Signs of progress..  As Columbus and his companions
sailed westward across an unknown ocean they were
cheered whenever they saw birds.  They regarded a bird
as a favourable sign, indicating the nearness of land. But
in this they were repeatedly disappointed.  They watched
for other signs too.  They thought that floating seaweed
or low banks of cloud might indicate land, but they were
again disappointed.  One day, however, the signs multi-
plied.  On Thursday, the 11th of October, 1492, "they saw
sandpipers, and a green reed near the ship.  Those of the 
caravel Pinta saw a crane and a pole, and they took up 
another small pole which appeared to have been worked
by iron; also another bit of cane, a land-plant and a 
small board.  The crew of the caravel Nina also saw signs
of land, and a small branch covered with berries.  Every-
one breathed afresh and rejoiced at these signs."  And in
fact the next day they sighted land, the first island of a 
New World.
     Our undertaking may be important or unimportant,
our problem of any kind--when we are working in-
tensely, we watch eagerly for signs of progress as 
Columbus and his companions watched for signs of approaching 
land. We shall discuss a few examples in order to 
understand what can be reasonably regarded as a sign of
approaching the solution.
1. Examples. I have a chess problem.  I have to mate
the black king in, say, two moves.  On the chessboard
there is a white knight, quite a distance from the black
king, that is apparently superfluous.  What is it good for?
I am obliged to leave this question unanswered at first.
Yet after various trials, I hit upon a new move and observe
that it would bring an apparently superfluous
white knight into play.  This observation gives me a new
hope. I regard it as a favourable sign: that new move has
some chance to be the right one. Why?
     In a well-constructed chess problem there is no 
superfluous piece.  Therefor, we have to take into account all
chessmen on the board; we have to use all the data. The
correct solution does certainly use all the pieces, even 
that apparently superfluous white knight.  In this last
respect, the new move that I contemplate agrees with the
correct move that I am supposed to find. The new move
looks like the correct move; it might be the correct
move. 
     It is interesting to consider a similar situation in a 
mathematical problem. My task is to express the area of a 
triangle in terms of its three sides, a, b, and c. I have
already made some sort of plan.  I know, more or less
clearly, which geometrical connections I have to perform.
     Yet I am not quite sure whether my plan will work.
If now, proceeding along the line prescribed by my plan,
I observe that the quantity

     sqrt(b+c-a)

enters into the expression of the area I am about to construct,
I have good reason to be cheered.  Why?
     In fact, it must be taken into account that the sum of
any two sides of a triangle is greater than the third side.
This involves a certain restriction. The given lengths, 
a, b, and c cannot be quite arbitrary; for instance, b+c
must be greater than a.  This is an essential part of the
condition, and we should use the whole condition.  If
b+c is not greater than a the formula I seek is bound
to become illusory. Now, the square root displayed above
becomes imaginary if b+c-a is negative--that is, if
b+c is less than a--and so the square root becomes unfit
to represent a real quantity under just those circumstances
under which the desired expression is bound to
become illusory. Thus my formula, into which that
square root  enters, has an important property in common
with the true formula for the area.  formula
looks like the true formula; it might be true 
formula.
    Here is one more example. Some time ago, I wished to
prove a theorem in solid geometry. Without much trouble I found a
first remark that appeared to be pertinent;
but then I got stuck. something was missing to
finish the proof. When I gave up that day I had a much
clearer notion than at the outset how the proof should
look, how the gap should be filled; but I was not ale to
fill it. The next day, after a good night's rest, I looked
again in the question and soon hit upon an analogous 
theorem in plane geometry. In a flash I was convinced that 
now I had got hold of the solution and I had, I think, good reason
to be convinced, why?
    In fact, analogy is a great guide. The solution of a 
problem in solid geometry often depends on an analogous
problem in plane geometry(see ANALOGY 370.
Thus, in my case, there was a chance from the on set that
the desired proof would use a lemma from some theorem 
of plane geometry the kind of which actually came to my
mind.  This theorem looks like the lemma we need; it
might be the lemma I need" ... such was my reasoning.
    If Columbus and his men had taken the trouble to reason explicitly
they would have reasoned in some similar way. They knew how the sea looks 
near the shore.  They knew that, more often than on the open sea, there
there are birds in the air,  coming from the land 
and objects floating in the water, detached from the seashore.
Many of the men have observed such things when from former
voyages they had returned to their home port. The day before that memorable 
date which they sighted the land of San Salvador, as the floating objects
in the water became so frequent , they thought, "it looks 
as if we were approaching some land, and and "everyone 
breathed fresh air and rejoiced at these signs."

 2. Heuristic character of signs of progress let us insist upon a 
point which is perhaps already clear to everyone
but it is very important and, therefor it should be completely
clear. 
     The type of reasoning illustrated by the foregoing 
example deserves to be enticed and taken into account seriously,
although it yields only a plausible indication and no an unfailing
certainty. Let us restate pedantically, at full length, in rather 
unnatural detail, one of these reasonings:

If we are approaching land, we often see birds.
Now we see birds
therefor we are approaching land

     Without the word "probably" the conclusion would
be in outright fallacy.  In fact Columbus and his
companions saw birds many times but were disappointed
later. Just once came the day on which the saw sand-pipers
followed by the day of discovery.
    With the word "probably" the conclusion is reasonable
and natural, but by no means a proof, demonstrative 
conclusion; it is only an indication of a heuristic suggestion. 
It would be a great mistake to forget that such a conclusion
is only probable and to regard it as certain but to 
disregard such conclusions entirely would
be a still great mistake. If you take a heuristic conclusion
as certain you may be fooled and disappointed; but
if you neglect heuristic conclusions altogether you will
make no progress at all. The most important signs of
progress are heuristic.  Should we trust them? Should 
we follow them? follow, but keep your eyes open. Trust but
look. and never renounce your judgement.
3.  Clearly expressible signs can look at the foregoing
examples from another point of view.
    In one of these examples, we regarded as a favourable
sign that we succeeded in bringing in to play a datum not
used before. (the white knight). we were quite right to 
so regard it. In fact, to solve a problem is,essentially to 
find the connection between the data and the unknown 
moreover we should at least in well stated problems use
all the data, connect each of them with the unknown.
Thus, bringing one more datum into play is quite 
properly felt as progress, as a step forward.
    In another example, we regarded as a favourable sign
that an essential clause of the condition was appropriately
taken into account by our formula. We were quite
right to regard it. In fact, we should Use the whole
condition thus, taking into account one more clause of 
the condition I justly felt as progress, as a move in the 
right direction.
    In still another example, we regard as a favourable sign 
the emergence of a simpler analogous problem. This is
also justified. Indeed, analogy is one of the main
sources of invention. If other means fail, we should try
to imagine an analogous problem Therefor, if such a problem 
emerges spontaneously, by its own accord we 
naturally feel elated-we feel that we are approaching the
solution.
    After these examples, we cannot easily grasp the general
idea. There are certain mental operations typically 
useful in solving problems.  (the most usual operation
of this kind are listed in this book. If such a typical
operation succeeds (if one more datum is connected with
the unknown --- one more clause of the condition is taken
into account -a simpler analogous problem is introduced ) 
its success is felt as a sign of progress. Having 
understood this essential point, we can express with some 
clearness the nature of still other signs of progress.  All
we have to do is to read down our list and look at the
various questions and suggestions from our newly acquired
 point of view.
   Thus, understanding clearly the nature of the un-
known means progress. Clearly disposing the various data
so that we can easily recall any one also means progress.
Visualising vividly the condition as a whole may mean an
essential advance; and separating the condition into appropriate
 parts may be an important step forward. When
we have found a figure that we can easily imagine, or a 
notation that we can easily retain, we can reasonably 
believe that we have made some progress.  Recalling a 
problem related to ours and solved before may be a decisive
 move in the right direction.
     And so on, and so forth. To each mental operation
clearly conceived corresponds a certain sign clearly 
expressible. Our list, appropriately read, lists also
signs of progress.
     Now, the questions and suggestions of our list are 
simple, obvious, just plain common sense. This has been 
said repeatedly and the same can be said of the 
connected signs of progress we discuss here. To read such
signs no occult science is needed, only a little common
sense and, of course, a little experience.
4.    Less clearly expressible signs  When we work 
intently, we feel keenly the pace of our progress: when it is
rapid we are elated; when it is slow we are depressed. We
feel such differences quite clearly without being able to 
point out any distinct sign. Moods, feelings, general 
aspects of the situation serve to indicate our progress.
They are not easy to express. "It looks good to me," or
"It is not so good," say the unsophisticated. More 
sophisticated people express themselves with some nuance:
"This is a well-balanced plan," or "No, something is still
lacking and that spoils the harmony."  Yet behind 
primitive or vague expressions there is an unmistakable feeling
which we follow with confidence and which leads us
frequently in the right direction. If such feeling is very 
strong and emerges suddenly, we speak of inspiration.
People usually cannot doubt their inspirations and are 
sometimes fooled by them. In fact, we should treat guiding
 feelings and inspirations just as we treat the more 
clearly expressible signs of progress which we have 
considered before.  Trust, but keep your eyes open.
      Always follow your inspiration--with a grain of doubt.
[What is the nature of those guiding feelings?  Is there
some less vague meaning behind words of such aesthetic
nuances as "well-balanced," or "harmonious"?  These
questions may be more speculative than practical, but
the present context indicates answers which perhaps deserve
to be stated: Since the more clearly expressible
signs of progress are connected with the success or failure 
of certain rather definite mental operations, we may 
suspect that our less clearly expressible guiding feelings
may be similarity connected with other, more obscure, 
mental activities-perhaps with activities whose nature
is more "psychological" and less "logical."]
5.     How signs help  I have a plan. I see pretty clearly
where I should begin and which steps I should take first. 
Yet I do not quite see the lay-out of the road farther on;
I am not quite certain that my plan will work; and, in
any case, I have still a long way to go. Therefor, I start
out cautiously in the direction indicated by my plan and
keep a lookout for signs of progress. If the signs are rare
or indistinct, I become more hesitant. And if for a long
time they fail to appear altogether, I may lose courage, 
turn back, and try another road. On the other hand, if 
the signs become more frequent as I proceed, if they
multiply, my hesitation fades, my spirits rise, and I move
with increasing confidence, just as Columbus and his
companions did before sighting the island of San 
Salvador.
     Signs may guide our acts. Their absence may warn us
of a blind alley and save us time and useless exertion;
their presence may cause us to concentrate our effort
upon the right spot.
      Yet signs may also be deceptive. I once abandoned a 
certain path for lack of signs, but a man who came after 
me and followed the path a little farther made an important
 discovery-to my great annoyance and long-lasting-regret.
He not only had more perseverance than I did but he also
read correctly a certain sign which I had failed to notice.
 Again, I may follow a road cheerfully, encouraged by
favourable signs, and run against an unsuspected and
insurmountable obstacle.
     Yes, signs may misguide us in any single case, but they 
guide us right in the majority of them. A hunter may
misinterpret now and then the traces of his game but he
must be right on the average, otherwise he could not
make a living by hunting.
     It takes experience to interpret the signs correctly.
Some of Columbus's companions certainly knew by experience
 how the sea looks near the shore and so they were able to
 read the signs which suggested that they  were approaching
land. The expert knows by experience how the situation looks
 and feels when the solution is near and so he is able to 
read the signs which indicate that he is approaching it. 
The expert knows more signs than the inexperienced, and he knows
them better; his  main advantage may consist in such knowledge. 
An expert hunter notices traces of game and appraises even
their freshness or staleness where the inexperienced one
is unable to see anything.
    The main advantage of the exceptionally talented may
consist in a sort of extraordinary mental sensibility.  With
exquisite sensibility, he feels subtle signs of progress or
notices their absence where the less talented are unable
to perceive a difference.
[6.   Heuristic syllogism.  In section 2 we came across a 
mode of heuristic reasoning that deserves further consideration 
and a technical term.  We begin by restating that 
reasoning in the following form: 

     If we are approaching land, we often see birds.
     Now we see birds.
-------------------------------------------------------
     Therefor, it becomes more credible that we are ap-
     proaching land.

     The two statements above the horizontal line may be 
called the premises, the statement under the line, the 
conclusion. And the whole pattern of reasoning may be 
termed a heuristic syllogism. 
     The premises are stated here in the same form as in 
section 2, but the conclusion is more carefully worded.
An essential circumstance is better emphasised. Colum-
bus and his men conjectured from the beginning that
they would eventually find land sailing westward; and 
they must have given some credence to this conjecture, 
otherwise they would not have started out at all. As they
proceeded, they related every incident, major or minor, 
to their dominating question: "Are we approaching
land?"  Their confidence rose and fell as events occurred
or failed to occur, and each man's beliefs fluctuated more
or less differently according to his background and character.
 The whole dramatic tension of the voyage is due to
such fluctuations of confidence.
      The heuristic syllogism quoted exhibits a reasonable
ground for a change in the level of confidence. To occasion
 such changes is the essential role of this kind of 
reasoning and this point is better expressed by the wording
 given here than by the one in section 2.
      The general pattern suggested by our example can be 
exhibited thus:
 
     If A is true, then B is also true, as we know.
     Now, it turns out that B is true.
-------------------------------------------------------
     Therefore, A becomes more credible.
 
Still shorter:
    
     If A then B
     B true
     ------------
    A more credible

In this schematic statement the horizontal line stands for 
the word "therefore" and expresses the implication, the 
essential link between the premises and the conclusion.]
[7.  Nature of plausible reasoning.  In this little book 
we are discussing a philosophical question. We discuss it
as practically and informally and as far from high-brow
modes of expression as we can, but nevertheless our
subject is philosophical. It is concerned with the nature 
of heuristic reasoning and, by extension, with a kind of 
reasoning which is non-demonstrative although important
and which we shall call, for lack of a better term, 
plausible reasoning.
     The signs that convince the inventor that his idea is 
good, the indications that guide us in our everyday
affairs, the circumstantial evidence of the lawyer, the inductive
 evidence of the scientist, statistical evidence
invoked in many and diverse subjects-all these kinds of 
evidence agree in two essential points. First, they do not
have the certainty of a strict demonstration. Second, they
are useful in acquiring essentially new knowledge, and
even indispensable to any not purely mathematical or 
logical knowledge, to any knowledge concerned with the 
physical world. We could call the reasoning that underlies
this kind of evidence "heuristic reasoning" or "inductive 
reasoning" or (if we wish to avoid stretching the 
meaning of existing terms) "plausible reasoning." We
accept here the last term.
     The heuristic syllogism introduced in the foregoing
may be regarded as the simplest and most widespread
pattern of plausible reasoning.  It reminds us of a classical
pattern of demonstrative reasoning, of the so-called
"modus tollens of hypothetical syllogism."  We exhibit
here both patterns side by side:

     Demonstrative                  Heuristic
       If A then B                     If A then B
         B false                         B true
      --------------                  ---------------
         A false                        A more credible

The comparison of these patterns may be instructive.  It 
may grant us an insight, not easily obtainable elsewhere, 
into the nature of plausible (heuristic, inductive) reas-
soning. 
     Both patterns have the same first premise:
                 If A then B.
 
     They differ in the second premise. The statements:
 
     B false                           B true

     are exactly opposite to each other but they are of "similar
logical nature," they are on the same "logical level."
The great difference arises after the premises. The conclusions
 
     A false                           A more credible

are on different logical levels and t heir relations to their
respective premises are of a different logical nature.
     The conclusion of the demonstrative syllogism is of the 
same logical nature as the premises. Moreover, this conclusion
is fully expressed and is fully supported by the premises. If
my neighbour and I agree to accept the premises, we cannot
reasonably disagree about accepting also the conclusion,
however different our tastes or other convictions may be.
    The conclusion of the heuristic syllogism differs from 
the premises in its logical nature; it is more vague, not so
sharp, less fully expressed.  This conclusion is comparable
to a force, has direction and magnitude. It pushes us in a 
certain direction: A becomes more credible. The conclusion 
is not fully expressed and is not fully supported by 
the premises.  The direction is expressed and is implied
by the premises, the magnitude is not For any reasonable
person, the premises involve that A becomes more credible
(certainly not less credible) Yet my neighbour and I 
can honestly disagree how much more credible A becomes, 
since our temperaments, our backgrounds, and our unstated 
reasons may be different.
     In the demonstrative syllogism the premises constitute
a full basis on which the conclusion rests. If both 
premises stand, the conclusion stands too. If we receive some
new information that does not change our belief in the 
premises, it cannot change our belief in the conclusion. 

    In the heuristic syllogism the premises constitute only 
one part of the basismathematics and plausible reasoning.]
    Heuristic reasons are important although they prove
nothing. To clarify our heuristic reasons is also important
 although behind any reason clarified there are many 
others that remain obscure and are perhaps still more 
important.