Measurement by Paul Lockhart
My rating: 2 of 5 stars

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...

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