From Archimedes to Chris Rust:
Reflections on Heuristic Research
in Physics, Mathematics, Engineering,
and Design

[was: "Kens brief treatise - look at the evidence first."]


Dear Chris,

Thanks for your note (Rust 2003, [see 1, below]). Thanks for the link
and the picture. I appreciate the empirical input. I was inclined to
change my view in the face of empirical evidence until I puzzled on
this more deeply. The nature of my puzzling is that the empirical
evidence involves the shape of Old Joe and not the behavior of the
engineer. Working out the problem of the joke - as distinct from the
problem of working out the height of clock tower - involves the
interaction of engineering, art, and design.

If the engineer (Reese 2003 [see 2, below]) did, indeed, take the
trouble to determine the likelihood of accuracy, he may well have had
an accurate figure. According to the story, he didn't.

"The engineer, relishing no exercise at all, simply went to the
caretaker, gave him the watch, and asked him how high Joe was, then
retired to the Students Union for a pint." There is no description of
any method.

The issue of methodology - the comparative analysis of method -
features prominently in the description of both the physicist and the
mathematician:

"The physicist, knowing that the acceleration due to gravity is
32ft/sec/sec, dropped the weight from the top of the tower several
times, timed its fall with the watch, and thus was able to calculate
the height.

"The mathematician, however, not relishing the prospect of many trips
up the tower to drop the weight, created a simple pendulum using the
string tied to the top of Joe and the weight at the other end by the
ground, and knowing that the periodic swing is a function of its
length and gravity, was able, using the watch, to calculate the
height having made only one trip up the tower."

The challenge involves accounting for the differential between the
tower lip and the upper roof with flagpole. There are ways to account
for the difference.

My view is that the mathematician and physicist can account for
possible error far more reasonably than the engineer. At least, this
is so if the engineer "relishing no exercise at all, simply went to
the caretaker, gave him the watch, and asked him how high Joe was."
The operating phrase in this analysis is "simply." If that means he
did nothing more than the account describes, then the physicist and
mathematician are more likely to reach a decent approximation.

Working out the height of Old Joe would involve applying a scientific
formula to the distance from the ground (a) up to measuring point
(b). Physical or mathematical methods should provide the height of
(b). Then, completing the work would involve accounting for the
distance between (b) and the top of the flagpole (c). This can be
done by determining the distance from (b) to (c) as an approximate
proportion of the distance from (a) to (b).

The solution to full problem would be the results of the formula +/-
a margin of error added to the results of the approximation +/- a
margin of error. Because the margin of error for each of these will
vary slightly, each margin must be accounted for separately before
giving the distance from the ground to the top of flagpole +/- a
margin of error.

There are other ways to get a reasonable approximate height for Old
Joe. These methods render the lip problem irrelevant. This work can
be done using Euclidean geometry. Eratosthenes of Cyrene (284-192
BC), director of the library at Alexandria, used geometrical methods
to compute the radius and polar circumference of the earth.
Eratosthenes's result was 39,690 km as compared with modern values of
about 40,009 km (Lloyd 1973: 48-49). Eratosthenes measured the earth
using Greek stades. Several different lengths of stades were used in
antiquity, and there is no way to know today which value Eratosthenes
used for his computations. Lloyd argues that Eratosthenes used a
stade of 157.5 meters. Mason (1962: 53) gives ten stades to the mile,
making Eratosthenes accurate to within fifty British miles. Other
known values for stades could yield a circumference ranging between
17% too great and 6% too little.

Eratosthenes used simple geometry and the shadow of an obelisk or a
rod of known height for his work. Interestingly, one step in his
calculations involved determining the difference in latitude between
Aswan and Alexandria, for which he derived a figure essentially the
same as our own (Boorstin 1983: 95-96). Everyone agrees that
Eratosthenes's methods were more importance than the specific result.
Whether he was spot-on as Lloyd and Mason believe or 15% over as
Boorstin argues, the method he devised provided the basis for methods
still in use. The main difference involves better sources of
empirical data.

While Eratosthenes did his calculation using empirical data, this did
not require him to The same methods that enabled a Greek
mathematician sitting in Alexandria to determine the circumference of
the earth to within a few hundred kilometers while getting the
latitude essentially right would enable the mathematician to work out
the height of Old Joe.

The physicist could have used this method as well. Many physicists
study Euclidean geometry at some point in their lives. Newton's
Principia (1999 [1687, 1726]) used geometrical proofs rather than
mathematical proofs for many of the propositions on which modern
physics has ever since relied. Most modern physicists also understand
Euclidean geometry and they can perform different kinds of
geometrical calculations in addition to physical calculations. Nobel
laureate Richard Feynman (Goodstein and Goodstein 1997) once
delivered a lecture using geometrical proofs in the manner of Newton.

It is likely the physicist could have used geometry as well as the
mathematician.

For me, the question of critical inquiry is as important as the
result. I want to know how each of these three individuals reached a
result. That enables me to know which of the three answers is most
likely to be reliable.

Some consider these methodological issues to be distractions from the
practical work of engineering and design. I do not. I believe that
the foundation of practical judgment requires a stock of background
information and the ability to apply appropriate theory.

If we are to make up our own minds on important issues in
professional life - or, for that matter, in the larger world - we
must have a basis for reasoned judgment. The mathematician and the
physicist each offer a reasonable basis for their views. In contrast,
the engineer says, "I trust the watchman and you can trust me." There
is no description of any method. His method is a black box.

My problem in trusting the engineer's black box is that his main goal
seems to be avoiding work and getting his pint: "The engineer,
relishing no exercise at all, simply went to the caretaker, gave him
the watch, and asked him how high Joe was, then retired to the
Students Union for a pint."

Practical reasoning and value judgments - I will say more on axiology
another time - requires that we place the problem in context. If our
goal is a quick pint, the engineer's solution is best. If our goal is
good laugh over a pint, all three methods are equivalent. If we are
being asked to stake anything or risk anything on a reasonable
answer, the evidence given in the story suggests that I would do
better to trust the physicist or the engineer.

Let us recast the story in practical terms. Imagine that you are
placed in a situation where you must take a bet and you have no way
to avoid the bet. You have only the evidence of the story to go on.
You have three envelopes among which you must choose. One envelope
contains the solution as worked out by the physicist, one by the
mathematician, and one by the engineer. You must select among three
envelopes knowing whose answer it contains, but you do not know what
the answer is. You will receive a benefit of some kind if you select
the closest answer to the true height among the three. If you select
one of the other two answers, you will be obliged to pay a penalty of
some kind.

If the bonus and penalty are each a pint of beer, it probably will
not matter very much which you choose. If the nonuse and penalty are
a thousand pounds each, you will be far more careful.

Imagine yet another scenario. Imagine that you will find the correct
answer written on a check for ten thousand pounds along with the
closest figure to the true height of Old Joe, but while the other two
choices will be written on ordinary paper.

If you have anything at stake in choosing among these three, the
issue takes on a different perspective.

Can we trust the engineer? Only to the degree that we trust the
watchman. Can we trust the watchman? Who knows? We do not know if the
watchman knows anything at all. We may not even know if the watchman
has any reason to believe that his answer is correct. Even if the
watchman does believe that his answer is correct, however, he may
nevertheless be wrong. The difficulty with so many of these judgments
is the ease with which people overestimate their own likelihood of
having a correct answer (see, for example, Fischhoff, Slovic, and
Lichtenstein 1977).

The problem in this story is not which answer is correct. The problem
is which method is most likely to give the answer closest to the true
height of Old Joe. As I see it, the engineer is least likely for the
simple reason that he relies on a second-hand figure. Since knowing
how correct the watchman's figure is likely to be also requires
undertaking some work to determine the reliability and accuracy of
the watchman. We already know that the engineer relishes no exercise
at all. According to the story, therefore, he simply asked for an
answer without undertaking the different kinds of inquiries or
estimates he would have required if he needed a reason to believe
that the watchman's answer was correct. To be sure, the engineer may
well have used a different method if he could have won ten thousand
pounds with the closest answer or if he were required to pay a
penalty for an answer less accurate than the others derived. He might
just as well have asked the watchman, but he would have undertaken
some form of inquiry or checking estimation to ensure that the
watchman's answers were useable.

In this story, however, only the mathematician and the physicist give
us a reason to believe that they are able to reach a reasonable
approximation. The engineer's answer might come close to the true
height of Old Joe, but it may well be far off. We have no way of
knowing. In contrast, we will be able to estimate the likely
reliability of the other two answers within some reasonable measure.

This is precisely where robust theory informs good practice.

We've all hard the old riddle about whether it is better to have a
clock that is right precisely twice a day or a clock that is always
five minutes off. The physicist and the mathematician are likely to
be five minutes off. Rather like Eratosthenes, they will probably
miss the exact height of Old Joe by a small percentage of the true
height. In contrast, the engineer will only be right to the degree
that the watchman is right, and he has no way to know whether the
watchman is right or how close he is. He may hit the answer spot on
or be reasonably close. Since any way to know how close the
watchman's answer is involves some form of exercise, the engineer has
no way of knowing if he has derived a reasonable answer and he has no
way of knowing the conditions under which he can find out.

I liked Bruce Tharp's story better (Tharp 2003 [see 3, below]) than
the joke from the Telegraph precisely because it illustrates the
circumstances of reasonable pragmatism and useful approximation.
Bruce's story resembles the story my chemistry teacher used to tell
to explain the value of reasoned approximation.

He, too, began his story with the halving problem. He couched it as
an encounter between two sweethearts who could never meet because
they could never do more than halve the distance between them. This
story is best known as the story of Achilles and the tortoise, a
paradox attributed to Zeno of Elea.

According to Zeno, the two sweethearts could never meet. According to
my chemistry teacher, "they could get close enough for all practical
purposes."

As a young skeptic - nullius in verba! - I wanted to test this for
myself. I learned several things from my tests.

First, I discovered the heuristic principle. It is possible to
construct quick experiments to outline the nature of a problem and to
develop sensitizing concepts for further exploration. The kind of
experimenting that scientists often call "quick and dirty" involves
different kinds of probes to develop information. Quick, multiple
approaches helps one to develop information at different levels of
analysis. These provide a useful framework for the next stages of
research.

The second thing I discovered is the cybernetic principle. The method
of approximation works well in projects that include rich empirical
feedback. The feedback cycle permits the researcher to judge whether
the approximation is promising based on empirical data. This, in
turns, provides heuristic guidelines for future probes.

The third thing I learned about was the concept of tolerance limits.
For some kinds of projects, the margin of error may operate at an
entirely different scale than other kinds of projects. The margin of
error for a test to determine whether my sweetheart and I could get
close enough for all practical purposes was far less rigorous than
the margin of error in chemistry class, where I was a notorious
failure. I learned more about philosophy of science from my chemistry
teacher than I learned about chemistry. Either that, or my partners
in experimentation had a higher level of tolerance.

The first three lessons may be usefully applied to all research. This
particular series of experiments yielded two additional lessons.

Lesson four involves replication. In some fields of research, we
replicate to test earlier findings, to verify or falsify data, or to
falsify theory. In some fields of research, we replicate an
experiment as often as possible because we like the way it works. I
continued my applied research into Zeno's paradox long after I had
gathered enough data to reach a conclusion.

Lesson five involves one particularly successful series of
experiments that took place in a bathtub. The term heuristics is
derived from the Greek word, "heuriskein," to discover or to find
out. The famous exclamation - "Eureka!" - that Archimedes shouted as
he leaped from the bath is a variation on this word. What Archimedes
said simply means, "I found it!"

Archimedes is not the only researcher to joyfully shout "Eureka" in
the bath. My research partner shouted "Veni!" at about the same time
in our experiment. She felt that Greek science was too austere,
preferring the Latin classics. That was fine with me. I shared
Archimedes's interest in research on floating bodies and I wanted
someone to disturb my circles.

It is worth adding a few words on heuristics at this point.

Mario Bunge (1999: 120) defines a heuristic as a "nonalgorithmic aid
in problem finding and problem solving. Examples. Finding a few cases
of As being Bs suggests that all As are Bs; an analogy between two
problems suggests using the same method to investigate them; what-for
questions trigger research into biological and social functions.
Heuristic devices belong in the scaffolding of a construction and
they must be discarded after use. Their role is strictly midwifery."
While the heuristic may be nonalgorithmic in essence, heuristics may
also involve theory and embed algorithmic equations and well
understood concepts in their development. A famous case of this is
Einstein's (1905) article opening the way to quantum theory, "On a
Heuristic Point of View Concerning the Production and Transformation
of Light."

In the tale of the physicist, the mathematician, and the engineer,
all three developed a heuristic for solving the problem. The
mathematician and the physicist did not stop with their heuristic
approach. Then, they discarded their heuristic and moved forward to a
satisfactory approximate solution. Since substituting the watchman's
judgment for his own work, the engineer's heuristic method became his
first solution, his method, and his final answer. Lacking any method
for testing the answer, it was his only answer. Rather than building
on his inspired heuristic as a first step toward a better solution,
and then discarding it, the engineer stopped with his first answer
and moved on to a beer.

Determining the reliability of an information source is one central
heuristic in many kinds of research. According to the narrative, the
engineer did not do this. It is possible that he already knew the
watchman as a reliable soul who only offered facts on firm evidence
as contrasted with extreme confidence in wrong facts. If this were
the case, then his answer would have been the best of the three.

My question is: do we know enough to have any reason to believe this to be so?

My answer is, knowing no more than we are told, we do not have any
reason to believe this to be so.

Therefore, I would place my bet on the envelope containing the
answers of either the physicist or the mathematician.

There is, of course, an exception. Had the engineer in this story
been Chris Rust, I would have bet on you. This would also be the case
if you had been the mathematician or the engineer. Since experience
is an excellent heuristic guide to reliability, particularly when
other variables are uncertain, I would feel safe with my bet on the
envelope containing your answer.

I'll buy the first round when next we meet.

Cheers!

Ken



References

Boorstin, Daniel J. 1985. The Discoverers. New York: Random House.

Bunge, Mario. 1999. The Dictionary of Philosophy. Amherst, New York:
Prometheus Books.

Einstein, Albert. 1905. Reprinted in Einstein's Miraculous Year. Five
Papers that Changed the Face of Physics. Edited and introduced by
John Stachel. Princeton, New Jersey: Princeton University Press,
177-198.

Fischoff, Baruch, Paul Slovic, and Sarah Lichtenstein. 1977. Knowing
with Certainty. The Appropriateness of Extreme Confidence." Journal
of Experimental Psychology: Human Perception and Performance, 3,
552-564.

Goodstein, David L., and Judith R. Goodtsein. 1997. Feynman's Lost
Lecture. The Motion of the Planets Around the Sun. London: Vintage.

Lloyd, G. E. R. 1973. Greek Science After Aristotle. New York: W. W.
Norton and Company.

Mason, Stephen F. 1962. A History of the Sciences. New York:
Macmillan USA, Collier Books.

Newton, Isaac. 1999 [1687, 1726]. The Principia. Mathematical
Principles of Natural Philosophy. A New Translation by I. Bernard
Cohen and Anne Whitman assisted by Julia Budenz. Preceded by A Guide
to Newton's Principia by I. Bernard Cohen. Berkeley: University of
California Press.

Reese, R. Allan. 2003. "Subject: Role of engineers." PhD-Design.
Date: Thu, 13 Feb 2003 16:41:26 +0000.

Rust, Chris. 2003. "Subject: Kens brief treatise - look at the
evidence first." PhD-Design. Date: Sat, 15 Feb 2003 17:24:20 +0000.

Tharp, Bruce M. 2003. "Subject: Design Axiology." PhD-Design. Date:
Fri, 14 Feb 2003 19:32:52 -0600



Excerpts

[1]

Chris Rust wrote:

-snip-

The story was an amusing anecdote from a popular newspaper and the
editors, quite reasonably had to compress the story, inevitably
losing some of the detail. So let's look at the possibilities:

First of all, as Ken says, we don't know if the engineer conducted
his enquiries with any rigour, but we can see that he had the
opportunity to do so, and there are many ways in which the caretaker
could have provided him with reliable information about the height of
the tower.

So let's look at the other players, but first I suggest that you have
a quick look at this web page,
http://www.cs.bham.ac.uk/~whe/clock.html which contains an excellent
quality photograph of "Old Joe" the clock tower in question. If
anybody doubts the provenance of the photo they should note that the
web page is maintained by an employee of Birmingham University and,
by way of triangulation, I was a resident of the city of Birmingham
for 10 years and can assure readers that this is the actual tower.

The tower has an elegant, steep-sloping pyramidal roof, topped by a
flagpole or something similar. Now I don't want to labour this point
but, given the equipment available, neither investigator has any hope
of finding the height of the tower using the methods proposed. In
each case they will need to drop their weight, or fasten their
pendulum rope, at a point in space offset from the tip of the
flagpole (or the tip of the roof if you prefer) sufficiently to allow
a clear drop to the ground. This calls for some serious engineering
work (I daresay the engineer thought that through and decided that
there had to be a better way)

So we do not know how well each of the investigators thought through
their plan but we can see that the engineer had every chance of
success while the mathematician and physicist were hopelessly
impractical, even with some serious resources it would have been a
difficult task.

I felt also that, while the engineer was perfectly able to understand
the methods proposed by his colleagues, the other two would be very
unlikely to understand the practical engineering requirements for
executing their plans.

-snip-

[2]

Allan Reese wrote:

-snip-

The continuing debate suggests that list members may be interested
in the following old chestnut that appeared again in the letters column of
the Daily Telegraph this week.

Daily Telegraph Letters

Date: 11 February 2003

SIR -

Steve Devine's elegant solution to the weighing problem (letter, Feb.
10) reminded me of my (very short) time at Birmingham University.

A mathematician, a physicist and an engineer as part of a project
were each given a ball of string, a lead weight and a watch and told
to work out the height of "Joe", the large campus clock tower named
after Joseph Chamberlain.

The physicist, knowing that the acceleration due to gravity is
32ft/sec/sec, dropped the weight from the top of the tower several
times, timed its fall with the watch, and thus was able to calculate
the height.

The mathematician, however, not relishing the prospect of many trips
up the tower to drop the weight, created a simple pendulum using the
string tied to the top of Joe and the weight at the other end by the
ground, and knowing that the periodic swing is a function of its
length and gravity, was able, using the watch, to calculate the
height having made only one trip up the tower.

The engineer, relishing no exercise at all, simply went to the
caretaker, gave him the watch, and asked him how high Joe was, then
retired to the Students Union for a pint.

From:

David Lowe BSc (Engineering - Failed), Malvern, Worcs

-snip-

[3]

Bruce Tharp wrote:

-snip-

In the corners of a large triangular room stand an engineer,
mathematician, and an attractive male or female of a sexual
orientation complimentary to that of the engineer and mathematician
(politically correct sanitization mine). The engineer and
mathematician are told that they can move half way toward the
attractive being. After stopping they are told that they can again
move half way. After stopping again they are told that they may
continue with successive half-way moves. The mathematician stops,
saying that this was a ridiculous venture as he would never actually
reach the attractive being as there are an infinite number half-ways;
he self-righteously leaves. The engineer however agrees to continue
knowing that while not perfect he can get close enough to make it
work.

-snip-