Reading this book about the philiosophy of math, and he’s talking about the number 3 like it’s not obvious what it is. What do you mean by “3”? The sets of all things of which there are “3” elements? That’s kind of redundant. So he’s trying to build an understanding of “3” from first principles, and I don’t understand what he’s talking about specifically. But I get the gist.
Discussion
Is it Bertrand Russell? He wrote a great short book on the philosophy of mathematics that does the same thing.
The book cites Russell, but it's "Lectures on the Philosophy of Mathematics”, recommended to me by nostr:npub162zpxufpw8pnuytaf0gfxzkqtvk9rvcwkvppa7x57y3n7qkfpg4shatdhy. Dense going so far, but he says it picks up.
I love how weird seemingly obvious ideas get as we try to drill down on them.
I promise it picks up somewhat, and I'm sure you'll enjoy the later sections on proofs, comparability, infinity, and set theory (cardinals; large ones touched on too).
I warned the beginning was a bit slow, especially if you've read such things before, but it does a great job of describing the various schools of thought at the time.
The arc of math focused on is interesting historically, as it's arguably these foundational questions which pushed more formalism and rigor in late 19th century-present (my opinion based on others' opinions, I'm no expert). Calculus/diff equations were being used all over the place, but the exact understanding of limits and infinitely were shaky until all this stuff came around.
nostr:npub1t49ker2fyy2xc5y7qrsfxrp6g8evsxluqmaq09xt7uuhhzsurm3srw4jj5 a fun "comic" on the characters involved, told through Russel as the main character called "logicomix" may interest you for a light read.
Sorry nostr:npub1dtf79g6grzc48jqlfrzc7389rx08kn7gm03hsy9qqrww8jgtwaqq64hgu0 if the book doesn't live up to my hype for it 😬
So far so good! Hard reading at points but I will get a lot out of it, I’m sure. I might (if so inclined) ask AI for clarity on some points. Have found you can get it to dumb down almost anything and keep asking until it makes sense, which is what I did for the Busy Beaver function.
If you two galaxy brains are reading this, I'm going to have to pick up a copy. Thanks nostr:nprofile1qy2hwumn8ghj7ct8vaezumn0wd68ytnvv9hxgqgdwaehxw309ahx7uewd3hkcqpq62zpxufpw8pnuytaf0gfxzkqtvk9rvcwkvppa7x57y3n7qkfpg4sx8w3k8
nostr:nprofile1qy8hwumn8ghj7efwdehhxtnvdakqz8rhwden5te0dehhxarj95crztnev94kj6r0dehx2tnrdaksqgr26032xjqck9fus86gck85fegenea5ljxmuduppgqqmn3ujzmhgq430dvk nostr:nprofile1qy2hwumn8ghj7ct8vaezumn0wd68ytnvv9hxgqgdwaehxw309ahx7uewd3hkcqpq62zpxufpw8pnuytaf0gfxzkqtvk9rvcwkvppa7x57y3n7qkfpg4sx8w3k8 Holy shit this book is expensive. I might need to find it at the library. Where did you two source this bejeweled tome?
35 bucks digital I think
Yeah, I just looked back at the chapter after the exchange and realize it's a bit more mathy than I realized (some notation and formulas and terminology), but he keeps it pretty light especially later on. Push past any detail that slows you down too much and it'll click or prove unimportant :)
I'll admit I sometimes also lose the nuance of some long-winded philosophy comparisons he makes in attempting to cover all the POVs at various times.
gO;.;Od advice YODL/*&GM
Morning carnal
bien vida/*amigO*****YOnostr:npub162zpxufpw8pnuytaf0gfxzkqtvk9rvcwkvppa7x57y3n7qkfpg4shatdhy
redundant buttbut *B*rO ima feeling generous lOlz/*\*****/*\ a tu
my note to you is not showing up/? nostr:npub1v7k63c6y2vktlqhsuupywt3yc7ykursujc34at964f9cv9s9y9csjutfk0
Weird. I dont think I see it.
noted below/idklooking into Logicomix now
Oh, and one more thing, he's leading up to actually constructing set theoretic canonical structures of the number systems (up to isomorphism), which if you haven't seen before is kinda deep imo. Sure you might think you know what 3 is, but what's the set of real numbers exactly (dedekind and Cauchy gave equivalent definitions, the former being the more popular).
reading that exact part now. I feel like he's just saying real numbers are points on a line or lines in a plane, and all continuous (gapless) structures are essentially the set of real numbers. But I am fine to just read, partially understand, and wait for the light bulb to go on.
Maybe the construction is a bit underwhelming at first lol. There's something solid in making an actual concrete construction of them from earlier building blocks, and then showing the model is "categorical" (unique) model of a set of rules you'd like to think _define_ the real numbers. You don't often get this result when trying to nail things down precisely, as you see in later sections on set theory for example; attempts to write the rules of a system you think you understand fully often allow unexpected/non-standard models, or leave important questions unanswerable, provably. 🤷♂️
So glad you gave the book a shot! Now to get the author on nostr (he's quite active on forums for a professor)...
I get it's big to be able to connect these abstractions that are hard to pin down, but am still hazy on exactly *how* it’s pinned down, even if I can grasp the concept which seems intuitive, i.e., no gaps. I mean it’s hard even to define 3, let alone irrational numbers which are more abstract, and then to connect it as “isomorphic” with these geometrical concepts, but it’s still like I’m reading in Portuguese and translating it in my head to English.
Well, I believe it took mankind centuries to come to terms with things like irrational numbers (think I read some guy almost got killed by Greek clique for putting forth idea of diagonal of a square), and infinities, and even zero ha!
You won't get a better meaning of 3, I guess it's not possible beyond the schools of thought he lists out, but the formal def as {0, {0,1}, {0,1,2}} is pretty pure and we can at least all understand and communicate about such things much more precisely
That fuck ruined everyone’s tidy understanding with the square root of two.
{0, {0}, {0, {0,1}}}*
Typo was bugging me. Believe this is right def
I guess I’m missing the breakthrough in using that notation rather than 1, 2, 3. I get if you numerals, it’s circular, but starting with empty set and then the set itself, etc. just feels like counting. But counting with *what*?
I can't offer much more insight (if I have at all this far). The above def is how ordinal numbers are built, and I think THERE it will prove more valuable (probably worth a quick Wikipedia read when you see the term used, as I don't think he went into them as much as you'll want).
It's an attempt to put the standard math systems we know and love into a unified set theory model, and then you can worry about what exactly set theory is I guess.
Wish I had better answers
Ha, no, I’m cool with figuring it out myself. Just trying to wrap my head around it.
FYI, can always do this:
Not bad, though a bit verbose. I think the question may be a bit off for getting the best response. One of the more important properties, and what distinguishes them from the rationals the most, is the properties of them as a linear order (every bounded subset contains a least upper bound inside the real set itself). And I think something like containing a dense countable subset (the rationals) yields their unique/categorical property. This is mentioned in the book, as you know.
So you have these desirable properties your intuition wants them to have, and are able to prove that the description given uniquely captures it. So if two people were to discuss them, you can be sure that their conceptions are identical. There's a little nuance im hesitant to even mention, but I will, and that's that these canonical model proofs rely on second order logic. Which is fine, but when you get to completeness theorem it might confuse a bit since there you are restricting to first order logic.
Ok, thanks. wil dig a little deeper in that case.
Enjoy discussing, helps me learn too.
Another way to phrase it, is you have a short set of rules: linear order, upper bounds, dense countable subset..., that you hone in on as you begin to formalize what you imagine reals to be, and then there are two interesting questions. Are these rules consistent (is there at least one mathematical model satisfying them, or do they somehow lead to contradictions), and are they narrow they don't allow for other interpretations. Ie, have you nailed down a set of rules such that you can declare they unambiguously define THE real numbers.
For instance, if you drop the countable dense subset requirement, you get nonstandard models that also satisfy the rest of the rules.
I'm blathering...
will stop posting these after this, but it’s amazing how you can drill down these things with AI when you don’t fully understand something.
Not bad. I've played with AI for math stuff before and it does ok. Been a few months since I have, so maybe it's gotten better, but early experience was that it quickly gets lost if you go too deep; mine started spewing incoherent sounds a few times lol
I had this problem too with very large numbers. Kept making errors, and once it “corrected”, I couldn’t be sure it was really right since it might still be making errors that just weren’t obvious to me. But for concepts with known explanations, think it’s pretty useful.
Yeah, same concerns lol. Been using it a little recent for coding (nothing fancy, just like helping me write a "fuzzy join" on some text fields), and it's damn useful there. Coding has never been so easy