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"That’s because to understand zero, our mind must create something out of nothing. It must recognize absence as a mathematical object." - I don't see how that f

"That’s because to understand zero, our mind must create something out of nothing. It must recognize absence as a mathematical object." - I don't see how that follows from anything. So 0 in counting would account for "no item", while 0 in the decimal system would account for "multiply by 10"? Are there different neurons for that? :) I've also read the linked article "https://www.quantamagazine.org/why-the-human-brain-perceives-small-numbers-better-20231109/ https://www.quantamagazine.org/why-the-human-brain-perceives... which discusses the discovery of "number neurons" and goes on to discuss that there might be no "number neurons" after all.

t-3 2y ago
I feel like zero in the context of counting is very simple and intuitive, and in addition, subtraction, and multiplication as well (aren't those just extensions of counting anyway?). In division and exponentiation things can be a bit harder to comprehend - why is dividing by zero impossible? Why is the zeroth power always one? I reckon the barrier to understanding there is that division and exponentiation are not counting operations but meta-operations relating to multiplication.
- jon_richards 2y ago
  Exponentiation to zero annoyed me as a kid because I had equated "no operations" as zero from the pattern 2x2=2+2, 2x1=2, 2x0= Declaring 2x2=0+2+2, 2x1=0+2, 2x0=0 while 2xx2=1x2x2, 2xx1=1x2, 2xx0=1 seemed arbitrary. What helped was learning about negative exponentiation and exponentiation simplification, so 2xx0 = 2xx2 x 2xx(-2) = 2xx2 / 2xx2 = 1. That said, I clearly still take issue with unintuitive interpretations of "nothing" https://stackoverflow.com/questions/852414/how-to-dynamically-compose-an-or-query-filter-in-django/59762852#59762852 https://stackoverflow.com/questions/852414/how-to-dynamicall...
  - whatshisface 2y ago
    Another good way of thinking about it is that if 10^a 10^b = 10^(a+b), you can set a=3 and b=0 to get 10^3 10^0 = 10^3. If you divide both sides by 10^3, 10^0 = 1. QED.
  - anon291 2y ago
    I mean the reality is that in the same way that 2 * 0 is ONE. Or if I were using latex: 2 \times 0 = 1_{+} Because multiplication, being repeated addition and exponentiation repeated multiplication both behave the same way. When asked to repeat the operation zero times, they return the unit or 1 of the underlying group, which is typically denoted as 1_{whatever} However, the 1 of addition is 0 when using the standard notation for integers
  - bmacho 2y ago
    > Exponentiation to zero annoyed me as a kid Your teacher didn't tell you (or told you, but you didn't recognize it as something valuable and forgot it) that exponentiation to zero is a new definition over exponentiation to positive integers. Exponentiation to positive integers is defined somehow, and that definition says nothing about exponentiation to zero. It is a new definition, not something that you deduce. The same holds for 0^0 or 0/0 (with some amounts of confusion, lies, and hypocriticism).
    - jon_richards 2y ago
      Declaring it as a new definition was what annoyed me... Which I found incredibly silly once I leaned about negative exponentiation and could now deduce the pattern. Similarly, 0/0 became much more tractable to me once I learned L'Hôpital's rule https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule
      - deleted 2y ago
        [deleted]
  - n_plus_1_acc 2y ago
    Now, you probably now that this is because 0 and 1 are the identity elements of the additive and multiplicative groups of real numbers, respectively. Probably wouldn't've helped you as kid though
    - jon_richards 2y ago
      I think it's much more productive to teach children 2x0 = 2x1 + 2x(-1) = 2-2 = 0 and 2xx0 = 2xx1 x 2xx(-1) = 2/2 = 1 Inverting the concept and having those patterns stem from a fundamental identity can wait until its not seen as the mathematical equivalent of "Because I said so".
      - Etherlord87 2y ago
        "Children" at what age do you want to teach this, and how many children are you in contact with? Because that one kid probably suffers from autism.
- anon291 2y ago
  Dividing things by zero is not impossible for some definition of impossible. In the sense that you can always make progress in pulling a group of zero from a group of something and increasing your counter, you can be very productive when dividing by zero. This has to do with rings and the relationship between the two identities of the two underlying groups. It ultimately stems from the distributive property between multiplication and addition
- Ferret7446 2y ago
  Dividing by zero is impossible because it's not defined. Simple as that. Same as undefined behavior in any other language or system. It's like playing chess and saying "but what if I lay my pawn sideways and place it on top of my head? What happens then?". It's not defined in the rules.
  - bubblyworld 2y ago
    I think this point of view is a little bit backwards - it's true that it's not defined (for the reals), but I don't think that's a convincing causal explanation. I think we invent abstractions because they allow us to reason about patterns in reality in a consistent way. The fact that division by zero is undefined is (to my mind) because it doesn't correspond to any useful pattern in the parts of reality we typically apply arithmetic to (accounting, estimation, etc). What I like about this point of view is that it encourages thinking about what you are trying to accomplish, rather than fixating on formal rules. In some contexts "division by zero" can correspond to a meaningful pattern - look up the geometry of the projective line, for instance. In such cases you might want to include it in your model, rather than declaring it to be undefined as a convention!
    - Ferret7446 2y ago
      If you want to go further, it's because division is defined as the inverse of multiplication. It doesn't really have anything to do with: > it doesn't correspond to any useful pattern in the parts of reality we typically apply arithmetic to (accounting, estimation, etc). You can of course re-define it, but then we aren't talking about the same thing any more. The operation of inverting multiplication, is not defined for zero.
      - bubblyworld 2y ago
        But this is my point - the definitions are the way they are because it's a good model of counting (and other counting-like things). They weren't invented in a vacuum. Similarly the fact that multiplication and division are inverses is a property of this model. Conceptually you can imagine splitting and copying groups of objects quite independently of one another (and which one you view as fundamental is really a post-hoc choice). In general these days we mostly see clean mathematical abstractions because all the scaffolding has already been removed by mathematicians past. And as a result people come to believe that this is how mathematics is done. But always there is an initial period of exploration (which eventually gets forgotten) as people try to work out how to axiomatise the various systems they are interested in.

t-3 2y ago

I feel like zero in the context of counting is very simple and intuitive, and in addition, subtraction, and multiplication as well (aren't those just extensions of counting anyway?). In division and exponentiation things can be a bit harder to comprehend - why is dividing by zero impossible? Why is the zeroth power always one? I reckon the barrier to understanding there is that division and exponentiation are not counting operations but meta-operations relating to multiplication.

jon_richards 2y ago
Exponentiation to zero annoyed me as a kid because I had equated "no operations" as zero from the pattern 2x2=2+2, 2x1=2, 2x0= Declaring 2x2=0+2+2, 2x1=0+2, 2x0=0 while 2xx2=1x2x2, 2xx1=1x2, 2xx0=1 seemed arbitrary. What helped was learning about negative exponentiation and exponentiation simplification, so 2xx0 = 2xx2 x 2xx(-2) = 2xx2 / 2xx2 = 1. That said, I clearly still take issue with unintuitive interpretations of "nothing" https://stackoverflow.com/questions/852414/how-to-dynamically-compose-an-or-query-filter-in-django/59762852#59762852 https://stackoverflow.com/questions/852414/how-to-dynamicall...
- whatshisface 2y ago
  Another good way of thinking about it is that if 10^a 10^b = 10^(a+b), you can set a=3 and b=0 to get 10^3 10^0 = 10^3. If you divide both sides by 10^3, 10^0 = 1. QED.
- anon291 2y ago
  I mean the reality is that in the same way that 2 * 0 is ONE. Or if I were using latex: 2 \times 0 = 1_{+} Because multiplication, being repeated addition and exponentiation repeated multiplication both behave the same way. When asked to repeat the operation zero times, they return the unit or 1 of the underlying group, which is typically denoted as 1_{whatever} However, the 1 of addition is 0 when using the standard notation for integers
- bmacho 2y ago
  > Exponentiation to zero annoyed me as a kid Your teacher didn't tell you (or told you, but you didn't recognize it as something valuable and forgot it) that exponentiation to zero is a new definition over exponentiation to positive integers. Exponentiation to positive integers is defined somehow, and that definition says nothing about exponentiation to zero. It is a new definition, not something that you deduce. The same holds for 0^0 or 0/0 (with some amounts of confusion, lies, and hypocriticism).
  - jon_richards 2y ago
    Declaring it as a new definition was what annoyed me... Which I found incredibly silly once I leaned about negative exponentiation and could now deduce the pattern. Similarly, 0/0 became much more tractable to me once I learned L'Hôpital's rule https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule https://en.wikipedia.org/wiki/L'H%C3%B4pital's_rule
    - deleted 2y ago
      [deleted]
- n_plus_1_acc 2y ago
  Now, you probably now that this is because 0 and 1 are the identity elements of the additive and multiplicative groups of real numbers, respectively. Probably wouldn't've helped you as kid though
  - jon_richards 2y ago
    I think it's much more productive to teach children 2x0 = 2x1 + 2x(-1) = 2-2 = 0 and 2xx0 = 2xx1 x 2xx(-1) = 2/2 = 1 Inverting the concept and having those patterns stem from a fundamental identity can wait until its not seen as the mathematical equivalent of "Because I said so".
    - Etherlord87 2y ago
      "Children" at what age do you want to teach this, and how many children are you in contact with? Because that one kid probably suffers from autism.
anon291 2y ago
Dividing things by zero is not impossible for some definition of impossible. In the sense that you can always make progress in pulling a group of zero from a group of something and increasing your counter, you can be very productive when dividing by zero. This has to do with rings and the relationship between the two identities of the two underlying groups. It ultimately stems from the distributive property between multiplication and addition
Ferret7446 2y ago
Dividing by zero is impossible because it's not defined. Simple as that. Same as undefined behavior in any other language or system. It's like playing chess and saying "but what if I lay my pawn sideways and place it on top of my head? What happens then?". It's not defined in the rules.
- bubblyworld 2y ago
  I think this point of view is a little bit backwards - it's true that it's not defined (for the reals), but I don't think that's a convincing causal explanation. I think we invent abstractions because they allow us to reason about patterns in reality in a consistent way. The fact that division by zero is undefined is (to my mind) because it doesn't correspond to any useful pattern in the parts of reality we typically apply arithmetic to (accounting, estimation, etc). What I like about this point of view is that it encourages thinking about what you are trying to accomplish, rather than fixating on formal rules. In some contexts "division by zero" can correspond to a meaningful pattern - look up the geometry of the projective line, for instance. In such cases you might want to include it in your model, rather than declaring it to be undefined as a convention!
  - Ferret7446 2y ago
    If you want to go further, it's because division is defined as the inverse of multiplication. It doesn't really have anything to do with: > it doesn't correspond to any useful pattern in the parts of reality we typically apply arithmetic to (accounting, estimation, etc). You can of course re-define it, but then we aren't talking about the same thing any more. The operation of inverting multiplication, is not defined for zero.
    - bubblyworld 2y ago
      But this is my point - the definitions are the way they are because it's a good model of counting (and other counting-like things). They weren't invented in a vacuum. Similarly the fact that multiplication and division are inverses is a property of this model. Conceptually you can imagine splitting and copying groups of objects quite independently of one another (and which one you view as fundamental is really a post-hoc choice). In general these days we mostly see clean mathematical abstractions because all the scaffolding has already been removed by mathematicians past. And as a result people come to believe that this is how mathematics is done. But always there is an initial period of exploration (which eventually gets forgotten) as people try to work out how to axiomatise the various systems they are interested in.