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Comment by yekbun123 | Hacker News Reader

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Martin Gardner’s The Annotated Alice explains the subtleties, mathematical and otherwise, of both Alice books in engaging detail. It explained things I never understood, and showed me things I missed (and then explained them). Besides the mathematics there are references to politicians and events of the era, and jokes that would be known only to people at Oxford.

Oh excellent, my library has a copy available right now. I will go check this out :)

I would suggest you checkout all of Martin Gardner from that library if you can. He is sorely missed, by me at the least.

Lewis Carroll, an Oxford Don with a double first degree, was such an interesting person, right from his study [0] of geometry, algebra [1], to his incessant word play and pioneering forays in the then new field of photography.

Anyone interested in this ought to read Martin Gardner's book Annotated Alice https://en.wikipedia.org/wiki/The_Annotated_Alice

Let me add a few links that might be interesting:

Real Life Adventures of an Oxford Don https://www.csmonitor.com/1995/1120/20122.html

In the Shadow of the Dreamchild https://en.wikipedia.org/wiki/In_the_Shadow_of_the_Dreamchil...

How Lewis Carroll computed determinants https://www.johndcook.com/blog/2023/07/10/lewis-carroll-dete...

Condensation of Determinants, Being a New and Brief Method for Computing their Arithmetical Values https://www.gutenberg.org/files/37354/37354-pdf.pdf (cited by John D Cook in his article)

[0] Lewis Carroll's Mathematical works https://en.wikipedia.org/wiki/Lewis_Carroll#Mathematical_wor...

[1] Dodgson condensation https://en.wikipedia.org/wiki/Dodgson_condensation

One of your favourite “real characters”? Do you just mean… people? Or does this mean something else I’m missing?

I meant people. Let me correct it. He was quite a character too, although with sad undertones. I doubt he ever felt understood.

EDIT: please don't downvote parent comment. He pointed out a grammatical error in my previous comment that I have now edited.

This article has no references to back up its claims, some of which seem like a stretch without further evidence (e.g., "The Cheshire Cat is a property without a carrier" being a critique of group theory). Are there references that back these claims up?

I believe it's because this is random AI garbage. I swear ChatGPT can be used to justify anything. The "jokes" are nonsensical and unfunny, even for early-19th-century Oxford professor inside jokes...

I'm surprised to see, on HN of all places, so many people taking the AI bait so often. Seems like more than half of blog posts and articles posted here are AI generated, but people skip reading the content and just go straight to the comments to discuss the title out of context.

Several others have mentioned Martin Gardner's book "Annotated Alice". It has a section on the Cheshire cat's grin, at https://archive.org/details/agt-annotated-alice-5807b6/page/...

> "Grin like a Cheshire cat" was a common phrase in Carroll's day. Its origin is not known. The two leading theories are: (1) A sign painter in Cheshire (the county, by the way, where Carroll was born) painted grinning lions on the signboards of inns in the area (see Notes and Queries, No. 130, April 24, 1852, page 402); (2) Cheshire cheeses were at one time molded in the shape of a grinning cat (see Notes and Queries, No. 55, Nov. 16, 1850, page 412). "This has a peculiar Carrollian appeal," writes Dr. Phyllis Greenaere in her psychoanalytic study of Carroll, "as it provokes the fantasy that the cheesy cat may eat the rat that would eat the cheese." The Cheshire Cat is not in the original manuscript, Alice's Adventures Under Ground.

It continues with a full page on the topic, none of which are anything to do with math jokes.

Not even the first time I've seen random AI garbage from this domain on HN. I share your sense of despair.

He did actually work with permutations and cycles in voting ( https://en.wikipedia.org/wiki/Dodgson's_method ) but it's combinatorial and not very group'y. Agree with comments that this is AI generated and that the highlighted stuff isn't necessarily interesting, deep, or correct.

As for the cat's smile, analytic philosophy substance/property stuff goes back to Leibniz if not Aristotle. Dodgeson basically predates much of Russel's career, but he could have been an influence on his idea of "bundles" and he definitely influenced Quine. He wrote a few textbooks if you want to dig into his research interests but IIRC it's more along the lines of Boole and DeMorgan, even if fictional fun is arguably anticipating the next wave. I linked the haddock's eyes elsewhere in thread.. good fun but also some rich implications. Since he's preoccupied with self-reference you could argue it anticipates Godel. https://en.wikipedia.org/wiki/Use%E2%80%93mention_distinctio...

A lot of these are very reminiscent of asking ai to find references to a subject. You end up with very tenuous links that take multiple steps to get there and i’m never convinced that the author really did intend that meaning.

Of course this could also be traditional literary analysis. It’s hard to tell.

Exactly, even having a father in the 1800s reading this to their kid and instantly getting that it's a base 4 joke is such a stretch that the article lost me right there. Even after knowing what it is, it's pretty obscure to spot.

Re: "The multiplication that does not work", nothing in the quoted text seems to indicate that each multiplication should be interpreted in a different base, or anything like this. Certainly not that "four times [n]" should always have its result read in base 3n + 3 specifically.

It seems more likely to just be an absurd joke where Alice finds herself with an altered version of multiplication where 4n is interpreted as n + 7, causing multiplication to grow more slowly than normal, causing her to exclaim "I shall never get to twenty at that rate!" (a common exaggerated but non-literal use of "never", similar to "This is taking forever!" meaning "This is taking a long time!", not "This will literally never end").

The idea that we're instead supposed to think Alice thinks "four times 13 (decimal)" is to have its output read in base 42 (decimal), thus as "1A", considered distinct from "20", the latter being what would be "twenty", and thus she will literally never get to "twenty"... This just doesn't seem well-supported by anything in the text.

The leading theory is they are quaternions.

https://www.tandfonline.com/doi/full/10.1080/26375451.2022.2...

https://www.newscientist.com/article/mg20427391-600-alices-a...

These links' mentions of quaternions are about a different part of the book (the tea party). Furthermore, even regarding that different part, your first link explicitly debunks the second link and disavows the quaternion connection the latter alleges. Your first link's whole point is to conclude "it is indeed very unlikely that Dodgson had the quaternions in mind when writing the tea-party chapter."

Why would you need to presuppose some inexplicably shifting number base to get the result of "four times [n]" always equaling n + 7? What does that get you over just more simply observing "For Alice, four times [n] has come to be n + 7"? Shifting number bases are a pointless supposition here. They don't explain anything better than what is already happening without them.

I agree. She’ll never get to 20 because kids only do the times table up to 12.

The book "The Annotated Alice", mention by a couple of people here, says that is one valid interpretation, at https://archive.org/details/agt-annotated-alice-5807b6/page/... :

> The simplest explanation of why Alice will never get to 20 is this: the multiplication table traditionally stops with the twelves, so if you continue this nonsense progression—4 times 5 is 12, 4 times 6 is 13, 4 times 7 is 14, and so on—you end with 4 times 12 (the highest she can go) is 19—just one short of 20.

Gardner then writes "A. L. Taylor, in his book The White Knight, advances an interesting but more complicated theory" which is the changing base theory.

He ends with "For another interpretation of Alice's arithmetic, see "Multiplication in Changing Bases: A Note on Lewis Carroll," by Francine Abeles, in Historia Mathematica, Vol. 3 (1976), pages 183-84."

Available at https://www.academia.edu/download/122551204/82113901.pdf .

Up to 12? Is that a British/Anglosphere/Victorian thing? In Poland they teach up to 10, which is suffinient for arbitrarily large numbers because they also teach long division and how to combine it with times table. Technically up to 9 would be sufficient but 10 is such a nice round number.

Yeah, every US sixth-grader can instantly tell you 12x12=144 but will have to puzzle out 2x13 the long way.

Gross!

It is yes. The anglosphere has historically been somewhat base 12 in currency, time and units of measurement.

Currency is now metric but there’s still a few base 12 things in common usage (feet and inches) in the us at least. Nobody’s gone to metric time yet and base 12 transfers smoothly to base 60 too.

Of course it's because of imperial units. TIL, thanks. But on a sidenote, I question the utility of knowing x11 and x12 when working with time. x15 could be useful, unfortunate they don't teach that (but I think most people with higher education learn it on their own).

Feet and inches long predate imperial units, and the US has never used the imperial system, btw. “Imperial” has a specific meaning and isn’t just “anything not metric”.

Anyway, base 12 is also built into most Germanic languages which have unique names for 11 and 12 (rather than something along the lines of “one-teen” and “two-teen”, which is more common in Romance languages IIRC.

Out of the most spoken romance languages, Spanish and Portuguese have distinct names up to 15, French and Italian up to 16, while Romanian does stop at 10. This suggests hexadecimal influence to me.

It's definitely a UK thing. And 12 is a nicer round number than 10 - ask the Babylonians!

What's so nice about twelve-and-two (12)? Twelve (10) is a much nicer round number.

Though programmers may prefer base two (10) or base twelve-and-four (10).

If you say it in German, it doesn't even sound out of place. Zweiundzwölf, vierundzwölf.

You're absolutely correct, the base is not specified. That's the joke. 1-1=0 would not be a joke. Perhaps it's better not to think of it as a joke. When mathematician reads what seems like nonsense, questions like "hmm is there a base where this would be true?" and "which bases is this true in?" pop up

I'm a mathematician, and my reaction here is somewhat different.

So what's a good math joke?

I like this one: Young mathematician goes to first grade and the teacher asks who knows what is 1+2. She stands up and says "I don't know what is it, but I do know that it's the same as 2+1 as addition is commutative in the monoid of natural numbers"

People said the same thing about a joke Douglas Adams made in his Hitch Hikers series -- that the (corrupted) Ultimate Question to which the answer was 42 ("what do you get if you multiply six by nine?") was a maths joke because 6x9=42 in base 13. Douglas Adams said this was nonsense.

Reminds me of one of my favorite joke programs.

  #include <stdio.h>

  #define SIX 1+5
  #define NINE 8+1
  
  int main() {
      printf("%d\n", SIX * NINE);
  }

Didn't Linus #define THREE as something else in Linux source code.

Was that the 2-3-4 tree? Can't seem to find it now.

It was from some filesystem code. https://news.ycombinator.com/item?id=7296497

  unsigned three = 1;
  unsigned five = 5;
  unsigned seven = 7;

These actually get changed through pointers to consecutive powers of 3, 5 and 7 respectively. `three` is initialized to the 0th power of 3, but because only a single 1 is needed by the algorithm, `five` and `seven` are initialized to the 1st powers instead.

Ah right. Thank you so much. At least I wasn't hallucinating.

And by accident, 42 happens to be the first base after her multiplication gives the answer 19 here (when 20 would be expected), although it would produce an answer of "tenteen", not twenty.

I like the idea of explaining the math in his writing. I very much dislike changing people's writing to "adjust" the reading level. That's no longer their writing. Just use a different example or explain what was actually written rather than dilute famous prose.

graded readers has always been a resource for language learning.

Regarding the Tea Party: whilst what is written is not incorrect, there are some details missing which make it unsatisfactory. The Tea Party was Dodgson's attempt to mock Quaternions, which were getting attention at the time. The 'ridiculous rotations' through space and time were his interpretation of W.R. Hamilton's theories. Note: Hamilton was bald and wore a top hat, you can see one in the statue of him at Broom Bridge in Cabra.

This is strange to me, with quartenions you model rotations in 3d space, not 4 dim rotations.

One of my favorite things to come out of Alice in Wonderland is the Red Queen's Race, which has been used as a metaphor in many many fields for the concept of needing to work as hard as you can just to keep up with others, not even to get ahead.

> "Well, in our country," said Alice, still panting a little, "you'd generally get to somewhere else—if you ran very fast for a long time, as we've been doing."

> "A slow sort of country!" said the Queen. "Now, here, you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run at least twice as fast as that!"

https://en.wikipedia.org/wiki/Red_Queen%27s_race

Also related, recently it's been used to describe involution (neijuan) in the chinese economy https://en.wikipedia.org/wiki/Neijuan

> If you multiply in base 18, the answer to 4 times 5 is 20, which is written "12" (one eighteen plus two). In base 21, the answer to 4 times 6 is 24, written "13" (one twenty-one plus three). The pattern continues. In base 24, 4 times 7 is 28, written "14." Each step up advances the multiplication base by three. The product, written in that new base, always falls one short of twenty.

The product is always 1 and some other digit but not one short of 20

And we only have two multiplications with their result, that’s hardly a pattern

bad python incoming:

    digits = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ"

    def base_conv(num, base):
        if num == 0:
            return "0"
        if base > 36:
            return "too big of base"
        result = ""
        while num > 0:
            result = digits[num%base] + result
            num //= base
        return result

    i = 5;
    b = 18;
    while b < 37:
        print(base_conv(4\*i,b))
        i += 1
        b += 3

results: 12 13 14 15 16 17 18

Beyond this you need to get more creative with your digit symbols.

*edit: bad formatting

> If you multiply in base 18

Why would anyone be expected to multiply in base 18? Why 18? Where does it come from?

It comes from the result

4*5 = 12 is either wrong or base 18

Makes more sense to me that 4*n here means n+7, which fits the pattern and would make her stop at 12+7=19, one short of 20, because Anglosphere countries teach multiplication tables up to 12.

(I definitely didn't get this until I read the rest of the comments here. This is apparently explained in The Annotated Alice.)

To paraphrase Tom Lehrer: Base 18 is just like base 20... if you're missing two toes!

At least that mathematical interpretation does hold up logically all the way to the punch line (even if the interpretation wasn't obviously intended by Carroll). The following section on the Tea Party is just nonsensical slop.

https://en.wikipedia.org/wiki/Haddocks'_Eyes

That Wikipedia article is called Spring Violets, by the way.

I believe they missed the following one:

I can't remember it exactly, but in the tea-time scene, the guy gives Alice a quiz regarding the topology of a chair or something...

She tries to solve it, upon which they burst out laughing at her saying "there is no solution!"

Does anyone else remember it or am I trippin'?

Brilliant book by a genius writer, super impressive.

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