TT's Maths Thread

[TrueTears]

An interesting question:-

"There are only about 20 people in Victoria that can prove this." - Dr. Jianming He

Prove that 1 + 1 = 2.

Are the 20 people among the VN community?

Where in history do we get to start?

[brightsky]

"The proof starts from the Peano Postulates, which define the natural numbers N."

[TrueTears]

"The proof starts from the Peano Postulates, which define the natural numbers N."

One could go further back in history before Peano.

[/0]

An interesting question:-

"There are only about 20 people in Victoria that can prove this." - Dr. Jianming He

Prove that 1 + 1 = 2.

Are the 20 people among the VN community?

That's ridiculous. Show the proof to people and they will know how to prove it. Show it to 100 people and 100 people will know how to prove it.

[brightsky]

The proof starts from the Peano Postulates, which define the natural
numbers N. N is the smallest set satisfying these postulates:

  P1.  1 is in N.
  P2.  If x is in N, then its "successor" x' is in N.
  P3.  There is no x such that x' = 1.
  P4.  If x isn't 1, then there is a y in N such that y' = x.
  P5.  If S is a subset of N, 1 is in S, and the implication
       (x in S => x' in S) holds, then S = N.

Then you have to define addition recursively:
  Def: Let a and b be in N. If b = 1, then define a + b = a'
       (using P1 and P2). If b isn't 1, then let c' = b, with c in N
       (using P4), and define a + b = (a + c)'.

Then you have to define 2:
  Def:  2 = 1'

2 is in N by P1, P2, and the definition of 2.

Theorem:  1 + 1 = 2

Proof: Use the first part of the definition of + with a = b = 1.
       Then 1 + 1 = 1' = 2  Q.E.D.

Note: There is an alternate formulation of the Peano Postulates which
replaces 1 with 0 in P1, P3, P4, and P5. Then you have to change the
definition of addition to this:
  Def: Let a and b be in N. If b = 0, then define a + b = a.
       If b isn't 0, then let c' = b, with c in N, and define
       a + b = (a + c)'.

You also have to define 1 = 0', and 2 = 1'. Then the proof of the
Theorem above is a little different:

Proof: Use the second part of the definition of + first:
       1 + 1 = (1 + 0)'
       Now use the first part of the definition of + on the sum in
       parentheses:  1 + 1 = (1)' = 1' = 2  Q.E.D.

Here's the proof but I still don't understand it.

[brightsky]

"The proof starts from the Peano Postulates, which define the natural numbers N."

One could go further back in history before Peano.

Lolol, definitely agreed. Edit: Since mathematical concepts were around?

[/0]

It's all definitions and postulates, there's hardly a proof in it

[TrueTears]

It's all definitions and postulates, there's hardly a proof in it

Exactly.

[kamil9876]

I'm quite interested in proving the fundamental properties of numbers, and constructing the reals from the rationals, dedekind cuts etc. I recently read Edmund Landau's "Foundations of Analysis", a book that is completely devoted to this and he proves everything starting from the Peano Postulates. However "proving" "1+1=2" is useless. He never mentions the number 2, but denotes it always as 1+1. For example: "Theorem: 1+1 is irrational" or when refering to the midpoint of a and b he writes it as "". The only times he ever mentions the number 2 is in the preface:

"The multiplication table is not to be found in this book, not even the theorem 2*2=4; but i would recommend, as an exercise in connection with chapter 1, that you make the following definitions:

2=1+1
4=(((1+1)+1)+1)

and then prove the theorem"

Also found in the preface:

"Forgive me for 'theeing' and 'thouing' you. One reason for my doing so is that this book is written partly for my daugheters who have been studying chemistry at the University several semesters alreadyand think that they have learnt the differential and integral calculus in College; and yet they still don't know why:

x*y=y*x"

Having said that, this is what mathematicains are really interested in when it comes to Peano axioms: proving these general laws of arithmetic like xy=yx. They simply treat "2" as a denotation for "1+1" or "the succesor of 1", not as some deep theorem.

[brightsky]

One reason for my doing so is that this book is written partly for my daugheters who have been studying chemistry at the University several semesters alreadyand think that they have learnt the differential and integral calculus in College; and yet they still don't know why:

x*y=y*x"

That's probably true for 99.99999% of people. Great post! +1

[zzdfa]

An interesting question:-

"There are only about 20 people in Victoria that can prove this." - Dr. Jianming He

Prove that 1 + 1 = 2.

Are the 20 people among the VN community?

who is dr jianming he??

[TrueTears]

An interesting question:-

"There are only about 20 people in Victoria that can prove this." - Dr. Jianming He

Prove that 1 + 1 = 2.

Are the 20 people among the VN community?

who is dr jianming he??

Overrated tutor I believe.

[zzdfa]

ohh, dr he

[brightsky]

An interesting question:-

"There are only about 20 people in Victoria that can prove this." - Dr. Jianming He

Prove that 1 + 1 = 2.

Are the 20 people among the VN community?

who is dr jianming he??

Overrated tutor I believe.

Have you been there before? Lol, yeah, I think he is a bit overrated, but he is really really good at doing his job. I think his methods and ways of teaching help explain a lot of things that normal teachers can't accurately and aptly explain to a student.

[brightsky]

Lolol. Differs from person to person I guess. 可谓仁者见仁，智者见智!