This guy would not be happy to learn about the 1+1=2 proof
One part of the 360 page proof in Principia Mathematica:
It’s not a 360 page proof, it just appears that many pages into the book. That’s the whole proof.
A friend of mine took Introduction to Real Analysis in university and told me their first project was “prove the real number system.”
That assumes that 1 and 1 are the same thing. That they’re units which can be added/aggregated. And when they are that they always equal a singular value. And that value is 2.
It’s obvious but the proof isn’t about stating the obvious. It’s about making clear what are concrete rules in the symbolism/language of math I believe.
This is what happens when the mathematicians spend too much time thinking without any practical applications. Madness!
Isn’t 1 and +1 well defined by the Peano Axioms by using the intersection of all infinite successor functions and starting at the empty set?
Using the Peano axioms, which are often used as the basis for arithmetic, you first define a successor function, often denoted as •’ and the number 0. The natural numbers (including 0) then are defined by repeated application of the successor function (of course, you also first need to define what equality is):
0 = 0
1 := 0’
2 := 1’ = 0’’
etc
Addition, denoted by •+• , is then recursively defined via
a + 0 = a
a + b’ = (a+b)’
which quickly gives you that 1+1=2. But that requires you to thake these axioms for granted. Mathematicians proved it with fewer assumptions, but the proof got a tad verbose
The “=” symbol defines an equivalence relation. So “1+1=2” is one definition of “2”, defining it as equivalent to the addition of 2 identical unit values.
2*1 also defines 2. As does any even quantity divided by half it’s value. 2 is also the successor to 1 (and predecessor to 3), if you base your system on counting (or anti-counting).
The youtuber Vihart has a video that whimsically explores the idea that numbers and operations can be looked at in different ways.
A lot of things seem obvious until someone questions your assumptions. Are these closed forms on the Euclidean plane? Are we using Cartesian coordinates? Can I use the 3rd dimension? Can I use 27 dimensions? Can I (ab)use infinities? Is the embedded space well defined, and can I poke a hole in the embedded space?
What if the parts don’t self-intersect, but they’re so close that when printed as physical parts the materials fuse so that for practical purposes they do intersect because this isn’t just an abstract problem but one with real-world tolerances and consequences?
until someone questions your assumptions
Oh, come on. This is math. This is the one place in the universe where all of our assumptions are declared at the outset and questioning them makes about as much sense as questioning “would this science experiment still work in a universe where gravity went the wrong way”. Please just let us have this?
yea this is one of those theorems but history is studded with “the proof is obvious” lemmas that has taken down entire sets of theorems (and entire PhD theses)
It’s fucking obvious!
Seriously, I once had to prove that mulplying a value by a number between 0 and 1 decreased it’s original value, i.e. effectively defining the unary, which should be an axiom.
Mathematicians like to have as little axioms as possible because any axiom is essentially an assumption that can be wrong.
Also proving elementary results like your example with as little tools as possible is a great exercise to learn mathematical deduction and to understand the relation between certain elementary mathematical properties.
So you need to proof x•c < x for 0<=c<1?
Isn’t that just:
xc < x | ÷x
c < x/x (for x=/=0)
c < 1 q.e.d.
What am I missing?
This isn’t a rigorous mathematic proof that would prove that it holds true in every case. You aren’t wrong, but this is a colloquial definition of proof, not a mathematical proof.
Only works for a smooth curve with a neighbourhood around it. I think you need the transverse regular theorem or something.