I’ve written before about the importance and methodology of thought experiments, and how they are often screwed up even by professional philosophers (see On Hosing Thought Experiments). Today I’m going to pull a page out of the history of science to explain three thought experiments Galileo famously fucked up. We often forget all the ways scientists of his era actually got things wrong, sometimes even advocated harebrained nonsense. They were not the perfect paragons of scientific method legend has crafted them to be. We just “forget” all their screw-ups and praise their successes, and in result craft the illusion that they only had successes. In fact they often didn’t use the scientific method, or hosed its use, to reach conclusions about the world they remained nevertheless adamantly certain about. Newton, famously, was an alchemist and fanatical biblical numerologist. Harvey scoffed at the idea of capillaries and advanced kooky ideas about animal and human reproduction. And so on.

Well, so did Galileo. And it’s important to know this, because his hosed thought experiments are sometimes still held up as good examples of the practice that in fact they aren’t. And moreover, we ourselves can learn things from his mistakes. That’s my point here today.

Imagining How Gravity Works

There are actually a lot of modern misconceptions about the history of gravitational theory. The Romans, for instance, were much more diverse in their hypotheses about it than just “what Aristotle said.” In fact Aristotle’s thoughts on this were already obsolete within two generations. Medieval Christians just threw in the trash all subsequent works on gravity, and then falsely convinced themselves Aristotle was the pinnacle of ancient scientific achievement, and converted his obsolete musings into gospel. By the time Galileo got a hold of them, they were easy to refute. But really, that had already happened before. We just don’t get to know about it.

That’s a different story I will tell some other time (some of the details you’ll already find, though, in my book The Scientist in the Early Roman Empire). For the present, let’s look at what Galileo got wrong about it (rather than what he got right; as the latter story is already well known). I already covered this in my previous article (see “Galileo’s Goof”), and there are more things amiss than even I discussed there. But here I want to call attention to the central error in his thought experiment about gravity, so as to illustrate how we can derive a general rule from it. Which this time is, to always obey The Law of Excluded Middle: when you are concluding what possibilities remain, never leave any out. Otherwise you commit the Fallacy of False Dilemma. This is a common fallacy driven by an inborn cognitive bias innate to all human brains: our tendency toward black-and-white thinking, to assume there are only two options and they are at odds with each other and we have to choose one—which often reflects a natural abhorrence of ambiguity, complexity, and nuance. And this happens when we fail to properly test a competing theory to ours by first actually working out in what ways it could be true, rather than trying to come up with ways to insist it has to be false.

I’ll follow the analysis and translation used by Maarten Van Dyck, who quotes Galileo making his famed argument as follows:

Let us first make this assumption: if there are two bodies of which one moves more swiftly than the other, a combination of the two bodies will move more slowly than that part which by itself moved more swiftly, but the combination will move more swiftly than the part which by itself moved more slowly. …

On the basis of this assumption, I argue as follows in proving that bodies of the same material but of unequal volume move with the same speed. Suppose there are two bodies of the same material, the larger a and the smaller b, and suppose, if it is possible, as asserted by our opponent, that a moves more swiftly than b. We have, then, two bodies of which one moves more swiftly. Therefore according to our assumption, the combination of the two bodies will move more slowly than that part which by itself moved more swiftly than the other. If, then, a and b are combined, the combination will move more slowly than a alone. But the combination of a and b is larger than a is alone. Therefore, contrary to the assertion of our opponents, the larger body will move more slowly than the smaller. But this would be self-contradictory.

When people retell this story today as somehow a “brilliant” use of a thought experiment, a true reductio ad absurdum, they usually leave out the first part I just included. Because that should put everyone into a face-palm. Yes, Galileo simply states his conclusion from the experiment as a premise. Sigh. Of course that does make the thought experiment correct, in the trivial sense that if you assume that which entails the conclusion is true, then you’ll find the conclusion is true. But that doesn’t actually teach you anything about the world. And in this case, contrary to how Galileo pulls this trick, it doesn’t teach you anything about Aristotle’s theories of natural motion, either. It certainly isn’t capable of refuting them.

Aristotle never said what Galileo presumes (“if there are two bodies…”). To the contrary, Aristotle’s entire theory of natural fall entails not assuming what Galileo says. So in no way could Galileo have ever refuted it by adopting that assumption. This is an enormous failure to deploy a thought experiment correctly. Galileo simply wasn’t responding to his opponent’s argument at all. As such this is worse than a straw man. Galileo is literally hurling his bat at thin air. He’s not even contacting a straw man of Aristotle’s theory, much less, as one must, a steel man of it. Obviously if Aristotle had stated the assumption Galileo started with, then Aristotle’s theory of natural fall would entail contradictions. But since Aristotle rejected that assumption, that’s simply of no use knowing. Galileo comes across as pretty much a total dolt here—not the genius everyone has mistaken him for when making this argument. He hosed this.

Let’s Count the Ways

There are actually many respects in which Galileo’s conclusion that all objects fall at the same rate regardless of mass is actually in fact false (indeed, his mythical experiment of dropping objects from the tower of Pisa would have mysteriously refuted Galileo had he ever actually tried it). But his argument is also conceptually false. His empirical mistake is a different error to address; it relates to what happens to have turned out to be the case empirically. I’ll come to that shortly. But first I am only concerned with the logical space of competing models Galileo is falsely claiming victory in. Galileo is claiming not merely that Aristotle’s theory is refuted by observation (it is, though not as obviously as people think: see the analysis of Rovelli, which pertains to my coming point), but that it is not even logically possible. And Galileo is simply wrong about that.

Ironically (and we’ll be seeing a lot of irony in Galileo’s mistakes before we conclude here), Galileo acknowledged one respect in which objects do not fall at the same speeds—but completely missed the actual point of that for correctly understanding Aristotle’s model. Note I shall use the word “speed” here, as Galileo did, to mean inclusively of acceleration. Some seem not to be aware that Aristotle actually argued that objects fall at an accelerating rate (and not at a constant speed), which means (contrary to the usual assumption) Aristotle must have derived his theory from observation and not the arm chair after all—an important point we’ll get back to.

That one respect Galileo admitted Aristotle was right about, was “resistance of the medium.” He knew air resistance changed things, causing objects to fall at different rates after all. So his theorizing abstracted that away, in effect as if presuming we are dropping objects in a vacuum, or assuming ceteris paribus, i.e. that in a given case we construct, air resistance is the same regardless of mass. Galileo is thus stating not that objects all fall at the same rate, but that they do so but for air resistance (or the resistance of any other medium they fall through) impeding them differently. Galileo thought this was irrelevant and so could be abstracted out of his thought experiment. Galileo was wrong. The resistance of the medium that objects fall through is fundamental to Aristotle’s conclusion that objects fall at different speeds. And by failing to grasp that, Galileo completely failed to address Aristotle’s actual theory of fall.

Because take note: when we start with the de facto assumption that we are talking about objects falling in a vacuum, Aristotle actually said objects would all fall at the same rate. This is often forgotten; as it was by Galileo. Now, to be fair, Aristotle incorrectly inferred that the rate that all objects would then fell at was “infinite” (and then argues a vacuum cannot exist). But the important point here is that this means Aristotle only believed objects fall at different rates according to their mass because mass allowed them to penetrate a medium with greater force. He did not believe, as Galileo thought (and pretty much everyone else today who keeps getting Aristotle wrong), that it was the mass of the object alone that caused it to fall faster. The only specifically-stated role mass plays in having that effect in Aristotle’s system is in respect to its ability to push harder through a medium. And Aristotle was largely correct about that.

As Rovelli points out, it’s highly unlikely Aristotle didn’t come to this conclusion from conducting experiments: dropping objects of different weights in different mediums, particularly those in which differences of motion are more readily observable, such as water and oil. This explains how Aristotle knew about acceleration, and also about terminal velocity. It is extremely unlikely he could have just “arm chaired” his way to those conclusions accidentally. But Aristotle’s work on gravity was cursory and hardly constitutive of completed research. So he simply hadn’t done the experiment in air.

Aristotle never really assigns figures to anything, but he implied roughly that similarly shaped objects will fall at a rate equal to some unstated constant times weight divided by density. Which is indeed wrong—in air. It’s actually roughly correct in thicker mediums—the ones Aristotle was most likely making observations with. Aristotle was wrong about a lot of things. But he was soon corrected by subsequent scientists on quite a few things; and we know Strato, the third provost of Aristotle’s school, was updating Aristotle’s work by conducting experiments by dropping objects in the air; and we know Strato’s work was furthered a century or so later by Hipparchus using the empirical research from the post-Aristotelian engineering of torsion artillery. We are never told much about their works or what they found (all we have are some random snippets out of context by other authors who poorly understood them). Their resulting books publishing their experimental and theoretical findings on falling objects were simply thrown in the trash by disinterested Christians.

But back to Galileo’s conceptual failure: what happens when we apply his thought experiment to a scenario Aristotle probably was observing? What if we drop a heavy and a light object, of identical size and shape, into a jar of oil? We will see the heavier object fall to the bottom faster. Disproving Galileo. So his conceptual model was clearly messed up, and wholly incapable of refuting Aristotle. What if we applied his thought experiment to this possible Aristotelian experiment? What indeed happens when you connect the heavy and light object together, vertically so they retain the same shape and size penetrating the medium? Galileo’s thought experiment would predict—well, that the combined object must both fall faster and slower! “Therefore” they must always fall at the same rate; but they don’t. So his reductio is…invalid.

What does actually happen? Well, it depends on the density of the final combined object. Per Archimedes. Who corrected Aristotle on this point a hundred years later. There is an upward force slowing an object increasing with the object’s volume, and a downward force speeding the object that increases with the object’s weight (up to a terminal velocity, as observationally recognized even by Aristotle). So if you combine a heavy and a light object, but don’t change their volume (and change nothing else, such as the effects of friction on the falling object), then you double the volume without quite doubling the weight, so you increase the upward force more than you increase the downward force. The result is that the object moves more slowly than the heavier object alone would have (it has been rendered more buoyant by the attached lighter object), but more quickly than the light object alone would have (as it has been rendered less buoyant by the attached heavier object). Galileo’s thought experiment tells us this is impossible. But observation refutes Galileo’s thought experiment.

The mistake Galileo made was in assuming Aristotle’s theorized differences in rate of fall applied even in a vacuum. But it did not. His theory was entirely about how heavier masses push through resisting mediums faster. His attempt to represent this mathematically was an unempirical mess. But it couldn’t be conceptually defeated by Galileo’s thought experiment. And it’s weird that Galileo never noticed this. It’s weirder that people today who praise the “genius” of his failed thought experiment (mostly) haven’t noticed this. Galileo’s starting assumption in that experiment simply isn’t true in a vacuum in Aristotle’s conceptual system; so contrary to Galileo’s supposedly brilliant thought experiment, there could have been a world where Aristotle’s physics held true, where objects fall at a rate equal to some constant multiplied by W (weight) and divided by R (resistance). And what would happen if you connected two objects together would depend on why that rule was being obeyed. In the actual case of resistant mediums (in the actual world we live in), it has to do with Archimedes’ principle (the interaction of weight, volume, and impeding force), and (we now know) aerodynamics and friction and a whole stew of other complexities. Like the similar fate of Aristotle’s theory of elements, reality tends to be far less simple than people think at first. “People” here including Galileo.

Galileo’s thought experiment was incorrect in its results in other ways as well. For instance, it turns out, in actual fact heavier objects do fall faster than lighter objects—if we drop them at separate times (and reset everything after each drop), and measure as the “time of fall” the entire fall, that is, how long it takes for the object to connect with the ground (as Galileo was doing) and not how long it takes for the object to reach the “center of mass” between it and the earth (which Galileo wasn’t doing). As Newton would later prove: all objects attract each other; it’s not just objects being attracted by the earth, but the earth as well by them.

This would be more obvious if we went all Little Prince on Galileo and imagined the earth was vastly smaller than it is. Suppose the earth had a mass of 1000kg and we dropped a 1000kg weight on it: the earth would fly toward it in half the time, and it toward the earth in half the time, each exerting the same force on the other simultaneously. Then if we took that 1000kg weight away and started over by dropping a 1kg weight onto the earth, it would take much longer for that weight and the earth to collide, because the sum of forces drawing them together would be halved (we have gone from a 2000kg attractive system to a 1001kg attractive system). Note how Galileo’s thought experiment was completely incapable of discovering this fact, because he ruled out any possibility of it—yet concluded he had reached an accurate conclusion about gravity. He didn’t. It took Newton to do that. Galileo’s “experiment” was so disastrously wrong it actually blocked him from ever making Newton’s discovery.

But that’s how things actually turned out—which, contrary to scientific mythology, Galileo failed to anticipate. Things could have turned out very differently, also refuting Galileo’s reductio. For instance, one could imagine Aristotle’s theory (or some variant of it) in which each object is like a rocketship: every unit of mass it possesses is equal to strapping one rocket to that ship (the ship’s “mass” corresponding to Aristotle’s assumed constant, such that absent additional mass every object has the same “core” mass, and thus the same “base” velocity). This would accelerate every object toward the earth, exactly in accord with Aristotle’s theory: light objects (one-rocket ships) would accelerate more slowly than heavy objects (two-rocket ships), and contrary to Galileo’s broken thought experiment, if we tied the two together what we’d have is a heavier object (a three-rocket ship), which would go faster than either object would by itself, not “at the same time” slower as Galileo’s thought experiment purports (a similar point is made by Markus Schrenk).

Another model is the Le Sage’s theory of gravitation, which (suitably tweaked) could conceivably have been true had the cosmos been more as Aristotle imagined, and may even be true in some other universe: objects fall because they are “pushed” by unseen particles colliding with them, particles that are shielded by mass, creating a partial vacuum between objects and the earth. In this model, the earth blocks these particles, so an object has fewer of them striking its underside, but the same fixed amount striking it from all other directions, creating a net downward accelerating force. If the amount of (let’s call them) “gravatoms” shielded by a mass was a function of that mass’s density and not the mass itself, heavier objects of the same size would fall faster than light ones; and attaching a heavier object to a light one would produce a combined motion according to the resulting density of the combined mass. Had this been the case, once again, Galileo would have blinded himself to discovering it, by falsely declaring it to be impossible.

In reality, of course, we’ve found gravity is far weirder than either of these models, weirder than even Galileo imagined. And the earth is so much more massive than anything we are usually experimentally or even hypothetically dropping on it, that the difference in “time to collide” (what Galileo would have measured as “time of fall,” had he instruments accurate enough) for heavy and light objects in a vacuum, even when real, would be imperceptible to the human eye. It’s also negated when we drop one object and leave it there before dropping the next, as its mass is now adding to the earth’s gravitational pull, so the next object we drop will indeed connect with the earth in the same time. Or if we drop a heavy and light object right next to each other simultaneously, in which case both objects are pulling the earth closer at the same time. And so on. And still, add a medium, and objects typically fall at different rates in accordance with mass, but even Galileo conceded that. So there are respects in which Galileo was right. But he wasn’t right because of his thought experiment. That was actually terrible. It launched from assumptions not in evidence, and not even proposed by his opponents, and was thus incapable of exploring the actual possibility space available.

In short, Galileo failed to do the first thing any critical thinker should do when critiquing a theory: work very hard to steel man it, as in, work out all the ways it could in fact be true, and then deduce from that what observations we could make to test it. This is, essentially, the scientific method. And contrary to legend, Galileo wasn’t very good at it.

Imagining Causes for the Oceans’ Tides

Galileo was totally confident he had proved heliocentrism by appeal to the tides. His reasoning was hopelessly flawed and almost a textbook example of hosing the scientific method, replacing facts with speculation, treating armchair deduction as equal to experimentation or observation, cherry-picking data—almost every single thing you can get wrong in science. There is a decent brief on his mistake at Wikipedia. There is a lengthier article at NOVA. And a series of articles by physics teacher Christopher Graney. For a thorough treatment see Paolo Palmieri’s Re-examining Galileo’s Theory of Tides.

The gist of it is that Galileo thought the earth’s spinning and swinging around the sun must “slosh” the oceans back and forth, and therefore the exact match between what this weird theory predicts as to tidal periods confirmed the earth “must” be spinning and swinging around the sun. In fact the earth’s motion has next to no effect whatever on the tides. Its rotation does affect phenomena like large-scale currents in the sea and air, called now the Coriolis effect, but this makes little difference to the tides. Galileo did not have access to the massive global data one would have needed to prove heliocentrism by appeal to Coriolis effects (contrary to popular mythology, the Coriolis effect cannot be observed in sinks and toilets). And by itself the earth’s motion around the sun has no effect at all on the movement of objects on earth—the earth is in fact in free fall around the sun, it thus experiences no acceleration effects.

Galileo’s error here illustrates another general principle: that merely being a scientist doesn’t make you right—only a correct application of scientific method does, and scientists can often fail at that, and non-scientists sometimes succeed at it. This requires putting evidence (experiment and observation) before speculation and deduction; not cherry picking data, but looking hard for data that refutes your theory (not trying hard to avoid finding any); and actually testing your theory against the best alternative explanations.

Galileo well knew of the earth-sun-pull theory of the tides, but instead of devising ways to test that—the theory most directly competing with his—he dismissed it out of hand, and thus missed an enormous opportunity to discover the best-ever actual proof of heliocentrism. That would have to await the work of Newton half a century later, who realized that “universal gravitation plus heliocentrism” explains all celestial movements and the tides and the natural falling behavior of all objects on earth, a triune conjunction too improbable to be a coincidence. That his theory then went on to predict other unexpected things—the true gold standard of the scientific method—only clinched it. Though that does so for essentially the same reason: the extreme improbability of that being an accident, producing an enormous Bayes factor in favor of your theory (see my discussion in Proving History, index, “old evidence, problem of”).

The astonishing irony is that the real cause of the tides had already been figured out within a hundred years of Aristotle, but all writings on it were subsequently tossed in the trash by disinterested Christians. As I relate in The Scientist in the Early Roman Empire, “at work in the middle of the 2nd century B.C. was the heliocentric astronomer Seleucus, the student of Aristarchus,” the originator of heliocentric thought, “who also discovered the combined lunar-solar effect on the tides,” and, it appears, even attributed it to a theory of universal gravitation. And this was known in Galileo’s time, thanks to the Renaissance. Because of which, the once-ignored (and very nearly lost) writings of Plutarch led scholars by Galileo’s time to re-discover what Plutarch had said about Aristarchus and Seleucus: Aristarchus proposed heliocentrism as “only a hypothesis,” but Seleucus “proved it” (Platonic Questions 8.1 = Moralia 1006c). If only we knew how he “proved” it. But one other thing Seleucus was famed for, another tidbit re-learned by Galileo’s time, was discovering lunisolar tide theory.

Galileo might have put these two facts together and assumed Seleucus proved the one by the other. He almost certainly didn’t, as Seleucus, unlike Galileo, acknowledged the moon’s role in causing the tides, and must have correctly predicted tidal periods by conjunctions of moon and sun, as otherwise he’d never have been credited with discovering their explanation. All of Seleucus’s work, and all subsequent work building on it (such as Athenodorus’s On the Oceans, which apparently included the most sophisticated account of lunisolar tide theory then known), was destroyed. But what those lost books contained remained widely attested in scattered mentions across several authors:

• Pliny the Elder, Natural History 2.99.212–218 and 2.102.221
• Cicero, On Divination 2.34 and On the Nature of the Gods 2.7.15–16
• Seneca, On Providence 1.4
• Cleomedes, On the Heavens 156
• Ptolemy, Tetrabiblos 1.2.3–6
• Strabo, Geography 3.5.8 and 1.1.8–12

Plutarch also attests to the existence of Roman philosophers and astronomers who rejected Aristotelian dynamics and were engaging sophisticated debates on the subject, even contemplating theories of inertia and universal gravitation (see On the Face that Appears in the Orb of the Moon 6–11 = Moralia 922f-926b; and for modern analysis see Liba Taub’s Aetna and the Moon: Explaining Nature in Ancient Greece and Rome). All of their writings were thrown in the trash—by Christians. And yet we know there were scientific works that proposed a gravitational force projected by the moon and sun combined to move the seas and create tides, that the moon remained in orbit owing to a combined effect of outward motion and inward gravitational pull (using even the analogy of a stone in a sling), and that these theories had made successful predictions. Galileo knew all of that, yet tossed it aside as rubbish, and obsessively tried to prove his own harebrained tidal theory instead. He never relented. He went to his grave certain he must be right. So much for the great scientist.

Imagining Humans as Implacable Robots

In Dialogue Concerning the Two Chief World Systems, at one point Galileo digresses on how he “ought” to have argued with sailors who told him it was too much harder to use a telescope atop the mast than at its base owing to the ship’s movement being so much more magnified above than below. Galileo takes them at their word but later pooh poohs the idea to his interlocutors in the Dialogue, insisting he knows better than the sailors. Galileo proceeds to do a bunch of geometry showing that because the objects they’d be observing are so distant, the difference in geometric deviations produced atop the mast and below is too small to have caused any problem.

Here Galileo shows off how smart he is with a lengthy geometry lesson (indulgently wasting several pages in the modern edition, none of which even relevant to his book’s thesis). But it’s clear it never even occurred to him to actually test his theory. So sure he was right, on mere armchair reasoning alone, he was ready to discount the direct personal experience of actual sailors—who already were well familiar with the problem after thousands of years of using the astrolabe, which requires steady sighting on stars, which are vastly more distant than the objects Galileo was certain they could easily spy with a glass. In other words, Galileo here acts exactly like the medieval scholastics he otherwise takes to task for reasoning from the armchair rather than checking facts in experience. Hypocrisy, thy name is Galileo.

This one struck me (and perhaps has not been noticed by most scientific readers of the text) because I was a sailor, and I actually have been in almost exactly the circumstance Galileo is pontificating about—attempting to range-find distant objects atop a mast in high seas vs. deckside. And I can speak from experience: he is 100% wrong. Or maybe I should say 99% wrong, insofar as all his mathematics is correct, but irrelevant to the problem. What he proves is that if the telescope were held (we would say) by a perfect, implacable robot (like, say, the servo-stabilized 76mm automatic cannon I routinely checked the magazines of while on rounds), then Galileo would be right: the changes in angle atop the mast are geometrically identical to the changes in angle at its base; while changes in mere parallel position are even atop the mast trivial relative to the visual field being observed. But humans aren’t implacable robots. We aren’t servo-stabilized automatic cannons.

And that is where Galileo went wrong. By not actually testing his theory, it looks perfect on paper, because he didn’t discover that it’s not the geometry that’s the problem—whereas as soon as he attempted to prove himself right by experiment, he’d have discovered something even he would surely have found interesting: humans have a hard time maintaining muscle position under extreme stresses of motion. At the base of a mast (on the deck), the amount of g-forces you are subject to, and their duration, is relatively small, and easily compensated for by our natural muscle-and-nerve system. The same motor-cognitive ability that keeps our heads level while marching or dancing can keep a telescope level to our eye in our hands just as well. But put us atop a mast, and the g-forces are magnified considerably: you accelerate and decelerate harder, and longer, as the pole whips you to and fro. Atop a goodly sized mast these volatile forces will overwhelm your body’s innate ability to compensate. It becomes extremely difficult to hold position and direction, even of your eyes, much less a telescope, which in effect being an Archimedean lever with your hands as fulcrum, even amplifies those swaying forces. The sailors were right: it’s too difficult to hold a telescope on target atop a mast; it is far easier at its base.

It is particularly ironic that Galileo would crow over refuting the real experience of sailors with an armchair bit of geometry, while in so doing forgetting (or never thereby discovering) the issue isn’t geometry, but the laws of motion. For a man who is famous for revolutionizing our theories of motion, and in so doing becoming a poster boy against armchair reasoning, it is admittedly humorous to see him use armchair reasoning to evade an empirical conclusion and thus a crucial discovery about the laws of motion. Newton would come to understand principles of acceleration better than Galileo, of course. And it would not be until a century or more after even that that we’d start to take seriously the kinematics of human bodies, such as would readily comprehend why it is harder to hold a telescope steady when being thrown every which way under intense accelerations.

The general lesson here is astonishingly one still most people haven’t learned: never assume you are right about anything you’ve reasoned out until you have tested it in practical experience. Because there is probably something about the real-world you haven’t taken into account, and that you can only discover by jumping in and finding out. Correspondingly, you should listen more to people who are telling you things based on years of actual experience with it. Other people, as a collective whole, know vastly more than you do—about almost everything, if not in fact everything. It would be foolish of you to refuse to benefit from that.

Conclusion

As I articulate in my article Advice on Probabilistic Reasoning the only way to know you are right about anything is to do your honest damnedest to prove yourself wrong—and fail. If you “tell” yourself you are doing that, but every step you take is designed to avoid finding evidence against you, you are engaging in delusional reasoning. You are doing exactly the opposite of what you need to do to have a justified true belief. Galileo’s three worse failures exemplify his not learning, or not consistently applying, that lesson. His theory of gravity ended up mostly correct largely by accident; his circular and logically invalid reasoning leading him to it remained largely unnoticed by him and even most relators of his legend today. His certainty that the tides proved heliocentrism was downright bonkers, yet produced by his refusal to even consider, much less look for, evidence against him. And his argument about spotters on masts lacked even an attempt at any test of it, much less any good one.

So one overarching lesson from Galileo is clear: this great advocate of the scientific method, again and again, abandoned the scientific method—and never admitted it. Don’t make the same mistake.