OF SOME OBJECTIONS WHICH HAVE BEEN MADE TO THE DOCTRINES
STATED IN THE PREVIOUS CHAPTER.[3\2]
[Note 3\2: In order to render the present chapter more
intelligible, it may be proper to state briefly the
arguments which gave occasion to the review. After noticing
Stewart's assertions, that the certainty of mathematical
reasoning arises from its depending upon definitions, and
that mathematical truth is hypothetical; I urged,--that no
one has yet been able to construct a system of mathematical
truths by the aid of definitions alone; that a definition
would not be admissible or applicable except it agreed with
a distinct conception in the mind; that the definitions
which we employ in mathematics are not arbitrary or
hypothetical, but necessary definitions; that if Stewart had
taken as his examples of axioms the peculiar geometrical
axioms, his assertions would have been obviously erroneous;
and that the real foundation of the truths of mathematics is
the Idea of Space, which may be expressed (for purposes of
demonstration) partly by definitions and partly by axioms.]
THE _Edinburgh Review_, No. cxxxv., contains a critique on a
work termed _The Mechanical Euclid_, in which opinions were
delivered to nearly the same effect as some of those stated
in the last chapter, and hereafter in Chapter xi. Although I
believe that there are no arguments used by the reviewer to
which the answers will not suggest themselves in the mind of
any one who has read with attention what has been said in
the preceding chapters (except, perhaps, one or two remarks
which have reference to mechanical ideas), it may serve to
illustrate the subject if I reply to the objections
directly, taking them as the reviewer has stated them.
I. I had dissented from Stewart's assertion that
mathematical truth is hypothetical, or depends upon
arbitrary definitions; since we understand by an {108}
hypothesis a supposition, not only which we may make, but
may abstain from making, or may replace by a different
supposition; whereas the definitions and hypotheses of
geometry are necessarily such as they are, and cannot be
altered or excluded. The reviewer (p. 84) informs us that he
understands Stewart, when he speaks of hypotheses and
definitions being the foundation of geometry, to speak of
the hypothesis that real objects correspond to our
geometrical definitions. '_If_ a crystal be an exact
hexahedron, the geometrical properties of the hexahedron may
be predicated of that crystal.' To this I reply,--that such
hypotheses as this are the grounds of our applications of
geometrical truths to real objects, but can in no way be
said to be the foundation of the truths themselves;--that I
do not think that the sense which the reviewer gives was
Stewart's meaning;--but that if it was, this view of the use
of mathematics does not at all affect the question which
both he and I proposed to discuss, which was, the ground of
mathematical certainty. I may add, that whether a crystal be
an exact hexahedron, is a matter of observation and
measurement, not of definition. I think the reader can have
no difficulty in seeing how little my doctrine is affected
by the connexion on which the reviewer thus insists. I have
asserted that the proposition which affirms the square on
the diagonal of a rectangle to be equal to the squares on
two sides, does not rest upon arbitrary hypotheses; the
objector answers, that the proposition that the square on
the diagonal _of this page_ is equal to the squares on the
sides, depends upon the arbitrary hypothesis that the page
is a rectangle. Even if this fact were a matter of arbitrary
hypothesis, what could it have to do with the general
geometrical proposition? How could a single fact, observed
or hypothetical, affect a universal and necessary truth,
which would be equally true if the fact were false? If there
be nothing arbitrary or hypothetical in geometry till we
come to such steps in its application, it is plain that the
truths themselves are not hypothetical; which is the
question for us to decide. {109}
2. The reviewer then (p. 85) considers the doctrine that
axioms as well as definitions are the foundations of
geometry; and here he strangely narrows and confuses the
discussion by making himself the advocate of Stewart,
instead of arguing the question itself. I had asserted that
some axioms are necessary as the foundations of mathematical
reasoning, in addition to the definitions. If Stewart did
not intend to discuss this question, I had no concern with
what he had said about axioms. But I had every reason to
believe that this was the question which Stewart did intend
to discuss. I conceive there is no doubt that he intended to
give an opinion upon the grounds of mathematical reasoning
in general. For he begins his discussions (_Elements_, vol.
ii. p. 38) by contesting Reid's opinion on this subject,
which is stated generally; and he refers again to the same
subject, asserting in general terms, that the first
principles of mathematics are not axioms but definitions.
If, then, afterwards, he made his proof narrower than his
assertion;--if having declared that no axioms are necessary,
he afterwards limited himself to showing that seven out of
twelve of Euclid's axioms are barren truisms, it was no
concern of mine to contest this assertion, which left my
thesis untouched. I had asserted that the proper geometrical
axioms (that two straight lines cannot inclose a space, and
the axiom about parallel lines) are indispensable in
geometry. What account the reviewer gives of these axioms we
shall soon see; but if Stewart allowed them to be axioms
necessary to geometrical reasoning, he overturned his own
assertion as to the foundations of such reasoning; and if he
said nothing decisive about these axioms, which are the
points on which the battle must turn, he left his assertion
altogether unproved; nor was it necessary for me to pursue
the war into a barren and unimportant corner, when the
metropolis was surrendered. The reviewer's exultation that I
have not contested the first seven axioms is an amusing
example of the self-complacent zeal of advocacy.
3. But let us turn to the material point,--the proper
geometrical axioms. What is the reviewer's account of {110}
these? Which side of the alternative does he adopt? Do they
depend upon the definitions, and is he prepared to show the
dependence? Or are they superfluous, and can he erect the
structure of geometry without their aid? One of these two
courses, it would seem, he must take. For we both begin by
asserting the excellence of geometry as an example of
demonstrated truth. It is precisely this attribute which
gives an interest to our present inquiry. How, then, does
the reviewer explain this excellence on his views? How does
he reckon the foundation courses of the edifice which we
agree in considering as a perfect example of intellectual
building?
I presume I may take, as his answer to this question, his
hypothetical statement of what Stewart would have said (p.
87), on the supposition that there had been, among the
foundations of geometry, self-evident indemonstrable truths:
although it is certainly strange that the reviewer should
not venture to make up his mind as to the truth or falsehood
of this supposition. If there were such truths they would
be, he says, 'legitimate filiations' of the definitions.
They would be involved in the definitions. And again he
speaks of the foundation of the geometrical doctrine of
parallels as a flaw, and as a truth which requires, but has
not received demonstration. And yet again, he tells us that
each of these supposed axioms (Euclid's twelfth, for
instance) is 'merely an indication of the point at which
geometry fails to perform that which it undertakes to
perform' (p. 91); and that in reality her truths are not yet
demonstrated. The amount of this is, that the geometrical
axioms are to be held to be _legitimate filiations_ of the
definitions, because though certainly true, they cannot be
proved from the definitions; that they are involved in the
definitions, although they cannot be evolved out of them;
and that rather than admit that they have any other origin
than the definitions, we are to proclaim that geometry has
failed to perform what she undertakes to perform.
To this I reply--that I cannot understand what is meant by
'legitimate filiations' of principles, if the {111} phrase
do not mean consequences of such principles established by
rigorous and formal demonstrations;--that the reviewer, if
he claims any real signification for his phrase, must
substantiate the meaning of it by such a demonstration; he
must establish his 'legitimate filiation' by a genealogical
table in a satisfactory form. When this cannot be done, to
assert, notwithstanding, that the propositions are involved
in the definitions, is a mere begging the question; and to
excuse this defect by saying that geometry fails to perform
what she has promised, is to calumniate the character of
that science which we profess to make our standard, rather
than abandon an arbitrary and unproved assertion respecting
the real grounds of her excellence. I add, further, that if
the doctrine of parallel lines, or any other geometrical
doctrine of which we see the truth, with the most perfect
insight of its necessity, have not hitherto received
demonstration to the satisfaction of any school of
reasoners, the defect must arise from their erroneous views
of the nature of demonstrations, and the grounds of
mathematical certainty.
4. I conceive, then, that the reviewer has failed altogether
to disprove the doctrine that the axioms of geometry are
necessary as a part of the foundations of the science. I had
asserted further that these axioms supply what the
definitions leave deficient; and that they, along with
definitions, serve to present the idea of space under such
aspects that we can reason logically concerning it. To this
the reviewer opposes (p. 96) the common opinion that a
perfect definition is a complete explanation of a name, and
that the test of its perfection is, that we may substitute
the definition for the name wherever it occurs. I reply,
that my doctrine, that a definition expresses a part, but
not the whole, of the essential characters of an idea, is
certainly at variance with an opinion sometimes maintained,
that a definition merely explains a word, and should explain
it so fully that it may always replace it. The error of this
common opinion may, I think, be shown from considerations
such as these;--that if {112} we undertake to explain one
word by several, we may be called upon, on the same ground,
to explain each of these several by others, and that in this
way we can reach no limit nor resting-place;--that in point
of fact, it is not found to lead to clearness, but to
obscurity, when in the discussion of general principles, we
thus substitute definitions for single terms;--that even if
this be done, we cannot reason without conceiving what the
terms mean;--and that, in doing this, the relations of our
conceptions, and not the arbitrary equivalence of two forms
of expression, are the foundations of our reasoning.
5. The reviewer conceives that some of the so-called axioms
are really definitions. The axiom, that 'magnitudes which
coincide with each other, that is, which fill the same
space, are equal,' is a definition of geometrical
_equality_: the axiom, that 'the whole is greater than its
part,' is a definition of _whole_ and _part_. But surely
there are very serious objections to this view. It would
seem more natural to say, if the former axiom is a
definition of the word _equal_, that the latter is a
definition of the word _greater_. And how can one short
phrase define two terms? If I say, 'the heat of summer is
greater than the heat of winter,' does this assertion define
anything, though the proposition is perfectly intelligible
and distinct? I think, then, that this attempt to reduce
these axioms to definitions is quite untenable.
6. I have stated that a definition can be of no use, except
we can conceive the possibility and truth of the property
connected with it; and that if we do conceive this, we may
rightly begin our reasonings by stating the property as an
axiom; which Euclid does, in the case of straight lines and
of parallels. The reviewer inquires (p. 92), whether I am
prepared to extend this doctrine to the case of circles, for
which the reasoning is usually rested upon the
definition;--whether I would replace this definition by an
axiom, asserting the possibility of such a circle. To this I
might reply, that it is not at all incumbent upon me to
assent to such a change; for I have all along stated that it
is indifferent {113} whether the fundamental properties from
which we reason be exhibited as definitions or as axioms,
provided the necessity be clearly seen. But I am ready to
declare that I think the form of our geometry would be not
at all the worse, if, instead of the usual definition of a
circle,--'that it is a figure contained by one line, which
is called the circumference, and which is such, that all
straight lines drawn from a certain point within the
circumference are equal to one another,'--we were to
substitute an axiom and a definition, as follows:--
_Axiom_. If a line be drawn so as to be at every point
equally distant from a certain point, this line will return
into itself or will be _one_ line including a space.
_Definition_. The space is called a _circle_, the line the
_circumference_, and the point the _center_.
And this being done, it would be true, as the reviewer
remarks, that geometry cannot stir _one_ step without
resting on an axiom. And I do not at all hesitate to say,
that the above axiom, expressed or understood, is no less
necessary than the definition, and is tacitly assumed in
every proposition into which circles enter.
7. I have, I think, now disposed of the principal objections
which bear upon the proper axioms of geometry. The
principles which are stated as the first seven axioms of
Euclid's _Elements_, need not, as I have said, be here
discussed. They are principles which refer, not to Space in
particular, but to Quantity in general: such, for instance,
as these; 'If equals be added to equals the wholes are
equal;'--'If equals be taken from equals the remainders are
equal.' But I will make an observation or two upon them
before I proceed.
Both Locke and Stewart have spoken of these axioms as barren
truisms: as propositions from which it is not possible to
deduce a single inference: and the reviewer asserts that
they are not first principles, but laws of thought (p. 88).
To this last expression I am {114} willing to assent; but I
would add, that not only these, but all the principles which
express the fundamental conditions of our knowledge, may
with equal propriety be termed laws of thought; for these
principles depend upon our ideas, and regulate the active
operations of the mind, by which coherence and connexion are
given to its passive impressions. But the assertion that no
conclusions can be drawn from simple axioms, or laws of
human thought, which regard quantity, is by no means true.
The whole of arithmetic,--for instance, the rules for the
multiplication and division of large numbers, the rule for
finding a common measure, and, in short, a vast body of
theory respecting numbers,--rests upon no other foundation
than such axioms as have been just noticed, that if equals
be added to equals the wholes will be equal. And even when
Locke's assertion, that from these axioms no truths can be
deduced, is modified by Stewart and the reviewer, and
limited to _geometrical_ truths, it is hardly tenable
(although, in fact, it matters little to our argument
whether it is or no). For the greater part of the Seventh
Book of Euclid's _Elements_, (on Commensurable and
Incommensurable Quantities,) and the Fifth Book, (on
Proportion,) depend upon these axioms, with the addition
only of the definition or axiom (for it may be stated either
way) which expresses the idea of proportionality in numbers.
So that the attempt to disprove the necessity and use of
axioms, as principles of reasoning, fails even when we take
those instances which the opponents consider as the more
manifestly favourable to their doctrine.
8. But perhaps the question may have already suggested
itself to the reader's mind, of what use can it be formally
to state such principles as these, (for example, that if
equals be added to equals the wholes are equal,) since,
whether stated or no, they will be assumed in our reasoning?
And how can such principles be said to be necessary, when
our proof proceeds equally well without any reference to
them? And the answer is, that it is precisely because these
are the {115} common principles of reasoning, which we
naturally employ without specially contemplating them, that
they require to be separated from the other steps and
formally stated, when we _analyse_ the demonstrations which
we have obtained. In every mental process many principles
are combined and abbreviated, and thus in some measure
concealed and obscured. In analysing these processes, the
combination must be resolved, and the abbreviation expanded,
and thus the appearance is presented of a pedantic and
superfluous formality. But that which is superfluous for
proof, is necessary for the analysis of proof. In order to
exhibit the conditions of demonstration distinctly, they
must be exhibited formally. In the same manner, in
demonstration we do not usually express every step in the
form of a syllogism, but we see the grounds of the
conclusiveness of a demonstration, by resolving it into
syllogisms. Neither axioms nor syllogisms are necessary for
conviction; but they are necessary to display the conditions
under which conviction becomes inevitable. The application
of a single one of the axioms just spoken of is so minute a
step in the proof, that it appears pedantic to give it a
marked place; but the very essence of demonstration consists
in this, that it is composed of an indissoluble succession
of such minute steps. The admirable circumstance is, that by
the accumulation of such apparently imperceptible advances,
we can in the end make so vast and so sure a progress. The
completeness of the analysis of our knowledge appears in the
smallness of the elements into which it is thus resolved.
The minuteness of any of these elements of truth, of axioms
for instance, does not prevent their being as essential as
others which are more obvious. And any attempt to assume one
kind of element only, when the course of our analysis brings
before us two or more kinds, is altogether unphilosophical.
Axioms and definitions are the proximate constituent
principles of our demonstrations; and the intimate bond
which connects together a definition and an axiom on the
same subject is not truly expressed {116} by asserting the
latter to be derived from the former. This bond of connexion
exists in the mind of the reasoner, in his conception of
_that_ to which both definition and axiom refer, and
consequently in the general Fundamental Idea of which that
conception is a modification.
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