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Testing: From Aristotelian to Galilean, Part 2

In this post I’ll continue on directly from part 1 where we ended up with a lot of models and a recognition of competing interpretations of quality along with a need for testability.

I should note that this post likely provides the culmination of all the “History and Science” posts up to this point. The goal throughout this series of posts was showing a bit of how testing had to develop so that everyone could understand the historical basis of how testing evolved within the context of the scientific method.

Creating Increasingly Complicated Systems

There’s a key lesson that people, like astronomers and other thinkers, were starting to learn as they got closer to actually doing science.

With all of our models, we can increase complexity. Not a problem. Well, as long as our testing methods are able to keep up with that complexity. And our testing methods must be capable of being related to observations that have value. But, ideally, not just value in how something works but also in terms of how the way something works aligns with our experiences. And maybe a little about why it all works the way it does to provide the experience it does.

Let’s head back to Copernicus and dig into a little more history along the same lines as we looked at in the previous post.

It wasn’t just Copernicus that found circle for orbits didn’t always work so great. Those early Greek astronomers that Copernicus was responding to also recognized that circles centered exactly on the Earth had problems. So they knew they had to abandon a literally geocentric model. (They couldn’t, as I’ve pontificated on before, be model literalists.) This wasn’t the same as leaving a geocentric model entirely, however. Meaning, they were willing to accept a situational view of quality.

So here we’ll look at how the Ptolemaic system evolved the Aristotelian system we mainly looked at in the last post. But, to do that, let’s make sure we have our understanding of Aristotle’s conception firmly in mind. Here we’re doing something I’ve talked about before: recovering context by test thinking.

Starting With Aristotle

The simplest first step in Greek mathematical astronomy was to place planets on circular orbits.

The second step was to move the center of the circular orbit a small distance away from the Earth.

This non-Earth centered circle was known as an eccentric.

The third step was to place the planet on an epicycle.

Once the epicycle was defined, the main circle that defined the planet’s orbit became known as the deferent. In this model, the planet moves in steady, circular motion around the epicycle while the center of the epicycle moves in steady, circular motion around the center of the deferent.

It’s worth noting that all actual qualities of the orbiting planet — size, brightness, speed — appear unaltered but those qualities are perceived to change as seen by an observer on Earth. Put another way, the observed size and brightness and angular speed across the sky of the planet, as seen from Earth, will change throughout an orbit although the actual size and brightness and angular speed along the circle will not change.

So there again is that idea of what we actually see and what is actually the case can differ. Think about — really think about — what it means for a discipline like testing to evolve in that kind of context.

Yet with all this, the predictions — if such they could be called — of Aristotle’s theory — if such it could be called — didn’t quite match the actual positions of the planets. Clearly some more work was needed.

Refining with Ptolemy

Well before Copernicus, but well after Aristotle, Claudius Ptolemaeus (better known as Ptolemy) proposed a change. Specifically, he suggested we imagine a point known as the equant.

I’ll spare you most of the details but the idea was that now, although the deferent is centered on the actual center of the whole model, the motion of the epicycle along the path of the deferent is measured with constant angular speed as seen from the equant.

Yeah, that’s not confusing at all, right? It’s a bit easier to have a working visual:

Needless to say it all gets a little complicated:

This is sort of like how in modern technology, we layer libraries on type of frameworks and construct something that seems to do something.

Anyway, by the year 1500, observations had shown that the predictions of Ptolemy’s theory, much like Aristotle’s, still didn’t quite match the actual positions of the planets. And yet the geocentric model was still pretty much accepted as the only model. So if you have a model that doesn’t match observations but you trust the model more than your observations, what do you do? Well, you fit the observations to the model.

In this case, the way the astronomers made it fit the observations was to add additional epicycles. Remember in the last post how it was a case of adding more spheres? Well, here we just add more epicycles.

So what you see here is a constant refinement; a form of testing, to be sure.

The goal was to get a model to be more and more accurate. But this was coming at the cost of some extreme complication. This complication was exacerbated by the fact that our testing methods weren’t evolving at the same pace.

So what ended up suffering were two aspects of testability: reproducibility under varying conditions and predictability.

Let’s Get Reductionist

Okay, so let’s consider the Ptolemaic system in a bit of a simplified form.

Now let’s contrast that with a somewhat simplified Copernican view.

What Copernicus did to get his own model to work was resort to epicyclets, or “little epicycles.” In effect, Copernicus got rid of the big epicycles of the Ptolemaic system and replaced the equants — which pretty much nobody liked — with little epicycles. These were small circles sort of attached to the planets’ orbits that could account for their varying speeds across the sky.

Wait, let’s do a check-in here. You see what happened, right?

In the course of trying to get things “correct,” Copernicus ended up really just shifting complexity a bit; his model retained epicycles. They were just centered on the Sun instead of the Earth!

Testing Shows Too Much Reduction Is Bad

Let’s consider the two models without all the complications I just talked about. Here’s the Ptolemaic system:

Here’s the Copernican:

With the exception of where the Sun and the Earth are, you could be forgiven for thinking there’s not a whole lot of difference there.

Yet this is extremely oversimplified.

It is true that the Copernican system was simpler than the Ptolemaic system, but it still wasn’t simple. In fact, you could argue Copernicus made things more complex. Copernicus required forty-eight epicycles in his heliocentric system. That’s as compared to forty in the Ptolemaic geocentric system. But, on the plus side, Copernicus’ model did not use the complicated equant.

In fact, as we now know, the Copernican system, with the planets following perfectly circular orbits around the Sun, actually worked less well, in terms of reproducing accurately the observed movement of the planets, than the vastly more complex but highly evolved Ptolemaic system. Put yet another way, the model of Copernicus is just about as complicated and not appreciably more accurate than the Ptolemaic system and, in fact, is a marginally less accurate approximation in certain cases.

Not all of this was recognized at the time because, remember, testability was still suffering.

Thus there was a period of time when these two systems competed. You had the inelegant accuracy of the Ptolemaic against the elegant inaccuracy of the Copernican. This contest is a model of the progress of science. The scientist always strives for accuracy but sometimes accuracy has to be seen as approximate and thus something can be “good enough.”

That was a key thought pattern that was necessary for testing to emerge as a discipline underlying the scientific method. And it’s a thought pattern that really came to the fore with Galileo, which I’ll get to shortly.

Testing Shows That Reduction Can Show Equivalencies

Now, here’s an interesting thing. The ancient system and the Copernican system were observationally equivalent.

In the Ptolemaic system, the Earth was at rest at the center, and the Sun and planets all went around it. Here’s a simplified version of the Ptolemaic model, showing just the Earth (blue dot), Sun (orange dot), and Venus (red dot):

In this model, the Sun goes around the Earth in a circle, but Venus has a more complicated motion, involving a deferent (the big red circle) and an epicycle (the little red circle).

This is necessary because, as we talked about above, Venus — like the other planets — goes through periods of retrograde motion during which its apparent path through the sky reverses direction. In this model, retrograde motion occurs when the epicycle carries Venus close to the Earth. At those times, the epicycle is making Venus go around backwards faster than the deferent is making it go forwards, so it appears to reverse direction.

Actually, as I’ve mentioned, Ptolemy’s system was somewhat more complicated than this: to get the motions of the planets right in detail, he needed extra epicycles, as well as things called eccentrics and equants. But the circles in the above visual are the most important ones. They’re all you need to get the basic features of the Sun’s and Venus’s apparent motion right.

Here’s a funny thing about this model: Venus’s epicycle goes around the Earth at exactly the same rate as the Sun — once per year. That’s Ptolemy’s way of accounting for the fact that Venus has something called “bounded elongation.” This is just a domain phrase that means Venus always appears near the Sun in the sky.

As an observational case in point that serves as evidence: you’ll never see Venus rising in the east when the Sun is setting in the west. One thing that Copernicus found unsatisfying about the Ptolemaic system was that there was no good reason for these two motions to be synchronized like this.

Ptolemy put Venus’s orbit inside the Sun’s orbit, as I’ve shown. But there’s no reason he had to. You could make the Sun’s orbit bigger or smaller by as much as you want, and everything an observer on Earth sees would remain exactly the same. In particular, you could shrink the Sun’s orbit until it was exactly the same size as Venus’s deferent:

Let’s call this a Modified Ptolemy model. This model has the exact same appearance as Ptolemy’s model: if you use the two models to predict where the Sun and Venus will appear in the sky on any given date, you’ll get the same answers.

From a test vocabulary perspective, with the same context and same action, you get the same observable with both conditions. Thus do we have an equivalence class.

As a matter of fact, a well-known astronomer of the medieval period, Tycho Brahe, advocated exactly this model, in which the Sun goes around the Earth while Venus — and the other planets — go around the Sun.

Now take the above picture and imagine what it would look like from the point of view of someone standing on the surface of the Sun. That person would see both Venus and Earth circling the Sun like this:

This picture is exactly the same as the previous one, but with a change of reference frame: everything is drawn from the point of view of the Sun rather than the Earth. Once again, the two models are observationally equivalent.

If you freeze the two pictures at any moment, the relative positions of Earth, Sun, and Venus will be exactly the same. That means that an observer on Earth in either of these two pictures will see the exact same motions of Venus and the Sun.

The last picture is how Copernicus explains the motions of the Sun and Venus.

The key point here is that, although the three pictures are conceptually quite different, they’re all observationally equivalent. They’re exactly equally good at predicting where Venus and the Sun will appear on any given day.

Predictability is one of the outcomes of testability! While I’ve simplified things here to keep this post from becoming a book, how this played out was a key evolution of testing.

Testability Suffers

But . . . wait. Actually, Aristotle’s thoughts and thus Ptolemy’s methods of building on those thoughts did describe things pretty accurately. I mean, sure, maybe it was complicated — but it worked, right?

Well, did it? Maybe, maybe not. I’ll let you read books about science to come to your own determination. But here is what it didn’t do. It didn’t really allow the system as a whole to evolve and accommodate new facts. New things could be added but at the cost of a great deal more complexity.

As such, it was also increasingly difficult to get Copernicus’s system to work precisely. Adjustments would resolve discrepancies in one place, only to generate new errors somewhere else.

Testing Had to Scale With Tooling

Let’s take a look at an intervening thing that happened between Copernicus and Galileo.

In October 1604, a new star appeared in the night sky. This new star, or nova, was first observed by Johannes Kepler. The star became brighter than any other star and was visible during the day for three weeks. Here’s a (very) rough idea for comparison of how this would have appeared.

A few things immediately stood out about this thing.

First, this new star maintained its position among the fixed stars of the sky, unlike the moon and planets, which moved daily across the stellar background. It was reasoned that this new star must therefore be farther away than any planet. Second, this new star was transient: it appeared out of nowhere, was prominent for a time, then over a period of months faded away.

That latter was a big problem!

Aristotelian astronomy held that everything beyond the moon is unchanging: the stars shine forever, the planets follow their prescribed orbits forever. The existence of a transient event in the so-called was philosophically untenable.

So a new feature was hard to incorporate. Why? Well, because the testing was inadequate. We didn’t have a way to test why something should be the case. Consider again Copernicus’s concern about why the bounded elongation for Venus was in place.

But alas! A particular testing tool was created around this time: the invention of the telescope in the Netherlands in 1608. Galileo started using this testing tool and, in fact, built some variations on it.

Galileo observed Venus. He found that it showed phases, just like our moon, waxing from thin crescent to full and back again.

This was exactly what the Copernican model predicted, with Venus going from full when it was on the other side of the Sun and — to an observer on Earth — fully illuminated, to a bare sliver when Venus was between Earth and Sun. Thus the Copernican system explained another puzzle, which was that the brightness of Venus didn’t vary all that much despite its phase. The reason for this — the why — is that when Venus is full, it’s furthest from Earth. When Venus is a crescent — reflecting less of the Sun’s light — it’s much closer.

So this is good, right? But notice what Galileo was doing here that was different from what Copernicus did. Galileo was using test tooling in a very specific way to refine his understanding.

The above facts really made Galileo pay attention because they depended on specific and calculable quantities. Further, the above was derived from a mathematical model that could be compared with what an observer actually saw. So the quality was now related much more to the experience of interaction.

Historically speaking, the crucial bit here is that Galileo realized that Copernicus’s model of the solar system wasn’t some mere geometrical hypothesis. Rather, the model was a picture of the solar system as it actually is. This gave a rationale, beyond philosophy, of choosing between the two systems.

Testing Helps Seek Explanations

Now even all of what I just talked about came after something else that is relevant and, in fact, is likely what influenced Galileo’s thinking.

Daniel Santbech, around 1561, published “Problematum astronomicorum et geometricorum sectiones septem” (“Astronomical and Geometrical Problems in Seven Sections”). This was a book of geometrical problems. The book included an analysis of how far a cannonball would travel, depending on the angle at which it was fired into the air.

Does anything immediately stand out to you about this?

It’s basically utter garbage, right? That is not how a cannonball moves. Not even a little bit. You have to imagine that any observer at all would have been able to know that this is not actually what happens when a cannonball is fired. Certainly people at that time would have noticed that cannonballs follow a certain path, an arc, and then relatively gradually fall back to the ground.

So wouldn’t anything like this have seemed ridiculous to publish?

Well, here’s some context to note: the purpose of the diagram was to convey the accepted orthodoxy of Aristotelian physics and match it to a simple trigonometrical calculation. The goal wasn’t to provide an explanation for what is actually observed.

At the time, the idea was that if there was an apparent conflict between orthodox teaching and observation of the world around you, your best bet was to stick with orthodoxy and conjure up some plausible reasons why someone shouldn’t necessarily believe what they saw.

Think about that context in terms of how a discipline like testing had to evolve.

The fundamental duty of a university teacher in those days was not to foster intellectual inquiry among students nor to push students towards seeking new knowledge. Instead the duty was to instruct students in the certainties conveyed in orthodoxy. And those certainties came from ancient philosophy. A philosophy teacher would thus deliver Aristotelian wisdom. Any commentary that might get added on would only be to perhaps better elaborate the philosophy.

Daniel Santbech certainly was well aware that cannonballs did not appear to follow the path prescribed by philosophy. That prescribed path was a straight line rising at some angle, followed by a vertical drop. So there was what philosophy said and what Santbech’s eyes told him. Santbech’s conclusion — as it was for just about everyone — was that since philosophy was incontrovertible, then our senses must deceive us in some way when they contradict philosophy.

Again, I ask you to imagine trying to be a tester in this context. Testing had its birth in a context where what you observed should, likely, be doubted. Imagine the chilling effect that has on actual experimentation. That’s more than a bit problematic since experiments are based on observation.

Why Do Things Happen?

In Aristotelian philosophy, it was said that objects fell toward the center of the Earth because they sought out the lowest place. To our modern way of thinking, this sounds like a ridiculous explanation, right? An inanimate object, like a rock, seeks out a place to be? Really?

Yet to the committed Aristotelian it made sense — it provided an explanation rooted in the idea of purpose. The idea of “purpose” was a key quality: things had a purpose and thus had some internal motive force that constrained them towards achieving that purpose. In the Aristotelian world, things happened in order for some end state to be achieved. The philosopher’s task was to figure out those purposes, no matter how specious the reasoning might seem and no matter how much that reasoning seemed to not match what we observed.

Still, how ridiculous, right? Well, consider that even Kepler’s idea that planets move faster the closer they are to the Sun seems perilously close to the Aristotelian principle that objects fall to Earth because they are drawn to the center of the universe.

The Sun, to Kepler, was the motive power of the universe and yet he had no explanation for why that might be the case. How did this motive power actually work? Kepler couldn’t really say; that’s just how it was. Planets moved faster the closer they got to the Sun because that was what the Sun, as the motive power of the universe, wanted them to do.

Kepler had undeniable mathematical skill but these skills were aligned in his mind with a fundamentally mystical view of the universe. Interestingly, Kepler’s views reflected an era in which there wasn’t a clear distinction between astronomy and astrology, which were aligned with mathematics. But there was a growing distinction between astronomy/astrology with physics, which was aligned with what came to be called natural philosophy.

Thus there’s a very defined and specific middle view between the Aristotelian and the Galilean, which we can call the Keplerian. There’s no way I can incorporate that into this post without it becoming even longer than it is!

How Do Things Happen?

Galileo, by contrast to Aristotle, said nothing about why objects fall. Galileo’s method was to observe closely how objects fall. Galileo wanted to construct an accurate quantitative description. Crucially, however, it’s not that Galileo thought the “why” question was irrelevant; rather, he simply understood it to be a question he could hold off on worrying about since the answer would not detract from his ability to make a mathematical statement about the how.

Think about the level of insight this showed for our early specialist tester! The “why” could be answered distinct from the “how.”

But isn’t that a leap of faith that perhaps might be a leap too far? After all, what if the “why” did have some impact on the “how”? Well, As Galileo was figuring all this stuff out, he knew he had to perform experiments. Further, Galileo had to recognize when certain experiments were equivalent to others and what insights could be drawn from these things. Here’s an example:

Galileo eventually recognized that dropping things off of a tall height would be the same as dropping them off of a very steep inclined plane. And that recognition, Galileo came to understand, lets you model things even more accurately once you can correctly abstract.

This let Galileo reason that, in the case of (a), a smooth ball on a smooth incline speeds up going down. In the case of (b), that same smooth ball on that smooth incline slows down going up. This was the case regardless of the level of the incline. The limiting case here, which would be (c), would be a perfectly smooth and level surface. Galileo had what I previously referred to as the intuition for abstraction.

Yet Galileo was the first to admit that experiments such as those he conducted might not be perfect: he was aware of subtleties such as air resistance, as well as constraints, such as the practical difficulty of letting two objects go exactly simultaneously from given heights or even imperfections in the inclined planes.

The crucial point was that Galileo ended up at a proposition: all objects fall at the same rate. This was much closer to the truth than Aristotle’s proposition that heavier objects fall faster.

Testing Helps Get Close to Truth

Galileo understood that getting closer to the truth was the important thing. I can’t convey enough how radical of a concept this was for people at this time. Philosophers liked to deal with absolutes. Philosophical statements are either true or false. That’s how they can become the basis of orthodoxy. Galileo was saying that a little inexactitude was actually okay and probably expected because, yeah, our senses aren’t perfect. But we shouldn’t believe that our senses are always entirely deluded either.

To the rank-and-file philosophers of the time, this was an unsettling and dangerous suggestion.

So what stops a cannonball from flying off forever? Well, because it falls to the ground. Indeed, it begins to fall, as Galileo eventually realized, as soon as it leaves the mouth of the cannon. The two motions — along the line of flight and down toward the ground — happen together. It’s not the case, as the Aristotelians believed, that you get one motion and then the other motion, as in Santbech’s illustration.

Galileo ended up having to use an imperfect shape (a parabola) to model the flight of the cannonball.

Remember in previous posts how we talked about those circles being the only shapes to consider (philosophically) because they were perfect. Well, Galileo challenged that too. And, as a side note of interest, Johannes Kepler would use another imperfect shape, the ellipse, to depict the path of planets as he worked to refine the model of Copernicus.

The Test Specialty Emerges

Galileo wanted to make accurate observations and measurements, reason carefully about their meaning, and thus find the right quantitative set of data, including the mathematics to explain that data, to fit what he discovered.

This was how testing grew up.

Crucially, Galileo’s view was that maybe you can’t have the perfect understanding; quantifiability has its limits. You have to approach some sort of truth as opposed to an absolute truth.

This is how the notion of quality, within the context of testing, grew up.


This article was written by Jeff Nyman

Anything I put here is an approximation of the truth. You're getting a particular view of myself ... and it's the view I'm choosing to present to you. If you've never met me before in person, please realize I'm not the same in person as I am in writing. That's because I can only put part of myself down into words. If you have met me before in person then I'd ask you to consider that the view you've formed that way and the view you come to by reading what I say here may, in fact, both be true. I'd advise that you not automatically discard either viewpoint when they conflict or accept either as truth when they agree.

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