12 ms·

In a certain sense, "1+1 is 2" is also merely a definition, in the same sense that "0^0 is 1" is a definition. Addition can be formally defined in mathematics;

by rntz 13y ago

In a certain sense, "1+1 is 2" is also merely a definition, in the same sense that "0^0 is 1" is a definition. Addition can be formally defined in mathematics; we habitually omit this definition because it is tedious, and because addition is such an intuitive operation that we do not require a definition in order to reason about it. Much as the question "what if the parallel axiom didn't hold?" leads to alternative geometries, the question "what if addition didn't work in the same way?" (or "how can we generalize addition?") leads to some basic notions in abstract algebra.

rickhanlonii 13y ago
I agree with the point you're making, but I want to be a little pedantic: It's true that addition is a definition, but 1+1=2 is not--it logically follows from the definition of addition.
- Retric 13y ago
  Not quite, just as in calculus you end up rediscovering some old geometry formulas there are branches of mathematics where you end up rediscovering 2 such that 1 + 1 equals it. Other options include 1 + 1 = 0 or 1+1 = 1.
  - gordaco 13y ago
    But 1+1=0 or 1+1=1 happens because you are redefining the operator + (sometimes subtly, like using it over Z2), so what rickhanlonii said still holds. 1+1 is 2 if you're using the usual definition of sum, i.e., addition over integers or a superset of integers.
    - Retric 13y ago
      You missed the point 1 + 1 = 2 depends on the definition of '1', '+', '=', and '2'. The first case where this maters is probably binary where '2' does not exist, but that's hardly the only time 1+1 != 2.
- mbrutsch 13y ago
  It took Alfred Whitehead and Bertrand Russell 379 pages to prove that it "logically follows", and that was before they even defined addition! http://quod.lib.umich.edu/u/umhistmath/aat3201.0001.001/401?page=root;rgn=full+text;size=100;view=pdf http://quod.lib.umich.edu/u/umhistmath/aat3201.0001.001/401?...
  - rntz 13y ago
    Whitehead & Russell didn't have the proof machinery we have today; they were inventing the wheel, and they did it badly. (Which is not to insult their achievement. Note that I say inventing the wheel, not reinventing it.)
- SamReidHughes2 13y ago
  It's a definition of what "2" means.
  - smsm42 13y ago
    It's one of the possible definitions. You could as well define next(x) = min(y such as y>x) and define 2 as next(next(0)) (0 can be defined as either x such as for all y y+x = y, or as x such that for all y x<y, depending on what set you are working with). Proving that 2 is 1+1 would be a theorem then. Of course, that works only for integers :)
  - trentmb 13y ago
    I was always under the impression that 2 is shorthand for s(s(0)).
    - Perseids 13y ago
      It depends on where you are coming from. If your context are ordinal numbers then you are right and a typical definition of 0 is {} and of s(x) is x∪{x}. But if you working with finite fields for example then you only have an addition operation. "Successor" does not make much sense there, since 1+1+…+1=0 for the right amount of additions (you are calculating modulo a prime). Since 0+1:=1 is trivial you usually start with 1 and define 2 as 1+1, 3 as 1+1+1…
      - pavelrub 13y ago
        You don't usually define 2 at all when talking about abstract structures (it makes little sense to call the polynomial 2 as being "the 2" of the field of rational functions, for example). 2 is something that exists only in N, and talking about it in other structures makes sense only when you are referring to a ring homomorphism Z->F or something similar.
        Perseids 13y ago
        That definitely is a more clean way to look at the matter, yes. Nonetheless I've seen the definition I stated above a few times and the merit is that you do not have to take an implicit indirection every time you state something like "2≠0". And as a ring homomorphism ℤ->R for any ring R with identity element is already completely defined and in effect identical to the definition "2:=1+1…" for every positive whole number, it is really just a different way of formulating the same idea.
placeybordeaux 13y ago
There are a couple of ways to define this part of mathematics, I believe I have seen some definitions that do in fact define `1 + 1 = 2`
_nedR 13y ago
Its funny how quickly the argument devolved into an argument over how (or even whether) 1 + 1 = 2
- Perseids 13y ago
  It is not a devolution. The discussion whether 1+1=2 is a definition, an axiom, or a theorem is very similar to the discussion of "0^0=1". In each case we are looking at different way to construct a useful mathematic out of different sets of definitions and axioms. Regarding 0^0 there is no clear winner, as - like the blog shows - there are many incompatible advantages one would get from 0^0:=1 or 0^0:=0. With "1+1=2" there is no controversy, as the many different definitions or axioms essentially result in the same nomenclature. But as an example of a rather unintuitive result regarding "1+1" look at the field F_2 (or GF(2)). It has two elements, 0 and 1. So when I'm writing "1" or "0" I am referring to these elements. Nonetheless with a typical definition used in many branches of mathematics I define 2:=1+1, 3:=1+1+1,… In F_2 the addition operation is defined as 0+0=0, 0+1=1, 1+0=1, and… 1+1=0. I thus get 0=1+1=2, 1=(1+1)+1=3, 0=((1+1)+1)+1)=4. Note that this is essentially calculation mod 2. But whereas outside of algebra this is often seen as a different set of equalities ("1+1≡0 mod 2") in algebra F_2 is just another field alongside of the rational, the real and the imaginary numbers.
pavelrub 13y ago
This is simply wrong. 1+1=2 is the same kind of statement as 2^1=2 - it follows from the definition of the operation and from a general rule (it would be absurd here to say that 2^1 is the definition of 2, or that 2^1=2 is part of the definition of ^). 0^0 is just a meaningless symbol unless given explicit definition. There is no popular set of axioms where 1+1=2 is taken as an axiom - it is always proven. The symbol 2 is not defined as 1+1. Conceptually this reflects the fact that knowing what 2 means is different from knowing what 1 means and what addition means. In principle a computer can correctly recognize 2 apples as being 2 apples without knowing anything about addition and without being able to recognize 1 apple.
- tomp 13y ago
  Maybe your parent meant that "1 + 1 = 2" is the definition of "2", in the sense that we write the symbol "2" to denote the Peano successor of "1", the Peano successor of "0", which is not a successor.
  - pavelrub 13y ago
    Maybe. Formally this is inaccurate as the successor function is not the addition function, and one has to prove that S(S(0))=S(0)+S(0). Edit: it should also be noted that this equality shouldn't seem "obvious": the name "successor" here is misleading, as it automatically links to our everyday understanding of a successor as being "the next number", in which case it is indeed obvious. But formally it's just a meaningless symbol that can be changed to other meaningless symbols by certain meaningless rules. Our everyday life association of (apple) -> S(0), (apple,apple)->S(S(0)) is then a kind of a model for those meaningless things, but from within the formal system - they have no meaning. In general I also do not like the invocation of formal arguments here, I think it misses the point. The bigger issue is that for us humans there is an obvious difference in kind between 1+1=2 and 0^0=1 - this should be obvious. Trying to hide this difference by appealing to different kinds of language trickery and formalism is an indication of doing something wrong, not of discovering anything new. The difference is still there, and we should understand why it's there, not force it to disappear (compare to the claim that every human act is egoistic in nature - what information does it really give us about the human nature?).

by rntz 13y ago

rickhanlonii 13y ago

I agree with the point you're making, but I want to be a little pedantic: It's true that addition is a definition, but 1+1=2 is not--it logically follows from the definition of addition.

Retric 13y ago
Not quite, just as in calculus you end up rediscovering some old geometry formulas there are branches of mathematics where you end up rediscovering 2 such that 1 + 1 equals it. Other options include 1 + 1 = 0 or 1+1 = 1.
- gordaco 13y ago
  But 1+1=0 or 1+1=1 happens because you are redefining the operator + (sometimes subtly, like using it over Z2), so what rickhanlonii said still holds. 1+1 is 2 if you're using the usual definition of sum, i.e., addition over integers or a superset of integers.
  - Retric 13y ago
    You missed the point 1 + 1 = 2 depends on the definition of '1', '+', '=', and '2'. The first case where this maters is probably binary where '2' does not exist, but that's hardly the only time 1+1 != 2.
mbrutsch 13y ago
It took Alfred Whitehead and Bertrand Russell 379 pages to prove that it "logically follows", and that was before they even defined addition! http://quod.lib.umich.edu/u/umhistmath/aat3201.0001.001/401?page=root;rgn=full+text;size=100;view=pdf http://quod.lib.umich.edu/u/umhistmath/aat3201.0001.001/401?...
- rntz 13y ago
  Whitehead & Russell didn't have the proof machinery we have today; they were inventing the wheel, and they did it badly. (Which is not to insult their achievement. Note that I say inventing the wheel, not reinventing it.)
SamReidHughes2 13y ago
It's a definition of what "2" means.
- smsm42 13y ago
  It's one of the possible definitions. You could as well define next(x) = min(y such as y>x) and define 2 as next(next(0)) (0 can be defined as either x such as for all y y+x = y, or as x such that for all y x<y, depending on what set you are working with). Proving that 2 is 1+1 would be a theorem then. Of course, that works only for integers :)
- trentmb 13y ago
  I was always under the impression that 2 is shorthand for s(s(0)).
  - Perseids 13y ago
    It depends on where you are coming from. If your context are ordinal numbers then you are right and a typical definition of 0 is {} and of s(x) is x∪{x}. But if you working with finite fields for example then you only have an addition operation. "Successor" does not make much sense there, since 1+1+…+1=0 for the right amount of additions (you are calculating modulo a prime). Since 0+1:=1 is trivial you usually start with 1 and define 2 as 1+1, 3 as 1+1+1…
    - pavelrub 13y ago
      You don't usually define 2 at all when talking about abstract structures (it makes little sense to call the polynomial 2 as being "the 2" of the field of rational functions, for example). 2 is something that exists only in N, and talking about it in other structures makes sense only when you are referring to a ring homomorphism Z->F or something similar.
      - Perseids 13y ago
        That definitely is a more clean way to look at the matter, yes. Nonetheless I've seen the definition I stated above a few times and the merit is that you do not have to take an implicit indirection every time you state something like "2≠0". And as a ring homomorphism ℤ->R for any ring R with identity element is already completely defined and in effect identical to the definition "2:=1+1…" for every positive whole number, it is really just a different way of formulating the same idea.

placeybordeaux 13y ago

There are a couple of ways to define this part of mathematics, I believe I have seen some definitions that do in fact define `1 + 1 = 2`

_nedR 13y ago

Its funny how quickly the argument devolved into an argument over how (or even whether) 1 + 1 = 2

Perseids 13y ago
It is not a devolution. The discussion whether 1+1=2 is a definition, an axiom, or a theorem is very similar to the discussion of "0^0=1". In each case we are looking at different way to construct a useful mathematic out of different sets of definitions and axioms. Regarding 0^0 there is no clear winner, as - like the blog shows - there are many incompatible advantages one would get from 0^0:=1 or 0^0:=0. With "1+1=2" there is no controversy, as the many different definitions or axioms essentially result in the same nomenclature. But as an example of a rather unintuitive result regarding "1+1" look at the field F_2 (or GF(2)). It has two elements, 0 and 1. So when I'm writing "1" or "0" I am referring to these elements. Nonetheless with a typical definition used in many branches of mathematics I define 2:=1+1, 3:=1+1+1,… In F_2 the addition operation is defined as 0+0=0, 0+1=1, 1+0=1, and… 1+1=0. I thus get 0=1+1=2, 1=(1+1)+1=3, 0=((1+1)+1)+1)=4. Note that this is essentially calculation mod 2. But whereas outside of algebra this is often seen as a different set of equalities ("1+1≡0 mod 2") in algebra F_2 is just another field alongside of the rational, the real and the imaginary numbers.

pavelrub 13y ago

This is simply wrong. 1+1=2 is the same kind of statement as 2^1=2 - it follows from the definition of the operation and from a general rule (it would be absurd here to say that 2^1 is the definition of 2, or that 2^1=2 is part of the definition of ^). 0^0 is just a meaningless symbol unless given explicit definition. There is no popular set of axioms where 1+1=2 is taken as an axiom - it is always proven. The symbol 2 is not defined as 1+1. Conceptually this reflects the fact that knowing what 2 means is different from knowing what 1 means and what addition means. In principle a computer can correctly recognize 2 apples as being 2 apples without knowing anything about addition and without being able to recognize 1 apple.

tomp 13y ago
Maybe your parent meant that "1 + 1 = 2" is the definition of "2", in the sense that we write the symbol "2" to denote the Peano successor of "1", the Peano successor of "0", which is not a successor.
- pavelrub 13y ago
  Maybe. Formally this is inaccurate as the successor function is not the addition function, and one has to prove that S(S(0))=S(0)+S(0). Edit: it should also be noted that this equality shouldn't seem "obvious": the name "successor" here is misleading, as it automatically links to our everyday understanding of a successor as being "the next number", in which case it is indeed obvious. But formally it's just a meaningless symbol that can be changed to other meaningless symbols by certain meaningless rules. Our everyday life association of (apple) -> S(0), (apple,apple)->S(S(0)) is then a kind of a model for those meaningless things, but from within the formal system - they have no meaning. In general I also do not like the invocation of formal arguments here, I think it misses the point. The bigger issue is that for us humans there is an obvious difference in kind between 1+1=2 and 0^0=1 - this should be obvious. Trying to hide this difference by appealing to different kinds of language trickery and formalism is an indication of doing something wrong, not of discovering anything new. The difference is still there, and we should understand why it's there, not force it to disappear (compare to the claim that every human act is egoistic in nature - what information does it really give us about the human nature?).