Domainname optional
Principia Mathematica is modern and insightful
151–160 of 175 posts
Re: Principia Mathematica is modern and insightful
#152Earlier quoted context omitted.
Why the belittling language? You actually can prove the completeness and consistency of portions of mathematics. While axioms were known in ancient times, only Hilbert started the whole "prove Mathematics" thing. How else would you prove mathematics and why would that be childish to use math? The limitations discovered were quite surprising back then.
Yikes, guys/girls. I got downvoted to -4 points for a misunderstanding or something. Because the author of the website would probably agree with my simple point that although the Principia Mathematica tried to do the impossible, there is still utility for its value as a programming self-teaching resource for serious students of computer science. Wow. Yeah. You guys ironically didn't just throw out the baby with the b…
That would have made more sense as a reply to your own posting, because it doesn't bring the discussion forward or address my points (if any).
Re: Principia Mathematica is modern and insightful
#153As a former analytic philosophy student, it's always a bit strange and encouraging for me to see stuff like this show up in CS/Tech forums. We need more philosophy now that we are dealing with the implication of "intelligent" machines.
Re: Principia Mathematica is modern and insightful
#154Earlier quoted context omitted.
You mean you don’t have a framed, signed, bug-bounty cheque from Alfred North Whitehead on your wall?? More seriously, there is indeed a huge logical error at the heart of the whole enterprise but it was not discovered until much later by Kurt Gödel.
This is commonly believed, but Gödel didn't identify a logical error at the heart of the whole enterprise, he proved astonishing theorems revealing limitations of any sufficiently powerful formal system. One can kind of think of the Principia as a science experiment to find the extent to which known mathematics could be proven from foundational axioms that could be thought of as "laws of logic". To make their system…
The flaw was the belief that it was possible (though they were not sure that they two could pull it off). Gödel showed that it was not an impossibly difficult task but an actually impossible task.
Re: Principia Mathematica is modern and insightful
#155Earlier quoted context omitted.
In what way do you think this is useful over parentheses?
It's better visually. No dot, ., :, :., ::, ::., :::, etc. have increasing visual weight, which shows which are the more top-level operators without having to match up parentheses.
I see. It’s not clear how it would interact with operator precedence, however. For example, what would
a * . b + c . + d * e
Re: Principia Mathematica is modern and insightful
#156Earlier quoted context omitted.
It's better visually. No dot, ., :, :., ::, ::., :::, etc. have increasing visual weight, which shows which are the more top-level operators without having to match up parentheses.
I see. It’s not clear how it would interact with operator precedence, however. For example, what would a * . b + c . + d * e do?
Yeah. I'm not sure how one would handle that. (Can't remember if Whitehead & Russell had any inherent operator precedence.)
So may be most useful if you don't have inherent precedence, as might be the case if you're not dealing with the standard math operators.
Alternatively, one might say the dots override inherent precedence. So
a *. b*c+d*e
a * (b*c+d*e)
Re: Principia Mathematica is modern and insightful
#157I have a copy and like it much. However, i was always partial to Frege's Begriffschrift. His notation was really creative. It's a shame Russel's deflation of that project has sentenced it to the rubbish heap of history.
You are confusing his book Begriffsschrift ("concept notation"), where he invented what became modern predicate logic, and his later work "Grundgesetze der Arithmetik" (I/II) which (unsuccessfully) tried to derive arithmetic from purely logical notions, and which made heavy use of the Begriffsschrift.
Re: Principia Mathematica is modern and insightful
#158Earlier quoted context omitted.
If you want to get to the meat of the issue, without the extraneous stuff, you might also tackle: Godel's Theorem Simplified. https://www.amazon.com/Godels-Theorem-Simplified-Harry-Gensl...
The extraneous stuff is honestly fabulous, IMO of course. Just being able to listen to a four part fugue sensibly is a rare but accessible pleasure. And amusing dialogs as an instantiated dialectic for showing the truth synthesized from apparent opposites, is a great pedagogical learning.
Extremely agree. The extraneous stuff is what makes the book magical.
Re: Principia Mathematica is modern and insightful
#159Earlier quoted context omitted.
This is commonly believed, but Gödel didn't identify a logical error at the heart of the whole enterprise, he proved astonishing theorems revealing limitations of any sufficiently powerful formal system. One can kind of think of the Principia as a science experiment to find the extent to which known mathematics could be proven from foundational axioms that could be thought of as "laws of logic". To make their system…
The flaw was the belief that it was possible (though they were not sure that they two could pull it off). Gödel showed that it was not an impossibly difficult task but an actually impossible task.
A failed hypothesis is not the same thing as a logical error. Would they have been more free of flaws if they believed it was impossible and thus didn't try? We'd all be the poorer for it
Re: Principia Mathematica is modern and insightful
#160Earlier quoted context omitted.
> My point is that each of the material above (and i forgot to add the original Quantum Mechanics papers) were major watershed moments in science/mathematics and hence are not easy to read/understand. I must dissent. Einstein's Annus Mirabilis papers are really quite approachable. You don't need a book to guide you through. Though if you are having fun with the book, more power to you! Enjoy!
We have to agree to disagree then. You definitely need a good background in Physics to understand them. For a general reader, even though they might not get the whole thing they can still get an appreciation for the whole from the commentary/details added by the editor. The book contains the English translations of the 5 papers with Einstein's original introduction and a short discussion on each, a informative introd…
Oh, the book is probably good and entertaining! It's just not necessary, if you paid attention in high school.