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:
… as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.
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”
The metaphor of a scale is at least a common teaching tool for algebra: see 1, 2, 3.
I was taught algebra with a scale in the sixth grade. We had little weights that said “X” on them and learned that you could add or take away “x” from both sides.
Yeah, we were taught in basically the exact same way—moving around different colored weights on plastic print-outs of balances. I’ll also note that this was a public (non-magnet) school—a reasonably good public school in the suburbs, to be sure, but not what I would think of as an especially advanced primary education.
I join lots of other commenters as being genuinely surprised that the content of this post is understood so little, even by mathematicians, as it all seemed pretty common sense to me. Indeed, my instinctive response to the first meditation was almost exactly what Eliezer went on to say, but I kept trying to think of something else for a while because it seemed too obvious.
But I don’t recall ever getting that in my classes. Also, the illustration of “first step going from true equation to false equation” I think is also important to have in there somewhere.
My data point: as an undergraduate mathematician at Oxford, the mathematical logic course was one of the most popular, and certainly covered most of this material. I guess I haven’t talked a huge number of mathematicians about logic, but I’d be pretty shocked if they didn’t know the difference between syntax and semantics. YMMV.
Another data point: in Cambridge the first course in logic done by mathematics undergraduates is in third year. It covers completeness and soundness of propositional and predicate logic and is quite popular. But in third year people are already so specialised that probably way less than half of us take it.
It terrifies me that I seem to be unique in having had pretty much all of this covered in my high school’s standard math curriculum (not even an advanced or optional class). Eliezer’s method of “find the point where it becomes untrue” wasn’t standard, but I think (p ~0.5) that my teacher went over it when I wrote a proof of 2=1 on the board. I knew he was a cool math teacher who made a point of tutoring flagging students, but I hadn’t realized he was this exceptional.
Judging just from your description, he’s probably more than two standard deviations of abnormal.
Your curriculum sounds at least a good deal above average, but the core problem is that most “curricula” are effectively nothing more than a list of things that teachers should make sure to mention, along with a separate, disjoint, often not correlated list of things that will be “tested” in an exam.
I expect many curricula would contain a good deal of the good parts of traditional rationality and mathematics, but there are many steps between a list on one sheet of paper that each teacher must read once a year and actual non-password understanding becoming commonplace among students.
I still have a copy of my Secondary 4 (US 10th / high school sophomore) curriculum somewhere, which my math teacher gave me secretly back then despite the threat of severe reprimand (our teachers were not even allowed to disclose the actual curriculum—that’s how bad things often are). We both verified back then that not even half of what’s supposed to be covered according to this piece of paper ever actually gets taught in most classes. That teacher really was that exceptional, but he only had so much time, split across several hundred students.
“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”
Eliezer, I very much like your answer to the question of what makes a given proof valid, but I think your explanation of what proofs are is severely lacking. To quote Andrej Bauer in his answer to the question “Is rigour just a ritual that most mathematicians wish to get rid of if they could?”, treating proofs as syntatic objects “puts logic where analysis used to be when mathematicians thought of functions as symbolic expressions, probably sometime before the 19th century.” If the only important thing about the steps of a proof are that they preserve balanced weights (or semantic implication), then it shouldn’t be important which steps you take, only that you take valid steps. Consequently, it should be either nonsensical or irrelevant to ask if two proofs are the same; the only important property of a proof, mathematically, validity (under this view). But this isn’t the case. People care about whether or not proofs are new and original, and which methods they use, and whether or not they help give a person intuition. Furthermore, it is common to prove a theorem, and refer to a constant constructed in that proof. (If the only important property of a proof were validity, this should be an inadmissible practice.) Finally, as we have only recently discovered, it is fruitful to interpret proofs of equality as paths in topological spaces! If a proof is just a series of steps which preserve semantic implication, we should be wary of models which only permit continuous preservation of semantic implication, but this interpretation is precisely the one which leads to homotopy type theory.
So while I like your answer to the question “What exactly is the difference between a valid proof and an invalid one?”, I think proof-relevant mathematics has a much better answer to the question you claimed to answer “What exactly is a proof?”
(By the way, I highly reccomend Andrej Bauer’s answer to anyone interested in eloquent prose about why proof-relevant mathematics and homotopy type theory are interesting.)
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,
My middle school algebra books discussed this on a very basic level. The problem in the book would show the work of a fictional child who was trying to solve a math problem and failing, and the real student would be told to identify where the first one went wrong and how the problem should have been solved instead.
I never saw it done with anything more complicated than algebra, though.
I’m not sure whether or not this is common, either.
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”.
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,
Robin Hanson sometimes complains that when he tries to argue that conclusion X follows from reasonable-sounding premises Y, his colleagues disagree with X while declining to say which premise Y they think is false, or pointing out which step of the reasoning seems like an invalid implication.
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.
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.)
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.
Robin Hanson sometimes complains that when he tries to argue that conclusion X follows from reasonable-sounding premises Y, his colleagues disagree with X while declining to say which premise Y they think is false, or pointing out which step of the reasoning seems like an invalid implication.
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.
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.
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.
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.
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.
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?
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.
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.
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.
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.
We don;t know that the riposte moved to a differnt point because we weren;t there and do not have the profs words.
It did one of moving to a different point, agreeing with TsviBT or being outright incorrect. (Again following your assumption that it was, in fact, a riposte.) Moving to a different point is the most likely (and most generous) assumption.
(I have expressed my point as much as I ought and most likely then some. It would be best for me to stop.)
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.
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.
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.
Rather, my philosophy professors were the people I most hoped would have less than the standard rudeness, but did not at all.
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 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).
to not overload the word ‘true’ meaning comparison-to-causal-reality, the validity of a mathematical theorem
Ok, deductively true things should be labeled valid to avoid confusion with inductively true things. That’s an excellent resolution of the overloading of the word “true” in popular vernacular.
But sometimes you aren’t clear in earlier writings about whether you are referring to true or valid. For example, one message of Zero and One are not Probabilities is a sophisticated restatement of the problem of induction. Still, I’m honestly uncertain whether you intended a broader point. I mostly agree with what I understand, but there is some uncertainty about what precise rents are paid by these specific beliefs.
FWIW, the undergraduate Philosophy department at Southern Connecticut State University has at least 2 logic courses, and the “Philosophical logic” special topics course covers everything here and then some in a lot of detail. The first half of the course goes through syntax and semantics for propositional logic separately, as well as the relevant theorems that show how to link them up.
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
The metaphor of a scale is at least a common teaching tool for algebra: see 1, 2, 3.
I was taught algebra with a scale in the sixth grade. We had little weights that said “X” on them and learned that you could add or take away “x” from both sides.
Yeah, we were taught in basically the exact same way—moving around different colored weights on plastic print-outs of balances. I’ll also note that this was a public (non-magnet) school—a reasonably good public school in the suburbs, to be sure, but not what I would think of as an especially advanced primary education.
I join lots of other commenters as being genuinely surprised that the content of this post is understood so little, even by mathematicians, as it all seemed pretty common sense to me. Indeed, my instinctive response to the first meditation was almost exactly what Eliezer went on to say, but I kept trying to think of something else for a while because it seemed too obvious.
GOOD. Especially this one: http://www.howtolearn.com/2011/02/demystifying-algebra
But I don’t recall ever getting that in my classes. Also, the illustration of “first step going from true equation to false equation” I think is also important to have in there somewhere.
I love this idea, so I’ve taken it to the next level: http://sphotos-a.xx.fbcdn.net/hphotos-ash4/405144_10151268024944598_356596037_n.jpg
Hanger, paper clips, dental floss, tupperware, pencil, ruler, and lamp. If we’re trying to be concrete about this, no need to do it only part way.
My data point: as an undergraduate mathematician at Oxford, the mathematical logic course was one of the most popular, and certainly covered most of this material. I guess I haven’t talked a huge number of mathematicians about logic, but I’d be pretty shocked if they didn’t know the difference between syntax and semantics. YMMV.
Another data point: in Cambridge the first course in logic done by mathematics undergraduates is in third year. It covers completeness and soundness of propositional and predicate logic and is quite popular. But in third year people are already so specialised that probably way less than half of us take it.
It terrifies me that I seem to be unique in having had pretty much all of this covered in my high school’s standard math curriculum (not even an advanced or optional class). Eliezer’s method of “find the point where it becomes untrue” wasn’t standard, but I think (p ~0.5) that my teacher went over it when I wrote a proof of 2=1 on the board. I knew he was a cool math teacher who made a point of tutoring flagging students, but I hadn’t realized he was this exceptional.
Judging just from your description, he’s probably more than two standard deviations of abnormal.
Your curriculum sounds at least a good deal above average, but the core problem is that most “curricula” are effectively nothing more than a list of things that teachers should make sure to mention, along with a separate, disjoint, often not correlated list of things that will be “tested” in an exam.
I expect many curricula would contain a good deal of the good parts of traditional rationality and mathematics, but there are many steps between a list on one sheet of paper that each teacher must read once a year and actual non-password understanding becoming commonplace among students.
I still have a copy of my Secondary 4 (US 10th / high school sophomore) curriculum somewhere, which my math teacher gave me secretly back then despite the threat of severe reprimand (our teachers were not even allowed to disclose the actual curriculum—that’s how bad things often are). We both verified back then that not even half of what’s supposed to be covered according to this piece of paper ever actually gets taught in most classes. That teacher really was that exceptional, but he only had so much time, split across several hundred students.
Eliezer, I very much like your answer to the question of what makes a given proof valid, but I think your explanation of what proofs are is severely lacking. To quote Andrej Bauer in his answer to the question “Is rigour just a ritual that most mathematicians wish to get rid of if they could?”, treating proofs as syntatic objects “puts logic where analysis used to be when mathematicians thought of functions as symbolic expressions, probably sometime before the 19th century.” If the only important thing about the steps of a proof are that they preserve balanced weights (or semantic implication), then it shouldn’t be important which steps you take, only that you take valid steps. Consequently, it should be either nonsensical or irrelevant to ask if two proofs are the same; the only important property of a proof, mathematically, validity (under this view). But this isn’t the case. People care about whether or not proofs are new and original, and which methods they use, and whether or not they help give a person intuition. Furthermore, it is common to prove a theorem, and refer to a constant constructed in that proof. (If the only important property of a proof were validity, this should be an inadmissible practice.) Finally, as we have only recently discovered, it is fruitful to interpret proofs of equality as paths in topological spaces! If a proof is just a series of steps which preserve semantic implication, we should be wary of models which only permit continuous preservation of semantic implication, but this interpretation is precisely the one which leads to homotopy type theory.
So while I like your answer to the question “What exactly is the difference between a valid proof and an invalid one?”, I think proof-relevant mathematics has a much better answer to the question you claimed to answer “What exactly is a proof?”
(By the way, I highly reccomend Andrej Bauer’s answer to anyone interested in eloquent prose about why proof-relevant mathematics and homotopy type theory are interesting.)
My middle school algebra books discussed this on a very basic level. The problem in the book would show the work of a fictional child who was trying to solve a math problem and failing, and the real student would be told to identify where the first one went wrong and how the problem should have been solved instead.
I never saw it done with anything more complicated than algebra, though.
I’m not sure whether or not this is common, either.
How mainstream is the Monte Carlo methods stuff, specifically the kitten-observing material?
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.
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.
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.
We don;t know that the riposte moved to a differnt point because we weren;t there and do not have the profs words.
It did one of moving to a different point, agreeing with TsviBT or being outright incorrect. (Again following your assumption that it was, in fact, a riposte.) Moving to a different point is the most likely (and most generous) assumption.
(I have expressed my point as much as I ought and most likely then some. It would be best for me to stop.)
TsviBT ’s point was not outright correct,, which leads to further options such as expounding on a subtle error.
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.
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 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).
In Hungary this (model theory and co.) is part of the standard curriculum for Mathematics BSc. Or at least was in my time.
Ok, deductively true things should be labeled valid to avoid confusion with inductively true things. That’s an excellent resolution of the overloading of the word “true” in popular vernacular.
But sometimes you aren’t clear in earlier writings about whether you are referring to true or valid. For example, one message of Zero and One are not Probabilities is a sophisticated restatement of the problem of induction. Still, I’m honestly uncertain whether you intended a broader point. I mostly agree with what I understand, but there is some uncertainty about what precise rents are paid by these specific beliefs.
FWIW, the undergraduate Philosophy department at Southern Connecticut State University has at least 2 logic courses, and the “Philosophical logic” special topics course covers everything here and then some in a lot of detail. The first half of the course goes through syntax and semantics for propositional logic separately, as well as the relevant theorems that show how to link them up.
Finally, an Einstein quote used on the internet that isn’t obviously false.