Stay tuned for later this week where I look at the writing career of Rod Dreher and compare his evolution as a writer to that of the scientific and superhero career of Hank McCoy (founding X-Men member, Beast) before and after the Scarlet Witch's "No More Mutants", and how that mirrors Rod's personal and professional life post-Obergefell.
One of my more enjoyable experiences in graduate school was building a mathematical theorem solver using the Peano axioms[1] to actually prove things such as "2 + 2 = 4" in a recursive Lisp program. The experience stuck with me and I sometimes wonder what the computing world would have looked like had it taken a declarative[2] road over the imperative[3] one we're on.
I have fun blowing clients' minds from time to time bringing stuff from the declarative realm into some of their projects (a dialog engine being the latest example).
"Essentially, Gödel said that any logical proof involving mathematics cannot be both consistent—provable within itself—and complete, meaning it can prove its own consistency."
Another way to think about it is this way: logical systems both have a language for expressing statements ("2 + 2 = 4") and rules for how those statements may be constructed so that the "truth" of any given statement can be evaluated to be "true" or "false". What Gödel showed was that language for expressing these statements is ALWAYS more "powerful" than the evaluation portions of the logical system, because in ANY logical system, you can say things (using the language) that your system cannot evaluate to determine if what you are saying is true or false.
This first(?) emerged when Georg Cantor was building a mathematics around the rules of set theory and encountered two sizes of infinity: the "smallest" infinity that is the size of of all countable numbers (integers) and a larger infinity, which is the size of all real numbers (continuous, infinitely divisible numbers, including irrational numbers). He was trying to prove if there were any OTHER infinities of a size between these two ("aleph-zero" for the smaller infinity, and "aleph-one" for the larger infinity). There's a fun pop-science book called "The Mystery of the Aleph"[1] that goes over Cantor's discovery and him pretty much driving himself crazy trying to figure out a proof about the relationship between these two sets of numbers.
In the absence of a proof one way of the other, mathematicians added a new "rule" to evaluate this special case (Zermelo–Fraenkel set theory), and Gödel showed that adding this rule enabled the expression of new mathematical statements that came along with the new rule that were similarly undecidable. Doug Hofstadter explores this in his books "Gödel, Escher, Bach" (GEB) and "I Am A Strange Loop", hypothesizing that this may be an emergent property of self-referential systems. (I'm still working my way through GEB, so don't take that as a complete or authoritative interpretation of Hofstadter's work.)
We run into a similar issue in computing (another logical system subject to Gödel's theorem) with something called the Halting Problem, which basically states that it's impossible to build a computer that can inspect at ALL programs that might it may be able to run and determine if it contains an infinite loop (hence the "halt"). You can build more powerful computers with capabilities that allow them to determine if all of prior generations' programs will halt or not, but in the process, the new architecture will allow NEW programs to be written that were not possible on the older architecture, and among these new programs will be NEW programs that the new computer cannot determine whether they halt or not. This is the fundamental reason antivirus and malware-related work will always be a running arms battle into the future.
Anyhoo - apologies for geeking out over this small point in Steve's post. This is a fun corner of math and computer science, and I love sharing how odd and interesting it is.
You make some good points about curiosity. We should try to question and understand other points of view and civilly debate (most likely to never reach a consensus) our differences. But I think you have mashed human nature, arithmetic, math, engineering and science into an unrecognizable lump.
If I were introducing math into a discussion about humans and their differences, it would not be pure math. It would be more like empirical equations that take into account friction and drag and gravity and velocity and factors that could be observed and measured in a lab. Defining and measuring the factors that affect humans' politics is much more difficult and probably impossible - especially since politics can change over time.
Applied differential equations can be used to predict the terminal velocity of a falling object in air but not without using a drag coefficient determined a laboratory. A good example of how to collect empirical data in this case is a MythBusters episode where they used a small wind tunnel to determine the air velocity required to suspend a bullet. You could also use a lab test to demonstrate that a sphere of equal mass would fall more slowly through air than the bullet while the opposite would be true for a viscous fluid such motor oil.
You get an A in geeking out.
Or outgeeking anyone else who comments here. I thought I could get pretty nerdy but Chris has put my best efforts to shame.
Stay tuned for later this week where I look at the writing career of Rod Dreher and compare his evolution as a writer to that of the scientific and superhero career of Hank McCoy (founding X-Men member, Beast) before and after the Scarlet Witch's "No More Mutants", and how that mirrors Rod's personal and professional life post-Obergefell.
(Totally not a joke.)
(If you really want geeky.)
One of my more enjoyable experiences in graduate school was building a mathematical theorem solver using the Peano axioms[1] to actually prove things such as "2 + 2 = 4" in a recursive Lisp program. The experience stuck with me and I sometimes wonder what the computing world would have looked like had it taken a declarative[2] road over the imperative[3] one we're on.
I have fun blowing clients' minds from time to time bringing stuff from the declarative realm into some of their projects (a dialog engine being the latest example).
[1] https://en.wikipedia.org/wiki/Peano_axioms
[2] https://en.wikipedia.org/wiki/Declarative_programming
[3] https://en.wikipedia.org/wiki/Imperative_programming
"Essentially, Gödel said that any logical proof involving mathematics cannot be both consistent—provable within itself—and complete, meaning it can prove its own consistency."
Another way to think about it is this way: logical systems both have a language for expressing statements ("2 + 2 = 4") and rules for how those statements may be constructed so that the "truth" of any given statement can be evaluated to be "true" or "false". What Gödel showed was that language for expressing these statements is ALWAYS more "powerful" than the evaluation portions of the logical system, because in ANY logical system, you can say things (using the language) that your system cannot evaluate to determine if what you are saying is true or false.
This first(?) emerged when Georg Cantor was building a mathematics around the rules of set theory and encountered two sizes of infinity: the "smallest" infinity that is the size of of all countable numbers (integers) and a larger infinity, which is the size of all real numbers (continuous, infinitely divisible numbers, including irrational numbers). He was trying to prove if there were any OTHER infinities of a size between these two ("aleph-zero" for the smaller infinity, and "aleph-one" for the larger infinity). There's a fun pop-science book called "The Mystery of the Aleph"[1] that goes over Cantor's discovery and him pretty much driving himself crazy trying to figure out a proof about the relationship between these two sets of numbers.
In the absence of a proof one way of the other, mathematicians added a new "rule" to evaluate this special case (Zermelo–Fraenkel set theory), and Gödel showed that adding this rule enabled the expression of new mathematical statements that came along with the new rule that were similarly undecidable. Doug Hofstadter explores this in his books "Gödel, Escher, Bach" (GEB) and "I Am A Strange Loop", hypothesizing that this may be an emergent property of self-referential systems. (I'm still working my way through GEB, so don't take that as a complete or authoritative interpretation of Hofstadter's work.)
We run into a similar issue in computing (another logical system subject to Gödel's theorem) with something called the Halting Problem, which basically states that it's impossible to build a computer that can inspect at ALL programs that might it may be able to run and determine if it contains an infinite loop (hence the "halt"). You can build more powerful computers with capabilities that allow them to determine if all of prior generations' programs will halt or not, but in the process, the new architecture will allow NEW programs to be written that were not possible on the older architecture, and among these new programs will be NEW programs that the new computer cannot determine whether they halt or not. This is the fundamental reason antivirus and malware-related work will always be a running arms battle into the future.
Anyhoo - apologies for geeking out over this small point in Steve's post. This is a fun corner of math and computer science, and I love sharing how odd and interesting it is.
[1] https://www.goodreads.com/book/show/5786.The_Mystery_of_the_Aleph
As I am very fond of saying: there is a natural, explainable cause for everything - we just might not have the ability to explain it at this time.
You make some good points about curiosity. We should try to question and understand other points of view and civilly debate (most likely to never reach a consensus) our differences. But I think you have mashed human nature, arithmetic, math, engineering and science into an unrecognizable lump.
If I were introducing math into a discussion about humans and their differences, it would not be pure math. It would be more like empirical equations that take into account friction and drag and gravity and velocity and factors that could be observed and measured in a lab. Defining and measuring the factors that affect humans' politics is much more difficult and probably impossible - especially since politics can change over time.
Applied differential equations can be used to predict the terminal velocity of a falling object in air but not without using a drag coefficient determined a laboratory. A good example of how to collect empirical data in this case is a MythBusters episode where they used a small wind tunnel to determine the air velocity required to suspend a bullet. You could also use a lab test to demonstrate that a sphere of equal mass would fall more slowly through air than the bullet while the opposite would be true for a viscous fluid such motor oil.