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There is not, and there cannot be. However, even if he was figuring this all out from scratch, he could not have come up with the hand-written charts without r
by escape_goat 10y ago
There is not, and there cannot be. However, even if he was figuring this all out from scratch, he could not have come up with the hand-written charts without realizing that.
I'm going to go out on a limb and guess that it is of some habitual relevance to a mathematician, since he mentions the identity function (not really relevant here either), but that the material he actually explains here doesn't do anything to justify its appearance.
However, I cannot present to you the crucial difference between the maximum of two numbers and the minimum of two numbers that explains what he was thinking. This must exist for my hypothesized explanation to make sense.
- thaumasiotes 10y ago> I'm going to go out on a limb and guess that it is of some habitual relevance to a mathematician, since he mentions the identity function (not really relevant here either), but that the material he actually explains here doesn't do anything to justify its appearance. It's worse than that; he mentions the identity function only to make a gross error about it. The identity function on (a, a) is (a, a), not a.
- coinomega 10y agoI don't think you need to be so unforgiving? If you want extra precision, a -> f(a, a) is Id. Knowing that helped me understand one property of the chart, never said that was the most stunning of properties, and the best articulated one :-)
- thaumasiotes 10y ago> If you want extra precision, a -> f(a, a) is Id. What does this mean?
- coinomega 10y agoThe function which to any integer a associates f(a, a) with f defined as in the article is the identity function on the natural integer set.
- thaumasiotes 10y agoGranted. g(a) = f(a,a) is indeed the identity function for all f such that f(a,a) = a. That seems pretty pointless as an observation. We have f, so we're observing that some other, unrelated function is the identity function?
- coinomega 10y agoAssuming you actually care about the answer, this just help me drew the exploratory graph, and this property was useful to fill in cells for which operands were equal. Finding obvious properties is a great way to learn a pattern without even computing things. Unlike what you may think, I had no idea what I was about to find - I had never seen this chart drawn before, and so this post is structured around how I thought about the exploration, not around how it may be presented to an audience who already knows way more than I do on bitwise operations. Sounds fair?
- thaumasiotes 10y agoSure it's fair. Bitwise operations produce pretty strange numeric results, so no, I wouldn't think you'd expect much going in. But I don't see where the identity function comes into it at all. It looks like the term "the identity function" stuck in your mind, and you reached for it when you found something subjectively similar. But shared semantic space isn't a good reason to say that a function which isn't the identity function, is. Your middle point is the observation "f(a,a) = a", which is perfectly valid, but is not the same finding as the finding you report. Using pretty-simple terms to state something true is generally preferable to using even simpler terms to state something false.
- jonsen 10y agoIt is fair enough, that you find the law f(a,a) = a useful. Just don't call it the identity function when it is not.
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- jonsen 10y agoThe law x AND 1 = x is called the identity law. The law x AND x = x is called the idempotence law. Nothing to do with the identity function.