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Maybe i am nitpicking but the existence of transcendentals was proven by Liouville 30 years before Cantor brought up the considerations about countability. (dis

Maybe i am nitpicking but the existence of transcendentals was proven by Liouville 30 years before Cantor brought up the considerations about countability. (disclaimer liouville was the advisor of the advisor of the advisor of the advisor of the advisor of the advisor of my advisor) Also if you like that sort of things you would like http://amzn.com/0201038129 http://amzn.com/0201038129 Nearly 30 years ago, John Horton Conway introduced a new way to construct numbers. Donald E. Knuth, in appreciation of this revolutionary system, took a week off from work on The Art of Computer Programming to write an introduction to Conway's method. Never content with the ordinary, Knuth wrote this introduction as a work of fiction--a novelette. If not a steamy romance, the book nonetheless shows how a young couple turned on to pure mathematics and found total happiness.

ColinWright 13y ago
Indeed, Liouville proved the existence of transcendentals, exhibiting specific numbers as examples. That was around the 1840s. Cantor showed in 1874 that pretty much every real number is transcendental, and the linked article is a re-casting/adaptation of one version of that paper. Although it shows how to construct a transcendental, it really shows that every list of numbers is incomplete. Taking the algebraics as the list then gives the result. And I actually have a copy of "The Surreal Numbers". Still working on how it/they can be presented to a non-technical audience.
- stephencanon 13y ago
  Both "Surreal Numbers" and "On Numbers and Games" do a not-terrible job of presenting surreal numbers to a (relatively) non-technical audience.
  - ColinWright 13y ago
    Not convinced. I've tried to follow their presentations when talking about these topics with different audiences and found that people really just don't follow. There's a difference between teaching material and exposing people to it - both of those works try to teach the topic to people, which isn't really appropriate in a single lecture format, or a math club, or a masterclass.
    - stephencanon 13y ago
      Well, they are full-length books (Ok, SN is shortish). There aren't many books that can be presented as a single lecture as is. The surreal numbers (as you know, I'm sure) are fairly subtle, and people don't have much intuition for how they should behave. Even non-technical people have lots of intuition for real numbers (even if some of it is wrong); they know that they can add them, multiply them, compare them, etc, and they even know algorithms to do so. When confronted with the surreal numbers, it's not at all obvious that you can perform those operations, much less how to do so, so there's an enormous weight of basics to build up before you can really do anything interesting (much like how the first week or two of an introductory linear algebra course can be pretty dull). Games are perhaps easier to motivate and make interesting (in my experience), as you can get to the point of analyzing and reducing simple positions and establishing the addition operation pretty quickly, without getting bogged down in "wait, why would I use this horribly complex approach to what seems at first to be not terribly unlike the real numbers except maybe I can't even do all the operations I know and love". Unfortunately, multiplication of games doesn't really make terribly much sense, so you need to restrict to numbers reasonably quickly to continue developing "conventional" mathematical structures. TL;DR: for a single lecture format, motivating and defining games and showing that there's a reasonable embedding the natural (or dyadic rational) numbers isn't a bad introduction in my experience.
ColinWright 13y ago
By the way, you might be interested in this: http://www.penzba.co.uk/images/LO_n_CW.png http://www.penzba.co.uk/images/LO_n_CW.png

ColinWright 13y ago

Indeed, Liouville proved the existence of transcendentals, exhibiting specific numbers as examples. That was around the 1840s. Cantor showed in 1874 that pretty much every real number is transcendental, and the linked article is a re-casting/adaptation of one version of that paper. Although it shows how to construct a transcendental, it really shows that every list of numbers is incomplete. Taking the algebraics as the list then gives the result. And I actually have a copy of "The Surreal Numbers". Still working on how it/they can be presented to a non-technical audience.

stephencanon 13y ago
Both "Surreal Numbers" and "On Numbers and Games" do a not-terrible job of presenting surreal numbers to a (relatively) non-technical audience.
- ColinWright 13y ago
  Not convinced. I've tried to follow their presentations when talking about these topics with different audiences and found that people really just don't follow. There's a difference between teaching material and exposing people to it - both of those works try to teach the topic to people, which isn't really appropriate in a single lecture format, or a math club, or a masterclass.
  - stephencanon 13y ago
    Well, they are full-length books (Ok, SN is shortish). There aren't many books that can be presented as a single lecture as is. The surreal numbers (as you know, I'm sure) are fairly subtle, and people don't have much intuition for how they should behave. Even non-technical people have lots of intuition for real numbers (even if some of it is wrong); they know that they can add them, multiply them, compare them, etc, and they even know algorithms to do so. When confronted with the surreal numbers, it's not at all obvious that you can perform those operations, much less how to do so, so there's an enormous weight of basics to build up before you can really do anything interesting (much like how the first week or two of an introductory linear algebra course can be pretty dull). Games are perhaps easier to motivate and make interesting (in my experience), as you can get to the point of analyzing and reducing simple positions and establishing the addition operation pretty quickly, without getting bogged down in "wait, why would I use this horribly complex approach to what seems at first to be not terribly unlike the real numbers except maybe I can't even do all the operations I know and love". Unfortunately, multiplication of games doesn't really make terribly much sense, so you need to restrict to numbers reasonably quickly to continue developing "conventional" mathematical structures. TL;DR: for a single lecture format, motivating and defining games and showing that there's a reasonable embedding the natural (or dyadic rational) numbers isn't a bad introduction in my experience.

ColinWright 13y ago

By the way, you might be interested in this: http://www.penzba.co.uk/images/LO_n_CW.png http://www.penzba.co.uk/images/LO_n_CW.png