MATHEMATICS AND PHILOSOPHY
=1401.= Socrates is praised by all the centuries for having
called philosophy from heaven to men on earth; but if, knowing
the condition of our science, he should come again and should
look once more to heaven for a means of curing men, he would
there find that to mathematics, rather than to the philosophy of
today, had been given the crown because of its industry and its
most happy and brilliant successes.--HERBART, J. F.
_Werke [Kehrbach], (Langensalza, 1890),
Bd. 5, p. 95._
=1402.= It is the embarrassment of metaphysics that it is able to
accomplish so little with the many things that mathematics offers
her.--KANT, E.
_Metaphysische Anfangsgründe der
Naturwissenschaft, Vorrede._
=1403.= Philosophers, when they have possessed a thorough
knowledge of mathematics, have been among those who have enriched
the science with some of its best ideas. On the other hand it
must be said that, with hardly an exception, all the remarks on
mathematics made by those philosophers who have possessed but a
slight or hasty or late-acquired knowledge of it are entirely
worthless, being either trivial or wrong.--WHITEHEAD, A. N.
_Introduction to Mathematics (New York,
1911), p. 113._
=1404.= The union of philosophical and mathematical productivity,
which besides in Plato we find only in Pythagoras, Descartes and
Leibnitz, has always yielded the choicest fruits to mathematics:
To the first we owe scientific mathematics in general, Plato
discovered the analytic method, by means of which mathematics was
elevated above the view-point of the elements, Descartes created
the analytical geometry, our own illustrious countryman
discovered the infinitesimal calculus--and just these are the
four greatest steps in the development of mathematics.
--HANKEL, HERMANN.
_Geschichte der Mathematik im Altertum
und im Mittelalter (Leipzig, 1874), pp.
149-150._
=1405.= Without mathematics one cannot fathom the depths of
philosophy; without philosophy one cannot fathom the depths of
mathematics; without the two one cannot fathom anything.
--BORDAS-DEMOULINS.
_Quoted in A. Rebière: Mathématiques et
Mathématiciens (Paris, 1898), p. 147._
=1406.= In the end mathematics is but simple philosophy, and
philosophy, higher mathematics in general.--NOVALIS.
_Schriften (Berlin, 1901), Teil 2, p.
443._
=1407.= It is a safe rule to apply that, when a mathematical or
philosophical author writes with a misty profundity, he is
talking nonsense.--WHITEHEAD, A. N.
_Introduction to Mathematics (New York,
1911), p. 227._
=1408.= The real finisher of our education is philosophy, but it
is the office of mathematics to ward off the dangers of
philosophy.--HERBART, J. F.
_Pestalozzi’s Idee eines ABC der
Anschauung; Werke [Kehrbach],
(Langensalza, 1890), Bd. 1, p. 168._
=1409.= Since antiquity mathematics has been regarded as the most
indispensable school for philosophic thought and in its highest
spheres the research of the mathematician is indeed most closely
related to pure speculation. Mathematics is the most perfect
union between exact knowledge and theoretical thought.--CURTIUS, E.
_Berliner Monatsberichte (1873), p.
517._
=1410.= Geometry has been, throughout, of supreme importance in
the history of knowledge.--RUSSELL, BERTRAND.
_Foundations of Geometry (Cambridge,
1897), p. 54._
=1411.= He is unworthy of the name of man who is ignorant of the
fact that the diagonal of a square is incommensurable with its
side.--PLATO.
_Quoted by Sophie Germain: Mémoire sur
les surfaces élastiques._
=1412.= Mathematics, considered as a science, owes its origin to
the idealistic needs of the Greek philosophers, and not as fable
has it, to the practical demands of Egyptian economics.... Adam
was no zoölogist when he gave names to the beasts of the field,
nor were the Egyptian surveyors mathematicians.--HANKEL, H.
_Die Entwickelung der Mathematik in den
letzten Jahrhunderten (Tübingen, 1884),
p. 7._
=1413.= There are only two ways open to man for attaining a
certain knowledge of truth: clear intuition and necessary
deduction.--DESCARTES.
_Rules for the Direction of the Mind;
Torrey’s The Philosophy of Descartes
(New York, 1892), p. 104._
=1414.= Mathematicians have, in many cases, proved some things to
be possible and others to be impossible, which, without
demonstration, would not have been believed.... Mathematics
afford many instances of impossibilities in the nature of things,
which no man would have believed, if they had not been strictly
demonstrated. Perhaps, if we were able to reason demonstratively
in other subjects, to as great extent as in mathematics, we might
find many things to be impossible, which we conclude, without
hesitation, to be possible.--REID, THOMAS.
_Essay on the Intellectual Powers of
Man, Essay 4, chap. 3._
=1415.= If philosophers understood mathematics, they would know
that indefinite speech, which permits each one to think what he
pleases and produces a constantly increasing difference of
opinion, is utterly unable, in spite of all fine words and even
in spite of the magnitude of the objects which are under
contemplation, to maintain a balance against a science which
instructs and advances through every word which it utters and
which at the same time wins for itself endless astonishment, not
through its survey of immense spaces, but through the exhibition
of the most prodigious human ingenuity which surpasses all power
of description.--HERBART, J. F.
_Werke Kehrbach (Langensalza, 1890), Bd.
5, p. 105._
=1416.= German intellect is an excellent thing, but when a German
product is presented it must be analysed. Most probably it is a
combination of intellect (I) and tobacco-smoke (T). Certainly I₃T₁,
and I₂T₁, occur; but I₁T₃ is more common, and I₂T₁₅ and I₁T₂₀ occur. In
many cases metaphysics (M) occurs and I hold that I_{a}T_{b}M_{c}
never occurs without b + c > 2a.
N. B.--Be careful, in analysing the compounds of the three, not to
confound T and M, which are strongly suspected to be isomorphic.
Thus, I₁T₃M₃ may easily be confounded with I₁T₆. As far as I dare
say anything, those who have placed _Hegel, Fichte_, etc., in the
rank of the extenders of _Kant_ have imagined T and M to be
identical.--DE MORGAN, A.
_Graves’ Life of W. R. Hamilton (New
York, 1882-1889), Vol. 13, p. 446._
=1417.= The discovery [of Ceres] was made by G. Piazzi of
Palermo; and it was the more interesting as its announcement
occurred simultaneously with a publication by Hegel in which he
severely criticized astronomers for not paying more attention to
philosophy, a science, said he, which would at once have shown
them that there could not possibly be more than seven planets,
and a study of which would therefore have prevented an absurd
waste of time in looking for what in the nature of things could
never be found.--BALL, W. W. R.
_History of Mathematics (London, 1901),
p. 458._
=1418.=
But who shall parcel out
His intellect by geometric rules,
Split like a province into round and square?
--WORDSWORTH.
_The Prelude, Bk. 2._
=1419.=
And Proposition, gentle maid,
Who soothly ask’d stern Demonstration’s aid, ....
--COLERIDGE, S. T.
_A Mathematical Problem._
=1420.= Mathematics connect themselves on the one side with
common life and physical science; on the other side with
philosophy in regard to our notions of space and time, and in the
questions which have arisen as to the universality and necessity
of the truths of mathematics and the foundation of our knowledge
of them.--CAYLEY, ARTHUR.
_British Association Address (1888);
Collected Mathematical Papers, Vol. 11,
p. 430._
=1421.= Mathematical teaching ... trains the mind to capacities,
which ... are of the closest kin to those of the greatest
metaphysician and philosopher. There is some color of truth for
the opposite doctrine in the case of elementary algebra. The
resolution of a common equation can be reduced to almost as
mechanical a process as the working of a sum in arithmetic. The
reduction of the question to an equation, however, is no
mechanical operation, but one which, according to the degree of
its difficulty, requires nearly every possible grade of
ingenuity: not to speak of the new, and in the present state of
the science insoluble, equations, which start up at every fresh
step attempted in the application of mathematics to other
branches of knowledge.--MILL, J. S.
_An Examination of Sir William
Hamilton’s Philosophy (London, 1878), p.
615._
=1422.= The value of mathematical instruction as a preparation for
those more difficult investigations, consists in the applicability
not of its doctrines, but of its methods. Mathematics will ever
remain the most perfect type of the Deductive Method in general; and
the applications of mathematics to the simpler branches of physics,
furnish the only school in which philosophers can effectually
learn the most difficult and important portion of their art, the
employment of the laws of the simpler phenomena for explaining and
predicting those of the more complex. These grounds are quite
sufficient for deeming mathematical training an indispensable
basis of real scientific education, and regarding, with Plato,
one who is ἀγεωμέτρητος, as wanting in one of the most essential
qualifications for the successful cultivation of the higher branches
of philosophy.--MILL, J. S.
_System of Logic, Bk. 3, chap. 24, sect.
9._
=1423.= In metaphysical reasoning, the process is always short.
The conclusion is but a step or two, seldom more, from the first
principles or axioms on which it is grounded, and the different
conclusions depend not one upon another.
It is otherwise in mathematical reasoning. Here the field has no
limits. One proposition leads on to another, that to a third, and
so on without end. If it should be asked, why demonstrative
reasoning has so wide a field in mathematics, while, in other
abstract subjects, it is confined within very narrow limits, I
conceive this is chiefly owing to the nature of quantity, ...
mathematical quantities being made up of parts without number,
can touch in innumerable points, and be compared in innumerable
different ways.--REID, THOMAS.
_Essays on the Powers of the Human Mind
(Edinburgh, 1812), Vol. 2, pp. 422-423._
=1424.= The power of Reason ... is unquestionably the most
important by far of those which are comprehended under the
general title of Intellectual. It is on the right use of this
power that our success in the pursuit of both knowledge and of
happiness depends; and it is by the exclusive possession of it
that man is distinguished, in the most essential respects, from
the lower animals. It is, indeed, from their subserviency to its
operations, that the other faculties ... derive their chief
value.--STEWART, DUGALD.
_Philosophy of the Human Mind; Collected
Works (Edinburgh, 1854), Vol. 8, p. 5._
=1425.= When ... I asked myself why was it then that the earliest
philosophers would admit to the study of wisdom only those who
had studied mathematics, as if this science was the easiest of
all and the one most necessary for preparing and disciplining the
mind to comprehend the more advanced, I suspected that they had
knowledge of a mathematical science different from that of our
time....
I believe I find some traces of these true mathematics in Pappus
and Diophantus, who, although they were not of extreme antiquity,
lived nevertheless in times long preceding ours. But I willingly
believe that these writers themselves, by a culpable ruse,
suppressed the knowledge of them; like some artisans who conceal
their secret, they feared, perhaps, that the ease and simplicity
of their method, if become popular, would diminish its importance,
and they preferred to make themselves admired by leaving to us, as
the product of their art, certain barren truths deduced with
subtlety, rather than to teach us that art itself, the knowledge
of which would end our admiration.--DESCARTES.
_Rules for the Direction of the Mind;
Philosophy of Descartes [Torrey], (New
York, 1892), pp. 70-71._
=1426.= If we rightly adhere to our rule [that is, that we should
occupy ourselves only with those subjects in reference to which
the mind is capable of acquiring certain and indubitable
knowledge] there will remain but few things to the study of which
we can devote ourselves. There exists in the sciences hardly a
single question upon which men of intellectual ability have not
held different opinions. But whenever two men pass contrary
judgment on the same thing, it is certain that one of the two is
wrong. More than that, neither of them has the truth; for if one
of them had a clear and precise insight into it, he could so
exhibit it to his opponent as to end the discussion by compelling
his conviction.... It follows from this, if we reckon rightly,
that among existing sciences there remain only geometry and
arithmetic, to which the observance of our rule would bring us.
--DESCARTES.
_Rules for the Direction of the Mind;
Philosophy of Descartes [Torrey], (New
York, 1892), p. 62._
=1427.= The same reason which led Plato to recommend the study of
arithmetic led him to recommend also the study of geometry. The
vulgar crowd of geometricians, he says, will not understand him.
They have practice always in view. They do not know that the real
use of the science is to lead men to the knowledge of abstract,
essential, eternal truth. (Plato’s Republic, Book 7). Indeed if
we are to believe Plutarch, Plato carried his feeling so far that
he considered geometry as degraded by being applied to any
purpose of vulgar utility. Archytas, it seems, had framed
machines of extraordinary power on mathematical principles.
(Plutarch, Sympos., VIII., and Life of Marcellus. The machines of
Archytas are also mentioned by Aulus Gellius and Diogenes
Laertius). Plato remonstrated with his friend, and declared that
this was to degrade a noble intellectual exercise into a low
craft, fit only for carpenters and wheelwrights. The office of
geometry, he said, was to discipline the mind, not to minister to
the base wants of the body. His interference was successful; and
from that time according to Plutarch, the science of mechanics
was considered unworthy of the attention of a philosopher.
--MACAULAY.
_Lord Bacon; Edinburgh Review, July,
1837._
=1428.= The intellectual habits of the Mathematicians are, in
some respects, the same with those [of the Metaphysicians] we
have been now considering; but, in other respects, they differ
widely. Both are favourable to the improvement of the power of
_attention_, but not in the same manner, nor in the same degree.
Those of the metaphysician give capacity of fixing the attention
on the subjects of our consciousness, without being distracted by
things external; but they afford little or no exercise to that
species of attention which enables us to follow long processes of
reasoning, and to keep in view all the various steps of an
investigation till we arrive at the conclusion. In mathematics,
such processes are much longer than in any other science; and
hence the study of it is peculiarly calculated to strengthen the
power of steady and concatenated thinking,--a power which, in all
the pursuits of life, whether speculative or active, is one of
the most valuable endowments we can possess. This command of
attention, however, it may be proper to add, is to be acquired,
not by the practice of modern methods, but by the study of Greek
geometry, more particularly, by accustoming ourselves to pursue
long trains of demonstration, without availing ourselves of the
aid of any sensible diagrams; the thoughts being directed solely
by those ideal delineations which the powers of conception and of
memory enable us to form.--STEWART, DUGALD.
_Philosophy of the Human Mind, Part 3,
chap. 1, sect. 3._
=1429.= They [the Greeks] speculated and theorized under a lively
persuasion that a Science of every part of nature was possible,
and was a fit object for the exercise of a man’s best faculties;
and they were speedily led to the conviction that such a science
must clothe its conclusions in the language of mathematics. This
conviction is eminently conspicuous in the writings of Plato....
Probably no succeeding step in the discovery of the Laws of
Nature was of so much importance as the full adoption of this
pervading conviction, that there must be Mathematical Laws of
Nature, and that it is the business of Philosophy to discover
these Laws. This conviction continues, through all the succeeding
ages of the history of the science, to be the animating and
supporting principle of scientific investigation and discovery.
--WHEWELL, W.
_History of the Inductive Sciences, Vol.
1, bk. 2, chap. 3._
=1430.= For to pass by those Ancients, the wonderful _Pythagoras_,
the sagacious _Democritus_, the divine _Plato_, the most subtle
and very learned _Aristotle_, Men whom every Age has hitherto
acknowledged as deservedly honored, as the greatest Philosophers,
the Ring-leaders of Arts; in whose Judgments how much these
Studies [mathematics] were esteemed, is abundantly proclaimed
in History and confirmed by their famous Monuments, which
are everywhere interspersed and bespangled with Mathematical
Reasonings and Examples, as with so many Stars; and consequently
anyone not in some Degree conversant in these Studies will in vain
expect to understand, or unlock their hidden Meanings, without the
Help of a Mathematical Key: For who can play well on _Aristotle’s_
Instrument but with a Mathematical Quill; or not be altogether
deaf to the Lessons of natural _Philosophy_, while ignorant of
_Geometry?_ Who void of (_Geometry_ shall I say, or) _Arithmetic_
can comprehend _Plato’s_ _Socrates_ lisping with Children
concerning Square Numbers; or can conceive _Plato_ himself
treating not only of the Universe, but the Polity of Commonwealths
regulated by the Laws of Geometry, and formed according to a
Mathematical Plan?--BARROW, ISAAC.
_Mathematical Lectures (London, 1734),
pp. 26-27._
=1431.=
And Reason now through number, time, and space
Darts the keen lustre of her serious eye;
And learns from facts compar’d the laws to trace
Whose long procession leads to Deity
--BEATTIE, JAMES.
_The Minstrel, Bk. 2, stanza 47._
=1432.= That Egyptian and Chaldean wisdom mathematical wherewith
Moses and Daniel were furnished, ....--HOOKER, RICHARD.
_Ecclesiastical Polity, Bk. 3, sect. 8._
=1433.= General and certain truths are only founded in the
habitudes and relations of _abstract ideas_. A sagacious and
methodical application of our thoughts, for the finding out of
these relations, is the only way to discover all that can be
put with truth and certainty concerning them into general
propositions. By what steps we are to proceed in these, is to be
learned in the schools of mathematicians, who, from very plain and
easy beginnings, by gentle degrees, and a continued chain of
reasonings, proceed to the discovery and demonstration of truths
that appear at first sight beyond human capacity. The art of
finding proofs, and the admirable method they have invented for
the singling out and laying in order those intermediate ideas that
demonstratively show the equality or inequality of unapplicable
quantities, is that which has carried them so far and produced
such wonderful and unexpected discoveries; but whether something
like this, in respect of other ideas, as well as those of
magnitude, may not in time be found out, I will not determine.
This, I think, I may say, that if other ideas that are the real as
well as the nominal essences of their species, were pursued in the
way familiar to mathematicians, they would carry our thoughts
further, and with greater evidence and clearness than possibly we
are apt to imagine.--LOCKE, JOHN.
_An Essay concerning Human
Understanding, Bk. 4, chap. 12, sect.
7._
=1434.= Those long chains of reasoning, quite simple and easy,
which geometers are wont to employ in the accomplishment of their
most difficult demonstrations, led me to think that everything
which might fall under the cognizance of the human mind might be
connected together in a similar manner, and that, provided only
that one should take care not to receive anything as true which
was not so, and if one were always careful to preserve the order
necessary for deducing one truth from another, there would be
none so remote at which he might not at last arrive, nor so
concealed which he might not discover.--DESCARTES.
_Discourse upon Method, part 2; The
Philosophy of Descartes [Torrey], (New
York, 1892), p. 47._
=1435.= If anyone wished to write in mathematical fashion in
metaphysics or ethics, nothing would prevent him from so doing
with vigor. Some have professed to do this, and we have a promise
of mathematical demonstrations outside of mathematics; but it is
very rare that they have been successful. This is, I believe,
because they are disgusted with the trouble it is necessary to
take for a small number of readers where they would ask as in
Persius: _Quis leget haec_, and reply: _Vel duo vel nemo._
--LEIBNITZ.
_New Essay concerning Human
Understanding, Langley, Bk 2, chap. 29,
sect. 12._
=1436.= It is commonly asserted that mathematics and philosophy
differ from one another according to their _objects_, the former
treating of _quantity_, the latter of _quality_. All this is
false. The difference between these sciences cannot depend on
their object; for philosophy applies to everything, hence also to
_quanta_, and so does mathematics in part, inasmuch as everything
has magnitude. It is only the _different kind of rational
knowledge or application_ of reason in mathematics and philosophy
which constitutes the specific difference between these two
sciences. For philosophy is _rational knowledge from mere
concepts_, mathematics, on the contrary, is _rational knowledge
from the construction of concepts_.
We construct concepts when we represent them in intuition _a
priori_, without experience, or when we represent in intuition
the object which corresponds to our concept of it.--The
mathematician can never apply his reason to mere concepts, nor
the philosopher to the construction of concepts.--In mathematics
the reason is employed _in concreto_, however, the intuition is
not empirical, but the object of contemplation is something _a
priori_.
In this, as we see, mathematics has an advantage over philosophy,
the knowledge in the former being intuitive, in the latter, on the
contrary, only _discursive_. But the reason why in mathematics we
deal more with quantity lies in this, that magnitudes can be
constructed in intuition _a priori_, while qualities, on the
contrary, do not permit of being represented in intuition.--KANT, E.
_Logik; Werke [Hartenstein], (Leipzig,
1868), Bd. 8, pp. 23-24._
=1437.= Kant has divided human ideas into the two categories of
quantity and quality, which, if true, would destroy the
universality of Mathematics; but Descartes’ fundamental
conception of the relation of the concrete to the abstract in
Mathematics abolishes this division, and proves that all ideas of
quality are reducible to ideas of quantity. He had in view
geometrical phenomena only; but his successors have included in
this generalization, first, mechanical phenomena, and, more
recently, those of heat. There are now no geometers who do not
consider it of universal application, and admit that every
phenomenon may be as logically capable of being represented by an
equation as a curve or a motion, if only we were always capable
(which we are very far from being) of first discovering, and then
resolving it.
The limitations of Mathematical science are not, then, in its
nature. The limitations are in our intelligence: and by these we
find the domain of the science remarkably restricted, in
proportion as phenomena, in becoming special, become complex.
--COMTE, A.
_Positive Philosophy [Martineau], Bk. 1,
chap. 1._
=1438.= The great advantage of the mathematical sciences over the
moral consists in this, that the ideas of the former, being
sensible, are always clear and determinate, the smallest
distinction between them being immediately perceptible, and the
same terms are still expressive of the same ideas, without
ambiguity or variation. An oval is never mistaken for a circle,
nor an hyperbola for an ellipsis. The isosceles and scalenum are
distinguished by boundaries more exact than vice and virtue,
right or wrong. If any term be defined in geometry, the mind
readily, of itself, substitutes on all occasions, the definition
for the thing defined: Or even when no definition is employed,
the object itself may be represented to the senses, and by that
means be steadily and clearly apprehended. But the finer
sentiments of the mind, the operations of the understanding, the
various agitations of the passions, though really in themselves
distinct, easily escape us, when surveyed by reflection; nor is
it in our power to recall the original object, so often as we
have occasion to contemplate it. Ambiguity, by this means, is
gradually introduced into our reasonings: Similar objects are
readily taken to be the same: And the conclusion becomes at last
very wide off the premises.--HUME, DAVID.
_An Inquiry concerning Human
Understanding, sect. 7, part 1._
=1439.= One part of these disadvantages in moral ideas which has
made them be thought not capable of demonstration, may in a good
measure be remedied by definitions, setting down that collection
of simple ideas, which every term shall stand for; and then using
the terms steadily and constantly for that precise collection.
And what methods algebra, or something of that kind, may
hereafter suggest, to remove the other difficulties, it is not
easy to foretell. Confident, I am, that if men would in the same
method, and with the same indifferency, search after moral as
they do mathematical truths, they would find them have a stronger
connexion one with another, and a more necessary consequence from
our clear and distinct ideas, and to come nearer perfect
demonstration than is commonly imagined.--LOCKE, JOHN.
_An Essay concerning Human
Understanding, Bk. 4, chap. 3, sect.
20._
=1440.= That which in this respect has given the advantage to the
ideas of quantity, and made them thought more capable of
certainty and demonstration [than moral ideas], is,
First, That they can be set down and represented by sensible
marks, which have a greater and nearer correspondence with them
than any words or sounds whatsoever. Diagrams drawn on paper
are copies of the ideas in the mind, and not liable to the
uncertainty that words carry in their signification. An angle,
circle, or square, drawn in lines, lies open to the view, and
cannot be mistaken: it remains unchangeable, and may at leisure
be considered and examined, and the demonstration be revised, and
all the parts of it may be gone over more than once, without any
danger of the least change in the ideas. This cannot be done in
moral ideas: we have no sensible marks that resemble them,
whereby we can set them down; we have nothing but words to
express them by; which, though when written they remain the same,
yet the ideas they stand for may change in the same man; and it
is seldom that they are not different in different persons.
Secondly, Another thing that makes the greater difficulty in
ethics is, That moral ideas are commonly more complex than those
of the figures ordinarily considered in mathematics. From whence
these two inconveniences follow:--First, that their names are of
more uncertain signification, the precise collection of simple
ideas they stand for not being so easily agreed on; and so the
sign that is used for them in communication always, and in
thinking often, does not steadily carry with it the same idea.
Upon which the same disorder, confusion, and error follow, as
would if a man, going to demonstrate something of an heptagon,
should, in the diagram he took to do it, leave out one of the
angles, or by oversight make the figure with an angle more than
the name ordinarily imported, or he intended it should when at
first he thought of his demonstration. This often happens, and is
hardly avoidable in very complex moral ideas, where the same name
being retained, an angle, i.e. one simple idea is left out, or
put in the complex one (still called by the same name) more at
one time than another. Secondly, From the complexedness of these
moral ideas there follows another inconvenience, viz., that the
mind cannot easily retain those precise combinations so exactly
and perfectly as is necessary in the examination of the habitudes
and correspondences, agreements or disagreements, of several of
them one with another; especially where it is to be judged of by
long deductions and the intervention of several other complex
ideas to show the agreement or disagreement of two remote ones.
--LOCKE, JOHN.
_An Essay concerning Human
Understanding, Bk. 4, chap. 3, sect.
19._
=1441.= It has been generally taken for granted, that mathematics
alone are capable of demonstrative certainty: but to have such an
agreement or disagreement as may be intuitively perceived,
being, as I imagine, not the privileges of the ideas of number,
extension, and figure alone, it may possibly be the want of due
method and application in us, and not of sufficient evidence in
things, that demonstration has been thought to have so little to
do in other parts of knowledge, and been scarce so much as aimed
at by any but mathematicians. For whatever ideas we have wherein
the mind can perceive the immediate agreement or disagreement that
is between them, there the mind is capable of intuitive knowledge,
and where it can perceive the agreement or disagreement of any two
ideas, by an intuitive perception of the agreement or disagreement
they have with any intermediate ideas, there the mind is capable
of demonstration: which is not limited to the idea of extension,
figure, number, and their modes.--LOCKE, JOHN.
_An Essay concerning Human
Understanding, Bk. 4, chap. 2, sect. 9._
=1442.= Now I shall remark again what I have already touched
upon more than once, that it is a common opinion that only
mathematical sciences are capable of a demonstrative certainty;
but as the agreement and disagreement which may be known
intuitively is not a privilege belonging only to the ideas of
numbers and figures, it is perhaps for want of application on our
part that mathematics alone have attained to demonstrations.
--LEIBNITZ.
_New Essay concerning Human
Understanding, Bk. 4, chap. 2, sect. 9
[Langley]._