by XOR
's hammer:
These pictures lead in to a nice explanation of étale spaces, sheaves, and stalks, followed by topological Kripke frames.
Posts tagged with modal logic
by XOR
's hammer:
These pictures lead in to a nice explanation of étale spaces, sheaves, and stalks, followed by topological Kripke frames.
Several years ago I sat (after yoga class) with some Zaa Zen practitioners. As I understood the practice from doing it once, Zaa Zen basically consists of sitting in good posture, staring at a blank wall, and clearing your mind.
It wasn’t my favourite meditation I’ve ever tried. (So far my favourite was something that into the continuum introduced me to: Vipassana meditation. The way I did it was to sit outdoors in nice weather and listen to the sounds and stop thinking about my own anxiety or problems. Something much like the John Cage lecture that until a single soliton survives posted. Being aware of the world around you and “listening” or “taking in” rather than “forcing” or “pushing out”.)
But I definitely remember the conversation I had with one of the practitioners (Tony) afterwards. Tony was maybe 20 or 30 years older than me but I felt we instantly connected on some mental level. He told me he had been a failure at pretty much everything he had tried in life. How he was a black sheep of his family; how he tried to be a biologist; there were a few other things he tried and he hadn’t been very good at any of them. But in some sense it didn’t matter (remember, this is the wisdom of years talking. According to economic research people tend to mellow, their aspirations and hopes drop to a realistic level, and they become intimately familiar with the passing of time—whatever you optimise, whatever you read, however much you drink, whatever you earn, however you train, however many relationships you destroy—that passing of time always clicks, click, click, tick, steady.) and he could always come back to his practice. A different meaning of “return to the breath”.
Anyway, we were talking about various I guess spiritual things. More like a mixture of the mental-ethereal and the sense-grounded. He was telling me how Zaa Zen was so great and I would really like it and I should read this book and so on. You know how people always do that—they’ve read a book and then they say you would love it. Well, no, I think just you liked it and I have my own stack of stuff that’s my to-read list already. So normally I would just keep that kind of thought to myself but since Tony and I had an unusual level of honesty and directness for perfect strangers who just met, I brought up what I see as the circular-logic problem of picking up any book.
This is why, I said, I won’t read the book you’re telling me I will like so well. From my outsider’s perspective I don’t trust enough in the Zaa Zen idea. Not to say that it is some hokey New Age crystals or whatever, but I don’t sense—from standing on the threshold—that this is a house I want to get comfortable in.
(This is also why I started reading so much mathematics. From an outsiders’ perspective it seemed like “This is where the truth is. Following Wolsey’s idea, with a hungry reification of Plato’s philosopher-kings, if I put in only veracity and earnest labour, the result should be something good.)
Tony told me this attitude was actually quite Buddhistic or Zen of me. So I felt very proud that in avoiding looking at the Zaa Zen I had apparently picked up something of it—and it’s a nice geometric shape now that I reflect on it.
So it’s a logical circular logic and a higher modal order than the standard model of choice—and it relates to two other themes I want to talk more about later:
❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤
☠☠☠☠☠☠☠☠☠☠☠☠
Oops, that last sentence wasn’t modal.
◻ ⋄ ∀ ∃ ⊢ (⊬¬)
Brought to you by into the continuum:
To paraphrase Boolos paraphrasing Gödel paraphrasing himself:
It is true that 2 plus 2 is 4. I hope you can at least believe that much. Of course, it can be proved ⊢ that 2 plus 2 is 4. It can also be proved that it can be proved ⊢⊢ that 2 plus 2 is 4.Moreover, it can be proved that it can be proved that it can be proved ⊢⊢⊢ that 2 plus 2 is 4. We could continue in this seemingly trivial manner, but what we are hoping for is that 2 plus 2 is not 5. Fortunately, it can be proved ⊢ that 2 plus 2 is not 5, and that too can be proved ⊢⊢.
Now just to be on the safe side what we really should ask is if it can be proved that it can’t be proved ⊢⊬ that 2 plus 2 is 5…. It can’t! ⊬ ⊢⊬… In fact, no claim of the form “claim X can’t be proved” can be proved.
∀X:(⊢⊬X)→⊥ or
∀X:⊢⊬⊬XSo thanks to Gödel … it can be proved that if it can be proved that it can’t be proved that 2 plus 2 is 5, then it can be proved that 2 plus 2 is 5.
⊢(⊢⊬X→⊢X)
[Which is a contradiction.] I would rather take an incomplete theory over [an unquestionably wrong, self-]inconsistent one any day. Wouldn’t you?… pretty tanpura drones in the pitches of
C#
andA
, respectively (which I suggest listening to with eyes closed and nice headphones).
If on reading that you thought: “the ⊢⊬⊢⊢ symbols…could you do math with those operators?” then here’s some more at the SEP.
The theory of universal algebras was well-developed in the twentieth century. [It] provides a basis for model theory, and [provides] an abstract understanding of familiar principles of induction, recursion, and freeness.
The theory of coalgebras is considerably [less] developed. Coalgebras arise naturally, as Kripke models for modal logic, as automata and objects for object oriented programming languages in computer science, and more.