He goes on: "Mathematicians and sometimes physicists prefer proofs that are beautiful.
But assumptions aren't good enough for mathematicians, who prefer proofs, and have long been looking for a proof that settles that NP problems will never yield to a fast solution.
Nadelmann cited infamous bootlegger Al Capone, who preferred higher proof spirits — which often bring more of a health risk.
Bolzano definitely preferred direct proofs, because in indirect proofs the "false conclusion could never have been produced if all the premises from which we derive it were true" (Bolzano 2004, 78).
According to Hafner, Bolzano prefers direct proofs for two reasons: 1) indirect proofs proceed by a detour [Umweg] and contain redundant premises, 2) indirect proofs always contain a false premise which cannot be admitted as the ground of other truths.
By this method one can find elementary proofs of sentences formulated in the language of that theory at least in principle, since, as Brumfiel remarks, "it certainly might be very tedious, if not physically impossible, to work out this elementary proof" (166) Brumfiel prefers a third proof method which aims at giving non-transcendental proofs of purely algebraic results.
"Well, we need to triple-seal airport security, like Tylenol," he said, preferring tamper-proof cockpit doors and armed marshals or police officers aboard all flights.
Insurance is costly and judges prefer to see visual proof of a motorist's damage claims.
On the other hand, we prefer to keep the proofs for the sake of completeness.
We first give a proof of Theorem 3 for d = 2, 3 and then we give another proof which is valid for all d ≥ 2. We could have given only the second proof but we prefer to give them both as the first is simpler (but breaks down at d = 4 ).
