MathFoundations17: Extremely big numbers

We look at extremely big numbers. This is the best way to get a feel for the immensity and complexity in the sequence of natural numbers. And why we have no right to talk about `all' of them as a completed `infinite set'. Our main tool is a cool inductive way of defining higher and higher operations, going beyond multiplication and exponentiation.

Category:

Science & Technology

Tags:

• Foundations
• mathematics
• natural
• numbers
• operations
• big
• exponentiation
• Wildberger

[POWLIHERE22]

I hope that you can, with your video lectures, help me understand how to make sense of the philosophical implications of quantum mechanics. As of now, I am not sure how to proceed given your stricture against the use of infinite sets and axiomatic systems. For example, how does one deal with the concept of diffeomorphism invariance between coordinate systems without Real analysis? I was hopping that you would cover Robinson's non-standard Arithmetic and that this would point me toward a solution

[njwildberger]

@POWLIHERE22 I will be talking about non-standard arithmetic at some later point in this series. It will certainly be a while before we get close to quantum mechanics!

[tommyrjensen]

In the first or second video he said that it is very easy to understand the natural numbers, they are just strokes on paper. Now he says that it is very difficult. So which is it?

[njwildberger]

@tommyrjensen To say what a natural number is relatively easy, at least in an informal way. But to talk about `all natural numbers'' is quite different. As I try to explain, the further you go in the natural numbers, the more complicated they get. Ultimately one is overwhelmed by the immensity of really big numbers, and arithmetic with them becomes virtually impossible. This is the reality of number theory, a reality that is rarely acknowledged!

[TanvirKaykobad]

i'd love to see a method to compare the two large numbers mentioned at the end of the vid. thanks!

[njwildberger]

The second of those two numbers is not so intractible. However if the 3 was changed to a four, it would be much harder.

[tothemesosphere]

I have just watched this for a second time and still perplexed.

It seems to me your argument is this: because the natural numbers is a very difficult collection of objects to understand, we should not think of them as a whole object.

What about finite groups? Can we consider these collections of objects as complete sets? Of course we can, but they can be extremely difficult to understand, just as difficult as large natural numbers.

[tothemesosphere]

Just because it is very difficult to comprehend and describe individual "extremely large" numbers does not mean we should not be able to think of the natural numbers as a completed set!

In this video you are going along just fine, convincing us that the natural numbers is indeed a very complicated collection of objects but then you use this to jump to the conclusion that the natural numbers does not exist as a set out of nowhere!

[relike868p]

Thinking about star minus one, minus two, etc....

So if we do it this way, do we get "extremely small numbers" perhaps?

[MooOfDoom]

@samruby82

Well, 2(*_100)3 = 2(*_99)(2(*_99)2), and it's easy to work out that 2(*_n)2 = 4 for any number n, so 2(*_100)3 = 2(*_99)4, and this is clearly less than 27(*_99)15 because both arguments are smaller, while the operator is the same.

[sftw009]

@njwildberger

"to talk about `all natural numbers'' is quite different"

Why is this different? There is nothing wrong with the statement that the sum of 2 natural numbers is always a natural number. (m+n is a natural number for all natural numbers m and n).

Some people are overwhelmed by a simple long division and doing arithmetic with numbers smaller then 10.

Also, The field of rational numbers is infinite. Does this cause problems as well in your opinion?

[tommyrjensen]

On the contrary, the reality of the difficulty of factoring large numbers is quite widely acknowledged and has important applications. The underlying reason is however not understood as due to the size of the numbers, but to the problems with understanding factoring well enough. Will you prove unique factorization?