I just posted something in the positive/neutral thing thread that may be of interest to fellow mathematicians!
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I think the specific field you’d be looking for is number theory, then…? That’s the branch of mathematics that interests me the most, at least.
Would completely dropping deeper layers of axioms make it any more or less difficult to work with complex numbers, beyond “They’re simply a thing; ponder no more deeply”?
Mostly wondering this because of this video (which I have not finished at the time of writing this, so I also might not be making the right connection here).
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If you’re into basic arithmetic… it’s pretty well done now (I wonder if they’ll start up another picture when the last results come in?) but Human Shader is amusing.
I am also reminded of the little page I made back in 2016 to help me finally memorize the pieces necessary to do mental calendar math…
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Seems like I should’ve been clearer in my little rant about the state of mathematics 
I have no problem with set theory by itself. Nor do I have an issue with people seeing it as a generalisation for number theory. What I don’t like is when people say that ZFC is the foundation of mathematics. It makes it seem (whether they intend this or not) that if you peel back the covers of numbers you’ll see sets hiding inside. Number-less sets can be an important part of maths, but they’re not the core of maths. In my humble non-mathematician’s opinion.
Choosing axioms is of course hard. And it’s a fine goal to want to minimise the number of axioms, and that may lead us to switch to different ways of doing maths with different axioms. But it doesn’t invalidate other axiom systems.
I’m somewhat familiar with Goedel’s incompleteness theorems. I guess I just don’t see the value in perfect axiomisation as you put it. Or alternatively, maybe we need to embrace the axiom of paradox, that sufficiently complex system must have at least one paradox. (I don’t know if that’s actually true, but it should be!)
I don’t think it’s taking it on faith. The Peano axioms for example are just a formalisation of the inherent understanding we all have by mid-primary school that the natural numbers are an (infinite) sequence where each number is uniquely generated by adding 1 to another number.
I think that’s really the value of formalisation - that it allows us to set down explicitly the tacit knowledge we share.
So I think part my problem with trying to explain everything through set theory is that some people are trying to do that in linguistics, and it’s just the worst. Well actually type theory instead of set theory, but isn’t type theory basically just set theory with the axiom that sets can’t be members of themselves? In any case, using type theory some linguists under the label of “formal semantics” have tried to formalise language, with the result that they have almost entirely removed semantics from language. But unlike in maths, it seems to have almost no explanatory power, and doesn’t seem to have resulted in a more rigid understanding of language. And they’re not formalising tacit knowledge. So… yeah. I don’t like it. Thank you for listening to my rant.
@Draconis might have other opinions about type theory as I think they’ve got a background in linguistics too.
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Is this like the “record that breaks record players” dialogue in GEB:EGB?
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Ahhh mental calendar math… That reminded me of my high school days! I basically worked out how to calculate the day of the week of any given date by observing that there was a pattern to the number of days in a month, assuming you would start counting from March:
31 - 30 - 31 - 30 - 31 …
So adding that up and taking the average got me to 2.6. with some additional stuff to take care of the fact we start counting in March (so March = 1, etc) and taking care of leap years and centuries I ended up with a formula like this:
DoW = (D + int(M * 2.6 - 0.2) + Y + int(Y / 4) + int(C / 4) - 2 * C) % 7
Where DoW is the day of the week, counting Sunday as 0 and up to Saturday as 6.
Also keep in mind we start counting March=1 etc, so if you calculate for e.g. January it would be month 11 of the previous year.
So for example today (21-07-2023) would end up as
DoW = (21 + int(5 * 2.6 - 0.2) + 23 + int(23 / 4) + int(20 / 4) - 2 * 20) % 7
= (21 + 12 + 23 + 5 + 5 - 40) % 7 = 26 % 7 = 5 (Friday)
I felt very pleased with myself to be able to do this pretty quick, without needing to memorize any tables etc… Until one day I visited the Science Museum in London where they had a computer doing the very same thing… pick a date, and I’ll tell you the day of the week. To my dismay they used the very same formula! I was like what the hell… Ah well, at least I could bask in the knowledge that I worked it out myself
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That, but then sometimes the tacit knowledge turns out to be wrong. Euclid’s axioms formalized everybody’s inherent knowledge of geometry, and that led to realizing all those other geometries. Peano’s axioms didn’t lead to any equally famous realizations about the integers (that I know of), but the attempts to formalize set theory sure did. (“Set of all sets” was an intuitively obvious concept which turned out not to work.)
(Not a mathematician, but I read GEB back in the day too.)
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Yes that’s exactly it!
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Yeah, I’m distinctly not a fan of that sort of formal semantics. I’ve recently been arguing that John Austin was right and sentences don’t really have truth conditions at all.
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There’s a great BBC Horizon documentary about Wiles’ struggle for the proof. You have to be in the UK to watch it on the BBC website, but there are ways around that… Simon Singh, who made the documentary, also published a book about the Proof, delving deeper into the history of the Theorem. Worth watching and reading.
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Yes, I’ve read Singh’s book. As you said, it’s really worth reading. Don’t know the TV docu but because it is from the same author, I recommend it to anyone. And good old BBC is great anyway!
I’ve also read a book about encryption by the same author and can recommend that, too.
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Ran across this regarding ChatGPT getting worse at simple math problems:
Assuming that’s because we don’t understand how the guts work, so when we tinker with something to improve one bit, it has unpredictable consequences for other bits, as it sort of wrote itself instead of being conceived and planned by a human mind.
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Aaaaaah! That’s painful. Scary that people are still writing articles and even doing “research” like that.
They’re just statistical models, they don’t “know” anything (about math or anything else). They just generate plausible extensions of the text that you feed in. So if you ask a question, then in their training data, most of the time what follows is text that looks like an answer, so that’s usually what you get.
If you give it back your question and its answer and then “why did you do that?” then you’ll probably get text that looks like a rationale, because that’s what usually follows that text in the examples that it’s built on. It’s not actually a rationale, it doesn’t have anything whatsoever to do with how it came up with the previous answer, it’s just words that are statistically likely to autocomplete the input you gave it…
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Once upon a time, I was going to be a mathematician. I got my BS in math, but eventually dropped out of graduate school. But I’ve never been able to leave the subject alone.
I’ve spent a lot of time on foundations myself. I’ve been drawn to set theory ever since I first grasped Cantor’s diagonal argument. The concept of infinite cardinaltiies fascinates me. I love to read papers on cardinal characterestics of the continuum, and cardinal functions in general topology. Most of that takes ZFC for granted, but it’s amazing how many inequalities and relationships can be proved regardless of which cardinal arithmetic holds.
I’ve also enjoyed reading about the consequences of systems weaker than full ZFC. FIguring out just which axioms are necesaary for which theorems is attractive. There’s a subject called “reverse mathematics” that does that by proving axioms from theorems.
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I do multiplication problems in my head sometimes just for basic mental exercise. I happened to notice that for any two numbers that add up to 50 (or a multiple of it), their squares will both have the same final two digits. Similarly, any number’s square will have the same last two digits as the number plus any multiple of 50. That may already be Barfnitz’s Second Law of DUH in the math world, but I found it an interesting discovery…
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Have any of you read The Housekeeper and the Professor by Yōko Ogawa? As the Wikipedia page says:
The story centers around a mathematician, “the Professor,” who suffered brain damage in a traffic accident in 1975 and since then can produce only 80 minutes’ worth of memories, and his interactions with a housekeeper (the narrator) and her son “Root” as the Professor shares the beauty of equations with them.
The mathematics stuff in the novel is possibly inspired by Erdős.
The memory loss side is very moving but the mathematics stuff is also in my opinion handled well. And seems to communicate things to non mathematicians too. We read it in a book club I’m in, and all enjoyed it.
I studied mathematics through to university, but only in my first year at uni, then focusing first on astronomy and computer science, and finally just computer science. I have always found applied maths more comfortable than pure maths. At uni we had different classes in each of those and I was very clear in my results about my preferences!
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Hmm, I checked it out…
(50-x)^2=2500-100x+x^2
since the first two terms are multiples of 100, if you modulo by 100, all that is left is x^2 so they share the same terms.
What’s cool is that the middle term is 2 times 50, so if you changed 50 to any other number less than 100, this trick wouldn’t work because it wouldn’t be a multiple of 100.
I’ve never heard to this result before, so I’m glad you showed it! I’m sure it’s known but it’s very cool to learn.
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A mathematician (and IF) friend reminds me of the news earlier this year of two high school students, Ne’Kiya Jackson and Calcea Johnson, presenting a novel (trigonometric) proof of the Pythagorean theorem to the AMS.
This was in the news in March (e.g. Guardian); their proof wasn’t public then, and I haven’t found anything more recently in their own words (in ten minutes of looking), but Youtuber MathTrain reconstructed the likely proof from slides visible in press reports.
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Oh, and vaguely on the topic, the same friend also enjoyed this video: Animation vs. Math - YouTube
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I love those ‘einsteins’! I’ve spent ages trying to fit ‘hats’ together on-screen. I hadn’t heard of the spectre. Thanks for the links!
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