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A chief point of
interest that has emerged from modern attempts to characterize philosophy is the
importance of distinguishing dialectical or analytical inquiries about meaning
from empirical inquiries about fact. A primary, traditional task of the
philosopher has been to present things in such a light that human feelings may
be reasonably grounded. The need for this is especially obvious in the case of
the moral philosopher or the aesthetician, whose work treats explicitly of
subjective concerns. But the need remains the same in all philosophical
inquiries, including even discussions of the foundations
of mathematics. For it is obvious to a careful observer that persons
who put forward theses about the nature of mathematics are involved not just
intellectually but also emotionally in their pursuits; while it must be supposed
that in some cases this involvement stems from or inevitably leads to
intellectual confusion, it must also be allowed that a certain emotional
commitment may perhaps be a necessary condition for the making of discoveries.
Thus it can scarcely be an accident that no great mathematician has ever
accepted the conventionalist view according to which mathematical truths are
man-made. (see also mathematics,
philosophy of) |
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The inquiry into
the nature, underlying assumptions, and scope of mathematics has emerged in the
20th century as a subdiscipline of mathematics itself, known as the study of
foundations. For a full historical treatment of this field, see the Macrop?ia
article MATHEMATICS, THE
FOUNDATIONS OF . The material
below, edited from an article originally written by Alfred North Whitehead for
the 11th edition of the Encyclop?ia Britannica, treats mathematics itself
as an object of philosophical investigation. |
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It has been usual
to define mathematics as "the science of discrete and continuous
magnitude." Even Leibnitz, who initiated a more modern point of view,
follows the tradition in thus confining the scope of mathematics properly so
called, while apparently conceiving it as a department of a yet wider science of
reasoning. A short consideration of some leading topics of the science will
exemplify both the plausibility and inadequacy of the above definition.
Arithmetic, algebra, and the
infinitesimal calculus, are
sciences directly concerned with integral numbers, rational (or fractional)
numbers, and real numbers generally, which include incommensurable numbers. It
would seem that "the general theory of discrete and continuous
quantity" is the exact description of the topics of these sciences.
Furthermore, can we not complete the circle of the mathematical sciences by
adding geometry? Now geometry
deals with points, lines, planes, and cubic contents. Of these all except points
are quantities. Also, as the Cartesian geometry shows, all the relations between
points are expressible in terms of geometric quantities. Accordingly, at first
sight it seems reasonable to define geometry in some such way as "the
science of dimensional quantity." Thus every subdivision of mathematical
science would appear to deal with quantity, and the definition of mathematics as
"the science of quantity" would appear to be justified. We have now to
see why the definition is inadequate. |
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8.1.1.1
Types relating to numbers.
What are numbers?
We can talk of five apples and 10 pears. But what are "five" and
"10" apart from the apples and pears? Also in addition to the cardinal
numbers there are the ordinal numbers: the fifth apple and the 10th pear claim
thought. What is the relation of "the fifth" and "10th" to
"five" and "10"? "The first rose of summer" and
"the last rose of summer" are parallel phrases, yet one explicitly
introduces an ordinal number and the other does not. Again, "half a
foot" and "half a pound" are easily defined. But in what sense is
there "a half," which is the same for "half a foot" as
"half a pound"? Furthermore, incommensurable numbers are defined as
the limits arrived at as the result of certain procedures with rational numbers.
But how do we know that there is anything to reach? We must know that {sqroot 2}
exists before we can prove that any procedure will reach it. (see also number
system) |
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Also in geometry,
what is a point? The straightness of a straight line and the planeness of a
plane require consideration. Furthermore, "congruence" is a
difficulty. For when a triangle "moves," the points do not move with
it. So what is it that keeps unaltered in the moving triangle? Thus the whole
method of measurement in geometry as described in the elementary textbooks and
the older treatises is obscure to the last degree. Lastly, what are
"dimensions"? All these topics require thorough discussion before we
can rest content with the definition of mathematics as the general science of
magnitude; and by the time they are discussed the definition has evaporated. An
outline of the modern answers to questions such as the above will now be given.
A critical defense of them would require a volume. |
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A one-one relation
between the members of two classes
 and
 is any
method of correlating all the members of
to all
the members of
, so
that any member of
has one
and only one correlate in
, and
any member of
has one
and only one correlate in
. Two
classes between which a one-one relation exists have the same cardinal
number and are called cardinally similar; and the cardinal number of
the class
is a
certain class whose members are themselves classes--namely, it is the class
composed of all those classes for which a one-one correlation with
exists.
Thus the cardinal number of
is
itself a class, and furthermore
is a
member of it. For a one-one relation can be established between the members of
and
by the
simple process of correlating each member of
with
itself. Thus the cardinal number one is the class of unit classes, the cardinal
number two is the class of doublets, and so on. Also a unit class is any class
with the property that it possesses a member x such that, if y is
any member of the class, then x and y are identical. A doublet is
any class which possesses a member x such that the modified class formed
by all the other members except x is a unit class. And so on for all the
finite cardinals, which are thus defined successively. The cardinal number zero
is the class of classes with no members; but there is only one such class,
namely--the null class. Thus this cardinal number has only one member. The
operations of addition and multiplication of two given cardinal numbers can be
defined by taking two classes
and
,
satisfying the conditions (1) that their cardinal numbers are respectively the
given numbers, and (2) that they contain no member in common, and then by
defining by reference to
and
two
other suitable classes whose cardinal numbers are defined to be respectively the
required sum and product of the cardinal numbers in question. |
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With these
definitions it is now possible to prove the following six premises
applying to finite cardinal numbers, from which Peano has shown that all
arithmetic can be deduced:-- |
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i. Cardinal numbers
form a class. ii. Zero is a cardinal number. iii. If a is a cardinal
number, a + 1 is a cardinal number. iv. If s is any class and zero
is a member of it, also if when x is a cardinal number and a member of s,
also x + 1 is a member of s, then the whole class of cardinal
numbers is contained in s. v. If a and b are cardinal
numbers, and a + 1 = b + 1, then a = b. vi. If a
is a cardinal number, then a + 1 = 0. |
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It may be noticed
that (iv.) is the familiar principle of mathematical induction. Peano in a
historical note refers its first explicit employment, although without a general
enunciation, to Maurolycus in his work, Arithmeticorum libri duo (Venice,
1575). |
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But now the
difficulty of confining mathematics to being the science of number and quantity
is immediately apparent. For there is no self-contained science of cardinal
numbers. The proof of the six premises requires an elaborate investigation into
the general properties of classes and relations that can be deduced by the
strictest reasoning from our ultimate logical principles. Also it is purely
arbitrary to erect the consequences of these six principles into a separate
science. They are excellent principles of the highest value, but they are in no
sense the necessary premises that must be proved before any other propositions
of cardinal numbers can be established. On the contrary, the premises of
arithmetic can be put in other forms, and, furthermore, an indefinite number of
propositions of arithmetic can be proved directly from logical principles
without mentioning them. Thus, while arithmetic may be defined as that branch of
deductive reasoning concerning classes and relations that is concerned with the
establishment of propositions concerning cardinal numbers, the introduction of
cardinal numbers makes no great break in this general science. It is merely a
subdivision in a general theory. |
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We must first
understand what is meant by "order," that is, by "serial
arrangement." An order of a set of things is to be sought in that relation
holding between members of the set that constitutes that order. The set viewed
as a class has many orders. Thus the telegraph posts along a certain road have a
space-order very obvious to our senses; but they have also a time-order
according to dates of erection, perhaps more important to the postal authorities
who replace them after fixed intervals. A set of cardinal numbers has an order
of magnitude, often called the order of the set because of its insistent
obviousness to us; if they are the numbers drawn in a lottery, their time-order
of occurrence in that drawing also ranges them in an order of some importance.
Thus the order is defined by the "serial" relation. A relation (R) is
serial when (1) it implies diversity, so that, if x has the relation R to
y, x is diverse from y; (2) it is transitive, so that if x
has the relation R to y, and y to z, then x has the
relation R to z; (3) it has the property of connexity, so that if x
and y are things to which any things bear the relation R, or which bear
the relation R to any things, then either x is identical with y, or
x has the relation R to y, or y has the relation R to x.
These conditions are necessary and sufficient to secure that our ordinary
ideas of "preceding" and "succeeding" hold in respect to the
relation R. The "field" of the relation R is the class of things
ranged in order by it. Two relations R and R' are said to be ordinally similar,
if a one-one relation holds between the members of the two fields of R and R',
such that if x and y are any two members of the field of R, such
that x has the relation R to y, and if x' and y' are
the correlates in the field of R' of x and y, then in all such
cases x' has the relation R' to y', and conversely, interchanging
the dashes on the letters; i.e., R and R', x and x', etc.
It is evident that the ordinal similarity of two relations implies the cardinal
similarity of their fields, but not conversely. Also, two relations need not be
serial in order to be ordinally similar; but if one is serial, so is the other.
The relationship-number of a relation is the class whose members are all those
relations that are ordinarily similar to it. This class will include the
original relation itself. The relation-number of a relation should be compared
with the cardinal number of a class. When a relation is serial its
relation-number is often called its serial type. The addition and multiplication
of two relation-numbers is defined by taking two relations R and S, such that
(1) their fields have no terms in common; (2) their relation-numbers are the two
relation-numbers in question, and then by defining by reference to R and S two
other suitable relations whose relation-numbers are defined to be respectively
the sum and product of the relation-numbers in question. We need not consider
the details of this process. Now if n be any finite cardinal number, it
can be proved that the class of those serial relations, which have a field whose
cardinal number is n, is a relation-number. This relation-number is the ordinal
number corresponding to n; let it be symbolized by n.
Thus, corresponding to the cardinal numbers 2, 3, 4 . . . there are the ordinal
numbers 2., 3., 4.. . . . The definition of the ordinal number 1 requires some
little ingenuity owing to the fact that no serial relation can have a field
whose cardinal number is 1; but we must omit here the explanation of the
process. The ordinal number 0. is the class whose sole member is the null
relation--that is, the relation that never holds between any pair of entities.
The definitions of the finite ordinals can be expressed without use of the
corresponding cardinals, so there is no essential priority of cardinals to
ordinals. Here also it can be seen that the science of the finite ordinals is
merely a subdivision of the general theory of classes and relations. |
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Owing to the
correspondence between the finite cardinals and the finite ordinals, the
propositions of cardinal arithmetic and ordinal arithmetic correspond point by
point. But the definition of the cardinal number of a class applies when the
class is not finite, and it can be proved that there are different infinite
cardinal numbers, and that there is a least infinite cardinal, now usually
denoted by [Hebrew transliteration follows]A[End Hebrew transliteration]0,
where [Hebrew transliteration follows]A[End Hebrew transliteration] is the
Hebrew letter aleph. Similarly, a class of serial relations, called well-ordered
serial relations, can be defined, such that their corresponding
relation-numbers include the ordinary finite ordinals, but also include
relation-numbers which have many properties like those of the finite ordinals,
though the fields of the relations belonging to them are not finite. These
relation-numbers are the infinite ordinal numbers. The arithmetic of the
infinite cardinals does not correspond to that of the infinite ordinals. It will
suffice to mention here that Peano's fourth premise of arithmetic does not hold
for infinite cardinals or for infinite ordinals. Contrasting the above
definitions of number, cardinals and ordinals, with the alternative theory that
number is an ultimate idea incapable of definition, we find that our procedure
exacts greater attention and less credulity. (see also transfinite
number, finite set)
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Rational
numbers and real numbers in
general can now be defined according to the same general method. If m and
n are finite cardinal numbers, the rational number m/n is
the relation that any finite cardinal number x bears to any finite
cardinal number y when n ?x = m ?i>y. Thus the
rational number one, which we will denote by 1r, is not the
cardinal number 1; for 1r is the relation 1/1 as defined
above, and is thus a relation holding between certain pairs of cardinals.
Similarly, the other rational integers must be distinguished from the
corresponding cardinals. The arithmetic of rational numbers is now established
by means of appropriate definitions, which indicate the entities meant by the
operations of addition and multiplication. But in order to obtain general
enunciations of theorems without exceptional cases, mathematicians employ
entities of ever-ascending types of elaboration. These entities are not created
but are employed by mathematicians, and their definitions should show the
construction of the new entities in terms of the old. The real numbers,
including irrational numbers, have now to be defined. Consider the serial
arrangement of the rationals in their order of magnitude. A real number is a
class (
, say)
of rational numbers that satisfies the condition that it is the same as the
class of those rationals each of which precedes at least one member of
. Thus,
consider the class of rationals less than 2r; any member of
this class precedes some other members of the class--thus 1/2 precedes 4/3, 3/2
and so on; also the class of predecessors of predecessors of 2r
is itself the class of predecessors of 2r. Accordingly this
class is a real number; it will be called the real number 2R. Note
that the class of rationals less than or equal to 2r is not a
real number. For 2r is not a predecessor of some member of the
class. In the above example 2R is an integral real number, which is
distinct from a rational integer, and from a cardinal number. Similarly, any
rational real number is distinct from the corresponding rational number. But now
the irrational real numbers have all made their appearance. For example, the
class of rationals whose squares are less than 2r satisfies
the definition of a real number; it is the real number {sqroot 2}. The
arithmetic of real numbers follows from appropriate definitions of the
operations of addition and multiplication. Except for the immediate purposes of
an explanation, such as the above, it is unnecessary for mathematicians to have
separate symbols, such as 2, 2r, and 2R, or 2/3 and
(2/3)R. Real numbers with signs (+ or -) are now defined. If a
is a real number, +a is defined to be the relation that any real number
of the form x + a bears to the real number x, and -a
is the relation that any real number x bears to the real number x
+ a. The addition and multiplication of these "signed" real
numbers is suitably defined, and it is proved that the usual arithmetic of such
numbers follows. Finally, we reach a complex number of the nth order.
Such a number is a "one-many" relation which relates n signed
real numbers (or n algebraic complex numbers when they are already
defined by this procedure) to the n cardinal numbers 1, 2, . . . n
respectively. If such a complex number is written (as usual) in the form x1e1
+ x2e2 + . . . + xnen,
then this particular complex number relates x1 to 1, x2
to 2, . . . xn to n. Also the "unit" e1
(or es) considered as a number of the system is merely a
shortened form for the complex number (+ 1)e1 + 0e2
. . . + 0en. This last number exemplifies the fact that one
signed real number, such as 0, may be correlated to many of the n
cardinals, such as 2 . . . n in the example, but that each cardinal is
only correlated with one signed number. Hence the relation has been called above
"one-many." The sum of two complex numbers x1e1
+ x2e2 + . . . + xnen
and y1e1 +y2e2
+ . . . + ynen is always defined to be the complex
number (x1 + y1)e1 + (x2
+ y2)e2 + . . . + (xn + yn)en.
But an indefinite number of definitions of the product of two complex numbers
yield interesting results. Each definition gives rise to a corresponding algebra
of higher complex numbers. We will confine ourselves here to algebraic complex
numbers--that is, to complex numbers of the second order taken in connection
with that definition of multiplication that leads to ordinary algebra. The
product of two complex numbers of the second order--namely, x1e1
+ x2e2 and y1e1
+ y2e2, is in this case defined to mean the
complex (x1y1 + x2y2)
e1 + (x1y2 + x2y
1)e2. Thus e1 ?i>e1
= e1, e2 ?e2 = -e1,
e1 ?i>e2 = e2 ?i>e1
= e2. With this definition it is usual to omit the first
symbol e1, and to write i or {sqroot -1} instead of e2.
Accordingly, the typical form for such a complex number is x + yi,
and then with this notation the above-mentioned definition of multiplication is
invariably adopted. The importance of this algebra arises from the fact that in
terms of such complex numbers with this definition of multiplication the utmost
generality of expression, to the exclusion of exceptional cases, can be obtained
for theorems that occur in analogous forms, but complicated with exceptional
cases, in the algebras of real numbers and of signed real numbers. This is
exactly the same reason as that which has led mathematicians to work with signed
real numbers in preference to real numbers, and with real numbers in preference
to rational numbers. |
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It has now become
apparent that the traditional field of mathematics in the province of discrete
and continuous number can only be separated from the general abstract theory of
classes and relations by a wavering and indeterminate line. Of course a
discussion as to the mere application of a word degenerates into the most
fruitless logomachy. But on the assumption that "mathematics" is to
denote a science well marked out by its subject matter and its methods, and that
at least it is to include all topics habitually assigned to it,
"mathematics" is employed in the general sense of the "science
concerned with the logical deduction of consequences from the general premises
of all reasoning." |
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The typical
mathematical proposition is: "If x, y, z . . . satisfy such and such
conditions, then such and such other conditions hold with respect to them."
By taking fixed conditions for the hypothesis of such a proposition a definite
department of mathematics is marked out. For example, geometry is such a
department. The "axioms" of geometry are the fixed conditions that
occur in the hypotheses of the geometrical propositions. It is sufficient to
observe here that they are concerned with special types of classes of classes
and of classes of relations, and that the connection of geometry with number and
magnitude is in no way an essential part of the foundation of the science. |
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We now must deduce
the general properties of classes and relations from the ultimate logical
premises. In the course of this process, some contradictions have become
apparent. That first discovered is known as Burali-Forti's
contradiction and consists in the proof that there both is and is not a greatest
infinite ordinal number. But these contradictions do not depend upon any theory
of number, for Russell's contradiction does not involve number in any form. This
contradiction arises from considering the class possessing as members all
classes that are not members of themselves. Call this class w; then to
say that x is a w is equivalent to saying that x is not an x.
Accordingly, to say that w is a w is equivalent to saying that w
is not a w. An analogous contradiction can be found for relations. It
follows that a careful scrutiny of the very idea of classes and relations is
required. Note that classes are here required in extension, so that the class of
human beings and the class of rational featherless bipeds are identical;
similarly for relations, which are to be determined by the entities related. Now
a class in respect to its components is many. In what sense then can it be one?
This problem of "the one and the many" has been discussed continuously
by the philosophers. All the contradictions can be avoided, and yet the use of
classes and relations can be preserved as required by mathematics, and indeed by
common sense, by a theory that denies to a class--or relation--existence or
being in any sense in which the entities composing it--or related by it--exist.
Thus, to say that a pen is an entity and the class of pens is an entity is
merely a play upon the word "entity"; the second sense of
"entity" (if any) is indeed derived from the first but has a more
complex signification. Consider an incomplete proposition, incomplete in the
sense that some entity that ought to be involved in it is represented by an
undetermined x, which may stand for any entity. Call it a propositional
function; and, if
 x
be a propositional function, the undetermined variable x is the argument.
Two propositional functions
x
and
 x
are "extensionally identical" if any determination of x in
x
that converts
x
into a true proposition also converts
x
into a true proposition, and conversely for
and
. Now
consider a propositional function F
 in which
the variable argument
is itself
a propositional function. If F
is true
when, and only when,
is
determined to be either
or some
other propositional function extensionally equivalent to
, then
the proposition F
is of
the form which is ordinarily recognized as being about the class determined by
x
taken in extension--that is, the class of entities for which
x
is a true proposition when x is determined to be any one of them. A
similar theory holds for relations that arise from the consideration of
propositional functions with two or more variable arguments. It is then possible
to define by a parallel elaboration what is meant by classes of classes, classes
of relations, relations between classes, and so on. Accordingly, the number of a
class of relations can be defined, or of a class of classes, and so on. This
theory is in effect a theory of the use of classes and relations and does
not decide the philosophic question as to the sense (if any) in which a class in
extension is one entity. It does indeed deny that it is an entity in the sense
in which one of its members is an entity. Accordingly, it is a fallacy for any
determination of x to consider "x is an x" or
"x is not an x" as having the meaning of propositions.
Note that for any determination of x, "x is an x"
and "x is not an x" are neither of them fallacies but
are both meaningless, according to this theory. Thus Russell's contradiction
vanishes, and the other contradictions vanish also. (see also Russell's
paradox)
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8.1.3.1
Selection of topics.
The selection of
the topics of mathematical inquiry among the infinite variety open to it has
been guided by the useful applications, and indeed the abstract theory has only
recently been disentangled from the empirical elements connected with these
applications. For example, the application of the theory of cardinal numbers to
classes of physical entities involves in practice some process of counting. It
is only recently that the succession of processes that is involved in any
act of counting has been seen to be irrelevant to the idea of number. Indeed, it
is only by experience that we can know that any definite process of counting
will give the true cardinal number of some class of entities. It is perfectly
possible to imagine a universe in which any act of counting by a being in it
annihilates some members of the class counted during the time and only during
the time of its continuance. A legend of the Council of Nicaea illustrates this
point: "When the Bishops took their places on their thrones, they were 318;
when they rose up to be called over, it appeared that they were 319; so that
they never could make the number come right, and whenever they approached the
last of the series, he immediately turned into the likeness of his next
neighbour." Such a story cannot be disproved by deductive reasoning from
the premises of abstract logic. We can only assert that a universe in which such
things are liable to happen on a large scale is unfitted for the practical
application of the theory of cardinal numbers. The application of the theory of
real numbers to physical quantities involves analogous considerations. In the
first place, some physical process of addition is presupposed, involving some
inductively inferred law of permanence during that process. Thus in the theory
of masses we must know that two pounds of lead when put together will
counterbalance in the scales two pounds of sugar, or a pound of lead and a pound
of sugar. Furthermore, the sort of continuity of the series (in order of
magnitude) of rational numbers is known to be different from that of the series
of real numbers. Indeed, mathematicians now reserve "continuity" as
the term for the latter kind of continuity; the mere property of having an
infinite number of terms between any two terms is called
"compactness." The compactness of the series of rational numbers is
consistent with quasi-gaps in it--that is, with the possible absence of limits
to classes in it. Thus the class of rational numbers whose squares are less than
2 has no upper limit among the rational numbers. But among the real numbers all
classes have limits. Now, owing to the necessary inexactness of measurement, it
is impossible to discriminate directly whether any kind of continuous physical
quantity possesses the compactness of the series of rationals or the continuity
of the series of real numbers. In calculations the latter hypothesis is made
because of its mathematical simplicity. But the assumption has certainly no a
priori grounds in its favour, and it is not very easy to see how to base it upon
experience. For example, the continuity of space apparently rests upon sheer
assumption unsupported by any a priori or experimental grounds. Thus the current
application of mathematics to the analysis of phenomena can be justified by no a
priori necessity. |
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In one sense there
is no science of applied mathematics. When once the fixed conditions that any
hypothetical group of entities are to satisfy have been precisely formulated,
the deduction of the further propositions, which also will hold respecting them,
can proceed in complete independence of the question as to whether or not any
such group of entities can be found in the world of phenomena. Thus rational
mechanics, based on the Newtonian Laws and viewed as mathematics, is independent
of its supposed application, and hydrodynamics remains a coherent and respected
science though it is extremely improbable that any perfect fluid exists in the
physical world. But this unbendingly logical point of view cannot be the last
word upon the matter. For no one can doubt the essential difference between
characteristic treatises upon "pure" and "applied"
mathematics. The difference is a difference in method. In pure mathematics the
hypotheses that a set of entities are to satisfy are given, and a group of
interesting deductions are sought. In "applied mathematics" the
"deductions" are given in the shape of the experimental evidence of
natural science, and the hypotheses from which the "deductions" can be
deduced are sought. Accordingly, every treatise on applied mathematics, properly
so-called, is directed to the criticism of the "laws" from which the
reasoning starts, or to a suggestion of results that experiment may hope to
find. Thus if it calculates the result of some experiment, it is not the
experimentalist's well-attested results that are on their trial but the basis of
the calculation. (A.N.W.) |
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