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Measurement Reprint edition by Lockhart, Paul (2014) Paperback
For seven years, Paul Lockhart’s A Mathematician’s Lament enjoyed a samizdat-style popularity in the mathematics underground, before demand prompted its 2009 publication to even wider applause and debate. An impassioned critique of K–12 mathematics education, it outlined how we shortchange students by introducing them to math the wrong way. Here Lockhart offers the positive side of the math education story by showing us how math should be done. Measurement offers a permanent solution to math phobia by introducing us to mathematics as an artful way of thinking and living.
In conversational prose that conveys his passion for the subject, Lockhart makes mathematics accessible without oversimplifying. He makes no more attempt to hide the challenge of mathematics than he does to shield us from its beautiful intensity. Favoring plain English and pictures over jargon and formulas, he succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable. His elegant discussion of mathematical reasoning and themes in classical geometry offers proof of his conviction that mathematics illuminates art as much as science.
Lockhart leads us into a universe where beautiful designs and patterns float through our minds and do surprising, miraculous things. As we turn our thoughts to symmetry, circles, cylinders, and cones, we begin to see that almost anyone can “do the math” in a way that brings emotional and aesthetic rewards. Measurement is an invitation to summon curiosity, courage, and creativity in order to experience firsthand the playful excitement of mathematical work.
In conversational prose that conveys his passion for the subject, Lockhart makes mathematics accessible without oversimplifying. He makes no more attempt to hide the challenge of mathematics than he does to shield us from its beautiful intensity. Favoring plain English and pictures over jargon and formulas, he succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable. His elegant discussion of mathematical reasoning and themes in classical geometry offers proof of his conviction that mathematics illuminates art as much as science.
Lockhart leads us into a universe where beautiful designs and patterns float through our minds and do surprising, miraculous things. As we turn our thoughts to symmetry, circles, cylinders, and cones, we begin to see that almost anyone can “do the math” in a way that brings emotional and aesthetic rewards. Measurement is an invitation to summon curiosity, courage, and creativity in order to experience firsthand the playful excitement of mathematical work.
Paperback
First published August 20, 2012
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About the author
Paul Lockhart
5 books209 followersPaul Lockhart became interested in mathematics when he was 14 (outside the classroom, he points out). He dropped out of college after one semester to devote himself exclusively to math. Based on his own research he was admitted to Columbia, received a PhD, and has taught at major universities, including Brown University and UC Santa Cruz. Since 2000 he has dedicated himself to "subversively" teaching grade-school math at St. Ann's School in Brooklyn, New York.
Librarian Note: There is more than one author in the Goodreads database with this name.
Librarian Note: There is more than one author in the Goodreads database with this name.
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January 11, 2020
Green Bananas: Quantification and Its Discontents
I’m at that stage in my life in which I hesitate to buy green bananas lest they go to waste: I might not be around to eat them before they’re fully ripe. The color green is a significant measure of not just the bananas but my life, at least what remains of it. I don’t know if anyone else uses this measure; it may have only personal appeal. Certainly Lockhart would reject it; he wouldn’t see the point. The title Lockhart has chosen for his book is somewhat misleading. He has nothing to say about the possible significance of green bananas, nor of many other scales of measurement. This is acutely disappointing. It also reflects a fundamental error in what he thinks constitutes measurement. This is acutely dangerous.
Lockhart makes a distinction early on between the imaginary world of maths and the physical world of objects and phenomena, like green bananas. He is only concerned with the former - Euclidean lengths, areas and volumes. Bananas of any kind, or the drop of curtains in the living room, and the road miles between Devon and Scotland - the kinds of things folk actually are concerned to measure - are not things that keep him up at night.
Lockhart does make geometry - the analysis of space - his focal point. But only after making geometry an entirely arithmetic affair. That is, he confines the things he is concerned about measuring to objects in the mathematical world. This conflation of maths and geometry is an old philosophical trick, and clearly has its uses - among others the capacity to develop all sorts of clever ways to estimate shapes, quantities, and complicated volumes.
But this capacity comes at a price: it makes it look as if measurement is a process of documenting the properties of an object - because that’s what happens in mathematical geometry. Since the objects of mathematical measurements are entirely fictional - composed of dimensionless points, lines without width and algebraically perfect dimensions - the properties of these objects are defined as part of their conceptual existence. A mathematical triangle actually does have internal angles summing to precisely 180 degrees - because that is part of its definition (Although triangles in non-Euclidian geometry can have less than 180 degrees, but that’s another story).
The fact that there are exactly half the angular arc-degrees of a circle in such a triangle is a property of mathematical triangles simply because they are defined to have that property. We make them that way. Just like we construct perfect circles, line segments, and conceptual three dimensional spaces. Such objects do not occur ‘naturally’, they are the sum of their defined properties. Their properties are limited to characteristics that can be derived from a set of axioms, which act as divine words of creation. Things don’t get any more knowable than that.
Obviously the physical world is different from this mathematical universe. The most important difference is one that is typically ignored in most discussions of measurement - namely that the ‘properties’ being measured in the physical world are not in any way inherent in the objects or phenomena being measured. Real objects and phenomena are not defined by their measurements, merely described. There are no axioms from which the properties of a real object, say the greenness of a banana, can be derived.
Unlike mathematical objects, real objects can be measured in an infinite number of ways but their ‘properties’ are in fact entirely unknown. To speak of such properties is an innocuous conceit in everyday life, but creates gross misconceptions when we act as if our words had the same power in the physical world. They don’t and we should know better.
To put the matter succinctly, physical measurements do not, indeed cannot, record the properties of phenomena. Rather, physical measurements are constituted by the assignment of the phenomena to a place on an ordinal scale (usually but not necessarily involving numbers) called a metric. A metric is merely a rule for ordering which we impose on things in the physical world. We Impose this ordering; it does not cut the world, as it were, at its joints. This we call quantification.
Quantification does not establish the properties of the object of the physical world in any sense. The object is in fact made a property of the metric through quantification. We adopt, as it were, things from the physical world into the mathematical world when we measure them. Being a mathematical object the properties of a metric are known precisely; and we can change the definition of this mathematical object to include the physical object as long as we have regard to the basic axioms on which the mathematical world is constructed. The position assigned to an object on the metric, its order, its rank, or its number, is not a property of the object but a consequence of the axioms of the metric.
I know from experience that this proposition of the measured object as a property of the metric on which it is measured is unfamiliar, disconcerting, and confusing. The remainder of this review (essay really) is my attempt to make the proposition less of all these. The reason for my making the effort, and for the perhaps greater effort of the reader, is that this proposition has significant practical as well as philosophical implications for metrology, the study of measurement. These implications are obscured to the point of disappearance in Lockhart’s book; so it is a convenient foil against which to establish the proposition as valid and useful.
The impossibility in principle of measuring the inherent properties in the physical world was shown by Immanuel Kant more the two centuries ago. No one has successfully refuted his proof that what he called the Thing-in-Itself is permanently and inevitably inaccessible to human language, including the language of mathematics. We are, as Plato expressed the situation poetically, forced to know the world as shadows on the wall of a cave.
This limitation isn’t a matter of perceptual ability, for example a lack of instrumentation or appropriate technology. Phenomena can be quantified at any level of technological development and they, that is their inherent properties, will nevertheless remain essentially unknown. No matter how accurate our observations, we are still observing shadows.
This is another way of saying that even an infinite description would not capture the essence of a rose, a poem, a star... or a banana. Even more disturbing is that the descriptions we make through measurement are not simply incomplete. They are, in a sense, lies, falsehoods which have the capacity to not just distort reality, but also to hide it entirely. We can end up investigating not the shadows but the cave wall on which the shadows appear. More importantly, measurement can be used to manipulate physical reality for hidden purposes - commercial, ideological and political. The way to prevent these potential falsehoods from affecting our judgments is to recognise what actually goes on when we measure, when we quantify something, in the physical world.
On the face of it, Kant’s claim appears counter-intuitive, if not just plain silly. Measurements may be erroneous sometimes, but that doesn’t make them lies, only corrigible mistakes. We can, we know, get more accurate by being more careful, eliminating bias and just generally paying closer attention to the operational procedures of observing, recording and reporting our measurements. What could justify the idea that we cannot know the properties of something we can measure? Isn’t that what we’re measuring - properties? We are able to estimate the depth of the ocean, the temperature of the air and the density of building materials.
Actually this isn’t what’s happening. The water, the air, the slab of marble are mute forces that act on our senses either directly or through our technology. But even the way we talk about measurement attributes our measurements to the ocean, the air, building material as something which is part of them, their attributes, their properties, not our senses.
This is the fundamental issue Kant was getting at. The issue is not the accuracy of a measurement on any metric - for example of depth, heat, or impenetrability - but of the suitability of the metric itself. Which metric to use to measure an object is the subject of what Kant called epistemology, the study of what metrics we use to impose on, colloquially to ‘represent’, a reality we cannot comprehend in any other way.
Epistemology got somewhat side-tracked over the last two centuries, concentrating on things like methods of research and the necessary rules for valid inference. It turns out that no one has been able to discover the singular methods or rules by which scientific advances take place. In fact, not infrequently, the biggest breakthroughs in science occur by violating established methods and ignoring apparently fixed rules of inductive logic. (But even that doesn’t constitute a rule - sometimes following the rules also generates startling results.)
One way to recast epistemology as a fruitful area of study is to recognise that the central problem which must be addressed is not one of method or procedure but one of numbers, specifically metrics, in their relation to the physical world. Lockhart ignores the non-numerical world entirely, thus avoiding the issues of epistemology. This is scientifically disingenuous, mainly because it makes the choice of the metric of measurement seem either trivial and therefore of no fundamental significance in our understanding of the universe.
Numbers are fictional entities. By this I mean no disrespect to numbers, nor do I deny their existence. By fictional I mean that numbers are stories. We have stories about how numbers have arisen, the evolution of their symbology and some very complex and sophisticated stories about how numbers are related to one another. These latter stories are called number theory, and they define the properties of numbers just as all mathematical objects are defined - by axioms and their implications.
So, although we don’t know everything there is to know about numbers - the axiomatic implications are infinite - we know quite a bit. Like their properties, the way they interact, their limitations and their capabilities. For example we know that numbers are infinite (in fact they are of various increasing orders of infinity). We also know that they are they are infinitely dense, that is, there is always a number to be found between any two numbers. We also know, perhaps somewhat disconcertingly, that there are many numbers (in fact there are infinitely more than other numbers) which are impossible to express entirely by numbers. These are called irrational numbers and they cannot be completely stated even with an infinite number of digits or decimal places.
Perhaps the most basic example of the error in presuming that quantitative measurement establishes properties of an object or phenomenon is provided by mathematical geometry itself. A fundamental proof in Euclidean geometry is that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. This applies to all right triangles. In particular it applies to the unit triangle, the two sides of which have the measure length of one (the numeraire - feet centimeters, finger-widths, etc - don’t matter). The hypotenuse of this triangle, as many of us will remember from our school days, is therefore the square root of two (the square root of (1 squared + 1 squared)).
The square root of two is one of those indeterminate irrational numbers. It can’t be stated exactly, only approximately. Yet any physical triangle of unit dimension will have a definite length. Whatever that real length is, it simply is not the same as that in the mathematical world, namely the indeterminate square root of two. In short the square root of two cannot be a property of a physical triangle. The triangle, whether mathematical or real, is a property of the metric of measurement - in both cases, even if less obviously so in the geometrical abstraction.
It gets worse however. Certain irrational numbers, like the number pi that links the diameter and circumference of a circle can’t even be expressed algebraically. They are called transcendentals. Transcendentals are of fundamental importance in mathematics even in basic geometric analysis. And there are an infinite number of these too. Yet they too are impossible to find in the physical world. They appear as sort of an alien presence to direct our attention around the cave but they never show up in it.
The difference between the mathematical world and the physical world has caused immense practical problems for the unwary thinker who forgets that mathematical properties don’t transfer to the physical world. The notorious paradox of Zeno is one of the oldest and most persistent confusions to plague the philosophically oriented scientist. Zeno’s infamous sprinter could apparently never reach his finish line for the purported reason that he would first have to run half the distance of his race, then half the remaining distance, the half that, and an infinite further number of further halved distances. In a finite time frame, therefore, the runner cannot ever finish the race. Entirely logical and obviously entirely incorrect.
There are various way to deflate the annoying Zeno. But the simplest is to merely point of that the infinite density of the points on the mathematical line from start to finish is not a property of the course itself. Zeno pretended it was and that his runner had the property of being at successive positions along this infinitely dense line. Once it is recognised that in fact the runner never has these positional properties but is himself being assigned as a property of Zeno’s metric of distance, the paradox disappears.
Advances in modern science show how the defined characteristics of the mathematical world don’t ‘map’ or correspond on a one to one basis with the physical world. Quantum theory for example posits the existence of shortest distance, smallest mass, and even briefest time periods. The infinite density of numbers means that there are effectively ‘gaps’ in physical reality which cannot be found in numbers.
Lockhart knows there is a fundamental discontinuity between the mathematical and the physical world: “Mathematical reality... is imaginary,” he says, “It can be as simple and pretty as I want it to be. I get to have all those perfect things I can’t have in real life. I will never hold a circle in my hand, but I can hold one in my mind. And I can measure it. Mathematical reality is a beautiful wonderland of my own creation, and I can explore it and think about it and talk about it with my friends.” But even beauty has its limits as a criterion for appropriate action.
Lockhart doesn’t take this discontinuity between the physical and the mathematical very seriously at all. “What is measuring? What exactly are we doing when we measure something?” He asks. “I think it is this: we are making a comparison. We are comparing the thing we are measuring to the thing we are measuring it with. In other words, measuring is relative. Any measurement that we make, whether real or imaginary, will necessarily depend on our choice of measuring unit. In the real world, we deal with these choices every day... The question is, what sort of units do we want for our imaginary mathematical universe?... One way to think of it is that we simply aren’t going to have any units at all, just proportions. Since there isn’t a natural choice of unit for measuring length, we won’t have one.”
By disregarding the physical world and its differences from the mathematical world, the issue of the right metric of measurement is first reduced to a question of the ‘units’ of measurement (the numeraire) and then to the simple procedure of comparison. While he’s certainly correct to point out the measurement is essentially comparison he can’t see that the choice of what to compare is of crucial importance. He can’t even see the metric.
For example when I was a child I collected British postage stamps. I was fascinated not just by their design and content but by their relationships to one another. The image below is of four such stamps compared in four different ways, that is on four different metrics. Although each metric does have a numeraire, a distinct unit of some monetary amount associated with it, these units are actually of trivial importance. Just as Lockhart suggests, each gets along just fine as a simple comparison. Metrics are rules for ordering. Here are several possible rules for ordering the stamps: [click on the link since I haven’t figured out how to make the html work. Apologies to all]
https://btcloud.bt.com/web/app/share/...
These are obviously four quite different comparisons and result in uniquely different ordering. All the comparisons are ‘real’ in the sense that there is nothing in philately which defines an inherent property or which comparisons are allowed. They just happen to be comparisons that someone might want to make. The position of stamps on each scale is clearly not anything to do with the inherent properties of postage stamps. Yet according to Lockhart (and me) these are measurements. All are correct but none pretend to anything but what they actually are - measures of relative value.
All physical measurements are measurements of relative value. We wouldn’t bother to make them unless they were. They are an ‘appreciation’ of an object or phenomenon in light of a specific intent not a statement of the character of the object or phenomenon in itself. This value is not expressed in terms of the units of an arbitrary numeraire (dollars, pounds, utils etc.), which as Lockhart points out might be irrelevant anyhow, but in the relative position of the stamps on each metric. This is another way of saying that all measurements are made for a purpose. This purpose is incorporated/expressed in/ approximated by the metric. And crucially, it has nothing to do with the object or phenomenon measured. Once again: it is not a property of these things; the things become a property of the metric.
Any number of further examples could be given but they would only sharpen the point not make it. The epistemological challenge is very real indeed. But Lockhart has the wrong end of the epistemological stick. Units of measurement matter orders of magnitude less than the metric - the ordinal scale of measurement - on which and through which measurements are made. The political and sociological as well as the scientific implications of this fact are beyond the scope of this review. Perhaps another book will pop up as an excuse for following these up.
Then again there’s the problem of the green bananas.
See for further analysis of the same subject: https://www.goodreads.com/review/show... and https://www.goodreads.com/review/show... which also contain further references. For an axample of the opposing and also erroneous view that measurements exist entirely in the head of the one measuring, see: https://www.goodreads.com/review/show...
I’m at that stage in my life in which I hesitate to buy green bananas lest they go to waste: I might not be around to eat them before they’re fully ripe. The color green is a significant measure of not just the bananas but my life, at least what remains of it. I don’t know if anyone else uses this measure; it may have only personal appeal. Certainly Lockhart would reject it; he wouldn’t see the point. The title Lockhart has chosen for his book is somewhat misleading. He has nothing to say about the possible significance of green bananas, nor of many other scales of measurement. This is acutely disappointing. It also reflects a fundamental error in what he thinks constitutes measurement. This is acutely dangerous.
Lockhart makes a distinction early on between the imaginary world of maths and the physical world of objects and phenomena, like green bananas. He is only concerned with the former - Euclidean lengths, areas and volumes. Bananas of any kind, or the drop of curtains in the living room, and the road miles between Devon and Scotland - the kinds of things folk actually are concerned to measure - are not things that keep him up at night.
Lockhart does make geometry - the analysis of space - his focal point. But only after making geometry an entirely arithmetic affair. That is, he confines the things he is concerned about measuring to objects in the mathematical world. This conflation of maths and geometry is an old philosophical trick, and clearly has its uses - among others the capacity to develop all sorts of clever ways to estimate shapes, quantities, and complicated volumes.
But this capacity comes at a price: it makes it look as if measurement is a process of documenting the properties of an object - because that’s what happens in mathematical geometry. Since the objects of mathematical measurements are entirely fictional - composed of dimensionless points, lines without width and algebraically perfect dimensions - the properties of these objects are defined as part of their conceptual existence. A mathematical triangle actually does have internal angles summing to precisely 180 degrees - because that is part of its definition (Although triangles in non-Euclidian geometry can have less than 180 degrees, but that’s another story).
The fact that there are exactly half the angular arc-degrees of a circle in such a triangle is a property of mathematical triangles simply because they are defined to have that property. We make them that way. Just like we construct perfect circles, line segments, and conceptual three dimensional spaces. Such objects do not occur ‘naturally’, they are the sum of their defined properties. Their properties are limited to characteristics that can be derived from a set of axioms, which act as divine words of creation. Things don’t get any more knowable than that.
Obviously the physical world is different from this mathematical universe. The most important difference is one that is typically ignored in most discussions of measurement - namely that the ‘properties’ being measured in the physical world are not in any way inherent in the objects or phenomena being measured. Real objects and phenomena are not defined by their measurements, merely described. There are no axioms from which the properties of a real object, say the greenness of a banana, can be derived.
Unlike mathematical objects, real objects can be measured in an infinite number of ways but their ‘properties’ are in fact entirely unknown. To speak of such properties is an innocuous conceit in everyday life, but creates gross misconceptions when we act as if our words had the same power in the physical world. They don’t and we should know better.
To put the matter succinctly, physical measurements do not, indeed cannot, record the properties of phenomena. Rather, physical measurements are constituted by the assignment of the phenomena to a place on an ordinal scale (usually but not necessarily involving numbers) called a metric. A metric is merely a rule for ordering which we impose on things in the physical world. We Impose this ordering; it does not cut the world, as it were, at its joints. This we call quantification.
Quantification does not establish the properties of the object of the physical world in any sense. The object is in fact made a property of the metric through quantification. We adopt, as it were, things from the physical world into the mathematical world when we measure them. Being a mathematical object the properties of a metric are known precisely; and we can change the definition of this mathematical object to include the physical object as long as we have regard to the basic axioms on which the mathematical world is constructed. The position assigned to an object on the metric, its order, its rank, or its number, is not a property of the object but a consequence of the axioms of the metric.
I know from experience that this proposition of the measured object as a property of the metric on which it is measured is unfamiliar, disconcerting, and confusing. The remainder of this review (essay really) is my attempt to make the proposition less of all these. The reason for my making the effort, and for the perhaps greater effort of the reader, is that this proposition has significant practical as well as philosophical implications for metrology, the study of measurement. These implications are obscured to the point of disappearance in Lockhart’s book; so it is a convenient foil against which to establish the proposition as valid and useful.
The impossibility in principle of measuring the inherent properties in the physical world was shown by Immanuel Kant more the two centuries ago. No one has successfully refuted his proof that what he called the Thing-in-Itself is permanently and inevitably inaccessible to human language, including the language of mathematics. We are, as Plato expressed the situation poetically, forced to know the world as shadows on the wall of a cave.
This limitation isn’t a matter of perceptual ability, for example a lack of instrumentation or appropriate technology. Phenomena can be quantified at any level of technological development and they, that is their inherent properties, will nevertheless remain essentially unknown. No matter how accurate our observations, we are still observing shadows.
This is another way of saying that even an infinite description would not capture the essence of a rose, a poem, a star... or a banana. Even more disturbing is that the descriptions we make through measurement are not simply incomplete. They are, in a sense, lies, falsehoods which have the capacity to not just distort reality, but also to hide it entirely. We can end up investigating not the shadows but the cave wall on which the shadows appear. More importantly, measurement can be used to manipulate physical reality for hidden purposes - commercial, ideological and political. The way to prevent these potential falsehoods from affecting our judgments is to recognise what actually goes on when we measure, when we quantify something, in the physical world.
On the face of it, Kant’s claim appears counter-intuitive, if not just plain silly. Measurements may be erroneous sometimes, but that doesn’t make them lies, only corrigible mistakes. We can, we know, get more accurate by being more careful, eliminating bias and just generally paying closer attention to the operational procedures of observing, recording and reporting our measurements. What could justify the idea that we cannot know the properties of something we can measure? Isn’t that what we’re measuring - properties? We are able to estimate the depth of the ocean, the temperature of the air and the density of building materials.
Actually this isn’t what’s happening. The water, the air, the slab of marble are mute forces that act on our senses either directly or through our technology. But even the way we talk about measurement attributes our measurements to the ocean, the air, building material as something which is part of them, their attributes, their properties, not our senses.
This is the fundamental issue Kant was getting at. The issue is not the accuracy of a measurement on any metric - for example of depth, heat, or impenetrability - but of the suitability of the metric itself. Which metric to use to measure an object is the subject of what Kant called epistemology, the study of what metrics we use to impose on, colloquially to ‘represent’, a reality we cannot comprehend in any other way.
Epistemology got somewhat side-tracked over the last two centuries, concentrating on things like methods of research and the necessary rules for valid inference. It turns out that no one has been able to discover the singular methods or rules by which scientific advances take place. In fact, not infrequently, the biggest breakthroughs in science occur by violating established methods and ignoring apparently fixed rules of inductive logic. (But even that doesn’t constitute a rule - sometimes following the rules also generates startling results.)
One way to recast epistemology as a fruitful area of study is to recognise that the central problem which must be addressed is not one of method or procedure but one of numbers, specifically metrics, in their relation to the physical world. Lockhart ignores the non-numerical world entirely, thus avoiding the issues of epistemology. This is scientifically disingenuous, mainly because it makes the choice of the metric of measurement seem either trivial and therefore of no fundamental significance in our understanding of the universe.
Numbers are fictional entities. By this I mean no disrespect to numbers, nor do I deny their existence. By fictional I mean that numbers are stories. We have stories about how numbers have arisen, the evolution of their symbology and some very complex and sophisticated stories about how numbers are related to one another. These latter stories are called number theory, and they define the properties of numbers just as all mathematical objects are defined - by axioms and their implications.
So, although we don’t know everything there is to know about numbers - the axiomatic implications are infinite - we know quite a bit. Like their properties, the way they interact, their limitations and their capabilities. For example we know that numbers are infinite (in fact they are of various increasing orders of infinity). We also know that they are they are infinitely dense, that is, there is always a number to be found between any two numbers. We also know, perhaps somewhat disconcertingly, that there are many numbers (in fact there are infinitely more than other numbers) which are impossible to express entirely by numbers. These are called irrational numbers and they cannot be completely stated even with an infinite number of digits or decimal places.
Perhaps the most basic example of the error in presuming that quantitative measurement establishes properties of an object or phenomenon is provided by mathematical geometry itself. A fundamental proof in Euclidean geometry is that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. This applies to all right triangles. In particular it applies to the unit triangle, the two sides of which have the measure length of one (the numeraire - feet centimeters, finger-widths, etc - don’t matter). The hypotenuse of this triangle, as many of us will remember from our school days, is therefore the square root of two (the square root of (1 squared + 1 squared)).
The square root of two is one of those indeterminate irrational numbers. It can’t be stated exactly, only approximately. Yet any physical triangle of unit dimension will have a definite length. Whatever that real length is, it simply is not the same as that in the mathematical world, namely the indeterminate square root of two. In short the square root of two cannot be a property of a physical triangle. The triangle, whether mathematical or real, is a property of the metric of measurement - in both cases, even if less obviously so in the geometrical abstraction.
It gets worse however. Certain irrational numbers, like the number pi that links the diameter and circumference of a circle can’t even be expressed algebraically. They are called transcendentals. Transcendentals are of fundamental importance in mathematics even in basic geometric analysis. And there are an infinite number of these too. Yet they too are impossible to find in the physical world. They appear as sort of an alien presence to direct our attention around the cave but they never show up in it.
The difference between the mathematical world and the physical world has caused immense practical problems for the unwary thinker who forgets that mathematical properties don’t transfer to the physical world. The notorious paradox of Zeno is one of the oldest and most persistent confusions to plague the philosophically oriented scientist. Zeno’s infamous sprinter could apparently never reach his finish line for the purported reason that he would first have to run half the distance of his race, then half the remaining distance, the half that, and an infinite further number of further halved distances. In a finite time frame, therefore, the runner cannot ever finish the race. Entirely logical and obviously entirely incorrect.
There are various way to deflate the annoying Zeno. But the simplest is to merely point of that the infinite density of the points on the mathematical line from start to finish is not a property of the course itself. Zeno pretended it was and that his runner had the property of being at successive positions along this infinitely dense line. Once it is recognised that in fact the runner never has these positional properties but is himself being assigned as a property of Zeno’s metric of distance, the paradox disappears.
Advances in modern science show how the defined characteristics of the mathematical world don’t ‘map’ or correspond on a one to one basis with the physical world. Quantum theory for example posits the existence of shortest distance, smallest mass, and even briefest time periods. The infinite density of numbers means that there are effectively ‘gaps’ in physical reality which cannot be found in numbers.
Lockhart knows there is a fundamental discontinuity between the mathematical and the physical world: “Mathematical reality... is imaginary,” he says, “It can be as simple and pretty as I want it to be. I get to have all those perfect things I can’t have in real life. I will never hold a circle in my hand, but I can hold one in my mind. And I can measure it. Mathematical reality is a beautiful wonderland of my own creation, and I can explore it and think about it and talk about it with my friends.” But even beauty has its limits as a criterion for appropriate action.
Lockhart doesn’t take this discontinuity between the physical and the mathematical very seriously at all. “What is measuring? What exactly are we doing when we measure something?” He asks. “I think it is this: we are making a comparison. We are comparing the thing we are measuring to the thing we are measuring it with. In other words, measuring is relative. Any measurement that we make, whether real or imaginary, will necessarily depend on our choice of measuring unit. In the real world, we deal with these choices every day... The question is, what sort of units do we want for our imaginary mathematical universe?... One way to think of it is that we simply aren’t going to have any units at all, just proportions. Since there isn’t a natural choice of unit for measuring length, we won’t have one.”
By disregarding the physical world and its differences from the mathematical world, the issue of the right metric of measurement is first reduced to a question of the ‘units’ of measurement (the numeraire) and then to the simple procedure of comparison. While he’s certainly correct to point out the measurement is essentially comparison he can’t see that the choice of what to compare is of crucial importance. He can’t even see the metric.
For example when I was a child I collected British postage stamps. I was fascinated not just by their design and content but by their relationships to one another. The image below is of four such stamps compared in four different ways, that is on four different metrics. Although each metric does have a numeraire, a distinct unit of some monetary amount associated with it, these units are actually of trivial importance. Just as Lockhart suggests, each gets along just fine as a simple comparison. Metrics are rules for ordering. Here are several possible rules for ordering the stamps: [click on the link since I haven’t figured out how to make the html work. Apologies to all]
https://btcloud.bt.com/web/app/share/...
These are obviously four quite different comparisons and result in uniquely different ordering. All the comparisons are ‘real’ in the sense that there is nothing in philately which defines an inherent property or which comparisons are allowed. They just happen to be comparisons that someone might want to make. The position of stamps on each scale is clearly not anything to do with the inherent properties of postage stamps. Yet according to Lockhart (and me) these are measurements. All are correct but none pretend to anything but what they actually are - measures of relative value.
All physical measurements are measurements of relative value. We wouldn’t bother to make them unless they were. They are an ‘appreciation’ of an object or phenomenon in light of a specific intent not a statement of the character of the object or phenomenon in itself. This value is not expressed in terms of the units of an arbitrary numeraire (dollars, pounds, utils etc.), which as Lockhart points out might be irrelevant anyhow, but in the relative position of the stamps on each metric. This is another way of saying that all measurements are made for a purpose. This purpose is incorporated/expressed in/ approximated by the metric. And crucially, it has nothing to do with the object or phenomenon measured. Once again: it is not a property of these things; the things become a property of the metric.
Any number of further examples could be given but they would only sharpen the point not make it. The epistemological challenge is very real indeed. But Lockhart has the wrong end of the epistemological stick. Units of measurement matter orders of magnitude less than the metric - the ordinal scale of measurement - on which and through which measurements are made. The political and sociological as well as the scientific implications of this fact are beyond the scope of this review. Perhaps another book will pop up as an excuse for following these up.
Then again there’s the problem of the green bananas.
See for further analysis of the same subject: https://www.goodreads.com/review/show... and https://www.goodreads.com/review/show... which also contain further references. For an axample of the opposing and also erroneous view that measurements exist entirely in the head of the one measuring, see: https://www.goodreads.com/review/show...
April 5, 2017
I believe that Measurement is meant to be more or less a math text book for students around the middle-school or high-school grades. I'm not in the intended audience, and I'm not a math teacher. I'm just a professional mathematician, so take my opinions with a grain of salt.
I first encountered Paul Lockhart (as many did) through A Mathematician's Lament, an essay highly critical of the status quo in math education. Lockhart's essay was controversial, but many people loved it. I was one that did.
Maybe this excerpt will give an idea of Lockhart's approach:
With Measurement, Lockhart demonstrates how much more difficult it is to do something right than to point out the flaws in how others are doing it. That being said, Measurement is a brave and important shot at doing math education right. It is far from being a failure. I really liked Part One: Size and Shape, which deals mostly with geometry. Part Two: Time and Space, which introduces calculus, seemed much weaker.
Lockhart included lots of great hand-drawn diagrams. (I'm assuming they are hand-drawn. Maybe they were just made to look that way.) He also has included lots of exercises. Here is an example of his diagrams and exercises in one:
I didn't actually do a lot of the exercises. They just looked too darn hard.
Like I said, Part One I enjoyed. I had a lot of fun with it. Two things that really sparked my interest were projective space and parabolas. Yes I've seen both of these things many times before, but Lockhart gave me some brand new perspectives on them. In fact, I became inspired (jointly by Lockhart and my good friend Jason Lee) to compile a list of mathematical properties of parabolas. You may hear more from me on that in the future...
Part Two was interesting, but it seemed much more contrived. When calculus was first introduced to me (in high school) I was blown away by it. Maybe calculus has just become too trite to me. Any careful introduction of it seems tedious. I'm afraid I've become a poor judge of how engaging an introduction to calculus will be to fresh minds.
In Part Two I did like Lockhart's discussion of space and dimension:
I would have eaten that kind of stuff up when I was in middle school or high school. Well, OK. I still will eat it up.
Here is something that I think Lockhart gets wrong. He says:
I say to that: "yes and no." In pure mathematics it is true that we are not modeling real-world phenomena with mathematical structures. We are studying the mathematical structures themselves. However, (and it took me a lot of years in pure mathematics to realize this) we can and do model mathematical structures with other mathematical structures. As Lockhart says, the model may be equivalent to the original structure. On the other hand it may not be! It may be that the original structure is just too complicated for us to make any progress, and we need to model it with a simpler structure. Then learning about the simpler structure can tell us things about the original structure, but there is room for error. See my post on the Eternity Puzzle (and the links that it points to) for more on this. In other words, there is (in pure mathematics) the possibility of conflating imagination with imaginary imagination.
Measurement is thought provoking. I hope that it finds its way into some young people's hands.
I first encountered Paul Lockhart (as many did) through A Mathematician's Lament, an essay highly critical of the status quo in math education. Lockhart's essay was controversial, but many people loved it. I was one that did.
Maybe this excerpt will give an idea of Lockhart's approach:
So no, I can't tell you how to do it, and I'm not going to hold your hand or give you a bunch of hints or solutions in the back of the book. If you paint a picture from your heart, there is no 'answer painting' on the back of the canvas. If you are working on a problem and you are stuck or in pain, then welcome to the club. We mathematicians don't know how to solve our problems either.
With Measurement, Lockhart demonstrates how much more difficult it is to do something right than to point out the flaws in how others are doing it. That being said, Measurement is a brave and important shot at doing math education right. It is far from being a failure. I really liked Part One: Size and Shape, which deals mostly with geometry. Part Two: Time and Space, which introduces calculus, seemed much weaker.
Lockhart included lots of great hand-drawn diagrams. (I'm assuming they are hand-drawn. Maybe they were just made to look that way.) He also has included lots of exercises. Here is an example of his diagrams and exercises in one:
I didn't actually do a lot of the exercises. They just looked too darn hard.
Like I said, Part One I enjoyed. I had a lot of fun with it. Two things that really sparked my interest were projective space and parabolas. Yes I've seen both of these things many times before, but Lockhart gave me some brand new perspectives on them. In fact, I became inspired (jointly by Lockhart and my good friend Jason Lee) to compile a list of mathematical properties of parabolas. You may hear more from me on that in the future...
Part Two was interesting, but it seemed much more contrived. When calculus was first introduced to me (in high school) I was blown away by it. Maybe calculus has just become too trite to me. Any careful introduction of it seems tedious. I'm afraid I've become a poor judge of how engaging an introduction to calculus will be to fresh minds.
In Part Two I did like Lockhart's discussion of space and dimension:
What about four-dimensional space? Is there such a thing? If we're asking whether four-dimensional space is real we might as well ask about three-dimensional space: Is there such a thing? I suppose it appears that there is. We're walking around (apparently), and things certainly look and feel as though they are part of a three-dimensional universe, but when you come right down to it, three-dimensional space is really an abstract mathematical object--inspired by our perception of reality, to be sure, but imaginary nonetheless. So I don't think we should put four-dimensional space in any special mystical category. Spaces come in all sorts of dimensions, and none are any more real than any other. There are no one-dimensional or two-dimensional spaces in real life, and the only thing that gives the number 3 any special status is that our senses appear to offer us that particular illusion.
I would have eaten that kind of stuff up when I was in middle school or high school. Well, OK. I still will eat it up.
Here is something that I think Lockhart gets wrong. He says:
A mathematical structure is what it is, and anything we discover about it is the truth. In particular, if we choose to model an imaginary curve or motion by a set of equations, we are not making any guesses or losing any information through oversimplification: our objects are already (for aesthetic reasons) as simple as they can be. There is no possibility of conflating reality and imagination if everything is imaginary in the first place.
I say to that: "yes and no." In pure mathematics it is true that we are not modeling real-world phenomena with mathematical structures. We are studying the mathematical structures themselves. However, (and it took me a lot of years in pure mathematics to realize this) we can and do model mathematical structures with other mathematical structures. As Lockhart says, the model may be equivalent to the original structure. On the other hand it may not be! It may be that the original structure is just too complicated for us to make any progress, and we need to model it with a simpler structure. Then learning about the simpler structure can tell us things about the original structure, but there is room for error. See my post on the Eternity Puzzle (and the links that it points to) for more on this. In other words, there is (in pure mathematics) the possibility of conflating imagination with imaginary imagination.
Measurement is thought provoking. I hope that it finds its way into some young people's hands.
October 30, 2016
Love, love, love this book. Most of the lower ratings seem to be people who need to show they are so advanced that they only need symbols for math and that this book is too basic; or contrariwise, that there are no answers to any problems or there is "too much math". What?! Anyone with any advanced technical background can just flip through the book and see a dearth of equations, symbols and formal rigor. Anyone with no math background can just read the preface and see that Lockhart is not going to chew your mathematical food for you.
If you like to explore by thinking then this book is for you *if* you like to think about mathematical ideas. If you need an analogy to understand what this book is about, then you are a Buddhist acolyte and have purchased a $20 mountaintop with a Zen guru at the top. You still have to start at the bottom and walk up. The path is straight (you can read it through without working on any problems) but along the way there are little shrines (problems) where you can stop and meditate. When you get to the mountaintop, Lockhart will point you to the rest of the chain of mountains, farther off, with higher peaks. There is just one question that you will answer at the summit: Do you like climbing mountains?
Lockhart states explicitly that the point of the book is not to teach you answers to problems. His goal is to liberate you from conventional, stultifying "problem-recipe-solution mind" to independent, personal "curiosity-exploration-elevation mind." So this is a mathematical self-help type of book. Just like a diet & exercise book, you can read it through and gloss the message, but to really gain benefit, you should actually try doing what the book suggests you do and hopefully you will ingest the meaning.
I believe something to the following is said in the book, but I include this advice so that you can avoid the masochism that modern education teaches people is the right way to learn.
Q: How do I know when to stop thinking about one of the problems in the book?
A: When it is no longer fun to think about the problem.
Q: But what if I don't get the answer?
A: Think about it again when you want to. Otherwise, ask someone or use the internet.
This is one of the few books I wish I had available when I was in high school or college.
If you like to explore by thinking then this book is for you *if* you like to think about mathematical ideas. If you need an analogy to understand what this book is about, then you are a Buddhist acolyte and have purchased a $20 mountaintop with a Zen guru at the top. You still have to start at the bottom and walk up. The path is straight (you can read it through without working on any problems) but along the way there are little shrines (problems) where you can stop and meditate. When you get to the mountaintop, Lockhart will point you to the rest of the chain of mountains, farther off, with higher peaks. There is just one question that you will answer at the summit: Do you like climbing mountains?
Lockhart states explicitly that the point of the book is not to teach you answers to problems. His goal is to liberate you from conventional, stultifying "problem-recipe-solution mind" to independent, personal "curiosity-exploration-elevation mind." So this is a mathematical self-help type of book. Just like a diet & exercise book, you can read it through and gloss the message, but to really gain benefit, you should actually try doing what the book suggests you do and hopefully you will ingest the meaning.
I believe something to the following is said in the book, but I include this advice so that you can avoid the masochism that modern education teaches people is the right way to learn.
Q: How do I know when to stop thinking about one of the problems in the book?
A: When it is no longer fun to think about the problem.
Q: But what if I don't get the answer?
A: Think about it again when you want to. Otherwise, ask someone or use the internet.
This is one of the few books I wish I had available when I was in high school or college.
December 1, 2018
Wow, so many new ways of thinking about what I thought I already knew! I like to revisit the foundations of mathematics and re-interpret them in terms of notions picked up from physics, such as symmetry and mathematical equivalence. Paul Lockhart actually understands mathematics, and is able to take my project far deeper than I imagined.
He is barely getting started when, in passing, he casually solves the millennia old problem of the arbitrary nature of the Parallel Postulate. Parallel lines are elegantly re-defined as two straight lines that have a half-rotation (180 degree) symmetry. Then he quickly moves on, because he would rather inspire the reader with interesting ideas than develop a formal mathematical framework. But wait a minute; can I re-define a single straight line (also problematic) in terms of symmetry? He does not say, but then the whole point of the book is to get readers to reach their own conclusions.
The title is based on the idea that geometry is about measuring shapes with our minds using philosophical arguments, rather than physical measurement that must always be imperfect. The art of mathematics is to “measure” in the simplest and most elegant way possible. The units of measurement are proportions, which leads directly leads to the concept scaling. I love how he makes geometry comes alive: shapes scale up and down, lines sweep out areas, areas sweep out volumes.
The second half, which develops calculus, proceeds at a slower pace but is still worthwhile. The ideas are developed in a different order than usual. We are working with differential equations and deriving the chain rule before we learn to actually differentiate.
There are even some philosophical musings. I liked this one – shall we blame math for what is wrong with art?
“I suppose what I’m really talking about here is modernism. The exact same issues—abstraction, the study of pattern for its own sake, and (sadly) the resulting alienation of the layperson—are all present in modern art, music, and literature. I would even venture to say that we mathematicians have gone the furthest in this direction, for the simple reason that there is nothing whatever to stop us. Untethered from the constraints of physical reality, we can push much further in the direction of simple beauty. Mathematics is the only true abstract art.”
My only complaint is that many of the problems for the reader to solve require more than the tools he has just given us, and no answers are provided. Other than that, I highly recommend this book for anyone interested in learning, re-learning, or teaching math at the high school level.
He is barely getting started when, in passing, he casually solves the millennia old problem of the arbitrary nature of the Parallel Postulate. Parallel lines are elegantly re-defined as two straight lines that have a half-rotation (180 degree) symmetry. Then he quickly moves on, because he would rather inspire the reader with interesting ideas than develop a formal mathematical framework. But wait a minute; can I re-define a single straight line (also problematic) in terms of symmetry? He does not say, but then the whole point of the book is to get readers to reach their own conclusions.
The title is based on the idea that geometry is about measuring shapes with our minds using philosophical arguments, rather than physical measurement that must always be imperfect. The art of mathematics is to “measure” in the simplest and most elegant way possible. The units of measurement are proportions, which leads directly leads to the concept scaling. I love how he makes geometry comes alive: shapes scale up and down, lines sweep out areas, areas sweep out volumes.
The second half, which develops calculus, proceeds at a slower pace but is still worthwhile. The ideas are developed in a different order than usual. We are working with differential equations and deriving the chain rule before we learn to actually differentiate.
There are even some philosophical musings. I liked this one – shall we blame math for what is wrong with art?
“I suppose what I’m really talking about here is modernism. The exact same issues—abstraction, the study of pattern for its own sake, and (sadly) the resulting alienation of the layperson—are all present in modern art, music, and literature. I would even venture to say that we mathematicians have gone the furthest in this direction, for the simple reason that there is nothing whatever to stop us. Untethered from the constraints of physical reality, we can push much further in the direction of simple beauty. Mathematics is the only true abstract art.”
My only complaint is that many of the problems for the reader to solve require more than the tools he has just given us, and no answers are provided. Other than that, I highly recommend this book for anyone interested in learning, re-learning, or teaching math at the high school level.
February 5, 2018
A book by the author of A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form, Paul Lockhart wants to describe mathematical reality. To make patterns, to figure out how they work. The issues he deals with have nothing to do with the real world in any way, nor with other applied sciencies. Purely theoretical maths.
This is personal, not scholar at all. It means no footnotes, no references. This is only Paul Lockhart explaining some ideas he particularly likes. Just that.
The book has an introduction and two chapters. The former relies on reality and imagination, and then on problems in their metaphysical view. He considers a problem like a test of mahtematical reality to see how it behaves poking it with our minds. He also gives us an advice (a good one, in fact): be open-minded and flexible, let the problem let you where it takes you.
So the solution to a math problem it's not a number but an argument, a proof. And Lockhart watches them as little poems of pure reason. He insists on a double road: in mathematics there is the significance, but also the beauty.
The two chapters are 'Size and Shape', 'Time and Space'. In the first one, we read about issues like conics, proportion, symmetry, projections, perspective. In the second one, we have motion, curves, calculus, logarithms.
In general, this book covers the last grades of high school curriculum (at least here, in Spain): trigonometry, logarithms, conics, differentation and integration... And barely more, althoug inserted between paragraphs there are many open questions (always in bold). They are questions related to what he explains, so that the reader can expand his knowledge about it. They are not difficult, each one takes only a little time... but they are a lot of them, though.
So it is a book for people that age, around 18 or so, in order to they can "grok" these topics. Lockhart offers a point of view slightly different, and more profound, of the usual one in school, he builds some ideas with more explanation than in class, for we don't have the time.
Anyway, I didn't really like the book. I don't see some examples as suitables, the explanations are frequently a little obscure. And sometimes I see clearly where he wants to go, but I think he beats around the bush for a person with a little less training. He needs to be more direct about what he is trying to build.
In a certain way this book complements his Lament, for there he complains about the mathematics taught in USA primary and secondary school, but doesn't propose a better system. Here there isn't anything about education or pedagogy, only maths, but he spreads his explanations focusing in pure ideas, intuition, patterns, reasoning and the intrinsecal beauty of the subject itself. Just the main lacks he thinks the current math teaching has.
This is personal, not scholar at all. It means no footnotes, no references. This is only Paul Lockhart explaining some ideas he particularly likes. Just that.
The book has an introduction and two chapters. The former relies on reality and imagination, and then on problems in their metaphysical view. He considers a problem like a test of mahtematical reality to see how it behaves poking it with our minds. He also gives us an advice (a good one, in fact): be open-minded and flexible, let the problem let you where it takes you.
So the solution to a math problem it's not a number but an argument, a proof. And Lockhart watches them as little poems of pure reason. He insists on a double road: in mathematics there is the significance, but also the beauty.
The two chapters are 'Size and Shape', 'Time and Space'. In the first one, we read about issues like conics, proportion, symmetry, projections, perspective. In the second one, we have motion, curves, calculus, logarithms.
In general, this book covers the last grades of high school curriculum (at least here, in Spain): trigonometry, logarithms, conics, differentation and integration... And barely more, althoug inserted between paragraphs there are many open questions (always in bold). They are questions related to what he explains, so that the reader can expand his knowledge about it. They are not difficult, each one takes only a little time... but they are a lot of them, though.
So it is a book for people that age, around 18 or so, in order to they can "grok" these topics. Lockhart offers a point of view slightly different, and more profound, of the usual one in school, he builds some ideas with more explanation than in class, for we don't have the time.
Anyway, I didn't really like the book. I don't see some examples as suitables, the explanations are frequently a little obscure. And sometimes I see clearly where he wants to go, but I think he beats around the bush for a person with a little less training. He needs to be more direct about what he is trying to build.
In a certain way this book complements his Lament, for there he complains about the mathematics taught in USA primary and secondary school, but doesn't propose a better system. Here there isn't anything about education or pedagogy, only maths, but he spreads his explanations focusing in pure ideas, intuition, patterns, reasoning and the intrinsecal beauty of the subject itself. Just the main lacks he thinks the current math teaching has.
October 12, 2013
Lockhart's Measurement is a work of mathematical beauty. From the very humble beginnings of the mathematics of shape, he develops a mathematical edifice including ideas of Plane Geometry, Solid Geometry, Higher-Dimensional Geometry, Coordinate Geometry, Projective Geometry, Vector Geometry, Elementary Algebra, and Differential and Integral Calculus. He even hints at the heights of modern mathematics (with its emphasis on structure) - and he does so from an aesthetic perspective. He shows the inherent beauty of mathematical truth without sacrificing the challenging nature of its methodology. Furthermore, his approach leaves behind the messiness of mathematical application. While mathematics is certainly applicable to various "real world problems", that is certainly not the point of studying mathematics. It is an unfortunate error of modern education that the point of education is to prepare oneself for work in the modern factories (which nowadays take the form of desk jobs in corporate offices), rather than the proper disposal of the students for truth and beauty. The approach of this book seems to contain at least a partial remedy for this disease within modern mathematical education. As such, it is a must-read for all mathematical educators and, in my opinion, all aspiring mathematicians.
May 14, 2025
A personal book about the process of mathematical discovery. Starting from ideas in classical geometry, the author brings us on a marvellous journey towards the foundations of analysis. We are guided by natural curiosity, and every step is well justified. I think this would make a good book for a keen high school student, or anyone interested in learning about mathematical thinking. Additionally, the book contains a great collection of elegant arguments for well-known theorems.
We are constantly reminded of the author's platonist leanings, and that mathematical reality is something separate from physical reality. This is an uncommon view to see in a book of this kind, but one which I welcome. Finally, a maths book that tells it like it is.
We are constantly reminded of the author's platonist leanings, and that mathematical reality is something separate from physical reality. This is an uncommon view to see in a book of this kind, but one which I welcome. Finally, a maths book that tells it like it is.
December 9, 2018
Investigation of the ideal universe of math, as opposed to the real world of discrete surfaces and discontinuities. Digs into patterns and describes the beauty of ideal mathematics, connecting concepts with curiosity. The author describes the book as personal, and I agree.
Like reality, though, it wasn't perfect. Bold sentences interrupt the text, describing what the author sees as "problems and questions that occur to me." These were occasionally interesting but nearly always jarring, breaking the narrative. Sections (chapters?) were quite short, I think it would have been better to collect these bold questions at the end of each. I ended up just skipping them, perhaps missing something of interest, but definitely finishing the book.
Other minor quibbles include a skipped step in the proof of Heron's formula and a bit of glossing over sine and cosine (and no mention of how they are related to tangents!). It's a good book, worthy of 3½ stars - and I really need to read the authors most important work, A Mathematician's Lament.
Like reality, though, it wasn't perfect. Bold sentences interrupt the text, describing what the author sees as "problems and questions that occur to me." These were occasionally interesting but nearly always jarring, breaking the narrative. Sections (chapters?) were quite short, I think it would have been better to collect these bold questions at the end of each. I ended up just skipping them, perhaps missing something of interest, but definitely finishing the book.
Other minor quibbles include a skipped step in the proof of Heron's formula and a bit of glossing over sine and cosine (and no mention of how they are related to tangents!). It's a good book, worthy of 3½ stars - and I really need to read the authors most important work, A Mathematician's Lament.
February 7, 2016
Very fascinating and well written book. The author essentially tells you a story and guides you through mostnot the mathematics that you learn in junior high and high school. He very skillfully explains these concepts in such a way that I didn't even realize we were switching from geometry to algebra to trigonometry and beyond (as the author skillfully points out, there really isn't a difference between these ideas, but that's how They get broken up in school, so we tend to segregate their concepts in our mind).
The author also drives home an important point over and over again: that mathematics as we know it is simply a language we've invented to describe the world around us and measure the patterns around us. (hence the title)
Very well done, and I enjoyed it. I even found myself getting out a paper and pencil and sketching shapes and trying some of the math "challenges" the author gives. It was great!
The author also drives home an important point over and over again: that mathematics as we know it is simply a language we've invented to describe the world around us and measure the patterns around us. (hence the title)
Very well done, and I enjoyed it. I even found myself getting out a paper and pencil and sketching shapes and trying some of the math "challenges" the author gives. It was great!
June 7, 2015
This book was amazing. A very personal view (which is really, really rare in maths) about some really interesting parts of very basic mathematics. And yet even if the topics covered are so basic (triangles, conics, lines, derivatives...) the book manages to open a completely new view on them!! A must-read for any teacher or student of math.
I suggest, before reading this book, to read the short essay written by the same author commonly known as "Lockhart's Lament" which is available online and circulated greatly among mathematicians, or the published book version "A Mathematician's Lament" which added some material, and is a personal criticism of the mathematical education system. It acts as a good introduction to the feeling and the tone of this more extensive and fascinating book.
I suggest, before reading this book, to read the short essay written by the same author commonly known as "Lockhart's Lament" which is available online and circulated greatly among mathematicians, or the published book version "A Mathematician's Lament" which added some material, and is a personal criticism of the mathematical education system. It acts as a good introduction to the feeling and the tone of this more extensive and fascinating book.
April 22, 2022
DNF at 31%
This just wasn’t working for me. I am not a big math person, admittedly, but I was hoping to see math in a new light, at least a little bit, from this book. Turns out, I understood exactly what I already knew and didn’t learn anything I didn’t already know. It now seems pointless for me to continue.
This just wasn’t working for me. I am not a big math person, admittedly, but I was hoping to see math in a new light, at least a little bit, from this book. Turns out, I understood exactly what I already knew and didn’t learn anything I didn’t already know. It now seems pointless for me to continue.
February 19, 2024
Did Not Finish -- Tapped out after about 150 pages. At that point I wasn't grasping much for 50 straight pages and saw that it was only getting more complex. There's no point to skimming this kind of book so I'm putting it down. This book attempts to teach fundamental geometry and arithmetic "elegantly" in both its content and style, but I was not able to follow the specifics of his teachings for long. Not necessarily a bad book, but it didn't click for me, and this book's entire purpose is trying to make math click for the reader.
There was one pretty bad aspect though. I skipped ahead to Part Two -- "Time and Space" -- to see if that was any better, and quickly came upon the author's hand-wavy, unrigorous assertion that three-dimensional space was not real (lol), which amounted to over two straight pages of babble. He didn't even leave a sentence for supporting evidence or a counterargument. It made feel better about putting this book down.
There was one pretty bad aspect though. I skipped ahead to Part Two -- "Time and Space" -- to see if that was any better, and quickly came upon the author's hand-wavy, unrigorous assertion that three-dimensional space was not real (lol), which amounted to over two straight pages of babble. He didn't even leave a sentence for supporting evidence or a counterargument. It made feel better about putting this book down.
May 23, 2023
This book is fascinating read that describes how mathematical concepts "evolve" and "relate" to reality. While a bit wordy at times, it conveys the relations between various concepts in geometry, algebra, trigonometry, and calculus in a gradual, incremental, and intuitive manner. I wish that I had read this book after my first learned calculus in high school. I think reading this book along with "The Mathematics Lover’s Companion: Masterpieces for Everyone" can help demystify math and abstraction for high school kids (after their first brush with calculus).
March 12, 2013
This is a book about understanding mathematics visually. It's about the enjoyment of solving problems and developing an intuition for mathematics, particularly in geometry/spatial situations. Measurement is divided broadly into two halves, the first focusing primarily on geometry ("Size and Shape") and the second giving an introduction to differential calculus (the sort you'd see in a first-semester college course).
It took me awhile to get into this book, especially since I didn't feel like the intended reader. I have had an intuition for math in the way Lockhart describes, and for quite awhile I didn't think I was going to learn anything in this book. About half an hour in, I saw a visual demonstration of the "difference of squares" formula (p. 57), which blew my mind---and then the book continued to show me new ways of thinking of things. In fact, my reaction to each concept the book presented was one of two extremes: "well, of course, that's the obvious/intuitive way to look at it", or "wow, I've never thought about it that way before, and that makes it so much clearer," which to me says that Lockhart achieved his goal, regardless of if the reader has had similar conclusions before.
At times, the limitations of this being a book are evident and frustrating. Diagrams and descriptions can only go so far, especially when trying to bring readers to a deep, intuitive understanding of visual concepts. As examples, Steven Wittens and Bret Victor have used interactive web applications to accomplish what Lockhart can't in printed form, and I found myself occasionally wishing for their insight in explaining some of the book's ideas.
There's a question that I think is really important for this book: who is the intended reader? It seems that this book's ideal reader is someone who:
- doesn't necessarily (yet) have an intuitive grasp of math
- is interested in math, or is open to the possibility of being shown that math is interesting
- is able to mentally imagine complex spatial situations that are only described on paper
If you've read Lockhart's A Mathemetician's Lament, you know who he's trying to reach. I'm not sure that enough people who didn't "get" math the first time have the patience for another try, and I can't accurately judge how well this book reaches them---but I suspect it does a pretty good job.
It took me awhile to get into this book, especially since I didn't feel like the intended reader. I have had an intuition for math in the way Lockhart describes, and for quite awhile I didn't think I was going to learn anything in this book. About half an hour in, I saw a visual demonstration of the "difference of squares" formula (p. 57), which blew my mind---and then the book continued to show me new ways of thinking of things. In fact, my reaction to each concept the book presented was one of two extremes: "well, of course, that's the obvious/intuitive way to look at it", or "wow, I've never thought about it that way before, and that makes it so much clearer," which to me says that Lockhart achieved his goal, regardless of if the reader has had similar conclusions before.
At times, the limitations of this being a book are evident and frustrating. Diagrams and descriptions can only go so far, especially when trying to bring readers to a deep, intuitive understanding of visual concepts. As examples, Steven Wittens and Bret Victor have used interactive web applications to accomplish what Lockhart can't in printed form, and I found myself occasionally wishing for their insight in explaining some of the book's ideas.
There's a question that I think is really important for this book: who is the intended reader? It seems that this book's ideal reader is someone who:
- doesn't necessarily (yet) have an intuitive grasp of math
- is interested in math, or is open to the possibility of being shown that math is interesting
- is able to mentally imagine complex spatial situations that are only described on paper
If you've read Lockhart's A Mathemetician's Lament, you know who he's trying to reach. I'm not sure that enough people who didn't "get" math the first time have the patience for another try, and I can't accurately judge how well this book reaches them---but I suspect it does a pretty good job.
August 5, 2018
I think this is one of my favorite books of all time. The illustrations are beautiful, and so are the proofs! The first section was stronger than the second section but I enjoyed almost every page. The first section is a refresher on geometry which makes the topic seem simpler than and more whimsical than it's ever been before. The second section (somehow) does the same for calculus and differential equations and logarithms.
He does all this not to teach you geometry or calculus (this would be a pretty bad textbook, for one he goes out of his way to not name things, like the fundamental theorem of calculus, even as he describes them) but to give you an idea of what it means to be a mathematician, of what it feels like to see a pattern and play with its structure. Interspersed among the proofs you'll find his philosophy of mathematics. It's beautiful:
He does all this not to teach you geometry or calculus (this would be a pretty bad textbook, for one he goes out of his way to not name things, like the fundamental theorem of calculus, even as he describes them) but to give you an idea of what it means to be a mathematician, of what it feels like to see a pattern and play with its structure. Interspersed among the proofs you'll find his philosophy of mathematics. It's beautiful:
The solution to a math problem is not a number; it's an argument, a proof. We're trying to create these little poems of pure reason. Of course, like any other form of poetry, we want our work to be beautiful as well as meaningful. Mathematics is the art of explanation, and consequently, it is difficult, frustrating, and deeply satisfying.
It's also a great philosophical exercise. We are capable of creating in our minds perfect imaginary objects, which then have perfect imaginary measurements. But can we get at them? There are truths out there. Do we have access to them? It's really a question about the limits of the human mind. What can we know? This is the real question at the heart of every mathematics problem.
So the point of making these measurements is to see if we can. We do it because it's a challenge and an adventure and because it's fun. We do it because we're curious, and we want to understand mathematical reality and the minds that can conceive it.
January 18, 2014
I thoroughly enjoyed The Mathematician's Lament. After reading Lockhart's critique of math education in that book, though, I was left wanting to know exactly what we (as math teachers) should do instead. Or, rather, what would effective math education look like to Lockhart? Measurement begins to answer that question. This is all about visual mathematics. The first half of the book on size and shape was much more accessible to me than the second half on time and space (I'm an elementary school teacher with very minimal math education beyond high school in the 90s). Lockhart's belief is that math instruction/education should be exploratory, imaginative, and fun - "creating little poems of pure reason" that are "beautiful and meaningful." Lockhart presents many questions, situations, and images to think about. But there are no answers (not even in the back of the book!). I didn't have time to pour myself into each and every question, but I look forward to purchasing this book and spending more time with it in the future.
October 8, 2013
I abandoned this book, and usually if I abandon a book it's a default one star rating. But it seems unfair to do that to this book, because my main criticism is that it's too mathy, and if you pick up a book that is all about math and then complain that it's too mathy then that's really your own damn fault.
What happened here is that I read Lockhart's first offering, A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form, and was thoroughly impressed with it and eager to read his next book. Even though A Mathematician's Lament was all about math, it was a fascinating and enjoyable read. Knowing what Lockhart is capable of writing, I just expected more out of this book than what I ended up encountering.
What happened here is that I read Lockhart's first offering, A Mathematician's Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form, and was thoroughly impressed with it and eager to read his next book. Even though A Mathematician's Lament was all about math, it was a fascinating and enjoyable read. Knowing what Lockhart is capable of writing, I just expected more out of this book than what I ended up encountering.
December 17, 2016
It's like having someone explain to me something that I already understand, except with very cumbersome language instead of abstract and symbolic representations. It's intended to help a layperson take a look at math in a different light, a more inviting light, perhaps. The author succeeded in achieving his goal with many ingenious ways to describe and deduce complex math concepts. However, I fear that not too many people will find this book entertaining or informative. Those who knew this stuff don't need his logocentric approach. Those who didn't understand these topics before will gain some new information but will gain little ground in breaking the code of the symbolic derivations.
August 5, 2013
This is a quite phenomenal tour de force by a talented and enthusiastic mathematician keen to break down the mystique and barriers to the subject. He does do spectacularly well: this little volume covers an amazing amount of ground, from simple measurement concepts and problem-solving all the way up to the ideas that underlie some of the most modern pieces of mathematics like differential forms and the calculus of variations. On the way everyone will learn something they didn't know, in my case the relationship between natural logarithms and conic sections.
February 7, 2017
This should maybe be required reading at the beginning of the high-school math curriculum. I did fine with the material in those courses, but much of it was introduced without much context to motivate why it mattered. For example, conic sections seemed like a pretty arbitrary topic. I didn't have any idea of why to care about them. I would much rather have had them introduced to me the way that Lockhart does in this book.
July 21, 2018
Kitap özellikle ilköğretim ve lise anlatılan matematik ve geometri konularını kapsamaktır.Sonrada üniversite düzeyine bir miktar yaklaşmaktadır.Kitabın en güzel yanı özellikle lisede ispatı yapılmadan sadece teorik verilen bilgileri detaylı ve güzel açıklamaktadır.Matematiğim küçüklükten beri kötüdür diyen kişilere özellikle tavsiye ederim
January 22, 2013
Here's my official vote: let's please stop teaching math in the traditional way and instead teach using the ideas put forward by Paul Lockhart.
Mathematics is beautiful and astounding. This book provides a glimpse of how we might use natural curiosity to teach mathematics.
Mathematics is beautiful and astounding. This book provides a glimpse of how we might use natural curiosity to teach mathematics.
September 25, 2016
A wonderful, wonder-filled, engaging book for mathematics for high school and up. I recommend that all math teachers read through and work through this book. They will gain many ideas for turning math from mindless training to the beginning of a life-long love of mathematics.
November 3, 2017
It had some really good ideas and connections. The only downside is that it is too "mathy" for some non-math people, who I think are the target audience. It is a tough line to walk and Lockhart did a pretty good job at it.
May 20, 2025
From fitting cones into cylinders, triangles into squares and connecting it all to differential equations. Lockhart opens the rift into the mathematical world for you and provides humorous commentary with it. Great read!
Want to read
October 12, 2012As seen in
Nature
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June 18, 2013
I need to re-read. I didn't find it as accessible as I was hoping.
Read
December 30, 2015516 LOC
Only half through.
Only half through.
June 11, 2014
I stopped reading around 40%. Being a math major, it wasn't providing enough insight-density to warrant further reading. Others with less math experience would likely get a lot out of it, however.
November 18, 2015
Helped me to develop intuition for math.
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