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James_Miller comments on Rationality Quotes April 2013 - Less Wrong
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Daniel Kahneman,Thinking, Fast and Slow
As far as I can tell this doesn't agree with my experience; a good chunk of every day is spent in groping uncertainty and confusion.
Come and take my herb?
Unless you took John Leslie's advice and Ankified the multiplication table up to 25.
I've read your link to John Leslie with both curiosity and bafflement.
17 x 24 is not perhaps the best example of a question for which no answer comes immediately to mind. Seventeen has the curious property that 17 x 6 = 102. (The recurring decimal 1/6 = 0.166666... hints to us that 17 x 6 = 102 is just the first of a series of near misses on a round number, 167 x 6 = 1002, 1667 x 6 = 10002, etc). So multiplying 17 by any small multiple of 6 is no harder than the two times table. In particular 17 x 24 = 17 x (6 x 4) = (17 x 6) x 4 = 102 x 4 = 408.
17 x 23 might have served better, were it not for the curious symmetry around the number 20, with 17 = 20 - 3 while 23 = 20 + 3. One is reminded of the identity (x + y)(x - y) = x^2 - y^2 which is often useful in arithmetic and tells us at once that 17 x 23 = 20 x 20 - 3 x 3 = 400 - 9 = 391.
17 x 25 has a different defect as an example, because one can hardly avoid apprehending 25 as one quarter of 100, which stimulates the observation that 17 = 16 + 1 and 16 is full of yummy fourness. 17 x 25 = (16 + 1) x 25 = (4 x 4 + 1) x 25 = 4 x 4 x 25 + 1 x 25 = 4 x 100 + 25 = 425.
17 x 26 is a better example. Nature has its little jokes. 7 x 3 = 21 therefore 17 x 13 = (1 + 7) x (1 + 3) = (1 + 1) + 7 x 3 = 2 + 21 = 221. We get the correct answer by outrageously bogus reasoning. And we are surely puzzled. Why does 21 show up in 17 x 13? Aren't larger products always messed up and nasty? (This is connected to 7 + 3 = 10). Any-one who is in on the joke will immediately say 17 x 26 = 17 x (13 x 2) = (17 x 13) x 2 = 221 x 2 = 442. But few people are.
Some people advocate cultivating a friendship with the integers. Learning the multiplication table, up to 25 times 25, by the means exemplified above, is part of what they mean by this.
Others, full of sullen resentment at the practical usefulness of arithmetic, advocate memorizing ones times tables by the grimly efficient deployment of general purpose techniques of rote memorization such as the Anki deck. But who in this second camp sees any need to go beyond ten times ten?
Does John Leslie have a foot in both camps? Does he set the twenty-five times table as the goal and also indicate rote memorization as the means?
I'm not sure exactly what he had in mind, but learning the multiplication tables using Anki isn't exactly rote.
Now, this may not be the case for others, but when I see a new problem like 17 x 24, I don't just keep reading off the answer until I remember it when the note comes back around. Instead, I try to answer it using mental arithmetic, no matter how long it takes. I do this by breaking the problem into easier problems (perhaps by multiplying 17 x 20 and then adding that to 17 x 4). Sooner or later my brain will simply present the answers to the intermediate steps for me to add together and only much later do those steps fade away completely and the final answer is immediately retrievable.
Doing things this way, simply as a matter of course, you develop somewhat of a feel for how certain numbers multiply and develop a kind of "friendship with the integers." Er, at least, that's what it feels like from the inside.
That's not the important point. Even if you have, you will still face the same problem when facing a question like, for example, say 34 × 57 = ?. The quote was using that particular problem as an example. If that example does not apply to you because you Ankified the multiplication table up to 25 or for any other reason, it is trivial to find another problem that gives the desired mental response. (As I just did with the 34 × 57 problem.)
Agreed. I'm not so much disagreeing with the thrust of the quote as nitpicking in order to engage in propaganda for my favorite SRS.
Of course, even if I have no complete answer to 34 × 57, I still have "intuitive feelings and opinions" about it, and so do you. For example, I know it's between 100 and 10000 just by counting the digits, and although I've just now gone and formalized this intuition, it was there before the math: if I claimed that 34 × 57 = 218508 then I'm sure most people here would call me out long before doing the calculation.
What has this got to do with the original quote? The quote was claiming, truthfully or not, that when one is first presented with a certain type of problem, one is dumbfounded for a period of time. And of course the problem is solvable, and of course even without calculating it you can get a rough picture of the range the answer is in, and with a certain amount of practice one can avoid the dumbfoundedness altogether and move on to solving the problem, and that is a fine response to give to the original quote, but it has no relevance to what I was saying.
All I was saying is that it is an invalid objection to object to the quote based on the fact that with a certain technique the specific example given by the quote can be avoided, as that example could have easily been replaced by a similar example which that technique does not solve. I was talking about that specific objection I was not saying the quote is perfect, or even that it is entirely right. You may raise these other objections to it. But the specific objection that Jayson_Virissimo raised happens to be entirely invalid.
I'm a little perplexed that I haven't got the multiplication table up to 25 memorized, given the number of times I've multiplied any two numbers under 25.
I'm curious - what advantage do you get from this?
So far, mostly the ability to perform entertaining parlor tricks (via mental arithmetic and a large body of facts about the countries of the world). I admit, it is not very impressive, but not useless either. In other words, nothing you couldn't do in a few minutes with a smartphone (although, I imagine, that would tend to ruin the "trick").
Those moments send me into panic attacks. (At least when they're on significant topics not on maths).
Math is a significant topic!
*Topics where my inability to work out the answer immediately implies a lack of ability or puts me at risk.