A critique of the Cartesian coordinate system

(copyright 2018 by Russ Hodge)

March 10, 2018

To the editor.

Dear Sir(s) and/or Madam(s),

My compliments on the zesty new editorial direction your journal is pursuing at a time when the print media are generally regarded as a horse so dead you could never beat it back to life. Not that a scientist would ever beat a horse, of course, unless there was some therapeutic purpose to it (e.g., cardiac massage). Which in this case would be senseless, because the horse is purely metaphorical – not that my colleagues hold a metaphorical horse in any less regard than a real one. But I digress.

It is bold indeed of you to publish in serial form the epistlary war that has been raging for so many months between Prof. Dr. Marius Linksunteraermer and Dr. Dr. Vincenzo Gambini. I am so caught up in it that I find myself waiting for each edition of your journal in a state of heightened anticipation bordering on erotic arousal (which is really saying something, at my age). There’s nothing better than watching two scientists go at each other tooth and nail, rapier and switchblade, particularly when they are arguing about something of no significance whatsoever.

To summarize: about a year ago you published a paper by Gambini’s lab. Linksunteraermer had no issue with the work or its results, with the exception of the legend to one figure. As customary, you gave Gambini the opportunity to respond and the two letters appeared side-by-side. Naturally Linksunteraermer felt compelled to respond to the response, and naturally Gambini provided a rebuttal. Now, of course, the Djinn had been let out of the bottle and there was no putting him back.

Here the figure and legend as they originally appeared (vol. 139, p. 1206):

Fig 6a.  We collected ca. 1 billion pieces of data as summarized above (for the complete list see the Supplemental Data). The results fell into four distinct groups which could be cleanly separated into the quadrants 1, 2, 3 and 4 as indicated. Since printing 1 billion dots would have taken about 6 years, we simplified the diagram by selecting the most representative dot in each cluster (i.e., the one closest to the middle), then rounded it off to the nearest integer for plotting on the chart. The result confirms our hypothesis that four distinct mechanisms are at work in the system. (Note: What appears to be a single point at position 0,0 is actually four median points, one in each quadrant: after rounding rounding off, they overlap.)

What followed is best captured by citing a few passages from the exchange. Linksunteraermer’s first letter expressed polite skepticism that 1 billion data points could be so cleanly sorted into four distinct groups. “There must have been outliers,” he protested.

Gambini provided the following “clarification”: “We could cleanly distinguish the datapoints appearing in quadrant 1 from those in quadrants 2, 3, and 4 because the data point with the highest value for y in quadrant 1 lay higher than any of the y values for the data points in quadrants 3 and 4, although not necessarily higher than the highest y in quadrant 2, nor any of the x values, of course, and the lowest point in quadrant 1 also lay higher than the highest points in quadrants 3 and 4, but not that of quadrant 2; and the lowest y value in 1 also lay higher than the lowest points in quadrants 3 and 4, although it was not necessarily lower than the lowest point in quadrant 2. Taken together, this implies that both the highest and lowest values for y in quadrant 1 were higher than the y values for either the highest or lowest points in quadrants 3 and 4, although the highest value for y in quadrant 1 may have been lower than the highest in quadrant 2 and the lowest higher than its lowest. That accounts for the y value. The values for x behaved exactly the same way, which spares me the task of having to repeat all of that – provided you make the following alterations: exchange the terms ‘highest’ and ‘lowest’ with ‘farthest right’ and ‘farthest left,’ respectively, and wherever the terms ‘quadrant 2’ ‘quadrant 3’ or ‘quadrant 4’ appear in the description above, replace the 2 with a 4, and replace the 3 with 2. Be careful about the order in which you do this so that you don’t change the 3 to a 2 and the 4 to a 3 and then change the resulting 2 to a 4 and 3 to a 2; only one transformation may be applied per quadrant.”

While mere mortals might have been daunted by this answer, Linksunteraermer’s response came a scant week later: “You’ve completely missed the point. My question is, how can you be sure that none of the points plotted in quadrant 1 actually belongs to one of the other experimental groups, which would mean you would have to cluster it with datapoints in quadrants 2, 3, or 4 rather than grouping it with the other points lying within quadrant 1?” To which Gambini replied, “Because by definition a point in quadrant 1 lies to the right of the vertical axis and lies above the horizontal axis (i.e., both x and y have positive values), and any point failing to meet both criteria must lie in one of the other quadrants, depending on whether it is positive or negative, also by definition,” to which Linksunteraermer retorted, “But that doesn’t address the question,” earning the following rather snarky reply from Gambini: “Perhaps you failed to understand the answer,” and from there the discussion deteriorated into an exchange of personal insults, including some rather colorful and highly creative references to anatomical features and their functions in reproductive biology, in some cases across a species barrier, which you faithfully printed. I will refrain from going farther to avoid spoilers, but I ensure anyone who cares to read the letters that they will find entertainment of the highest order.

I have not written this letter as a means of getting drawn into what will surely end in at least one homocide. Wagers are being made on who will survive, throughout the research community. My money’s on Gambini; the intellect responsible for that figure legend and the subsequent explanation will not go down quietly.

No, all of this reminded me that it was time to finish a little project I started a few years ago demonstrating that any attempt to plot data onto an x-y axis, the type that Gambini used, is doomed because of a fundamental flaw of reasoning that renders them all meaningless.

I enclose a copy along with this letter, which I humbly submit to your journal for consideration.

With warmest regards,

Wilford Terris, Prof. emeritus (at large)




An inherent flaw in Cartesian coordinate systems is associated with death by elevator and accurately predicts the end of civilization as we know it

by Wilford Terris, Prof. emeritus (at large)

Introduction: a brief history

Scientists and normal people, too, are familiar with the practice of plotting information on a system with an x-y axis. The formal designation is the Cartesian coordinate system (or CCS, fig. 1). It is named for half of the French mathematician René (Des)cartes (fig. 2), who invented it but later disowned it, at least partly – apparently the latter half, to judge by its name. He did not, as some claim, cry out on his deathbed that it had been inspired by the Devil; his disappointment was purely financial. The coordinate system became the Microsoft of the 17thcentury and would have made Descartes as rich as Bill Gates, if you imagine Bill in a powdered wig and pointed purple shoes with large buckles, if Descartes had remembered to patent it. As a result of his failure to do so, he never saw a penny (or sous, as the French call it) while others made a killing. By the time he came up with his second major invention, the “mind-brain dichotomy,” he’d learned his lesson and patented it right way. Unfortunately it turned out to have no practical value whatsoever, except as a sort of occupational therapy for philosophers. Descartes became pathologically bitter and died in poverty while teaching in Sweden on a visiting professorship contract in the company of a female robot that was either a life-sized replica of his deceased daughter or a mechanical sex toy.

Fig. 1.  Cartesian coordinate system


What made the CCS so popular was that a portable version (called the iCCS) could do just about everything that smartphones are used for today. You could do addition and multiplication on it, and if you were really clever subtraction and division. And what is GoogleMaps, really, other than an x-y axis with a few details filled in? You could use it as a chessboard, and even play the game “Pong” on it by calling out equations such as “y = (x-1)!” to describe the trajectory of the ball. The biggest-seller, of course, was Battleships, which hadn’t made much sense at all before the arrival of a coordinate system. When a few customers complained that you couldn’t watch cat videos on the iCCS, the King had them beheaded, to the delight of all.

Fig. 2  René Descartes, (1596, 1650) inventor of the Cartesian Coordinate System


First scientific applications

With the arrival of the Cartesian coordinate system, people began counting things they’d never paid attention to. A cause could now be connected to its effects, supported by actual data. On the x axis you could plot the number of beers a person drank, for example, and on the y axis the number of times they fell down. A situation in which both numbers always rose together or fell together, like the drinking-and-falling effect, was called a correlation (derived from the French word for incest) and was generally assumed to indicate a causal relationship. This led to landmark publications in journals such as Nature with titles such as, “Drinking alcohol makes people drunk.”

Another revolutionary scientific discovery to emerge from the CCS was the concept of being fat. Some people had always been thicker than others, but it didn’t seem to have any practical consequences, so no one particularly noticed. There had been a theory that if a rotund person jumped from a high place, he would bounce upon hitting the ground. Galileo proved this wasn’t true in the first human clinical trial on falling, carried out at the leaning Tower of Pisa. The two colleagues he used as the test subject (250 kg) and the control (50 kg) not only hit the ground at the same time but produced nearly identical blots of a gelatinous composition, considered the origin of the scientific practices of obtaining gels and blots. He concluded that a person’s mass had no practical consequences, in terms of its rate of falling or the degree of flattening upon impact, so science agencies struck research on body mass from their funding agendas, and it wasted away for more than a century.

Then a French physician named Jacques Derrière got tired of his wife asking – for the 1000thor 1,000,000thtime – “Does my butt look wide in this dress?” Before offering any conclusions he thought it might be wise to gather some scientific data. He began measuring the height and weight of everyone he met, plotting one against another on an x-y axis, leading to the first first Body Mass Index, or BMI. The results were interesting. Most people’s measurements fell within a particular zone of the chart. A few points lie far outside this zone, generally people of normal height but with the weight of a grand piano, or a small whale. Since these were frequently the same people who drank lots of alcohol and fell over outside in the street, Derrière called them outliers.

Derrière packed one of them in his wagon and carted him off to the hospital for dissection. After a few pokes with a needle to ensure the putative Scientific Breakthrough was dead, he made his first incision in the gut, looking for balloons or some other mechanism that would make a fellow swell up that way. What he found was a blubber-like tissue that he described, in his landmark paper, this way: “Upon close examination, the patient was determined to be Full of Adipose Tissue, or FAT.”

He extended his plot of BMIs to other species, and the next time his wife thrust her impressive posterior toward him and demanded an answer, he was ready with precise data. “Your butt has the BMI of a small elephant,” he told her, upon which she immediately rewarded him by elevating him to the rank of Martyr to the Cause of Science. His data, thankfully, survived.


A tool for engineering

The BMI charts from Derrière rapidly became valuable references for experts in disciplines beyond butts, such as the engineers who designed the second elevator ever to be constructed. Elevator design was not yet a science, because there was only dataset, from the first elevator, and to become a proper field of science it is generally considered necessary to have at least two. The results of the first elevator experiment are recorded in the Cartesian coordinate chart below (Fig. 3).

(Image being processed)

Fig. 3. First human trial of an elevator. The x axis represents
an individual’s lifespan (measured in seconds from the moment
of boarding the elevator); the y axis records the weight of each
person on board. Based on this chart, engineers concluded that
there was no association between an individual’s BMI and risk
of death by elevator.

Although later no definitive cause could be established for the premature termination of that experiment, the frayed ends of the rope were suggestive of some sort of separation event. While no one disputes that the effect on 50 passengers who had eagerly volunteered for the maiden voyage of an elevator was rather negative, on the whole the experiment was considered a success: their cabin had ascended nearly 3/4thsof the height of the Eiffel Tower before abruptly reversing directions.

The Assistant Head Engineer, whose social status had not allowed him on board, suspected that the thickness of the rope might have been a factor. He could have benefited from the calculations of the Head Engineer, who had been prominent enough to be given a spot on the historic flight, but his notes were an unreadable scrawl and the Head Engineer himself was no longer available for consultation. Had he provided an accurate estimate of the weight of the passengers into his calculations? Had he realized they might be carrying things in their pockets – loose change, car keys – or have body piercings that would add to the weight? For his new caculations, the Assistant Head Engineer could draw on Derrière’s BMI tables to estimate the total weight that a rope would have to hold. At the last moment he remembered to add the weight of the cabin, which was about a ton. This gave a figure that could be plugged into an equation to yield the guage of the rope that would be needed:

more weight = thicker rope


Looming clouds

The purpose of this piece is not to cause panic; if that had been my intent I would have started with alarming statistics about how many millions of people die in elevator accidents every year, then claim that there is currently no way to diagnose an individual’s risk of dying in an elevator accident based on molecular markers. And the number of deaths just keeps accumulating. (Indeed, how could it get smaller? You can’t subtract any from the bottom of the pile.) There are no effective treatments, because the first symptoms (a sensation of falling, usually accompanied by loud vocalizations) appear just moments before death. By that time it’s too late to unpack the parachutes.

Later experiments showed that parachutes don’t work in a falling elevator anyway, due a localized disturbance in the laws of physics that Newton called the “temporary exception for a parachute in an elevator.” So there is an urgent need for further research to design novel rational therapies for a condition that causes morbidity and mortality not only for the victims, but also for anyone who happens to be standing at the bottom of an elevator shaft at the wrong moment.

Today’s engineers draw on the Body Mass Index and a plethora of other Cartesian coordinate systems in the design of practical equipment such as elevators. Modern elevator science has progressed far beyond the state of affairs which reigned at the second clinical trial of an elevator, for which it was surprisingly hard to find volunteers. In the end they substituted cows, replacing the Body Mass Index with a Bovine Mass Index, in one of the first historical examples of replacing humans with animal subjects. That experiment was a success; all of the cows reached the top alive. What happened when they stampeded out of the cabin and found themselves on the rather small viewing platform at the top of the Eiffel Tower is another matter, but is irrelevant to the study’s results.

As I mentioned, I do not wish to cause panic. But let us imagine, purely hypothetically, that someone were to discover some inherent, fundamental flaw in the Cartesian coordinate system. Wouldn’t this call into question the structural integrity of every real-world device designed using a CCS? Every elevator, bridge, floor, cieling, zipper, button, gas mileage, the location of every street and town? Even the Earth itself wouldn’t be in the position we had assumed it to be. What about the social institutions, government, and the premises on which society is founded? Yes, CCSs have been used in creating these institutions as well. Thinks of the consequences? What would happen if the weaknesses in every CCS caused them all to fail at the same time, perhaps through the activity of a virus that has been lying dormant inside a glacier, and is suddenly revived through global warming? This would be likely to happen at a time we can’t predict, because all of our estimates depend on the very type of CCS that is hastening toward a collapse?

Problems with the Cartesian coordinate system have been known for many years, but papers demonstrating them rarely reach the pages of journals. The central dogma of the CCS is the foundation of the entire system of impact points by which editors and the other Plutocrats of science justify their power; any challenge to the model calls their authority into question. But evidence from outside the mainstream has now accumulated to the point that it is finally spewing through the cracks, while the status quo no longer has enough fingers to plug all the leaks. There is a major paradigm shift in the making. It is usually heralded by portents: a plague of locusts, a weird person who might be a zombie, inexplicable changes in your partner’s mood, your cell phone battery draining faster than it should… If you notice these or similar signs, take appropriate measures. You can never go wrong laying in a nice assortment of canned foods, but firearms are not advised. A paradigm shift cannot be countered by conventional weapons.



(coming soon, stay tuned!)

If you liked this article, you will probably like:

The Evolution of Pizza: Novel Insights into the Fourth Domain of Life

The Subtle Art of the Truly Vicious Review  and

Even God’s First Paper got Rejected

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I am a science writer at the Max Delbrück Center for Molecular Medicine in Berlin, author of fiction and popular science books, an artist, and a professional musician who performs on the viola da gamba and Medieval and Renaissance stringed instruments. I edit manuscripts of all types and teach the full range of scientific communication skills. I am doing theoretical work in this subject - see for example https://goodsciencewriting.wordpress.com/2018/03/11/ghosts-models-and-meaning-in-science/

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