R020700: Mathematics -- To See What Is General in What Is Particular

Whitehead, Alfred North.  An Introduction to Mathematics.  Oxford University Press (New York: 1911, 1948).  12th printing of paperback edition issued 1958.  ISBN 0-19-500211-3.
     I find this material of interest not so much for its mathematical content as for its effort to address just what mathematics is that it is important to us, and how casting abstractions in mathematical terms provides a way to form a general idea of the course of natural events, "to see what is general in what is particular and what is permanent in what is transitory [p.4]."  That is, to abstract the perceived orderliness of nature into mathematical theories that are interpreted to predict the course of events.  I am interested to learn how the very idea of mathematics as a device for marshalling abstract theories separated from particular interpretations was perceived by someone active in the period where this point of view was strengthened so much by revolutionary developments in logic and the mathematical language of science. -- dh:2002-07-05.

Last updated 2002-08-08-09:38 -0700 (pdt)

Notes on the Book

My interest is in the context of theorizing about theories.  I want to see the standing that Whitehead, at the turn of the past century, and afterwards, gives to mathematics as a theory building activity.

     1. The Abstract Nature of Mathematics
     2. Variables
     3. Methods of Application
     4. Dynamics
     5. The Symbolism of Mathematics
     6. Generalizations of Number
     7. Imaginary Numbers
     8. Imaginary Numbers (continued)
     9. Co-ordinate Geometry
     10. Conic Sections
     11. Functions
     12. Periodicity in Nature
     13. Trigonometry
     14. Series
     15. The Differential Calculus
     16. Geometry
     17. Quantity

1. The Abstract Nature of Mathematics

W starts out concerned about being able to disentangle the fundamental ideas of mathematics from the technical procedures which are used among mathematicians.  p.1

Whitehead sees mathematics as a science and "necessarily the foundation of exact thought as applied to natural phenomena."  p.2.

He gets into separation of feelings, emotions, and other subjective aspects of perception, and has mathematics dealing with things just because they are things, p.2.  He seems to suggest that numbers are things, and at least that arithmetic applies to everything: "Of all things it is true that two and two make four."

"It is worth while to spend a little thought in getting at the root reason why mathematics, because of its very abstractness, must always remain one of the most important topics for thought.  Let us try to make clear to ourselves why explanations of the order of events necessarily tend to become mathematical."  p.3.

Observes the orderliness of nature, and interconnection.  p.4.

"The laws satisfied by the course of events in the world of external things are to be described, if possible, in a neutral universal fashion, the same for blind men as for deaf men, and the same for beings with faculties beyond our ken as for normal human beings." p.5.

"Thus it comes about that, step by step, and not realizing the full meaning of the process, mankind has been led to search for a mathematical description of the properties of the universe, because in this way only can a general idea of the course of events be formed, freed from reference to particular persons or to particular types of sensation." p.5

Here it gets interesting.  I would not say we are describing the universe, but the idea of having a theory that stands apart from our individual experience and yet is verifiable intersubjectively would certainly be appropriate.

There seems to be a confusion with reduction, though, also on p.5.  "But in its final analysis, science seeks to describe an apple in terms of the positions and motions of molecules ... ."  I would say, if that is what science is seeking to do, it is remarkably unsuccessful.

To ponder: "This point has usually been misunderstood, from being thought of in too narrow a way.  Pythagoras had a glimpse of it when he proclaimed that number was the source of all things.  In modern times the belief that the ultimate explanation of all things was to be found in Newtonian mechanics was an adumbration of the truth that all science as it grows towards perfection becomes mathematical in its ideas." pp.5-6.

2. Variables

"Mathematics as a science commenced when first someone, probably a Greek, proved propositions about any thing or about some things, without specification of definite particular things."  p.7.

Talks about the use of letters for free variables in algebra.

Talks about the notion of infinity when "we generalize and say that if x be any number there exists some number (or numbers) y such that y > x."  p.7.

There is a tendency to see mathematics as offering descriptions and explanations.  I would assert that mathematics provides neither.

5. The Symbolism of Mathematics

"For our present purposes ... the history of the notation is a detail.  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. ... " p.39

"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."  pp.41-42  -- Lovely!

"The symbolism of mathematics is in truth the outcome of the general ideas which dominate the science.  We have now two such general ideas before us, that of the variable and that of algebraic form."  p.47.

6. Generalizations of Number

Whitehead does the construction on the denumerability of the rationals on pp. 53-54.

Have the index of m/n be m+n.  Then proceed through the indices, starting with 2 (1/1) and enumerate all of the rationals having that index. There are always m+n-1 of them.  Within an index, one can order the rationals, starting with 1/(m+n-1), ..., m/n, ... (m+n-1)/1, ordering by numerator.

He then discusses that Cantor showed the reals are not denumerable.

Explores generality, for example, working with the integral domain rather than the non-negative natural numbers, even though we end up with negative results of subtractions, having no counter part in the world.  This generalization is indispensible, and we have come not to question it.

Whitehead has his particular finesse -- how to look at the operations of arithmetic -- to allow the generalization for the real field, for example.  He describes the difficulties of not doing so:

"Here are difficulties become acute; for this form [x=b-a] can only be used for the numerical interpretation [of a quantity in the world] so long as b is greater than a, and we cannot say without qualification that a and b may be any constants.  ... Really prolonged mathematical investigations would be impossible under such conditions. Every equation would at last be buried under a pile of limitations."  pp.57-58.

Whitehead proposes a different interpretation that has the signed numbers be operations for adding or taking away.  Then x = -3 is fine, because it means it is the taking away of 3.

We can do that or not.  Here is how Whitehead leaves it:

"The idea of positive and negative numbers has been practically the most successful of mathematical subtleties."  p.60.  

After earlier acknowledging the power of introducing 0.

11. Functions

Looks at continuity.  Uses the notion of independent variable and the dependent one.

17. Quantity

At the end,

"As it is, physical science reposes on the main ideas of number, quantity, space, and time.  The mathematical sciences associated with them do not form the whole of mathematics, but they are the substratum of mathematical physics as at present existing."  p.186.

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