MATHEMATICS AND LOGIC
=1301.= Mathematics belongs to every inquiry, moral as well as
physical. Even the rules of logic, by which it is rigidly bound,
could not be deduced without its aid. The laws of argument admit
of simple statement, but they must be curiously transposed before
they can be applied to the living speech and verified by
observation. In its pure and simple form the syllogism cannot be
directly compared with all experience, or it would not have
required an Aristotle to discover it. It must be transmuted into
all the possible shapes in which reasoning loves to clothe
itself. The transmutation is the mathematical process in the
establishment of the law.--PEIRCE, BENJAMIN.
_Linear Associative Algebra; American
Journal of Mathematics, Vol. 4 (1881),
p. 97._
=1302.= In mathematics we see the conscious logical activity of
our mind in its purest and most perfect form; here is made
manifest to us all the labor and the great care with which it
progresses, the precision which is necessary to determine exactly
the source of the established general theorems, and the
difficulty with which we form and comprehend abstract
conceptions; but we also learn here to have confidence in the
certainty, breadth, and fruitfulness of such intellectual labor.
--HELMHOLTZ, H.
_Vorträge und Reden, Bd. 1
(Braunschweig, 1896), p. 176._
=1303.= Mathematical demonstrations are a logic of as much or more
use, than that commonly learned at schools, serving to a just
formation of the mind, enlarging its capacity, and strengthening
it so as to render the same capable of exact reasoning, and
discerning truth from falsehood in all occurrences, even in
subjects not mathematical. For which reason it is said, the
Egyptians, Persians, and Lacedaemonians seldom elected any new
kings, but such as had some knowledge in the mathematics,
imagining those, who had not, men of imperfect judgments, and
unfit to rule and govern.--FRANKLIN, BENJAMIN.
_Usefulness of Mathematics; Works
(Boston, 1840), Vol. 2, p. 68._
=1304.= The mathematical conception is, from its very nature,
abstract; indeed its abstractness is usually of a higher order
than the abstractness of the logician.--CHRYSTAL, GEORGE.
_Encyclopedia Britannica (Ninth
Edition), Article “Mathematics.”_
=1305.= Mathematics, that giant pincers of scientific logic....
--HALSTED, G. B.
_Science (1905), p. 161._
=1306.= Logic has borrowed the rules of geometry without
understanding its power.... I am far from placing logicians by
the side of geometers who teach the true way to guide the
reason.... The method of avoiding error is sought by every one.
The logicians profess to lead the way, the geometers alone reach
it, and aside from their science there is no true demonstration.
--PASCAL.
_Quoted by A. Rebière: Mathématiques et
Mathématiciens (Paris, 1898), pp.
162-163._
=1307.= Mathematics, like dialectics, is an organ of the higher
sense, in its execution it is an art like eloquence. To both
nothing but the form is of value; neither cares anything for
content. Whether mathematics considers pennies or guineas,
whether rhetoric defends truth or error, is perfectly immaterial
to either.--GOETHE.
_Sprüche in Prosa, Natur IV, 946._
=1308.= Confined to its true domain, mathematical reasoning is
admirably adapted to perform the universal office of sound logic:
to induce in order to deduce, in order to construct.... It
contents itself to furnish, in the most favorable domain, a
model of clearness, of precision, and consistency, the close
contemplation of which is alone able to prepare the mind to
render other conceptions also as perfect as their nature permits.
Its general reaction, more negative than positive, must consist,
above all, in inspiring us everywhere with an invincible aversion
for vagueness, inconsistency, and obscurity, which may always be
really avoided in any reasoning whatsoever, if we make sufficient
effort.--COMTE, A.
_Subjective Synthesis._
=1309.= Formal thought, consciously recognized as such, is the
means of all exact knowledge; and a correct understanding of the
main formal sciences, Logic and Mathematics, is the proper and
only safe foundation for a scientific education.--LEFEVRE, ARTHUR.
_Number and its Algebra (Boston, Sect.
222.)_
=1310.= It has come to pass, I know not how, that Mathematics
and Logic, which ought to be but the handmaids of Physic,
nevertheless presume on the strength of the certainty which they
possess to exercise dominion over it.--BACON, FRANCIS.
_De Augmentis, Bk. 3._
=1311.= We may regard geometry as a practical logic, for the
truths which it considers, being the most simple and most
sensible of all, are, for this reason, the most susceptible to
easy and ready application of the rules of reasoning.--D’ALEMBERT.
_Quoted in A. Rebière: Mathématiques et
Mathématiciens (Paris, 1898), pp.
151-152._
=1312.= There are notable examples enough of demonstration
outside of mathematics, and it may be said that Aristotle has
already given some in his “Prior Analytics.” In fact logic is as
susceptible of demonstration as geometry, .... Archimedes is the
first, whose works we have, who has practised the art of
demonstration upon an occasion where he is treating of physics,
as he has done in his book on Equilibrium. Furthermore, jurists
may be said to have many good demonstrations; especially the
ancient Roman jurists, whose fragments have been preserved to us
in the Pandects.--LEIBNITZ, G. W.
_New Essay on Human Understanding
[Langley], Bk. 4, chap. 2, sect. 12._
=1313.= It is commonly considered that mathematics owes its
certainty to its reliance on the immutable principles of formal
logic. This ... is only half the truth imperfectly expressed. The
other half would be that the principles of formal logic owe such
a degree of permanence as they have largely to the fact that they
have been tempered by long and varied use by mathematicians. “A
vicious circle!” you will perhaps say. I should rather describe
it as an example of the process known by mathematicians as the
method of successive approximation.--BÔCHER, MAXIME.
_Bulletin of the American Mathematical
Society, Vol. 11, p. 120._
=1314.= Whatever advantage can be attributed to logic in
directing and strengthening the action of the understanding is
found in a higher degree in mathematical study, with the
immense added advantage of a determinate subject, distinctly
circumscribed, admitting of the utmost precision, and free from
the danger which is inherent in all abstract logic,--of
leading to useless and puerile rules, or to vain ontological
speculations. The positive method, being everywhere identical, is
as much at home in the art of reasoning as anywhere else: and
this is why no science, whether biology or any other, can offer
any kind of reasoning, of which mathematics does not supply a
simpler and purer counterpart. Thus, we are enabled to eliminate
the only remaining portion of the old philosophy which could even
appear to offer any real utility; the logical part, the value of
which is irrevocably absorbed by mathematical science.--COMTE, A.
_Positive Philosophy [Martineau],
(London, 1875), Vol. 1, pp. 321-322._
=1315.= We know that mathematicians care no more for logic than
logicians for mathematics. The two eyes of exact science are
mathematics and logic: the mathematical sect puts out the logical
eye, the logical sect puts out the mathematical eye; each
believing that it can see better with one eye than with two.
--DE MORGAN, A.
_Quoted in F. Cajori: History of
Mathematics (New York, 1897), p. 316._
=1316.= The progress of the art of rational discovery depends in a
great part upon the art of characteristic (ars characteristica).
The reason why people usually seek demonstrations only in numbers
and lines and things represented by these is none other than that
there are not, outside of numbers, convenient characters
corresponding to the notions.--LEIBNITZ, G. W.
_Philosophische Schriften [Gerhardt] Bd.
8, p. 198._
=1317.= The influence of the mathematics of Leibnitz upon his
philosophy appears chiefly in connection with his law of
continuity and his prolonged efforts to establish a Logical
Calculus.... To find a Logical Calculus (implying a universal
philosophical language or system of signs) is an attempt to apply
in theological and philosophical investigations an analytic
method analogous to that which had proved so successful in
Geometry and Physics. It seemed to Leibnitz that if all the
complex and apparently disconnected ideas which make up our
knowledge could be analysed into their simple elements, and if
these elements could each be represented by a definite sign, we
should have a kind of “alphabet of human thoughts.” By the
combination of these signs (letters of the alphabet of thought) a
system of true knowledge would be built up, in which reality
would be more and more adequately represented or symbolized....
In many cases the analysis may result in an infinite series of
elements; but the principles of the Infinitesimal Calculus in
mathematics have shown that this does not necessarily render
calculation impossible or inaccurate. Thus it seemed to Leibnitz
that a synthetic calculus, based upon a thorough analysis, would
be the most effective instrument of knowledge that could be
devised. “I feel,” he says, “that controversies can never be
finished, nor silence imposed upon the Sects, unless we give up
complicated reasonings in favor of simple _calculations_, words
of vague and uncertain meaning in favor of fixed symbols.” Thus
it will appear that “every paralogism is nothing but _an error of
calculation_.” “When controversies arise, there will be no more
necessity of disputation between two philosophers than between
two accountants. Nothing will be needed but that they should take
pen in hand, sit down with their counting-tables, and (having
summoned a friend, if they like) say to one another: _Let us
calculate_.”--LATTA, ROBERT.
_Leibnitz, The Monadology, etc. (Oxford,
1898), p. 85._
=1318.= Pure mathematics was discovered by Boole in a work which
he called “The Laws of Thought”.... His work was concerned with
formal logic, and this is the same thing as mathematics.
--RUSSELL, BERTRAND.
_International Monthly, 1901, p. 83._
=1319.= Mathematics is but the higher development of Symbolic
Logic.--WHETHAM, W. C. D.
_Recent Development of Physical Science
(Philadelphia, 1904), p. 34._
=1320.= Symbolic Logic has been disowned by many logicians
on the plea that its interest is mathematical, and by many
mathematicians on the plea that its interest is logical.
--WHITEHEAD, A. N.
_Universal Algebra (Cambridge, 1898),
Preface, p. 6._
=1321.= ... the two great components of the critical movement,
though distinct in origin and following separate paths, are found
to converge at last in the thesis: Symbolic Logic is Mathematics,
Mathematics is Symbolic Logic, the twain are one.--KEYSER, C. J.
_Lectures on Science, Philosophy and Art
(New York, 1908), p. 19._
=1322.= The emancipation of logic from the yoke of Aristotle very
much resembles the emancipation of geometry from the bondage of
Euclid; and, by its subsequent growth and diversification, logic,
less abundantly perhaps but not less certainly than geometry, has
illustrated the blessings of freedom.--KEYSER, C. J.
_Science, Vol. 35 (1912), p. 108._
=1323.= I would express it as my personal view, which is probably
not yet shared generally, that pure mathematics seems to me
merely a _branch of general logic_; that branch which is based on
the concept of _numbers_, to whose economic advantages is to be
attributed the tremendous development which this particular
branch has undergone as compared with the remaining branches of
logic, which until the most recent times have remained almost
stationary.--SCHRÖDER, E.
_Ueber Pasigraphie etc.; Verhandlungen
des 1. Internationalen
Mathematiker-Kongresses (Leipzig, 1898),
p. 149._
=1324.= If logical training is to consist, not in repeating
barbarous scholastic formulas or mechanically tacking together
empty majors and minors, but in acquiring dexterity in the use of
trustworthy methods of advancing from the known to the unknown,
then mathematical investigation must ever remain one of its most
indispensable instruments. Once inured to the habit of accurately
imagining abstract relations, recognizing the true value of
symbolic conceptions, and familiarized with a fixed standard of
proof, the mind is equipped for the consideration of quite other
objects than lines and angles. The twin treatises of Adam Smith
on social science, wherein, by deducing all human phenomena first
from the unchecked action of selfishness and then from the
unchecked action of sympathy, he arrives at mutually-limiting
conclusions of transcendent practical importance, furnish for all
time a brilliant illustration of the value of mathematical
methods and mathematical discipline.--FISKE, JOHN.
_Darwinism and other Essays (Boston,
1893), pp. 297-298._
=1325.= No irrational exaggeration of the claims of Mathematics
can ever deprive that part of philosophy of the property of being
the natural basis of all logical education, through its
simplicity, abstractness, generality, and freedom from
disturbance by human passion. There, and there alone, we find in
full development the art of reasoning, all the resources of
which, from the most spontaneous to the most sublime, are
continually applied with far more variety and fruitfulness than
elsewhere;.... The more abstract portion of mathematics may in
fact be regarded as an immense repository of logical resources,
ready for use in scientific deduction and co-ordination.
--COMTE, A.
_Positive Philosophy [Martineau],
(London, 1875), Vol. 2, p. 439._
=1326.= Logic it is called [referring to Whitehead and Russell’s
Principia Mathematica] and logic it is, the logic of propositions
and functions and classes and relations, by far the greatest (not
merely the biggest) logic that our planet has produced, so much
that is new in matter and in manner; but it is also mathematics,
a prolegomenon to the science, yet itself mathematics in its most
genuine sense, differing from other parts of the science only in
the respects that it surpasses these in fundamentality,
generality and precision, and lacks traditionality. Few will read
it, but all will feel its effect, for behind it is the urgence
and push of a magnificent past: two thousand five hundred years
of record and yet longer tradition of human endeavor to think
aright.--KEYSER, C. J.
_Science, Vol. 35 (1912), p. 110._