Why is 1+1=2 True?

[StudentDriver]

From a kid's perspective he would say "one apple and one apple is 2 apples." From a mathematician perspective he would say "simply because I said so." Then the next question we should ask is "what is the difference between pulling a unicorn out of thin air vs a mathematician mathing?" The difference is that doing mathematics actually lead to somewhere, while pulling a unicorn out of thin air doesn't really lead to anywhere.

Moral of the story: By asking the question "Why is 1+1=2 true," it lead to the conclusion, that there is a right and wrong way to pull things out of thin air.

in my personal opinion this a 10/10 analysis. I would like to hear your guys input on this.

[StudentDriver]

if you look closer at the comment, I'm slandering all artists out there, and saying that math is the only way to create things correctly.

Additionally, who cares about axioms. Why do I need some algebraic axioms to justify that 1+1=2 is true? When I can plainly state that 1+1=2 is true?

[evening-calm]

Have you come across Peano Arithmetic?

Proving 1+1=2 takes a bit of work but it’s something rather satisfying to do. The point of doing so is to generalise the principles by which we can reason about mathematical objects. If we try to pick a small but tightly defined “rules of the game”, or axioms as they’re called then generate a maximal set of conclusions one can draw from those rules the beauty of mathematics is found. Like any other art the creativity is a function of its constraints.

If we agree that there’s something called 0, such that a+0=a for all a, and that 1 exists, a unit of ‘having something’, and that the number after any number n is called a successor (succ) then:

example:
1 = succ(0), and
2 = succ(succ(0));

Then our familiar arithmetic becomes a syntax atop the basic rules of the game I laid out above. Unrolling in this way a basic sum like 2+3=5 becomes a stack of succ’s of zero where the total number of successor calls will total the final number we expect to get. The Hindu Arabic numerals like 2, 3, or 5 are just a shorthand that obviate the long strings of succ(succ(succ(……)))) that you would otherwise have to do but the principle that even the most trivial to the most complex facts can be unravelled back to basic first principles and built ‘from the ground up’ of these very primitive basic rules and much much more can be derived going the other direction to ask what else is implied by our premises is one of the great joys in maths.

[StudentDriver]

Eyyyyyyy, fk you maiinnnne. You didn't answer my question. I dont give a sht about Peano arithmetic. Look at my question. "Why do I need some algebraic axioms to justify that 1+1=2 is true, When I can plainly state that 1+1=2 is true." ... The answer is: I dont... I dont need ur axioms. It's a Rhetorical Question dumbahh.

[StudentDriver]

Do y'all want to know how pure math is done?How to make real sht?

[StudentDriver]

Have you come across Peano Arithmetic?

Proving 1+1=2 takes a bit of work but it’s something rather satisfying to do. The point of doing so is to generalise the principles by which we can reason about mathematical objects. If we try to pick a small but tightly defined “rules of the game”, or axioms as they’re called then generate a maximal set of conclusions one can draw from those rules the beauty of mathematics is found. Like any other art the creativity is a function of its constraints.

If we agree that there’s something called 0, such that a+0=a for all a, and that 1 exists, a unit of ‘having something’, and that the number after any number n is called a successor (succ) then:

example:
1 = succ(0), and
2 = succ(succ(0));

Then our familiar arithmetic becomes a syntax atop the basic rules of the game I laid out above. Unrolling in this way a basic sum like 2+3=5 becomes a stack of succ’s of zero where the total number of successor calls will total the final number we expect to get. The Hindu Arabic numerals like 2, 3, or 5 are just a shorthand that obviate the long strings of succ(succ(succ(……)))) that you would otherwise have to do but the principle that even the most trivial to the most complex facts can be unravelled back to basic first principles and built ‘from the ground up’ of these very primitive basic rules and much much more can be derived going the other direction to ask what else is implied by our premises is one of the great joys in maths.

EveningCalm, after reviewing a little bit of Peano's Arithmetic, and seeing how you can prove the basic algebraic properties like commutativity, associativity, etc and knowing a bit of real analysis, and seeing how we can construct the real numbers from basically natural numbers, now I begin to see the significance in Peano's arithmetic. I just went with the perspective that the algebraic properties are already obvious, and I start with those as axioms.

HOWEVER, "the algebraic properties are already obvious" for me comes from a specific place. In MagiOmni words the (Visual AND Symbolic Reasoning) place.