An Introduction to Mathematics · Alfred North Whitehead
In the previous chapter we pointed out
that lengths are measurable in terms of some unit length, areas in term of a unit area, and volumes in terms of a unit volume.
When we have a set of things such as lengths which are measurable in terms of any one of them, we say that they are quantities of the same kind. Thus lengths are quantities of the same kind, so are areas, and so are volumes. But an area is not a quantity of the same kind as a length, nor is it of the same kind as a volume. Let us think a little more on what is meant by being measurable, taking lengths as an example.
Lengths are measured by the foot-rule. By transporting the foot-rule from place to place we judge of the equality of lengths. Again, three adjacent lengths, each of one foot, form one whole length of three feet. Thus to measure lengths we have to determine the equality of lengths and the addition of lengths. When some test has been applied, such as the transporting of a foot-rule, we say that the lengths are equal; and when some process
has been applied, so as to secure lengths being contiguous and not overlapping, we say that the lengths have been added to form one whole length. But we cannot arbitrarily take any test as the test of equality and any process as the process of addition. The results of operations of addition and of judgments of equality must be in accordance with certain preconceived conditions. For example, the addition of two greater lengths must yield a length greater than that yielded by the addition of two smaller lengths. These preconceived conditions when accurately formulated may be called axioms of quantity. The only question as to their truth or falsehood
which can arise is whether, when the axioms are satisfied, we necessarily get what ordinary people call quantities. If we do not, then the name "axioms of quantity" is ill-judged--that is all.
These axioms of quantity are entirely abstract, just as are the mathematical properties of space. They are the same for all quantities, and they presuppose no special mode of perception. The ideas associated with the notion of quantity are the means by which a continuum like a line, an area, or a volume can be split up into definite parts. Then these parts are counted; so that numbers can be used to determine the exact properties of a continuous whole.
Our perception of the flow of time and of
the succession of events is a chief example of the application of these ideas of quantity. We measure time (as has been said in considering periodicity) by the repetition of similar events--the burning of successive inches of a uniform candle, the rotation of the earth relatively to the fixed stars, the rotation of the hands of a clock are all examples of such repetitions. Events of these types take the place of the foot-rule in relation to lengths. It is not necessary to assume that events of any one of these types are exactly equal in duration at each recurrence. What is necessary is that a rule should be known which will enable us to express the relative durations of, say, two examples of some type. For example, we may if we like suppose that the rate of the earth's rotation is decreasing, so that each day is longer than the preceding by some minute fraction of a second. Such a rule enables us to compare the length of any day with that of any other day. But what is essential is that one series of repetitions, such as successive days, should be taken as the standard series; and, if the various events of that series are not taken as of equal duration, that a rule should be stated which regulates the duration to be assigned to each day in terms of the duration of any other day.
What then are the requisites which such a rule ought to have? In the first place it should lead to the assignment of nearly equal durations to events which common sense judges to possess equal durations. A rule which made days of violently different lengths, and which made the speeds of apparently similar operations vary utterly out of proportion to the apparent minuteness of their differences, would never do. Hence the first requisite is general agreement with common sense. But this is not sufficient absolutely to determine the rule, for common sense is a rough observer and very easily satisfied. The next requisite is that minute adjustments of the rule should be so made as to allow of the simplest possible statements of the laws of nature. For example, astronomers tell us that the earth's rotation is slowing down, so that each day gains in length by some inconceivably minute fraction of a second. Their only reason for their assertion (as stated more fully in the discussion of periodicity) is that without it they would have to abandon the Newtonian laws of motion. In order to keep
the laws of motion simple, they alter the measure of time. This is a perfectly legitimate procedure so long as it is thoroughly understood.
What has been said above about the abstract nature of the mathematical properties
of space applies with appropriate verbal changes to the mathematical properties of time. A sense of the flux of time accompanies all our sensations and perceptions, and practically all that interests us in regard to time can be paralleled by the abstract mathematical properties which we ascribe to it. Conversely what has been said about the two requisites for the rule by which we determine the length of the day, also applies to the rule for determining the length of a yard measure--namely, the yard measure appears to retain the same length as it moves about. Accordingly, any rule must bring out that, apart from minute changes, it does remain of invariable length; Again, the second requisite is this, a definite rule for minute changes shall be stated which allows of the simplest expression of the laws of nature. For example, in accordance with the second requisite the yard measures are supposed to expand and contract with changes of temperature according to the substances which they are made of.
Apart from the facts that our sensations are accompanied with perceptions of locality and of duration, and that lines, areas, volumes, and durations, are each in their way quantities, the theory of numbers would be of very subordinate use in the exploration of the laws of the Universe, 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.
· · ·
End of part 17 of 19
Continue readingNotes
Starvard