11–20 of 73 posts
What's the point of inserting untranslated French text into an English article? Is it supposed to just be window-dressing? Is it supposed to promote the (outdated and highly dubious) notion that all educated people speak French? Is it just for the author to show off? Whatever the reason, it's a highly obnoxious practice and it doesn't improve the article.
Counterpoint (from my blog): http://forwardscattering.org/post/7 http://forwardscattering.org/post/14 In summary i don't think AIT offers an absolute measure of complexity, due to having to choose the abstract machine. Not to say the ideas aren't interesting, and that this isn't a nice article, from one of the main figures in the field.
I don't think your claims are correct. If you read a formal definition of Chaitin-Kolmogorov, you can find that the concept are essentially normalized by recursive function theory. By using the Church-Turing thesis, you can see that any program in any language can be simulated up to a constant length-multiple by an abstract Turing machine (speed isn't considered in these definitions so architecture doesn't matter muc…
This is a good article, and an important topic, but it mischaracterizes the discipline of mathematics: > the principle that mathematical truth is black or white and provides absolute certainty. > Pure mathematicians like to think that they have absolute truth Formal mathematics has no concept of absolute truth; this is left for philosophy. It's just concerned with axioms and theorems (and their proofs). Whether an ax…
What's the point of inserting untranslated French text into an English article? Is it supposed to just be window-dressing? Is it supposed to promote the (outdated and highly dubious) notion that all educated people speak French? Is it just for the author to show off? Whatever the reason, it's a highly obnoxious practice and it doesn't improve the article.
Would you have preferred Latin or German ? Leibniz used them too.
So we are left with the weird conclusion:
1. It has the value it has for no reason.
2. It has the value it has for a reason that is impossible to formulate.
This is a good article, and an important topic, but it mischaracterizes the discipline of mathematics: > the principle that mathematical truth is black or white and provides absolute certainty. > Pure mathematicians like to think that they have absolute truth Formal mathematics has no concept of absolute truth; this is left for philosophy. It's just concerned with axioms and theorems (and their proofs). Whether an ax…
Earlier quoted context omitted.
I don't think your claims are correct. If you read a formal definition of Chaitin-Kolmogorov, you can find that the concept are essentially normalized by recursive function theory. By using the Church-Turing thesis, you can see that any program in any language can be simulated up to a constant length-multiple by an abstract Turing machine (speed isn't considered in these definitions so architecture doesn't matter muc…
have a look at the second post, i address that.
The fact that for any specific string S you can find a language L such that the Kolgomorov complexity K(L, S) is 0, is not that interesting.
1. For any specific L, you can only play this trick (hardcoding a target string) for a finite number of strings. That means for essentially all strings (all but a small finite number) the trick is irrelevant.
2. There is no free lunch: The trick you use to bring down the Kolgomorov complexity K(L, S) down to 0 increases the complexity of L. This can be taken into account, see "conditional Kolgomorov complexity".
The book "An Introduction to Kolmogorov Complexity and Its Applications" by Li and Vitányi explains this, and much more in great detail, and is highly recommended.
Earlier quoted context omitted.
have a look at the second post, i address that.
I'm sorry to say this, but your second post also doesn't really address the issue and continues to misunderstand the purpose and definitions of Kolgomorov complexity. The fact that for any specific string S you can find a language L such that the Kolgomorov complexity K(L, S) is 0, is not that interesting. 1. For any specific L, you can only play this trick (hardcoding a target string) for a finite number of strings.…
This is a good article, and an important topic, but it mischaracterizes the discipline of mathematics: > the principle that mathematical truth is black or white and provides absolute certainty. > Pure mathematicians like to think that they have absolute truth Formal mathematics has no concept of absolute truth; this is left for philosophy. It's just concerned with axioms and theorems (and their proofs). Whether an ax…
This is false. The axioms don't have to be true. You can still talk about their implications in absolute terms.
"Assuming a=0 implies a=0" is absolutely true. Regardless of whether a actually is 0.
This is a good article, and an important topic, but it mischaracterizes the discipline of mathematics: > the principle that mathematical truth is black or white and provides absolute certainty. > Pure mathematicians like to think that they have absolute truth Formal mathematics has no concept of absolute truth; this is left for philosophy. It's just concerned with axioms and theorems (and their proofs). Whether an ax…
"Formal mathematics has no concept of absolute truth" This is false. The axioms don't have to be true. You can still talk about their implications in absolute terms. "Assuming a=0 implies a=0" is absolutely true. Regardless of whether a actually is 0.
No, you can't. Two mathematicians using different rules of inference (say, those of classical vs. intuitionistic logic) will arrive at different theorems, even if they begin from the same axioms.
> "Assuming a=0 implies a=0" is absolutely true.
Assuming that “a = 0” and implication are both expressible in our formal system, it would indeed be very weird (though not a priori impossible) if “a = 0” didn't imply “a = 0”. But I have no problem imagining a formal system where “a = 0” isn't expressible (e.g., because equality is inexpressible), or implication isn't expressible (e.g., geometric logic), or implication has strange properties to someone only familiar with the classical and/or intuitionistic interpretation of “implication” (e.g., linear logic).