Prove 1+1 = 2 in Fewer Pages & Win US$5 at Evo & I

[SW VandeCarr]

You can't pretend it wasn't clear we were talking about ordinary arithmetic all along. Why would anyone say "prove 1+1=2" if he meant in arithmetic modulo 2, for example.

But you're asking for a proof. A proof is based on a certain set of assumptions which are (taken to be) consistent. Different kinds of arithmetic are equally valid according to the assumptions on which they are based.

EDIT: I didn't see the other posts that just preceded mine. I didn't intend to be redundant. However, consider this. In standard arithmetic we can say 1+1=10 in binary notation. It's still standard arithmetic. There's nothing special about base ten except habit and convenience.

 

[FlexGunship]

http://lmgtfy.com/?q=1+1

 

[qspeechc]

Yes, I think so.

From The Pocket Oxford Dictionary, 5th ed.: “finger. 1. n. Any of five or (excluding thumb) four terminal members of hand.”
That is the definition I used, and as far as I know, the only definition in use.

Why do they have to "come from" anywhere? They are postulated or defined axiomatically. When you start to attribute these abstract concepts onto "real world" objects like fingers and water drops you start doing science rather than mathematics.

Oh sigh! I was hoping I didn’t need to give this long-winded answer. I was trying avoid answering it without being noticed. Well, anyway!

From Principles of Mathematics, 2nd ed. 1963 (? not sure) written by Allendoerfer and Oakley, both former mathematics professors:

Virtually all the mathematics with which you are familiar had its roots somewhere in nature. Arithmetic and algebra grew out of men's needs for counting, financial management, and other simple operations of daily life; geometry and trigonometry developed from problems of land measurement, surveying and astronomy. [...] In recent years new forms of mathematics have been invented to help us cope with problems in social science, business,[...etc.]. Let us lump all these sources of mathematical ideas together and call them Nature.

At first our approach to nature is descriptive, but as we learn more about it and perceive relationships between its parts, we begin to construct a Mathematical Model of nature. [...] Perhaps you are familiar with this sort of process through your study of geometry, in which the axioms form an abstract description of what man saw when he began to measure the earth [italics mine].[...]

The next step in the process is to deduce the consequences of our collection of axioms. By applying logical methods of deduction we then arrive at theorems. These theorems are nothing more than logical conclusions from our axioms and must not be assumed to be firm statements about relationships which are necessarily true in nature.

You get the idea. There's a diagram in the book, which basically goes:
Nature --> Definitions, Axioms --> Theorems, Rules --> Nature (through applications of mathematics)

Unfortunately, the way mathematics is taught, many people think mathematics has nothing to do with reality, that there is no place in mathematics for intuition, insight, meaning, examples (which are like the scientist's observations and data), which is unfortunate. Does anyone think mathematics is just a big game of logic with no meaning or intuition at all? That mathematicians just suck theorems and definitions out of their thumbs? I think not.

 

[qspeechc]

http://lmgtfy.com/?q=1+1

Lol, thanks.

Is someone actually on my side in this discussion?

 

[phinds]

Since no one noticed this, let me expand.

1 finger + 1 finger = 2 fingers

So "1+1=2" is true for concrete, physical objects like fingers, therefore it cannot be false for numbers. Numbers are just abstractions of real things like fingers.

Furthermore, when you add 1 the result gives the next number, and "2" is simply the name of the number after 1, therefore "1+1=2" is a tautology.

This is an "engineering" kind of solution that utterly lacks mathematical rigor. It IS the kind of thing I personally agree w/ but I don't pretend that it is a rigorous proof.

 

[Dmobb Jr.]

First define 0: 0 = {}
Define 1: 1={{}}
Define 2: 2={{},{{}}}
Define =: A = B iff (for all x (x in A iff x in B)
In other words, two sets are equal if they have the same elements.
Define S(): S(x) = xU{x}
Define +:
1) A + 0 = A
2) A + S(B) = S(A) + B

Now for the proof:

Clearly S(0) = S({}) = {} U {{}} = {{}} = 1
and S(1) = S({{}}) = {{}}U{{{}}} = {{},{{}}} = 2
So 1 = S(0) and S(1) = 2
1 + 1 = 1 + S(0) = S(1) + 0 = S(1) = 2



Then again that is just as valid as defining 2 as 1+1.

 

[micromass]

Lol, thanks.

Is someone actually on my side in this discussion?

Yes. I agree with you. I agree that 1 finger + 1 finger = 2 fingers proves that 1+1=2 (when everything is interpreted normally).

A formal proof of this result can be given but is not necessary. In fact, if we invent axioms and we find out that $1+1\neq 2$, then our conclusion would not be that 1+1=2 is not true, but rather that our axioms are wrong and do not describe real life. Rigorouss proofs are not invented in order to show that 1+1=2, because we know that from experience. Rigorous proofs are invented for this in order to show that our axioms are right.

I think it's very silly to somehow pretend that mathematics is somehow disjoint from nature. We care about certain mathematics because it occurs in nature and in other fields. We don't just invent some arbitrary axioms that means nothing, our axioms are always grounded in things we can observe and feel. I admit that mathematics abstracts things so much that somehow it is not clear anymore what we are studying, but initially at least the axioms always have some purpose and are not arbitrary.

 

[ModusPwnd]

We don't just invent some arbitrary axioms that means nothing...

Why not? I think some mathematicians do do that.

If math must be informed by and inform us of nature then why is it distinguished from science? Empiricism is a requirement of science, not math..

Rigorous proofs are invented for this in order to show that our axioms are right.

Axioms cannot be right or wrong. They are assumed, by definition. What you are describing sounds more like a scientific postulate, like from physics.

 

[Dmobb Jr.]

Why not? I think some mathematicians do do that.

Well as far as I know mathemeticians sort of do it. They do it but the work isn't particularly respected or cared about by the mathematical community.

 

[SW VandeCarr]

Yes. I agree with you. I agree that 1 finger + 1 finger = 2 fingers proves that 1+1=2 (when everything is interpreted normally).

Well, the OP asked for a proof of a formal statement: 1+1=2. That implies a formal proof, no?
It seems that the OP asked for something and then objected when he/she got a discussion about what was asked.

[EDIT] Sorry, it was qspeechc that objected, but the argument is the same. The stated subject of the thread is a formal proof.

 

[D H]

It seems that the OP asked for something and then objected when he/she got a discussion about what was asked.

The OP has not said one single thing since the original post. And that's a problem.

 

[Greg Bernhardt]

The OP has not said one single thing since the original post. And that's a problem.

Might have had a busy day at work, give him time :)

 

[StevieTNZ]

It is currently proved in x number of pages. I'm asking for the proof to be in x-1 pages. So three lines doesn't meet the criteria, sorry, FlexGunship.

(don't know the figure x, but it is certainly more than 100 - refer Principia Mathematica)

 

[StevieTNZ]

Shows how much attention was paid when reading! Correct, I had not said anything - thus invaliding 'It seems that the OP asked for something and then objected when he/she got a discussion about what was asked.'

Plus, I asked for the proof to be in one page less than currently is, i.e. Principia Mathematica. Therefore a proof on 3 lines will not suffice for a US$5 discount coupon.

 

[AnTiFreeze3]

Why not spend the energy on something applicable to the real world instead?

Pure math is done because people find it enjoyable, not because there might be some later consequence or byproduct of its applicability that relates it to the real world. If this approach were taken from the very origin of mathematics then much of what we know today would be lost.

 

[AnTiFreeze3]

... Unfortunately, the way mathematics is taught, many people think mathematics has nothing to do with reality, that there is no place in mathematics for intuition, insight, meaning, examples ...

I'm not sure what classes or books you have gone through, but in my experiences, this couldn't be further from the truth. The purpose of a text in mathematics is to develop one's knowledge and problem solving abilities with regards to the topic of the book, and the only way to do this is to build one's mathematical intuition, often through examples.

I'm indifferent towards everything else you've said

 

[MathematicalPhysicist]

The "full" proof can be found in Russell and Whiteheads "Principia Mathematica", and no it can not be done on one page (it takes them about 300 pages to get to the proof).

But then Gödel came along and showed that the whole exericise was futile

I believe that most mathematicians laughed at them when they heard that it took them 300 pages to prove a definition... :-D

 

[collinsmark]

I believe that most mathematicians laughed at them when they heard that it took them 300 pages to prove a definition... :-D

No. From Wikipedia: "'From this proposition it will follow, when arithmetical addition has been defined, that 1+1=2.' —Volume I, 1st edition, page 379 (page 362 in 2nd edition; page 360 in abridged version). (The proof is actually completed in Volume II, 1st edition, page 86, accompanied by the comment, 'The above proposition is occasionally useful.')"

 

[qspeechc]

The point is that you don't "prove" 1+1=2. Russell and Whitehead were not trying to prove 1+1=2, they were trying to deduce it. They were trying to build a foundation from which all of mathematics would follow, including 1+1=2. If their system proved, or rather "deduced", anything else they would know it was wrong, and was not the foundation they were looking for.

Do you think that before 1+1=2 was "proven", all mathematicians went about saying "Well, dear me, if someone actually proves 1+1=2 is not true then all my life's work is rubbish"?

Edit: I think micromass has already made this point

 

[qspeechc]

I believe that most mathematicians laughed at them when they heard that it took them 300 pages to prove a definition... :-D

Yes, "1+1=2" is actually a definition. It's inconvenient to always go about saying "one plus one", so, for brevity, we call this "two". So we say "57" instead of "1+1+1+1+...+1"

 

[MathematicalPhysicist]

I don't understand why all this work took them something like three volumes of hundreds of pages, I read somewhere that Quine said their work could be much more brief in the number of pages, as in only 200 and something pages.

Philosophers...

 

[SW VandeCarr]

I don't understand why all this work took them something like three volumes of hundreds of pages, I read somewhere that Quine said their work could be much more brief in the number of pages, as in only 200 and something pages.

Philosophers...

What's wrong with this proof using Peano's axioms? If "one" is represented as the successor of zero, S(0), then it's sufficient to prove S(0)+S(0)=S(S(0)).

http://www.xamuel.com/proving-1-plus-1-equals-2/

 

[MathematicalPhysicist]

Well from the axioms it's just a matter of simple deduction.

But I hope you see the circularity of proving it, I mean the successor operator and its definition presuppose you know what are the natural numbers.

 

[SW VandeCarr]

Well from the axioms it's just a matter of simple deduction.

But I hope you see the circularity of proving it, I mean the successor operator and its definition presuppose you know what are the natural numbers.

I'm not sure what you mean by "circular". The Peano axioms give us a way to construct the natural numbers given the existence of zero. Does that mean it presupposes the existence of the natural numbers?

 

[MathematicalPhysicist]

You suppose that zero and one exist and that they are different, otherwise all the numbers will be equal to zero.

I mean the axiomatics just put in formal way what we know already from first grade or kindergarden, or whatever.

As Kronecker put it:"God gave us the natural numbers, all else is man made".

 

[SW VandeCarr]

You suppose that zero and one exist and that they are different, otherwise all the numbers will be equal to zero.

I mean the axiomatics just put in formal way what we know already from first grade or kindergarden, or whatever.

Be that as it may, most of mathematics is based on axioms, theorems and formal proofs. The OP asked for a proof, whether it was serious or not. 1+1=2 is a formal statement. That is, it's a well formed formula that admits a proof under the Peano axioms (using the Peano formalism). The proof may be simple, even trivial, but a proof was what was requested.

BTW, my understanding is that the Peano system only presupposes zero and a successor function exist. Under the Peano system 0 and S(0) are clearly not the same.

 

[Dmobb Jr.]

It is currently proved in x number of pages. I'm asking for the proof to be in x-1 pages. So three lines doesn't meet the criteria, sorry, FlexGunship.

(don't know the figure x, but it is certainly more than 100 - refer Principia Mathematica)

In Principia Mathematica, type theory is used, which is a very complicated set theory. In modern set theory it is quite easy to prove 1 + 1 = 2 as I showed in post 36. This is a formal proof like Russell's, except it uses the now standard ZFC axioms.Edit: I have not actually read Principia Mathematica so correct me if I am wrong.