Enjoy a stupidly long explanation that I copy pasta'd from a Google answers thing.

"I hope you like this brief answer to what is actually an incredibly

complex question.

The simple answer is that, in everyday mathematics, in number base

systems higher than ?4,? 2 + 2 = 4 because it is defined as such.

Based on the definitions of the number ?2?, the number ?4?, and the

mathematical operation of addition, the answer is always the same. It

is at the basis of all number theory and other branches of

mathematics.

Simply put, if you take a pile of objects that we designate as

consisting of ?2? objects, and place it with another identical pile,

the count of the resulting pile of objects is equal to what we label

as the number ?4.?

Of course, there are other number systems and other ways of doing math

where that definition is not used, but they don?t generally produce

very useful results for everyday applications.

You can find a longer explanation at:

http://www.mathmojo.com/interestinglessons/why%20do2plus2equal4/why%20do2plus2equal4.html,

but it says generally the same thing.

It may surprise you to learn that an entire book, Principia

Mathematica, (the one by Whitehead and Russell, 1910-1913, not the one

by Sir Isaac Newton in 1687) devotes several hundred pages to deriving

an explanation of just why 2 + 2 = 4 (actually, as I recall, it was 1

+ 1 = 2).

I don?t recommend that you pick up a copy of this 2,000-page,

three-volume set, since it consists almost entirely of equations.

You can find some excellent discussions about the nature of

mathematical proofs and in particular some 3,000 logic and set theory

proofs at:

http://us.metamath.org/mpegif/mmset.html

In particular, you can find links to a 122-level ?proof? of why 2 + 2 = 4 at:

http://us.metamath.org/mpegif/mmset.html#trivia

A complete proof, such as the one presented in Principia Mathematica,

involves 1,789 sub-theorems consisting of 19,731 individual steps.

As some others have already commented, the question about quantum

mechanics only applies to extremely small events at and below the

atomic level. The mathematics involved in solving questions in quantum

mechanics seldom even involves addition. In the simplest sense, as I

once heard an old friend, John Van Vleck, Nobel Laureate Physics,

1977, say, you should always think of any question involving quantum

mechanics as being a question of probabilities. In that sense, in

quantum mechanics 2 + 2 never = 4 with any certainty.

You will find a more extensive, but still reasonably accessible,

introduction to quantum mechanics and mathematics at:

http://www.math.rutgers.edu/~oldstein/quote.html

One thing to keep in mind is that when you start getting into the

details of advanced physics then you must switch languages. The reason

things don?t seem to make sense at the atomic and sub-atomic levels is

that non-physicists are forced to discuss things in English or some

other language, which leads to what appear to be contradictions. In

reality, you can only discuss such things in mathematics, which is why

you will find most scientists very reluctant to talk about their work

with non-scientists.

Think about it this way. You would never use chemistry notation to

show someone how to bake a cake. In the same way, you simply can?t

explain most advanced physics concepts in English or any spoken

language; the best you can do is summarize, simplify, and generalize ?

the language simply won?t support detailed explanations without

resulting in apparent paradoxes or contradictions."

www.google.com