This target article reports a case in which for a very long time the very best experts (Tycho and Kepler 400 years ago, Thomas Kuhn and many others in recent decades) have suffered an easily corrected illusion. It provides a striking counterexample to claims that cognitive illusions can be reasonably treated merely as effects that clever experimenters can elicit from naive or poorly motivated subjects, but otherwise as not really illusions. The present illusion is based on an apparent collision between the spheres of Mars and the sun in the "Tychonic" alternative to Copernicus in the early 17th Century. The perception of a collision permanently disappears when Tycho's own diagram is presented in a novel way. The illusion appears to come from an unconscious (hence not consciously noticed) mental rotation which is consistent with everyday habits of mind but flatly incompatible with the logic of the situation. This then produces the failure of even the very best experts to notice that what they are seeing in Tycho's diagram is something that is not there. However, once a person sees the cutout described in the article move (i.e., once the rotation becomes a physical rotation, not a mental rotation), the illusion disappears permanently. The present target article is the expanded version of a shorter summary that appeared in Nature (Margolis 1998); this fuller version is published in Psycoloquy so as to elicit Open Peer Commentary, to which the author will respond.
1. For two decades now cognitive illusions have been a focus of lively debate. Does the work led by Wason (1977) and by Kahneman & Tversky (1982) provide deep insight into the fallibility of human reason? Or (taking the opposite extreme) does such work mostly show only how far clever experimenters can trick naive subjects? For more carefully framed and nuanced versions of these arguments see Cohen (1981) and Gigerenzer (1996).
2. We will show evidence for a cognitive illusion of 400 years standing. In the early 17th Century, the main alternative to Copernicus was not Ptolemy but a compromise system proposed by Tycho Brahe. In this "Tychonic" system, the planets move relative to the sun exactly as Copernicus proposed, but the Earth remains fixed while the sun moves around it. Tycho's system seemed to embrace the elegance of Copernicus yet avoided his heresies. In the Tychonic system, however, the orbits of the sun and Mars intersected, and from that it seemed obvious that Tycho's system was inconsistent with the traditional view that the sun and planets are carried around the heavens by solid spheres.
3. The connection between Tycho's system and the cognitive debate is that the putative collision in Tycho's system turns out to be an illusion which has deceived the very best of experts for 400 years. From Kepler (1609) in Tycho's time to Kuhn (1957), Swerdlow (1996) and many others in our own, there has been no dissent (see End Note A). Yet the fact that the collision is only an illusion can be shown almost instantly, using nothing more than Tycho's own diagram.
4. The episode therefore provides a remarkable illustration of how far a cognitive illusion can actually influence significant beliefs. This case can hardly be dismissed as merely providing entertainment for academic psychologists, or as a trick psychologists that play on their subjects, or as in some other way merely an artifact of experimental procedures. Here we have real people, indeed very famous people, the most expert in their field, dealing with a matter which they take very seriously, who are mistaken over a period of 400 years. And in seeing how this happened and how little it takes to correct the illusion, we can also learn a sobering lesson about how far mere logic can be counted on to change belief when entrenched intuitions go against it.
5. To set the stage a bit of history is needed. Figures 1, 2, and 3 show the systems of the world that were in significant contention circa 1600.
ftp://coglit.psy.soton.ac.uk/pub/psycoloquy/1998.volume.9/Pictures/margolis.fig1.html ftp://coglit.psy.soton.ac.uk/pub/psycoloquy/1998.volume.9/Pictures/margolis.fig2.html ftp://coglit.psy.soton.ac.uk/pub/psycoloquy/1998.volume.9/Pictures/margolis.fig3.html
Figure 1 shows a part of Ptolemy's system of nested spheres, which had survived essentially unchallenged for 1400 years. His system is an astonishing jewel in the history of science. It worked about as well as it was possible for a system to work given the limitations of astronomy before the telescope. Until Kepler, even the Copernican proposal (Figure 2), and more trivially, the Tychonic one as well (Figure 3), had the relation to Ptolemy that one way of seeing a reversible figure has to its alternative: They provided radically different ways of seeing what, in terms of technical astronomy, was exactly the same thing (see End Note B).
6. In Ptolemy's setup, the space between the moon and the fixed stars was filled by six "spheres" like the one in Figure 1 (taken from a leading 15th Century text; reproduced in Swerdlow 1996). Nested one inside the other, there is one such sphere for each of the five visible planets, and the sixth for the sun. Within each sphere was a "deferent" (the white annulus in Figure 1), which rotated in place relative to the Earth. Inside the deferent, an epicycle rotated in place relative to the deferent, with the epicycle carrying the heavenly object. Note that nothing in this system moves at all relative to the object containing it. The deferent rotates in place, as was deemed natural for an eternal object in the heavens. The epicycle rotates in place inside the deferent. The actual planet merely sits, just inside the perimeter of the epicycle.
7. The relative sizes of the deferent and epicycle that will yield correct predictions can be determined from a few well-chosen observations. Ptolemy did all that, and then derived what were for 14 centuries believed to be real cosmic distances by making each successive deferent thick enough to contain its epicycle, with no overlaps and no gaps. The rotation of the deferent interacting with the rotation of its epicycle yielded the looping motions an astronomer could observe in the heavens. A possibility pointed out by Ptolemy was that Figure 1 could be inverted to yield identical planetary motion from a model in which what had been the epicycle and radius to the center of the epicycle were interchanged. This inverted setup is shown in Figure 1', which will play a substantial role in the subsequent discussion.
8. Figure 4 is Kepler's (1609) illustration of the geocentric path of Mars over the years 1580-1597.
For Kepler, what he called the "pretzel-shaped" orbits of geocentric planets, and further complications beyond that, showed why a sensible person would believe Copernicus. If Mars did indeed follow the path in Kepler's diagram, however, and if the Earth was indeed motionless at its center, and if the additional conditions were met, then what one saw from the Earth would be exactly what a Copernican would expect. Why God should prefer this contraption would be a deep mystery, but before the telescope, Ptolemy's system, or Tycho's (which generates the same pretzel-shaped orbits), was observationally indistinguishable from the Copernican system.
9. The sun also was carried on an epicycle. This was too small to produce loops in the sun's path, but just large enough to provide the acceleration during northern winters (and slowing during northern summers) which our calendar reflects by making February several days shorter than August. Copernicus retained that solar epicycle (and many other secondary features of the Ptolemaic view), but his proposal turned the major epicycles of Ptolemy's planets into mere reflections of a single, startling orbit of the Earth itself. Half a century later, Tycho proposed the compromise which is the focus of the discussion here. In his alternative (Figure 3), the planets move in Copernican orbits relative to the sun, but it is the sun and those orbits that go round the Earth, not the Earth which is thrown into the heavens to orbit the sun. So in place of the path of the Earth around the sun in the Copernican diagram, Tycho showed a circuit of the sun around the Earth.
10. If the planets are carried by solid spheres, then could the Tychonic system work? For 400 years it seemed obvious to everyone who wrote on the matter that there must be a collision between the orbits of the sun and Mars. No detailed argument was ever made, for no detailed argument was ever seen as necessary. From just noticing the intersection of the orbits in the Tychonic diagram, the collision seemed obvious.
11. That apparent collision turns out to be the illusion advertised in the title of this paper. A reader who will take a moment to make the cutout provided by Figure 5 will find that whereas the intersection (of course) remains, the collision disappears.
In a world of solid spheres a solid object rotating in place (relative to what contains it) must carry each planet on its path around the sun. So the orbit of Mars in Tycho's diagram will be a rotating solid object carrying Mars around the sun. But the path of the sun in Figure 3 does not carry anything. It only marks the mathematical path of the center of all the epicycles as the entire region between the Earth/moon and the fixed stars proceeds through its annual rotation. Mars could no more collide with that mathematical locus than a ship could collide with the equator.
12. Figure 5 separates Tycho's diagram into two components. The cutout (Figure 5a) contains everything that shares in the sun's annual motion and nothing else. The template (Figure 5b) is what is left after removing the cutout. Cut apart a copy of Figures 5a & 5b; fold Figure 5a to bisect the black circular region in its center. Cut the black center out and trim away the remaining black on the periphery, unfold the cutout and center its hole over the Earth/moon region at the center of the template. Turning the cutout produces the Tychonic orbit of the sun, accompanied (as Tycho and observations require) by the orbits of the planets.
13. In a Copernican world the annual component of each planet's motion is only a reflection of the Earth's actual orbit around the sun. For Ptolemy, since the Earth does not move, each planet must have an actual annual motion, but every astronomer who had access to Ptolemy's book could learn how to invert the annual component of the Ptolemaic models for the outer planets so that each would not merely track the apparent motion of the sun but would exactly coincide with that motion. No Ptolemaic astronomer seems actually to have done that, but when it is done, all the annual motions in Ptolemy's system are reduced to a single common motion. All the epicycles can then be carried within a single large deferent providing that single motion. If, in addition, the models are rescaled to set the radius of the annual component for each planet to equal the Earth/sun radius, then you have the Tychonic system. The cutout provided by Figure 5a is a perfectly adequate model (on a scale of about 1 to a trillion) of the single Tychonic deferent that results. On the other hand, if the models are inverted but not rescaled, this yields the alternative arrangement of Figure 1'. But on either the Tychonic or what I have called (Margolis, 1993) the "inverted Ptolemaic" arrangement, that single sphere of course cannot collide with itself.
14. Yet on the record of four centuries it is extremely difficult to come to realize that there is no collision in a solid spheres version of Tycho's system. So there is a psychological puzzle here, and an interesting one. Until you see the cutout move (but not after that), when you look at Tycho's diagram it is very hard to escape seeing a collision that is not there. When you see the cutout move, the collision disappears, although you are only seeing what you thought you already understood perfectly well: that in Tycho's system the sun and the planetary orbits must move together in an annual rotation around the Earth. You still (of course) see an intersection between the orbits of Mars and the sun, but it is now obvious that the intersection does not entail a collision, for even in a world of solid spheres, one of the intersecting paths (that of the sun) is only a mathematical locus, not a "bumpable" object.
15. Trying to persuade people who have not seen the cutout that the collision they so plainly sense is not really there is like trying to persuade someone, merely by a logical argument, that the two lines in the Muller-Lyer illusion (Figure 6) are the same length.
Our eyes tell us that the argument must be wrong, and it is only after we see a physical demonstration (first drawing two equal lines and then adding the arrows) that it becomes obvious that our very clear intuition is an illusion. Especially for individuals who are expert in this Tychonic matter -- people who are understandably very confident that they could not be grossly wrong about something they know so well -- the only efficient way to make the logical point is to construct the moving model of the Tychonic motions (Figure 5) and actually make it move.
16. The cutout emerged from just such a discussion. While visiting Harvard on another matter, I called Owen Gingerich, a leading expert on renaissance astronomy, and asked him to sit through my argument about Tycho's illusion one more time. Not too happily, I think, but cordially nevertheless, he invited me over. After a while, when I thought he was perhaps halfway persuaded, Gingerich recalled a new toy he had in his desk. It was a sort of draftsman's compass he had picked up a week earlier at a clearance sale in a book store in Iowa where he had gone to give a talk. This compass had a little knife where you would expect a little pencil, so instead of drawing circles, it would cut them. Gingerich proposed that we might settle the matter by actually cutting up a copy of Tycho's diagram. So for a few minutes, two professors in their 60's happily carried out this trivial grade-school project. The startled look on Gingerich's face when the cutout was moved told me right away that I at last had a way to make this simple argument convincing. This was four years after I first discussed the argument in print (Margolis 1993) and six after I first began my sporadic but never entirely abandoned effort to persuade historians of its validity.
17. Why is the illusion so stubborn without the cutout yet so easily resolved with it? Note that the illusion cannot be induced just by the intersection between the orbits of Mars and the sun. After all, there is also an intersection between the orbit of the sun and those of Venus and Mercury, but no one immediately sees a collision in either of these logically equivalent situations (see End Note C). So another answer is required that does not turn on the mere existence of an intersection. It is not hard to see what that answer might be.
18. What creates the illusion is that our experience in the world is overwhelmingly that little things circle big things, as flies circle a dish of food. We do not see big things circling smaller things, and indeed centrifugal forces make that physically difficult. In Tycho's scheme, however, the orbit of the sun is the rotating center of orbits (the inverted Ptolemaic orbits of Mars, Jupiter and Saturn) that are bigger than itself. In the Muller-Lyer illusion, our solidly entrenched experience with seeing things in perspective allows the "arrow" distance cues to distort our ordinarily reliable perception. It is a similar reflection of entrenched mental habits that when someone who has not yet seen the cutout move tries to envision the Tychonic motions, they are not seen moving as they logically should.
19. Although we know perfectly well that what must move to match observations is the entire system (the sun and all the orbits), what we sense during this mental rotation is apparently something different: the smaller orbits (Venus and Mercury) move with the sun as they should, whereas the larger orbits (Mars, Jupiter, Saturn), illogically, stay put, creating a powerful intuitive sense that a collision must occur. It is only after you see the cutout move that you readily discern that the Tychonic solar system would be carried by rotation of the entire region of the cutout. So now you see the motion as big carrying small rather than small carrying big, and the illusion accordingly disappears.
20. This suggests an odd psychological possibility. People seeing the illusory collision commonly think they are envisioning how the Tychonic system moves, but this can't be conscious mental imagery, of the sort that anyone who uses mathematics will be familiar with, where a person can "see" what will change in a diagram when parameters change. The imagery can't be conscious because if conscious imagery showed us what I just described (with the sun carrying the orbits of Venus and Mercury but the larger orbits staying in place), it would be obvious that it was mistaken. Confronted with a direct contradiction between what we know makes sense and what we seem to see, we would soon come to realize that there was no such collision, perhaps as quickly as we see it when the cutout is rotated. But the mental rotation is not consciously "seen," only felt.
21. What appears to be occurring is a kind of "imagery blindsight": blindsight with respect to the mental rotation of Tycho's system. The only cases in which we could notice such entirely-in-the-head blindsight would be when the blindsight was illusory. Only then could it become apparent that somehow, in the back of our mind, something is being seen that should not really be there. For a more detailed discussion of blindsight see Weiskrantz (1986).
22. How could the Tychonic illusion have gone unchallenged for such a very long time? Accounting for this calls for some consideration of affect. For Muller-Lyer, there would be natural illusions that were responses to cues in ambiguous conditions which prompt responses resembling what is going on in rudimentary form in the Muller-Lyer diagram (e.g., Margolis, 1987, Figure 7.1.) Occasionally, however, we would encounter unmistakable evidence that we had seen things wrong. This overt conflict arouses curiosity, or distress, or perhaps only amusement, any of which might prompt inquiry and checking. An alert psychologist noticing such a thing could make use of it, as Muller- Lyer famously did, and as (in the realm of cognitive rather than perceptual illusions) Wason and Kahneman & Tversky have so often done.
23. Yet even without an overt rivalry to prompt affect, if we are motivated to wish X true, this will prompt a visceral response that pushes us to look harder to see whether somehow X might indeed be true. On the other hand, suppose affect is aversive: for example, because "everyone knows" Tycho was right, why waste time taking seriously an argument that he was wrong? Then it is wanting to escape that aversiveness that is much more easily prompted. There will be little inclination to check, much less actively doubt, anything that comes to mind as a plausible dismissal of what you know is wrong.
24. This makes much more sense, however, for us (or for an early Copernican like Kepler) than for geocentric astronomers circa 1600, for whom Tycho's system was an extremely serious matter. Seeing through Tycho's illusion is ultimately so simple that some explanation is needed of how he, and all other astronomers who abandoned Ptolemy for Tycho's system, could have missed it.
25. The explanation I want to suggest is that astronomers in Tycho's time had apparently moved far more to the Copernican view than they themselves consciously recognized, so that those who consciously believed the Earth did not move were nevertheless tacitly committed to the Copernican way of seeing the world. Indeed, is it psychologically plausible that astronomers would carry around in their heads a sense of how the planets move like that of the pretzel-shaped orbits of Figure 4 when a simple circle would yield exactly the same observations if the world were Copernican? (see End Note D)
26. There is another case that seems to reveal an odd inattention to how the Tychonic system actually works. Accounts of Galileo and the Pope (Urban VIII) have routinely claimed that Galileo ignored the Tychonic system in his Dialogue, often criticizing Galileo for concealing from his readers the system that (as such writers see it) the Copernicans could not beat. A sufficiently careless reading of the Dialogue could support that view, but it is a puzzle how a careful reader could fail to notice that what Galileo always refers to as the "Ptolemaic system" is in fact not Ptolemaic but Tychonic. Throughout the book, Galileo calls the geocentric alternative Ptolemaic. But at each of the few points where he explicitly mentions features that could distinguish between the Ptolemaic and Tychonic versions of geocentric astronomy, Galileo's "Ptolemaic" system turns out to be Tychonic. Given Galileo's circumstances, it is not hard to suggest a reason for this odd treatment: it allowed him to avoid any frontal attack on what was by then the system of the Church's own astronomers. That, in turn, is at least relevant (and in fact seems to me central) to understanding both the Pope's approval of the book before it was published, and his eventual fierce belief that he had been tricked. Galileo explicitly (and in several different places) specifies his "Ptolemaic" system as one in which the planets (Mercury, Venus, Mars, Jupiter and Saturn) go round the sun. But no writer discussing the Dialogue seems to have noticed that such language unambiguously identifies the geocentric system Galileo has in mind as Tycho's, not Ptolemy's. For details see Margolis (1991).
27. So there is somehow a blind spot here with respect to how Tycho's system works. This is revealed by yet another Tychonic illusion, one with a much more direct bearing on the argument here. The literature uniformly reports that Tycho's system preserves nearly all the advantages of Copernicus over Ptolemy without abandoning a fixed Earth. (It rarely mentions that the disadvantage that remains is a whopper: the Tychonic orbits are still pretzel-shaped: the orbit Kepler shows in Figure 4 fits both Ptolemy and Tycho.) Abandoning Ptolemy is completely unnecessary to gain those advantages, however, since they have nothing whatever to do with the heliocentric orbits of the Tychonic planets. Any advantages come only from inverting the outer planets, which is not an innovation of Tycho at all, since Ptolemy himself had pointed out that possibility. The resulting "inverted Ptolemaic" system (invert the outer planets but do not rescale to make them heliocentric) would have all the purported advantages of Tycho's system while preserving the Ptolemaic distances. Yet for 1400 years no one showed any interest in that. For a more detailed discussion, see Margolis (1993), pp. 117-29.
28. So why should Tycho's arrangement be so attractive circa 1600 that after 1400 years Ptolemy simply faded away without any noticeable fight? I don't believe any credible logic for this can be found. Rather, astronomers who were consciously opposed to the Copernican idea revealed a powerful preference for a scheme whose only actual merit (relative to an inverted Ptolemaic scheme) is that it looks more or less Copernican! This preference is like that of experimental subjects subliminally exposed to a bit of music, who then report that they like that (consciously) unfamiliar music better than really unfamiliar music (Zajonc 1980). Such an astronomer could find -- and a fortiori a Copernican then or now could find -- a decided preference for the Tychonic system over the Ptolemaic one without any need at all to think about how the Tychonic system actually works.
29. The Ptolemaic system, which reigned for 1400 years, ended without even a whimper. By 1610, both Kepler and Galileo were treating Tycho as the man to beat, and Ptolemy as scarcely worth discussing (Margolis 1991). No evidence has survived of anything like a serious debate over why it would make sense for a geocentric astronomer to abandon 1400 years of tradition for Tycho's alternative. Ptolemy simply faded away. That seamless transition has been rationalized by casual reference to the purported (but in fact nonexistent) advantages of Tycho over Ptolemy. But it seems highly implausible that such casually defended, indeed unexamined claims could have been persuasive without some subliminal attachment to the Copernican way of seeing the world. Unsurprisingly, a Copernican like Kepler had that entrenched in his head. But so apparently did non-Copernicans, even though conscious belief remained explicitly anti-Copernican.
30. There is a further bit of evidence. For the Tychonic illusion itself is contingent on seeing the Tychonic diagram from a Copernican, not a Ptolemaic, perspective. From a Copernican perspective, it is natural to see the path of the sun around the Earth as replacing the path of the Earth around the sun. Starting from that, the tacit misperception of the motions is prompted by the big-on-small effect described earlier. But a person starting from a Ptolemaic perspective would see a motion, not like that of the Copernican Figure 2, but like that of the inverted Ptolemaic Figure 1', with the motion carrying the orbits and the sun together. For details, see Margolis 1993, chapter 8. Even Copernicans like us readily see this when confronted with the motion of the cutout. That we have trouble seeing what really happens in the Tychonic system without that help is easily explained by the fact that it is a Copernican sense that guides our intuition. But that astronomers who were resisting Copernicus circa 1600 would suffer the same illusion hardly makes sense unless those astronomers were somehow committed to the Copernican way of seeing the heavens despite their contrary conscious beliefs.
31. Who would be naive enough to suppose that today we are less vulnerable to illusions (but different ones, of course) than were our predecessors in Tycho's time, or that we would be more able (especially when motivated by adverse visceral prompting) to allow mere logic to challenge comfortable intuitions?
32. How I came to notice the illusion itself fits the argument just sketched. Writing a "Ptolemaic tutorial" for my 1993 book I wanted to take readers through the exercise of inverting Ptolemy's model for the outer planets, rescaling the planetary orbits, and thus obtaining the Tychonic system. Working that through, I noticed what I have here argued a contemporary of Tycho's would notice if he started from Ptolemy, rather than Copernicus (see End Note E).
33. There was for me an idiosyncratic but clear motivation to follow up any glimmer of doubt about Tycho's collision. In my 1987 book I had already argued, on other grounds, that by the late 16th Century astronomers' habits of mind must have shifted quite a way towards seeing things as Copernicus saw them. I nevertheless accepted and repeated the usual collision story (as I also accepted and repeated the usual story of Galileo ignoring the Tychonic system). But the second time around, given that motivation and the serendipitous occasion to see things from the Ptolemaic side, I managed to avoid missing the opportunity again.
A. For Kepler, see Chapter 1 of his Astronomia Nova (1609). For Kuhn, see the discussion of the Tychonic system in his Copernican Revolution (Harvard, 1957), pp. 205-6. For Swerdlow (1996), pp. 211-2.
B. This accounts for the dismissive view of Copernicus sometimes encountered in appraisals by writers who focus narrowly on Copernicus as a technical astronomer. Two influential exemplars are Neugebauer (1983) and Price (1958).
C. Kuhn (1957) noticed this possibility. In contrast to the Mars problem, however, this is "merely" a logical point, never an immediate intuition. If there is a problem with Mars, then indeed logically there should also be a problem with Venus and Mercury. Tycho, however, and nearly everyone since who has commented on the collision, was not troubled by that at all. In any case, when the main illusion is corrected the secondary one vanishes as well.
D. But aren't planetary orbits really ellipses? And didn't Copernicus use various secondary epicycles? The answer to both questions is yes: indeed the secondary epicycles of the latter were intended to take account of the ellipses of the former. These have consequences that are essential for calculating exact motions. But a flawless diagram of the elliptical orbit of Mars is indistinguishable to the eye from a simple circle.
E. Inverting a schematic Ptolemaic diagram is the usual way of showing readers the mathematical equivalence of the Ptolemaic and Copernican systems. This schematic shows the path of the center of the epicycle, but not the solid deferent that contains that path shown in the 15th Century Figure 1. Inverting that schematic is not enough to correct the illusion. What is required is to invert the "solid spheres" (Figure 1'), which then explicitly reveals the full deferent carrying the epicycle within it.
Many thanks to Psycoloquy's Assistant Editor, Marcus Munafo', who did a good deal of work on the figures, made the links, and finalized the text, Abstracts and Rationale.
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