A “proof by contradiction” is not a proof that ends with a contradiction

March 4, 2017

It is well-known that constructivists (supposedly) renounce “proof by contradiction”, and that classicists scoff at the critique.  “Those fools,” the criticism goes, “want to rule out proofs by contradiction.  How absurd!  Look, Pythagoras showed that the square root of two is irrational by deriving a contradiction from the assumption that it is rational.  There is nothing wrong with this.  Ignore them!”

On careful examination the critique fails miserably, because a proof by contradiction is not a proof that derives a contradiction.  Pythagoras’s  proof is perfectly valid, one of the eternal gems of mathematics.  No one, but no one, questions the validity of that line of argument.  Despite the righteous fury, no punch has landed.  We may as well laugh at the flat earthers, who are of course dead wrong, but also nonexistent.

Pythagoras’s Theorem expresses a negation: it is not the case that the square root of two can be expressed as the ratio of two integers.  Of course one proves negative statements by contradiction!  Assume that it can be so represented.  A quick deduction shows that this is impossible.  So the assumption is false.  Done.  This is a direct proof of a negative assertion; it is not a “proof by contradiction”, which is also known as an indirect proof.

What, then, is a proof by contradiction?  It is the affirmation of a positive statement by refutation of its denial.  It is a direct proof of the negation of a negative assertion that is then pressed into service as a direct proof of the assertion, which it is not.  Anyone is free to ignore the distinction for the sake of convenience, as a philosophical issue, or as a sly use of “goto” in a proof, but the distinction nevertheless exists and is important.  Indeed, part of the beauty of constructive mathematics is that one can draw such distinctions, for, once drawn, one can selectively disregard them.  Once blurred, forever blurred, a pure loss of expressiveness.

For the sake of clarity, let me rehearse a standard example of a proof by contradiction.  The claim is that there exists irrationals a and b such that a to the b power is rational.  Here is an indirect proof, a true proof by contradiction.  Move number one, let us prove instead that it is impossible that any two irrationals a and b are such that a to the b is irrational.  This is a negative statement, so of course one proves it be deriving a contradiction from it. But it is not the original statement!  This will be clear from examining the information content of the proof.

Suppose, for a contradiction, that every two irrationals a and b are such that a to the b power is irrational.  We know from Pythagoras that root two is irrational, so plug it in for both a and b, and conclude that root two to the root two power is irrational.  Now use the assumption again, taking a to be root two to the root two, and b to be root two.  Calculate a to the power of b, it is two, which is eminently rational.  Contradiction.

We have now proved that it is not the case that every pair of irrationals, when exponentiated, give an irrational.  There is nothing questionable about this proof as far as I am aware.  But it does not prove that there are two irrationals whose exponent is rational!  If you think it does, then I ask you, please name them for me.  That information is not in this proof (there are other proofs that do name them, but that is not relevant for my purposes).  You may, if you wish, disregard the distinction I am drawing, that is your prerogative, and neither I nor anyone has any problem with that.  But you cannot claim that it is a direct proof, it is rather an indirect proof, that proceeds by refuting the negative of the intended assertion.

So why am I writing this?  Because I have learned, to my dismay, that in U.S. computer science departments–of all places!–students are being taught, erroneously, that any proof that derives a contradiction is a “proof by contradiction”.  It is not.  Any proof of a negative must proceed by contradiction.  A proof by contradiction in the long-established sense of the term is, contrarily, an indirect proof of a positive by refutation of the negative.  This distinction is important, even if you want to “mod out” by it in your work, for it is only by drawing the distinction that one can even define the equivalence with which to quotient.

That’s my main point.  But for those who may not be familiar with the distinction between direct and indirect proof, let me take the opportunity to comment on why one might care to draw such a distinction.  It is entirely a matter of intellectual honesty: the information content of the foregoing indirect proof does not fulfill the expectation stated in the theorem.  It is a kind of boast, an overstatement, to claim otherwise.  Compare the original statement with the reformulation used in the proof.  The claim that it is not the case that every pair of irrationals exponentiate to an irrational is uncontroversial.  The proof proves it directly, and there is nothing particularly surprising about it.  One would even wonder why anyone would bother to state it.  Yet the supposedly equivalent claim stated at the outset appears much more fascinating, because most people cannot easily think up an example of two irrationals that exponentiate to rationals.  Nor does the proof provide one. Once, when shown the indirect proof, a student of mine blurted out “oh that’s so cheap.”  Precisely.

Who cares about the information content of proofs?  You may not, but I do.  It matters, at least in computer science, because proofs are programs and propositions are specifications.  You may well not care, as you are well entitled to do.  But you may also find one day that you do care.  For example, Vladimir Voevodsky’s univalence principle, which codifies classical mathematical practices, is only sensible in a constructive framework in which the distinction cannot be disregarded.  Don’t be so sure you don’t care!