Here's my solution. A statement is a grouping of symbols(which is what words are) and can, itself, be treated as a symbol. Symbols are used to refer to something other than themselves, but when you have the statement refer to itself, you get a paradox. When it refers to itself it becomes infinitely interpretable causing "This statement is false." to cycle between True and False for as long as you try to interpret it.
Just a minor point:
"when you have the statement refer to itself, you get a paradox" is not necessarily true. For example, the statement "this statement has five words" is self-referential and true. No paradox. Even a self-referential statement that includes its own truth value can be non-paradoxical: "This statement is true and has two words" is merely false.
By the way, this leads me to consider Prior's resolution as somewhat problematic:
"This statement is true and has eight words" "This statement has eight words"
The first statement is true and the second false, hence they cannot be equivalent. Nevertheless, adding "This statement is true and " to any statement should not change the statement's truth value if we accept that every statement implicitly states its own truth.
I do not like "steep learning curve" the way people use it. It raises my probability estimate that person using it has done no study of learning. This reduces my probability estimate that they have sound insight about learning.
Sometimes I work with learning curves for machine learning or human learning. These curves are plots of a measure of learning (say, correct score, or number of mistakes) versus time or number of trials or number of training examples or number of iterations. When topic is hard, curve is shallow! It takes more learning to improve score or reduce mistakes. Steep learning curve means topic is easy. Mastery comes with very few repeats or little time or little data.
Does anyone know why so many use it wrong, to mean topic is hard and progress is slow? I believe Wikipedia article is right and "learning curves" must come from scientific study. How did meaning get reversed? Wikipedia article says "Arguably, the common English use is due to metaphorical interpretation of the curve as a hill to climb." but has no citation.
That's what I understand by a "learning curve" too. I would tend to agree with the wikipedia proposed explanation: "steep" is, in some contexts, synonymous with "difficult". Saying "you have to climb a shallow learning curve" would certainly be interpreted wrongly by most people.
Concerning the OP question, I wish I had learnt something interesting at school, instead of the thoroughly irrelevant, utterly boring, mindboggingly wrong pseudoknowledge.
To zedzed and D_Malik: I think what you have in mind is learning to play an instrument as a profession. I that case I agree that the rewards are probably not commensurate to the effort, unless you are exceedingly talented and lucky. But for everybody else it's probably a good idea. If done for its entertainment value it's surely worth it, and it helps you understand and enjoy music at a deeper level. On the minus side, it also makes you realize that most popular music sucks. Really really sucks, to the point of irritating me.
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I think (it's not my post) that it's supposed to be evidence that numbers are real because the set of all possible minds is vast. Because there are so many possible minds, it's unlikely that a mind chosen randomly from that set would have similar intuitions about numbers to mine. It's even more unlikely that a third mind would also have those intuitions. Yet, for some reason, this vastly unlikely thing happens anyway. This implies that there is some reason which is responsible for all the minds feeling the same way. One such reason would be "the intuition is correct, and the minds have correctly figured it out".
(Other possible reasons could be "all the minds are biased in the same way" or some other reason unrelated to truth of the idea, but nobody's saying it's proof, it's just evidence.)
Though the number of possible minds is vast, I think the likelihood of two minds sharing an intuitive concept of number is high, because minds (or perhaps I should say consciousnesses) process information sequentially. Perhaps it is akin to the shared perception of rhythm, which is not limited to human minds. I suppose you have already seen it, but this video is amazing: the dancing cockattoo.