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byrnema comments on Thinking without words? - Less Wrong Discussion
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Do you mean you have a subsymbolic concept of 3,568 or that you seem to arrive at it in a subsymbolic way?
I'll normally double things starting at the right and going left. So first 4*2=8, no problem. Then I need 8*2. As soon as I think the number 8 my brain instantly returns the number 16 as its double, but I still feel like I need to consciously think to myself "What's eight times two? Sixteen". Then the same thing happens for 7*2=14. Now I feel like I need to add up the work I've done so far. At this point my brain instantly returns the number 1568 (The number 1568 appears in my mind as a sequence of digits, not visualised visually, but just the concepts "one" "five" "six" "eight" in that order.) However I still feel the need to think the thought "16 represents 160. 160 + 8 = 168, 14 represents 1400, and 1400 +168 =1568" even though I already know that the answer will be 1568. Then 1*2+1=3 so we have 3568.
I don't think I visualise the numbers at every stage as their visual symbols, I just have concepts for each of the digits which I shuffle around. If I need to store a number to remember it later then I sub-vocalise it and then later I can just recall what I "heard".
There's this known result that people make decisions intuitively and later come up with verbal rationalizations for them but when all of that gets assembled into a stream of consciousness, it can feel like the rationalizations came first and were the deciding factor. Your experiecne could be something like this, only in the other direction. Or you have a really high-definition number sense. Do you still get correct results if you try to surpress the verbalizations while doing mental arithmetic?
If I don't verbalise the multiplications as I carry them out then I often forget or misremember the running total when I want to add another number to it. I feel like practice would improve this.
Wow, I'm glad I asked, since that is interesting. I think I can see how the computation could be automatized and regulated to subconscious components of your brain through practice, but its neat that your brain can keep track of those products (8 .. 16 .. 14 .. 2) and their place value (8 .. 160 .. 1400 .. 2000) and add them.