In a world governed by the uncertainty principle of Quantum Mehanics,
Why should 2 + 2 always  = 4 in Mathematics?

Hi, thank you for your question.

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.

I hope you consider this an adequate answer for the price.

Google search term: why does 2+2=4
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Quantum mechanic as a principle only applies to physical processes in
the sub-atomar region, not for macroscopic phenomena.

till-ga

It doesn't.

In base 3, 2 + 2 = 11.

Question ID: 44464 asked: How much is 4 + 4?...

Aw, C'mon, Poe

You are pulling our legs ...

How can the addition of two even numbers make an odd one?

What have you been drinking?

In base 3, 11 isn't an odd number - it's divisible by 2. Mine's a pint!

Ian G.

So, there, Poe & Ian G

I was right!

I was, of course, relying on Principia Mathematica by Whitehead and
Russell (1910-1913) for my observation and it's nice to know that
Siliconsamurai is also on our same wavelength.

I've never come across Siliconsamurai before but he/she is certainly one to watch.

> "You would never use chemistry notation to show someone how to bake a cake"

But nethertheless, some people still try!:

+++++++++++++++++++++++++++++++++++++++++++++++
Chocolate Chip Cookies:

Ingredients:

1. 532.35 cm3 gluten
2. 4.9 cm3 NaHCO3
3. 4.9 cm3 refined halite
4. 236.6 cm3 partially hydrogenated tallow triglyceride
5. 177.45 cm3 crystalline C12H22O11
6. 177.45 cm3 unrefined C12H22O11
7. 4.9 cm3 methyl ether of protocatechuic aldehyde
8. Two calcium carbonate-encapsulated avian albumen-coated protein
9. 473.2 cm3 theobroma cacao
10. 236.6 cm3 de-encapsulated legume meats (sieve size #10)

To a 2-L jacketed round reactor vessel (reactor #1) with an overall
heat transfer coefficient of about 100 Btu/F-ft2-hr, add ingredients
one, two and three with constant agitation. In a second 2-L reactor
vessel with a radial flow impeller operating at 100 rpm, add
ingredients four, five, six, and seven until the mixture is
homogenous. To reactor #2, add ingredient eight, followed by three
equal volumes of the homogenous mixture in reactor #1. Additionally,
add ingredient nine and ten slowly, with constant agitation. Care must
be taken at this point in the reaction to control any temperature rise
that may be the result of an exothermic reaction.

Using a screw extrude attached to a #4 nodulizer, place the mixture
piece-meal on a 316SS sheet (300 x 600 mm). Heat in a 460K oven for a
period of time that is in agreement with Frank & Johnston's first
order rate expression (see JACOS, 21, 55), or until golden brown. Once
the reaction is complete, place the sheet on a 25C heat-transfer
table, allowing the product to come to equilibrium

Wow, fj, I bet Pinkfreud (who knows a good recipe when she sees one)
will try this one, for sure.

Incidentally, I really do like the idea of 2 + 2 = 11.

Being an accountant, I can think of some great applications.

2 subatomic particles plus 2 subatomic doesn't equal 4 subatomic
particles.  You're right about that, and many believe that that
uncertainty extends to everything on the macrosopic level (via a
sort've butterfly effect) . . . but 2 + 2 still equals 4 every time
because mathematics is not a physical thing which can be measured.  It
is a concept (like a language) which we use for measuring things.  You
can't go visit the number 2, or hold "2" in your hand.  But if you
have a couple pairs of cookies in your hand, you can apply the concept
of 2 and 2 and then use the concept of mathematics to deduce 2 + 2 =
4.   So, the entire process of arriving at the number 4 is free from
the influence of quantum uncertainty (OK, so maybe quantum
indeterminacy in some microtubule in your brain could cause a glitch
in your thinking process, but that doesn't affect the concept of math,
just your understanding of it in that precise moment)

kamal3r...

I don't know about 2 + 2, but here's an interesting proof
that 1 = 0. Of course it's not true, but it should offer
you some food for thought:

Given a = b

multiply both sides by a:

a^2 = ab

subtract b^2 from both sides:

a^2-b^2 = ab-b^2

factor:

(a+b)(a-b) = b(a-b)

divide both sides by (a-b):

(a+b)(a-b)/(a-b) = b(a-b)/(a-b)

a+b = b

substitute b for a, since a = b

2b = b

divide by b:

2 = 1

subtract 1 from each side:

1 = 0

I notice this question is being asked on income tax
deadline day, so maybe this will be of some help to
you in preparing your returns...  : )

I think what it comes down to at it's simplest, is 2+2=4 because we
defined it to, just like we defined the word hot to be hot.  We made
up the number 2 and we made up the number 4, the idea of it and the
shape of the symbol for it and presto.

I say "resilono" means 100.  Why does it mean 100?  Because I said so.

I have just searched for 'resilono' and NOTHING!

Sorry, Bastian, but your marketing of this word leaves something to be desired.

I DO like it though.

Well done!

As for the 1=0 proof, you cant divide both sides by a-b because a=b
and so a-b = 0 and you cant divide by 0.

WONDERFUL!!!
Although I'm not taking the time to assess that recipe, it looks
pretty good but I'm not sure about the thermodynamics - I would
probably go with 425 or 400 degrees.

Actually I got pretty good grades in chemistry at Harvard (AHH, the
old Mallinckrodt building, does anyone know if it's still there?) but
I learned to  bake mostly around Radcliff, at least from The French
Cheff (Julia) who I worked with at WGBH back in the 60's. (NOTE, Julia
Child lived nearby in Cambridge back then and was often seen around
some "Cliffie" dorms.) Me, I lived on a boat out in Winthrop.

My vote for the best answers goes to poe.

He answered the question, correctly, with proper explanation.

Other may have added some wisdom and folklor ..

 which however may obscure the basic mathematical fact,
that validity of an aquation depends on the base system.

An aquation, Hedgie?

Surely you mean aquestrian ... Something to do with horses.