exponential growth and geometric progression

Here we find a rare opportunity to spank the authorities. Fowler's discussion of geometric and arithmetic progressions as "popularized technicalities", retained nearly verbatim* by Gowers in the second edition, supports Bill Walsh's observation that "word people are notorious for being uneducated in matters of mathematics." Burchfield, as always, only makes matters worse in his so-called The New Fowler's ... Third Edition. Although the topic is omitted by Follett from his updated list of popularized technicalities, it reappears in Wensberg's edition under the now more fashionable title exponential growth, but again, the treatment is unsatisfactory.

A continuously increasing quantity is said to exhibit exponential growth if the ratio of its rate of growth to the quantity itself is a constant. If such a quantity, Q, is considered as a function of time, t, and the constant ratio is denoted by r, then this property may be expressed as a differential equation that admits the unique solution Q(t) = aert, where a = Q(0). Perhaps with the benefit of a semester of calculus to motivate the terminology, Wensberg would not have taken quite such a suspicious view of the expression exponential growth, which he classifies as "mathematical jargon".

A geometric progression is a sequence in which the ratio of successive terms is a constant. Any set of periodic measurements of an exponentially increasing quantity constitutes a geometric progression. For example, the values of the quantity Q defined above corresponding to t=0,1,2,3,... form the geometric progression a,ak,ak2,ak3,...., where k = er. Thus, the notions of exponential growth and geometric progression are, respectively, continuous and discrete versions of the same phenomenon.

A similar relationship holds between linear growth and arithmetic progression . A continuously increasing quantity is said to grow linearly if its rate of change is constant. In an arithmetic progression, the difference between successive terms is constant. If Q grows linearly at a rate r from an initial value Q(0) = a, then its value at time t is given by Q(t) = a + rt. In particular, its values for t=0,1,2,3,... form the arithmetic progression a,a+r,a+2r,a+3r,....

The usage issues regarding these terms that are addressed by Fowler et al. stem from a general confusion over their implications with respect to relative rates of increase. On the subject of exponential growth, Wensberg offers this absurd claim:

Both the rate and magnitude of such change are usually beyond imagining.
A surefire get-rich-quick scheme, it would follow, is to invest in a savings account, at pretty much any rate of compound interest.

In his article on geometric and arithmetic progressions, Fowler expresses the opposite view:

These are in constant demand to express a rapid rate of increase, which is not involved in either of them, and is not even suggested by a.p. Those who use the expressions should bear in mind ... that every rate of increase that could be named is slower than some rates of a.p. and of g.p., and faster than some others, and consequently ... that the phrases `better than a.p., than g.p.', `almost in a.p., g.p.', are wholly meaningless.
As an illustration of the common and supposedly erroneous assumption that a geometric progression necessarily increases faster than an arithmetic progression, Fowler quotes the following unhappy corollary of Malthus's doctrine on population and subsistence:
The healthy portion of the population is increasing by a.p., and the feeble-minded by g.p.
Burchfield's commentary on this subject is noteworthy, not so much for its muddled logic, which is to be expected, but because it is a rare instance in which he essentially agrees with the author of the work that he pretends to be editing (but then, it is also rare to find Fowler so far from the mark):
Both expressions, but especially geometric progression, in popular use tend to be employed to suggest a rapid rate of increase. But both terms are relative. If the rate of increase is very small both sequences can be used to indicate a relatively slow rate of increase, e.g. (arithmetical) 10,000, 10,001, 10,002, 10,003, etc; (geometrical) .00001, .00002, .00004, .00008, etc.
What is meant here by "a relatively slow rate of increase"? If the intention is to describe a progression in which the increment is small in comparison to the terms of the sequence, as strongly suggested by the first example, then by no means does the second example fit the description.

The source of all this confusion is that the usage experts have conspired to ignore the question of what it means to say that a quantity increases rapidly, or that one grows faster than another. According to standard mathematical terminology, given two increasing functions of a common independent variable, one is considered to increase faster than the other if the ratio of the second to the first approaches zero as the independent variable increases without bound. For example, if Q1(t) = a1er1t and Q2(t) = a2 + r2t, where a1, r1, a2, and r2 are positive constants, it may be shown that the ratio Q1(t)/Q2(t) approaches zero as t approaches infinity, regardless of the choices of constants. Thus, according to this notion, every exponential function (or geometric progression) increases faster than every linear function (or arithmetic progression). In fact, the same is true if the second function is replaced by Q2(t) = a2 + r2tn, for any fixed n, a result expressed by the familiar statement that exponential growth is faster than polynomial growth.

This is a reasonable definition from the perspective of pure mathematics, as it guarantees that if a function increases faster than another, then the value of the first exceeds that of the second everywhere except on a finite interval of the domain. Moreover, it is a notion that has proved to be consistently relevant in practical applications. In the field of computing, for example, the complexity of an algorithm is characterized by a function that expresses the time required for its execution in terms of the size of the input data (the latter being the independent variable in this case). The effectiveness of an algorithm is naturally determined by its performance in the solution of relatively large problems. Hence, competing algorithms may be usefully compared in the limit of increasingly large input data. By convention, an algorithm that may be executed in polynomial time is considered to be tractable; one that requires exponential time is intractable.

Similarly, demographic analysis is generally concerned with patterns of population growth that are exhibited over a long period. In particular, the startling mathematical consequence of Malthus's observation, as quoted above, which seems to have escaped Fowler, is that regardless of the unspecified rates of growth, the entire human population will eventually be dominated by the feeble-minded.

Since these concepts are essentially mathematical in nature, any of the authorities cited would have done better to consult someone educated in the exact sciences. After all, if a mathematician were concerned with a matter properly belonging to the realm of English usage, would he attempt to resolve it on his own or have the good sense to defer to qualified experts?