I’ll read the paper. Looks cool and completely different from the approaches to QFT that I’ve seen before.
Okay sure. Here’s how I should have begun an introduction to trigonometry (rather than regurgitating SOHCAHTOA in 20 seconds):
As you go around a circle, your position on the circle is entirely determined by the angle theta that you’ve swept out. Just like how on a clock, you could tell what time it is if only you knew what angle the hour hand and minute hand are each making with the vertical “12” mark. The sine and cosine functions allow us to convert between an angle theta, and the x,y (“cartesian”) coordinates of the point on the circle which is found at angle theta. In the clock analogy, imagine that we put the hands of the clock on some graph paper. Then if we knew the angle of the minute hand and its length, the sine and cosine functions would allow us to determine which square cell the tip of the minute hand is in. [Insert picture here]
To define these functions precisely, we first choose as a convention the following way of defining the angle: we measure the angle between a ray going out from the origin and the +x axis, and an increase in the angle corresponds to a rotation of the ray in the counterclockwise direction. [Insert picture here]. Then, we define two functions sin(theta) and cos(theta) as follows: the x,y coordinates of the point on the circle centered at the origin of unit radius at angle theta are ( cos(theta), sin(theta) ). Note that these functions are well-defined (they take a single value at any given theta), and that they are necessarily 2*pi periodic. Also note that if we had defined theta in a different way (say by letting it increase in the clockwise direction and start at the -y axis, or by letting the circle have radius 2 and be centered at (1,1)), we’d get different functions, though we wouldn’t lose any essential information; they would just be translations or inversions or rescalings of what we just defined.
Why do we care? Speaking very generally, knowing how to convert between these two descriptions of geometry is important because sometimes things are easier to understand or compute in the angular description, and sometimes it’s easier in the cartesian description. Also, there are a lot of problems that you didn’t know how to solve in geometry that trig will enable you to solve easily (or to say it a bit more precisely, it will allow you to easily reduce such geometry problems to already-solved problems like finding the first three digits of cos(7 * pi / 16)). There are many more reasons why trig is important (like how (cos,sin) is the fundamental example of a covering map, like Fourier transforms, etc), but they won’t come up until later in your education so I can’t motivate them properly now.
I think that most of the standard trig facts can be derived from this starting point. The next thing I’d do would be to explain how to determine the cartesian coords of a point on a circle of any radius by rescaling, and in the same breath I could derive the SOHCAH rules for right triangles. Then I’d explain how to compute sine and cosine for certain special values of theta like pi/4, how to derive certain properties like antisymmetry under translation by pi, then I might show them a numerical plot of sine and cosine and check that it is consistent with the properties we’ve derived, etc etc.
I think it would be good for me to state the following facts explicitly so that I can’t evade them later if someone criticizes my explanation:
- Writing this explanation took much more effort than the last one (in total I think it was about ~1.5 hours of writing, plus something like ~1hr of thinking about it over the past couple days).
- My least favorite paragraph of the explanation I gave is definitely the first one (where I describe the general idea), but I don’t know exactly what I dislike about it or how to improve it.
- Overall I am fairly satisfied with what I wrote.
Would you agree that that amount of time indicates lack of mastery? And that it’s partly lack of mastery of trig concepts, rather than only of writing?
BTW, placing a low value on writing is a systemic error in math and physics related to anti-conceptual attitudes. Writing in a language like English is humanity’s best general tool for dealing with concepts/ideas. Being able to explain concepts in words is a big part of how people learn them, think about and discuss them, improve them, etc. It also helps with remembering without memorizing.
[Insert picture here]
What’s the reason for omitting pictures? That also seems like it indicates lack of relevant mastery. In other words, you find something hard or burdensome about including the pictures. The process is not easy/intuitive/automatic for you. That seems problematic because working with diagrams on computers is a relevant useful skill for your fields. (Related, I noticed you didn’t use any MathJax in this post or IIRC previously, which could indicate a lack of mastery of LaTeX, which also seems relevant to your fields.)
Also, do you agree that trig is a relevant building block to your field, and something you should be good at?
The big picture is that, while you know some stuff about how it works, I think the concepts and writing are both kinda a mess. It’s not well-organized, streamlined knowledge. It has “code smells” if you know that term. This is different than wrong and is compatible with doing the calculations correctly for tests.
I want to start with the above, not explain this part more yet, but I didn’t want to leave out an overall comment entirely.
Yes, the fact that I spent so much time indicates lack of mastery of something.
I don’t know how to separate out lack of mastery of trig concepts from lack of other sorts of mastery, so I can’t answer the second question.
This lack of interest in clear writing is something that has always bothered me about my field. People are much better rewarded for getting the right answer than they are for giving clear explanations. I can see how that’s an anti-conceptual attitude.
This may surprise you, but I think I have always placed a higher value on explanations than any of my math/physics peers (the ones that I’ve met at least), and many of my professors (though that’s harder to judge). Now I’m starting to think that I haven’t done it enough.
I think you are wrong about me lacking relevant mastery here.
I use TeX all the time, I take notes in it, I do my homework in it, and I am comfortable with it. I figured that this forum might support TeX equations because I think I saw Max using it earlier, but I didn’t care enough to look into it because none of the math I’ve had to type here is notationally complicated.
Regarding pictures (edit: new thoughts caused me to make significant changes to this section like an hour after I wrote it),
- I had resolved not to look anything up when I wrote my explanation, so I didn’t want to go find suitable images from Wikipedia or somewhere else on the internet. I think that they would be easy to find b/c it’s such a common topic, and if I had an image I wanted I could insert it into my forum post in like 20 seconds with no thought.
- I normally could make a digital drawing really easily too, but I got a new computer a few days ago and still have to set some things up, and I guess I didn’t think it was worth the bother in this case.
- TikZ diagrams are relevant for my field, but I haven’t learned TikZ yet because I have not yet written anything that requires me to make a custom diagram (so there is a lack of mastery here but not an important one—yet).
Yes.
Okay, I await your elaboration.
Actually, I guess I could have drawn something on a piece of paper and taken a picture really easily, but I didn’t do that. The question I then need to ask myself is: If it’s easy for me to put pictures in my post, and if my goal in writing that post was to make as clear an explanation as possible, why didn’t I put pictures in my post?
I think that the answer to that question is that I just didn’t think much about the costs or benefits of adding pictures. I don’t think I even considered adding real pictures. I was being irrational, because in hindsight I do think that pictures would add a lot to the explanation I gave.
This is still not a lack of mastery, though. At least not a lack of mastery wrt computer images or TeX.
When I parenthetically wrote:
could indicate a lack of mastery of LaTeX
that is not wrong if, in this case, there is no lack of mastery.
The thing I was claiming was that it was worth mentioning (as an aside) given the evidence available to me. After seeing your reply, I stand by that.
What I said about images could also be accurate even if your reply is correct. So I’m concerned that you’re not noticing the difference when I say one thing instead of another kinda similar thing.
Not considering something does show lack of mastery. It means the ideas aren’t integrated well enough into your life/thinking. Easily recognizing opportunities to use something is part of mastering it.
In short, mastery includes everything necessary to use something in your life and to connect it with your other ideas and your actions.
Why didn’t you mention this earlier when I was saying there were systemic issues in your field and you said that I was biased instead of volunteering this opinion of yours that a major systemic issue exists in the field?
BTW, you didn’t respond to my denial re bias, so I don’t know if you changed your mind or what.
Why would it surprise me? It is on-theme/on-message for what I have been saying in this topic. It’s the kind of thing I was trying to tell you. Did I say something that clashes with it or shows potential ignorance of it?
I think I had it filed away as a different category of error, so it just never came to mind. I think I saw it as like “here’s this slightly annoying cultural thing in my field that makes lectures more boring and makes some papers harder to read,” rather than “here’s this evidence of an anti-conceptual attitude in my field that could affect the actual physics.”
Sure, I changed my mind. By the way, I think I did respond in some sense because I liked the post. If there was something you said in your post that I had a significant disagreement with, I wouldn’t have liked it.
Because in the whole thread you have been talking about how I don’t put enough value on explanations and concepts. I’m saying that you might be surprised to know that I think I’m better about it than so many others in my field, as opposed to me being an average representative.
(I also figure you probably aren’t actually all that surprised, given that we have had conversations and that you know that I’m interested in philosophy. I was saying “this may surprise you” in kind of a tongue-in-cheek way.)
I’m posting as a practice activity (improve posting frequency) and to check my understanding. Not meant to disturb the discussion.
I think that @lmf could value explanations and concepts more than most in his field but still undervalue them. Like a person could be bad at X and the entire field could be really bad at X. In that case the person would be better than most in his field at X but still bad at X.
1 Like
I don’t know if you want this feedback, but this wording reads to me as passive-aggressive. It’s a cumbersome wording (“seems odd to me”) and an indirect (passive) way to criticize an error (aggressive).
A more direct version, just as an example for comparison, would be: You say you don’t understand this, so it doesn’t make sense to comment on it, particularly with a strong opinion. Commenting is implicitly premised on understanding the thing you’re commenting on. A better way to deal with not understanding something is to ask questions about it.
Another reason I thought it could be passive-aggressive is that I saw a trigger for that type of response. The trigger was the arrogant, aggressive, non-tentative comment that you were replying to:
Deutsch is just dead wrong about this.
And that was actually part of a pattern. Earlier in this thread, lmf was similarly dismissive of DD’s ideas about physics, in favor of his own:
but that’s simply false.
If you were offended by these comments, or disliked or disagreed with them, there are better ways to handle them, like commenting directly on what you think is objectionable about them.
If there was some other cause in this particular case rather than being passive-aggressive, then your text miscommunicates, which is also bad. One enabler of this kind of miscommunication is being capable of being passive-aggressive, rather than having automatized awareness of the issue and its problems – that enables writing stuff that sounds passive-aggressive, with the wording intuitively seeming normal instead of you noticing the problem, even when actually something else is going on.
A difficulty with this kind of writing (both the passive-aggressive sentence and the arrogant sentences) is people often intuitively know what it means but don’t explicitly, consciously know what they are reacting to and why. People often get impressions, emotions and intuitions, based on their subconscious and automatized knowledge, without knowing exactly what happened. That makes it hard for them to address the issues directly or specifically. That’s one of the things that happens to me: People feel like I’ve been mean to them, and then when I ask for a quote with an explanation of what was mean about it, they can’t give one. But they still have the feeling and general negative impression. They often fail to problem solve that themselves and also don’t ask for help with that problem solving process (they often also reject help if I offer it, claiming that the problem is solved as a way to end the discussion. They say this when they have not automatized better knowledge; they just reached a conscious conclusion that they were wrong and then tried to suppress their feelings that they regard as irrational and perhaps also static memes). This kind of thing causes a lot of trouble and is one of the ways people end up failing.
BTW, I am interested to see some continuation/resolution to the debate about string theory. There appears to be a substantial disagreement between two people who both have some relevant expertise.
1 Like
I finally got around to doing this. I recorded every “trivial” mistake I made in a ~2hr period of doing calculations. To be clear, these are the types of things I was referring to earlier as “arithmetic mistakes,” even though strictly speaking they are mostly not arithmetic mistakes.
For all the things I’m about to mention, I did them in my head, wrote the answer down, and then noticed an error a little bit later. These were small parts of much longer calculations. I happened to notice all these errors before they propagated very far so they were easy to fix.
I was doing what I consider a medium amount of error correction. I wasn’t rushing, but I wasn’t being particularly careful either.
Here’s my log of all my “trivial” mistakes:
-
On one line, I had a 1/2 factor sitting next to some quantity X, and then I evaluated X and it contained a factor of 1/4. On the next line I intended to combine these factors, which should have resulted in a 1/8, but for some reason I wrote 1/4 instead.
-
There was a b^{-5/2} in my answer and I accidentally wrote b^{5/2}
-
There was a b^{-1} that I accidentally wrote as b (not directly related to above error)
-
I had to do evaluate an integral of the form \int_0^\infty e^{-(a + b)x}dx, and I forgot about the a and wrote the integral as 1/b instead of 1/(a+b)
-
I took the x derivative of e^{a x} and wrote it as a rather than a e^{a x}
-
I had an expression of the form (e^{-a x} - e^{-b x}) that I needed to Taylor expand to first order in x, and I wrote (a - b)x rather than (b - a)x
I would be very interested to know if you (@Elliot presumably) can think of a way that a conceptual deficiency could be causing some of these errors, or if there is some extra info I could provide (or extra experiments I could do) that could reveal that. My operating theory up til now has basically been that the unaided human brain (or at least my unaided human brain) can’t avoid errors like these.
There’s a related joke btw.
Q: What’s the difference between a good theoretical physicist and a bad theoretical physicist?
A: Good theoretical physicists make an even number of sign errors.
Context: Are you doing these with pencil and paper? Is the paper blank, lined or graph/grid paper? Only one color of pencil? Do you erase much? Are you good at handwriting? Do you have neat handwriting which writes each character in a consistent way, or is there much variation in how stuff looks so that the same character can look different (including size differences that aren’t intended for e.g. superscript, not just shape differences) when written in different places? Is the final result kinda messy or clearly organized? Are you using a physical calculator (or a calculator app on some device?) for some of the math? How much do you use mental math or shortcuts vs. writing stuff out in small steps? Do you have experience using smaller or larger steps for your math writing or thinking processes? Do you find it burdensome to write a lot of stuff out or to duplicate some parts of a formula repeatedly with no changes? And whatever the answers, did you put thought into these things, and have reasons for them, in the past?
Yes
Lined
Yes
I think so?
I think I have decent handwriting. No one has given me negative feedback about my handwriting, and I can always read it.
I think I have neat handwriting in this regard.
Clearly organized I think.
Not usually, no. I don’t have to do many calculations where a calculator would help.
It’s hard to quantify. I do a lot in my head and write a lot down, and the exact balance depends on a lot of specifics.
I suppose so. For example, in math, if I’ve made an error and it’s not immediately clear where the error started, I start to think through things in smaller steps.
I’m not sure. Is it more burdensome than copy pasting a line in a TeX file? Yes. Is it something that I dread doing or is it a place where I’m noticeably much more likely to make an error? No.
Yes and no.
I think I have put some thought into the handwriting and neatness stuff before.
Something which I have not put much thought into is how much I use mental math or shortcuts vs writing stuff out in small steps.
Those look like process errors. And I think this is a conceptual error:
My operating theory up til now has basically been that the unaided human brain (or at least my unaided human brain) can’t avoid errors like these.
And I think there are conceptual errors related to processes/methods. Those don’t look particularly related to conceptual errors in “math” (e.g. the stuff they try to teach in math classes), but they are relevant to doing math well.
One way to approach process errors is by trying different processes and monitoring for changes in error rate, types of errors, etc.
For example, it looks like sign errors are a common category of error generally and for you. If you wrote all negative signs in a different color which stands out, you might make fewer sign errors.
I haven’t tried them, but a quick web search indicates that many erasable colored pencils are available.
Other ways to vary process include using apps (there are many types, like free drawing, vector art, whiteboard, visual apps meant specifically for doing math work, regular text editor, math-oriented text-based app, etc.) and input devices (e.g. mouse, keyboard, stylus).
Another process variation would be to use pen instead of pencil. Or to use graph paper and consistently write one character per box (one result is minus signs would get a whole box to themselves). Many others could be brainstormed and then tested for advantages and disadvantages.
A different sort of process change is to go through math steps more explicitly and write more out.
A different sort of process change, which is harder to test, is paying more attention to detail. It requires a sort of mindset change. Finding a way to pay a lot of attention to detail, and to actually somewhat enjoy it, is important to doing math well. It’s kind of similar to the attention to detail needed to read and write accurately in discussions. Coding in languages with complex syntax is another thing that requires attention to detail (dealing with lots of nesting also requires attention to detail). Dealing with shortcuts can also require a sort of attention to detail, because you have to accurately track more stuff in your head to make up for the skipped stuff. With writing, sometimes when people write less stuff and make it sloppy and don’t explain some parts, that requires more attention to detail to read, despite it being imprecise. That may be counter-intuitive. But if the author gives less information, then the reader has less information to go on, which means he needs to squeeze more value out of the information available – and getting more out of a given information is a type of paying more attention to detail.
Another process change is going more slowly and carefully. Can you actually stop making all these errors if you go slowly enough? What about with no backtracking or double checking, just being more methodical?
Another process change is doing different types of practice when learning skills.
Some errors may be related to doing exploration during solving, rather than knowing the whole path to the solution from the start. In that case, one of the fixes is to consciously know, and perhaps label (e.g. different color writing) which stuff is exploration and which are a final, organized solution. One way people make a mess is they do something that might work, but they aren’t sure if it’ll work. So it’s exploration. But if it does work, then they recharacterize it as non-exploration and continue instead of redoing that part. This is similar to, when coding, writing a first draft of a function to try out an idea, and then finding out it produces the correct answer so you just call it done with no refactoring, no step where you consider how it should be organized now that you’ve got the (this part of) the solution worked out.
Broadly, fixing one type of error helps find other types. Sometimes people think “I could do X to fix this; it’s just not worth the effort.” They may be right. But they may not. If they actually did X, they might find the error still isn’t fixed. Why? Besides them being just plain wrong, they might have had two or more problems. They knew what was going on with the most visible problem, but it obscured another problem. If you fix these sorts of arithmetic errors, you’ll either do much better at math or find that it’s now easier to do problem solving for remaining issues because then there are fewer factors involved.
I’m confident that if you wrote out many different math solving process flowcharts, and followed them, you could find one which prevents these particular errors.
Also, I take “unaided” to allow a desk, a chair, pencil, paper, written notes, looking stuff up in books, and I don’t know what else. BTW, that ambiguous use of “unaided” is the sort of seemingly small error in philosophy that is somewhat parallel to the math errors we’re discussing, and which partly looks like an attention to detail issue. Also, a lot of attention to detail issues are related to automatization – programming what details your subconscious deals with and how it deals with them – and can’t be solved in a satisfactory way just with conscious attention to detail (which is too slow). For my flowchart point above, I think that if you could follow a flowchart with full conscious attention and get that to work reliably, then it’d also be possible to practice and automatize those steps.
fwiw I wasn’t actually upset by @alanforr’s comment, though I do see how it (and also my preceding post) could be interpreted as aggressive.
I never responded to this, sorry.
I assumed that when Deutsch speaks of “fully quantum mechanical” entities he is being vague, so I didn’t (and still don’t) think it mattered that I moved on to criticizing other parts of the paragraph. Having read some of the paper you linked, I still think he was being vague with that term.
When I wrote that paragraph I definitely presumed that Deutsch isn’t aware that particles in QFT are merely things that show up in a quasi-classical approximation, which is unfair. Maybe Deutsch doesn’t think that’s the right way to think about particles, or maybe he meant something else by “fully quantum mechanical.” I should have been more humble.
same here.
@alanforr would you be interested in this?
I read some of the paper you linked, but I probably would need to ask you some questions if I’m to understand it better.
I think the difference is important. These errors involve more complexity, so there’s scope for complexity management errors that you might not make when doing plain arithmetic.
Complexity management errors are extremely common in philosophy discussions. One type is when people lose track of some part of the discussion tree. People forget or some nodes said by any participant, or change them in their memory. Or they do the same thing with connections between nodes.
Some stuff people should be doing about this, in philosophy discussions, is using conscious attention, organizational methods (like tree diagrams) and frequent rereading (rereading allows you to check for memory errors, which if done a lot helps you develop a sense of when there’s enough risk that you should reread stuff). Do this enough correctly enough and you can automatize stuff, e.g. accurately knowing small trees without writing them down or trying very hard consciously, and with more practice it can be done with large trees.
With math, similar approaches can be used. E.g. you could double check stuff (parallel to rereading) constantly until you start to develop a good intuitive feel for which things are 100% safe and don’t need double checking, and which are only 99% safe or less. That intuitive feel is an automatized subconscious calculation. You can develop it with an aim towards very high reliability from the start – make it very conservative. If you already developed it in a more unreliable way, you can change habits in the usual way: practice something new, and use conscious attention to override the old habit, until the new thing becomes your habit. (It’s also possible to learn things as different modes and flexibly switch between them, rather than overwriting/deleting the old habit. That’s harder but sometimes useful.)
You could make a tree or other organized diagram that keeps track of all the math elements and then represents transformations/changes/conversions to them. If the organizational technique manages the complexity well enough so nothing gets lost, and none of the transformations are wrong, then you shouldn’t make errors, or at least very few. If some transformations are wrong, then it may require smaller steps, more attention to detail, or more formal steps (there are some things that can only be done as a big step or at least are too inconvenient to break up, so they can be handled carefully, e.g. writing out the whole formula or technique being used, and write out all the inputs and outputs generically, then mapping nodes from your diagram to every input, so no piece of the correct process gets forgotten, skipped, etc. Similarly this prevents some element from being double used – used in the big process but also still left elsewhere.). If this doesn’t work, it can at least help pinpoint errors. And one can try it with simpler math until it does work. Like you ought to be able to reliably do a basic arithmetic problem involving a dozen or more operations if you write things out in an organized way. If you can’t, there’s a problem there (complexity management/organization or attention to detail would be first guesses). If you can, then you can escalate the math difficulty until the errors start appearing and try to understand what made them start appearing at that point.
It’s common that you subconsciously see part of what’s going on, and it makes a difference to you and your reactions, without conscious awareness of that happening or a conscious feeling like being upset.
How can it affect your reactions if it isn’t in your conscious mind? One way is that your conscious mind delegates some sub-steps of your reactions to your subconscious. Another way is that your subconscious does something (e.g. writing some data somewhere in memory or changing a setting on something) that your conscious mind then interacts with, possibly at a later time. You may never connect that particular data or setting to a negative reaction to that particular text. You may not even know that it was changed because your conscious just has access to the current value, but no changelog, no last updated date, etc.
Can a human (or group of humans) execute assembly code unaided (meaning no computer help), with near-zero errors?
I think so. It just requires an appropriate process. (And it’s slow.) Do you disagree?
You need a reliable way to keep track of the current values of memory locations. It is possible to come up with a process using paper, pens, pencils, tables, tape, etc. which would do this ok.
You need a reliable way to load into your human memory what the current instruction is and what steps to do to execute it (this can involve using your longterm memory, but requires an adequately reliable judgment process to know when your memory might be inadequate and you need to look in a book – you’ll also need to have that book).
You’d need a reliable way to pay attention and actually follow the process and do all the correct steps in order. Things like pointing and calling could help. You’d have to avoid getting bored or distracted – or detect those states and change them or take a break. You could try to detect and handle those states indirectly via detecting computation errors and then backtracking, but there are various problems and downsides with that approach.
You need some other stuff too but that gives some idea of it.
What they did at Los Alamos (while working on atomic bombs) to get people (who were not great mathematicians – closer to secretaries) to participate in complex math calculations – with adequate reliability/accuracy to be useful – is a relevant example.