9 ms·

Why isn't 2+2==4 obvious? http://us.metamath.org/mpegif/mmset.html#trivia http://us.metamath.org/mpegif/mmset.html#trivia

amelius 10y ago
Perhaps you should read Principia Mathematica [1], which proves on page 379 that 1+1=2. [1] https://en.wikipedia.org/wiki/Principia_Mathematica https://en.wikipedia.org/wiki/Principia_Mathematica
pavelrub 10y ago
2+2=4 is obvious. The axiomatic proofs are mostly a meaningless and boring exercise that mathematicians invented when they wanted to axiomatize everything. They have nothing to do with whether something is obvious or not. It isn't as if it was possible to doubt that 2+2=4 before the invention of the Peano axioms.
- ingenter 10y ago
  I believe you're mistaken. There is value in axioms and axiomatic proofs: two different people will most definitively have a different notion of "obvious", and even have a different understanding of a mathematical problem. So a proof may be accepted by one person and rejected by another. Given a set of axioms and proofs it's possible to mechanically check a proof. It's not quite possible to reliably check proofs otherwise.
  - hugh4 10y ago
    But are people more likely to accept the axioms of Principia Mathematica (and the soundness of every logical step from page 1 to 300) than they are to accept the notion that 2 + 2 = 4 based on intuitive notions of what twoness, fourness and plusness are?
    - CarolineW 10y ago
      Of course they are more likely to believe their intuitions. They also believe that it makes no difference whether or not you swap doors in the Monty Hall problem, and don't believe that with only 23 people the odds of a shared birthday are more than 50%. To some extent, there is the problem. People trust their intuitions, and their intuitions are often wrong. That's why for some things we need proper proofs.
      - hugh4 10y ago
        But proofs always come back to axioms, and on what basis do we accept axioms? That they sound intuitively right. So we've just kicked the problem upstairs a bit, we can't avoid using our intuition. Personally I'm more likely to believe 2 + 2 = 4, something I can easily check to my own satisfaction using four objects, than I am to believe the Axiom of Choice.
        kosievdmerwe 10y ago
        We still use our intuitions, but now everyone knows the starting set of assumptions. As for the axioms in use, I think the big reasons they were chosen is: They lead to results we already wanted/proved to be true. Another thing to keep in mind, not everyone works with the same sets of axioms. Which, as someone with a formalist[2] view on mathematics, I find interesting. For example, not everyone studying logic assumes the principle of the excluded middle[1]. One of the consequences of this is that you can no longer do proofs by contradiction. The axiom of choice is another example of this where two groups of mathematicians accept it or not. I'm a formalist, so I don't have issues with this (as long as both sets of axioms are interesting and "intuitive"), other philosophies of maths might. [1] https://en.wikipedia.org/wiki/Law_of_excluded_middle https://en.wikipedia.org/wiki/Law_of_excluded_middle [2] https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathe...
        mnx 10y ago
        Well, the whole idea is to pick simple axioms, so it's harder to get wrong. And also, every successful prediction that math makes based on the axioms, is in a sense a verification of them.
  - pavelrub 10y ago
    I'm not saying anything about the ability to check proofs, or the value of axiomatic proofs in general, only that 2+2=4 specifically doesn't require an axiomatic proof in order to convince anybody that it is true. This is like saying that we need a rigorous theory of color in order to be convinced that black is darker than red. Mathematicians didn't axiomatize natural numbers in order to show that 1+1=2 or 2+2=4, or any other trivial arithmetical fact. They have never doubted it, and I don't know what "doubting 2+2=4" even means. In fact the entire process is reversed: they invented axioms that can form a formal basis for what we already know to be true. If Peano axioms proved that 2+2 = 6 - they wouldn't be a valid axiomatization of the natural numbers. One cannot axiomatize the natural numbers without already assuming that all the basic arithmetic facts we know about them are true (or else he wouldn't be axiomatizing the natural numbers, but something else). Somebody who rejects 2+2=4 has a problem understanding human language, not proofs.
    - thaumasiotes 10y ago
      > This is like saying that we need a rigorous theory of color in order to be convinced that black is darker than red. You do, if you want to be right. The fact that you can get people to agree with you doesn't make you right, and red is frequently darker than black by some pretty normal definitions of "darker". Red and black are differentiated by the shape of their reflective spectrum, not the amplitude.
      - jessriedel 10y ago
        You guys are basically arguing over Moore's here-is-one-hand problem. https://en.wikipedia.org/wiki/Here_is_one_hand https://en.wikipedia.org/wiki/Here_is_one_hand pavelrub's point is that you sometimes have less reason to believe the axioms of your formalization than their derived consequences. We have better reason to believe the intuitive idea that 2+2=4 than we do any putative axioms of arithmetic. If we derived that 2+2=5 from some particular axioms of arithmetic, we would conclude those axioms were wrong (or rather, were not the proper system for formalizing 2-plus-2-ness) rather than conclude that 2+2=5.
        deleted 10y ago
- thaumasiotes 10y ago
  > It isn't as if it was possible to doubt that 2+2=4 before the invention of the Peano axioms. It certainly was; primitive cultures frequently lack words for medium-high numbers like 10, and have been known to lack 4. Unsurprisingly, those people are generally uncomfortable when asked to manipulate quantities that high. (They may use other methods, like having a collection of stones which is known to match the number of sheep in a flock, and "counting" sheep as they arrive by moving a stone from one pile to the other. If you failed to move a stone, you're missing a sheep.)
  - pavelrub 10y ago
    Not having a word for 4 and doubting that 2+2=4 aren't the same thing. The former means that you cannot understand what the proposition means, while the latter means that you do understand what it means, but aren't convinced that it's true.
- czinck 10y ago
  There's a lot of historical context you're missing if you think axiomatic proofs are meaningless. Around the late 1800s, a few contradictory proofs started popping up because people weren't being rigorous enough (the example I know involved proofs about pointwise vs uniformly continuous functions just being referred to as "continuous"). Then a few paradoxes were discovered (like Russel's paradox, the set of all sets that don't contain themselves) and the Mathematics community realized formalizing their assumptions and reproving everything from the ground up was necessary. So Whitehead and Russel started to write Principa Mathematica, and everyone was happy in the Math world until Godel came along and proved that Principa Mathematica would either have contradictions or have unprovable theorems.
  - pavelrub 10y ago
    I think I wasn't clear enough. I never meant that axiomatic proofs in general are meaningless, only that axiomatic proofs of trivial arithmetic facts (1+1=2, 2+2=4...) are meaningless.
    - mcguire 10y ago
      You have two choices here: 1. You assume "trivial arithmetic facts" as axioms. Result: You have an infinite number of axioms. (Whee!) The likelihood that you have snuck in non-trivial assumptions is pretty high, unless you are very strict about how you define "trivial" (which is probably as much work as just proving the trivial facts), and in that case, there's a high probability that some of your trivial facts are false. 2. You demonstrate that you can prove "trivial facts" in your system and you do so when needed by more complex proofs. The proofs of trivial facts are not necessarily trivial. In neither case is your handling of "trivial facts" meaningless. Quod erat demonstrandom.
      - jerf 10y ago
        From a mathematical point of view, one must be very careful, and we have abundant evidence of that. However, we are fully justified in saying that if anybody came up with a mathematical system in which 2 + 2 != 4, we can dismiss it without having to do some sort of deep analysis of it. 2 + 2 = 4 is obvious. We can literally do it with 4 little objects right in front of us. If we can not accept that as obvious, we are hopelessly ignorant and have no reason to trust our fancy proofs, either. (Italicized to emphasize my main point.) If you can't trust that, you certainly can't trust the significantly more complicated number theory axioms do anything useful. Note that 2 + 2 = 4 carries some implicit context when we say it without qualification, and subtly sliding in a context change is not a disproof. 2 + 2 = 1 modulo 3, but that's not what anybody means without qualification. Clearly we're operating on "the numbers I can hold in my hand" here, or some superset thereto. Note how I'm not even willing to say "the natural numbers" necessarily; it isn't obvious to me what some billion digit number added to some other billion digit number is. It's actually crucial to my point here that I'm not extending "obvious" out that far; I can only run an algorithm on that and trust the algorithm. But I'm just being disingenuous if I claim ignorance of 2 + 2. And being disingenuous like that tends to turn people off, and doesn't encourage them to try to learn more.
fiatjaf 10y ago
Why is that chain of axiomatic proofs obvious? How is it obvious that one step follows the previous?
- serge2k 10y ago
  2 + 2 = S(S(0)) + S(1) = S(S(S(0)) + 1) = S(S(S(0)) + S(0)) - S(S(S(S(0 + 0)))) = S(S(S(S(0)))) = 4 Definiton of Successor (S) and addition. It's trivial.
  - fiatjaf 10y ago
    How is the definition of Successor so obvious, even if you explain it? How can I understand what it so obviously means?
    - serge2k 10y ago
      It's an axiom. We are talking specifically about 2+2=4 being trivial because it falls out of the axioms. https://en.wikipedia.org/wiki/Peano_axioms https://en.wikipedia.org/wiki/Peano_axioms

amelius 10y ago

Perhaps you should read Principia Mathematica [1], which proves on page 379 that 1+1=2. [1] https://en.wikipedia.org/wiki/Principia_Mathematica https://en.wikipedia.org/wiki/Principia_Mathematica

pavelrub 10y ago

2+2=4 is obvious. The axiomatic proofs are mostly a meaningless and boring exercise that mathematicians invented when they wanted to axiomatize everything. They have nothing to do with whether something is obvious or not. It isn't as if it was possible to doubt that 2+2=4 before the invention of the Peano axioms.

ingenter 10y ago
I believe you're mistaken. There is value in axioms and axiomatic proofs: two different people will most definitively have a different notion of "obvious", and even have a different understanding of a mathematical problem. So a proof may be accepted by one person and rejected by another. Given a set of axioms and proofs it's possible to mechanically check a proof. It's not quite possible to reliably check proofs otherwise.
- hugh4 10y ago
  But are people more likely to accept the axioms of Principia Mathematica (and the soundness of every logical step from page 1 to 300) than they are to accept the notion that 2 + 2 = 4 based on intuitive notions of what twoness, fourness and plusness are?
  - CarolineW 10y ago
    Of course they are more likely to believe their intuitions. They also believe that it makes no difference whether or not you swap doors in the Monty Hall problem, and don't believe that with only 23 people the odds of a shared birthday are more than 50%. To some extent, there is the problem. People trust their intuitions, and their intuitions are often wrong. That's why for some things we need proper proofs.
    - hugh4 10y ago
      But proofs always come back to axioms, and on what basis do we accept axioms? That they sound intuitively right. So we've just kicked the problem upstairs a bit, we can't avoid using our intuition. Personally I'm more likely to believe 2 + 2 = 4, something I can easily check to my own satisfaction using four objects, than I am to believe the Axiom of Choice.
      - kosievdmerwe 10y ago
        We still use our intuitions, but now everyone knows the starting set of assumptions. As for the axioms in use, I think the big reasons they were chosen is: They lead to results we already wanted/proved to be true. Another thing to keep in mind, not everyone works with the same sets of axioms. Which, as someone with a formalist[2] view on mathematics, I find interesting. For example, not everyone studying logic assumes the principle of the excluded middle[1]. One of the consequences of this is that you can no longer do proofs by contradiction. The axiom of choice is another example of this where two groups of mathematicians accept it or not. I'm a formalist, so I don't have issues with this (as long as both sets of axioms are interesting and "intuitive"), other philosophies of maths might. [1] https://en.wikipedia.org/wiki/Law_of_excluded_middle https://en.wikipedia.org/wiki/Law_of_excluded_middle [2] https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathematics) https://en.wikipedia.org/wiki/Formalism_(philosophy_of_mathe...
      - mnx 10y ago
        Well, the whole idea is to pick simple axioms, so it's harder to get wrong. And also, every successful prediction that math makes based on the axioms, is in a sense a verification of them.
- pavelrub 10y ago
  I'm not saying anything about the ability to check proofs, or the value of axiomatic proofs in general, only that 2+2=4 specifically doesn't require an axiomatic proof in order to convince anybody that it is true. This is like saying that we need a rigorous theory of color in order to be convinced that black is darker than red. Mathematicians didn't axiomatize natural numbers in order to show that 1+1=2 or 2+2=4, or any other trivial arithmetical fact. They have never doubted it, and I don't know what "doubting 2+2=4" even means. In fact the entire process is reversed: they invented axioms that can form a formal basis for what we already know to be true. If Peano axioms proved that 2+2 = 6 - they wouldn't be a valid axiomatization of the natural numbers. One cannot axiomatize the natural numbers without already assuming that all the basic arithmetic facts we know about them are true (or else he wouldn't be axiomatizing the natural numbers, but something else). Somebody who rejects 2+2=4 has a problem understanding human language, not proofs.
  - thaumasiotes 10y ago
    > This is like saying that we need a rigorous theory of color in order to be convinced that black is darker than red. You do, if you want to be right. The fact that you can get people to agree with you doesn't make you right, and red is frequently darker than black by some pretty normal definitions of "darker". Red and black are differentiated by the shape of their reflective spectrum, not the amplitude.
    - jessriedel 10y ago
      You guys are basically arguing over Moore's here-is-one-hand problem. https://en.wikipedia.org/wiki/Here_is_one_hand https://en.wikipedia.org/wiki/Here_is_one_hand pavelrub's point is that you sometimes have less reason to believe the axioms of your formalization than their derived consequences. We have better reason to believe the intuitive idea that 2+2=4 than we do any putative axioms of arithmetic. If we derived that 2+2=5 from some particular axioms of arithmetic, we would conclude those axioms were wrong (or rather, were not the proper system for formalizing 2-plus-2-ness) rather than conclude that 2+2=5.
      - deleted 10y ago
thaumasiotes 10y ago
> It isn't as if it was possible to doubt that 2+2=4 before the invention of the Peano axioms. It certainly was; primitive cultures frequently lack words for medium-high numbers like 10, and have been known to lack 4. Unsurprisingly, those people are generally uncomfortable when asked to manipulate quantities that high. (They may use other methods, like having a collection of stones which is known to match the number of sheep in a flock, and "counting" sheep as they arrive by moving a stone from one pile to the other. If you failed to move a stone, you're missing a sheep.)
- pavelrub 10y ago
  Not having a word for 4 and doubting that 2+2=4 aren't the same thing. The former means that you cannot understand what the proposition means, while the latter means that you do understand what it means, but aren't convinced that it's true.
czinck 10y ago
There's a lot of historical context you're missing if you think axiomatic proofs are meaningless. Around the late 1800s, a few contradictory proofs started popping up because people weren't being rigorous enough (the example I know involved proofs about pointwise vs uniformly continuous functions just being referred to as "continuous"). Then a few paradoxes were discovered (like Russel's paradox, the set of all sets that don't contain themselves) and the Mathematics community realized formalizing their assumptions and reproving everything from the ground up was necessary. So Whitehead and Russel started to write Principa Mathematica, and everyone was happy in the Math world until Godel came along and proved that Principa Mathematica would either have contradictions or have unprovable theorems.
- pavelrub 10y ago
  I think I wasn't clear enough. I never meant that axiomatic proofs in general are meaningless, only that axiomatic proofs of trivial arithmetic facts (1+1=2, 2+2=4...) are meaningless.
  - mcguire 10y ago
    You have two choices here: 1. You assume "trivial arithmetic facts" as axioms. Result: You have an infinite number of axioms. (Whee!) The likelihood that you have snuck in non-trivial assumptions is pretty high, unless you are very strict about how you define "trivial" (which is probably as much work as just proving the trivial facts), and in that case, there's a high probability that some of your trivial facts are false. 2. You demonstrate that you can prove "trivial facts" in your system and you do so when needed by more complex proofs. The proofs of trivial facts are not necessarily trivial. In neither case is your handling of "trivial facts" meaningless. Quod erat demonstrandom.
    - jerf 10y ago
      From a mathematical point of view, one must be very careful, and we have abundant evidence of that. However, we are fully justified in saying that if anybody came up with a mathematical system in which 2 + 2 != 4, we can dismiss it without having to do some sort of deep analysis of it. 2 + 2 = 4 is obvious. We can literally do it with 4 little objects right in front of us. If we can not accept that as obvious, we are hopelessly ignorant and have no reason to trust our fancy proofs, either. (Italicized to emphasize my main point.) If you can't trust that, you certainly can't trust the significantly more complicated number theory axioms do anything useful. Note that 2 + 2 = 4 carries some implicit context when we say it without qualification, and subtly sliding in a context change is not a disproof. 2 + 2 = 1 modulo 3, but that's not what anybody means without qualification. Clearly we're operating on "the numbers I can hold in my hand" here, or some superset thereto. Note how I'm not even willing to say "the natural numbers" necessarily; it isn't obvious to me what some billion digit number added to some other billion digit number is. It's actually crucial to my point here that I'm not extending "obvious" out that far; I can only run an algorithm on that and trust the algorithm. But I'm just being disingenuous if I claim ignorance of 2 + 2. And being disingenuous like that tends to turn people off, and doesn't encourage them to try to learn more.

fiatjaf 10y ago

Why is that chain of axiomatic proofs obvious? How is it obvious that one step follows the previous?

serge2k 10y ago
2 + 2 = S(S(0)) + S(1) = S(S(S(0)) + 1) = S(S(S(0)) + S(0)) - S(S(S(S(0 + 0)))) = S(S(S(S(0)))) = 4 Definiton of Successor (S) and addition. It's trivial.
- fiatjaf 10y ago
  How is the definition of Successor so obvious, even if you explain it? How can I understand what it so obviously means?
  - serge2k 10y ago
    It's an axiom. We are talking specifically about 2+2=4 being trivial because it falls out of the axioms. https://en.wikipedia.org/wiki/Peano_axioms https://en.wikipedia.org/wiki/Peano_axioms