ABSTRACT
Look at line 5 of the proof. Where does it come from? The answer is, from the logical theorem (or necessary truth): ‘Either p or non-p’, where ‘p’ is called a sentential variable. But how do we get line 5 from this theorem? The answer is, by using the rule of inference known as the “Rule of Substitution for Sentential Variables,” according to which a statement can be derived from another containing such variables by substituting any statement (in this case, ‘y is prime’) for each occurrence of a distinct variable (in this case, the variable ‘p’). The use of these rules and logical theorems is, as we have said, frequently an all but unconscious action. And the analysis that exposes them, even in such relatively simple proofs as Euclid’s, depends upon advances in logical theory made only within the past one hundred years.5 Like Molière’s M.Jourdain, who spoke prose all his life without knowing it, mathematicians have been reasoning for at least two millennia without being aware of all the principles underlying what they were doing. The real nature of the tools of their craft has become evident only within recent times.
