Yeah, the four color problem becomes obvious to the brain if you try to place five territories on a plane (or a sphere) that are all adjacent to each other. (To require four colors, one of the territories has to be surrounded by the others)
But this does not make for a mathematical proof. We have quite a few instances where this is frustratingly the case.
Then again, I thought 1+1=2 is axiomatic (2 being the defined by having a count of one and then another one) So I don’t understand why Bertrand Russel had to spend 86 pages proving it from baser fundamentals.
Then again, I thought 1+1=2 is axiomatic (2 being the defined by having a count of one and then another one) So I don’t understand why Bertrand Russel had to spend 86 pages proving it from baser fundamentals.
Well, he was trying to derive essentially all of contemporary mathematics from an extremely minimal set of axioms and formalisms. The purpose wasn’t really to just prove 1+1=2; that was just something that happened along the way. The goal was to create a consistent foundation for mathematics from which every true statement could be proven.
Of course, then Kurt Gödel came along and threw all of Russell’s work in the trash.
Saying it was all thrown in the trash feels a bit glib to me. It was a colossal and important endeavour – all Gödel proved was that it wouldn’t help solve the problem it was designed to solve. As an exemplar of the theoretical power one can form from a limited set of axiomatic constructions and the methodologies one would use it was phenomenal. In many ways I admire the philosophical hardball played by constructivists, and I would never count Russell amongst their number, but the work did preemptively field what would otherwise have been aseries of complaints that would’ve been a massive pain in the arse
Yeah, the four color problem becomes obvious to the brain if you try to place five territories on a plane (or a sphere) that are all adjacent to each other.
I think one of the earliest attempts at the 4 color problem proved exactly that (that C5 graph cannot be planar). Search engines are failing me in finding the source on this though.
But any way, that result is not sufficient to proof the 4-color theorem. A graph doesn’t need to have a C5 subgraph to make it impossible to 4-color. Think of two C4 graphs. Choose one vertex from each- call them A and B. Connect A and B together. Now make a new vertex called C and connect C to every vertex except A and B. The result should be a C5-free graph that cannot be 4-colored.
Only works for a smooth curve with a neighbourhood around it. I think you need the transverse regular theorem or something.
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.
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 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.
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 friend of mine took Introduction to Real Analysis in university and told me their first project was “prove the real number system.”
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)