My impression is they’re similar.
The specific procedure I described works because if you assume the premises, then for any natural number n, I can give you a proof of P(n). I’d start by establishing P(1), if n=1 I’d be done, and if not I’d use P(1) => P(2) to establish P(2), then continue onwards like that. I could write an algorithm to produce the proof for any n.
Speaking conceptually (and more generally than the rigorous definition I gave, so as to also include strong induction or structural induction or whatever),
Mathematical induction is a rigorous procedure for taking some set of propositions about some of the objects in some sort of data structure, and some sort of logical relationship between those propositions, then generalizing them to facts that are true for all the objects in the data structure.
Could you compare these? Which explanation is better? Why? What flaws do they have? How do they compare to yours?
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I like the mathisfun article a lot more than the purplemath article. I don’t see major problems with the mathisfun article. I like that it is concise, and I like that it embeds exercises for the students to do into the article. The problems that I see with the mathisfun article all have to do with the fact that there are some parts where their language is quite vague. However, I think that’s unavoidable given the tough spot that they are in: they need to write something very concise because most of the people reading their article are not good at reading, and yet they can’t use concise/precise math terminology (e.g. they can’t talk about propositions) because they can’t assume much math background from their readers.
There are a lot of major problems I noticed with the purplemath article. Here are some of them:
I think students are going to find this sentence in the purplemath article to be very confusing:
Induction proofs have four components: (1) the thing you want to prove, (2) the beginning step (usually “let n = 1”), (3) the assumption step ("let n = k "), and (4) the induction step (“let n = k + 1”).
This whole thing is unclear. Especially (3) and (4), because he didn’t explain the logic of how you assume it’s true for k and then show that implies that it’s true for k+1. Worse yet, he said “let n = k + 1” which makes it sound like the induction step is just another assumption.
I noticed a mistake in the “adding fractions” line of the table he made: he is missing a pair of parentheses.
The purplemath article has a very social tone, e.g. “If you’re anything like me, then you probably […]”. I find this off-putting. I thought this part was especially bad:
I hope these examples, in showing that induction cannot prove things that are not true, have increased your confidence in the technique. Induction is actually quite powerful and clever, and it would be a shame for you not to have caught a glimpse of that.
I think the purpose of these two sentences is that he’s trying to socially pressure the readers into believing that he did a good job and that they understand his article (“he” hopes, and “it would be a shame” if the reader doesn’t get it). I also dislike that he hopes to increase their “confidence” in the technique, because I think his aim should have been to convince them completely.
These explanations are intended for a different audience than the one I gave, so it’s hard to compare. The purplemath/mathisfun ones are designed to teach induction to people who are learning it for the first time. Mine wasn’t intended to be pedagogical, it was just a quick summary designed to demonstrate that I understand induction to someone who already knows induction.
Do you see the point of Career, Physics and Goals (was: Artificial General Intelligence Speculations) - #83 by Elliot (stuff about how to post-mortem arithmetic errors)
given my clarifying followup: Career, Physics and Goals (was: Artificial General Intelligence Speculations) - #90 by Elliot
? Is that actually getting somewhere and you’re starting to get my point, or not really?
I think the Purple Math article is better. There’s an underlying disagreement (a theme behind a bunch of separate exchanges in this topic) which, from my pov, is basically that you don’t value or understand concepts and explanations (at least in math). This is a systemic issue that’s widespread in the field.
I think that anti-conceptual or instrumentalist mentality is related to both your belief in the extreme simplicity of mathematical induction and your not really taking on board my point, re arithmetic (that also applies to other areas) that if you don’t know the (underlying) cause of an error, then you don’t know the consequences of that error, so it’s not safe to ignore it.
There also appear to be elements of defensiveness or bias in your replies. I get that a lot in similar contexts – when people think they’re already really good at something, and they’re invested in it, then they’re broadly unwilling to reconsider (which, from my pov, prevents fixing the problems and becoming better at it). This is often partly related to relative standards – e.g. they are correct that they’re similar or better than some peers they’re familiar with. But that is compatible with my claims. A second reason for resistance to these kinds of discussions is that people often think that I’m straying outside my field into their field, where they’re better than me at stuff, and they stop respecting my knowledge and start expecting to actually win debates (people do this who have lost many debates before, without winning one with me, by their own admission – but they think this time, on this topic, the pattern is going to break – as if I’d suddenly stopped carefully making only claims with the same quality and confidence that I usually do, and I was just saying whatever opinions I pleased in their field without studying all the details. Related to this, people sometimes seem conveniently and incorrectly to assume I don’t know much about their field, or have much history with it. Part of the answer, btw, is that I’m not straying outside my field – issues like what conceptual explanations are, and their importance and use, are philosophical issues).
Due to the large perspective gap re how to think about math, I’m having a difficult time finding a short way to explain it to you that will resonate with you. I could tell you that the way you think about math is very different than DD or Feynman (or myself, but I don’t think you’ll believe that actually matters), but I’m not sure that you’d believe I actually know what I’m talking about, and even if you did, I think you might categorize DD and/or Feynman as special cases and not infer that you’re doing it wrong, or you might think there are multiple good approaches and yours must be fine too.
You might also jump to arguments like “I can’t be doing it all wrong because I [or some other person like one of your professors] have success at X.” I don’t claim it’s all wrong. I believe you that you can calculate a bunch of things. Poor conceptual understanding has consequences like making it much harder to do creative work in the field, but it doesn’t mean you can’t do anything effectively or couldn’t go through a whole career without changing.
One way to approach it is by considering what training you have to evaluate the mathisfun and purplemath articles. How many similar activities have you done in school? What have you practiced to be good at judging this?
Another way to continue would be to followup on your claim about the extreme simplicity of induction. Are you ready to concede that people (who can do the calculations) can think about it in different ways, or do you still think there’s essentially only one set of concepts that could go with the skill and no room for different qualities of conceptual knowledge about it?
Not really. I still believe what I wrote in this post.
!!! I find this shocking
I have been getting a bit defensive, but I’m trying to fight against it consciously because I want to be proven wrong if I’m wrong.
Regarding bias, I don’t know. The reasons why I chose my field are that I think people in math and theoretical physics are way smarter and better than people in any other mainstream field, and I think that the stuff we do is difficult and interesting and creative. I didn’t just stumble into it.
I think you and DD are biased against mainstream culture, and while I tend to agree with this perspective, I think my field is basically an exception to the rule.
I would definitely be curious to find out how those people think about math, and how it (allegedly) differs from the way I think about math. I obviously can’t predict ahead of time how I would respond to such information.
None at all that I can recall. I don’t think I’ve ever done a school activity like that. It seems plausible that you are better equipped than I am to evaluate such things.
Yes. I concede that induction is more complicated and more possible to misunderstand than I was giving it credit for.
If you don’t know the (underlying) cause of an error, then you don’t know the consequences of that (underlying) error, so it’s not safe to ignore it.
Did you respond to this issue somewhere?
How did you determine that I’m biased? Did you find (and say anything about?) an error? Why is this (IIRC) the first I’m hearing of it?
Then why are you shocked that I reached a different conclusion about it than you did?
Could you see how this is a systematic weakness or bias in how schools approach and train math? They’re not even trying to address some aspects of how to think about math.
Is this another mistake that you will dismiss as not important, or do you see a major concern here? In general, I find it very hard to talk to people productively when they dismiss errors as unimportant. If a general abstract explanation (e.g. about bad philosophy causing systemic errors in a field) doesn’t work, and also particular examples don’t work, then what would work? It’s hard enough to convince someone that I’m right about any examples when there’s a large underlying disagreement / perspective difference. But even when I succeed at that, people often don’t really care.
No, I didn’t. Let me address it head-on.
Suppose I run a factory, and I have a machine that produces some gizmos with errors every once in a while. I don’t know what causes the errors: Maybe it’s thermal fluctuations, maybe it’s a faulty machine, maybe the operator is using it wrong.
If a broken gizmo makes its way into one of our final products, that’s bad. When this happens, we can usually tell by just seeing that the final product manifestly doesn’t work, but we never know for sure. If we want to be more certain, we can put more quality checks in front of our gizmo machine or later down the line, or we can even check the final product using a computer algebra system (I couldn’t think of an analogy sorry), but that would slow things down so we do different amounts of quality control in different contexts.
Despite not having a great understanding of what causes the gizmo errors, I have a lot of empirical experience with these gizmo errors. I find that they are relatively rare and don’t mess up the final product very often, and that when they do mess up the final product, I have a systematic way to find the broken gizmos and fix them that takes some time but always works.
Why am I running my factory wrong, or why is it “unsafe” (unsafe for what)?
I think I’m using the word “bias” in a looser way than what you have in mind. I just mean that I think your point of view in general inclines you towards seeing academic physics as really bad, and that’s what you’re going to think until proven otherwise. I think my usage is consistent with the first definition I found when I searched for it, “prejudice in favor of or against one thing, person, or group compared with another, usually in a way considered to be unfair.”
You haven’t heard of it until now because I thought it was obvious, and because it’s not a good argument that you are wrong.
I do not claim to have identified an error, other than DD’s error regarding string theory not being motivated by problems (though I have been wondering to what extent that error skews his view of physics in a negative direction, especially since string theory is what most of the smartest physicists seem to be doing).
My expectation was that your opinion would differ from mine but not immensely, because I saw myself as being pretty good at evaluating these sorts of things but figured you’re better at it. In hindsight this seems wrong, because you being better at it means you make fewer errors in this area, and I don’t know how my errors will manifest themselves.
Since most math students don’t go into the field of education, it’s not obviously a weakness that universities do not equip their math students to learn how to evaluate pedagogical math material.
I’m not going to answer your question for now because I think our real disagreement might be about what I wrote in the first part of this post.
Oh, I see. Well, I can say that I believe I’m expressing a considered, serious opinion that draws on my expertise at reasoning and on a lot of information. I’m not coming to a conclusion based on vague general principles or based on taking a worldview and then expecting everything to fit it. I’m not expressing a default opinion that I’d hold unless/until proven otherwise. I deny the bias rather than considering it obvious. I consider making that error ~incompatible with being a great philosopher (who ought to be able to deal with bias well, know what he does and doesn’t know, qualify claims appropriately, and only assert stuff he has rational confidence about).
Re string theory: DD is inconsistent about being rational so it could be one of his weaknesses. Your rebuttal to him seemed reasonable to me but I don’t know enough about the details. Another potential issue with string theory – which I have some loose impressions about but don’t really know and didn’t investigate – is possible connections with attitudes to math similar to yours.
If anything biases DD against many other physicists, my first guess would be its their rejection of MWI. Which I think he has good points about. And related to the MWI issue is instrumentalism, positivism, disrespect for conceptual explanations, “shut up and calculate”, etc.
I’m actually curious if DD is/was as good of a physicist as his opinion or public reputation says. When I find a lot of errors with someone, I get suspicious of their other stuff, and all those misquote errors DD made in his books show poor attention to detail (and/or some alternatives like dishonesty), which is relevant to doing physics well. But I’m not in a position to check all of it, plus I have other stuff to do. I know of some errors in some stuff he wrote in physics papers but not in the core physics itself which I don’t have the math/physics background to check without a ton of work. There was some problematic stuff he wrote about Popper in one paper (search for “The logic of experimental tests, particularly of Everettian quantum theory” here) and his Turing misquote in another paper (btw I contacted the publisher, who doesn’t care about the error and won’t even publish some errata/correction/retraction on a website or do anything at all, which says something bad about academic journals).
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I like MWI, and I don’t like instrumentalism, positivism, disrespect for conceptual explanations, or the “shut up and calculate” attitude, all of which I do see in physics (though I think string theory is one of the better subfields in that regard). If you’re correct that I’ve internalized these attitudes to some extent, I would definitely want to change that.
My error was that I treated a concept as being atomic and I neglected its complexity. I can’t just say whether or not this error is “important.” That’s not a complete thought. I can only answer the question, Important for what?
Important for teaching induction to other people? Sure.
Important for answering your questions the way you wanted me to answer them? Sure.
Important for doing the proofs I do? No, I don’t think so. I don’t see the causal relationship.
Important for understanding more advanced math concepts? No, I don’t think so. I don’t see the causal relationship.
Maybe a better way to characterize my issue with this abstract point is to ask: not safe for what?
Like, my operating system occasionally has some glitch where everything freezes and I have to restart my computer. I don’t really know the causes of those glitches. I know that they must be due to bugs in the code, I know they tend to happen when I’m using most of the memory on my computer, but I couldn’t say much more than that.
If my computer is hosting mission control software for NASA, it’s not safe that I don’t know the cause of those errors. If my computer crashes in that scenario, terrible things could happen, and I can’t give a good explanation for why it won’t crash.
But if the context is that all I’m trying to do with my computer is write CF posts, it’s perfectly safe. I doubt my computer will crash right now, but if it does, it’s not a big deal: I’ll just restart it.
I don’t think that asking “for what” is instrumentalist or anti-conceptual. I think that ignoring questions of “for what” is Platonic and contradicts CF and Objectivism stuff (e.g. your IGC idea). I assume that you agree, but I thought I’d state this belief explicitly just in case.
What then, is instrumentalist or anti-conceptual about my mentality? All I can think of is how I said that I don’t agree that arithmetic errors matter for the purpose of me solving the problems I want to solve. You seem to agree that they aren’t relevant for test scores. I then explained why I think they are also not relevant for understanding math and physics well,
and I don’t think you’ve given a rebuttal to that (unless your rebuttal is the abstract argument, but then my counter-argument to that needs to be answered).
Yeah that’s not the issue.
More later.
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It seems like your concept of learning a new piece of physics is becoming able to calculate it, with limited concern for what I consider a conceptual, explanatory understanding.
Your point seems to be it’s not a problem since you’re succeeding, but I’m questioning the premise – the quality of that success.
There’s major overlap between what you’re thinking of as pedagogy/education and conceptual understanding.
In general, if you understand something well conceptually, you can explain it well. Being able to teach something is a decent proxy for conceptual understanding. Getting someone to try to explain something is a way to find out how they think about it – what concepts are in their mind and how are those concepts organized?
Also explanations aren’t necessarily about teaching. You can explain stuff in other contexts, e.g. with a collaborator or in a debate.
There are some aspects of teaching which are separate from understanding the field (e.g. how to handle a group of people, or how to avoid questions/prompts/activities that give away the answer to a student who doesn’t actually understand), but I don’t think I brought them up or criticized your answers regarding those teaching-only things. For example, your comments on trig brought up things like “theta” and “opposite side” without saying what theta is (an angle) or the opposite side of what (a right triangle). Although there could be an important conceptual issue there directly about math, it could also just be inexperience at explaining stuff, e.g. not being used to thinking about other people’s contexts and what information they have and don’t have. While I think that’s a good thing to work on that is relevant to being good at math or physics, it wasn’t the main issue. I think I got the general idea that basically you would teach SOHCAHTOA (or equivalent without the memory aid) and show how to calculate some information about right triangles using other information. Although presenting the formulas out of context could seem like you expected a student to memorize them, I charitably took that as kinda an accident and figured you actually do understand some context about their use with triangles and that you could give some real world examples. I think not doing that may reflect explanations not being at the forefront of your math thinking, though, and also I disagree with what I think is your way of conceptualizing trig. (Let me know if I got something wrong, but otherwise I wasn’t really trying to get into this more now.)
The purplemath article did have some teaching-only stuff in it (aimed at a target audience in middle or high school), which I thought was OK – not great nor awful – but that wasn’t what I cared about.
The factory workstations are analogous to math concepts in your head.
One reason it’s unsafe is the situation could be the following:
- Workstation A has an error.
- Workstation B and C are downstream of A.
- Quality control finds and fixes all the errors downstream of C.
- The errors due to workstation B are not being detected.
This may seem implausible if you think of a factory that produces tangible goods which can be easily checked for correctness after they’re done.
However, it becomes more plausible with more complex scenarios.
For example, the product may have tiny internal parts, e.g. electronics, which sometimes break within 5 years. It can be very hard to track down where that problem is or do anything about it based on customer complaints and warranty repairs/replacements.
Or the production line could have side effects, e.g. wasted labor time or scrap materials, which are not being monitored. Or they are being monitored in aggregate but the A+B combination is causing inefficiencies that have not been traced back to the source.
Fortunately, A+C is creating visible errors! That is good!
In general, with complex systems, you get some visible errors and some invisible errors. The visible errors, if you take them seriously, help you find root causes which cause a mix of visible and invisible errors. In this case, investigating the A+C issue thoroughly will help you find out about the A+B issue too.
Another problem is that you may take an existing workstation and then use it in a new way. If it has an error, that can cause unpredictable results like:
- it doesn’t work for the new purpose (visible error, but big problem)
- it has an invisible error for the new purpose
Note: Visibility is contextual. It depends on the ways that you’re looking.
With ideas, invisible errors – errors we don’t know how to detect, understand, judge as an error and/or correct – are common.
Rather than a factory, think of a really complex software project with hundreds of libraries. If you find a bug and put a bandaid over it (e.g. a particular function gives some invalid outputs, so you add an extra function to detect and correct those invalid outputs), it could easily be caused by a bug in lower level function which is breaking other stuff that hasn’t been reported, or which has been reported but people haven’t figured out the underlying cause. Also, occasionally the bug causes the wrong output but it looks like a valid output so it doesn’t get corrected. Minds are like this complex software project but even worse/harder to deal with.
If an arithmetic error is due to an underlying conceptual error, then that same error can ruin attempts to use that concept in integrations in the future. In can screw up attempts to build new knowledge on the incorrect concept. This particularly gets in the way of discovery and creativity. It can become much harder to invent stuff, and many people think the invention they do is normal (or even good) – invention or pioneering is hard and slow – and they never realize what’s going on. If they weren’t blocked by problems like this, perhaps they would be as creative and Feynman instead of thinking he’s a special genius and having no idea what the difference between them and him is.
In short: Visible errors are the tip of the iceberg. Finding them is precious. People often lose this attitude (if they ever had it) due to overreaching and becoming overwhelmed by errors. In that case, they urgently need to fix stuff until visible errors become uncommon enough for each one to be investigated.
Some investigations are short. It’s possible to determine some partial but adequate information about an error (like it’s type without full details – in fact we’re never infinitely precise about anything, we just need the right information to make judgments and the required information varies by context). And then you may decide that the error is a low priority or not worth the effort of fixing. That can be the outcome of an investigation sometimes. But it’s also important not to make excuses or rationalizations in that way. And that kind of conclusion of an investigation requires more information than “it was an arithmetic error”, because that’s inadequate to have a reasonable conceptual model of what’s going on. In general, the more something will be reused, the more important it is to investigate errors thoroughly and to fix them. The more you want to build on knowledge, the higher quality that knowledge needs to be. This makes basics like arithmetic really high priorities to get really high quality knowledge about – rather than seeing them as simple/easy, they should be seen as foundations that are trying to support a large amount of weight, so they need to be especially sturdy.
The most important arithmetic errors are due to conceptual errors. These errors can cause all sorts of trouble including problems when trying to be creative by e.g. varying or integrating concepts. They can also make it hard to learn a complicated math idea from a book, because the advanced idea might only be compatible with the correct concept but not the incorrect concept. One thing that may happen next is the student figures out how to do the calculation anyway, but has a poor conceptual understanding of it.
~All arithmetic errors are important because they distract attention when trying to think about other things that build on arithmetic. The easier and more reliable your arithmetic and other building blocks are, the easier it is to direct your attention to higher level issues.
I don’t think I understand a new piece of physics just because I can calculate it. I think conceptual understanding is more important! I and use calculations as a test of my conceptual understanding.
I think I understand what you’re saying now. You’re saying in practice, the only objective standard I’m using for judging whether or not I’ve actually been successful at understanding concepts are these indirect proxies like my ability to calculate things, or non-objective criteria like how I compare to my peers. In contrast, it sounds like you are saying that you—as a philosopher—know of a way to objectively evaluate explanations of math concepts.
Okay sure. That definitely seems plausible to me, and I see how schools are not testing for that stuff at all.
I agree with this.
I think we had a misunderstanding, because I had a very different thing in mind when you asked me to explain induction and trig. I wasn’t even considering trying to fully write something out as if I was trying teaching it to you, I was trying to produce an extremely brief summary to quickly prove that I know it to someone who also knows it. Maybe it’s a problem in and of itself that I interpreted your request in the way that I did, but idk.
In my defense though, I think I could have made something way more like the purplemath or mathisfun article if I realized that was what you wanted me to do. E.g. I am well aware that what I was saying about trig was in the context of right triangles; it took me like 20 seconds to write what I wrote, so I left out a lot of stuff.
If you want to get a better idea of my real conceptual understanding, I think you should ask me to explain a 3rd math concept (or let me re-write what I wrote about induction or trig), given my new understanding of what sort of thing you’re requesting that I write.
That doesn’t really work. It’s a little like saying that you think code is well-organized because it passes lots of unit tests.
I don’t think brevity (not “fully” writing something out) was the issue. E.g. this quote is brief and shows some conceptual understanding:
Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis ) and that from each rung we can climb up to the next one (the step ).
— Concrete Mathematics , page 3 margins.
The paragraph after that on Wikipedia also has some good features, e.g. the terms “base case” and “next step” which help explain what some of the math things mean/represent – what concepts they map to (it’s not an adequate explanation for someone with no prior familiarity with those concepts, so there’s a teaching issue depending on the target audience. However, I understand those concepts and, if there aren’t red flags, I can charitably assume that the author does too even if he doesn’t go into details on what a base case is.)
Similarly, purple math says:
You’d like to know if (*) is true for all whole numbers, but you obviously can’t check every single whole number, since the whole numbers never end. You only know that it’s true for the relatively few numbers that you’ve actually checked. Of course, adding up all those numbers quickly becomes tedious, so you’d really like for (*) to work! But to use (*) for whatever number you’d like, you somehow have to prove that (*) works everywhere. Induction proofs allow you to prove that the formula works “everywhere” without your having to actually show that it works everywhere (by doing the infinitely-many additions).
This provides an explanation of what induction is for – what problem it solves, when and why to use it – which a student can understand. It’s providing some context for the math. This is different knowledge than the calculation itself. And people who can calculate induction think about the context or concepts in different ways. E.g. I’ve seen several pages mention dominoes as an attempt to conceptually explain it, which I don’t think works very well without additional explanation that wasn’t present (it’s maybe OK as a really lose comment that isn’t meant to explain much, as compared to nothing). I have no recollection of anyone mentioning dominoes when I learned induction, or later, so a comparison to dominoes was not part of my conception of induction a week ago. Dominoes could be really central to how one person thinks about it, and tertiary or unknown for another person. If you don’t know the dominoes concept, or deprioritize it, then it’d be harder to make a mental connection between induction and a different concept that has some connection to dominoes.
I also didn’t previously know the ladder comparison, which intuitively strikes me as better than the dominoes comparison.
I don’t know if anyone ever explained it to me or I figured it out myself, but I did previously understand the concept that you can use induction to prove many (even infinitely many) cases at once instead of doing them one by one, which I think is more important. The dominoes and ladders concepts strike me as more useful for helping someone understand induction initially than for thinking about it after you already understand it. They seem more like initial learning aids than long-term useful concepts to me (I think they’re worth knowing – it’s good to know many ways to approach a topic, which sometimes helps with making connections to other stuff – but they shouldn’t get much long term attention). If someone could only understand induction that way – like they kept thinking about dominoes and had trouble viewing it abstractly with no dominoes – that would be a warning sign to me about a problems to investigate.
Does that make sense? And did you get the point about underlying errors and arithmetic?
Yes, I agree now that it doesn’t really work. That’s what I was saying in the paragraph below what you quote. Not sure if it was clear.
I still do not accept that I couldn’t have given better explanations than the induction/trig ones. I don’t think they were as representative of my conceptual understanding as you think. I didn’t understand what you wanted from me.
I am thinking about it.