Can i prove 2 + 2 = 5 mathematiically?

Knowing what a proof is, is the first step in answering this question. From what I am familiar with...

A proof is: a set of logical steps acquired through deductive (therefore, not making any giant leaps in logic, unless by definition), and hence, empirically (from the evidence provided) resulting in a direct equivalence (being, among other types of equivalence, but primarily, in permutation, multiplicative/additively & negatively/positively & even/odd... meta-mathematically) of states, that's shortest distance is (in absolute terms), either infinity, zero, and/or, also, one.

Really, the attempted 'proof' of 2 + 2 = 5 is based on a distorted type of Trigonometry, which was in essence the source of today's Calculus (just try to draw Tangent or Secant without running into the idea of Calculus' derivative & integral, respectively), and actually is the result of any additive equavalence of any two numbers' to being alike to any number, (because measuring hypotenuse of a given sides is essentially multiplicative, hence partially irrational). 

(Which makes me wonder... is there a 2 * 2 = 5 equivalent? and the answer is a resounding, yes! But first the 'proof' as written by Charles Seife.)

Let a & b each be equal to 1. Since a ^ b are equal,

b^2 = ab ...(eq.1)

Since a equals itself, it is obvious that 

a^2 = a^2 ...(eq.2)

Subtract equation 1 from equation 2. This yeilds

(a^2) - (b^2) = (a^2)-ab ...(eq. 3)

We can factor both sides of the equation; (a^2)-ab equals a(a-b). Likewise, (a^2)-(b^2) equals (a + b)(a - b) (Nothing fishy is going on here. Ths statement is perfectly true. Plug in numbers and see for yourself!) Substituting into the equation 3 , we get

(a+b)(a-b) = a (a-b) ...(eq.5)

So far, so good. Now divide both sides of the equation by (a-b) and we get

a + b = a ...(eq.5)

b = 0 ...(eq.6)

But we set b to 1 at the very beginning of this proof, so this means that

1 = 0 ...(eq.7)

...Anyways, getting that far gives us the jist of the proof, later in the proof, Charles Seife goes on to prove that Winston Churchill was a carrot! if you want to know how that is possible, I recommend you read the book.

From equation 7, add a number to either side and get it equal to any other number, one greater than itself.

Multiplying equation 7 after adding to it, and one can get: any number is equal to any other number. 

Hence, conceptually, any number is equal to zero, and, theoretically, that includes infinity. But that's also the reason why when you divide by zero, it is 'Undefined.' Which, consequentially, is what is happening in this equation... just subsistute 1 into equation 3 and one will see that we are dividing by zero in equation 5. 

This is what lead to the invention of calculus. Really, from here this segways into Hilbert Space... but that is best left for another entry, hopefully, on the actual subject of quantazation.

That's all I have time for... 

THIS PROOF IS BY DEFINITION INCORRECT, but it provides a good tool as of why we define things in mathematics the way we do.

A good question to ask from here would be (based on my previous tangent):

Does 1/3 plus 1/3 plus 1/3 = 1?
Or, does it equal just zero point nine repeating?