Can anyone please represent an abstract concept of "Sum" of two numbers differing from its definition in linear algebra?

For example, why 1 + 1 equals 2, and How?

Your answers should not be necessarily mathematical and rational !

Thanks.

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- Jul 28th 2012, 07:59 AM #1

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## Why 1 + 1 = 2 ?

Can anyone please represent an abstract concept of "Sum" of two numbers differing from its definition in linear algebra?

For example, why 1 + 1 equals 2, and How?

Your answers should not be necessarily mathematical and rational !

Thanks.

- Jul 28th 2012, 09:21 AM #2

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## Re: Why 1 + 1 = 2 ?

the simplest answer is: "2" is the name we give for the expression "1+1", which is usually interpreted to mean: "one more than 1 (the next number after 1)"

this has more to do with how we define "+" (addition of natural numbers) than it does with "1".

- Jul 28th 2012, 09:25 AM #3

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## Re: Why 1 + 1 = 2 ?

I am afraid you will have to tell us what you mean by the 'abstract definition of "Sum" of two numbers' and what you mean by 'its definition in Linear Algebra". We do not normally define addition of two numbers in Linear Algebra- we define a vector space over a field (rational, real, or complex numbers) and use the definition of sum already given in that field.

And what 'abstract definition of "Sum of two numbers' you want to use depends upon how you are defining**numbers**. The most basic (and abstract) definition that I know is for whole numbers, N, and defines them as a collection of sets. We define 0 to be the empty set, {}, 1 to be the set whose only member is the empty set: {{}} or {0}, 2 to be the set whose members are 0 and 1: {{}, {{}}} or {0, 1}, etc. One can show that these satisfy the these satisfy the "Peano axioms": there exist a mapping s (the "successor" function) from N to N such that

1) s maps N one-to-one and onto N\0. (That is, s maps every set to another and every set except 0 has something mapped onto it).

2) if a set of whole numbers contains 0 and, whenever it contains a specific whole number k, it also contains s(k), then the set**is**the set of all whole numbers. (induction).

(Essentially, s allows us to put a linear order on the set: s(n) is the "next" whole number: s(0)= 1, s(2)= 2, etc..)

Now, for h and k any two such whole numbers, we define h+ k by

a) If k= 0, then h+ k= h.

b) If k is not 0, then there exist m such that k= s(m) and we define h+ k to be s(h+ m).

One can use the induction axiom to show that this**is**a valid definition- that it assigns a unique whole number to each pair of whole numbers, h and k.

In particular, 1+ 1 is defined as s(1) which, as before, is 2. To find "2+ 2" we would have to argue that 2= s(1) so 2+ 2= s(2+ 1)= s(s(2)). s(2) is, by definition of s, 3, and then s(3)= 4, proving that 2+ 2= 4.

How you would then define "addition" for the integers, the rational numbers, the real numbers, the complex numbers, then depends on exactly how you define those sets.

For example, we can define the integers as "equivalence classes of**pairs**of whole numbers with the equivalence relation [tex](a,b)\equiv (c,d)[tex] if and only if a+ d= b+c."

In that case, we define addtion of two integers, i and j, by "choose a pair (a, b) from i and a pair (c, d) from j. The i+j is the equivalence class that contains the pair (a+c, b+e)."

Abstract enough for you?

- Jul 28th 2012, 09:28 AM #4
## Re: Why 1 + 1 = 2 ?

- Jul 28th 2012, 09:28 AM #5

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## Re: Why 1 + 1 = 2 ?

Well, this is a math forum, so can only expect to find mathematical and rational answers here. You could post this question in the Math Philosophy subforum if you are interested in a discussion that is not mathematical in nature.

By "represent an abstract concept of 'Sum'," do you mean to give an example? You can define 1 + 1 to be anything. Such definition of addition may not have nice properties like commutativity and associativity, though. One nice example is addition modulo 2, which has all the usual properties and where 1 + 1 = 0.

When we are trying to justify very basic statements that are usually covered by intuition, we go to axioms. In Peano Arithmetic, the basic concepts are 0 and the successor S. The number 2 is defined as S(S(0)). The fact that S(0) + S(0) = S(S(0)) is a corollary of the two axioms that define addition. In a ring, there is no such concept as 2, so one can*define*it as 1 + 1.

- Jul 28th 2012, 09:55 AM #6

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## Re: Why 1 + 1 = 2 ?

alternatively, if one has a notion of cardinality and disjoint union of sets, then one can define:

which for A = B = {*} (a one-element set...{Ø} can be used if one needs to be reassured such a set actually "exists") leads to:

1 + 1 = 2 (technically, one has to display a set C for which |C| = 2 (so we know what "2" means). the usual example is {Ø,{Ø}}).

while this definition of + agrees with our intuition for finite cardinals, it is not quite the same thing as "ordinary addition" for non-finite cardinals. and there is no clear way to express what we might mean by "negative cardinality" or "fractional cardinality".

- Jul 28th 2012, 11:05 AM #7

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- Jul 28th 2012, 11:49 AM #8

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## Re: Why 1 + 1 = 2 ?

We do not normally define addition of two numbers in Linear Algebra- we define a vector space over a field (rational, real, or complex numbers) and use the definition of sum already given in that field.

Thanks to Emakarov for noticing me, actually what I mean by an 'abstract definition' is more referring to the math philosophy.

The reason as to why I asked this question is to find a human solution to implement this concept for a non-intelligent machine. Actually, I want to know when we add up two numbers, do we need anything else such as 'intelligence' or similar mental concepts to do this operation? Or, it's just about a "+" sign and an agreement on a mathematical rule?

For example, from a kid's perspective, what is "Sum" or "Counting"?

Or assume someone asking: "Why for the first time somebody thought about numbers, adding them up, counting and sequence?"

By the way, thank you for your outstanding answer.

- Jul 29th 2012, 04:23 PM #9

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## Re: Why 1 + 1 = 2 ?

By perception two different "somethings", be it a solid object or an image of a concept in your head is separable. Humans as most animals have short term memory, so we are able to backtrack our actions, in this case avoiding counting the same objects more than once. The reason why we count objects - either consciously or subconsciously is to place one "something" relative to another "something". If we were unable to count everything would be equally sized and be of the same quantity, like a whiteout of all your surroundings.

If you want to design a machine to count, it will need a set of rules describing the different objects to count (neural networks, perceptron etc) and it will need memory to avoid counting objects twice - or to not count objects that are equal. If the machines operates in a domain where objects disappear similar to objects on a conveyor belt then it will not need to memorize objects. A non-intelligent machine can count objects going down a conveyor belt by simply looking at a laser beam, if it's cut - count up, if not - wait for it to be cut again.

The constraints and rules for the counting machine can be anything, but obviously the simpler the domain/area the simpler the machine.

But the concept of counting is basically just placing something relative to something else and it is quite similar to the concept of measuring size. In humans this is taken care of in the parietal lobe, more specifically the intraparietal sulcus.

Edit: I'm sorry if I went completely out of the box on this one

- Jul 29th 2012, 06:13 PM #10

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## Re: Why 1 + 1 = 2 ?

By perception two different "somethings", be it a solid object or an image of a concept in your head is separable. Humans as most animals have short term memory, so we are able to backtrack our actions, in this case avoiding counting the same objects more than once. The reason why we count objects - either consciously or subconsciously is to place one "something" relative to another "something". If we were unable to count everything would be equally sized and be of the same quantity, like a whiteout of all your surroundings.

Personally, I believe that the difference between two entities can express the beauty. However, there would be a condition and that is learning the difference consciously.

That difference, in my mind, is not an integer. It is an entity itself that fills any gap between two others in which you can have a feeling of continuity. At no points you can find a disconnection in that gap.

That's why what you are thinking about a "whiteout" is the "actual truth". Let's think about a piece of music you hear. With an assumption of discontinuity in terms of hearing the notes, can you call what is interpreted to your brain the so-called "music"?

Thanks to short-term memory that makes the incidents overlapped, we are able to have a feeling of listening to a piece of music (not 100 pieces/notes !!!).

Now, we know why a musician can compose a music which delivers passion and sensation. He/She actually knows how to deal with the differences between notes and octaves and also the emotion of hearing those differences.

As we know, in digital era, numbers are everything in order to model algorithms and procedures. On the basis of what I have said, do you think it is possible to implement an integral algorithm to teach computer to learn the difference between entities "CONSCIOUSLY"?

An example I stared from is "why 1 + 1 = 2 ?".

I was thinking that it would be a good idea to share this simple question with mathematician and who are interested in maths to study their natural reflection and interpretation of the "Sum" operation or "Counting".

How did they learn to count, and what is the difference between 1 and 2 for each individual with the hope of finding a suitable thought to implement as an algorithms for computer.

Thanks to you again for your thoughtful answer.