Heuristic Reasoning  is reasoning not regarded as final
and strict but as provisional and plausible only, whose 
purpose is to discover the solution of the present problem.
We are often obliged to use heuristic reasoning. We shall
attain complete certainty when we shall have obtained
the complete solution, but before obtaining certainty
we must often be satisfied with a more or less
plausible guess.  We may need the provisional before
we attain the final. We need heuristic reasoning when we 
construct a strict proof as we need scaffolding when we
erect a building. 
     See SIGNS OF PROGRESS 
Heuristic reasoning is often based on induction or on analogy;
see INDUCTION AND MATHEMATICAL INDUCTION  and 
ANALOGY,8,9,10 6(footnote: see also a paper by the author
in American Mathematical Monthly, vol 48, pp 450-465).
     Heuristic reasoning is good in itself. What is bad is 
to mix up heuristic reasoning with rigorous proof.
What is worse is to sell heuristic reasoning for rigorous 
proof.
     The teaching of certain subjects, especially the 
teaching of calculus of engineers of physicists, should be essentially
improved if the nature of heuristic reasoning were
better understood, both its advantages and its limitations
openly recognised, and if the textbooks would present
heuristic arguments openly.  A heuristic argument presented
with taste and frankness may be useful; it may 
prepare for the rigorous argument o which it is usually
contains certain germs. But a heuristic argument is likely
to be harmful if presented ambiguously with visible
hesitation between shame and pretencion. See 
WHY PROOFS?