Only the mediocre are supremely confident of their ability. The better you are, the higher the standards you set yourself—you can see beyond your immediate reach.
— Sir Michael Atiyah. The Princeton Companion to Mathematics, §VIII.6 Advice to a Young Mathematician (pdf).
You want a proof. I guess that means that you want to be more convinced that the Propositional Calculus is consistent than you are convinced of your own sanity. Any proof I could think of would involve mental operations of a greater complexity than anything in the Propositional Calculus itself. So what would it prove? Your desire for a proof of consistency of the Propositional Calculus makes me think of someone who is learning English and insists on being given a dictionary which defines all the simple words in terms of complicated ones…
—
The character of Imprudence in the dialogue between Prudence and Imprudence in Chapter Seven of Douglas Hofstadter’s Gödel, Escher, Bach.
This criticism is very spot-on.
All real problems are word problems.
— at least it’s an ethos… - how does higher education actually apply?
Proof of 1 + 1 = 2 in Principia Mathematica.
“The above proposition is occasionally useful.”
“From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2.” Principia Mathematica theorem 54-43 - Wikipedia
Up All Night, Doin’ Proofs
Techniques thus far:
- Mathematical induction ✔
- Reductio ad absurdum ✔
- Direct proof by contraposition ✔
- Proof by cases ✔
- Constructive proof ✔
- Vigorous handwaving ✔
- Direct proof by definition
- Bidirectional proof
- Proof by counterexample
This Theory of Computation homework is sheer horror.
The Infinities In Between (via CellarAcademic)
Unexpected hanging paradox - Wikipedia →
A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day. Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the “surprise hanging” can’t be on a Friday, as if he hasn’t been hanged by Thursday, there is only one day left - and so it won’t be a surprise if he’s hanged on a Friday. Since the judge’s sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday. He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn’t been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all. The next week, the executioner knocks on the prisoner’s door at noon on Wednesday — which, despite all the above, will still be an utter surprise to him. Everything the judge said has come true.
Like all good myths, the Mersenne prime cryptography myth is so widespread because it is so close to being true. The most widely-used form of encryption used on the internet is RSA encryption, which works by multiplying two huge prime numbers together to form an even larger number with exactly two prime factors. Since factoring numbers is believed to be computationally difficult, reversing this process is currently a very difficult problem, which leads to RSA providing reasonably strong encryption. The thing is, RSA typically uses primes that have a few hundred digits, not a few million digits.
— Nathaniel Johnston » No, Primes with Millions of Digits Are Not Useful for Cryptography
