Haha, I came across something quite funny on Math Stackexchange just now. :) Someone was asking for a proof that \(2+2 =4\). Well, it's kind of intuitive because we see it in our every day life.

But let's go down deeper and explore this statement more rigourously. In other words, how do we really know for sure that that statement is true?!

Let me post that question to you now. Try it before you look at the link below. :)

Here's the link. :)

Enjoy!

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## Comments

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TopNewestIt's shocking that \(2+2=4\) can be proven in only 2 pages, when it took 300 pages in

Principia Mathematicaby Russell and Whitehead to prove that \(1+1=2\).Log in to reply

300 pages!! I wonder what they wrote so much to prove that 1+1=2

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Wow! 300 pages??? Another link here to prove using only field axioms (using 0s) to prove that \(-1 \times -1 = 1\). :D INTERESTING! :) (Refer to the first answer provided).

To Prove \(-1 \times -1 = 1\)

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@Michael Mendrin I don't understand why it would need \(300\) pages to proof that \(1 + 1 =2\). By definition of multiplication \(1 + 1 = 1 \times 2\) and since \(1\) is the identity element for multiplication over all numbers (excluding zero) \(1 + 1 = 1 \times 2 = 2 \times 1 = 2\). Can you explain to me why my proof is less "Rigorous"? Notice that I only used the definition of multiplication and identity element since both of them was used in "Rigorous" proof of \(2 + 2 = 4\).

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Well the whole point of the proof is how do you prove the definition of multiplication, not just we know by the definition of multiplication that this multiplied by this is this.I think that is why you would need a 300 page proof right? @Michael Mendrin and @Siam Habib

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Principia Mathematicaeffort, but the problem is much like what happens when you answer a child's question of "why?", he immediately asks the next "why?" in an infinite regression. Kurt Godel, after reading the efforts of Russell and Whitehead, finally came out with his [first of his incompleteness theorems]On Formally Undecidable Propositions in Principia Mathematica and Related Systems. He essentially showed that perhaps Russell and Whitehead were fruitlessly chasing their own tails, and that, after all, the idea of putting all of mathematics on a "consistent and complete" foundation was doomed from the beginning. Hence, we pretty much go back to simply saying, "1+1=2 because it just is.". @Siam Habib and @Mardokay MosazghiLog in to reply

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If the whole point of it is about defining multiplication, why did they want to write about proving \(1+1=2\) instead?

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hey michael is this book available on internet ? i am eager to see this

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OMG 300 pages

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Yes. Such is mathematics. Arithmetic requires so much information to compute.

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Such an easy question but no one can prove it!!!

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2+2=(2)+(1+1)=3+1=4 (This is a fact that adding 1 is moving right to the number line which increases the count by 1)

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Mind = Blown! :D

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Haha Ikr :)

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