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It wouldt fit so there are multiple replies

Buckle up! Our quest to prove 1+1=21+1=2will prove (haha!) to be a rather long one…

Where do we even begin? Let’s start with notation.

Mathematics is largely concerned with notation, which is used to help package an idea up into a single unit that can be more easily manipulated later on.

It’s a really wonderful thing that takes advantage of the “chunking” abilities of the brain. This is the same chunking your brain uses to touch type, memorise phone numbers, and understand new learning material, along with millions of other things.

But what we need to do here is “unchunking”, or what mathematicians sometimes call “unfolding”. When you’re given any problem in maths you need to unfold the definitions. And in fact you do it all the time! Not just in maths but whenever you communicate in a language. The goal is to unfold words or notation to an intuitive level–into the unique native language of your brain, if you will.

Now the problem here is that we’ve been presented with notation that we don’t really know how to unfold. In a conversation this wouldn’t usually matter, since the primary goal is to communicate efficiently without worrying about what words really mean. But in mathematics it does matter, and very much so.

So let’s consider what 1+1=2 means… well, um… heck, what really is 1+1? Or ==? How do we define ++? Is the first 11 and the second 11 even the same thing? (actually, the last question is more of a philosophical one)

To answer those questions, we first need to understand that mathematics is built from statements which we call axioms, and which are accepted as self-evident truths (at least temporarily). You can liken this to a game where, while playing the game, you try to do as much as possible while still abiding to some given rules.

In theory, we could simply come up with any random collection of axioms and shoot off to see what we can derive! But of course, over time, mathematicians have agreed that there are some sets of axioms that are simple and yet produce extremely complex behaviour, which is exactly a mathematician’s cup of tea.

Sometimes, those axioms are very good at modelling things that we’re familiar with in the real world, which brings us to Peano Axioms, a set of nine axioms presented by Giuseppe Peano in 1889 which described numbers very well. We’ll only be needing the following five axioms today:

(1) 00 is a natural number.

(4) For all natural numbers xx, yyand zz, if x=yx=y and y=zy=z, then x=zx=z.

(5) For all aa and bb, if bb is a natural number and a=ba=b, then aa is also a natural number.

(6) For every natural number nn, S(n)S(n) is a natural number.

(7) For all natural numbers mm and nn, m=nm=n implies S(m)=S(n)S(m)=S(n)and S(m)=S(n)S(m)=S(n) implies m=nm=n.