Yearly Archives: 2016

The Worth of Measure

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By The Metric Maven

Fourth Anniversary Edition

When the well’s dry, we know the worth of water

— Benjamin Franklin

In the Winter of 2014 the “Polar Vortex” event brought marrow chilling cold to my father’s town. For months the water pipes which fed his house were frozen, and he had to fill large containers of water each day for household needs. There was some irony in the fact that new water meters had been installed not long before, and my father noticed a measurable change in his usage and water bill. The water company’s explanation was that these new water meters are much more accurate than the old ones and so his bill reflected a more accurate quantity of water. It would be hard to verify this assertion and life went on.

This interaction between water user and municipal water supplier is not new. L. Sprauge de Camp in his excellent book The Ancient Engineers relates a similar situation in ancient Rome (units converted to metric):

The unit used in measuring water was a calix or standard nozzle. The standard calix was the quinaria, a length of bronze pipe 1 1/4 digits (= 18.5 mm) in diameter and 12 digits (= 222 mm) long, connecting the distributing tank to the user’s pipeline.

Users of larger calices were charged in a rough proportion to the cross-sectional area of their nozzles, as nearly as the Romans could calculate these areas with their awkward system of numerals. These charges were made on the assumption that doubling the cross-sectional area would double the flow, when in fact it would more than double it.

The Romans also knew that the flow of water through an orifice is greater if the the hydraulic head or water pressure is higher. But they did not know how much greater. So, not having water meters, they could not adjust their charges accordingly.

***

Still, for want of effective water meters, this system of measuring water supply long continued in use; it was employed in Paris as late as the 1850s. When  large discrepancies appeared in Frontinus‘ figures, he thought that these were due entirely to theft of water and leakage. In fact, however, they were also due to his crude methods of reckoning.

A modern person might wince when thinking about doing calculations using Roman numerals, or, considering the ubiquity of electronic calculators (physical or computer simulated) doing the calculation by hand with Hindu-Arabic numbers. An expression in modern Hindu-Arabic numerals is easily comprehended, whereas Roman numerals require some manipulation. For instance Super Bowl XLIX is what? Why Super Bowl 49 of course!  How about a simple subtraction: MMMMMMMMMMMMCCCXLV – MMCCCXLV = ? Perhaps it might be easier in Hindu-Arabic: 12345 – 2345 = ?  Yes it’s 10 000.

Modern persons immediately note, that the Romans poor numerical system was further confounded with poor measurement techniques. It is often stated that the world around us is mathematical, but this statement without qualification is not true. We cannot make the claim that the world is mathematical, without an idea that is generally invisible to the psyche. Worse yet, when it does make its way into the minds of engineers and scientists, it is often a concept which is thought to be of minimal importance. This attitude was brought into high relief when I read this quotation in Basic Concepts of Measurement by Brian Ellis:

Measurement is the link between mathematics and science. The nature of measurement should therefore be a central concern of the philosophy of science. Yet, strangely, it has attracted little attention. If it is discussed at all in works on the philosophy of science, it is usually dismissed in a fairly short and standard chapter.

Without the intermediary of measurement between our observed world and mathematics, science and engineering as we know them would not be so obviously mathematical. Pat Naughtin realized this. It was obvious to him, but it was difficult for Naughtin to conceive that it was not obvious to the greater technical community in the US:

During the 2008 election campaign in the USA, ‘Scientists and Engineers for America and fifteen other science organizations‘ united to ask the congressional candidates seven questions in preparation for the Federal elections in the USA.
                                                               ***
The highly original, smart, clever, and creative scientists and engineers who wrote the seven questions had not begun to address the most basic issue in science–how to measure things. They simply didn’t seem to see that measurement was at all important to their nation.

One might ask “what would the world be like if measured quantities and mathematical quantities had no relationship?” I can safely say that engineering and science as we know it would not exist. We would be no better off than the Romans—at best. We might desire the existence of a  computational relationship between the size of a pipe and its flow rate, but if we lived in a world where our mathematics and measures could not be related, we would be left to rely only on our perceptions of large and small, hot and cold, heavy and light. We would know the worth of measurements, if they were suddenly not mathematically expressible. This quantitative absence would leave us intellectually naked and left without recourse to deal with the vicissitudes of nature. Before mathematics was combined with measurement through quantitative experimentation, the entire world was filled with unexplained mysterious forces. To deal with this un-quantified world, we would instead call upon supernatural agency and sympathetic magic. Lord Kelvin clearly understood this when he stated:

When you can measure what you are speaking about and express it in numbers you know something about it; but when you cannot express it in numbers your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge but you have scarcely progressed in your thoughts to the stage of science whatever the matter may be.

People often understand this at an intuitive level when discussing research. The quantification of physical properties like mass, energy, volume and so on are all fixed. When measured quantities are mathematically expressed, these relationships can often be verified to the limits of our measurement capability. When we attempt to relate sociological quantities like happiness, sadness and anger in mathematical form we often feel the reluctance that Lord Kelvin did. How could we possibly measure these quantities? The devil is in the measurements, and without repeatable and exacting measurements, which are also mathematically expressible, what we are doing just does not seem like science and in fact is not.

When I insist on the importance of the metric system, and its clearest use, I’ve often been dismissed by other engineers and scientists. In some cases a visceral ejaculation of impatient prose follows that claims: “it doesn’t matter what measurement units one uses.” Sometimes this is followed with a dismissive hand gesture. The analogy of computation with Roman numerals versus Hindu-Arabic numerals is brushed aside as trivial and irrelevant. The metric system was developed, from its inception, to be the easiest  and clearest measurement system for relating mathematics to definite physical quantities, which is the very definition of “hard” science. There is often an insinuation that people who can “master” more than one measurement system are somehow intellectually superior to those who “cannot.” I had a chemistry instructor in my Fundamentals of Engineering review course say this during one class: “American engineers and scientists are better than the rest of the world because we understand two systems.” This proliferation of measurements does not make us “better” it makes us slaves to thoughtless tradition, similar to the tradition of Roman numerals. It took over 1000 years for humanity to wean itself from them.

Michael Faraday (1791–1867) gave a fascinating public lecture called The Chemical History of a Candle. This lecture was published in 1861 and it is a very interesting read. When he was investigating the constituents of water in this lecture, Faraday offered this table:

Faraday_Table

It is hard to image that this switching of units demonstrates the superiority of using multiple units when looking for correlation. This is the pre-metric world and it is not insight-friendly. I would convert the table to metric, but I’m not sure which pint or which ounce or which grains are used for certain.

It has struck me as a sad commentary that modern public figures who aspire to be scientific ambassadors and popularizers, have such a trivializing attitude toward the metric system. Neil deGrasse Tyson, when asked by a Canadian to explain why he did not use metric in Nova Science Now argued that if you understood the metric system, perhaps you did not need to watch the show! “I’m really just trying to reach the people that need the science.” This statement indicates the metric system is difficult, when in fact it is designed for elegance and simplicity. When the metric system is used in thousands (triads), the gram, milliliter and millimeter are all integers for expressing everyday quantities. deGrasse Tyson reinforces the corrosive American prejudice that “metric is for scientists and engineers” and regular folk don’t need to worry their pretty little heads with that complicated stuff. He later trivializes the audience member’s request for metric by stating: “it does put a greater burden on you, the Canadian to adapt to our mysterious ways in America.” Yes, they are mysterious, and not really funny.

Bill Nye explains the metric system by offering up both centimeters and millimeters and the idea that the metric system is only “better by ten” instead of better by one-thousand. No hectometers or decameters or decimeters are offered to demonstrate the better by ten assertion.

Neil deGrassee-Tyson and Bill Nye do excellent work explaining and defending science. They, like most engineers and scientists seem to have given the metric system very little thought or investigation. Is this their fault? I would argue that it is probably not. This metric ignorance is a systemic problem within our educational institutions from kindergarten through graduate school. I only recall a quick statement by a professor, stating that we will be using the metric system in our science and engineering courses, and then, without any reflection or background, we were often introduced to physics and engineering problems with both Ye Olde English, and metric, and metric was used without context. One way that scientists often trivialize metric is by using scientific notation and not bothering with metric prefixes at the end of their computation.

Our modern scientific world as we know it would not exist if measurements could not be made and then related to mathematics. The late Pat Naughtin seemed absolutely baffled when he could not get US engineers and scientists to make the adoption of the metric system in the US an intellectual priority. It is generally the examination of basic assumptions that propels engineering and science forward, but often it is the basics that are assumed and ignored. This lack of interest in the metric system by those who would communicate science to the public, and their indulgence of tradition over clarity, does not serve the public well, or demonstrate a deep understanding of science.

If you liked this essay and wish to support the work of The Metric Maven, please visit his Patreon Page and contribute. Also purchase his books about the metric system:

The first book is titled: Our Crumbling Invisible Infrastructure. It is a succinct set of essays  that explain why the absence of the metric system in the US is detrimental to our personal heath and our economy. These essays are separately available for free on my website,  but the book has them all in one place in print. The book may be purchased from Amazon here.


The second book is titled The Dimensions of the Cosmos. It takes the metric prefixes from yotta to Yocto and uses each metric prefix to describe a metric world. The book has a considerable number of color images to compliment the prose. It has been receiving good reviews. I think would be a great reference for US science teachers. It has a considerable number of scientific factoids and anecdotes that I believe would be of considerable educational use. It is available from Amazon here.


The third book is called Death By A Thousand Cuts, A Secret History of the Metric System in The United States. This monograph explains how we have been unable to legally deal with weights and measures in the United States from George Washington, to our current day. This book is also available on Amazon here.

Insignificance

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By The Metric Maven

Bulldog Edition

Some years back I became involved in evaluating the results of an experiment that clearly had scientific issues. I assisted two other volunteers, and was mostly there to critique the experimental methods. Early on I asked if the answer was as simple as the data had been cooked. One of the volunteers was a graduate student in mathematics. He looked at me and said “no, it’s fine, I already looked at the data.” I was a bit puzzled and wanted to know why he had such confidence that the data had not been altered. The mathematician said “the data is consistent with Benford’s Law.” I had no idea what that was, and surprised to hear that generally the first significant digit of numerical data is not random. The distribution of 1’s, 2’s, 3’s and so on up to 9 is not uniform. The probability of a one is higher than a two and they all follow a statistical pattern.

My mind had a very difficult time accepting this statement. The mathematician told me to “go look it up” which I did.

The story goes that Frank Benford (1883-1948), while working as an electrical engineer at General Electric, had obtained a well-used copy of a book of logarithms. He noticed that the beginning pages were the most soiled and worn. The idea that people would most often look up logarithms of numbers that begin with number one, and then those with two, and so on up to nine was surprising. Benford wrote up his observation, which is often called Benford’s Law. However Simon Newcomb (1835-1909) had earlier published the same observation in 1881.

I had a hard time accepting this because it meant that the first significant digit of a number is not statistically independent. The mathematical analysis to derive Benford’s law is beyond my expertise.[1] Sven pointed out that Warren Weaver (1874-1978) in his book Lady Luck has a reasonably intuitive explanation of how Benford’s Law comes to be. The relevant section is called The Distribution of Significant Digits, and does not mention Benford directly. Weaver makes this statement: “Although it remained unsuspected or at least unidentified for centuries, this distribution law for first integers is a built-in characteristic of our number system.”

Here is a nice graph from Wikipedia showing the distribution of the first significant digit for numerical data:

Benford-2

Thirty percent of the time, the first significant digit of commonly used physical constants found in an elementary physics textbook is one. Census populations follow Benford’s Law, as do income tax data, one-day returns on the Dow-Jones industrial average and Standard and Poors indexes. Benford’s law is often used in forensic accounting to screen for fraud.

At this point, I want you to note something about this plot of Benford’s Law: what is the probability of zero for the first significant digit? Well, there isn’t one. If you add up all the probabilities you end up with 100%, so no probability is assigned to zero for the first significant digit, or should it be called the first insignificant digit?

I had given thought to discussing significant digits in the past, but there are differing views about how to go about determining significant figures in calculations, and so I tended to shy away from any discussion of the topic. Not until a reader took me to task over a statement I made in a blog about the 100th anniversary of the USMA did I decide it was worth some examination:

Also, the Maven writes: “The world record eyebrow hair is touted as 9 centimeters (90 millimeters for those with a refined measurement sense).”
In this case, since the measurement apparently was not to the nearest 0.1 cm, writing it as 90 millimeters would be false precision. (Of course if it were given as, say 9.2 cm, then 92 mm would be better.)
Thus, centimeters should be considered in such circumstances to avoid any indication of false precision; otherwise, centi-, and deci-, deka, and hecto-, should be considered as sort of “informal prefixes”…

While there is a lot of disagreement about how to determine significant digits, the one statement about them which is generally accepted is that adding zeros on the right side of a whole number does not constitute adding significant digits.

Here is a statement from Learn How To Determine Significant Figures:

If no decimal point is present, the rightmost non-zero digit is the least significant figure. In the number 5800, the least significant figure is ‘8’.

Another university website has:

…Trailing zeros in a whole number with no decimal shown are NOT significant. Writing just “540” indicates that the zero is NOT significant, and there are only TWO significant figures in this value.

Wikipedia has this to say:

The significant figures of a number are digits that carry meaning contributing to its measurement resolution. This includes all digits except:[1]

Wikipedia in its rules for identifying significant figures states:

In a number without a decimal point, trailing zeros may or may not be significant. More information through additional graphical symbols or explicit information on errors is needed to clarify the significance of trailing zeros.

If a whole number is encountered without any context, the trailing zeros should be assumed as insignificant unless the text specifies otherwise.  Clearly when I pointed out that 9 cm would be better written as 90 mm, I did not conjure up an extra significant digit and imply more measurement resolution. The commentator made an unwarranted assumption about the 9 cm value: “since the measurement apparently was not to the nearest 0.1 cm, writing it as 90 millimeters would be false precision.”  He is acting as a psychic and divining what precision was implied by the person who measured the value and offered it as 9 cm. What was offered up by the commentator is actually an example of false precision. There is no reason to assume the measurement was or was not to the nearest 0.1 cm or 0.05 cm or 0.025 cm. Only a single integer with a single significant figure of 9 is offered. Adding a zero on the end and expressing it in millimeters without providing any additional information altered that fact not one bit.

In this situation the zero is just a final place holder, therefore when an extra zero is added to the end, it does not introduce any increase in implied precision or become a significant figure. The first significant digit is still nine for 90 mm as it was for 9 cm.

“Trapped zeros” are considered significant. In the case of 402, the zero between the 4 and the 2 is significant, but a trailing zero such as that found on 420 is not. Adding an infinite number of trailing zeros to an integer number does not increase the number of significant digits. When I pointed out that Metric Today should change centimeter values to exclusively millimeter values, I only changed 9 cm to 90 mm. I can equivalently write 9 cm as 90 mm, or 90 000 µm or 90 000 000 nm without introducing any extra “implied precision.” Unless I tell you that 90 mm is a value with two significant figures, you should assume the zero is not significant. The centimeter is a coarse enough measurement length, that when implemented for everyday measure, any useful value will have a decimal point, and is more appropriately written in millimeters.

The “implied precision” argument against using millimeters exclusively in everyday life is one that has an appearance of technical relevance, but is no more than an ad hoc truthiness statement. Everyday it is empirically demonstrated as vacuous by those who construct metric buildings in Australia, Bangladesh, Botswana, Cameroon, India, Kenya, Mauritius, New Zealand, Pakistan, South Africa, United Kingdom, and Zimbabwe. It is also theoretically superficial when examined carefully. Adopting knee-jerk contrarianism mantled in truthiness does not contribute to human understanding, it only attempts to squelch it.

Why is this question worthy of an entire blog? Because we probably get more flak on the millimeter vs centimeter question than any other. And the flak comes from metric advocates. Occasionally, it comes from a metric advocate whom we admire. And yet, the argument for keeping the centimeter hanging around like an albatross is always based on a misunderstanding of precision: the notion that that extra zero has some meaning beyond establishing scale.  It doesn’t. Scientists, engineers, and mathematicians are all in accord that it doesn’t.  It really isn’t even a metric question, but it’s only metric advocates that aren’t on the same page here. Odd, that.

[1] Hill, Theodore P., “A Statistical Derivation of the Significant-Digit Law” 1996-03-20 Georgia Institute of Technology

If you liked this essay and wish to support the work of The Metric Maven, please visit his Patreon Page and contribute. Also purchase his books about the metric system:

The first book is titled: Our Crumbling Invisible Infrastructure. It is a succinct set of essays  that explain why the absence of the metric system in the US is detrimental to our personal heath and our economy. These essays are separately available for free on my website,  but the book has them all in one place in print. The book may be purchased from Amazon here.


The second book is titled The Dimensions of the Cosmos. It takes the metric prefixes from yotta to Yocto and uses each metric prefix to describe a metric world. The book has a considerable number of color images to compliment the prose. It has been receiving good reviews. I think would be a great reference for US science teachers. It has a considerable number of scientific factoids and anecdotes that I believe would be of considerable educational use. It is available from Amazon here.


The third book is called Death By A Thousand Cuts, A Secret History of the Metric System in The United States. This monograph explains how we have been unable to legally deal with weights and measures in the United States from George Washington, to our current day. This book is also available on Amazon here.