Question

According to N J Wildberger the number 23+(10^10^10^10^10^10^10^10^10^10) doesn't have a prime factorization, can we prove him wrong? Here's the video where he claims it: https://www.youtube.com/watch?v=WabHm1QWVCA&feature=youtu.be&t=719

Answer 1

i saw that video and i agree with him

Answer 2

he says "according to our current understanding of size of universe" that number doesn't exist

Answer 3

I already know of at least one prime factor it has, also I'm not really agreeing/disagreeing with him I just think it would be fun to try to solve.

Answer 4

All numbers have a prime factorization.

Source: Fundamental Freaking Theorem of Arithmetic.

Answer 5

He's basically saying that the number is prime, isn't he?

Answer 6

"doesn't exist" as he says it just means that this number has not manifested as anything we've observed. It kind of depends on what you mean by "exist" I think.

For example, the number 5 doesn't exist, it's just a concept we use for looking at things, you can never separate "5" and "cows" from a collection of "five cows".

@ParthKohli the number is divisible by 3, so it's not prime.

Answer 7

he is saying that number doesn't exist because thats too big to fit in our universe

and anything thats not part of our universe is not our concern

Answer 8

OK, sorry for not watching the video. Hold on...

Answer 9

I don't see how a number can ever be too big to fit in our universe since numbers don't have volume.

Answer 10

we can prove 3 divides that number
im just quoting what he says - if you cannot store that number in digital form or cannot have any other means to access using our current technology, it is as good as it doesn't exist

Answer 11

How many digits would that number have?

Answer 12

Just cause you can't hold something in a computer doesn't mean it doesn't exist. Can you store happiness in a computer?

Answer 13

Wouldn't that be done using logarithms?

Answer 14

we cannot debate hapiness/feelings
we can only debate math

Answer 15

Why?

Answer 16

As far as I'm concerned, 5 stones or 5 stored as bits in a computer are both just physical realizations of a concept of 5 but they are not "5" itself. There doesn't ever have to be 5 of anything for there to exist the concept of 5. I feel like we're arguing past each other though, I don't really understand how being able to store it in a computer or something very physical like this is a requirement for a number to exist.

Answer 17

@bibby just like 1000=10^3 has 3+1=4 digits, 10^10^...^10 has 10^...^10+1 digits haha

Answer 18

so why can't it be represented in text form?

Answer 19

it's huge

Answer 20

my issue is with the word can't

Answer 21

Cause I'm too lazy to write a program to write it out for me, that's why lol.

Answer 22

Showing that 3 is a factor of that number proves it is a composite number. When saying this, I think we're agreeing to some extent that the number exists.

However we don't have any access to the actual number itself. We cannot express its prime factorization using any of our current tools. This is precicely the reason he says the prime factorization doesn't exist i think..

Answer 23

well I guess can't can also mean not feasible

Answer 24

yeah, that's a better way to put it

Answer 25

is there a problem with writing it in this form 23+(10^10^10^10^10^10^10^10^10^10) that makes it somehow inferior to writing it out in base 10 form expanded?

Does pi not exist because we can't write out all the digits?

Answer 26

I see, so existence means "has not been discovered by a human somewhere at some time" rather than "it could never be discovered ever"

Answer 27

you will have to email him but i saw many of his videos and i believe he has good reasons for saying "the decimal expansion of pi doesnt exist"

Answer 28

"doesn't exist" means it is not part of our universe

Answer 29

...based on my interpretation of his lectures

Answer 30

universe = everything that exists

if something is not part of our universe then it doesn't exist...

his reasoning goes something along those lines..

Answer 31

In that sense, I have no problems with saying the complete prime factorization "does not exist" but I think "has not been found" is more appropriate.

As far as pi is concerned I'm not interested because I am not concerned that we have an infinite decimal representation in base 10. Pi is a well defined concept, it's the radius of a circle's circumference to its diameter, and the fact that it can be expressed as an approximation at all is convenient but not necessary. So as long as ratios exist, that's all that matters haha.

Answer 32

I agree

Answer 33

Interesting stuff

Answer 34

So ... pi does exist, even if we can't write out the digits, we can express it in other ways, right? But, for the main prime factorization problem, it might be true, but it would take a little bit of thinking to prove that it can't be factoredd

Answer 35

pi exists
all he says is that decimal expansion of pi doesnt exist which we all seem to agree too

Answer 36

Well, I'm not sure if we can disprove that a prime factorization exists, but I guess we could start like this ...

Answer 37

Let's suppose that some factorization exists
Then we could write the expression 23+ 10^10^10.. etc, as a multiple of two factors, a and b

Answer 38

We might try a modular arithmetic approach instead, but it's such an ugly number to work with no matter how you approach it >.<

Answer 39

Counting is just as subjective as any other human experience. Just like there is no happiness on the moon there is also no "3" on the moon because it is only in our minds. We can only ever label things as having a property of "3" but we can never actually have a "3".

In this sense, no numbers exist... But that sort of ruins the meaning of the word "exist". So as long as we accept that happiness and numbers exist in our minds and that this is some form of acceptable existence, and since our minds exist in reality I am certain numbers exist in this sense of the word.

Answer 40

I agree with that philosophical position about numbers xD

Answer 41

I'll have to think about it, but I sort of feel like math is kind of purely manipulating adjectives.

Answer 42

That's some Jaden Smith level deep.

Answer 43

Yeah you're right hahaha

Answer 44

Its deeper than the ocean, young man.

Of course, if we're only concerned with coming up with a prime factorization, then the number itself should be a part of that factorization if it is prime. So I suppose you could argue that a prime factorization must exist simply because either the number is prime, in which case the prime factorization would simply be the number itself, or the number is composite, in which case a prime factorization exists but may not be known.

I would say that by definition it must contain a prime factorization, even if the knowledge of what that prime factorization is could never be known

Answer 45

I notice that in the video he emphasizes that it's not _computationally_ true that all numbers had a prime factorization

I think I would say that mathematics is really about logic and defining arbitrary systems, and then exploring what those definitions imply --- That's the way I look at it, although I know different people have different theories

Answer 46

Yeah I think you're right. It's sort of confusing the difference of existence with computationally found. For instance, you might be in a prison unable to escape but you have a map of the world. You don't have to go to Australia as an example to know it exists, you just lack the means cause you're trapped in a jail cell.

Answer 47

Right, I think that's a good metaphor!

Answer 48

I have also been watching the video you linked, and it's definitely interesting.
And really, our knowledge that it exists is in a sense even stronger, because the idea follows from how we are defining things...

But I guess it comes from the difference he is talking about between the axiom of choice and the algorithm type of approach that he prefers. I think I would say though, with regard to the question of infinity and whether it's a useful concept, that a lot of problems really can't be resolved sensibly without it. I think the problem with his algorithm approach to mathematics is that all algorithms are potentially subject to chaotic effect.
To chaotic effects.. usually there is some marginal case that you can feed into an algorithm to get it to behave really unpredictably

Answer 49

It's interesting since a lot of things that are infinite are proved with an algorithm approach. Give me x, then x+1 so we can always go higher. Or how about infinite number of primes? Suppose there is a finite number of primes, multiply them altogether and subtract that by 1 and you will have a number that's not divisible by any of the primes you currently have so it is impossible to have all the primes.

But it's kind of weird, something seems fishy and I'm not sure what it is. It's using finite operations to somehow leapfrog our way into infinity because we have a secret extra bit that's not really even part of the argument it seems, and it's the fact that "we can always show it's not finite" but maybe because it's not finite does that necessarily mean it has to be infinite or is it possible to have something in between or an alternative we haven't considered some how?

Answer 50

Yes, that's a very interesting observation, you're definitely right that lots of things involving infinity are proven through algorithm approaches!
I think another interesting topic to consider with respect to this is questions about cardinality, for instance, how different "infinities" can compare to one another, or be larger or smaller. And, how could we articulate that through a system designed only around algorithms? I guess the extra part that combines with the algorithms that we are using to prove comes from our abstract definition of infinity and what it has to imply, or how it relates to other math concepts

Answer 51

You know, you asking about there being something in between got me thinking about cardinality and different types of 'sets', such as countable or uncountable sets (natural numbers vs the reals for instance)

Answer 52

Yeah, actually it's because of this kind of stuff I could never take real analysis very seriously. I just disagree with a lot of set theory things and proof things.

For instance I wondered if there were any two numbers next to each other and my teacher replied you can always take their average and get a number in between. But this is again a finite operation that doesn't seem to work because you could never do this. I think this argument sort of relates to the 1=.999... thing and I am honestly not sure I'm convinced in any direction by finite arguments for infinity, including things like mathematical induction and convergence, etc... but that's just how the "game is played" by everyone else and I don't completely disagree nor completely accept it.

I think one thing about cardinality that really messes me up is the idea that there are just as many squares of numbers as there are numbers since you can establish a 1 to 1 mapping between them. Yet it seems just as plausible to continue reasoning that we can now remove the squares from the natural numbers and we'd still have numbers left over... So maybe I'm just not seeing the reasoning. It's kind of like why I don't really care about the Banach-Tarski paradox... it's not a paradox it's just fake haha.

Answer 53

Yes, it's definitely an interesting topic. I agree with you that the set-theoretic basis is just "how the game is played" and I think it's fine to just go along with it
There are interesting alternate definitions of finiteness and infiniteness in wikipedia and elsewhere, for instance Dedekind-infinite. There is definitely a lot to explore on this topic.
Even if it doesn't have a necessary correspondence to reality as we live in it, etc

I think for numbers being next to each other, there's definitely an interesting question there in a lot of senses, because you could also try to apply different ordering concepts to numbers to explore whether they are next to each other and what that implies

The point about a 1-to-1 mapping between squares of numbers and numbers themselves is especially interesting because they are bouth countable infinities, but I agree that it seems you are losing numbers to say they have equal cardinality.
I guess that's a perfect example of where you might apply dedekind-infinity, since it's mapping betwen the sets that is surjective but doesn't seem to be injective

Answer 54

Anyway, as fascinating as it all is, we probably shouldn't talk forever!

Answer 55

I think we agree on most things about math philosophy. Hehe

Answer 56

Shouldn't talk forever or can't talk forever? Ok bad joke about infinity. This is fun though, I like hearing different opinions it helps me think differently. I rarely get the chance to talk about what I feel is important in mathematics since most people feel like math is boring and they hate it hahaha.

Answer 57

Haha, I know what you mean, it's great that you are thinking and working about these things though!

Answer 58

I don't see any reason to hate a subject like math, I mean c'mon we use it in everyday life and if you have probs with numbers then you really need to put your acts together, because you'll be using math forever!