# Proving 1 + 1 = 2: An Exploration

• Trail_Builder
In summary, the conversation touches on the topic of 'proof' in mathematics and the idea of using axioms to simplify and prove mathematical statements. The conversation also mentions the importance of defining terms and concepts, such as addition and numbers, before attempting to prove them. The conversation ends with a suggestion to study axiomatic systems, Euclidean geometry, and the Peano axioms to gain a better understanding of mathematical proofs.

#### Trail_Builder

hi

lately i have been reading in maths a lot and started to look more deeply into 'proof'. however, i have come across a few proofs that seem a little 'silly'. for example, taking to an extreme, the extremely long proof (many hundred of pages) for 1 + 1 = 2. now i havn't actually read this proof, but I rekon i could reason it quite simply (probably my lack of experience and nievity saying this haha). anyways, here goes:

1 more than something, is one integer more than something. so, using the arithemetic counting (1, 2, 3, etc...), surely it is easily to see that the way to prove 1 + 1 = 2 is: start at 1, and count 1 number higher (adding 1 is moving one up the number line), therefore, 1 + 1 = 2.

now, i think my proof may not be rigorous because maybe I am assuming that "adding 1 is moving one up the number line", but surely using axiomatics (is that a word?) and going further back is unnecesary, because then you also get into semantics more and more?

now, i know i am fundamentally wrong somewhere, and because I've just started id be grateful if someone corrected me where i was going wrong...

thnx

o yeh, by "axiomatics" I am backing trying to get the idea across of using the original axioms and further simplification.

You haven't defined 1, 2, or +, justified why they have anything to do with the number line, or defined the number line, for that matter.

No one would argue that Russell's long proof is something you should read or take seriously.

so youd have to define numbers and addition and stuff lol.

i wouldve thought that addition is simply a word, and doesn't require proof lol. anyways, might have to glance through the 1 + 1 = 2 proof outa curiosity lol

Trail_Builder said:
so youd have to define numbers and addition and stuff lol.

i wouldve thought that addition is simply a word, and doesn't require proof lol. anyways, might have to glance through the 1 + 1 = 2 proof outa curiosity lol

I wouldn't recommend looking at the Russell '1+1=2' proof, if you were referring to this specific proof. You may start by looking at what an axiomatic system is, what is truth in mathematics with regard to a specific set of axioms, study some basic Ecleidian Geometry, so that you can see how a set of axioms can produce all these nice and easy theorems you know, check the Peano axioms, then procede to some Set Theory and the road goes on... It depends on what you want to learn, and what interests you.

However, this is just my opinion and my suggestions, there are much more than these...

o rite, yeh i read a book that mentioning euclidian geomtry and where are the theorems are dervied from - the 5 euclid axioms. guess the peano axioms are my next step then.

Trail_Builder said:
so youd have to define numbers and addition and stuff lol.

i wouldve thought that addition is simply a word, and doesn't require proof lol. anyways, might have to glance through the 1 + 1 = 2 proof outa curiosity lol

I didn't say 'addition required a proof', I said it requires a definition. In my normal system of mathematics, it is an axiom that 1+1=2, that is the definitions of 1, +, = are such that 2 is defined to satisfy 1+1=2.

There are no such things as inherently true statements - we work with not obviously inconsistent sets of statements.

The pointer to Euclidean geometry is important. The parallel postulate is a good example of an axiom where negating it leads to interesting mathematics. Mathematics that was for a long time thought not to exist because people often thought that the parallel postulate must follow from the other axioms, but they couldn't prove it. Turns out it is independent of the other axioms, hence spherical geometry and hyperbolic geometry.

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Trail_Builder said:
o yeh, by "axiomatics" I am backing trying to get the idea across of using the original axioms and further simplification.

For exercise prove axiomaticaly that:

$$0^x=0;x\in\mathbb{R}_{+}$$

tehno said:
For exercise prove axiomaticaly that:

$$0^x=0;x\in\mathbb{R}_{+}$$

lol, nah looks too easy to bother with hehe :P

Trail_Builder said:
lol, nah looks too easy to bother with hehe :P

You say that's too easy when you're bothering yourself with 1+1=2. :grumpy:

## What is the significance of proving 1 + 1 = 2?

The proof of 1 + 1 = 2 is significant because it is a fundamental concept in mathematics and serves as the basis for many other mathematical operations. It also highlights the importance of logical reasoning and proof in scientific and mathematical fields.

## What is the history of the proof of 1 + 1 = 2?

The proof of 1 + 1 = 2 dates back to ancient civilizations such as the Egyptians and Babylonians who used basic counting methods. It was later formalized by Greek mathematicians such as Euclid and Pythagoras and further developed by renowned mathematicians like Leibniz and Gauss.

## Is there more than one way to prove 1 + 1 = 2?

Yes, there are multiple ways to prove 1 + 1 = 2 using different mathematical techniques and concepts. Some of the most common methods include using basic arithmetic operations, mathematical induction, and set theory.

## How does the proof of 1 + 1 = 2 relate to real-world applications?

The proof of 1 + 1 = 2 has various real-world applications, such as in computer programming, finance, and physics. It is used in calculations involving quantities, measurements, and mathematical models.

## Are there any challenges or controversies surrounding the proof of 1 + 1 = 2?

While the proof of 1 + 1 = 2 is widely accepted and considered a basic truth, there have been some controversies and debates surrounding its validity. Some philosophers argue that it is a tautology and lacks true mathematical insight, while others believe it is a necessary and foundational concept in mathematics.