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Comment by rntz | Hacker News Reader

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.

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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.

This is a good point. Your grandparent's original distinction was that 1+1=2, just as 2^1=2, follows from a definition while 0^0=1 is immediately defined. However, the parent's still valid point was clearly that even the plus operator is in the larger sense just as arbitrary as the definition of 0^0, concurring with the overall theme of this thread that mathematics ultimately gives no credibility to intuition.

What do you think the definition of ^ is then?

The ^ operator is defined as:

  0^0 = 1
  x^y = exp(ylog(x))  if  x != 0
  0^y = 0             if  y != 0

Or similarly set theoretically.

I'm happier starting with the natural numbers, and just defining m^n as m multiplied by itself n times. Everyone gets that this is the point of exponentiation of natural numbers, and it has the obvious recursive definition:

    m^0 = 1
    m^n = m^(n-1).

And now 0^0 = 1 follows immediately, just as it does for 0+0 = 0 and the recursive definition of addition in terms of successor.

After that, I consider the question of what x^y means for non naturals as motivated largely by considerations of algebraic and analytic extension. We get negative exponents simply by extending the operation with the requirement that

    m^0 = 1
    m^(n+p) = m^n*m^p.

We get rational exponents by extending the operation with the requirement that

    (m^n)^p = m^(n*p).

We get real exponents by extending the operation to the limits where they exist. We get complex exponents by extending the operation to the largest analytic extension.

The tension then is that the real and complex extensions don't "confirm" the original reasonable setting of 0^0=1, but I personally find it more objectionable poking a hole in the function, when it was so clear in the case of natural numbers that 0^0 should be 1 (and see the set theoretic and category theoretic accounts for more reasons why this is clear).

Most real numbers only exist in fever dreams and acid trips, anyway.

Whatever it is, 2^1=2 is not part of it - it follows from it.

Uhm, anything that is part of a definition follows from that definition.

`A and B implies A' and so on.

In the end a function, in the mathematical sense, is just a long (possibly infinite) table of `from' and `to' values. We could argue that one definition of ^ is more or less complex than another, for some definition of complexity, but I'm not sure which would win, and if it would be useful.

"Follows" in the sense that it is not an axiom - it has a proof of more than one line in the sense of the Hilbert system. Seeing functions as collections of ordered pairs tells us nothing here.

Your definition of ^ doesn't directly tell me that 2^1=2. It does tell me however that 0^0=1.

Isn't that a circular definition, since, exp is e^x?

We can just define exp as:

  exp(x) = 1 + x/1! + x*x/2! + x*x*x/3! + ...

If one defines x^y this way, one usually defines exp by a power series, and then defines e as the number such that e^x = exp(x).

Or of course just e = exp(1)

>There is no popular set of axioms where 1+1=2 is taken as an axiom - it is always proven.

I would love to see a proof of 1+1=2. For example in the case of elliptic curves, as far as I know addition is simply an axiom. E.g. A+A'=0=inf , where inf is the point at infinity and A' is the reflection of A.

Math is not related to nature. Try to reason your way to matrix multiplication with apples: A * B != B * A

Using the Peano axioms, we need to prove that S(0)+S(0)=S(S(0)).

From the definition of +: S(0)+S(0)=S(S(0)+0) Again using the definition of +: S(0)+0=S(0). And we get: S(0)+S(0)=S(S(0))

Q.E.D.

There is also a famous proof by Whitehead and Russell on page 379 of Principia Mathematica: http://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhist...

Thank you, I didn't know you could prove that!

I believe my disagreement was specifically about this (quoted):

> 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.

I have a hard time seeing this. If I am not mistaken again, there is no definition of the symbol 2 that does not include addition. The proof above doesn't really prove this either as Peano's axiom rely on the definition of "successor". Successor definition: "a+1 is the successor of a".

http://mathworld.wolfram.com/PeanosAxioms.html

http://mathworld.wolfram.com/Successor.html

See the edit in my reply to tomp. In the Peano axioms, 2 is defined as S(S(0)), not as S(0)+1. The correct way to think about this is to ignore any inclination to give those symbols any "real world" meaning. From the point of view of the formal system - they are just strings, and the only thing we know about them is how to manipulate them to form other strings. Under this perspective, the connection between S(S(0)) and S(0)+1, or in general between S(a) and a+1, is again something which requires proof: S(a) = S(a+0) = a + S(0).

A computer can be given an explicit map between the symbols S(0), S(S(0)), s(S(S(0))), ... and 1,2,3..., so to identify S(S(S(S(S(S(0)))))) with 6 - it wouldn't need to know anything at all about + or about 1.

Edit: also consider the following quote from Wittgenstein's Philosophical Investigations and whether the person described needs to have any concept of 'addition' in order to correctly use numbers: "Now think of the following use of language: I send someone shopping. I give him a slip marked 'five red apples'. He takes the slip to the shopkeeper, who opens the drawer marked 'apples', then he looks up the word 'red' in a table and finds a colour sample opposite it; then he says the series of cardinal numbers--I assume that he knows them by heart--up to the word 'five' and for each number he takes an apple of the same colour as the sample out of the drawer.--It is in this and simlar ways that one operates with words--"But how does he know where and how he is to look up the word 'red' and what he is to do with the word 'five'?" ---Well, I assume that he 'acts' as I have described. Explanations come to an end somewhere.--But what is the meaning of the word 'five'? --No such thing was in question here, only how the word 'five' is used."

This is quite beyond my knowledge, I like the Wittgenstein philosophy, but I would still argue:

You cannot calculate S^(x+1)(0) before the result of S^x(0) is known.

To calculate S^6(0) you start by calculating S(0) ----> It's not possible to "identify S^6(0) with 6" if you haven't calculated S(0) first, because you cannot know S^6(0) at this point.

The computer has a table in memory that tells it S^6(0) = 6 (or equivalently O O O O O O = 6 apples). When it sees a symbol - for example S^6(0) - it searches this table for the same symbol, and then outputs the corresponding number. In no stage of this process does the computer "calculate" anything - it doesn't even need to know that S^5(0)=5 in order to find that S^6(0)=6. In fact S^5 might not even be in the table.

Here is a simple definition of the natural numbers and addition, in Haskell:

    data Nat = Zero | Suc Nat

    plus :: (Nat, Nat) -> Nat
    plus (Zero, y) = y                  -- axiom 1
    plus (Suc x, y) = Suc (plus (x, y)) -- axiom 2

    one = Suc Zero
    two = Suc one

Here is a proof that plus (one, one) = two:

    plus (Suc Zero, Suc Zero)
    = Suc (plus (Zero, Suc Zero))  [by axiom 2]
    = Suc (Suc Zero)               [by axiom 1]

Can you please translate that to mathematics? My request was regarding a formal mathematical proof.

The code I gave translates easily into Agda, a computerized proof checker, and as such more formal than most mathematics. However, here's the same thing in a modernized version of Peano arithmetic. We assume all the usual properties of equality: reflexivity, symmetry, transitivity, and substitution.

    -- Axioms (only 1 and 2 are relevant)
    1. 0 ∈ N
    2. ∀ x∈N. S(x) ∈ N
    3. ∀ x∈N. 0 ≠ S(x)
    4. ∀ x∈N, y∈N. S(x) = S(y) ⊃ x = y
    5. P(0) ∧ (∀ x∈N. P(x) ⊃ P(S(x))) ⊃ ∀ x∈N. P(x)

    -- Definition of addition
    6. ∀ a∈N. a + 0 = a
    7. ∀ a,b ∈ N. a + S(b) = S(a+b)

    -- Proof that S(0) + S(0) = S(S(0))
    9. S(0) ∈ N                     [from 2 and 1]
    10. S(0) + S(0) = S(S(0) + 0)   [from 7 and 9]
    11. S(0) + 0 = S(0)             [from 6 and 9]
    12. S(S(0) + 0) = S(S(0))       [substitution of equals, from 11]
    13. S(0) + S(0) = S(S(0))       [transitivity from 10 and 12]

Edit: Ah, I see I was beaten to it by pavelrub.

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.

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?).

I did not in fact mean that 2 is defined as 1+1; you (and several others) are correct in saying that 1+1=2 requires proof for all reasonable definitions of +. 0^0 = 1 follows much more directly from the definitions in which it is true. My overall point has been articulated by arketyp in a sibling post to tomp's; like ^, + is "just a definition" which we arranged so as to line up with our intuitions.

I disagree with your assertion that 1+1=2 and 0^0=1 are of completely diferent kinds. That depends on what intuitive angle you approach them from; my intuition for ^ is that it is the iteration of multiplication, and in this context the only sensible definition is that x^0 = 1 (the unit of multiplication), for any x. The generalization to continuous bases and exponents should preserve this property.

But this is certainly a matter of taste, and certainly you'll find more disagreement here than in the matter of 1+1=2. What this tells us about human nature, I think, is not much: humans will disagree more when things get more complicated (^ is more complicated/abstract than +) and there is more room for disagreement.

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.

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?...

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.)

It's a definition of what "2" means.

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 :)

I was always under the impression that 2 is shorthand for s(s(0)).

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…

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.

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.

My point is that when talking about the definition of 2, it is enough to restrict ourselves to the natural numbers - since that is where 2 is coming from. If we then want to extend this symbol to other places - the meaning of such an extension will be given by a map, not by requiring a different definition. That is - there is only one 2, everything else is ψ(2) for the homomorphism ψ:Z->R.

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.

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.

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.

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

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.

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`

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