= 27d567bc2d48d9f40e9ccc20ea370b15)
TsviBT comments on Proofs, Implications, and Models - Less Wrong
You are viewing a comment permalink. View the original post to see all comments and the full post content.
= 783df68a0f980790206b9ea87794c5b6)
Loading…= b715d02e85f7b3b4ae65ca3d3e450868)
Subscribe to RSS Feed
Powered by Reddit
= f037147d6e6c911a85753b9abdedda8d)
You are viewing a comment permalink. View the original post to see all comments and the full post content.
Comments (209)
Mainstream status:
This is meant to present a completely standard view of semantic implication, syntactic implication, and the link between them, as understood in modern mathematical logic. All departures from the standard academic view are errors and should be flagged accordingly.
Although this view is standard among the professionals whose job it is to care, it is surprisingly poorly known outside that. Trying to make a function call to these concepts inside your math professor's head is likely to fail unless they have knowledge of "mathematical logic" or "model theory".
Beyond classical logic lie the exciting frontiers of weird logics such as intuitionistic logic, which doesn't have the theorem ¬¬P → P. These stranger syntaxes can imply entirely different views of semantics, such as a syntactic derivation of Y from X meaning, "If you hand me an example of X, I can construct an example of Y."
I can't actually recall where I've seen someone else say that e.g. "An algebraic proof is a series of steps that you can tell are locally licensed because they maintain balanced weights", but it seems like an obvious direct specialization of "syntactic implication should preserve semantic implication" (which is definitely standard). Similarly, I haven't seen the illustration of "Where does this first go from a true equation to a false equation?" used as a way of teaching the underlying concept, but that's because I've never seen the difference between semantic and syntactic implication taught at all outside of one rare subfield of mathematics. (AAAAAAAAAAAAHHHH!)
The idea that logic can't tell you anything with certainty about the physical universe, or that logic is only as sure as its premises, is very widely understood among Traditional Rationalists:
--Albert Einstein
Great post!
Hrmm... I don't think that's right at all, at least for professors who teach proof-based classes. Doing proofs, in any area, really does feel like going from solid ground to new ground, only taking steps that are guaranteed to keep you from falling into the lava. My models of my math professors understand how an argument connects its premises to its conclusions - but I haven't gotten into philosophical discussions with them. My damned philosophy professors, on the other hand, are the logically-rudest people I've ever spoken to.
BTW,
I parsed this as "...his colleagues disagree with X while declining ... or while pointing out which step of the reasoning seems like an invalid implication", which is the opposite of what you meant.
Tried an edit.
Math professors certainly understand instinctively what connects premises to conclusions, or they couldn't do algebra. It's trying to talk about the process explicitly where the non-modern-logicians start saying things like "proofs are absolutely convincing, hence social constructs".
A math professor who failed to get a solid course in formal logic sounds unlikely... (If it's not what you are saying, it's not clear to me what that is.)
I go to a Big Ten university where the graduate-level sequence in formal logic hasn't been taught in six years.
Update: It's being taught again in the fall semester!
I never got a course in formal logic, and could probably have been close to being a math professor by now.
Yes, well, the problem is that many courses on "logic" don't teach model theory, at least from what I've heard. Try it and see. I could've just been asking the wrong mathematicians - e.g. a young mathematician visiting CFAR knew about model theory, but that itself seems a bit selected. But the modern courses could be better than the old courses! It's been known to happen, and I'd sure be happy to hear it.
(I'm pretty sure mathbabe has been through a course on formal logic and so has Samuel R. Buss...)
When I went to Rutgers, the course on proof theory and model theory was taught by the philosophy department. (And it satisfied my humanities requirement for graduation!)
At Yale, the situation is similar. I took a course on Gödel's incompleteness theorem and earned a humanities credit from it. The course was taught by the philosophy department and also included a segment on the equivalence of various notions of computability. Coolest humanities class ever!
I shudder to think of what politics were involved to classify it as such, though.
Probably it was that a Phil professor wanted to teach the class, and no one cared to argue. It's not things like which classes are taught that are the big political fights, to my knowledge; the fights are more often about who gets the right to teach a topic of their choosing, and who doesn't.
Have you read the book of Marker? I love that thing to pieces.
I think the correct syntax here is "...his colleagues disagree with X while declining to say which premise..., or to point out which step...", which links the two infinitives "to say" and "to point out" to the verb "declining". ETA: The current edit works too.
Really? I find people who have done just first year philosophy worse. Maybe I was lucky and had the two philosophy professors who philosophize well.
Ok, fair enough. My philosophy professors were the logically-rudest adults I've spoken to. Actually, that's not even true. Rather, my philosophy professors were the people I most hoped would have less than the standard rudeness, but did not at all.
An anecdote: spring quarter last year, I tried to convince my philosophy professor that logic preserves certainty, but that we could (probably) never be absolutely certain that we had witnessed a correct derivation. He dodged, and instead sermonized about the history of logic. At one point I mentioned GEB, and he said, I quote, "Hofstadter is something of a one trick pony". Here, "one trick" refers to "self-reference". I was too flabbergasted to respond politely.
Or tried to tell you something you didn't get.
From the description TsviBT gives it is far more likely that the professor was stubbornly sermonizing against a strawman.
If TsviBT failed to get something, it is quite likely that from TsviBT 's perspective the professor was waffling pointlessly, and that TsviBTs account would reflect that. We cannot appeal to TsviBT's subjective perspective as proving the objective validity of itself, can we?
I can look at the specific claim TsviBT says he made and evaluate whether it is a true claim or a false claim. It happens to be a true claim.
Assuming you accept the above claim then before questioning whether TsviBT failed to comprehend you must first question whether what TsviBT says he said is what he actually said. It seems unlikely that he is lying about what he said and also not especially likely that he forgot what point he was making---it is something etched firmly in his mind. It is more likely that the professor did not pay sufficient attention to comprehend than it is than that Tsvi did not say what he says he said. The former occurs far more frequently than I would prefer.
Edit:
Is it? I think it's a bit misleading. Logic would preserve certainty if there were any certainty. But there probably isnt. Maybe the prof was trying to make that point.
No, that isn;t the issue. TsviBT only offered a subjective reaction to the professor's words, not the words themselves. We cannot judge from that whether the professor was rudely missing his birlliant point, or making an even more birlliant riposte, the subteleties of which passed TsviBT by.
I disagree regarding the accuracy of the claim as stated (you seem to be making the mistake a professor may carelessly make by considering a different more trivial point). I also disagree that the alleged "brilliant riposte" could be 'brilliant' as more than an effective social move given that it moved to to a different point (as a riposte to the claim and not just an appropriate subject change) rather than acknowledging the rather simple technical correction.
You are giving the professor the benefit of doubt that can not exist without TsviBT's claim of what he personally said being outright false.
I hope that means you refrain from responding at all. You can't fix broken high status people!
Wait. Oh bother. I try to do that all the time. But I at least tend to direct my efforts towards influencing the social environment such that the incentives for securing said status further are changed so that on the margin the behavior of the high-status people (including, at times, logical rudeness) is somewhat altered. "Persuasion" of a kind.
Name three ways of you performing said persuasion.
No. Not at this time. (I would prefer to be not believed than to give examples of this right now.)
Actually it was more on the line of "give me practical examples so I can extrapolate the rule better than from an abstract summary", but, sure, suit yourself.
To be fair to the professor, conflating deductive and inductive reasoning is a basic error that's easy to pattern match.
To be fair to TsviBT it is pattern matching to the nearest possible stupid thing that is perhaps the most annoying logical rude tactics there is (whether employed deliberately or instinctively according to social drives).
This does seem correct as stated. I wonder if he deliberately avoided the point to save face regarding a previous error or, as Tim suggested, pattern matched to something different.
Now that is far less surprising and on average I can agree there (although I personally had exceptions!) It was the absolute scale that I perhaps questioned.
To be fair, Hoftstadter is basically a one-trick pony, in that he published one academically-relevant book and then more or less jumped straight to Professor Emeritus in all but name, publishing very little and interacting with academia even less.
Just wishing I had read GEB sooner. Reading it now and it seems to be getting ruined by politics.
Politics? I don't understand how.
Also, above comment should in no way be taken as criticism of GEB. It's great. It's just that that's pretty much all he has to his credit.
I mean people want to tear chunks out of it for status reasons... ...I think.