This preprint was written by a researcher at an accredited university with a PhD in physics. I'm sure they know what a vector valued function is.
The point of this paper is not to revolutionize how a scientific calculator functions overnight, its to establish a single binary operation that can reproduce the rest of the typical continuous elementary operations via repeated application, analogous to how a NAND or NOR gate creates all of the discrete logic gates. Hence, "continuous mathematics" as opposed to discrete mathematics. It seems to me you're being overly negative without solid reasoning.
its to establish a single binary operation that can reproduce the rest of the typical continuous elementary operations via repeated application,
But he didn't show this though. I skimmed the paper many times. He creates multiple branches of these trees in the last page, so it's not truly a single nested operation.
Well, it is still the case, even if not explicitly shown. Personally I think it almost boils down to school math, with some details around complex logarithms; the rest seems to be simpler.
Some of us had the wondrous epiphany as children that we could build any digital device we could dream of (yes, up to and including a full computer, CPU, RAM, peripherals, and all) out of SN7400 NAND gates that we could buy down at the local Radio Shack, if only we could scrape together enough change to buy the requisite number of parts and sockets and Tefzel wire.
Obviously, I can't speak for all of us who had that epiphany, but I strongly suspect that most of us who had that epiphany would find this result joyous.
In numerical analysis, elementary function membership, like special function membership, is ambiguous. In many circumstances, it’s entirely reasonable to describe the natural logarithm as a special function.
This is an incredibly ironic comment. "Freestyle" chess was an attempt to do exactly this with Magnus's support, and it failed to secure funding after its initial run. This event is them running back to FIDE in shambles to salvage their tour. Kasparov attempted something similar in the 90s, making his own world championship title, and similarly failed horribly.
The stability of a 100+ year old international organization that's led by serious politicians with connections in every major country is hard to contend with. FIDE's current president was Russia's Deputy Prime Minister for 6 years.
But notice they changed the game AND started a league. I'm suggesting they ignore fide, start a new league based on vanilla chess, and try to get sponsors and modernized it. Relative to how many people love following the drama, it should be cheap to start a league compared to racecars or even bike racing. There's only travel, lodging and broadcasting as expenses.
> Also his current rating is higher than either Karpov’s or Kasparov’s were when they first won the title. His rating when he first won was about the same as Fischer’s when Fischer first won.
This doesn't really mean anything. Rating is a purely relative system, as in the other thing that matters when performing Elo calculations is the difference in Elo between the two players. The absolute value of an Elo rating carries no real meaning and drifts over time based on the volume, skill level, and initial rating of lower level players. Since these change frequently, it's pretty much useless to compare ratings separated in time by more than a decade or so, maybe less. 50+ years is certainly far too long.
My views on this, which are mature and have been held for many years now, are mostly informed by the results obtained by Kenneth Regan and Guy Haworth in their paper “Intrinsic Chess Ratings” which, unless you have intelligence to the contrary, is the only rigorous treatment of this issue that has yet been performed and is yet the only argument that has any persuasive hold over me.
You say that ratings drift over time to such an extent that to use them in comparisons across long time spans is meaningless yet their analysis determined that chess ratings as a measure of intrinsic quality of move choice (which must be highly correlated with playing strength) is stable over several decades with only some indications that a small amount of deflation has occurred.
Your argument in comparison amounts to informal speculation. If I were to share my own I would say that those potentially error-inducting considerations, are statistically insignificant compared to the sheer number of games, that is to say corrective and informative exchanges of points, that occur. Further, I would add that the absolute values of ratings were defined by the playing strengths of the original players and that this definition has been well preserved even as the player pool has evolved.
I have heard many such arguments in my time yet not a single proponent cares to demonstrate them. What I find amusing is that those same proponents will often readily accept a comparison across time of a single player (often themselves) across similar time spans without controversy, as evidence of their progress as a player, for instance using Carlsen’s rating today and comparing it with one from early in his career, say from 2003 or 2004, which at this point was more than 20 years ago.
Obviously a board game will be easier for a child to compete at than a physical sport. Tons of Rubik's cube world records are held by 9 year olds. I don't see why any of this is relevant in answering the question "is it impressive to be winning at 35 in chess?"
Is your point that young kids have an advantage in chess, making it harder to keep up as an adult? They clearly don't. No 12 year old has ever been able to seriously compete with top players, at best they can hold a few draws or win a blitz game here and there. As far as I'm aware Judit Polgar was the only 12 year old to even break into the top 100, and she's an outlier among outliers. Right now the top 3 players in the world are all in their 30s, and there's only one player in the top 50 who's younger than 18.
To be clear, "came last in Speed Chess Championship" actually means he came in 4th out of 16. He still made it to the semifinals. Even then he barely lost to Alireza, who is pretty universally considered a top 3 speed chess player. The loss to Lazavik was a lot worse, but it was still a close match against a strong player. He hasn't won a Titled Tuesday this year but he hasn't scored worse than 8/11 and he's still made the top 10. That's not as much of a slump as you imply IMO.
Sure he's still one of the top players, but he's not as strong this year and OP is suggesting he still has an edge against the GOAT, who this year:
- Has won Freestyle WC
- Has won SCC
- Has won 2x Titled Tuesday's
- Has won a Freestyle Friday
Hikaru can snipe a win off Magnus here and there, but I don't think there's any time control or format where he could win a long series of chess matches against Magnus.
He could win bullet. No increment means his years of streaming bullet will let an edge when moving in the endgame, so he just needs to draw out the game long enough to get Carlsen either to 0 or in trouble. Somehow we got a chess format where mechanics matter :)
His record in bullet in the Speed Chess Championship against Carlsen is rather unremarkable, although that is 1+1. Perhaps he would fair better at 1+0.
It isn't a slump at all, really. He had his first kid in December. He's preparing for the Candidates in March. Weekly chess.com tournaments are just, you know, going to be relegated to streaming content for a bit.
He's up there for sure, but not clearly the best. According to him both he and Magnus think Alireza Firouzja is the best in longer matches of multiple bullet games.[0] I suspect he would give the edge to Magnus in a shorter match, but I haven't found evidence for this.
No, even the best prodigies typically aren't winning super tournaments until 17 or 18, and we haven't really had one of those since Gukesh won candidates last cycle. The youngest player in this event was a 20 year old who placed last. (Though to be fair to the youngsters, 3rd and 4th place are both 21 years old.)
Generally speaking it's expected that chess players will peak around their late 20s and slowly decline from there, with sharp declines around age 50. It's unusual but not unheard of for players in their 40s to win major tournaments. 42 year old Levon Aronian won several last year, but it was considered a notable example of longevity every time he won.
In terms of raw numbers, there are currently 30 players in their 30s, 15 players in their 40s, 4 players in their 50s, and no players older that in the top 100. The youngest is 14-year old Yagiz Kaan Erdogmus, who is considered the greatest chess prospect of all time.
For some concrete numbers, there are only four players over 50 years of age in the top 100 at the moment by live ratings[0]. They are ranked #13 (age 56), #89 (age 53), #95 (age 54), and #97 (age 57). In their primes these players were ranked #1, #10, #4, and #3 respectively.
Isn't he playing Chess960 because he started finding standard chess boring? And wasn't that why Fischer worked on it in the first place? Experts might get bored of it by the time they're 50.
The reason the top pros like chess960 is because they don’t need to spend hundreds of hours of opening preparation, they can just sit down and play.
Caruana (the guy who lost to Magnus), mused in a podcast that chess960 feels strange as a competitor because he doesn’t really prepare (because there are far too many openings to study) and said it feels like he’s getting paid for much less work.
There are 960 possible starting positions and the chosen one is known at the start of the tournament where players are given 15m to prepare. I have observed that GMs aren't surprised when they see the board. They usually go "ah it's this one with the opposite bishops" or something similar.
When a chess player means "no prep" it probably still means more prep than any normal person would consider reasonable, because what would require you to sit down and take notes, move pieces and memorize, they can just do in their head getting coffee by now. So yeah they recognize almost all the patterns, it's just harder justify spending 1 month on an opening you won't even be able to use, but they still know how to play certain patterns.
Oh, totally, I just wanted to highlight what beasts these players are and how wonderous it is to see them recognize so many starting positions that they already started showing familiarity despite how new the tournament format is.
Candidates prep and also the entire Freestyle chess experiment has been a bit of a mess. Here's what he told chess.com[0]:
A few months ago I was invited to the first leg of the 2026 Freestyle Tour with the same format and prize fund. I let everyone know that I'd be playing there.
Just a few days ago I received news that there will be no year-long tour for Freestyle. The format for the only event to be held will be only three days and only rapid formats. Instead of the tour that was planned, Freestyle has joined forces with FIDE and are now calling it a World Championship. I think it might hold the record for most rushed arrangement for a World Championship title in history.
I truly enjoyed the first event in Weissenhaus in 2025, and it's a shame that the classical length format wasn't continued. Furthermore, this all feels like a hastily arranged tournament with less than 1/3rd the prize fund it originally had, and now it's attached to FIDE, which isn't a positive development in my opinion.
Despite many phone calls and messages from the organizer, I have decided to decline my slot in this event. I have an important tournament in the end of March/April to focus on, and that is where my attention will be.
The point of this paper is not to revolutionize how a scientific calculator functions overnight, its to establish a single binary operation that can reproduce the rest of the typical continuous elementary operations via repeated application, analogous to how a NAND or NOR gate creates all of the discrete logic gates. Hence, "continuous mathematics" as opposed to discrete mathematics. It seems to me you're being overly negative without solid reasoning.
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