What is 1+1? And why?

So you don't know, thanks

No that is evidence of one peanut and one peanut.

prove 2 exists

1 + 1
= 1 + successor(0) [definition of 1, 1 = successor(0)]
= successor(1 + 0) [2nd Peano axiom for addition, a + successor(b) = successor(a + b)]
= successor(1) [1st Peano axiom for addition, a + 0 = a]
= 2 [definition of 2, 2 = successor(1)

QED.

>Now I add another (1) peanut to the location that contains my previous peanut.
Sorry, but in our 4-dimensional spacetime two objects cannot occupy the same location in space

2+2 =4
-1 = 3

>Now I add another (1) peanut to the location that contains my previous peanut.
[math]Peanut^2[/math]

>
>>Now I add another (1) peanut to the location that contains my previous peanut.
>Sorry, but in our 4-dimensional spacetime two objects cannot occupy the same location in space
Further to that...two objects cannot occupy the same location in space...atthe same time: thus if you indeed placed another peanut in the same location occupied to the previous peanut, it means that you displaced the previous peanut and therefore, there is still only one peanut at that location.

1+1 is a mathematical expression that evaluates to:
>2 (in ordinary arithmetic)
>1 (in Boolean algebra with a notation where '+' denotes a logical disjunction)
>0 (in Boolean algebra with a notation where '+' denotes 'exclusive or' operation, or in a quotient ring of numbers modulo 2)

so you've proven it for peanuts, but what about the general case?

x+x=2x

nat : ∀α.(α→α)→(α→α)

zero : nat
zero : ∀α.(α→α)→(α→α)
zero : ∀α.τ τ = (α→α)→(α→α)
Λα.m ∶ ∀α.τ zero = Λα.m
Γ,α ⊢ m:τ
Γ,α ⊢ m:(α→α)→(α→α)
Γ,α ⊢ m:t1→t2 t1, t2 = α→α
Γ,α ⊢ λf∶t1.M:t1→t2 m = λf∶t1.M
Γ,f∶t1 ⊢ M∶t2
Γ,f∶α→α ⊢ M∶α→α
Γ,f∶α→α ⊢ M∶t4→t5 t4, t5 = α
Γ,f∶α→α ⊢ λx∶t4.N:t4→t5 M = λx∶t4.N
Γ,f∶α→α,x∶t4 ⊢ N∶t5
Γ,f∶α→α,x∶α ⊢ N∶α
Γ,f∶α→α,x∶α ⊢ x∶α N = x
no remaining proof goals

zero = Λα.m = Λα.λf∶t1.M = Λα.λf∶α→α.M = Λα.λf∶α→α.λx∶t4.N = Λα.λf∶α→α.λx∶α.N = Λα.λf∶α→α.λx∶α.x

We have proved that 0 is a natural number. The rest is left as an exercise to the reader.

>nat : ∀α.(α→α)→(α→α)

>zero : nat
>zero : ∀α.(α→α)→(α→α)
>zero : ∀α.τ τ = (α→α)→(α→α)
>Λα.m ∶ ∀α.τ zero = Λα.m
>Γ,α ⊢ m:τ
>Γ,α ⊢ m:(α→α)→(α→α)
>Γ,α ⊢ m:t1→t2 t1, t2 = α→α
>Γ,α ⊢ λf∶t1.M:t1→t2 m = λf∶t1.M
>Γ,f∶t1 ⊢ M∶t2
>Γ,f∶α→α ⊢ M∶α→α
>Γ,f∶α→α ⊢ M∶t4→t5 t4, t5 = α
>Γ,f∶α→α ⊢ λx∶t4.N:t4→t5 M = λx∶t4.N
>Γ,f∶α→α,x∶t4 ⊢ N∶t5
>Γ,f∶α→α,x∶α ⊢ N∶α
>Γ,f∶α→α,x∶α ⊢ x∶α N = x
>no remaining proof goals

>zero = Λα.m = Λα.λf∶t1.M = Λα.λf∶α→α.M = Λα.λf∶α→α.λx∶t4.N = Λα.λf∶α→α.λx∶α.N = Λα.λf∶α→α.λx∶α.x

>We have proved that 0 is a natural number. The rest is left as an exercise to the reader.

Proof in the surreal numbers

1 + 1
= {0|} + {0|}
= {0+{0|},{0|}+0|}
= {{0|}|}
= {1|}
= 2

...

QED

this fails in the general case
consider this counterexample:
you take a pile of sand, and add it to a pile of sand
you are left with a single pile of sand

Because what you're defining as a "pile of sand" isn't being upheld. It would be more logical to use "grains of sand" instead of piles. It's less the fault of "1+1=2" and more a fault of definition.

>Let 2 be a natural number defined as 1+1
>Therefor, 1+1=2

The only correct answer.

1+1=2

when dealing with addition and subtraction, we are simply making one entity bigger and in the end, only one product is affected by the sum. in this instance "1" is adding another value of "1" to itself thus creating "2". the "2" is still the same "1" but the "1" has now changed due to adding the extra "1".

This.

Basically simplified peanut math. 1 + 1 is defined as the same thing as 2, we just write 2 because who wants to write out 1 + 1 every time.

Proof by Church encoding

+ 1 1
⇒ (λm.λn.λf.λx.m f (n f x)) (λf.λx.f x) (λf.λx.f x)
⇒ (λn.λf.λx.(λi0.λi1.i0 i1) f (n f x)) (λf.λx.f x)
⇒ λf.λx.(λi0.λi1.i0 i1) f ((λi0.λi1.i0 i1) f x)
⇒ λf.λx.(λi0.f i0) ((λi0.λi1.i0 i1) f x)
⇒ λf.λx.f ((λi0.λi1.i0 i1) f x)
⇒ λf.λx.f ((λi0.f i0) x)
⇒ λf.λx.f (f x)
⇒ 2

...

>Now I add another (1) peanut to the location that contains my previous peanut.

That violates the Pauli exclusion principle.

Maths is independent of physics, bro.

No, it isn't.

1+1=2
Because:
3-1=2=1+1+1-1=1+1

Hope I helped.

Exactly!

>>Maths is independent of physics, bro.

If this is true than an illustration using an example in the physical universe is invalid and therefor the Peanut Exercise" is invalid.

>
>
> (You)
>
> (You)
>
Actually this proves nothing unless one can "prove" the unfailing correlation between peanuts and real numbers. In point of fact the "Peanut Exercise" proves nothing from a mathematical standpoint...it is a simple "illustration" and not a mathematical "proof". Thus far there have been no complete proofs offered here that 1+1=2, only summaries. A complete proof would offer definitions, assumptions, proofs to support them and the equations for same. The complete proof takes up the space of a sizable book. What I have seen here thus far only "proves" that anons here can cut and paste with a fair degree of skill.

...

Consider the following, if you have a single grain of sand, is that a pile of a sand? Is two? Is three? Define a pile.

>What is 1+1? And why?
Silly answers so far because they are attempts by wannabees to show off their maths knowledge; however they absolutely FAIL to correctly answer the question from a maths standpoint. All seemed to want to infect the question with their pesky = sign. Kids: READ the problem CORRECTLY and do NOT add anything! I shall proffer that the most logical and correct answer is that "1+1" is an arithmetic expression or numerical expression. "QED!" :grin: SOLVED!

Not that user, but I can get around the objection by using clouds instead of piles of sands, i.e. adding one cloud to another cloud equals one cloud.

And pre-emptively, the objection that clouds are collections of distinct water molecules can be met with the refutation that molecules are collections of identical fermions.

1 + 1 = undefined

Proof

1 × 1 = 2/1

>All seemed to want to infect the question with their pesky = sign.

Looks like you need some homotopy type theory.

FAILS to correctly answer the question from a maths standpoint. Why does everyone insist on infecting the question with their pesky = sign? The simple and elegant answer has correctly been given: "1+1" is, by definition, an arithmetic expression....period.

>homotopy
Sorry but the answer has nothing to do with math proofs or homotropy. It is a question of semantics since the OP did not proffer this as a function or equation to find a solution for one of more variables nor did he ask for such a solution. The addition of the = sign is superfluous. Simply put...Question: "What is 1+1?" The crucial words being "What is..."Answer: "1+1" "is" an arithmetic expression or numerical expression. Your moot requires moving further from the original semantics of the question.

Indeed.

Show me a quantified unit and Ill show you infinity.

True the only person that did it right ITT is

>True the only person that did it right ITT is
>
However, once you claim what it evaluates to it complicates the answer because the original question did not require the "evaluation" of 1+1. I you are to evaluate in different systems why not add that it resolves to 10 in the binary system...etc? That answer is an expansion that is also superfluous. In addition the 355 response does not give the reasons "why".

2 is 1 + 1 by definition
3 is 2 + 1 by definition
4 is 3 + 1 by definition and so forth

succ(my dick)

>The complete proof takes up the space of a sizable book.

That sounds implausible. What's your background theory?

>The complete proof takes up the space of a sizable book.

Also inb4 Principia that doesn't count because not all of it is necessary to define and evaluate 1 + 1.

>What is 1+1?

It is an arithmetical expression denoting the sum of 1 and 1.

>And why?

Because we say so.

>2 is 1 + 1 by definition
10 is also 1+1 by definition...dpending on your chosen number system

>>>The complete proof takes up the space of a sizable book.
>That sounds implausible. What's your background theory?

Whitehead and Russell spend over 300 pages on a comprehensive proof that 1+1=2 For starters they took space to to actually define what the the symbols "1", "+", "2", and "=" were. Why? Because these symbols had not been as formally defined until the book was written. It took until page 300 just to define the symbols and to prove that they could put the symbols 1+1= together before proving that 1+1=2. They went to the basics of basic in this proof--including defining what a "1" is. From there the final formulae were relatively simple. That was a formal and comprehensive proof. What most students learn as a "proof" is only a summary that depends on many assumptions but those assumptions are not supported because it is beyond the paygrade of most maths instructors to provide that support.

>Also inb4 Principia that doesn't count because not all of it is necessary to define and evaluate 1 + 1.
While it may not be "necessary" in order to "evaluate" 1+1, it certainly IS necessary to provide support for all assumptions if you are going to "prove" that 1+1=2. A one page summary formula is not a formal proof in this case. This is where skills in mathematical logic come into play.

Admittedly, I interpreted the question as "what is 1 + 1 equal to", which is commonly what people mean when they ask "what is [some expression]"?

HoTT deals with 'equality' and 'equivalence' in general (c.f. the univalence axiom) which is why it might come in handy.

The numbers 1-9 are all numerical values of 1. And anything that 10 and above is expressed by the intial numeral of 10(s) plus the quotient of 1(s)

The number two is the expression of one twice, the number three is the expression of one added 3 times, and so on

Lets say we have the number 32. 32 can be expressed using 1’s and 10’s such that 3 tens and 2 ones =32

Think of it like a tally . Each tally is equal to one. If you had one tally and then added abother you would have two. And if you added 3 groups of 10 tallies you would have 32

>Admittedly, I interpreted the question
Exactly. However, maths is an exacting science and if one is going to go off into incorporating the superfluous one must have good reason. There is a broad gulf between what "common people commonly mean by "What is..." and what a mathematician skilled in his science would mean. Actually if anything tangible rode on the correctness of the answer, the correct thing to do would be to question the semantics of the OP's post and ask if he meant that he was asking us to provide an evaluation of the term or if he was indeed asking for the answer to the simple plain English question "What is..." Perhaps OP was seeking the answer "10" or "010" ? How to know? However, if the simple answer I proffered was marked as incorrect by an instructor, an academic review would quickly correct his mistake. Homotropy type theory would not come into play at all. Also note that if we accept 355's Wiki paste see that" 1+1 is a Ukrainian TV channel
1+1 a 1972 album by Grin
1+1 a 1997 duet album by Herbie Hancock and Wayne Shorter
"1+1" a song by Beyoncé Knowles
Why? Because WikiP sed soh!

Empiricism is plebian n00b

>what a mathematician skilled in his science would mean
>implying OP is a skilled mathematician

And...by the way: 1+1 also = 0 Professor E is outta here for a while and gunna listen to 1+1 by Beyonce! Oh...one more thing...just for kicks...dare ya... provide a proof, that, in an ordinary decimal system 2+2=4 ;-} Have fun!

>implying OP is a skilled mathematician
It is not relevant whether he is or is not. Perhaps he is the common person. How to know? [Ask!] :grin:

>Empiricism is plebian n00b
No, not in all cases: for if we take the definition of Empiricism as being the theory that all knowledge is birthed in sensory experience, you must accept that you would have no ability to participate here absent Empirical knowledge. Plebian! ;-}

>Whitehead and Russell spend over 300 pages on a comprehensive proof that 1+1=2
The dictionary spends over 300 pages defining what zebra means.

Is this a proof or an opinion? Just sayin'...

FUCK MATH BE SOME WHITEY SHEIT!

Descartes plz go

>>Whitehead and Russell spend over 300 pages on a comprehensive proof that 1+1=2
>The dictionary spends over 300 pages defining what zebra means.
Actually W&R spent 300 pages on many common fundamentals and definitions--upon which a formal proof of 1+1=2 must be based. So, yeah..pretty much so.

>Descartes plz go
Why?

>The dictionary spends over 300 pages defining what zebra means.
So, now provide a formal proof of what "zebra" means. ;-}

Why do yops make fun of americans for using the imperial system when yops literally count everything in base10. Like do you not think aliums look down from outer space and think america is the only worthwhile country because they bothered to count in ways beyond HOW MANY FUCKING FINGERS YOU CAN SEE ON YOUR HANDS YOU IMMENSE BRAINLET TROG

America for hexidecimal system fuck everyone else and fuck decimal.

Descartes was *against* empiricism, not for it.

...

>Descartes was *against* empiricism, not for it.
It is impossible to be against empiricism except in the most esoteric sense. All humans depend upon empirical acquisition of knowledge as babies and children prior to developing higher reasoning skills. So...being "against" empiricism in the broadest sense is simple not supportable.

Descartes' failure was the assumption of "grades" of reality. His attempt at proof of God was dopey. I prefer not to prove the existence of God and leave such as my personal belief system and therefore not subject to outside scrutiny, questions, or the need to answer for my faith lest I flip a finger and utter a hearty Fcuk you! Ok...so 1+1=0 [And 2...and 10...and...pick one...] And the Cheshire Cat keeps smiling...

you sound like the teacher that made me despise mathematics in general

>you sound like the teacher that made me despise mathematics in general
This happened with you as a child? I despised maths as a child because--and I did not realize this until later--most maths teachers are morons. So if you so despise maths, why are reading a thread dealing largely in math? Seems a tad masochistic and warped to moi. I would never read an opera thread unless in the last few seconds before I slit my throat to add more self punishment to my suicide.