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!

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“Because we have conscious access to our neo-cortical thought processes, we tend to regard them as “logical,” often dismissing the actions of the amygdala as providing (“mere,” “animal-like”) “instinctive reactions.” But that misses the point that, because that “instinctive reaction organ” has evolved to ensure its owner’s survival in a highly complex and ever changing environment, it does in fact operate in an extremely logical fashion, honed by generations of natural selection pressure to be in synch with its owner’s environment.”

Good point. To be effective, amygdala (which is said to be responsible for some emotional reactions) has to work in a logical fashion consistent with its evolved structure and function. Consistency is the operational word in here. To me to be consistent means being coherent hence logical.

Our emotional reactions appear irrational not because they are illogical, but because we are unable to observe them in action. Monitoring unconscious processes would slow them down and make them useless for the survival of the organism.

It is not fully appreciated by the general public that the purpose of learning and education in general is not different to that of amygdala – we go to school to automate our thoughts to such an extent that we are no longer able to say why something is true or how this something was arrived at. Mathematics is full of such “learned reflexes”. So is our everyday language.

Wes

]]>(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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