topics: |  mathematics | philosophy |

This book, originally written in 1911, remains a very valuable read for anyone interested in mathematics, particularly for those who are trying to tell others about why they should study mathematics.

It also describes very lucidly the relation between mathematics and logic.

Excerpts

Why mathematics seems hard, but need not be

The study of mathematics is apt to commence in disappointment. The important applications of the science, the theoretical interest of its ideas, and the logical rigour of its methods, aU generate the expectation of a speedy introduction to processes of interest.

We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet's father, this great science eludes the efforts of our mental weapons to grasp it - " 'Tis here, 'tis there, 'tis gone'"...

The reason for this failure of the science to live up to its reputation is that its fundamental ideas are not explained to the student disentangled from the technical procedure which has been invented to facilitate their exact presentation in particular instances. Accordingly, the unfortunate learner finds himself struggling to acquire a knowledge of a mass of details which are not illuminated by any general conception.

The object of the following chapters is not to teach mathematics, but to enable students from the very beginning of their course to know what the science is about, and why it is necessarily the foundation of exact thought as applied to natural phenomena. p.8

The idea of "any" and "some" - variables and equations

The ideas of any and of aome are introduced into algebra by the use of
letters, instead of the definite numbers of arithmetic.  Thus, instead of
saying that 2+3 = 3+2, in algebra we generalize and say that, if x and y
stand for any two numbers, then x+y =y+x.
... 
it was not till within the last few years that it has been realized how
fundamental any and some are to the very nature of mathematics, with the
result of opening out still further subjects for mathematical exploration. 16

Let us now make some simple algebraic statements, with the object of
understanding exactly how these fundamental ideas occur.

	(1) For any number x, x+2 =2+x; 
	(2) For some number x, x+2 = 3;   [one number only]
	(3) For some number x, x+2 > 3.   [infinite set of numbers]

It is natural to supersede the last two statements 
	(2') For what number x is  x+2 = 3;
	(3') For what numbers x is x+2 > 3. 

Equations are of great importance in mathematics, and it seems as though (2')
exemplified a much more thorough-going and fundamental idea than the
original statement (2). This, however, is a complete mistake. The idea of the
undetermined "variable" as occurring in the use of use of "some" or "any" is
the really important one in mathematics; that of the "unknown" in an
equation, which is to be solved as quickly as possible, is only of
subordinate use, though of course it is very important.

Relations between variables

But the majority of interesting formulae, especially when the idea of some is
present, involve more than one variable. For example, the consideration of
the pairs of numbers x and y (fractional or integral) which satisfy x+u = 1
involves the idea of two correlated variables, x and y. When two variables
are present the same two main types of statement occur. For example,

	(1) for any pair of numbers, x and y, x+y =y+x,  and 
	(2) for some pairs of numbers, x and y,  x+y = 1. 

The second type of statement invites con- sideration of the aggregate of
pairs of num- bers which are bound together by some fixed relation-in the
case given, by the relation x+y = 1. One use of formulae of the first type,
true for any pair of numbers, is that by them formuLe of the second type can
be thrown into an indefinite number of equivalent forms. For example, the
relation x +y=1 is equivalent to the relations

	y+x = 1;  (x-y) +2y = 1; 6x+6y =6, 

and so on. Thus a skilful mathematician uses that equivalent form of the
relation under consideration which is most convenient for his immediate
purpose.

Again there is another important point to be noticed. If we restrict
ourselves to positive numbers, integral or fractional, in considering the
relation x+y = 1, then if either x or y be greater than 1, there is no
positive number which the other can assume so as to satisfy the relation.
[AM: Again, if we restrict x and y to the natural integers [>0], then, there
are no values to satisfy the relation. ]

Thus the "field" of the relation for x [is of utmost importance.]

Given the relation y^2 = x, if y is an integer, postive or negative, ...
the "field" for x is restricted in two ways. In the first place x must be
positive, and in the second place, since y is to be integral, x must be a
perfect square. Accordingly, the "field" of x is restricted to the set of
integers 1, 4, 9, 16, and so on. p.20

---About the Hindu numeral system--
The interesting point to notice is the admirable illustration which this
numeral system affords of the enormous importance of a good notation.

By relieving the brain of all unnecessary work, a good notation sets it free
to concentrate on more advanced problems, and in effect increases the mental
power of the race. p.59

The relevance of symbols

If anyone doubts the utility of symbols, let him write out in full, without
any symbol whatever, the whole meaning of some equations [e.g.]:

		z+y=y+z (1) 

Without symbols, (1) becomes: If a second number be added to any given number
the result is the same as if the first given number had been added to the
second number.  

This example shows that, by the aid of symbolism, we can make transitions in
reasoning almost mechanically by the eye, which otherwise would call into
play the higher faculties of the brain.

It is a profoundly erroneous truism,repeated by all copy-books and by eminent
people when they are making speeches, that we should cultivate the habit of
thinking of what we are doing. The precise opposite is the case.
Civilization advances by extending the number of important operations
which we can perform without thinking about them. Operations of thought
are like cavalry charges in a battle - they are strictly limited in number,
they require fresh horses, and must only be made at decisive moments. p.61


--bio

Alfred North Whitehead, who began his career as a mathematician, ranks as the
foremost philosopher in the twentieth century to construct a speculative
system of philosophical cosmology. After his graduation from Cambridge
University, he lectured there until 1910 on mathematics. Like Bertrand
Russell (see also Vol. 5), his most brilliant pupil, Whitehead viewed
philosophy at the start from the standpoint of mathematics, and, with
Russell, he wrote Principia Mathematica (1910--13). This work established the
derivation of mathematics from logical foundations and has transformed the
philosophical discipline of logic. 

From his work on mathematics and its logical foundations, Whitehead
proceeded to what has been regarded as the second phase of his career. In
1910 he left Cambridge for the University of London, where he lectured
until he was appointed professor of applied mathematics at the Imperial
College of Science and Technology. During his period in London, Whitehead
produced works on the epistemological and metaphysical principles of
science. The major works of this period are An Enquiry Concerning the
Principles of Natural Knowledge (1919), The Concept of Nature (1920), and
The Principles of Relativity (1922). In 1924, at age 63, Whitehead retired
from his position at the Imperial College and accepted an appointment as
professor of philosophy at Harvard University, where he began his most
creative period in speculative philosophy. In Science and the Modern World
(1925) he explored the history of the development of science, examining its
foundations in categories of philosophical import, and remarked that with
the revolutions in biology and physics in the nineteenth and early
twentieth centuries a revision of these categories was in order. Whitehead
unveiled his proposals for a new list of categories supporting a
comprehensive philosophical cosmology in Process and Reality (1929), a work
hailed as the greatest expression of process philosophy and
theology. Adventures of Ideas (1933) is an essay in the philosophy of
culture; it centers on what Whitehead considered the key ideas that have
shaped Western culture.

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