it seems like asking for trouble to say this:...if you actually mean this:
I really don’t think there’s much of a difference. Yes, there was some rhetorical exaggeration in the first quote; to make it literally accurate I should have written about a necessary prerequisite for comfortably or reliably solving the problem. But since I take it for granted that a solid understanding is the goal, the difference between barely being able to solve it and not being able to solve it at all isn’t particularly significant from my point of view.
There’s a real psychological phenomenon having to do with difficulty that I’m pointing to here, something that goes beyond mere aesthetic preference (as important as that might also be, to me). I have a history of having trouble with problems of this type, at multiple points during my life. Believe me, I’ve been exposed to the standard explanations—the three-row tables and so on. I’ve tried to learn them. I’ve even had sporadic success. But it just doesn’t work reliably. There is some piece of knowledge or cognitive habit that these explanations presuppose that is always left implicit, but that I don’t actually possess. To the extent that I can understand them, it is always by figuring out the solution in my own way and then “translating”—a process which greatly increases the effort required. Failing that, I’m stumbling in the dark and guessing—which sometimes works, and sometimes doesn’t.
If I introduced a new concept every time, my picture of the world would get pretty baroque. It might be better to go in and knock down old, wrong concepts instead, or just clarify the ones I have, or remind myself that yes, they really do apply here
I think most people’s picture of the world is probably not “baroque” enough. However, yes, it’s always possible to overdo anything; the way you avoid having too many concepts is by constantly upgrading the ones you have—making sure they’re as general and powerful as possible.
This is my conception of mathematical research: “concept R&D.” Whereas for most people, it’s about solving problems (by any means, and then you’re done once they’re solved), for me, it’s about developing concepts that make the solution easy.
The idea of solving for a different variable (like you’re doing here with your inverse rates, taking time as a function of distance instead of distance as a function of time) often seems to confuse my students. I wonder if it comes from introducing the concepts of dependent and independent variables too early, and teaching them how to do all kind of operations with functions, but only spending a couple of days talking about how to find the inverse of a function.
My personal saying is bite the abstraction bullet early. My experience is that students have trouble with the very idea of a function, and will try to avoid actually learning it (and similarly abstract ideas) for as long as they can—instead seeking lower-level “shortcuts” that allow them to get the right answer (sometimes). They need to be broken out of this as early as possible, lest they show up in a calculus class incapable of understanding the meaning of “f(x+h)”, as all too often happens.
That’s what I suspect is going on here: what’s confusing them is that they’re not used to thinking about independent and dependent variables at all: they think of d = rt as a rule for manipulating symbols, not as a representation of a relationship between (abstract) quantities. It’s like confusing numbers with numerals. (You can ultimately think of mathematics as rules for maniplulating symbols, but in that case the rules are much more complicated than “d = rt”! That’s their problem: they want the rules to be both concrete and simple, but they can’t have it both ways!)
There is some piece of knowledge or cognitive habit that these explanations presuppose that is always left implicit, but that I don’t actually possess. To the extent that I can understand them, it is always by figuring out the solution in my own way and then “translating”—a process which greatly increases the effort required. Failing that, I’m stumbling in the dark and guessing—which sometimes works, and sometimes doesn’t.
Hm, hell if I know. I can tell you that your talk about speed as a “mapping of times to distances” seemed downright weird to me—I intuitively think of the relationship as being two-way, and to say that you’re going faster means both that you’ll go further in the same length of time and that you’ll travel the same distance in a shorter period. If the price of chocolate bars goes down, I can buy the same number of chocolate bars for less, or I can buy more chocolate bars for the same amount of money.
Is it possible that you love functions too much? Not every situation has a natural choice of independent and dependent variables, after all. It’s not any more meaningful to say that pressure depends on volume than that volume depends on pressure; pV just equals nRT.
It always fascinates me to find how different people are in the types of explanations that work for them. For example, all that “distracting detail” in my mileage story would be motivating detail for me, and for many of my students I’ve found that the more concrete I can make a problem, the more sense it makes to them.
I intuitively think of the relationship as being two-way, and to say that you’re going faster means both that you’ll go further in the same length of time and that you’ll travel the same distance in a shorter period.
I might have thought the same, before the experience of being confused by this problem revealed otherwise.
Do you find the following to be equally easy to answer, intuitively?
(1) You spend one hour going 10 mph and then one hour going 20 mph. What’s your average speed?
(2) You go one mile at 10 mph and then one mile at 20 mph. What’s your average speed?
Perhaps you do; but (at least prior to this discussion) I wouldn’t have.
Not every situation has a natural choice of independent and dependent variables, after all. It’s not any more meaningful to say that pressure depends on volume than that volume depends on pressure; pV just equals nRT.
However, you cannot talk about rates—that is, derivatives—without making a choice: dp/dV is as different from dV/dp as speed is from inverse speed.
Which brings me to the following:
I can tell you that your talk about speed as a “mapping of times to distances” seemed downright weird to me
Well, as it turns out, it’s inherent in the very definition!
The derivative of a function at a point is defined to be the linear map that best approximates the function near that point. So if we have a function x = f(t) that maps times t to distances x, the derivative f’(t) -- the “speed”—at time t is by definition also a mapping from times dt to distances dx (given by the formula dx = f’(t)dt).
Hence, there’s nothing idiosyncratic about my way of thinking. It might be “sophisticated”, but it’s hardly “weird”. Of course, it has been my repeated experience that perspectives labeled “sophisticated”, “advanced”, or “abstract” are those that I tend to find most natural.
However, I think the exoticity here is actually pretty minimal. Consider how people visually represent speed: they usually draw arrows whose length represents the distance traveled in a fixed time interval. To represent a speed that is twice as fast, they will make the arrow twice as long, not half as long.
Do you find the following to be equally easy to answer, intuitively?
I am equally confident that I can give a right answer to them both, but one of them makes the calculations easier to do in my head. Here’s what I might say if you sprung each of these on me:
(1) You spend one hour going 10 mph and then one hour going 20 mph. What’s your average speed?
“I go ten miles and then twenty miles. 30 miles/2h = 15 mph”
On the SAT, problems are arranged from easiest to hardest, not by the difficulty of the concepts involved, but according to how many students get them wrong. If two questions use the same concepts and procedures, but one gives an answer that “looks right” (is a whole number, for example), there will be a difficulty difference between them. This one would be right at the beginning of the SAT, because it’s the same answer you get by doing the problem in a naive way: you see two numbers and the word “average”, so you just average them.
(2) You go one mile at 10 mph and then one mile at 20 mph. What’s your average speed?
“I take 1⁄10 of an hour and then 1⁄20 of an hour, so that’s two miles over, um, 3⁄20 hours… so 40⁄3 mph? Yeah, I guess that’s between 10 and 20.”
The math is a little trickier, and the answer isn’t a whole number, so I’m sure it would take a few more seconds to come up with, but I did the problem in basically the same way, by dividing distance by time. (Of course, I’m assuming that given the distances involved, you know how to get the times, but any high school chemistry student knows you can flip your conversion factors if you need to.) This one would definitely go at the end of the SAT, not only because of the weirdness of the answer(+), but because it requires you to recognize exactly what question is being asked.
So intuitively I find neither problem harder to understand. I know that going an hour at 20 mph is totally different from going a mile at 20 mph. Just about everybody knows that, if they think about it. The difference is that you can get a right answer on the first problem without understanding it.
However, you cannot talk about rates—that is, derivatives—without making a choice: dp/dV is as different from dV/dp as speed is from inverse speed.
Well, yes, you would have to differentiate with respect to one or the other variable, but you can do either just as well; the relationship doesn’t force you. And having found your dp/dV, you could flip it over to get dV/dp. This seems like it might be a pitfall of function notation, actually; if I tell you that V(p) = nRT/p, you can tell me that V’(p) = -nRT/p^2, but you’re forced to differentiate with respect to p, and it’s probably not so easy to make the jump to seeing that dV/dp = -V/p and dp/dV = -p/V. Maybe it’s no coincidence that my Calc I students sometimes learn how to perform the chain rule, but don’t figure out what it actually means until they learn to do implicit differentiation? I dunno, just thinking aloud here. (thinking a-type?)
Well, as it turns out, it’s inherent in the very definition!
Is that the only way to define a derivative? I know it’s one way, and it works, but is that the only way?
However, I think the exoticity here is actually pretty minimal. Consider how people visually represent speed: they usually draw arrows whose length represents the distance traveled in a fixed time interval. To represent a speed that is twice as fast, they will make the arrow twice as long, not half as long.
Not sure this is a good example. It’s a lot more natural to have lengths of arrows correspond to distances than to times… since, you know, they actually are distances. But if you consider that people often say “coming quick” to mean “coming soon”(++), it seems like there’s an instinctive association between higher speeds and shorter times as well.
(+)You have no idea how much kids hate fractions. When they see fractions they just don’t even try.
(++)Is this a Southern thing? When I was a kid people would say “Christmas is coming quickly!” and I would think “It’s not coming any more quickly than it was before. It’s coming at a rate of one day per day.”
(2) You go one mile at 10 mph and then one mile at 20 mph. What’s your average speed?
“I take 1⁄10 of an hour and then 1⁄20 of an hour, so that’s two miles over, um, 3⁄20 hours...
Very interesting. When I read this, it struck me as a “good-at-math” person’s thought process, and after reflecting on it, I think I know why:
You went directly from “one mile at 10 mph” to “1/10 of an hour”—skipping right over what is for me the most important step in the whole solution: the conversion from 10 mph to 1⁄10hpm. I’m guessing you didn’t even realize there was a step missing here, did you?
It’s a fairly abstract step, of course: it involves explicitly performing an operation on rates, which as discussed previously, are mappings (functions). But the point is, if you talk to me about “one mile at 10 mph”, my natural, intuitive reaction is “ERROR: SYNTAX”. The operation “10 mph” does not accept “one mile” as an input (nor vice-versa: “one mile” doesn’t accept “10 mph” either). A quantity with “mph” needs a number of “hours”; a quantity with “miles” needs something with “miles” in the denominator.
(Strictly speaking, thanks to a mathematical construction known as the tensor product, anything can operate on anything else—but the result will in general be a new kind of thing. For example, if mph acts on miles, the result will be labeled miles^2/hour.)
Now, you write:
Of course, I’m assuming that given the distances involved, you know how to get the times, but any high school chemistry student knows you can flip your conversion factors if you need to.
but “knowing that you can” do something (or even “knowing how”) is different from being able to do it without explicitly thinking about it as a separate step!
It’s interesting that this parenthetically-mentioned assumption of yours is, for me, the entire sticking point, and the subject of this post.
Now that you mention high-school chemistry, let me tell you another interesting thing: I used to be on the other side of this discussion, once upon a time—or so it would have appeared. That is, I used to ridicule high-school chemistry for this “dimensional analysis” business, satirizing it by elaborately solving problems such as “if there are 5 apples in each barrel, and you have 6 barrels, how many apples do you have?” via “conversion factors” and cancellation of “barrels”. It seemed to me that this was just a technique for mechanizing these problems for the benefit of slow students who couldn’t just see that obviously if you have 6 barrels of 5 apples, you must have 30 apples in total. (Perhaps exactly analgously to the way that you, unlike me, can just “see” that if you go one mile at 10 mph, you took 1⁄10 of an hour.)
I now realize, however, that that wasn’t my true rejection. What I actually objected to about “dimensional analysis” was that it was an ad-hoc, discipline-specific kind of mathematics that chemistry people were using which lacked a theoretical justification in math class. The latter, you see, had never provided any conceptual foundation for treating “5″, “5 apples”, and “5 barrels” as different kinds of mathematical objects. Sure, there were expressions with “unlike terms” (such as x, y, and xy) that you couldn’t just “add together”, but those unlike terms always stood for different amounts of the same kind of thing: abstract numbers, or numbers reprenting one particular kind of quantity. So where did these chemistry people get the idea that they were allowed to perform symbolic algebra on units, which after all aren’t numbers at all?
It was for the same reason that I resisted vectors, when they were introduced in physics class before I had been properly exposed to the mathematical subject of linear algebra: you’re not allowed to invent new mathematics outside of mathematics class (which in my mind serves as the Department of Anti-Compartmentalization).
Now if you say “What? How crippling that would be to physics and chemistry!”, you’re missing the point. The problem wasn’t with physics and chemistry, the problem was with math class. (Indeed, often physics and chemistry were too accomodating to the lack of mathematical prerequisites, such as in avoiding calculus, which is utterly silly.) The logical foundation for “dimensional analysis” is multilinear algebra, and so I should have learned multilinear algebra in math class before being asked to do “dimensional analysis” in chemistry class.
So, you can see that my apparently having been on the other side, once upon a time, was in fact nothing other than an instance of the same thing: a need for the proper theoretical foundations to be in place before I can “understand” something.
Is that the only way to define a derivative? I know it’s one way, and it works, but is that the only way?
It’s the most general way(+), hence the best. All other ways are either equivalent to this (and just as abstract) or don’t make sense outside of a restricted setting.
(+) Perhaps not technically true, but close enough to the truth for our purposes here.
So to summarize, basically komponisto needs to learn to always think of bijections as always accompanied by their inverses, in particular when that bijection is given by multiplication by a nonzero real number[0], as will always be the case when the mapping in question is a nonzero derivative and you’re only working in one dimension, and more generally to not always think of relations as one-way functions?
OK, but it’s still important to understand how this plays out in the 1-dimensional case. These aren’t incompatible, one’s just a special case. Though I’m not seeing the relevance of that particular isomorphism here, as I don’t see just what it is here that would naturally be interpreted as an element of that first space in the first place?
OK, but it’s still important to understand how this plays out in the 1-dimensional case
Well, yes! That’s what I seek to do, as opposed to regarding the 1-dimensional case as a separate magisterium, compartmentalized away from the general case.
I don’t see just what it is here that would naturally be interpreted as an element of that first space in the first place?
Here V is distances, and W is times. If something has the label “distance”, it’s an element of V; if it has the label “time”, it’s an element of W; and if it has the label “time^-1”, it’s an element of W. Something with the label “distance/time” is then an element of
 .
Here V is distances, and W is times. If something has the label “distance”, it’s an element of V; if it has the label “time”, it’s an element of W; and if it has the label “time^-1”, it’s an element of W*.
Oh, OK. For some reason I was thinking the scaling was wrong for that to work. Of course, if you travel 3 miles in 2 hours, that’s 3 mi \otimes 1⁄2 h^-1, not 3 mi \otimes 2 h^-1...
That’s right: (1/2)h^-1 is the map that takes a time and gives its coordinate with respect the basis {2h}, which is the one being used here to define the speed.
(General rule: a/b means you input b to get a. So, since our coordinate-computing map should input 2h and output 1, it is written 1/(2h), or (1/2)h^-1.)
I really don’t think there’s much of a difference. Yes, there was some rhetorical exaggeration in the first quote; to make it literally accurate I should have written about a necessary prerequisite for comfortably or reliably solving the problem. But since I take it for granted that a solid understanding is the goal, the difference between barely being able to solve it and not being able to solve it at all isn’t particularly significant from my point of view.
There’s a real psychological phenomenon having to do with difficulty that I’m pointing to here, something that goes beyond mere aesthetic preference (as important as that might also be, to me). I have a history of having trouble with problems of this type, at multiple points during my life. Believe me, I’ve been exposed to the standard explanations—the three-row tables and so on. I’ve tried to learn them. I’ve even had sporadic success. But it just doesn’t work reliably. There is some piece of knowledge or cognitive habit that these explanations presuppose that is always left implicit, but that I don’t actually possess. To the extent that I can understand them, it is always by figuring out the solution in my own way and then “translating”—a process which greatly increases the effort required. Failing that, I’m stumbling in the dark and guessing—which sometimes works, and sometimes doesn’t.
I think most people’s picture of the world is probably not “baroque” enough. However, yes, it’s always possible to overdo anything; the way you avoid having too many concepts is by constantly upgrading the ones you have—making sure they’re as general and powerful as possible.
This is my conception of mathematical research: “concept R&D.” Whereas for most people, it’s about solving problems (by any means, and then you’re done once they’re solved), for me, it’s about developing concepts that make the solution easy.
My personal saying is bite the abstraction bullet early. My experience is that students have trouble with the very idea of a function, and will try to avoid actually learning it (and similarly abstract ideas) for as long as they can—instead seeking lower-level “shortcuts” that allow them to get the right answer (sometimes). They need to be broken out of this as early as possible, lest they show up in a calculus class incapable of understanding the meaning of “f(x+h)”, as all too often happens.
That’s what I suspect is going on here: what’s confusing them is that they’re not used to thinking about independent and dependent variables at all: they think of d = rt as a rule for manipulating symbols, not as a representation of a relationship between (abstract) quantities. It’s like confusing numbers with numerals. (You can ultimately think of mathematics as rules for maniplulating symbols, but in that case the rules are much more complicated than “d = rt”! That’s their problem: they want the rules to be both concrete and simple, but they can’t have it both ways!)
Hm, hell if I know. I can tell you that your talk about speed as a “mapping of times to distances” seemed downright weird to me—I intuitively think of the relationship as being two-way, and to say that you’re going faster means both that you’ll go further in the same length of time and that you’ll travel the same distance in a shorter period. If the price of chocolate bars goes down, I can buy the same number of chocolate bars for less, or I can buy more chocolate bars for the same amount of money.
Is it possible that you love functions too much? Not every situation has a natural choice of independent and dependent variables, after all. It’s not any more meaningful to say that pressure depends on volume than that volume depends on pressure; pV just equals nRT.
It always fascinates me to find how different people are in the types of explanations that work for them. For example, all that “distracting detail” in my mileage story would be motivating detail for me, and for many of my students I’ve found that the more concrete I can make a problem, the more sense it makes to them.
I might have thought the same, before the experience of being confused by this problem revealed otherwise.
Do you find the following to be equally easy to answer, intuitively?
(1) You spend one hour going 10 mph and then one hour going 20 mph. What’s your average speed?
(2) You go one mile at 10 mph and then one mile at 20 mph. What’s your average speed?
Perhaps you do; but (at least prior to this discussion) I wouldn’t have.
However, you cannot talk about rates—that is, derivatives—without making a choice: dp/dV is as different from dV/dp as speed is from inverse speed.
Which brings me to the following:
Well, as it turns out, it’s inherent in the very definition!
The derivative of a function at a point is defined to be the linear map that best approximates the function near that point. So if we have a function x = f(t) that maps times t to distances x, the derivative f’(t) -- the “speed”—at time t is by definition also a mapping from times dt to distances dx (given by the formula dx = f’(t)dt).
Hence, there’s nothing idiosyncratic about my way of thinking. It might be “sophisticated”, but it’s hardly “weird”. Of course, it has been my repeated experience that perspectives labeled “sophisticated”, “advanced”, or “abstract” are those that I tend to find most natural.
However, I think the exoticity here is actually pretty minimal. Consider how people visually represent speed: they usually draw arrows whose length represents the distance traveled in a fixed time interval. To represent a speed that is twice as fast, they will make the arrow twice as long, not half as long.
I am equally confident that I can give a right answer to them both, but one of them makes the calculations easier to do in my head. Here’s what I might say if you sprung each of these on me:
“I go ten miles and then twenty miles. 30 miles/2h = 15 mph”
On the SAT, problems are arranged from easiest to hardest, not by the difficulty of the concepts involved, but according to how many students get them wrong. If two questions use the same concepts and procedures, but one gives an answer that “looks right” (is a whole number, for example), there will be a difficulty difference between them. This one would be right at the beginning of the SAT, because it’s the same answer you get by doing the problem in a naive way: you see two numbers and the word “average”, so you just average them.
“I take 1⁄10 of an hour and then 1⁄20 of an hour, so that’s two miles over, um, 3⁄20 hours… so 40⁄3 mph? Yeah, I guess that’s between 10 and 20.”
The math is a little trickier, and the answer isn’t a whole number, so I’m sure it would take a few more seconds to come up with, but I did the problem in basically the same way, by dividing distance by time. (Of course, I’m assuming that given the distances involved, you know how to get the times, but any high school chemistry student knows you can flip your conversion factors if you need to.) This one would definitely go at the end of the SAT, not only because of the weirdness of the answer(+), but because it requires you to recognize exactly what question is being asked.
So intuitively I find neither problem harder to understand. I know that going an hour at 20 mph is totally different from going a mile at 20 mph. Just about everybody knows that, if they think about it. The difference is that you can get a right answer on the first problem without understanding it.
Well, yes, you would have to differentiate with respect to one or the other variable, but you can do either just as well; the relationship doesn’t force you. And having found your dp/dV, you could flip it over to get dV/dp. This seems like it might be a pitfall of function notation, actually; if I tell you that V(p) = nRT/p, you can tell me that V’(p) = -nRT/p^2, but you’re forced to differentiate with respect to p, and it’s probably not so easy to make the jump to seeing that dV/dp = -V/p and dp/dV = -p/V. Maybe it’s no coincidence that my Calc I students sometimes learn how to perform the chain rule, but don’t figure out what it actually means until they learn to do implicit differentiation? I dunno, just thinking aloud here. (thinking a-type?)
Is that the only way to define a derivative? I know it’s one way, and it works, but is that the only way?
Not sure this is a good example. It’s a lot more natural to have lengths of arrows correspond to distances than to times… since, you know, they actually are distances. But if you consider that people often say “coming quick” to mean “coming soon”(++), it seems like there’s an instinctive association between higher speeds and shorter times as well.
(+)You have no idea how much kids hate fractions. When they see fractions they just don’t even try.
(++)Is this a Southern thing? When I was a kid people would say “Christmas is coming quickly!” and I would think “It’s not coming any more quickly than it was before. It’s coming at a rate of one day per day.”
Very interesting. When I read this, it struck me as a “good-at-math” person’s thought process, and after reflecting on it, I think I know why:
You went directly from “one mile at 10 mph” to “1/10 of an hour”—skipping right over what is for me the most important step in the whole solution: the conversion from 10 mph to 1⁄10 hpm. I’m guessing you didn’t even realize there was a step missing here, did you?
It’s a fairly abstract step, of course: it involves explicitly performing an operation on rates, which as discussed previously, are mappings (functions). But the point is, if you talk to me about “one mile at 10 mph”, my natural, intuitive reaction is “ERROR: SYNTAX”. The operation “10 mph” does not accept “one mile” as an input (nor vice-versa: “one mile” doesn’t accept “10 mph” either). A quantity with “mph” needs a number of “hours”; a quantity with “miles” needs something with “miles” in the denominator.
(Strictly speaking, thanks to a mathematical construction known as the tensor product, anything can operate on anything else—but the result will in general be a new kind of thing. For example, if mph acts on miles, the result will be labeled miles^2/hour.)
Now, you write:
but “knowing that you can” do something (or even “knowing how”) is different from being able to do it without explicitly thinking about it as a separate step!
It’s interesting that this parenthetically-mentioned assumption of yours is, for me, the entire sticking point, and the subject of this post.
Now that you mention high-school chemistry, let me tell you another interesting thing: I used to be on the other side of this discussion, once upon a time—or so it would have appeared. That is, I used to ridicule high-school chemistry for this “dimensional analysis” business, satirizing it by elaborately solving problems such as “if there are 5 apples in each barrel, and you have 6 barrels, how many apples do you have?” via “conversion factors” and cancellation of “barrels”. It seemed to me that this was just a technique for mechanizing these problems for the benefit of slow students who couldn’t just see that obviously if you have 6 barrels of 5 apples, you must have 30 apples in total. (Perhaps exactly analgously to the way that you, unlike me, can just “see” that if you go one mile at 10 mph, you took 1⁄10 of an hour.)
I now realize, however, that that wasn’t my true rejection. What I actually objected to about “dimensional analysis” was that it was an ad-hoc, discipline-specific kind of mathematics that chemistry people were using which lacked a theoretical justification in math class. The latter, you see, had never provided any conceptual foundation for treating “5″, “5 apples”, and “5 barrels” as different kinds of mathematical objects. Sure, there were expressions with “unlike terms” (such as x, y, and xy) that you couldn’t just “add together”, but those unlike terms always stood for different amounts of the same kind of thing: abstract numbers, or numbers reprenting one particular kind of quantity. So where did these chemistry people get the idea that they were allowed to perform symbolic algebra on units, which after all aren’t numbers at all?
It was for the same reason that I resisted vectors, when they were introduced in physics class before I had been properly exposed to the mathematical subject of linear algebra: you’re not allowed to invent new mathematics outside of mathematics class (which in my mind serves as the Department of Anti-Compartmentalization).
Now if you say “What? How crippling that would be to physics and chemistry!”, you’re missing the point. The problem wasn’t with physics and chemistry, the problem was with math class. (Indeed, often physics and chemistry were too accomodating to the lack of mathematical prerequisites, such as in avoiding calculus, which is utterly silly.) The logical foundation for “dimensional analysis” is multilinear algebra, and so I should have learned multilinear algebra in math class before being asked to do “dimensional analysis” in chemistry class.
So, you can see that my apparently having been on the other side, once upon a time, was in fact nothing other than an instance of the same thing: a need for the proper theoretical foundations to be in place before I can “understand” something.
It’s the most general way(+), hence the best. All other ways are either equivalent to this (and just as abstract) or don’t make sense outside of a restricted setting.
(+) Perhaps not technically true, but close enough to the truth for our purposes here.
So to summarize, basically komponisto needs to learn to always think of bijections as always accompanied by their inverses, in particular when that bijection is given by multiplication by a nonzero real number[0], as will always be the case when the mapping in question is a nonzero derivative and you’re only working in one dimension, and more generally to not always think of relations as one-way functions?
[0]Or in other words, “division is available”...
Who said I think of relations as one-way functions? I think of them as what they are, namely subsets of the Cartesian product.
As for division, I’m very happy to trade it in for an intuitive understanding of the canonical monomorphism
(which, in concrete terms, means the ability to view something labeled “mph” as a linear map from the space of times to the space of distances).
OK, but it’s still important to understand how this plays out in the 1-dimensional case. These aren’t incompatible, one’s just a special case. Though I’m not seeing the relevance of that particular isomorphism here, as I don’t see just what it is here that would naturally be interpreted as an element of that first space in the first place?
Well, yes! That’s what I seek to do, as opposed to regarding the 1-dimensional case as a separate magisterium, compartmentalized away from the general case.
Here V is distances, and W is times. If something has the label “distance”, it’s an element of V; if it has the label “time”, it’s an element of W; and if it has the label “time^-1”, it’s an element of W. Something with the label “distance/time” is then an element of  .
Oh, OK. For some reason I was thinking the scaling was wrong for that to work. Of course, if you travel 3 miles in 2 hours, that’s 3 mi \otimes 1⁄2 h^-1, not 3 mi \otimes 2 h^-1...
That’s right: (1/2)h^-1 is the map that takes a time and gives its coordinate with respect the basis {2h}, which is the one being used here to define the speed.
(General rule: a/b means you input b to get a. So, since our coordinate-computing map should input 2h and output 1, it is written 1/(2h), or (1/2)h^-1.)