This is one of the finest proofs in mathematics as it completely exposes the non-constructive philosophy of mathematics as propounded by David Hilbert.
Theorem: There exists irrational numbers p and q such that pq is a rational number.
Theorem: There exists irrational numbers p and q such that pq is a rational number.
Proof: Consider the number a = √2 √2 .
Case I:
a is a rational number. We are done. Just take p = √2 and q = √2. Therefore, pq is a rational number.
Case II:
a is an irrational number. In this case, take p = a and q = √2.
Thus, by the laws of arithmetic, pq = (√2 √2)√2 = (√2 ) 2 = 2, a rational number.
QED.
The beauty of this proof is that at the end of the day (or the proof) we really don't know if a is a rational number or an irrational number. Note that this is technically a valid proof without any fallacies.
Mathematicians who harp on constructivism will frown at this proof and the law of excluded middle is the real culprit. For more details about constructivism (and Brower-Hilbert controversy) look into this link.
Note: It turns out that a is indeed an irrational number as a consequence of Gelfand-Schneider Theorem since √2 is an algebraic number (as it is the solution of the polynomial equation x2-2 = 0).
No comments:
Post a Comment