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!

## 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 \(1+1=2\). – Michael Mendrin · 2 years, 11 months agoLog in to reply

– Anuj Shikarkhane · 2 years, 11 months ago

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

To Prove \(-1 \times -1 = 1\) – Happy Melodies · 2 years, 11 months ago

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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 \(2 + 2 = 4\). – Siam Habib · 2 years, 11 months ago

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@Michael Mendrin and @Siam Habib – Mardokay Mosazghi · 2 years, 11 months ago

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?Log in to reply

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 Mosazghi – Michael Mendrin · 2 years, 11 months agoLog in to reply

– Happy Melodies · 2 years, 11 months ago

Yea I think so.... Like referring to the proof of \(-1 \times -1 = 1\) as I have posted as a comment above, the lemma proved that \(0 = 0 \times 0\) which most of us think is common sense....Log in to reply

If the whole point of it is about defining multiplication, why did they want to write about proving \(1+1=2\) instead? – Samuraiwarm Tsunayoshi · 2 years, 11 months ago

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– Rishabh Jain · 2 years, 11 months ago

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

– Aman Sharma · 2 years, 11 months ago

OMG 300 pagesLog in to reply

– Sharky Kesa · 2 years, 11 months ago

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

Such an easy question but no one can prove it!!! – Anuj Shikarkhane · 2 years, 11 months ago

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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) – Anshul Gupta · 2 years, 1 month ago

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Mind = Blown! :D – Sharky Kesa · 2 years, 11 months ago

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– Happy Melodies · 2 years, 11 months ago

Haha Ikr :)Log in to reply