Number systems - open or not? (1 Viewer)

[buchanan]

A few days ago, njwildberger posted the following on youtube:

We think of natural numbers. We think of defining objects called fractions, integers, rational numbers. So we think of the concept of ''number'' in quotes as something that's open, that keeps on going. There's no bound. So we for example in our definitions may get to a certain stage. But in the next generation someone who's smarter who thinks more about these things, will be able to go further. But nobody will ever get to the end of the notion of number. There will always be a better way of thinking, a more embracive way of thinking, a more general way of thinking. The system is open.

http://www.youtube.com/watch?v=hTMGuBO-Hss

(from 6:40-7:22)

BUT …..

All division algebras have dimension 1, 2, 4, or 8.

So it's not as open as njwildberger would have us believe.

Proofs:

Kervaire:

http://www.pnas.org/content/44/3/280.full.pdf

Milnor:

http://www.ams.org/bull/1958-64-03/S0002-9904-1958-10166-4/S0002-9904-1958-10166-4.pdf

 

[Cazic]

Why are you associating division algebra with the word number? Surely you consider the natural numbers to be numbers?

I'm with Cayley and Dickson on this one.

 

[Cazic]

And isn't dimension dependant on the field you're working over? For example, the complex numbers are a one dimensional algebra over themselves, but a two dimensional algebra over the reals.

Either way, I don't think dimension is a useful property to try and attach to the concept of number. I'm not even sure it's worth trying to define the word, but I'd like to hear any thoughts on that.

 

[buchanan]

It presents a problem for extending number systems beyond the octonions (8 dimensional).

For example the sedenions (16 dimensional) cannot be a division algebra. Two nonzero elements can be multiplied to give 0. All higher-dimensional hypercomplex numbers likewise contain zero divisors.

So we must stop at 8 dimensions.

This presents a problem for njwildberger when he wants numbers to be "open, keeps going, no bound, go further, can't get to the end, more embracive, more general ...."

 

[Cazic]

But I don't see why we need the requirement: a number system must be a division algebra?

 

[buchanan]

Are you comfortable with zero divisors and are happy with sedenions, etc as numbers? Most people are not.

We could start with natural numbers, then integers, rationals, reals, complex numbers, quaternions and LASTLY the octonions.

Furthemore, we should ask what these numbers are useful for. Accepting numbers as entities in their own right in mathematics isn't enough. We should also ask what we can do with them.

All these numbers have been found to be useful.

But sedenions are useless!

 

[Cazic]

Many many many people said exactly the same thing when it came to "a number such that its square is negative".

Cool factoid: Lewis Carroll was one of those people, and he expressed this in disdain in Alice in Wonderland.

Anyway, why must numbers not have zero divisors? I'm not arguing with you, I'm just curious. Surely this is as arbitrary a requirement as requiring the square of a non-zero number to always be positive?

 

[buchanan]

Didn't your teacher teach you that you can't divide by 0?

Suppose we ignore the teacher and decide we are going to use sedenions. Then we can divide by 0.

Every sedenion constructed with the Cayley–Dickson construction is a real linear combination of the unit sedenions 1, e₁, e₂, e₃, ... , e₁₅, which form a basis of the vector space of sedenions.

Then (e₃+e₁₀)(e₆-e₁₅)=0.

Nasty. Really nasty. Especially if you want your numbers to be useful. Which sedenions aren't. And never will be!

 

[Cazic]

My teacher told me that we can divide by zero, we just can't divide zero by zero. In this case, we get the real projective line or the Riemann sphere, and both of these definitely are useful. The real projective line, for instance, can be associated with the ratios that we learn in early high school.

I also don't see anything wrong with the equation you write. We see this happen all the time, for instance, matrix multiplication. I argue that matrices are also a useful concept.

 

[buchanan]

Sedenions cannot be represented by matrices because they are not associative.

The real projective line and Riemann sphere are one-point compactifications of the real numbers and complex numbers. They are not generalisations to higher dimensions.

If you think number systems more general than octonions, such as sedenions are useful, where is the evidence for this? I claim no such evidence exists.

 

[tywebb]

i dunno about useful.

but does pretty count?

here are some sedenion fractals:

http://www.hyperfract.com/sedenion_gallery1/index.htm

 

[Cazic]

Fair enough buchanan, it's just that I don't think we should be so keen to throw away any mathematical object as useless when we can't predict the future so well, and when preconcieved notions of useless have been shown to be wrong in the past (eg. complex numbers)

 

[Schoey93]

A few days ago, njwildberger posted the following on youtube:

We think of natural numbers. We think of defining objects called fractions, integers, rational numbers. So we think of the concept of ''number'' in quotes as something that's open, that keeps on going. There's no bound. So we for example in our definitions may get to a certain stage. But in the next generation someone who's smarter who thinks more about these things, will be able to go further. But nobody will ever get to the end of the notion of number. There will always be a better way of thinking, a more embracive way of thinking, a more general way of thinking. The system is open.

YouTube - Reconsidering `functions' in modern mathematics: MF43

(from 6:40-7:22)

BUT …..



All division algebras have dimension 1, 2, 4, or 8.

So it's not as open as njwildberger would have us believe.

Proofs:

Kervaire:

http://www.pnas.org/content/44/3/280.full.pdf

Milnor:

http://www.ams.org/bull/1958-64-03/S0002-9904-1958-10166-4/S0002-9904-1958-10166-4.pdf

Why is it that philosophy of pure mathematics is always fascinating to me...like this Philosophy of Real Numbers/Complex Number yet the philosophy of the applied sciences, except for Feyerabend's Anything Goes methodology... is just, bland boring and...sorreey?

I'm not too sad about it, I just think two similar things should yield a similar result... but that's philosophy+mathematics/science for you... they just don't!