Prove 1+1 = 2 in fewer pages

[StevieTNZ]

I dare anyone to prove 1+1 = 2, in one less page than already is; not reducing the font size or changing font, either.

Whoever does so will receive US$5 off their next psychic appointment/end-of-world date prediction email subscription. Only redeemable at Evo and I's business to be set up at some point in the future.

 

[zoobyshoe]

How can you prove a stipulation? 1+1=2 is a statement that stipulates 1+1 may also be referred to as "2". There's nothing to prove.

 

[phinds]

How can you prove a stipulation? 1+1=2 is a statement that stipulates 1+1 may also be referred to as "2". There's nothing to prove.

Actually, I seem to recall that in an advanced math graduate class I took some 50 years ago, one of the exercises was to use Peano's Postulates to prove exactly that. 1+1=2

I'm an engineer, not a mathematician and I remember thinking the whole thing seemed stupid from my practical point of view but I understood that as an exercise in mathematical formalism it could be useful to folks who care about that sort of thing.

Stevie, is there some reason for your rant or did you just need to get that off your chest?

 

[Nikitin]

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

 

[FlexGunship]

i dare anyone to prove 1+1 = 2, in one less page than already is; not reducing the font size or changing font, either.

Whoever does so will receive us$5 off their next psychic appointment/end-of-world date prediction email subscription. Only redeemable at [Evo's and my] business to be set up at some point in the future.

2-1=1
2-1+1=1+1
1+1=2

(Edit in bold)

 

[qspeechc]

If I hold up one finger, then another one finger, I get two fingers.

 

[f95toli]

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

 

[Greg Bernhardt]

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

Some people like reading Harry Potter. Some people like doing math exercises.

 

[D H]

I dare anyone to prove 1+1 = 2, in one less page than already is; not reducing the font size or changing font, either.

What, exactly, do you mean by "1", "+", "=", and "2"? The proof is incredibly trivial or incredibly complex depending on how deep you want to go.

Some claim you cannot prove a definition. That's not quite true. Better said, the proof of a definition is trivial: Cite the definition. End of proof. Given that, here's the trivial proof:

Proof that 1+1=2: The definition of 2 is that it is the natural number that satisfies 1+1=2. QED.​

Defining "1", "+", "=" as per the Peano axioms and defining 2 as the successor of 1 requires a tiny bit more work, but not much. On the other hand, going the full nine yards as was done by Whitehead requires an immense amount of work and won't fit on a page.

So what do you want? The proof is either trivial and requires but a line or two, or it's incredibly non-trivial and cannot be reproduced without writing a book.

 

[jim hardy]

This one has been kicking around since at least 1966 when I saw it on a math dep't bulletin board at U of Miami.
Today I found it many places by a search on "unknown but astute source"

This is a pdf from U of Chicago:

http://www.oekonometrie.uni-saarland.de/oekonometrie/oeko2011/FirstLesson.pdf

 

[phinds]

http://www.oekonometrie.uni-saarland.de/oekonometrie/oeko2011/FirstLesson.pdf

That's funny

 

[FlexGunship]

This one has been kicking around since at least 1966 when I saw it on a math dep't bulletin board at U of Miami.
Today I found it many places by a search on "unknown but astute source"

This is a pdf from U of Chicago:

http://www.oekonometrie.uni-saarland.de/oekonometrie/oeko2011/FirstLesson.pdf

I was fine until we made zero a vector.

 

[WannabeNewton]

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

If this is the overarching mentality then much of pure math is a waste of energy. Regardless, intellectual stimulation is intellectual stimulation regardless of how "applicable" it is to the real world.

 

[ModusPwnd]

If this is the overarching mentality then much of pure math is a waste of energy.

Which is why there are so few workers in the world doing it. Still, its certainly funner than watching a basketball game (IMO).

 

[qspeechc]

If I hold up one finger, then another one finger, I get two fingers.

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.

 

[bp_psy]

If I hold up one finger, then another one finger, I get two fingers.

If you hold one finger then cut one of your other fingers of and hold it in another hand.Do you still have two fingers?

 

[D H]

So "1+1=2" is true for concrete, physical objects like fingers, therefore it cannot be false for numbers.

Nonsense. 1 drop of water + 1 drop of water = 1 bigger drop of water.

 

[ModusPwnd]

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.

You want to experimentally verify mathematics? I don't think so. It works the other way around. You experimentally verify science, not math.

 

[qspeechc]

Nonsense. 1 drop of water + 1 drop of water = 1 bigger drop of water.

And 1 bigger drop = 2 drops. You are just giving two drops another name

 

[ModusPwnd]

And 1 bigger drop = 2 drops. You are just giving two drops another name

Thats the same thing you did when you appealed to fingers... You arbitrarily choose a naming scheme that would support your preconceived conclusion.

 

[bp_psy]

And 1 bigger drop = 2 drops. You are just giving two drops another name

What if it was made up of 4 smaller drops? Your system does not give unique labels to your objects. How do define your '1' drop?

 

[qspeechc]

I thought about it, and the problem is actually that "drop" is a vague word, it describes something but doesn't define it. In "1 drop + 1 drop = 1 bigger drop" each time you use "drop" it actually has a different meaning.

Thats the same thing you did when you appealed to fingers... You arbitrarily choose a naming scheme that would support your preconceived conclusion.

I don't understand. Was my meaning of "finger" arbitrary and chosen to support my conclusion?

You want to experimentally verify mathematics? I don't think so. It works the other way around. You experimentally verify science, not math.

Then tell me where the abstract numbers "1", "2" etc. come from.

What if it was made up of 4 smaller drops? Your system does not give unique labels to your objects. How do define your '1' drop?

I think you're getting at the same thing as me.

And everyone has conveniently forgotten this:

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.

 

[ModusPwnd]

I don't understand. Was my meaning of "finger" arbitrary and chosen to support my conclusion?

Yes, I think so.

Then tell me where the abstract numbers "1", "2" etc. come from.

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.

 

[qspeechc]

Yes, I think so.




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.

Ha ha ha. Yes. Indeed. Ho Ho Ho. Good one. Yes. Very good.

 

[SW VandeCarr]

Afaik, Boolean arithmetic is just as valid as standard arithmetic. In Boolean arithmetic 1+1=1

 

[collinsmark]

It's also perfectly legitimate to specify a Galois field (GF) and use the corresponding arithmetic within it. For example, in GF(2)

1 + 1 = 0.​

 

[D H]

Afaik, Boolean arithmetic is just as valid as standard arithmetic. In Boolean arithmetic 1+1=1

And arithmetic modulo 2 is just as valid as either, in which 1+1 = 0.Bottom line: You can't prove 1+1=2 unless and until you define what "1", "+", "=", and "2" mean.

 

[qspeechc]

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.

 

[D H]

Ha ha ha. Yes. Indeed. Ho Ho Ho. Good one. Yes. Very good.

ModusPwnd was not joking.

You do realize that mathematics is not science, don't you?

Scientific theories cannot be proven to be true. Scientific theories are at best provisionally true. All it takes is one experiment to demonstrate that the theory does not comport with reality and kaboom! the theory is dead (or at least needs modification).

Mathematical theorems can be proven to be true, and once proved to be so, they remain true for all time. Mathematics is not necessarily connected with reality. That the universe is non-Euclidean does not mean that Euclid's geometry has been falsified. Euclid's theorems are as valid now as they were 2300 years ago.

 

[collinsmark]

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.

Galois fields (finite number systems) have plenty of real-world applications. Any time you watch Blu-ray or DVD,or listen to a CD, finite number systems are involved. The information on these media are encoded with a forward error correction technique called Reed-Solomon codes. Reed-Solomon codes can be considered a subset of BCH codes (short for Bose, Chaudhuri, Hocquenghem). That way, if you scratch the disk, all is not necessarily lost.

So these types of number systems are not too terribly removed from practical life after all.