Could someone explain why Priest, in discussing the 'four corners', presents the Hasse diagram with {F} at the bottom? Why would it not be {}? I'm not too familiar with posets, but it seems the order matters, right?
The logic of Buddhist philosophy
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Re: The logic of Buddhist philosophy
#32If you only want to go partway to catuskoti, consider the intuitionistic logic of Heyting and Brouwer. This is the actual form of logic which has the Curry-Howard isomorphism with lambda calculus. In particular, it rejects PEM but not PNC.
What does this world feel like? Well, let's assume we've formulated a proposition. In type theory syntax we'd write the body of the proposition `B` and give it a name `p` like so
p : B
It's important to note that while we've immediately written this proposition but we've not established whether it is true or false. PEM would imply that the state we are in currently is unstable—p must be either true or false already!
Intuititionistic Logic, however, admits that the state of almost all propositions is far closer to what p is now—neither proven nor refuted. This neither-true-nor-false state is natural and only through a process of communication and work can we move forward. For instance, we could construct a proof of p
p : B
p = proof1 -- note how this looks like a typed program now
notp : not B
notp = proof2
What's immediately nice about this system is that it can model internally the notion of an independent statement—all statements are assumed independent until proven otherwise even!
It also reflects the nature of (functional) programming (of a certain style) where we define the type we're hoping to achieve and then work to create a program satisfying that type.
Note finally that Intutionistic Logic still holds that PNC is true. In particular, we can trivially prove the following proposition
notPNC : forall prop . not (prop AND not prop)
notPNC = /\_ -> \(p, np) -> np p -- this is a legitimate proof term written in (System F) lambda calculusRe: The logic of Buddhist philosophy
#33This is nice. I'm wondering how much our restricted thought environment (x can only be true or false) has hindered our development in the last thousands of years?
A multi-valued logic (at least of the sort described in the article) is not stronger than a single-valued one. They can be transformed into one another.
For instance lets say give the proposition "the sky is blue", the truth value (0.78, right here, right now), corresponding to being quite blue, but not necessarily always or everywhere.
truth value(the sky is blue) = (0.78, right here, right now)
Since a function is just a special case of a relation, we could say
the sky is blue ~ (0.78, right here, right now)
And a relation can be described as function from the composite space to {false, true}, where false denotes not related, and true denotes related. So we could transform the whole thing into
truth value((the sky is blue, (0.78, right here, right now)) ) = true
Now another question is which model is most convenient.
Re: The logic of Buddhist philosophy
#34This is nice. I'm wondering how much our restricted thought environment (x can only be true or false) has hindered our development in the last thousands of years?
It's been thrown away in some (growing) part of modern mathematics and has lead to the creation of automated provers as logic rejecting PEC and not PNC models lambda calculus. See my top level response in this thread for more detail.
Re: The logic of Buddhist philosophy
#35Could someone explain why Priest, in discussing the 'four corners', presents the Hasse diagram with {F} at the bottom? Why would it not be {}? I'm not too familiar with posets, but it seems the order matters, right?
Intuitively, it is to indicate degree of 'truthiness'. Certainly false {F} is less truthy than either both true and false {T,F} and neither true nor false {}, which are in turn less truthy than certainly true {T}. The Hasse diagram indicates that there is no relationship between {} and {T,F} in terms of level of truthiness.
More mathematically, you can think of the Hasse diagram as a lattice, with the meet and join operations being AND and OR. For example, we are used to thinking of the expression
p AND q
{F} AND {T} = {F}
{F} OR {T} = {T}
{F} AND {} = {F}
{F} AND {T,F} = {F}
{F} OR {} = { }
{F} OR {T,F} = {T,F}
{T} OR {T,F} = {T}
{T} OR {} = {T}
{ } OR {T,F} = {T} // I think..!
{ } AND {T,F} = {F}
{ } OR { } = { }
{ } AND { } = { }
We're used to seeing Hasse diagrams for subsets of a set S used to indicate inclusion, in which case you would have {} < {F}, but you can have a valid poset structure that's not based on inclusion.
Re: The logic of Buddhist philosophy
#36Paradox is not really so alien to Western philosophy, as this article initially suggests. It does end up referencing a number of modern Western philosophers, but the idea is older than that. It's ironic he refers to "Western orthodoxy" as being strictly anti-paradox, as traditional Christian theology is full of official paradoxes; a major example would be the doctrine that Jesus is simultaneously fully human and full…
> Another way to think of paradox is simply rejecting false dichotomies in favor of a more complex and uncertain reality.
That, as it so happens, is an excellent shorthand for Buddhism as well. Which I believe is a fruitful avenue for discussion, as the author does here.
Re: The logic of Buddhist philosophy
#37Central to his teachings is the view that things are ‘empty’ (sunya). This does not mean that they are non-existent; only that they are what they are because of how they relate to other things.
Immediately I thought of Category Theory as this statement is a not terrible expression of what CT tries to teach. I'll immediately recommend the paper Numbers Can Be Just What They Have To (http://www.cwru.edu/artsci/phil/NumbersCanBeJustWhattheyHave...) for more exploration.
Briefly, while CT is usually "bootstrapped" off having a notion of objects and their relations, it becomes quickly obvious that focusing on the objects themselves is useless—they are given no emphasis in CT and thus wither away to being nothing at all. Instead, absolutely every interesting property of the objects must be expressed in the relations they take with other objects.
This is formalized in the Yoneda Lemma which, in a not terrifically generalized form, can be written
forall a . (forall r . (a -> r) -> r) a
So what's the point? Well, as the paper linked above suggests, sometimes the "intrinsic nature" of things makes them very difficult. We'd often like to equate two things which "ought to be the same" but aren't because their intrinsic nature differs. If you work in standard mathematics (set theory) you run into this problem because everything is described as having an intrinsic nature derived from sets. If you work in CT-based math you have no such issue because objects fail to have intrinsic natures altogether. They only have their "extrinsic" natures, their relations to other things.
Re: The logic of Buddhist philosophy
#38I would have enjoyed this article if it dropped the Buddhism and kept the math. Alternate logics are interesting, but you tend to be able to reduce them into each other in the same way that universal turing machines can simulate each other. They don't add new functionality, they add succinctness. So it's really strange to me to frame them as totally different philosophies . For example, it is the case that self-refer…
> I would have enjoyed this article if it dropped the Buddhism and kept the math. May I ask you why? Personally I enjoy reading about the origin of some particular idea/philosophy, even when it's purely anecdotal.
It's interesting, but the problem I have with the presentation is that the modern manifestation of this idea has nothing to do with Buddhism. While we can maybe say that Buddhism came up with some similar-looking stuff, there's no causal link, and the philosophical systems are not working within the same paradigm. Like seeing a human face on the surface of Mars, this is an example of coincidence more than any deep connection.
Well, OK, that's unfair, because Buddhism and formal logic are both human endeavours, and have to deal with the same facts. But this disconnect is even true with parts of Greek philosophy. "Atoms" as we understand them are very different from "atoms" as Aristotle thought of them, even though there is a historical connection. Today's chemists would do badly if they relied on Aristotle for anything other than historical curiosity.
Re: The logic of Buddhist philosophy
#39The logic of Buddhist philosophy: emotional attachment to and/or desire of impermanent things has the potential to result in dissatisfaction, as impermanent things are impermanent. It's hardly rocket science. Desire nothing, and you'll never never feel the dissatisfaction of not getting what you want. Be attached to nothing, and you'll never feel the pain of losing something.
Some quotes by emperor Marcus Aurelius, of the Stoics, expressing a similar idea:
"You have power over your mind - not over outside events. Realize this, and you will find strength."
"If you are distressed by anything external, the pain is not due to the thing itself, but to your estimate of it; and this you have the power to revoke at any moment."
Re: The logic of Buddhist philosophy
#40Very interesting.
It reminds me that I think an early start on programming made some Buddhist concepts easier for me to get. In particular for me there's a strong connection between using software to model the world and the Buddhist doctrine of emptiness, the notion that nothing exists in its own right: https://suite.io/matthew-bingley/1z8s2kx
I started coding young, and over time I gradually came to see that any computational representation of the world was always false. The act of writing software was always an editing of the real world, a discarding of everything that wasn't apparently material to my particular purpose. Changing software was sometimes a recognition of ignorance or fallibility, but often showed me how changing purpose changed what the maximally useful model was.
Years of that experience, combined with things like Douglas Hofstadter's work, made the apparent contradictions of Zen much easier for me to follow.
I think that also primed me to be ready for the Agile movement. Early on, I had strong BDUF tendencies. But once I gave up believing that there was one right model, one right design, I lost my taste for BDUF. So when people claimed we could keep our software as flexible as our understanding, that was very exciting for me.