Math is Understanding, the Continuum hypothesis and computational proof.

Cool Math, Math is Understanding, the Continuum hypothesis and computational proof

Math is understanding

Math relies on understanding to move forward.

If you don't understand 1+1=2 then how could you get to the point where you understand 2+2=4?

If you did not understand 1+1=2 you simply could not move forward. Unless someone has stated that it is true by way of proof. You should be able to take it as fact and deduce with logic that 1+2 would = 3. and from there you could move further.

Well, maths has moved a lot further than simple addition. But it is still not totally understood.

Usually, something is either true or false and that is it.

But, there is 1 problem that as far as we can tell has another answer. Undecided.

This is the Continuum hypothesis

without getting into it too deeply, the basic idea is this,

We have a number line, that runs to infinity in either direction 1234 etc

and -1-2-3-4-5 etc. these are natural numbers.

Lets call the infinity that the numbers go to 'Infinity set A'.

Now when we look between 1+2 we see ½ and 1/4 and thirds etc etc these are called real numbers there is also an infinite number of these numbers just between 1 and 2.

so lets call the sets between natural numbers infinity set B.

It is agreed that infinity B is bigger than infinity A.

Now the question.

Is there an infinity set that is smaller than B but bigger than A?

Its a tough question to answer. Infact it is so hard that the only proofs we have are

that it has been proven that we cannot prove it to be false.

and we can not prove it to be true.

Because we cant properly comprehend infinities we just cant get an answer either way.

But, what if a computer could comprehend infinities and actually answer the question for us?

and say Yes. there is an infinity between A and B.

Well, then we would have an answer. But how can we tell if the answer is right or wrong? Maybe get another computer to tell us, but then, we go down a never ending spiral of 'But how do we know for sure?'

Well, math is heading in this direction. We are looking at computers to solve problems that are so deep and complex that we can hardly understand them. Take the information it spits out and then go on to use it in another equation. for deeper and more complex answers.

But What's the value of an answer without the understanding?

Well, with these answers we will start to be able to do some pretty crazy things.

Who knows, maybe one day we will meet aliens. and the alien will ask us, how the heck did you get all the way out here.

and our answer will be...I dont know.

This video is completely open source

[JacksonAbascal]

This video isn't entirely accurate, the undecidability of the continuum hypothesis has nothing to do with our understanding of infinities, but rather our imprecise definition of them. There is no way a computer could prove or disprove the continuum hypothesis; it's literally impossible to do so within the standard set of axioms in which we perform mathematics (ZFC). In fact, we could add on the definitive rule "there is a set between A and B" or "there is not a set between A and B" and either one wouldn't interfere with any of the other standard axioms.

[thekkl]

I've seen positive integers called the naturals and I've seen all non-negative integers called the naturals but I think it's rare that someone would call negative integers natural numbers.

[jefferyleath]

"How did you get all the way out here?"
"I don't know."
That's a really powerful statement, speaks a lot to how we learn and teach now.

[stenoch]

Alien: How did you get here?
Me: First, how did you get here?
Al: I was born here. I have always been here. How did you get here?
Me: I was born coming here, I have always been coming here. Now I've arrived. Where are you going from here?
Al: I don't know.

[TheDavddd]

great job (i loved the ending), although you should change natural numbers for integers

[TimJSwan]

If a computer answers the question, that means there's a finite deduction (formal proof) of the statement and therefore an understanding.

[abhimanyushekhawat]

Negative numbers are not part of natural numbers they are integers @1:40

[Euquila]

in fact, the infinite set of reals between real numbers and is also a bigger infinity than the infinite set naturals

[Mike]

The last line of this video is amazingly deep. Well done.

[GrothenDitQue]

CH is definitely not the only assertion that is left undecided by ZFC! 😅
And this is completely independent of whether it is a computer or a human trying to decide it, it IS undecidable by ZFC...

[cuboidalswim]

You used to have more than 1ksubs, now you have less than 200, please continue making vids, it would make so many people happy

[Fizx]

Well done video. I am a college math student and found this very helpful in aiding me to understand the Continuum Hypothesis.

[seanspartan]

1:40 those are integers. Natural numbers are nonnegative.

[davidm.johnston]

Are you the one who's made this video or did you just upload it? I enjoyed it.

[jsdsparky]

lol. "I don't know!" Great video by the way!

[WeLoveMusicStudio]

Is there a set of every odd natural number? Does it even make sense to say that?

[benjaminbrady]

This is such a good video for its length. I wish you would make more videos...

[clermeil]

Dude, thank you.  I still don't fully understand the continuum hypothesis, but you brought me closer than others.

[brandonw]

But if we had a computer that could prove the CH then we could look at the computer's instructions and have a proof. Which violates the proof that we cannot prove that CH is true.

[dudeB]

You would love my professor. His name is Doron Zeilberger. He has a pretty entertaining blog where he makes a case for experimental mathematics and computer usage being the future of pure mathematics. Furthermore that mathemtatics ( number theory, set theory, topology etc) are statistics in disguise.