Re: Exponential growth

From: Stevan Harnad <>
Date: Wed, 8 Nov 2000 09:19:26 +0000

On Tue, 7 Nov 2000, Greg Kuperberg wrote:

> Exponential and linear are examples of mathematical terms whose
> lay connotations have strayed somewhat from their rigorous meanings.
> Many people say "exponentially" when they really mean "quickly", as in
> "journal prices are rising exponentially". If journal prices rose by
> 0.5% per year (for the sake of argument, after adjusted for inflation),
> that would be exponential. But I assume that libraries would prefer
> that to seeing journal prices rise by 10 cents per page per year,
> even though that is linear. If the exponent varies over time, as it
> usually does in the real world, then exponentiation is only a point of
> view and not a predictive law. Any trajectory is exponential with a
> time-dependent exponent.

Never venture to use technical language with a mathematician! Mea
culpa. But nothing whatsoever of substance changes:

My posting was not about journal price-rises. I was talking about the
rate at which the refereed literature is being freed online in the
Physics archive. That rate is linear, and will take us another decade
to free the entire (current and future) corpus in Physics. In other
fields the growth rate is even slower than that, mostly infinitesmal.

Hence what I meant was nothing more precise than a sharp curve upward,
now, so as to free the entire (finite) literature within a year or
sooner, instead of a straight line for another decade in Physics (and
even longer in the rest of the disciplines).

Too ambitious or impatient? History will be the judge. The fact is,
that the means are already in place to free it all almost overnight;
there is no reason not to do it now, and every reason (lost research
impact, lost research accessibility) to do it today (indeed, to have
done it yesterday).

So, no, it is not the loose language of journalists and doom-sayers I
was using, it was the simple language of taking that slow growth curve
and turning it sharply upward.

> Often a system described in exponential language actually follows a
> power law. One common reason is that the system expands first in the
> locales where it can expand quickly, and then later where it expands
> more slowly. For example, HIV/AIDS never spread in the United States
> with a constant exponent; I have heard that the curve of total infections
> was, at the beginning, closer to a cubic law. A more relevant example
> is new submissions per month to the arXiv, whose growth is strikingly
> close to linear:
> It is also germane to call this a power law, because if new submissions
> grow linearly, total submissions grow quadratically. And I suspect
> the usual reason, because the first research areas in the arXiv were
> turbulent ones such as string theory and quantum computation. More sedate
> topics such as enumerative combinatorics and granular materials only came
> much later. I don't see why an alternative model, such as distributed
> interoperability, would be exempt from the general principle.
> Scientifically, then, I can't accept claims that a new standard or a
> new project for e-prints will grow exponentially. Mathematically such
> claims do not entirely imply the intended hype anyway.

Greg, please give me some technically irreproachable words for saying
what I am saying, so we can get on with it. I think you know what I

Stevan Harnad
Professor of Cognitive Science
Department of Electronics and phone: +44 23-80 592-582
             Computer Science fax: +44 23-80 592-865
University of Southampton
Highfield, Southampton

NOTE: A complete archive of the ongoing discussion of providing free
access to the refereed journal literature online is available at the
American Scientist September Forum (98 & 99 & 00):

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Received on Mon Jan 24 2000 - 19:17:43 GMT

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