[Phys-l] trends in "significant figures" ... or, the Structure of Scientific Devolutions

[John Denker <jsd@av8n.com>]

I made a small effort to look into 
 a) the history of "significant figures" in general, and
 b) the abuse of "significant figures" as a way of representing 
  the uncertainty associated with a number.

Here's what set me off:

> Once upon a time, back before the 1970's, almost all scientific
> calculations were done by using a slide rule.  A slide rule could
> only display three significant digits, sometimes, rarely four.  So
> all calculations were automatically rounded off to three significant
> digits.  The invention of the electronic calculator changed all that.
> Calculators can easily display many – perhaps eight to ten –
> significant digits.  Realistically though, it would be extremely rare
> to have data that justifies that many digits in the answer.
>
> So, its largely the calculator that is responsible for the emphasis
> on significant digits in science classes.
  From: http://www.digipac.ca/chemical/sigfigs/contents.htm

That just cracks me up.  Almost every point it makes is wrong.

People have been doing calculations to high accuracy for a long
time, and have been using the term "significant figures" in
non-abusive ways since 1500 if not earlier, although the meaning 
has drifted significantly over the years.  Reference:
  David Eugene Smith
  _History of mathematics_
  Courier Dover Publications (1958)
  volume II page 15ff
  courtesy of Google Books.


In the Subject: line I referred to scientific devolution, because
it appears that the community is getting dumber as time goes on.
Abuse is becoming widespread.

So the timeline of non-abusive significant figures, as far as I 
can tell, includes the following:
 -- 1500 : Use of the term;  details unclear at the moment.
 -- 1831 : Non-abusive use of the term.  W. & M. Yale.
 -- 1892 : Non-abusive use of the term.  Holman.
 -- 1970 : Clear exposition of methods that work.  Acton.

So far so good.

In parallel with that, there is a long list of what I consider 
"old school" physics books to which I give an "incomplete" grade, 
i.e. an indeterminate score of 0/0, because they say little or
nothing about uncertainty at all.  This is IMHO greatly preferable 
to saying wrong things about "significant figures".  In this 
category are:
 -- 1906 : Millikan and Gale
 -- 1949 : Sears and Zemansky
 -- 1960 : Halliday and Resnick
 -- 1960 : PSSC _Physics_
 -- 1964 : _The Feynman Lectures on Physics_
 -- 1960 : PSSC _Physics_
 -- 1973 : Berkeley Physics series

All dates refer to the 1st edition.

I find it peculiar, to say the least, that the people that think
the notion of "significant figures" is important think it is
very, very important ... yet there is a long string of IMHO
pretty respectable physics authors who didn't think it was
worth mentioning.

The bad news is that lots of recent texts promote the abuse of
"significant figures", to the exclusion of any methods that might 
actually make sense.  The problem of course is not with the sig 
figs themselves, but rather with the abuse of sig figs, namely 
the doctrine that you must keep rounding off until the roundoff 
error dominates (or is at least comparable to) the overall 
uncertainty.

I have not yet figured out exactly when / where / how people 
lost their grasp of the distinction between roundoff error and 
overall uncertainty.  Some stars in the hall of shame include:

 -- 1922 : Hoyt, Bareuther, Chamberlain
 -- 1969 : Bevington
 -- 1988 : Segal
 -- 1988 : Giancoli

Status report:  I don't know for sure, but my working hypothesis
goes like this:
 a) The mathematical community has known how to do things right 
  for hundreds of years, and this extends into the present-day 
  computer science / numerical analysis communities.
 b) Meanwhile, says the hypothesis, abusive ideas grew up somewhere
  else, perhaps in some engineering fields.
 c) Over time the abusive ideas have started to leak into the physics 
  and chemistry texts.

Also, it seems clear that the community is fractured.  In some
experiments described at
  http://www.ncsu.edu/PER/Articles/DeardorffDissertation.pdf
at one institution, 11% of the students reported the uncertainty
separately and explicitly, while at another nearby institution 73% 
did.  On the other hand, there are so many things in that document 
that don't add up that it is hard to trust any of it.  On the third
hand, the 73%/11% ratio is consistent with my personal anecdotal 
observations, so I'm gonna run with it, at least for now.

So ... If anybody has any better data, or any better theories, I'd 
be interested to hear it.

Also !!! If anybody knows of an introductory-level physics text that
has something sensible to say about uncertainty, I would very much
like to hear about it.

In case you are wondering what I consider sensible, see
  http://www.av8n.com/physics/uncertainty.htm



==========================

Since I was thinking about historical trends, I decided to look
at recent trends, using
  http://www.google.com/trends?q="significant+figures";
I was impressed by how spiky the curve was.  It appears that the
main spike corresponds to the start of the school year, and the
secondary spike corresponds to the start of the second semester.
There is a summer trough and a Christmas trough.

This is not quite the spikiest trendline I have ever seen;  you
can for instance try something obviously seasonal such as:
  http://www.google.com/trends?q="jingle+bells";

However, the point remains that I consider "significant figures"
to be academic and artificial, and if you compare that with a 
non-academic non-artificial topic such as "turbocharger" the 
contrast is striking:
  http://www.google.com/trends?q=turbocharger,+"significant+figures";