Can math prove things with 100% certainty?

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I'll start things off with this simple little example:

1 is defined as s0. (sn means "the successor of n")
2 is defined as s1.

Addition is defined as follows:

a + 0 = a for any a by definition.
a + b for b > 0 is defined recursively by a + sb = s(a + b).

With definitions out of the way:

1 + 1 = 1 + s0 = s(1 + 0) = s1 = 2

Do you accept this proof as providing 100% certainty that 1 + 1 = 2? Why or why not?

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@Tradesecret
You may be interested in this, based on our recent discussion.
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1 pile of sand. 
+
1 pile of sand. 
=
?
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@Deb-8-a-bull
Define "pile of sand" mathematically. Is it not a math term? Then it is outside of the scope of this topic. You'll need to find a counterargument that doesn't simply rely on flaws in the English language. (i.e. what counts as a pile, what counts as one pile, etc.)
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  • The proof you presented is not a direct proof of the equation 1 + 1 = 2.
    • You've defined the principle of succession and 1+1=2 is just your case in point.
  • Your proof indicates  how addition can be defined recursively.
  • Traditionally, don't  proofs for 1+1=2 rely on Zermelo-Fraenkel set theory or the Peano axioms?

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OP: Can math prove things with 100% certainty?

Math can prove some things with certainty, cannot possibly prove other things, and can improve the probability of other things without certainty.
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Numbers are infinite so 1+1 can never equal 2. Math,  like everything else can ultimately be reduced to semantics. 1+1=2 when it comes to paychecks, hours worked  and taxes, that you can be sure of. You can argue 1+1 doesn't equal 2 and you would be right when you reduce it to semantics.

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Essentially everything comes down to semiotics when we want to understand them. What math does with 100% certainty cannot be conveyed in a way that also is 100% certain to interpreters due to the involvements of actual symbols needed to represent them, which further need to be defined separately, which is as important IMO.
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@Math_Enthusiast
It all depends upon what 1 equals.

1 bag of apples  and I bag of apples is 2 bags of apples.

Though there might be 3 apples in 1 bag and 2 apples in the other bag.

So therefore the 2 bags are not equal in either content or weight.

What you have is 5 relatively similar apples

A 2 relatively similar bags.
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Sooooooooo.

What we are all collectively saying isssssss. 
There ain't  no fucking way dumb dumb , that one could be 100% certain that 1  + 1 = 2.   

And having to ummm " respect " this. (   "ultimate truth" ' echos ' truth thruth truth ,,, ) 
Has us not passing GO and not collect  $200.
A Stall if you will. Is that the word ? 

Long short.
1 + 1 = 2  is not 100% truth. 
And this i find .   pisses me off. 

' I look at the chick across from me booobs ' 
I thought i had something but ummm,  yeah . 
No. 
Continue. 

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@zedvictor4
What about Boobs zed. 
Like 
One boob  +  one boob =  two boobs , orrr a set , a pair. 
Boobs are easy to add up hey ? 
Maths gives boobs sizes A to Big ones.

Testicles would add up much the same. 

1 tit plus 1 tit. =

No but
Mmmmm,  boobs . 
' holds up hand for hi5 ' 
Nice. 
 
Im a ( c cup ) kind of guy.
.
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@Deb-8-a-bull
Yep. 

I would say, 1 set relative to the host agent.

Like one bum has two cheeks.

Are you tit man?

I'm a bum man.

Let me clarify that:

There's nothing quite as aesthetically pleasing as a well formed female arse in a tight pair of leggings.

Obviously nothing below 16....For legal reasons.

Though driving up the High Street in the sunshine it's not easy to differentiate.

Unfortunately bigger females try to emulate but just don't cut the mustard, whatever Lizzo might think.

There's well formed and then there's over-developed....If you know what I mean.

Nor to old.....Just gets saggy....Unless they work out.

Not sure how old my next door neighbour is....certainly older than me.

But she's a regular gym goer and runner.....And certainly still cuts the mustard.

But don't tell Mrs Zed.
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@Deb-8-a-bull
Maybe this will help .99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 there are no whole numbers.
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@Math_Enthusiast
I'll start things off with this simple little example:

1 is defined as s0. (sn means "the successor of n")
2 is defined as s1.

Addition is defined as follows:

a + 0 = a for any a by definition.
a + b for b > 0 is defined recursively by a + sb = s(a + b).

With definitions out of the way:
Certainty in a formal system is amatter of definition, which is circular reasoning.  100% certainty becomes a matter of 100% agreementon definitions, which can never be universal or absolute.

1 + 1 = 1 + s0 = s(1 + 0) = s1 = 2

Do you accept this proof as providing 100% certainty that 1 + 1 = 2? Why or why not?
To the extent that mathematics is abstract and completelydetached from objective reality, and if we are talking about simple arithmetic,then yes, because mathematical certainty is a matter of definition, but only when you assume 100% agreement about said definitions. 

As it relates to reality, or even more complex mathematics, no,there is no 100% certainty.  As itrelates to reality, certainty is a matter of inductive reasoning, which isnever 100% certain. As it relates to mathematics that is more complex thansimple arithmetic, certainty is a matter of deductive reasoning and thevalidity of the associated axioms.

Euclid’s 5th postulate was alwaysquestionable because it cannot be derived from the other four, nevertheless, it was accepted that deductive reasoning led to certainty for centuries. With thediscovery, or invention, depending on your point of view, of non-EuclideanGeometries three of four times in the mid-19th century, the two-thousand-yearbelief that deductive reasoning led to Truth was lost.  With multiple deductively perfect geometries,the only way to determine which is true of space, becomes a matter ofexperiment and measurement, which gets fuzzy and eliminates any chance of 100% certainty. 


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Bertrand Russell started the Mathematica Principia with a proof that 1+1=2, it took him 360 pages.
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@sadolite
nien nien nien nien nien . 

But whats The next number in that sequence Sadolite. 
I mean , what number follows the 999999999999 one ? 
But i get what you are getting at. 
Well i kind of dont , but ive heard smart people talking about it like that so i ummm .
I pretend .
 
I go like this Sadolite.  

Of course there is no such thing as whole numbers. 
I understand completely.  
Then i like bite my lip and angrily make spit come out me mouth. 
I breath deep .   And just try to forget about it.  
It pisses me off. 

Its exactly the same way i deal with "  evolution "   
Of course i believe in evolution. 
NOT
Fucking stupifd fucking shit that i feel like i have to believe in. 

BUT , I ain't going to say i don't believe in evolution infront of you guys   
Sooooooooooo.
Its real. 
END OF FUCKING STORY. 

It took me over 3 years to try learning  the ( < bigger then )  and (  smaller then > )  
Or is it  ( >  lesser then )  symbols 
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@Sidewalker
The definition I used is the one that is generally accepted by mathematicians. For that matter, it doesn't really matter of the definitions are agreed upon, because that point is that 1 + 1 = 2 for the objects that I am using 1 and 2 to describe. Symbols and terminology are human constructs, but it is possible to prove things about the objects that they represent with 100% certainty. In fact, every mathematical statement can be reduced to a tautology. Axioms are never thought of as objectively true, but rather the resulting theorems can be seen as combinations of those axioms. It is objectively true that 1 + 1 = 2 follows from the axioms of ZFC. Since 1 and 2 are defined within ZFC, this is what we mean by 1 + 1 = 2.

Bertrand Russell started the Mathematica Principia with a proof that 1+1=2, it took him 360 pages.
I'm afraid that that is a myth. He did not spend 360 pages proving 1 + 1 = 2, he proved 1 + 1 = 2 360 pages in. The pages prior to that built up the foundations of Principia Mathematica. In my proof I considered those foundations to already be in place.
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@oromagi
  • Traditionally, don't  proofs for 1+1=2 rely on Zermelo-Fraenkel set theory or the Peano axioms?
Yes. This one can be done within ZFC or PA. Observe that I did not defined 0, s, or =. In ZFC, 0 is defined to be the empty set, sn is defined by n U {n}, and = is defined axiomatically. In PA, all three are defined axiomatically. (In case you didn't know, "defined axiomatically" means that they aren't actually defined. Instead, their meaning comes from axioms.)

Math can prove some things with certainty, cannot possibly prove other things, and can improve the probability of other things without certainty.
Precisely. Math can prove mathematical facts with certainty, no more, no less. These mathematical facts in turn apply to reality, where knowing things with 100% certainty is effectively impossible, but math can still help to improve our certainty. I must say, this is my favorite response of anyone yet.
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@sadolite
Numbers are infinite so 1+1 can never equal 2. Math,  like everything else can ultimately be reduced to semantics. 1+1=2 when it comes to paychecks, hours worked  and taxes, that you can be sure of. You can argue 1+1 doesn't equal 2 and you would be right when you reduce it to semantics.
What is your argument? You say that 1 + 1 = 2 can be argued against, but you don't provide any way to argue against it other than that "numbers are infinite, so 1 + 1 can never equal 2," which is a non-sequitur.
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1 + 1 = 2, but it also equals window
_________
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|— — | — —|
|____|____|
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When we crack this one plus one thing equalling two. 
And are 100% certain.  
We can move on to the more important questions.  
Like ummmm, 
2 + 2
And then we can find out what  3 + 3  is. 

Its just on and up from there. 
Soon we will be able to 100% know FOR CERTAIN.    

Then comes.  
(  Double digits proving  )     Or at least assuming 
So Like.
10  +  10  =  ? 

Cray cray right? 

I reckon in less then 100 years we will be able to starg proving for certain.  Subtraction. 
Thats like " take away " 
So 
1 - 1 = ?
Then diff numbers. 

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@Math_Enthusiast
The definition I used is the one that is generally accepted by mathematicians.
“Generally accepted”, as opposed to “universal or absolute” acceptance, which is to say the definition is not accepted with100% certainty.

For that matter, it doesn't really matter of the definitions are agreed upon, because that point is that 1 + 1 = 2 for the objects that I am using 1 and 2 to describe. Symbols and terminology are human constructs, but it is possible to prove things about the objects that they represent with 100% certainty.
Yep, “Certainty in a formal system isa matter of definition”.   BTW, I’m 100% certain Sherlock Holmessmoked a pipe.

In fact, every mathematical statement can be reduced to a tautology. Axioms are never thought of as objectively true, but rather the resulting theorems can be seen as combinations of those axioms. It is objectively true that 1 + 1 = 2 follows from the axioms of ZFC. Since 1 and 2 are defined within ZFC, this is what we mean by 1 + 1 = 2.

Bertrand Russell started the Mathematica Principia with a proof that 1+1=2, it took him 360 pages.
I'm afraid that that is a myth. He did not spend 360 pages proving 1 + 1 = 2, he proved 1 + 1 = 2 360 pages in. The pages prior to that built up the foundations of Principia Mathematica.
It took Russell and Whitehead 360pages to define and give meaning to the terms “1”, “+”, “=”, “2” and to lay thelogical foundation from which they could consider 1+1=2 to be proven.  They couldn’t have been more tedious, andwent off on a lot of tangents, apparently they themselves didn’t believe theywere there until page 362, I think most mathematicians think they hadn’t adequately defined "addition" yet, many believe it as actually took them 379 pages.   

Then ZFC and Godel came along andsquashed Logicism like a bug.

In my proof I considered those foundations to already be in place.
So...in your proof you considered the proof to already be inplace?
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@Sidewalker
“Generally accepted”, as opposed to “universal or absolute” acceptance, which is to say the definition is not accepted with100% certainty.
Definitions are arbitrary anyway, so I'm not sure why this is a point of contention. The point is that 1 + 1 = 2 is known with 100% certainty for the objects that most mathematicians use the symbols 1 and 2 to describe.

Yep, “Certainty in a formal system isa matter of definition”.   BTW, I’m 100% certain Sherlock Holmessmoked a pipe.
I'm not sure what your point is.

It took Russell and Whitehead 360pages to define and give meaning to the terms “1”, “+”, “=”, “2” and to lay thelogical foundation from which they could consider 1+1=2 to be proven.  They couldn’t have been more tedious, andwent off on a lot of tangents, apparently they themselves didn’t believe theywere there until page 362, I think most mathematicians think they hadn’t adequately defined "addition" yet, many believe it as actually took them 379 pages.   
Exactly. The proof that 1 + 1 = 2 was rather short, it's just that prior to that things like "1" and "2" weren't even defined yet.

Then ZFC and Godel came along andsquashed Logicism like a bug.
Did they? I'm not sure why there are so many misconceptions about Gödel's incompleteness theorem. All it says is that in any formal system F which contains basic arithmetic, there exists a statement A such that neither A nor its negation is a theorem in F. That's it. It doesn't mean that math is broken or anything like that. Math studies theorems and their proofs, most commonly within ZFC, but also within PA and other formal systems, and is in turn merely an extension of basic logic.

So...in your proof you considered the proof to already be inplace?
No. What?
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@Deb-8-a-bull
Hey Deb.

1 x 1 = 1

1 divide 1 = 1

2 divide 2 = 1

What the f**k?
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@Deb-8-a-bull
All numbers are infinite. Every number can  be infinite within itself. This is the semantics of math.  1+1 = 2 when the semantics are removed.  I to appeal to your frustration as I lost a debate about thinking 1+1 = 2. It does and it doesn't.  You must define the terms when asking if 1+1 = 2. Are you asking in simple terms or semantical mathematical terms. Numbers are infinite or rounded off.  Everyone lives in the simple rounded of version of 1+1 = 2. The person who argues for the sake of argument will argue that 1+1 doesn't = 2 and they would be right.
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@Math_Enthusiast
.99999999999999999999999999999999999999999999 for infinity. you will never get to 1.  Just as 1.999999999999999999999999999999999999999999 for infinity will never get to 2 all numbers are infinite within themselves. Its math reduced to semantics. I agree 1=1= 2 in the world we live in but I also accept that 1+1 can never equal 2 when mathematical semantics are used. Both are correct when the terms are established when asking if 1+1=2. You must come to terms with the term infinite. 
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@sadolite
.99999999999999999999999999999999999999999999 for infinity is equal to 1

It's a repeating decimal, the fact that the word infinoty can be used doesn't make it mysterious.

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@Sidewalker
.9999999999 infinitely  is equal to one. Just as 1.888888888888888888 infinitely would be equal to 2. You are the final word. You are right I am wrong.
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@sadolite
Just as 1.888888888888888888 infinitely would be equal to 2
Nope, 1.99999999999999 infinitely would be equal to 2.
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@Sidewalker
You are the final word, I wont debate you on it. I've already debated it and been proven wrong using mathematical semantics. 1.99999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 is not 2

Just one question is 1.899999999999999999999999999999999999999999999999999999999999999999999999999999999999999999 the same as 1.9 since 1.9999999999999999999999999999999999999999999999999999999999999999999999999999999999999 is the same as 2?