Conversation started Sep 6, 2017 at 6:17.
user131753
Sep 6, 2017 06:17
Although unrelated, recently while reading a significant amount of literature on Russell's logic (or more specifically the logic of Principia Mathematica), I found that your viewpoint as stated in the following,
user131753
Jul 21 at 16:00, by user21820
@user170039 Thanks, but I think I'll pass. Unless someone shows a flaw with the incompleteness theorems (which I've myself proven), I just don't see a need to know what exactly the authors of PM were thinking. If their system is really not susceptible to the incompleteness theorems, then it necessarily must either be imprecise or unable to prove at least one true Π1-sentence of arithmetic, which is quite serious a problem, to say the least.
user21820
user21820
@user170039 Then I suppose that there's not going to be much that we disagree on, because my main gripe with philosophers is just the lack of precision. Anyway let me respond to your comments. =)
@user170039 Sorry it was page 23. You can also find it by searching for "barred". =)
user131753
Sorry, in my previous comments, that is in the following,
user131753
4 mins ago, by user 170039
Although unrelated, recently while reading a significant amount of literature on Russell's logic (or more specifically the logic of Principia Mathematica), I found that your viewpoint as stated in the following,
user131753
4 mins ago, by user 170039
Jul 21 at 16:00, by user21820
@user170039 Thanks, but I think I'll pass. Unless someone shows a flaw with the incompleteness theorems (which I've myself proven), I just don't see a need to know what exactly the authors of PM were thinking. If their system is really not susceptible to the incompleteness theorems, then it necessarily must either be imprecise or unable to prove at least one true Π1-sentence of arithmetic, which is quite serious a problem, to say the least.
user21820
user21820
Sep 6, 2017 06:23
@user170039 Not every English sentence that looks like a factual statement can be assumed to have a truth value. Quine's paradox is the most undeniable instance of that. However, one cannot from that fact infer that not every mathematical statement can be assumed to have a truth value. In other words, it can still be consistent (even if meaningless) to assume that every mathematical statement has a truth value.
user131753
I forgot to add that "..although correct but not doing justice to Russell and Whitehead's original aim of pursuing the projects of PM." (Anyway, I am going to discuss this with you in the Philosophy of Mathematics room, if you are willing.)
user21820
user21820
More strongly, in my view detailed in that post about paradoxes, it is in fact sound to assume that every statement about reality has a truth value. Then the important question now is which mathematical statements are about reality.
Most logicians will say that all arithmetical statements can be interpreted to be about reality. A minority will say that only those with bounded quantifiers can.
This distinction is related to the 2 defensible positions I mentioned earlier.
@user170039 I don't really know the details, but from what I've read Russell himself gave up the original aim of PM, and later invented type theory as a way out of the Russell paradox. But it isn't satisfying to me. Neither is ZFC though.
I don't mind whichever room you'd like to discuss it in, by the way, so you could copy any of our comments over and continue there if you wish.
 
Conversation ended Sep 6, 2017 at 6:29.