On the other hand, Euclid seems to be trying to convince the reader (or someone): “I say that …. therefore …” ]]>

As for “proofs”, this always gives me a chuckle:

“The above proposition is occasionally useful.”

I haven’t read Russell & Whitehead’s Principia, but Russells Introduction to Mathematical Philosophy inspired me to broaden my mathematical thinking.

Best!

]]>(Proof by Appeal to Authority: A U.S. President came up with a proof. Therefore it must be true.)

]]>You can read a proof and conclude, with certainty, “This proof is wrong, and here is why.” But there is no way you can read a proof and say “This is correct.” All a proof can do on the positive side is increase your confidence in the truth of its conclusion. Proofs are evidence that you weigh. Nothing more.

Using computers cannot eliminate this inherent uncertainty. For instance, in 1994 computers with the Intel Pentium 5 chip gave this answer

4,195,835/3,145,727 = 1.333739068902037589

This identity was thus “proved” by a computer. Indeed, it was “proved” by means of an extremely simple argument, namely an arithmetic computation. But a human spotted that it was wrong at the fourth decimal place.

“[I]t makes no difference whether the argument is checked by a human or a machine, we can never achieve 100% certainty that it is logically correct.”

So when you say this, are you making a passing reference to Gödel’s incompleteness theorems–that the consistency of a formal system can’t be checked without going outside the system?

Is this also why you say, “you’ve never seen a proof that truly fits the standard definition”?

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