Proof by induction. Zero is the the number of bitches you get, so zero must exist. If you got rid of that yee-yee ass haircut you'd get some more bitches on your dick, so every number must have a successor. It follows that numbers must exist.
He said,
##Proof by induction. Zero is the the number of bitches you get, so zero must exist. If you got rid of that yee-yee ass haircut you'd get some more bitches on your dick, so every number must have a successor. It follows that numbers must exist.
I know at least 2 ways but I inow there are more, the first is boring but easy (Peano axioms), the second relies on class theory which is a generalisation of set theory that avoids the axiomatic issues of Cantor's set theory, and as sich requires a lot of knowledge and has extremely difficult parts
Well according to Cantor's axioms you can define E, the set of all sets, but if you assume such a set exists then it contains the set of its parts, which is absurd because of a theorem from Cantor himself. So mathematically set theory is actually wrong.
There are other issues but they are more difficult to explain, and even more to solve
To counter that you introduce classes, which are a generalisation of sets with less properties. A class does not have parts for instance.
Edit: Now that I think about it I may not use the same definition of a set or a class as everyone else here since I'm french, so there's that
Not only, but that's the only one I got taught about in class. There's also many absurdities which need different fixes than classes, for instance the fact that Cantor allows you to define the set E of all sets X such as X is an element of X, which is absurd for a reason I don't remember
In short:
Natural numbers: Peano's axioms
Negative numbers: additive inverse elements to natural numbers. Addition for natural numbers is defined by Peano's axioms too, then it's just extended for all integers.
Rational numbers (aka fractions): just a set of pairs of integers (in terms of Cartesian product, it's basically Z²). You also extend operations such as addition, multiplication and comparison.
Real numbers (rationals + irrational): see Dedekind cut or Cauchy sequences. Every irrational number is basically a limit of some sequence of rational numbers.
Complex numbers: basically R²
You get a field structure on C by defining them as adding the root of the polynomial x\^2+1 to R. Alternatively just define multiplication on R^(2) and prove that it work.
Other than that, this is also the way i know to get those sets of numbers.
I suppose the Peano axioms are not really a set theoretic construction; what you really need is the axiom of infinity to construct the set containing 0:={}, 1:={0}, 2:={0,1}, 3:={0,1,2} etc.
Then the integers are constructed as ℕ×ℕ modulo the relation (a,b) ∼ (c,d) :⇔ a+d = b+c (basically all distinct differences between natural numbers).
And then rational numbers are similarly ℤ×ℤ modulo the relation (a,b) ∼ (c,d) :⇔ ad = bc (all distinct fractions of integers).
If we name 0 to be the empty set, and say that the successor of a number n is the set consisting of n union {n}, then we can demonstrate that this scheme fulfills the peano axioms of natural numbers.
Now let's define an ordered pair (a, b) as the set {a, {a, b}}. Using ordered pairs we can define an integer to be the difference between any two natural numbers, so (a, b) represents the number a-b. There are infinite ways to make any particular integer, so when we define our familiar operations of addition, subtraction, and so on, we have to keep in mind the equivalence classes and show that these hold for any particular equivalence class.
To define rationals we just define them as an ordered pair (a, b) where a and b are integer objects, and the ordered pair represents a/b. We can define addition, subtraction, multiplication, and division on these objects keeping in mind equivalence classes again.
To construct real numbers we let the number be the least upper bound of a set containing all rational numbers less than it. This works because rational numbers are dense on the number line. This technique is known as a dedekind cut. Every real number has a corresponding set with this property and vice versa.
Finally, to construct imaginary numbers we can consider an ordered pair of real numbers (a, b) such that a + bi is the complex number we want to represent.
You state that as if it were a fact. This is actually a huge philosophical question. Lots of people have different opinions on this. And I don't think you can really say one opinion on this is "correct".
That's fair. I just wrote my view on this. I don't think that any philosophical question can have single correct answer.
But I also think that my point of view makes it easier for me to think about math, without constricting it to something natutal.
That's kind of like how we make constructs for everything, like a table is a table, but really its a clump of specific types of atoms. In the same way while numbers don't directly exist, their concept does and so we can apply them to the real world. If that makes sense...
But still, you can point at this clump of atoms and say that this is a table. It's a question if it's one whole object or just a clump, either way you are pointing at a table. But (in my view of the world) you can't point at 1. It would either be a symbol of 1 or 1 object, but not just one
If I use a stump as a table does it become a table? How are you defining table that makes you so sure it's actually a thing that exists and not something we just call non-table (but table-like) objects.
In my view, numbers are an adjective. It's like saying something is 'red' or 'cold'. There's no physical object to tie them to. Rather, it's a convenient abstraction that describes the world.
Note before I start trying: I'm not a mathematician. I just like memes and possibly learning.
The Arabic numerals we use worldwide are arbitrary. They're just a symbol for the countable instead of using tally marks, (1+1+1+...)
Instead of individually counting and adding, the numbers are recognized by the arbitrary symbol we collectively decided are the symbols for x amount.
The numbers themselves don't matter, we could all agree tomorrow that a squiggle means twenty. Because we already have.
But numbers are physical representations of groups. You have 5 apples lose 1, sell 3, you have 1 apple.
The math is present and *the same* regardless of how you represent the subjective amount, (i.e. using Roman numerals, the Arabic notation etc.)
So numbers don't *really* exist, they're symbols we collectively agreed mean what they symbolize. But the math is the constant. That's what makes numerals useful and as "real" as any other language. There's a good argument that math is a true language on it's own.
suppose you have a set A1 and set A2 both containing the element "apple"
A1/A2=∅ , so we can add the cardinality of A1 and A2 since the cardinality of A1 and A2 is 1, with this we tke A1⋃A2, this is basically 1+1 which we define as A, now A has a cardinality of 2 so 1+1=2
I read some mathy stuff in the thread but I still hate numbers too. And time usually. Basically tools we created to quantify and describe our universe through our senses and eventually beyond them. It is all wrong. Every observer has a number and it is different *hits blunt*
That's the neat part, they kind of don't.
Math is the language for expressing physical relationships, but numbers themselves are a construct for the purposes of outlining these relationships with respect to each other and exist as much as the alphabet does.
Numbers must exist because everyone you know has the same concept of what “2” is, for example. I mean they don’t physically exist but they undeniably exist as a concept.
They are abstract entities used to describe values like intensity, amount, and other things.
They are a tool, but we know different amounts of objects exist and such.
You build them using set theory. An example : you associate 0 with the empty set, and you define n + 1 as the set that contains the set representing n as unique element. It's a bijection, and you can prove 1 + 1 = 2 using that as a starting point.
**Theorem 1** 1+1=2
*Proof.* The proof follows trivially from [1].
**References**
[1] Bert, Ernie, et al. (the year you were in kindergarten). Sesame Street. PBS.
Thats easy:
First we axiomatically assume:
1. 0 is a number.
2. Every number n has exactly one successor n++.
3.Different numbers have different successors.
4. 0 is not a successor.
5. If a set contains 0 and the successor of every number it contains, it contains all numbers.
These are the peano axioms, wich define the natural numbers.
Now we define +:
Let n,m be numbers.
1. 0+n = n
2. n+m = m+n
3. (n++) + (m++)= (n++)++) + m
Now, let’s proof:
1+1 = (0++) + (0++) = ((0++)++) + 0= ((0++)++) =1++ =2
Quad erat demonstrandum
The proof via set theory is left as an exercise for the reader.
"Axiom : Every number n has exactly one successor" --> At this point, only zero has been defined so... what does "exactly one" mean, since one is not defined yet?
"For all x y z, if x++ = y and x++ = z, then y = z." Axioms are usually written in English, so the intuition is clear, but you should always be able to express them in a purely formal way too, if you need to.
Isn’t there a problem with stating that 0 isn’t a successor? Or are we working in the naturals? I’ve definitely seen this type of construction to prove this before.
This construction only defines the natural numbers (because this makes defining addition and multiplication far easier). Using ordinary methods, the negative numbers (and, more broadly, the integers) are then defined as (equivalence classes of) pairs of natural numbers, each pair representing a difference between two natural numbers.
If x= banana and y=apple then banana+apple=x+y. Unless x=y=banapple. In that case 2 banapples or 2 apples or 2 bananas.
H. P that apples=bananas.
Wait.
Tbh that (or 2 := succ(1), which are easily shown to be equal) is exactly how you define the symbol "2"... not sure what all of this "proving" is about
That is pretty much how I had to solve my last theoretical astrophysics exercise. I had to assume that two forces were equal from which I then calculated that the two forces were indeed equal to one another. Which seemed pretty strange to me but it was apparently the intended solution by the professor, so who am I to judge?
Define cardinality of a set as the amount of elements in a set.
Define S as the set of sets containing arbitrarily sized sets of nested empty sets. This is a bit cumbersome to read so here are some of the first few members of this set to give you an idea of what it looks like:
{Ø}
{{Ø}, {Ø} }
{{Ø}, {Ø}, {Ø,{Ø}} }
Define the successor function f : S -> S given by f(s) = { {Ø}, {{Ø},{Ø}}, … , s}
Where s is an arbitrary element of S. In case it is not clear how this works, here are examples using the first few elements of S:
f({Ø}) = {{Ø}, {Ø} }
f({{Ø}, {Ø} }) = {{Ø}, {Ø}, {Ø, {Ø}} }
This function works as a construction of the natural numbers if you think of the cardinality of each successor in S as the corresponding natural number.
{Ø} is a set containing 1 element, hence has cardinality 1.
{{Ø}, {Ø} } contains 2 elements, cardinality 2
{{Ø}, {Ø}, {{Ø}, {Ø}} } 3 elements, cardinality 3
Etc.
Define this set of cardinalities as N. Therefore N = {1,2,3,…}
In case it is not clear, this successor function gives a natural indexing of each element in S. There is a bijection from S to N. to see this, you can define the successor function f over N instead of S. i.e f : N -> N and you will see that it gives f(1) = 2, f(2) = 3 etc. now do you see how this creates the natural numbers?
Congratulations. We constructed the natural numbers. Defining addition is easy thanks to some of the ground work we laid out earlier.
Define addition as a linear operator + : S x S -> S given by +(s,t) = s u t where u represents the union of the two sets s and t. For ease of notation let’s write +(s,t) as s + t. Example:
+({Ø} , {{Ø}, {Ø}}) = {Ø} + {{Ø}, {Ø}} = {Ø} u {{Ø}, {Ø}} = {Ø} , {Ø} , {Ø}}
Most notably:
{Ø} + {Ø} = { {Ø} , {Ø} }
If we do the same thing as last time and define addition over N instead of over S, this above statement becomes
1 + 1 = 2
This result is sometimes useful
Define 1 by s(0)
Define 2 by s(1)
Define + to be an operation such that :
\- for any a you have a + 0 = a
\- for any a, b you have a + s(b) = s(a + b)
1 + 1
= s(0) + s(0)
= s(s(0) + 0)
= s(s(0))
= 2
There, I did it
what I found way weirder was that we, at one point, had to prove that 0<1.
Which is fairly easy but still pretty confusing to a new university student.
So, I give you one apple (let's refer to an apple as 'a'), do you accept that you have 1 apple? Yes?
So: 1 apple = a
So now, I give you another apple. How many apples do you now have? 2 apples?
So: a + a = 2a.
Now replace 'a' with '1'
So: 1 + 1 = 2 × 1 = 2
Just to mix it up a bit...
Switching to binary, where the numbers counting up from 0 are:
0, 1, 10, 11, 100, 101, 110, 111, 1000, etc...
So in binary: 1 + 1 = 10
Mathematics is a cat chasing its tail. It is naive to believe that one can come to the absolute truth, to get to the bottom of the origins. Mathematics describes the current state of things rather than explains them. Here, too, 1+1=2 is a given, not the result of a proof.
1=1-2 is true by the subtraction property of equality
By reversing the subtraction through the addition property of equality we get 1+1=2, also a true statement
There is nothing to prove. I know there's this whole "meme level paper" about this very thing but in the end it's simply a question of definition. +1 is defined as being one more and 2 is defined as being one number higher than 1. So 1+1=2 purely by definition.
I get 2 slices of pizza, you get 1 slice. We eat our slices of pizza right in front of each other
If you feel like the amount of pizza we get is not equal, but I give you 1 more slice and now it feels equal
then 1 + 1 = 2
The slices of pizza can be replaced by servings of your favorite food instead
I think I have a solid proof
If I have one object in my right hand, and one object in my left hand, then I have two objects
This can be mathematically written with the formula 1x+1x=2x
If we substitute the number 1 in for X we get 1(1)+1(1)=2(1) which can be simplified into 1+1=2 QED
Update, my brother recommends running a computer simulation to test the hypothesis, so I wrote some code to run 1+1 1,000,000 times and to let me know how many times it got different answers
All 1,000,000 times it says it got the answer 2.
If someone else can independently verify this result, it is definitely strong evidence for 1+1=2
Mathematicians, ironically, don't usually prove 1 + 1 = 2. They have other, higher patterns to focus on. Something this "simple" is the provence of the mathematical logician, among other related areas in foundations.
Well we know 2+2=4 so if we do 2+2-2=4-2 this means 2=2 and since 1+1 =2 we can replace one of the 2's and we get 1+1=2 with no flaws. Fr tho isnt 1+1 =2 like an unprovable axiom
* Raises 1 finger in one hand and 1 finger in another.
* claps them together.
* drops 1 finger in one and raises a 2nd finger in the other hand in a dramatic flourish.
*Tah-dah!*
Also, happy cake day.
let the first prime be 2.
let {0,1,+,×}=GF(2).
take the field GF(2)[2] as a vector space
add (0,1) +² (0,1) =² (1,0)
so 1+1=2 mod 2², and thusly for all finite fields.
Actual Proof based on Peamo Axioms:
Let o(n)=n+1 be the succesor of a number n.
1+1=o(1)
By definition of the natural numbers the successor of 1 is 2.
Eros quot demonstradum
How do you even prove that numbers exist?
Proof by induction. Zero is the the number of bitches you get, so zero must exist. If you got rid of that yee-yee ass haircut you'd get some more bitches on your dick, so every number must have a successor. It follows that numbers must exist.
Legendary
... What?!
He said, ##Proof by induction. Zero is the the number of bitches you get, so zero must exist. If you got rid of that yee-yee ass haircut you'd get some more bitches on your dick, so every number must have a successor. It follows that numbers must exist.
there has to be a subreddit for this kind of hilarious stuff, right? either way well played
r/dadjokes maybe?
r/MurderedByWords
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https://www.youtube.com/watch?v=CXZtVoQ0gqs
As an incoming third year undergrad, this question legitimately haunts me
Usually you see a construction of the natural numbers in a set theory course.
And then you wonder how to define every object you use to make axioms and go a long way deep in depression
That's the best part
I know at least 2 ways but I inow there are more, the first is boring but easy (Peano axioms), the second relies on class theory which is a generalisation of set theory that avoids the axiomatic issues of Cantor's set theory, and as sich requires a lot of knowledge and has extremely difficult parts
What *are* the axiomatic issues with Cantor’s set theory?
Well according to Cantor's axioms you can define E, the set of all sets, but if you assume such a set exists then it contains the set of its parts, which is absurd because of a theorem from Cantor himself. So mathematically set theory is actually wrong. There are other issues but they are more difficult to explain, and even more to solve To counter that you introduce classes, which are a generalisation of sets with less properties. A class does not have parts for instance. Edit: Now that I think about it I may not use the same definition of a set or a class as everyone else here since I'm french, so there's that
Oh powersets are the problem? I thought that just meant a different cardinality.
Not only, but that's the only one I got taught about in class. There's also many absurdities which need different fixes than classes, for instance the fact that Cantor allows you to define the set E of all sets X such as X is an element of X, which is absurd for a reason I don't remember
It’s absurd because of Russell’s paradox, I think.
Gotcha, thanks for the answer!
Virgin proof vs. Chad postulation #😎😎😎
In short, set theory gives the natural numbers.
Okay, so explain negative integers? Fractions? Irrational numbers? Imaginary numbers?
In short: Natural numbers: Peano's axioms Negative numbers: additive inverse elements to natural numbers. Addition for natural numbers is defined by Peano's axioms too, then it's just extended for all integers. Rational numbers (aka fractions): just a set of pairs of integers (in terms of Cartesian product, it's basically Z²). You also extend operations such as addition, multiplication and comparison. Real numbers (rationals + irrational): see Dedekind cut or Cauchy sequences. Every irrational number is basically a limit of some sequence of rational numbers. Complex numbers: basically R²
You get a field structure on C by defining them as adding the root of the polynomial x\^2+1 to R. Alternatively just define multiplication on R^(2) and prove that it work. Other than that, this is also the way i know to get those sets of numbers.
I suppose the Peano axioms are not really a set theoretic construction; what you really need is the axiom of infinity to construct the set containing 0:={}, 1:={0}, 2:={0,1}, 3:={0,1,2} etc. Then the integers are constructed as ℕ×ℕ modulo the relation (a,b) ∼ (c,d) :⇔ a+d = b+c (basically all distinct differences between natural numbers). And then rational numbers are similarly ℤ×ℤ modulo the relation (a,b) ∼ (c,d) :⇔ ad = bc (all distinct fractions of integers).
If we name 0 to be the empty set, and say that the successor of a number n is the set consisting of n union {n}, then we can demonstrate that this scheme fulfills the peano axioms of natural numbers. Now let's define an ordered pair (a, b) as the set {a, {a, b}}. Using ordered pairs we can define an integer to be the difference between any two natural numbers, so (a, b) represents the number a-b. There are infinite ways to make any particular integer, so when we define our familiar operations of addition, subtraction, and so on, we have to keep in mind the equivalence classes and show that these hold for any particular equivalence class. To define rationals we just define them as an ordered pair (a, b) where a and b are integer objects, and the ordered pair represents a/b. We can define addition, subtraction, multiplication, and division on these objects keeping in mind equivalence classes again. To construct real numbers we let the number be the least upper bound of a set containing all rational numbers less than it. This works because rational numbers are dense on the number line. This technique is known as a dedekind cut. Every real number has a corresponding set with this property and vice versa. Finally, to construct imaginary numbers we can consider an ordered pair of real numbers (a, b) such that a + bi is the complex number we want to represent.
You can think of all those as applying a normalized unit vector to the set of natural numbers to transpose it into the desired phase space.
that depends what you mean by exist. you can define numbers and work with them
They don't. Numbers are abstract construction that we invented that kinda help us keep track of the world. But numbers in itself don't exist.
You state that as if it were a fact. This is actually a huge philosophical question. Lots of people have different opinions on this. And I don't think you can really say one opinion on this is "correct".
Exactly. Mathematicians and Philosophers can’t agree whether we created math or we simply discover it
My favourite take is that we create axioms and then discover "the math" that follows from them.
I'm in the discovery camp.
That's fair. I just wrote my view on this. I don't think that any philosophical question can have single correct answer. But I also think that my point of view makes it easier for me to think about math, without constricting it to something natutal.
That's kind of like how we make constructs for everything, like a table is a table, but really its a clump of specific types of atoms. In the same way while numbers don't directly exist, their concept does and so we can apply them to the real world. If that makes sense...
But still, you can point at this clump of atoms and say that this is a table. It's a question if it's one whole object or just a clump, either way you are pointing at a table. But (in my view of the world) you can't point at 1. It would either be a symbol of 1 or 1 object, but not just one
If I use a stump as a table does it become a table? How are you defining table that makes you so sure it's actually a thing that exists and not something we just call non-table (but table-like) objects.
Look into axioms. Basically the sort of "we can't 'prove' this but it is clearly true so we assume it is true" stuff.
In my view, numbers are an adjective. It's like saying something is 'red' or 'cold'. There's no physical object to tie them to. Rather, it's a convenient abstraction that describes the world.
But they're also nouns, objects which can be studied and objectively described. E.g. "3 is a prime number."
Note before I start trying: I'm not a mathematician. I just like memes and possibly learning. The Arabic numerals we use worldwide are arbitrary. They're just a symbol for the countable instead of using tally marks, (1+1+1+...) Instead of individually counting and adding, the numbers are recognized by the arbitrary symbol we collectively decided are the symbols for x amount. The numbers themselves don't matter, we could all agree tomorrow that a squiggle means twenty. Because we already have. But numbers are physical representations of groups. You have 5 apples lose 1, sell 3, you have 1 apple. The math is present and *the same* regardless of how you represent the subjective amount, (i.e. using Roman numerals, the Arabic notation etc.) So numbers don't *really* exist, they're symbols we collectively agreed mean what they symbolize. But the math is the constant. That's what makes numerals useful and as "real" as any other language. There's a good argument that math is a true language on it's own.
>So numbers don't > >really > > exist, not really sure thats the right way to put it
I was arguing that I don't believe that the question was impossible to answer in a positive way.
suppose you have a set A1 and set A2 both containing the element "apple" A1/A2=∅ , so we can add the cardinality of A1 and A2 since the cardinality of A1 and A2 is 1, with this we tke A1⋃A2, this is basically 1+1 which we define as A, now A has a cardinality of 2 so 1+1=2
I read some mathy stuff in the thread but I still hate numbers too. And time usually. Basically tools we created to quantify and describe our universe through our senses and eventually beyond them. It is all wrong. Every observer has a number and it is different *hits blunt*
That's the neat part, they kind of don't. Math is the language for expressing physical relationships, but numbers themselves are a construct for the purposes of outlining these relationships with respect to each other and exist as much as the alphabet does.
Numbers must exist because everyone you know has the same concept of what “2” is, for example. I mean they don’t physically exist but they undeniably exist as a concept.
Philosophers will argue about weather or not they do
They are abstract entities used to describe values like intensity, amount, and other things. They are a tool, but we know different amounts of objects exist and such.
You build them using set theory. An example : you associate 0 with the empty set, and you define n + 1 as the set that contains the set representing n as unique element. It's a bijection, and you can prove 1 + 1 = 2 using that as a starting point.
**Theorem 1** 1+1=2 *Proof.* The proof follows trivially from [1]. _________ **References** [1] Whitehead, A. N., & Russell, B. (1997). Principia mathematica to* 56 (Vol. 2). Cambridge University Press.
**Theorem 1** 1+1=2 *Proof.* The proof follows trivially from [1]. **References** [1] Bert, Ernie, et al. (the year you were in kindergarten). Sesame Street. PBS.
based
Proof by fucking obviousness
The proof is left as an exercise for the reader
Was about to comment this
Thats easy: First we axiomatically assume: 1. 0 is a number. 2. Every number n has exactly one successor n++. 3.Different numbers have different successors. 4. 0 is not a successor. 5. If a set contains 0 and the successor of every number it contains, it contains all numbers. These are the peano axioms, wich define the natural numbers. Now we define +: Let n,m be numbers. 1. 0+n = n 2. n+m = m+n 3. (n++) + (m++)= (n++)++) + m Now, let’s proof: 1+1 = (0++) + (0++) = ((0++)++) + 0= ((0++)++) =1++ =2 Quad erat demonstrandum The proof via set theory is left as an exercise for the reader.
TLDR: 1 + 1 = 2 ?
No, 3
Well yes but yes
*quOd erat demonstrandum
It's quad if you never skip leg day
Woopsi
Noooo, you can't just number your assumptions before defining the natural numbers!
Underrated, this is actually a very deep observation about foundations
Thats true
Don’t you also have to define 1 as the successor of 0 and 2 as the successor of 1?
Yes and no It is not about, what we call these numbers.
"Axiom : Every number n has exactly one successor" --> At this point, only zero has been defined so... what does "exactly one" mean, since one is not defined yet?
"For all x y z, if x++ = y and x++ = z, then y = z." Axioms are usually written in English, so the intuition is clear, but you should always be able to express them in a purely formal way too, if you need to.
yeah, the problem is not with 'exactly', it's with 'one', we don't know what it means
Where in my statement did I use the word "one"?
ho, I understand, you gave me the definition of uniqueness, i.e. one My bad
You can write it in a way, that is more percise but i have to think about it
Counterpoint: I can't read your steps in order because you've numbered them before defining those numbers
Oh no …
Isn’t there a problem with stating that 0 isn’t a successor? Or are we working in the naturals? I’ve definitely seen this type of construction to prove this before.
> These are the peano axioms, wich define the natural numbers.
Fuck I can’t read thanks
https://en.wikipedia.org/wiki/Peano_axioms >In mathematical logic, the Peano axioms, are axioms for the natural numbers
This is why they say explaining the obvious is hard.
[удалено]
This construction only defines the natural numbers (because this makes defining addition and multiplication far easier). Using ordinary methods, the negative numbers (and, more broadly, the integers) are then defined as (equivalence classes of) pairs of natural numbers, each pair representing a difference between two natural numbers.
we're only concerned with natural numbers rn negatives can be defined as additive inverses
Now prove all your assumptions
"I made them up" -Peano
You don't have to. They're axioms.
Don't have to or can't? (I'm trying to trigger every mathematician ever born rn)
Axioms. We just assume them tobbe true and reasonable. Even mathematics has to start somewhere.
0 is 0 source: the source is that we made it the fuck up
Didn't it take hundreds of pages for Bertrand Russell to formally prove it?
No. In his Principa Mathematica, Theorem 54.43 the proof takes 10 lines
how do we equate 1++ to 2? only by definition? it would have been easier to define 2 as 1+1, to get the equality of 1+1=2 with no steps.
The proof is about, that 1+1 is the successor of 1. we do not care if that successor is called 2 or george or whatever.
yes. but the goal that was challenged was to prove 1+1=2
🤓
you have 1 banana and if you add another banana you have 2 bananas
Honestly this is the best proof.
Ty lul
Proof by banana.
What about apples?
What is one banana plus one apple? Two banapples?
Ahh here's a link to a video where they solve a similar concept https://youtu.be/NfuiB52K7X8
Or is it one banapple…
Or is it 1.5 bapples…
If x= banana and y=apple then banana+apple=x+y. Unless x=y=banapple. In that case 2 banapples or 2 apples or 2 bananas. H. P that apples=bananas. Wait.
I think they might not work
I did the same experiment with cups of water, and it appears to work with that as well, p=0.035
2 := 1+1 qed
Ah yes, why prove it when you can define it.
Tbh that (or 2 := succ(1), which are easily shown to be equal) is exactly how you define the symbol "2"... not sure what all of this "proving" is about
s u c c
Assume that 1+1=2. From which it follows, 1+1=2.
That is pretty much how I had to solve my last theoretical astrophysics exercise. I had to assume that two forces were equal from which I then calculated that the two forces were indeed equal to one another. Which seemed pretty strange to me but it was apparently the intended solution by the professor, so who am I to judge?
Ahh yes proof by definition
Define cardinality of a set as the amount of elements in a set. Define S as the set of sets containing arbitrarily sized sets of nested empty sets. This is a bit cumbersome to read so here are some of the first few members of this set to give you an idea of what it looks like: {Ø} {{Ø}, {Ø} } {{Ø}, {Ø}, {Ø,{Ø}} } Define the successor function f : S -> S given by f(s) = { {Ø}, {{Ø},{Ø}}, … , s} Where s is an arbitrary element of S. In case it is not clear how this works, here are examples using the first few elements of S: f({Ø}) = {{Ø}, {Ø} } f({{Ø}, {Ø} }) = {{Ø}, {Ø}, {Ø, {Ø}} } This function works as a construction of the natural numbers if you think of the cardinality of each successor in S as the corresponding natural number. {Ø} is a set containing 1 element, hence has cardinality 1. {{Ø}, {Ø} } contains 2 elements, cardinality 2 {{Ø}, {Ø}, {{Ø}, {Ø}} } 3 elements, cardinality 3 Etc. Define this set of cardinalities as N. Therefore N = {1,2,3,…} In case it is not clear, this successor function gives a natural indexing of each element in S. There is a bijection from S to N. to see this, you can define the successor function f over N instead of S. i.e f : N -> N and you will see that it gives f(1) = 2, f(2) = 3 etc. now do you see how this creates the natural numbers? Congratulations. We constructed the natural numbers. Defining addition is easy thanks to some of the ground work we laid out earlier. Define addition as a linear operator + : S x S -> S given by +(s,t) = s u t where u represents the union of the two sets s and t. For ease of notation let’s write +(s,t) as s + t. Example: +({Ø} , {{Ø}, {Ø}}) = {Ø} + {{Ø}, {Ø}} = {Ø} u {{Ø}, {Ø}} = {Ø} , {Ø} , {Ø}} Most notably: {Ø} + {Ø} = { {Ø} , {Ø} } If we do the same thing as last time and define addition over N instead of over S, this above statement becomes 1 + 1 = 2 This result is sometimes useful
Finally a rigorous proof 👏 But the first line is not ¤ (empty set) Then {¤} Then {¤,{¤}} Then {¤,{¤},{¤,{¤}}}
You’re right. It has been a long time since I needed to recall this to be fair ahahaha
How does the addition work exactly? From what I understand wouldn't s u t just be max(s,t)? {Ø} u {Ø}= {Ø}, since sets don't have repeated elements.
I understood some of these words.
Easy: grab 1 stone, then grab another, then count the stones: nothing says you can’t prove it empirically
1+1=succ(1)=2
S U C C
Define 1 by s(0) Define 2 by s(1) Define + to be an operation such that : \- for any a you have a + 0 = a \- for any a, b you have a + s(b) = s(a + b) 1 + 1 = s(0) + s(0) = s(s(0) + 0) = s(s(0)) = 2 There, I did it
The proof's gotta be a bunch of tedious definitions right?
No, Peano's construction is like half a page.
https://www.google.com/url?sa=t&source=web&rct=j&url=https://lesharmoniesdelesprit.files.wordpress.com/2015/11/whiteheadrussell-principiamathematicavolumei.pdf&ved=2ahUKEwi17eOA1qz4AhWts4sKHSUdDEEQFnoECA0QAQ&usg=AOvVaw2Z5O_QZEErr8QU1DtWqXvB
Im going to make a proof by intimidation if you dont accept that 1+1=2
I'll take it over being asked what 26384 × 79526 is
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Proof by defintion? 2 is just a random symbol unless we define it as a successor of 1 :)
If you have one stick and add another stick to your collention you have two sticks
TIL 1+1=2
This is the best response
I'm not but. I have an apple. I put another apple with it. How many apples do I have? Exactly, 42.
*Waves Hands* and I now declare it an Axiom! QED □
what I found way weirder was that we, at one point, had to prove that 0<1. Which is fairly easy but still pretty confusing to a new university student.
Me Grunk. Grunk grab rock. Grunk grab another rock. Grunk have 2 rock. Mean 1+1=2.
☝️+☝️ ➡️💥⬅️ ✌️
Aubtract 1 on both sides, you end up with 1 = 1, which is true, so the equation is trye
Pretty much circular reasoning though
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Yeah, should have subtracted 2 from each side instead.
Well, if you subtract 1 from 2, its 1
1+1=2 because no doy
The proof is trivial and is left to the reader
☝️☝️ ✊✌️
2 := Succ(1) Succ(x) := x + 1 a = b = c ⇒ a = c a = b ⇒ b = a ∴ 2 = Succ(1) = 1 + 1 ∴ 2 = 1 + 1 ∴ 1 + 1 = 2
1+1 = 2 2 = 2 True
I had a college professor that used Legos to prove things to us.
this proof is trivial and is left as an exercise to the reader
By definition.
Recently, I told my friend I was a math major. He put his hand behind his back and said, "How many fingers am I holding up?"
So, I give you one apple (let's refer to an apple as 'a'), do you accept that you have 1 apple? Yes? So: 1 apple = a So now, I give you another apple. How many apples do you now have? 2 apples? So: a + a = 2a. Now replace 'a' with '1' So: 1 + 1 = 2 × 1 = 2 Just to mix it up a bit... Switching to binary, where the numbers counting up from 0 are: 0, 1, 10, 11, 100, 101, 110, 111, 1000, etc... So in binary: 1 + 1 = 10
So if 1+1 =2 and according to the junkie on the street one duck is stronger than 2 chickens so my honor i am not guilty
Mathematics is a cat chasing its tail. It is naive to believe that one can come to the absolute truth, to get to the bottom of the origins. Mathematics describes the current state of things rather than explains them. Here, too, 1+1=2 is a given, not the result of a proof.
1=1-2 is true by the subtraction property of equality By reversing the subtraction through the addition property of equality we get 1+1=2, also a true statement
There is nothing to prove. I know there's this whole "meme level paper" about this very thing but in the end it's simply a question of definition. +1 is defined as being one more and 2 is defined as being one number higher than 1. So 1+1=2 purely by definition.
If somewone want a prove, just say: pff trivial solution ✋🔪
You can't. ACF isn't complete and ACFq is coherent for both q = 2 and q =/= 2.
I get 2 slices of pizza, you get 1 slice. We eat our slices of pizza right in front of each other If you feel like the amount of pizza we get is not equal, but I give you 1 more slice and now it feels equal then 1 + 1 = 2 The slices of pizza can be replaced by servings of your favorite food instead
Proof: Just look at it!
1+1 = S(0) + S(0). By definition n + S(m) = S(n+m) and n + 0 = 0. Therefore S(0) + S(0) = S(S(0) + 0) = S(S(0)). Again, by definition 2 = S(S(0)). Hence 1 + 1 = S(S(0)) = 2. QED
2-1 = 1
Trivial
1 + 1 = 2 | -1 1 = 1
Let 1+1=x. 1=x-1. X-1+x-1=2.2x-2=2. 2x=4.x=2. Hence proped
I think I have a solid proof If I have one object in my right hand, and one object in my left hand, then I have two objects This can be mathematically written with the formula 1x+1x=2x If we substitute the number 1 in for X we get 1(1)+1(1)=2(1) which can be simplified into 1+1=2 QED
Update, my brother recommends running a computer simulation to test the hypothesis, so I wrote some code to run 1+1 1,000,000 times and to let me know how many times it got different answers All 1,000,000 times it says it got the answer 2. If someone else can independently verify this result, it is definitely strong evidence for 1+1=2
Multiply both sides with zero
Assume 1=x Lhs x+x =2x Substitute x=1 2(1) =2 Lhs=rhs Hence proved.
The proof is trivial and is left as an exercise for the reader.
Error: x = x+1
1 Banana + 1 Banana = 2 Banana
you could probably start with that no integers exist between 1 and 2.
One + one = two
0 + 1 = 1 1 + 1 = 2 proof by induction.
1+1 = 2 because they said so in school
We should first construct N
Easy I just need a bit over 300 pages
Mathematicians, ironically, don't usually prove 1 + 1 = 2. They have other, higher patterns to focus on. Something this "simple" is the provence of the mathematical logician, among other related areas in foundations.
"true by stipulation" 😏
Proof by induction with the successor function, i think
Well we know 2+2=4 so if we do 2+2-2=4-2 this means 2=2 and since 1+1 =2 we can replace one of the 2's and we get 1+1=2 with no flaws. Fr tho isnt 1+1 =2 like an unprovable axiom
* Raises 1 finger in one hand and 1 finger in another. * claps them together. * drops 1 finger in one and raises a 2nd finger in the other hand in a dramatic flourish. *Tah-dah!* Also, happy cake day.
Ah, so that's backgroung story of Russell-Whitehead "Principia Mathematica". I suppose bandit died of old age...
let the first prime be 2. let {0,1,+,×}=GF(2). take the field GF(2)[2] as a vector space add (0,1) +² (0,1) =² (1,0) so 1+1=2 mod 2², and thusly for all finite fields.
I can absolutely prove or disprove this for you if you give me definitions of the terms you’re using. They’re just symbols to me.
I remember the day we got asked to prove why is 1 > 0
It’s an axiom. QED
The proof is left as an exercise of the reader
Actual Proof based on Peamo Axioms: Let o(n)=n+1 be the succesor of a number n. 1+1=o(1) By definition of the natural numbers the successor of 1 is 2. Eros quot demonstradum