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June 30, 2024 4:38 PM Subscribe
The first explanation of a Dedekind Cut that I’ll actually remember
posted by Going To Maine at 5:11 PM on June 30 [2 favorites]
posted by Going To Maine at 5:11 PM on June 30 [2 favorites]
They were good friends, and they hated each other. They cooperated, and they ignored each other
Mathematicians! Why can’t they just kiss?
posted by GenjiandProust at 6:31 PM on June 30 [3 favorites]
Mathematicians! Why can’t they just kiss?
posted by GenjiandProust at 6:31 PM on June 30 [3 favorites]
I was going to be a math major until I hit college. Aggressively terrible math professors. And then all this kind of weirdness that I could never quite grok. I got why those things had to exist for math to work, but imaginary numbers and such were too much. I did spend a lot of time however, looking for a way to make division by zero mean something mathematical.
posted by Windopaene at 7:03 PM on June 30 [1 favorite]
posted by Windopaene at 7:03 PM on June 30 [1 favorite]
"Imaginary" is just a really uncharitable and unkind name for a truly amazing group of numbers. ("Real" is another bad name, real numbers are just as real or imaginary as imaginary numbers. Only rational numbers really exist!!!) Gauss's name for them is much better: lateral numbers.
Truth in advertising: I used to be a math professor.
posted by phliar at 7:40 PM on June 30 [9 favorites]
Truth in advertising: I used to be a math professor.
posted by phliar at 7:40 PM on June 30 [9 favorites]
Emmy Noether is amazing! thank you for another (integral?) way towards understanding her work [PJ Olver, U.Minnesota, 85 page (slideshow) pdf]
posted by HearHere at 9:13 PM on June 30
posted by HearHere at 9:13 PM on June 30
This is the fascinating bit about what I recall of my own terrible mathematical education (partly, as I’ve freely admitted before, my own fault as a little shit of a young man). These irrational numbers are metaphors, substitutions for concepts that are very hard to express any other way, which was never explained: they were simply rote memorised as elements of a process, called ‘show your working’ to get a result that was either right or wrong. ‘Broad and flexible’ maths as the article says was almost exactly not what was going on.
Yet the teachers who were teaching lit were doing precisely the same thing, teaching metaphorical and abstract thinking. The only difference is that the English teachers embraced the fundamental oddness and humanity of the concept of metaphor, and rolled with it
posted by Fiasco da Gama at 12:03 AM on July 1
Yet the teachers who were teaching lit were doing precisely the same thing, teaching metaphorical and abstract thinking. The only difference is that the English teachers embraced the fundamental oddness and humanity of the concept of metaphor, and rolled with it
posted by Fiasco da Gama at 12:03 AM on July 1
> Only rational numbers really exist!
God made the integers; all else is the work of man. -- Kronecker.
posted by are-coral-made at 2:19 AM on July 1
God made the integers; all else is the work of man. -- Kronecker.
posted by are-coral-made at 2:19 AM on July 1
Suppose I only believe in 1, and addition. I see a tree containing 1 apple. I see another tree containing 1 apple and 1 apple and 1 apple. Together there are 1 apple and 1 apple and 1 apple and 1 apple. At some point we figure out enough abstract concepts and reasoning so we can reason about this more succinctly as 1 apple + 3 apples = 4 apples.
Suppose you put 3 apples in the bowl. Then I put 1 apple in the bowl. Then we look at the bowl: it contains 4 apples. Very well.
Suppose the bowl is empty, and you are standing next to it. Suppose I put 3 apples in the bowl. Then I leave the room for a while. Then I come back into the room. Now the bowl only contains 2 apples. How many apples did you put in the bowl? You didn't put 1 apple in the bowl, as that would give 4 apples, but there are only 2. You didn't put 2 apples in the bowl, as that would give 5 apples. I have no concept for how many apples you put in the bowl, but I do not think you put 1 or 2 or 3 or 4 or ... apples into the bowl.
I am filled with alarm and distress! We started with concepts I believed in, and then we did something slightly different with them, but now my concepts don't let me describe what is going on!
Suppose I write "nom!" for this new concept of how many apples you put into the bowl when I left the room, where "nom!" is not any of 1 or 2 or 3 or 4 or ... apples in the bowl.
With my familiar notation of + and 1, 2, 3 and this hazy new desired concept of "nom!", I want "nom!" to behave like so: 3 + "nom!" = 2.
I could begin asking things like 'does any such "nom!" exist?' and 'suppose some "nom!" does exist, is there exactly one unique "nom!", or could there be many distinct "nom!"s that each satisfying the property that 3 + "nom!" = 2 ?'.
It might be difficult for us to make much further progress unless we go back and formalise what we actually mean by "1" and "+".
posted by are-coral-made at 2:33 AM on July 1
Suppose you put 3 apples in the bowl. Then I put 1 apple in the bowl. Then we look at the bowl: it contains 4 apples. Very well.
Suppose the bowl is empty, and you are standing next to it. Suppose I put 3 apples in the bowl. Then I leave the room for a while. Then I come back into the room. Now the bowl only contains 2 apples. How many apples did you put in the bowl? You didn't put 1 apple in the bowl, as that would give 4 apples, but there are only 2. You didn't put 2 apples in the bowl, as that would give 5 apples. I have no concept for how many apples you put in the bowl, but I do not think you put 1 or 2 or 3 or 4 or ... apples into the bowl.
I am filled with alarm and distress! We started with concepts I believed in, and then we did something slightly different with them, but now my concepts don't let me describe what is going on!
Suppose I write "nom!" for this new concept of how many apples you put into the bowl when I left the room, where "nom!" is not any of 1 or 2 or 3 or 4 or ... apples in the bowl.
With my familiar notation of + and 1, 2, 3 and this hazy new desired concept of "nom!", I want "nom!" to behave like so: 3 + "nom!" = 2.
I could begin asking things like 'does any such "nom!" exist?' and 'suppose some "nom!" does exist, is there exactly one unique "nom!", or could there be many distinct "nom!"s that each satisfying the property that 3 + "nom!" = 2 ?'.
It might be difficult for us to make much further progress unless we go back and formalise what we actually mean by "1" and "+".
posted by are-coral-made at 2:33 AM on July 1
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posted by jozxyqk at 5:04 PM on June 30 [2 favorites]