§ 1. In the examination which formed the subject of the last chapter, into the nature of the evidence of those deductive sciences which are commonly represented to be systems of necessary truth, we have been led to the following conclusions. The results of those sciences are indeed necessary, in the sense of necessarily following from certain first principles, commonly called axioms and definitions; that is, of being certainly true if those axioms and definitions are so; for the word necessity, even in this acceptation of it, means no more than certainty. But their claim to the character of necessity in any sense beyond this, as implying an evidence independent of and superior to observation and experience, must depend on the previous establishment of such a claim in favor of the definitions and axioms themselves. With regard to axioms, we found that, considered as experimental truths, they rest on superabundant and obvious evidence. We inquired, whether, since this is the case, it be imperative to suppose any other evidence of those truths than experimental evidence, any other origin for our belief of them than an experimental origin. We decided, that the burden of proof lies with those who maintain the affirmative, and we examined, at considerable length, such arguments as they have produced. The examination having led to the rejection of those arguments, we have thought ourselves warranted in concluding that axioms are but a class, the most universal class, of inductions from experience; the simplest and easiest cases of generalization from the facts furnished to us by our senses or by our internal consciousness.

While the axioms of demonstrative sciences thus appeared to be experimental truths, the definitions, as they are incorrectly called, in those sciences, were found by us to be generalizations from experience which are not even, accurately 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 mere fiction, or supposition, made for the purpose of excluding the consideration of those modifying circumstances, when their influence is of too trifling amount to be worth considering, or adjourning 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 virtue of the peculiar character of one indispensable portion of the general formulæ 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 demonstration.

What we have now asserted, however, cannot be received as universally true of Deductive or Demonstrative Sciences,
until verified by being applied to the most remarkable of all those sciences, that of Numbers; the theory of the
Calculus; Arithmetic and Algebra. It is harder to believe of the doctrines of this science than of any other, either that
they are not truths *a priori*, but experimental truths, or that their peculiar certainty is owing to their being
not absolute but only conditional 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 *a priori* philosophers on
one side; and on the other, a theory the most opposite to theirs, which was at one time very generally received, and is
still far from being altogether exploded, among metaphysicians.

§ 2. This theory attempts to solve the difficulty apparently inherent in the case, by representing the propositions of the science of numbers as merely verbal, and its processes as simple transformations of language, substitutions of one expression for another. The proposition, Two and one is 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 doctrine, the longest process in algebra is but a succession of changes in terminology, 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 peculiarities in the processes of arithmetic and algebra which render the
theory in question very plausible, and have not unnaturally 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 reconcilable 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 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
banished from the mind during every intermediate part of the process, between the beginning, when the premises 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 any thing more? We seem to have come to one of Bacon’s Prerogative 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 induction, 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 any thing. Propositions, therefore, concerning numbers, have the remarkable peculiarity 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 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*)= 2 *a* + 2 *b*,
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 of 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 drawn, 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 entirely mechanical.
Hence the general language of algebra comes to be used familiarly without exciting ideas, as all other general language
is prone 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 derived, 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 plausibility 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 is equal to three, considered as an assertion respecting objects, as for instance, “Two pebbles and one pebble 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 insignificant, it seems but natural to consider the proposition, Two and one is 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 aggregation 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
*de*note the same things, their *con*notation 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 alteration 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 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
sums 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, [Symbol: three circles, two above one], may be separated into two parts, thus, [Symbol: two circles, a space, and a third circle]. This proposition being granted, we term all such parcels Threes, after which the enunciation 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 previously arrived at, that the processes even of deductive sciences are altogether inductive, and that their first principles are generalizations from experience. It remains to be examined whether this science resembles geometry in the further circumstance, that some of its inductions are not exactly true; and that the peculiar certainty ascribed to it, on account of which its propositions are called Necessary Truths, is fictitious and hypothetical, being true in no other sense than that those propositions legitimately follow from the hypothesis of the truth of premises which are avowedly mere approximations to truth.

§ 3. The inductions of arithmetic are of two sorts: first, those which we have just expounded, such as One and one are
two, Two and one are three, etc., which may be called the definitions of the various numbers, in the improper or
geometrical sense of the word Definition; and secondly, 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 demonstrations is independent of all hypothesis.

On more accurate investigation, however, it will be found that, even in this case, there is one hypothetical element
in the ratiocination. In all propositions concerning numbers, a condition is implied, without which none of them would be
true; and that condition is an assumption which may be 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 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 such cases in statistics; as, for instance, an inquiry 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 ascertain 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 inquiries 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 unconditional truth and perfect accuracy, is not an attribute of all mathematical truths, but of those only which relate to pure Number, as distinguished from Quantity in the more enlarged sense; and only so long as we abstain 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 can not 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 exactly true, whether they are a sufficiently 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 purely 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. Deduction, 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 inaccessible 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 synthetically the phenomena 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 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 assent 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 premise; 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 syllogism 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 maxim 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 any thing, 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 syllogistic 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 philosophic 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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