[G.Polya]
Terms, old and new, Describing the activity of solving
problems are often ambiguous. The activity itself is
familiar to everybody and it is often discussed but, as
other mental activities, it is difficult to describe. In the
absence of a systematic study there are no technical terms
to describe it, and certain usual half technical terms
often add to the confusion because they are used in different
meanings by different authors.
The following short list includes a few new terms used
and a few old terms avoided in the present study, and
also some old terms retained despite their ambiguity.
The reader may be confused by the following
discussion of terminology unless his notions are well anchored
in examples.
1. analysis is neatly defined by
PAPPUS and it is a useful term
describing a typical way of devising a plan,
starting from the unknown (or the conclusion) and
working backwards, toward the data (or the hypothesis).
Unfortunately the word has acquired very different
meanings (for instance, of mathematical, chemical, logical
analysis) and therefor it is regretfully avoided in
the present study.
2. Condition links the unknown of a "problem to
find" to the data (see
PROBLEMS TO FIND, PROBLEMS TO PROVE,3) In this meaning it is clear,
useful and unavoidable term. It is often necessary to decompose the
condition into several parts [into parts (I) and (II) in the
examples DECOMPOSING
AND RECOMBINING?,7,]. Now, this ambiguity which is sometimes
embarrassing could be easily avoided by introducing some technical
term to denote the parts of the whole condition; for instance,
such a part could be called a "clause."
3. Hypothesis denotes an essential part of a mathematical
theorem of the more usual kind(see PROBLEMS TO FIND, PROBLEMS TO PROVE, 4). The term, in this meaning,
is perfectly clear and satisfactory. The difficulty is that
each part of the Hypothesis is also called a hypothesis so
that the hypopthesis mat consist of several hypotheses
The remedy would be to call each part of the whole
hypothesis a "clause," or something similar. (Compare
the foregoing remark on "condition.")
4. Principal partsPROBLEMS TO FIND, PROBLEMS TO PROVE,3,4.)
5. Problem to find, problem to prove are a pair of new
terms, introduced regretfully to replace historical terms
whose meaning, however, is confused beyond redemption
by current usage. In Latin versions of Greek mathematical
texts, the common name for both kinds of problems
is "proposition" a "problem to find" is called "problem"
and a "problem to prove" "theorem" In old-fashioned
mathematical language, the words proposition, problem
theorem have still this "Euclidean" meaning, gut this is
completely changed in modern mathematical language;
this justifies the introduction of new terms.
6. Progressive reasoning was used in various meanings
by various authors, and in the old meaning of "synthesis"
(see 9)by some authors. The latter usage is defensible
but the terms avoided here.
7. Regressive reasoning was used in the old meaning of
"analysis" by some authors (compare 1,6). The term is
defensible but avoided here.
8. Solution is a completely clear term if taken in its
purely mathematical meaning; it denotes any object satis-
fying the condition of a "problem to find." Thus, the
solutions of the equation x^2-3x+2=0 are its roots,
the numbers 1 and 2. Unfortunately, the word has also
other meanings which are not purely mathematical and
which are used by mathematical along with its mathematical
meaning. Solution may also mean the "process of
solving the problem or the "work done in solving the
problem"; we use the word in this meaning when we talk
about a "difficult solution." Solution may also mean the
result of the world done in solving the problem; we may
use the word in this meaning when we talk about a
"beautiful solution. Now, it may happen that we have to
talk in the same sentence about the object satisfying the
condition of the problem, about the work obtaining it
and about the result of this work; if we yield to the
temptation to call all three things "solution" the sentence
cannot be too clear.
9. Synthesis is used by PAPPUS
in a well defined meaning which would deserve to be conserved,
The term is however, regretfully avoided in the present study,
for the same reasons as its counterpart "analysis" (see under 1).