute of all mathematical truths, but of those only which relate to pure Number, as distinguished59 from Quantity in the more enlarged sense; and only so long as we abstain60 from supposing that the numbers are a precise index to actual quantities. The certainty usually ascribed to the conclusions of geometry, and even to those of mechanics, is nothing whatever but certainty of inference. We can have full assurance of particular results under particular suppositions, but we cannot have the same assurance that these suppositions are accurately true, nor that they include all the data which may exercise an influence over the result in any given instance.
§ 4. It appears, therefore, that the method of all Deductive Sciences is hypothetical. They proceed by tracing the consequences of certain assumptions; leaving for separate consideration whether the assumptions are true or not, and if not [Pg 291]exactly true, whether they are a sufficiently61 near approximation to the truth. The reason is obvious. Since it is only in questions of pure number that the assumptions are exactly true, and even there, only so long as no conclusions except purely62 numerical ones are to be founded on them; it must, in all other cases of deductive investigation, form a part of the inquiry, to determine how much the assumptions want of being exactly true in the case in hand. This is generally a matter of observation, to be repeated in every fresh case; or if it has to be settled by argument instead of observation, may require in every different case different evidence, and present every degree of difficulty from the lowest to the highest. But the other part of the process—namely, to determine what else may be concluded if we find, and in proportion as we find, the assumptions to be true—may be performed once for all, and the results held ready to be employed as the occasions turn up for use. We thus do all beforehand that can be so done, and leave the least possible work to be performed when cases arise and press for a decision. This inquiry into the inferences which can be drawn from assumptions, is what properly constitutes Demonstrative Science.
It is of course quite as practicable to arrive at new conclusions from facts assumed, as from facts observed; from fictitious, as from real, inductions. Deduction63, as we have seen, consists of a series of inferences in this form—a is a mark of b, b of c, c of d, therefore a is a mark of d, which last may be a truth inaccessible64 to direct observation. In like manner it is allowable to say, suppose that a were a mark of b, b of c, and c of d, a would be a mark of d, which last conclusion was not thought of by those who laid down the premises. A system of propositions as complicated as geometry might be deduced from assumptions which are false; as was done by Ptolemy, Descartes, and others, in their attempts to explain synthetically65 the phenomena66 of the solar system on the supposition that the apparent motions of the heavenly bodies were the real motions, or were produced in some way more or less different from the true one. Sometimes the same thing is knowingly [Pg 292]done, for the purpose of showing the falsity of the assumption; which is called a reductio ad absurdum. In such cases, the reasoning is as follows: a is a mark of b, and b of c; now if c were also a mark of d, a would be a mark of d; but d is known to be a mark of the absence of a; consequently a would be a mark of its own absence, which is a contradiction; therefore c is not a mark of d.
§ 5. It has even been held by some writers, that all ratiocination rests in the last resort on a reductio ad absurdum; since the way to enforce assent67 to it, in case of obscurity, would be to show that if the conclusion be denied we must deny some one at least of the premises, which, as they are all supposed true, would be a contradiction. And in accordance with this, many have thought that the peculiar nature of the evidence of ratiocination consisted in the impossibility of admitting the premises and rejecting the conclusion without a contradiction in terms. This theory, however, is inadmissible as an explanation of the grounds on which ratiocination itself rests. If any one denies the conclusion notwithstanding his admission of the premises, he is not involved in any direct and express contradiction until he is compelled to deny some premise29; and he can only be forced to do this by a reductio ad absurdum, that is, by another ratiocination: now, if he denies the validity of the reasoning process itself, he can no more be forced to assent to the second syllogism68 than to the first. In truth, therefore, no one is ever forced to a contradiction in terms: he can only be forced to a contradiction (or rather an infringement) of the fundamental maxim69 of ratiocination, namely, that whatever has a mark, has what it is a mark of; or, (in the case of universal propositions,) that whatever is a mark of anything, is a mark of whatever else that thing is a mark of. For in the case of every correct argument, as soon as thrown into the syllogistic70 form, it is evident without the aid of any other syllogism, that he who, admitting the premises, fails to draw the conclusion, does not conform to the above axiom.
We have now proceeded as far in the theory of Deduction as we can advance in the present stage of our inquiry. Any further insight into the subject requires that the foundation shall have been laid of the philosophic71 theory of Induction itself; in which theory that of deduction, as a mode of induction, which we have now shown it to be, will assume spontaneously the place which belongs to it, and will receive its share of whatever light may be thrown upon the great intellectual operation of which it forms so important a part.
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imperative
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| n.命令，需要；规则；祈使语气；adj.强制的；紧急的 | |
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decided
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| adj.决定了的，坚决的；明显的，明确的 | |
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rejection
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| n.拒绝，被拒，抛弃，被弃 | |
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induction
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| n.感应，感应现象 | |
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inductions
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| 归纳（法）( induction的名词复数 )； （电或磁的）感应； 就职； 吸入 | |
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generalization
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| n.普遍性，一般性，概括 | |
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generalizations
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| 一般化( generalization的名词复数 )； 普通化； 归纳； 概论 | |
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accurately
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| adv.准确地，精确地 | |
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mere
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| adj.纯粹的；仅仅，只不过 | |
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trifling
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| adj.微不足道的；没什么价值的 | |
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adjourning
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| （使）休会， （使）休庭( adjourn的现在分词 ) | |
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virtue
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| n.德行，美德；贞操；优点；功效，效力 | |
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peculiar
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| adj.古怪的，异常的；特殊的，特有的 | |
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demonstration
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| n.表明，示范，论证，示威 | |
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applied
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| adj.应用的；v.应用，适用 | |
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remarkable
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| adj.显著的，异常的，非凡的，值得注意的 | |
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calculus
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| n.微积分；结石 | |
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algebra
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| n.代数学 | |
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doctrine
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| n.教义；主义；学说 | |
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doctrines
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| n.教条( doctrine的名词复数 )；教义；学说；（政府政策的）正式声明 | |
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conditional
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| adj.条件的，带有条件的 | |
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apparently
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| adv.显然地；表面上，似乎 | |
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transformations
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| n.变化( transformation的名词复数 )；转换；转换；变换 | |
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terminology
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| n.术语；专有名词 | |
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peculiarities
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| n. 特质, 特性, 怪癖, 古怪 | |
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plausible
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| adj.似真实的，似乎有理的，似乎可信的 | |
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unnaturally
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| adv.违反习俗地；不自然地；勉强地；不近人情地 | |
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banished
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| v.放逐，驱逐( banish的过去式和过去分词 ) | |
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premise
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| n.前提；v.提论，预述 | |
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premises
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| n.建筑物，房屋 | |
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prerogative
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| n.特权 | |
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peculiarity
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| n.独特性，特色；特殊的东西；怪癖 | |
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drawn
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| v.拖，拉，拔出；adj.憔悴的，紧张的 | |
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entirely
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| ad.全部地，完整地；完全地，彻底地 | |
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prone
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| adj.(to)易于…的，很可能…的；俯卧的 | |
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derived
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| vi.起源；由来；衍生；导出v.得到( derive的过去式和过去分词 )；（从…中）得到获得；源于；（从…中）提取 | |
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plausibility
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| n. 似有道理, 能言善辩 | |
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pebbles
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| [复数]鹅卵石； 沙砾； 卵石，小圆石( pebble的名词复数 ) | |
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pebble
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| n.卵石，小圆石 | |
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insignificant
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| adj.无关紧要的，可忽略的，无意义的 | |
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aggregation
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| n.聚合，组合；凝聚 | |
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alteration
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| n.变更，改变；蚀变 | |
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enunciation
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| n.清晰的发音；表明，宣言；口齿 | |
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previously
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| adv.以前，先前（地） | |
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remains
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| n.剩余物，残留物；遗体，遗迹 | |
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fictitious
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| adj.虚构的，假设的；空头的 | |
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legitimately
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| ad.合法地；正当地，合理地 | |
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avowedly
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| adv.公然地 | |
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expounded
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| 论述，详细讲解( expound的过去式和过去分词 ) | |
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improper
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| adj.不适当的，不合适的，不正确的，不合礼仪的 | |
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51
secondly
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| adv.第二，其次 | |
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demonstrations
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| 证明( demonstration的名词复数 )； 表明； 表达； 游行示威 | |
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investigation
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| n.调查，调查研究 | |
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ratiocination
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| n.推理；推断 | |
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inquiry
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| n.打听，询问，调查，查问 | |
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ascertain
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| vt.发现，确定，查明，弄清 | |
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inquiries
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| n.调查( inquiry的名词复数 )；疑问；探究；打听 | |
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unconditional
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| adj.无条件的，无限制的，绝对的 | |
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distinguished
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| adj.卓越的，杰出的，著名的 | |
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60
abstain
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| v.自制，戒绝，弃权，避免 | |
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sufficiently
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| adv.足够地，充分地 | |
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purely
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| adv.纯粹地，完全地 | |
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deduction
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| n.减除，扣除，减除额；推论，推理，演绎 | |
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inaccessible
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| adj.达不到的，难接近的 | |
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synthetically
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| adv. 综合地，合成地 | |
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phenomena
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| n.现象 | |
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assent
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| v.批准，认可；n.批准，认可 | |
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syllogism
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| n.演绎法，三段论法 | |
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maxim
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| n.格言，箴言 | |
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syllogistic
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| adj.三段论法的，演绎的，演绎性的 | |
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philosophic
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| adj.哲学的，贤明的 | |
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