In any case, what exactly are these more accurate predictions about the world that pop-physics enables you to make? I would be very curious to hear some examples.
I will never get a ping time to American servers from my home here in Melbourne of less than the distance times two divided by c.
If I drop a really heavy rock and a somewhat lighter rock from a moderate height there will be only a slight difference in how long they take to fall to the ground.
If I find some stuff that is really, really heavy and leave it in my pocket I will probably die of cancer.
Cars traveling towards me will sound slightly higher in pitch than after they go past me.
If I buy bullets that are designed to travel slower than sound they will probably make less noise than the bullets that go faster than the speed of sound.
If you give me some charts that show how much light of various wavelengths there is coming from two different stars and it so happens that they look really, really similar except that one is kind of ‘stretched out’ over the ‘wavelength’ axis I can tell you that the stretched out one is farther away from us.
And, the critical one:
If I make a bet with someone that a survey done ten years from now of prominent physicists fewer will declare affiliation with the Copenhagen Interpretation than if an equivalent survey was done today then I am more likely to make money than to lose money.
You might believe that my lack of status as a mathematical physicist doesn’t give me the right to make claims about Quantum Mechanics implications but the universe doesn’t care. I can apply basic principles of rational thinking to filter large swathes of evidence from those who popularize physics and, particularly, the verbal, non-mathematical claims of physicists in order to work out whether or not a specific claim is likely to be correct.
I will never get a ping time to American servers from my home here in Melbourne of less than the distance times two divided by c.
Funny you should mention that. I spent years working in IT, and this knowledge was actually useful once. I tried to ping a DNS router in Europe (I forget where) from California, and it came back in 1ms and I thought “Ummmmmm… no. You lie.” It turned out one of the smart switches on the local network was fucked up and was somehow returning all pings itself.
Behold! Even a pop-sci understanding of physics controlled my anticipations in a way that was useful for accomplishing goals in the world.
That is pretty awesome, but I also don’t think it’s necessary to think about light speed to solve that problem. Anyone who spends a lot of time debugging networking problems knows that 1ms is unreasonably fast for any communication with a machine more than a couple router hops away, even if it’s physically nearby.
I see that this is getting upvoted, but your example sounds like someone who realizes that a device doesn’t work because it’s not plugged in, and then makes a self-satisfied comment that his knowledge of electromagnetic theory usefully controlled his anticipations in this situation.
In other words, it’s about simple conventional nuts-and-bolts technical knowledge, not an improvement on such knowledge brought by more advanced understanding of anything. There’s no way someone who works in network administration wouldn’t know that a “<1ms” ping coming from around the world is anomalous.
it’s about simple conventional nuts-and-bolts technical knowledge, not an improvement on such knowledge brought by more advanced understanding of anything. There’s no way someone who works in network administration wouldn’t know that a “<1ms” ping coming from around the world is anomalous.
Actually, yes: I think this is correct.
But, I think I would have noticed something was wrong even if I hadn’t worked in IT before. But I’m not certain of that.
Fair enough. There are indeed many ways in which the folk physics intuition can be improved by internalizing a rule that’s simple enough to explain without math. I admit that my question was too aggressive and snarky.
However, I don’t think any such simple insights will move you any close to understanding either QM or relativity (let alone more advanced topics such as cosmology or the controversies over QM interpretations), which was the topic of the original dispute. I must also point out that your rules are either from classical physics (and thus reasonably close to the relevant folk physics intuitions) or in the form of entirely opaque rules for which you can’t find any justification except for appeal to authority. And I’m certainly not saying that as someone who has “status as a mathematical physicist”; I’m a complete amateur in physics, as I pointed out in an earlier comment. (Also, if you’ve read my earlier comments on LW, you’ll know that I don’t put much inherent weight on the official credentials of expertise bestowed by the present academic system.)
Also, regarding the trends in the Copenhagen vs. MW debate, how much of your opinion is based on understanding of the issues involved, and how much on mere perception of the social dynamics in the field?
However, I don’t think any such simple insights will move you any close to understanding either QM or relativity
Then, like Luke, I must observe that your definition of ‘understanding’ is completely incompatible with mine. Actually, no. I think we’ve moved beyond that now. You are just wrong.
Look at the first example for a start. That claim is based on understanding of (part of) relativity. But that example was only included coincidentally. May I remind you that the examples you demanded were for physics in general (and, for that matter, that the demand was in response to professed understanding of evolution.)? Space considerations stopped me before I also threw in predictions about “two really good clocks, one of which you fly around really fast!”, the fellow who gets into a rocket and outlives his great grandchildren and what happens when you shoot not very many very small things through very small slits.
Question: If the world were wiped out tomorrow leaving a few people and you were the only survivor with any minimal scientific training, could you on your own reconstruct special relativity? If the answer to this is “no” then you don’t understand it.
Note that by this standard I don’t understand QM and I really don’t understand GR and I’m a math grad student. It is ok to say that one doesn’t understand things.
Question: If the world were wiped out tomorrow leaving a few people and you were the only survivor with any minimal scientific training, could you on your own reconstruct special relativity? If the answer to this is “no” then you don’t understand it.
Yes. I already declared this, in a comment you replied to no less.
Additional claim: It has been too long since I studied physics for me to remember all of the mathematics of special relativity. Yet when I did the aforementioned study I also gained a solid grasp of the fundamental principles. With that understanding I could recreate the interesting mathematics from first principles. I am confident of this because I’ve done it before, just for kicks. Because memorizing the math wasn’t enough for me and I wanted to really grasp the science in depth. The way you do that is by knowing the concepts well enough that you could work out the equations for yourself. Because just memorizing them is detail work. (I haven’t got a chance in hell of doing this with GR.)
Look at the first example for a start. That claim is based on understanding of (part of) relativity.
As I pointed out in my reply to Luke, knowledge of that fact constitutes “understanding of relativity” to the same extent that knowledge of the fact that you can get hurt by sticking your fingers into electrical installations constitutes understanding of electromagnetic theory. It’s just a single fact you know in complete isolation, not a fact that is a part of some broader framework for understanding the world.
To build on this particular example, consider that some things like shadows, reflections, etc., can indeed move faster than light. Or, if you just spin on an office chair, in your rest frame the celestial objects are spinning around you way faster than c. Unless you can explain why such motions are consistent with the “no faster than light” principle, you have nothing more than a literally memorized fact that if it might be useful if something could move faster than c, it can’t happen. It’s a true fact, to be sure, and even a potentially useful one, but still.
May I remind you that the examples you demanded were for physics in general (and, for that matter, that the demand was in response to professed understanding of evolution.)?
The context was about the pop-scientific treatments of modern physics. If you insist on full precision, I will gladly admit that my wording about physics in general was imprecise, since there are indeed simple topics in classical physics where insight can be gained without math, building only on a folk-physics intuition. So please read my statements in their original context.
As I pointed out in my reply to Luke, knowledge of that fact constitutes “understanding of relativity” to the same extent that knowledge of the fact that you can get hurt by sticking your fingers into electrical installations constitutes understanding of electromagnetic theory. It’s just a single fact you know in complete isolation, not a fact that is a part of some broader framework for understanding the world.
We aren’t talking about the memorization of a simple fact about pings. We are talking about all the understanding you can have about relativity without having memorized a mathematical equation. This can be used to make the prediction that ping times will never be lower than the aforementioned limit.
You asked for examples of predictions about the world that can be made based on understanding physics minus the math. It would be disingenuous in the extreme to then dismiss all examples of predictions about the world given because they are, in fact, mere predictions about the world and therefore could have been memorized without real understanding.
To build on this particular example, some things like shadows, reflections, etc., can indeed move faster than light. Or, if you just spin on an office chair, in your rest frame the celestial objects are spinning around you way faster than c. Unless you can explain why such motions are consistent with the “no faster than c” principle, you have nothing more than a literally memorized fact that if it might be useful if something could move faster than c, it can’t happen. It’s a true fact, to be sure, and even a potentially useful one, but still.
My first prediction: If you (who I believe professed understanding of at least this much understanding of math) and lukeprog were locked in rooms disconnected from the outside world and given the task of answering this question not only would Luke be able to give an explanation, his explanation would be better than yours. You both know enough about physics to answer and Luke is better at explaining things.
My second prediction: There are many students who, in their physics exams, get all the questions that require mathematics correct and who, when encountering a question like this one, can’t give an answer. Because not only is knowing the math not strictly necessary to answer this question, it isn’t even sufficient.
Additional claim: It has been too long since I studied physics for me to remember all of the mathematics of special relativity. Yet when I did the aforementioned study I also gained a solid grasp of the fundamental principles. With that understanding I could recreate the interesting mathematics from first principles. I am confident of this because I’ve done it before, just for kicks. Because memorizing the math wasn’t enough for me and I wanted to really grasp the science in depth. The way you do that is by knowing the concepts well enough that you could work out the equations for yourself. Because just memorizing them is detail work. (I haven’t got a chance in hell of doing this with GR.)
In the various responses you have been given your claim that you can’t have any understanding of physics without math has been overwhelmingly refuted. All you are left with is “But that understanding isn’t true understanding, true understanding means you remember the math!” To that I reply “No, not all Scotsmen like haggis. You can only tell a true Scotsman by the kilt they are wearing!”
It is time to retreat from a complete rejection of all non-mathematical understanding so that you can express an actually tenable position regarding the limits of how much you can know about physics sans math. Because there really are such limits and I would love to be able to support you in declaring them. But right now you’ve gone overboard and tried to reject even that understanding which can exist. And that brings you to the realm of the absurd and I just can’t support a position which is just obviously factually incorrect.
You asked for examples of predictions about the world that can be made based on understanding physics minus the math. It would be disingenuous in the extreme to then dismiss all examples of predictions about the world given because they are, in fact, mere predictions about the world and therefore could have been memorized without real understanding.
However, there is indeed a difference between rules memorized in isolation without any additional understanding and, on the other hand, real understanding of physical theories and generating predictions based on them. Just like there’s a difference between understanding electromagnetic theory and knowing a few common-knowledge technical facts on how to deal with electrical devices, there is also a difference between understanding relativity and knowing a few facts that follow from it in isolation.
You are now employing a rhetorical tactic where you try to make this obvious and relevant point look like weaseling, but in reality it is a pertinent and adequate response to your example.
In the various responses you have been given your claim that you can’t have any understanding of physics without math has been overwhelmingly refuted. All you are left with is “But that understanding isn’t true understanding, true understanding means you remember the math!” To that I reply “No, not all Scotsmen like haggis. You can only tell a true Scotsman by the kilt they are wearing!”
This is sheer rhetoric. You latch onto one point I made, completely ignoring the context and making the most extreme uncharitable interpretation of it (one that is in fact bordering on caricature), all for the greatest rhetorical effect. Instead, a rational approach would be to see if there may be some validity behind my point even if its original statement was imprecise, especially since I readily admitted this imprecision on first objection. Not to mention that your own example is largely irrelevant in the original context, which was about lengthy pop-science works purporting to explain whole physical theories to lay audiences, not about isolated examples such as yours.
In any case, if you think the distinction I outlined above is invalid, or that I am applying it incorrectly, please go ahead and explain why you believe that. If you’re going to latch onto a caricature of what I wrote while treating the discussion as a rhetorical context, I have no further interest in continuing this exchange.
I think it might be helpful for you to taboo the word “real understanding”. It seems like a lot of the disagreement stems from luke and wedrifid being unable to understand what you mean when you use that phrase. To be honest, while I agree with many of your points, I also don’t think I understand what “real understanding” is supposed to mean. Perhaps you could restate your original point without use of the word “understanding”?
I think it might be helpful for you to taboo the word “real understanding”.
Even understanding isn’t well-defined. I at least don’t know of any agreed upon conceptual or mathematical definition. Does Wolfram Alpha understand math? Does a lookup table understand anything?
I think that you really understand a subject if you are able to transfer, teach and artificially recreate the skill or heuristic that is necessary to make useful predictions about the subject. The skill or heuristic needs to be utilizable given limited resources. You also have to be able to abstract and generalize your knowledge about the subject to solve new problems that are not similar to problems previously encountered. Furthermore, you have to be able to prove fundamental mathematical statements and relationships about the subject.
An example would be the ability to teach the game of Go to other agents, transfer your knowledge by writing books on how to play it, and create a Go AI that can play the game by predicting the success of its strategies.
My second prediction: There are many students who, in their physics exams, get all the questions that require mathematics correct and who, when encountering a question like this one, can’t give an answer. Because not only is knowing the math not strictly necessary to answer this question, it isn’t even sufficient.
No one else is arguing that it is sufficient, but they are arguing that it is necessary. In this context, I’m going to make a counter prediction: if one did give a test on SR to physics students just learning about it, the ones who answer the chair question correctly will be a proper subset of the ones who can do the mathematical manipulation.
In the more restricted context of highschool calculus or more basic physics this is (from my experience both teaching and tutoring) very much the case. There are students who say things like “I can’t do the math but I understand the concepts” and this just nonsense. The ones who answer the conceptual questions correctly are almost always those who can do the math. There may be kids who can do the formal manipulation and can’t connect to it conceptually but there’s almost no one who can’t handle the symbol pushing who can answer the conceptual level questions. They are too interrelated.
There are students who say things like “I can’t do the math but I understand the concepts” and this just nonsense. The ones who answer the conceptual questions correctly are almost always those who can do the math.
Why do the they believe what they believe? The simplest two explanations I can think of is that they are mistaken about their grasp on the concepts (when you ask them conceptual-level questions, they answer incorrectly), or they are mistaken about their inability to do the math (they feel insecure before quantitative tests, but score high). Is one of these the case?
There’s a combination of both, but there seem to be far more in this first category. To some extent it seems like it is an example of the Dunning-Kruger effect. And I suspect that some of the people here who are convinced that one can understand physics without understanding the math may be also running into Dunning-Kruger issues.
Given that the Dunning-Kruger effect may be a combination of noisy estimates and hidden, different scales (how do we weight the 1500-2500 separate skills involved in driving when deciding who’s in the top 10%), could we extend competence from both ends toward the middle?
What I mean is that an essentially correct but very fuzzy understanding may constrain your anticipations in many different circumstances, but may not tightly constrain them—anybody knows you’re not going to get a <1ms ping from another continent, and a car coming towards you makes a higher pitched noise than it does going away from you, but most people couldn’t diagnose the 500 mile email or calculate the speed of the car from the frequency change.
On the other side, many people just learning physics can carry out the calculus, as long as they have the formula in front of them and the problem’s explicitly stated; but would have trouble generalizing the formula to all the real-life situations in which it’s applicable.
As the intuition becomes more and more precise, it gets closer to math. As the math becomes more and more intuitive, the situations in which its applications are evident grow.
An engineer is never going to be successful with just intuition, because he needs very precise results. But, depending on your use case, generality may be more useful than precision. This seems more useful than trying to decide which one is “real” understanding.
No one else is arguing that it is sufficient, but they are arguing that it is necessary.
Did I leave out the part where students who couldn’t do all the math answered it correctly? Pardon me.
In this context, I’m going to make a counter prediction: if one did give a test on SR to physics students just learning about it, the ones who answer the chair question correctly will be a proper subset of the ones who can do the mathematical manipulation.
Then I would make counterfactual money by betting against you.
I will never get a ping time to American servers from my home here in Melbourne of less than the distance times two divided by c.
If I drop a really heavy rock and a somewhat lighter rock from a moderate height there will be only a slight difference in how long they take to fall to the ground.
If I find some stuff that is really, really heavy and leave it in my pocket I will probably die of cancer.
Cars traveling towards me will sound slightly higher in pitch than after they go past me.
If I buy bullets that are designed to travel slower than sound they will probably make less noise than the bullets that go faster than the speed of sound.
If you give me some charts that show how much light of various wavelengths there is coming from two different stars and it so happens that they look really, really similar except that one is kind of ‘stretched out’ over the ‘wavelength’ axis I can tell you that the stretched out one is farther away from us.
And, the critical one:
If I make a bet with someone that a survey done ten years from now of prominent physicists fewer will declare affiliation with the Copenhagen Interpretation than if an equivalent survey was done today then I am more likely to make money than to lose money.
You might believe that my lack of status as a mathematical physicist doesn’t give me the right to make claims about Quantum Mechanics implications but the universe doesn’t care. I can apply basic principles of rational thinking to filter large swathes of evidence from those who popularize physics and, particularly, the verbal, non-mathematical claims of physicists in order to work out whether or not a specific claim is likely to be correct.
Funny you should mention that. I spent years working in IT, and this knowledge was actually useful once. I tried to ping a DNS router in Europe (I forget where) from California, and it came back in 1ms and I thought “Ummmmmm… no. You lie.” It turned out one of the smart switches on the local network was fucked up and was somehow returning all pings itself.
Behold! Even a pop-sci understanding of physics controlled my anticipations in a way that was useful for accomplishing goals in the world.
That is pretty awesome, but I also don’t think it’s necessary to think about light speed to solve that problem. Anyone who spends a lot of time debugging networking problems knows that 1ms is unreasonably fast for any communication with a machine more than a couple router hops away, even if it’s physically nearby.
I see that this is getting upvoted, but your example sounds like someone who realizes that a device doesn’t work because it’s not plugged in, and then makes a self-satisfied comment that his knowledge of electromagnetic theory usefully controlled his anticipations in this situation.
In other words, it’s about simple conventional nuts-and-bolts technical knowledge, not an improvement on such knowledge brought by more advanced understanding of anything. There’s no way someone who works in network administration wouldn’t know that a “<1ms” ping coming from around the world is anomalous.
Actually, yes: I think this is correct.
But, I think I would have noticed something was wrong even if I hadn’t worked in IT before. But I’m not certain of that.
This great story is related.
Fair enough. There are indeed many ways in which the folk physics intuition can be improved by internalizing a rule that’s simple enough to explain without math. I admit that my question was too aggressive and snarky.
However, I don’t think any such simple insights will move you any close to understanding either QM or relativity (let alone more advanced topics such as cosmology or the controversies over QM interpretations), which was the topic of the original dispute. I must also point out that your rules are either from classical physics (and thus reasonably close to the relevant folk physics intuitions) or in the form of entirely opaque rules for which you can’t find any justification except for appeal to authority. And I’m certainly not saying that as someone who has “status as a mathematical physicist”; I’m a complete amateur in physics, as I pointed out in an earlier comment. (Also, if you’ve read my earlier comments on LW, you’ll know that I don’t put much inherent weight on the official credentials of expertise bestowed by the present academic system.)
Also, regarding the trends in the Copenhagen vs. MW debate, how much of your opinion is based on understanding of the issues involved, and how much on mere perception of the social dynamics in the field?
Then, like Luke, I must observe that your definition of ‘understanding’ is completely incompatible with mine. Actually, no. I think we’ve moved beyond that now. You are just wrong.
Look at the first example for a start. That claim is based on understanding of (part of) relativity. But that example was only included coincidentally. May I remind you that the examples you demanded were for physics in general (and, for that matter, that the demand was in response to professed understanding of evolution.)? Space considerations stopped me before I also threw in predictions about “two really good clocks, one of which you fly around really fast!”, the fellow who gets into a rocket and outlives his great grandchildren and what happens when you shoot not very many very small things through very small slits.
Question: If the world were wiped out tomorrow leaving a few people and you were the only survivor with any minimal scientific training, could you on your own reconstruct special relativity? If the answer to this is “no” then you don’t understand it.
Note that by this standard I don’t understand QM and I really don’t understand GR and I’m a math grad student. It is ok to say that one doesn’t understand things.
Yes. I already declared this, in a comment you replied to no less.
As I pointed out in my reply to Luke, knowledge of that fact constitutes “understanding of relativity” to the same extent that knowledge of the fact that you can get hurt by sticking your fingers into electrical installations constitutes understanding of electromagnetic theory. It’s just a single fact you know in complete isolation, not a fact that is a part of some broader framework for understanding the world.
To build on this particular example, consider that some things like shadows, reflections, etc., can indeed move faster than light. Or, if you just spin on an office chair, in your rest frame the celestial objects are spinning around you way faster than c. Unless you can explain why such motions are consistent with the “no faster than light” principle, you have nothing more than a literally memorized fact that if it might be useful if something could move faster than c, it can’t happen. It’s a true fact, to be sure, and even a potentially useful one, but still.
The context was about the pop-scientific treatments of modern physics. If you insist on full precision, I will gladly admit that my wording about physics in general was imprecise, since there are indeed simple topics in classical physics where insight can be gained without math, building only on a folk-physics intuition. So please read my statements in their original context.
We aren’t talking about the memorization of a simple fact about pings. We are talking about all the understanding you can have about relativity without having memorized a mathematical equation. This can be used to make the prediction that ping times will never be lower than the aforementioned limit.
You asked for examples of predictions about the world that can be made based on understanding physics minus the math. It would be disingenuous in the extreme to then dismiss all examples of predictions about the world given because they are, in fact, mere predictions about the world and therefore could have been memorized without real understanding.
My first prediction: If you (who I believe professed understanding of at least this much understanding of math) and lukeprog were locked in rooms disconnected from the outside world and given the task of answering this question not only would Luke be able to give an explanation, his explanation would be better than yours. You both know enough about physics to answer and Luke is better at explaining things.
My second prediction: There are many students who, in their physics exams, get all the questions that require mathematics correct and who, when encountering a question like this one, can’t give an answer. Because not only is knowing the math not strictly necessary to answer this question, it isn’t even sufficient.
Additional claim: It has been too long since I studied physics for me to remember all of the mathematics of special relativity. Yet when I did the aforementioned study I also gained a solid grasp of the fundamental principles. With that understanding I could recreate the interesting mathematics from first principles. I am confident of this because I’ve done it before, just for kicks. Because memorizing the math wasn’t enough for me and I wanted to really grasp the science in depth. The way you do that is by knowing the concepts well enough that you could work out the equations for yourself. Because just memorizing them is detail work. (I haven’t got a chance in hell of doing this with GR.)
In the various responses you have been given your claim that you can’t have any understanding of physics without math has been overwhelmingly refuted. All you are left with is “But that understanding isn’t true understanding, true understanding means you remember the math!” To that I reply “No, not all Scotsmen like haggis. You can only tell a true Scotsman by the kilt they are wearing!”
It is time to retreat from a complete rejection of all non-mathematical understanding so that you can express an actually tenable position regarding the limits of how much you can know about physics sans math. Because there really are such limits and I would love to be able to support you in declaring them. But right now you’ve gone overboard and tried to reject even that understanding which can exist. And that brings you to the realm of the absurd and I just can’t support a position which is just obviously factually incorrect.
However, there is indeed a difference between rules memorized in isolation without any additional understanding and, on the other hand, real understanding of physical theories and generating predictions based on them. Just like there’s a difference between understanding electromagnetic theory and knowing a few common-knowledge technical facts on how to deal with electrical devices, there is also a difference between understanding relativity and knowing a few facts that follow from it in isolation.
You are now employing a rhetorical tactic where you try to make this obvious and relevant point look like weaseling, but in reality it is a pertinent and adequate response to your example.
This is sheer rhetoric. You latch onto one point I made, completely ignoring the context and making the most extreme uncharitable interpretation of it (one that is in fact bordering on caricature), all for the greatest rhetorical effect. Instead, a rational approach would be to see if there may be some validity behind my point even if its original statement was imprecise, especially since I readily admitted this imprecision on first objection. Not to mention that your own example is largely irrelevant in the original context, which was about lengthy pop-science works purporting to explain whole physical theories to lay audiences, not about isolated examples such as yours.
In any case, if you think the distinction I outlined above is invalid, or that I am applying it incorrectly, please go ahead and explain why you believe that. If you’re going to latch onto a caricature of what I wrote while treating the discussion as a rhetorical context, I have no further interest in continuing this exchange.
I think it might be helpful for you to taboo the word “real understanding”. It seems like a lot of the disagreement stems from luke and wedrifid being unable to understand what you mean when you use that phrase. To be honest, while I agree with many of your points, I also don’t think I understand what “real understanding” is supposed to mean. Perhaps you could restate your original point without use of the word “understanding”?
Even understanding isn’t well-defined. I at least don’t know of any agreed upon conceptual or mathematical definition. Does Wolfram Alpha understand math? Does a lookup table understand anything?
I think that you really understand a subject if you are able to transfer, teach and artificially recreate the skill or heuristic that is necessary to make useful predictions about the subject. The skill or heuristic needs to be utilizable given limited resources. You also have to be able to abstract and generalize your knowledge about the subject to solve new problems that are not similar to problems previously encountered. Furthermore, you have to be able to prove fundamental mathematical statements and relationships about the subject.
An example would be the ability to teach the game of Go to other agents, transfer your knowledge by writing books on how to play it, and create a Go AI that can play the game by predicting the success of its strategies.
No one else is arguing that it is sufficient, but they are arguing that it is necessary. In this context, I’m going to make a counter prediction: if one did give a test on SR to physics students just learning about it, the ones who answer the chair question correctly will be a proper subset of the ones who can do the mathematical manipulation.
In the more restricted context of highschool calculus or more basic physics this is (from my experience both teaching and tutoring) very much the case. There are students who say things like “I can’t do the math but I understand the concepts” and this just nonsense. The ones who answer the conceptual questions correctly are almost always those who can do the math. There may be kids who can do the formal manipulation and can’t connect to it conceptually but there’s almost no one who can’t handle the symbol pushing who can answer the conceptual level questions. They are too interrelated.
Why do the they believe what they believe? The simplest two explanations I can think of is that they are mistaken about their grasp on the concepts (when you ask them conceptual-level questions, they answer incorrectly), or they are mistaken about their inability to do the math (they feel insecure before quantitative tests, but score high). Is one of these the case?
There’s a combination of both, but there seem to be far more in this first category. To some extent it seems like it is an example of the Dunning-Kruger effect. And I suspect that some of the people here who are convinced that one can understand physics without understanding the math may be also running into Dunning-Kruger issues.
Given that the Dunning-Kruger effect may be a combination of noisy estimates and hidden, different scales (how do we weight the 1500-2500 separate skills involved in driving when deciding who’s in the top 10%), could we extend competence from both ends toward the middle?
What I mean is that an essentially correct but very fuzzy understanding may constrain your anticipations in many different circumstances, but may not tightly constrain them—anybody knows you’re not going to get a <1ms ping from another continent, and a car coming towards you makes a higher pitched noise than it does going away from you, but most people couldn’t diagnose the 500 mile email or calculate the speed of the car from the frequency change.
On the other side, many people just learning physics can carry out the calculus, as long as they have the formula in front of them and the problem’s explicitly stated; but would have trouble generalizing the formula to all the real-life situations in which it’s applicable.
As the intuition becomes more and more precise, it gets closer to math. As the math becomes more and more intuitive, the situations in which its applications are evident grow.
An engineer is never going to be successful with just intuition, because he needs very precise results. But, depending on your use case, generality may be more useful than precision. This seems more useful than trying to decide which one is “real” understanding.
Did I leave out the part where students who couldn’t do all the math answered it correctly? Pardon me.
Then I would make counterfactual money by betting against you.