like most commentary on Gödelian incompleteness, you got the right “vibe” but you’re not exactly correct.
for example, geometry is not an incomplete system, euclidean geometry is famously complete and thus not subject to incompleteness properties unless viewed strictly from the perspective of reality. in that case, all systems and models we build are necessarily presumed incomplete because of what people like Kant and Descartes said - all you know is known filtered through your perception, it isn’t what is actually real.
that is where people get the “it’s impossible to completely describe anything” argument from. it isn’t true flat out like that, though. you need some qualifiers, like, it’s impossible to completely describe reality solely through the human experience. this makes sense if you think about it. how much data lives just in a grain of sand? how many atoms are there in it? molecules? where might it fall in 2.76463 seconds? what does cathy juniper, born 1973, think this grain of sand would be named if it worked at the dollar store?
just because information that is incomprehensible or unreal to us lives in the world doesn’t mean it isn’t there. one iota of the universal computer has more computational power than all humanity combined over all history, many times over. our minds are so much smaller compared to the amount of information that actually lives in the world that we necessarily can only ever shine our spotlight of focus on tiny pieces of it at a given time, and not for very long. that is what the incompleteness theorem is describing…
this doesn’t mean that “it’s impossible to create any consistent set of math statements that completely describes everything,” if you’re willing to be a bit clever about it… after all, as humans, our big schtick is recording information in the world for later use, in reality… and there is proper suspicious to believe that reality itself could be a complete system, understandable from both the outside and inside if only viewed at the right angle…
hilbert’s dream is not dead yet, the early neoplatonist were just doing the dirty work of finding bounds.
Nah, euclidean geometry was not complete. Tarski didn’t come up with a complete version until the 20th century. I’m not sure how famous Tarski geometry is, but it doesn’t seem very famous in USA outside of math depts.
this doesn’t mean that “it’s impossible to create any consistent set of math statements that completely describes everything,”
It says far less than that: “It’s impossible for a mathematical system containing the natural numbers to be both complete and consistent.”
In itself it has very little to do with physical reality. I think it’s more about how we think about math and then its applications.
reality itself could be a complete system, understandable from both the outside and inside if only viewed at the right angle…
like most commentary on Gödelian incompleteness, you got the right “vibe” but you’re not exactly correct.
for example, geometry is not an incomplete system, euclidean geometry is famously complete and thus not subject to incompleteness properties unless viewed strictly from the perspective of reality. in that case, all systems and models we build are necessarily presumed incomplete because of what people like Kant and Descartes said - all you know is known filtered through your perception, it isn’t what is actually real.
that is where people get the “it’s impossible to completely describe anything” argument from. it isn’t true flat out like that, though. you need some qualifiers, like, it’s impossible to completely describe reality solely through the human experience. this makes sense if you think about it. how much data lives just in a grain of sand? how many atoms are there in it? molecules? where might it fall in 2.76463 seconds? what does cathy juniper, born 1973, think this grain of sand would be named if it worked at the dollar store?
just because information that is incomprehensible or unreal to us lives in the world doesn’t mean it isn’t there. one iota of the universal computer has more computational power than all humanity combined over all history, many times over. our minds are so much smaller compared to the amount of information that actually lives in the world that we necessarily can only ever shine our spotlight of focus on tiny pieces of it at a given time, and not for very long. that is what the incompleteness theorem is describing…
this doesn’t mean that “it’s impossible to create any consistent set of math statements that completely describes everything,” if you’re willing to be a bit clever about it… after all, as humans, our big schtick is recording information in the world for later use, in reality… and there is proper suspicious to believe that reality itself could be a complete system, understandable from both the outside and inside if only viewed at the right angle…
hilbert’s dream is not dead yet, the early neoplatonist were just doing the dirty work of finding bounds.
Nah, euclidean geometry was not complete. Tarski didn’t come up with a complete version until the 20th century. I’m not sure how famous Tarski geometry is, but it doesn’t seem very famous in USA outside of math depts.
It says far less than that: “It’s impossible for a mathematical system containing the natural numbers to be both complete and consistent.”
In itself it has very little to do with physical reality. I think it’s more about how we think about math and then its applications.
This has been largely debunked.
I dunno what his dream was, but Hilbert’s program is very much dead.