Is 1+1 a Provable Concept in Mathematics?

[lewis198]

Can 1+1 be proven? If so, then is mathematics a science?

 

[HallsofIvy]

No, 1+ 1 is not an observation. It is not even a statement! Did you mean "1+ 1= 2"? If so then it can be proven from the "axioms" and definitions of, say, the Peano model for the natural numbers. You asked two things: "Is 1+ 1 an observation?" and "can 1+ 1 be proven?". I don't know which of those the "if so" refers to. No, 1+ 1 (= 2) is not an observation, yes, it can be proven. Therefore, mathematics is not a science, at least not in the strict sense of being based on the "Scientific Method". The scientific method is itself based on observation and experimentation.

More fundamentally, all science is necessarily based on a "Realist" philosophy in that the "truth" of a theory depends on correspondence with reality (it matches experimental results) while mathematics is based on an "Idealist" philosophy in that the "truth" of a theory depends on consistency.

 

[lewis198]

Is the Peano model based on the assumption that 1+1=2?

 

[arildno]

I'd like to add something to Halls' post:

Tnere is nothing intrinsically wrong when defining a number system to define what your symbols are meant to mean.
Thus, the symbol "2" can be connected axiomatically as a true statement in the symbol expression 2=1+1, which might be regarded as the DEFINITION of 2.

But, what "is" our number system, really?
In particular, can we show that all those axioms we define a number system with can also define a model that we can use for something?

Can we CONSTRUCT from something more basic than number system something that PROVABLY behaves as our abstract "mumbers"?

Indeed we can, regarding "sets" as the most basic concept to develop maths from.

But, the axioms in SET theory doesn't specify "1" "+" "2" and so on in an entirely trivial way. Therefore, if we are to formulate something solely by aid/constraint of the axioms in set theory that will mimick our "numbers", then we must PROVE statements like 1+1=2, with suitable set definitions of 1,+,2 and =.

 

[Hurkyl]

Is the Peano model based on the assumption that 1+1=2?

In the usual formulation of Peano arithmetic, that is the definition of the number represented by '2'.

Peano arithmetic is a theory, not a model. Peano arithmetic is merely a formal language together with a list of axioms. Any mathematical structure that happens to satisfy those axioms is a model of peano arithmetic, and we often call such a thing a "set of natural numbers".

 

[arildno]

When the OP posted his question, I was thinking along the lines of Russell&Whitehead, that you know a lot more about than me, Hurkyl.

Hope my nonsense fraction was acceptably small..

 

[HallsofIvy]

Is the Peano model based on the assumption that 1+1=2?

No, it is not. That is a theorem (although a very easy one).

The "Peano Axioms" essentially say there exist a set of objects, N, called "numbers", together with a function, s, from N to itself such that:
1. There exist a unique member of N, called "1", such that s is a one-to-one function from N onto N-{1}.
(That essentially says that for y any member of N except 1, there exist a member of N, x, such that s(x)= y. There is NO x in N such that s(x)= 1. s is often called the "successor function". Every number has a successor and every number except 1 is the successor of some number.)

2. If a subset, X, of N as the properties that:
a) 1 is in X
b) whenever x is in X, s(x) is also in X
then X= N.
(This is the "principle of induction".)

We then define addition, +, by
For any a in N, a+ 1= s(a). If b is also in N, b not equal to 1, then, (by (1) above) there exist c such that b= s(c) and a+ b= s(a+ c).

One needs to show that this is "well defined"- that is given any a and b, this defines a member of N- that's not too difficult but tedious.

Finally, we define 2 to be s(1). Now it is easy to prove that 1+1= 2. From (1) above, and the the first part of the definition of addition, 1+ 1= s(1)= 2. We have proved, from the axioms and the definitions of "addition" and 2, that 1+ 1= 2. (Notice that 1 is not "defined"- it is "given" in the first axiom.)

A little more complicated is the proof that 2+ 2= 4. We have already defined 2 as s(1). Now we define 3 to be s(2) and 4 to be s(3).

Then, by the second part of the definition of addition, 2+ 2= s(2+1). By the first part of the definition of addition, 2+ 1= s(2)= 3 so 2+ 2= s(3)= 4.

 

[Hurkyl]

When the OP posted his question, I was thinking along the lines of Russell&Whitehead, that you know a lot more about than me, Hurkyl.

Hope my nonsense fraction was acceptably small..

I know of Principia Mathematica, but I've not actually studied it. But judging from how I hear it described, I'm quite content to stick with modern styles.

 

[Feldoh]

Basically any mathematical field is started by axioms, or things that you hold to be true that don't need to be proven. Everything else is build up from the axioms in a particular field. Or at least that's how I came to understand it...

 

[eastside00_99]

)while mathematics is based on an "Idealist" philosophy in that the "truth" of a theory depends on consistency.

Yay, Kant!

 

[mcampbell]

1+1=2 might be argued by many to be a report of an observation, but a statement like 123456789012345678+1=123456789012345679 could not be an observation made by a human (perhaps a counting machine?)

 

[HallsofIvy]

Argued, yes, but incorrectly. "1 apple + 1 apple= 2 apples" is an observation. "1 rock+ 1 rock= 2 rocks" is an observation. "1+ 1= 2" is not. You can "observe" 1 apple or 1 rock. You can not "observe" 1!

 

[Werg22]

Though when thinking about it, it seems impossible, at least to me, to define addition by other ways than mapping the natural numbers to objects, either of our thought or physical experience.

Edit: Perhaps defining addition in these terms would work: we define 0 + 0 = 0, 0 + 1 = 1 and 1 + 1 = 2. We can then refer to the binary addition algorithm for all other additions. Whenever using the addition algorithm in another number system, we'd need a higher number of defined additions, but we can define these by converting to binary, using the addition algorithm, and converting back.