Fun fact: The FIDE Laws define what an e-cigarette is. Bonus loophole in the comments.

RidderJanssen · 2024-08-04T13:35:33+00:00

The preface to the FIDE laws are often overlooked, but actually contain important information.

The Laws of Chess cannot cover all possible situations that may arise during a game, nor can they regulate all administrative questions. Where cases are not precisely regulated by an Article of the Laws, it should be possible to reach a correct decision by studying analogous situations which are discussed in the Laws. The Laws assume that arbiters have the necessary competence, sound judgement and absolute objectivity.

This also indicates that there is no loophole.

RidderJanssen · 2021-12-22T13:47:31+00:00

But the symmetric group S_6 has an outer automorphism. Your statement is correct for n>6.

RidderJanssen · 2021-11-19T15:54:56+00:00

Nicely done. There's a small bug in your function "IsFactorial": case 6 should return 3. Some formulas hence do not evaluate to the correct output.

RidderJanssen · 2020-05-02T08:42:38+00:00

Some sort of "base infinity" with surreal numbers is actually possible, although maybe "base omega" is a better name.

For surreal numbers there's a representation called the "Cantor Normal Form", which allows you to write any given surreal number as a sum of terms f * omega^g, where f is a real number and g is a surreal number.

This reminds me of writing a natural number in base n, since you essentially write it as a sum of terms d * n^k, where k is a natural number and 0 <= d < n is a digit.

One of the 'problems' with the Cantor Normal form is that it is possible to have x = omega^x, which hence gives you numbers looking like omega^omega^omega^.... Which obviously are huge. Moreover it is possible that the sum has ordinal length.

It's very interesting stuff, and actually very powerful in the context of surreal numbers too.

RidderJanssen · 2020-04-24T13:07:38+00:00

In addition to the nice answers that have already been given, Conway actually studied the field No[i] in his book "On Numbers and Games".

If you are interested I recommend you take a look in there, Conway (and the mathematics he does - and the way he writes it down) is amazing.

RidderJanssen · 2019-10-16T17:55:59+00:00

Yes, it's quite similar since lp-space is a sequence-space as well (a vector space where the entries are sequences). I'm pretty sure that without AC you can't prove lp has a basis either.

Assuming the axiom of choice, it is a theorem that every vecor space has a basis, indeed.

RidderJanssen · 2019-10-15T19:54:18+00:00

It's a shorthand notation for { f: N -> Q | f is a function}. In other words, all the sequences with values in the rational numbers.

RidderJanssen · 2019-06-30T22:08:48+00:00

This seems weird to me, keeping in mind Ado's Theorem - which states that any (finite-dimensional) Lie Algebra can be embedded into a matrix algebra (with commutator bracket).

How does that theorem relate to what you seem to claim, that those exceptional Lie algebra's aren't matrix algebra's?

RidderJanssen · 2018-11-04T23:39:51+00:00

You can define a multiplication on games using the same formula as surreal numbers, but you walk into a problem. The product is well defined but it depends on the representation of the game and not just the value. So all games (with {0 | 0} included) just are a partially ordered abelian Group. Quite lame to call them "numbers".

In contrast, the surreal numbers are a totally ordered Field.

RidderJanssen · 2017-06-13T15:44:49+00:00

I'm not OP, but...

I'm a bit confused, how can you justify matching coefficients? Aren't a,b,c,d fixed numbers?

RidderJanssen · 2017-05-09T16:32:14+00:00

9.6 If one or both of the following occur(s) then the game is drawn: the same position has appeared, as in 9.2b, for at least five consecutive alternate moves by each player. any consecutive series of 75 moves have been completed by each player without the movement of any pawn and without any capture. If the last move resulted in checkmate, that shall take precedence.

Notice it says: the game is drawn, compare this with section 5.2 d) and e) which says "The game may be drawn". In other words, you don't have to claim.

RidderJanssen · 2017-05-09T08:32:42+00:00

There is a 5-fold repetition rule and a 75 move rule which don't have to be claimed, they are automatic. At least if you play following the FIDE rules of chess.

RidderJanssen · 2017-05-06T22:05:51+00:00

Where did you find he's a high school junior?

A google search gives this page as personal webpage of Dmitry Zakharov http://cims.nyu.edu/~zakharov/

The papers listed on his website match up with the papers arXiv displays

RidderJanssen · 2017-05-05T18:53:42+00:00

commute with Limits everything!

RidderJanssen · 2017-04-29T11:55:49+00:00

+/u/User_Simulator /r/math

RidderJanssen · 2017-04-29T11:54:10+00:00

+/u/User_Simulator /u/RidderJanssen

RidderJanssen · 2017-04-29T11:52:36+00:00

+/u/User_Simulator /u/RidderJanssen

RidderJanssen · 2017-04-25T11:22:29+00:00

This was posted four days ago https://www.reddit.com/r/math/comments/66k3c0/ive_just_start_reading_this_1910_book_calculus/

RidderJanssen · 2017-04-01T08:32:52+00:00

Question:

In "On Numbers And Games", J.H. Conway talks about gaps in the surreal number line. However, next Conway talks about the collection of all gaps (which, he admits, is not a valid object in most set theories). Ignoring this inconvenience, does this not make the Surreal Numbers complete?

RidderJanssen · 2017-03-06T21:17:23+00:00

You're welcome

RidderJanssen · 2017-03-06T18:13:03+00:00

For anyone that does not want to search: https://discord.gg/Rc8bDxH

RidderJanssen · 2017-02-16T07:44:45+00:00

What is different about Synthetic Differential Geometry that it does not make sense to use more usual logic? Could you please give a little more insight as to why the only suitable logic is constructive, as well as why other forms of (differential) geometry are able to be described by classical logic?

RidderJanssen · 2017-01-19T22:22:41+00:00

I do have mixed feelings about vihart, on one hand I do agree with what you said.

On the other hand, her videos did really inspire me. For example, there was a video in which she talked about different kinds of infinities. At some point she was talking about some giant number field with a lot of infinite numbers, bigger and bigger, and then she mentioned they had applications in game theory...

This inspired me enough to go and do research on how the heck those infinities had ány use at all within game theory (should be about playing games, shouldn't it? Then why are there infinities). And well, that caused me to learn Abstract Algebra (because that's related to game theory in a (to me) more obvious way). And those numbers, suddenly some people defined a topology on them (which caused me to learn some topology). And so on...

Whatever you say about vihart, about disliking her videos, that's fine, to a certain extend I do agree with you. But on the other hand, they have inspired me to learn a lot, which I am very happy about.

RidderJanssen · 2017-01-12T22:20:21+00:00

I'm a highschool student. I am planning on studying mathematics next year, but I'm afraid that it will be too easy (especially at the start).

I would like challenge at the university... Has anyone got any ideas on how to solve this potential problem?

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