An explanation which conforms to the facts does not imply that the hypotheses are real and exist. … [Astronomers] have been unable to establish in what sense, exactly, the consequences entailed by these arrangements are merely fictive and not real at all. So they are satisfied to assert that it is possible, by means of circular and uniform movements, always in the same direction, to save the apparent movements of the wandering stars.[Simplicius, <cit.>.]
Instrumentalism, as opposed to realism, was supposedly the accepted philosophy of science among “the Greeks”:
[Ancient Greek astronomers] balked at the idea that the eccentrics and epicycles are bodies, really up there on the vaults of the heavens. For the Greeks they were simply geometrical fictions requisite to the subjection of celestial phenomena to calculation. If these calculations are in accord with the results of observation, if the `hypotheses' succeed in `saving the phenomena', the astronomer's problem is solved.[<cit.>.]
An astronomer who understands the true purpose of science, as defined by men like Posidonius, Ptolemy, Proclus, and Simplicius, … would not require the hypotheses supporting his system to be true, that is, in conformity with things. For him it will be enough if the results of calculation agree with the results of observation—if appearances are saved.[<cit.>.]
The Greek geometer in formulating his astronomical theories does not make any statements about physical nature at all. His theories are purely geometrical fictions. That means that to save the appearances became a purely mathematical task, it was an exercise in geometry, no more, but, of course, also no less.[<cit.>.]
Galileo, by contrast, brought “a radically new mode of realist-mathematical nature knowledge.”[<cit.>.]
Galileo endorsed a view that was [contrary to] that of the Greeks but was also much more creative … It is a crippling restriction to hold that no theory about reality can be in mathematical form; the Renaissance rejected this restriction, holding that it was a worthwhile enterprise to search for mathematical theories which also—by metaphysical criteria—could be supposed `real'. … The most eloquent and full defence of this process was given by Galileo.[<cit.>.]
Hence the Scientific Revolution owes much to “the novel quality of realism that the abstract-mathematical mode of nature-knowledge acquired in … Galileo's hands.”[<cit.>.]

In reality, no mathematically competent Greek author ever advocated instrumentalism. The notion that “the Greeks” were instrumentalists relies exclusively on passages by philosophical commentators. The notion that Ptolemy believed his planetary models were “fictional … combinations of circles which could never exist in celestial reality”[<cit.>.] is demonstrably false. First of all Ptolemy opens his big book with physical arguments for why the earth is in the center of the universe—a blatantly realist justification for this aspect of his astronomical models. Furthermore, Ptolemy has a detailed discussion of the order and distances of the planets that obviously assumes that the planetary models, epicycles and all, are physically real. “The distances of the … planets may be determined without difficulty from the nesting of the spheres, where the least distance of a sphere is considered equal to the greatest distance of a sphere below it.”[Ptolemy, Planetary Hypotheses, I, <cit.>.] That is to say, according to Ptolemy's epicyclic planetary models, each planet sways back and forth between a nearest and a furthest distance from the earth. The “sphere” of each planet must be just thick enough to contain these motions. Ptolemy assumes that “there is no space between the greatest and least distances [of adjacent spheres],” which “is most plausible, for it is not conceivable that there be in Nature a vacuum, or any meaningless and useless thing.”[Ptolemy, Planetary Hypotheses, I, <cit.>.] Clearly this is based on taking planetary models to be very real indeed, and not at all mathematical fictions invented for calculation.

Nor was Ptolemy an exception in his realism. His colleague Geminos “was a thoughtful realist” as well.[<cit.>.] Hipparchus too evidently chose models for planetary motion on realist grounds. His works are lost, but we know that he proved the mathematical equivalence of epicyclic and equant motion. In other words, he showed that two different geometrical models of planetary motion are observationally equivalent; they lead to the exact same visual impressions seen from earth, but they are brought about by different mechanisms. How should one choose between the two models in such a case? If Hipparchus was an instrumentalist, he wouldn't care one way or the other, or he would just pick whichever was more mathematically convenient. But if he was a realist he would be interested in which model could more plausibly correspond to actual physical reality. So what did he do? Here is what Theon says: “Hipparchus, convinced that this is how the phenomena are brought about, adopted the epicyclic hypothesis as his own and says that it is likely that all the heavenly bodies are uniformly placed with respect to the center of the world and that they are united to it in a similar way.”[Theon, Astronomia 34, <cit.>.] So Hipparchus decided between equivalent models based on physical plausibility. This is quite clearly a realist argument.

Historians have brought up other “evidence” that “the Greeks” were instrumentalists. One thing they point to is the alleged compartmentalisation of Greek science.
Phenomena [such as] consonance, light, planetary trajectories and the two states of equilibrium [i.e., statics and hydrostatics] are investigated separately. There is no search for interconnections, let alone for an overarching unity.[<cit.>.]
This would indeed make sense if mathematical science was just instrumental computation tools with no genuine anchoring in reality. The only problem is that the claim is false. Greek science is in fact full of interconnections, just as one would expect if they were committed realists. Ptolemy uses mechanics to justify geocentrism; Archimedean hydrostatics explains shapes of planets and “casts light on the earth’s geological past”;[<cit.>. See <ref>.] Archimedes used statical principles to compute areas in geometry. Ptolemy applies “consonance” (that is, musical theory) to “the human soul, the ecliptic, zodiac, fixed stars, and planets.”[Ptolemy, Harmonics, <cit.>.] He also applies the law of refraction of optics to atmospheric refraction, noting its importance for astronomical observations.[<cit.>.]

In Galileo's time, the same pattern as among the Greeks prevails: mathematically competent people are unabashed realists, while philosophers and theologians often find instrumentalism more appealing for reasons that have nothing to do with science. Copernicus's book, for example, is unequivocally realist. Spineless philosophers and theologians could not accept this. One even resorted to the ugly trick of inserting an unsigned foreword in the book without Copernicus's authorisation, in which they espoused instrumentalism:
It is the job of the astronomer to use painstaking and skilled observation in gathering together the history of the celestial movements, and then—since he cannot by any line of reasoning reach the true causes of these movements—to think up or construct whatever causes or hypotheses he pleases such that, by the assumption of these causes, those same movements can be calculated from the principles of geometry for the past and for the future too. … It is not necessary that these hypotheses should be true. … It is enough if they provide a calculus which fits the observations.[Osiander, foreword to Copernicus's book, <cit.>. For the attribution to Osiander, see <cit.>.
This surely fooled no one who actually read the book, with all its blatant realism. Giordano Bruno, for one, thought “there can be no question that Copernicus believed in this motion [of the earth],” and hence concluded that the timid foreword must have been written “by I know not what ignorant and presumptuous ass.”[Bruno, <cit.>.] But then again the mathematically incompetent people whom this foreword was designed to appease could not read the book anyway.

In medieval and renaissance philosophical texts it is not hard to find many assertions to the effect that “real astronomy is nonexistent” and what passes for astronomy “is merely something suitable for computing the entries in astronomical almanacs.”[Achillini (16th century), <cit.>.] There were many instrumentalists at the time,[<cit.>.] to be sure, but the challenge is to find a single serious mathematical astronomer among them. They were exclusively theologians and philosophers.

All historians nowadays recognise that “Copernicus clearly believed in the physical reality of his astronomical system,” but their inference that he “thus broke down the traditional disciplinary boundary between astronomy (a branch of mixed mathematics) and physics (or natural philosophy)”[<cit.>.] is dubious. This was “the traditional” view only in a very limited sense. It was traditional among the particular sect of Aristotelians that occupied the universities, but outside this narrow clique it had no credibility or standing whatsoever. Among mathematicians, Copernicus's view was exactly the traditional one.

All mathematically competent people continued in the same vein, long before Galileo entered the scene. Already in the 16th century, “Tycho and Rothman, Maestlin, and even Ursus … openly deploy a wide range of physical arguments in debating the issue between the rival world-systems.”[<cit.>.] Kepler puts the matter very clearly:
One who predicts as accurately as possible the movements and positions of the stars performs the task of the astronomers well. But one who, in addition to this, also employs true opinions about the form of the universe performs it better and is held worthy of greater praise. The former, indeed, draws conclusions that are true as far as what is observed is concerned; the latter not only does justice in his conclusions to what is seen, but also … in drawing conclusions embracing the inmost form of nature.[<cit.>.]
As Kepler notes, this was all obviously well-known and accepted since antiquity, for “to predict the motions of the planets Ptolemy did not have to consider the order of the planetary spheres, and yet he certainly did so diligently.”[<cit.>.]

§.§ Mechanical philosophy

Some say that “the mechanization of the world-picture” was the defining ingredient of “the transition from ancient to classical science.”[<cit.>.] A paradigm conception at the heart of the new science was that of the world as a machine: a “clockwork universe” in which everything is caused by bodies pushing one another according to basic mechanical laws, as opposed to a world governed by teleological purpose, divine will and intervention, anthropomorphised desires and sympathies ascribed to physical objects, or other supernatural forces. Galileo was supposedly a pioneer in how he always stuck to the right side in this divide.
Galileo possessed in a high degree one special faculty. … That is the faculty of thinking correctly about physical problems as such, and not confusing them with either mathematical or philosophical problems. It is a faculty rare enough still, but much more frequently encountered today than it was in Galileo's time, if only because nowadays we all cope with mechanical devices from childhood on.[<cit.>.]
Of course, this “special faculty” is precisely what led Galileo to reject as occult the correct explanation of the tides and propose his own embarrassing nonstarter of a tidal theory based on an analogy with “mechanical devices.”[<ref>.] But let us put that aside.

There is nothing modern about the mechanical philosophy. “We all cope with mechanical devices from childhood on,” but so did the Greeks, who built automata such as entirely mechanical puppet-theatres, self-opening temple doors, a coin-operated holy water dispenser, and so on.[[Humphrey et al. , 2003], <cit.>.] Pappus notes that “the science of mechanics” has many applications “of practical utility,” including machines for lifting weights, warfare machines such as catapults, water-lifting machines, and “marvellous devices” using “ropes and cables to simulate the motions of living things.”[Pappus, Collection, VIII.1, <cit.>.] Clearly, then, “Ancient Greek mechanics offered working artifacts complex enough to suggest that … organisms, the cosmos as a whole, or we ourselves, might `work like that'.”[<cit.>.] Thus we read in ancient sources that “the universe is like a single mechanism” governed by simple and deterministic laws that ultimately lead to “all the varieties of tragic and comedic and other interactions of human affairs.”[Theodorus, as paraphrased by Proclus, On providence 2, <cit.>.] This line of reasoning soon lead to a secularisation of science. “Bit by bit, Zeus was relieved of thunderbolt duty, Poseidon of earthquakes, Apollo of epidemic disease, Hera of births, and the rest of the pantheon of gods were pensioned off” in the same manner.[<cit.>.]

Mechanical explanations are widespread in Greek science. The Aristotelian Mechanics uses the law of lever to explain “why rowers who are … in the middle of the ship … move the ship the most,” and “how it is that dentists extract teeth more easily by … a tooth-extractor [forceps] than with the bare hand only.”[(Pseudo-)Aristotle, Mechanics, <cit.>.] Greek scientists explained perfectly clearly that sound is a “wave of air in motion,” comparable to the rings forming on a pond when when one throws in a stone.[<cit.>.] Atomism—a widely espoused conception of the world in Greek antiquity—is of course in effect a plan to “make material principles the basis of all reality.”[<cit.>.]

Greek astronomy went hand in hand with mechanical planetaria that directly reproduced a scale model of planetary motion. And not just basic toy models, but “complex and scientifically ambitious instruments” that could generate all heavenly motions mechanically from a single generating motion (the turn of a crank, as it were).[<cit.>.]

The possibility that even biological phenomena worked on the same principle immediately suggested itself and was eagerly pursued.
Just as people who imitate the revolutions of the wandering stars by means of certain instruments instill a principle of motion in them and then go away, while [the devices] operate just as if the craftsman was there and overseeing them in everything, I think in the same way each of the parts in the body operates by some succession and reception of motion from the first principle to every part, needing no overseer.[Galen, On the Use of the Parts 12.5, <cit.>.]
Indeed, ancient medical research put this vision into practice. “The use of what we should call mechanical ideas to explain organic processes”—such as digestion and other physiological functions—is “the most prominent feature” of the work of Erasistratus in medicine,[<cit.>.] who also tested his ideas experimentally.[<cit.>.]

In sum, the world did not need Galileo to tell them about the mechanical philosophy, since it had been widely regarded as common sense already thousands of years before.

§.§ Continuity thesis

Many have tried to stress commonalities between Galileo and the Aristotelian philosophers who preceded him. That is to say, they argue for the “continuity thesis” which says that the so-called “Scientific Revolution” was not a radical or revolutionary break with previous thought.
Galileo essentially pursued a progressive Aristotelianism [during the first half of his life—the period of] positive growth that laid the foundation … for the new sciences.[<cit.>.]
A particular school of Renaissance Aristotelians, located at the University of Padua, constructed a very sophisticated methodology for experimental science; … Galileo knew this school of thought and built upon its results; … this goes a long way toward explaining the birth of early modern science.[<cit.>, summarising [Randall , 1940].]
The mechanical and physical science of which the present day is so proud comes to us through an uninterrupted sequence of almost imperceptible refinements from the doctrines professed within the Schools of the Middle Ages.[<cit.>.]
Galileo was clearly the heir of the medieval kinematicists.[<cit.>.]
I agree with these authors that “those great truths for which Galileo … received credit” are not his.[<cit.>.] But the notion that they were first conceived in Aristotelian schools of philosophy is wrongheaded.

The argument of these historians is based on a simple logic. First they show that various concepts of “Galilean” science are prefigured in earlier sources. Then they want to infer from this that these sources marked the true beginning of the scientific revolution. But in order to draw this inference they need two assumptions: first, that Galileo was the father of modern science; and second, that the Greeks were nowhere near the same accomplishments. These two assumptions are simply taken for granted by these authors, as a matter of common knowledge. But in reality both assumptions are dead wrong, and therefore the inference to the significance of the Aristotelian sources is unwarranted.

The continuity thesis, then, devalues the contributions of Galileo, yet at the same time desperately needs to reassert the traditional view that “Galileo has a clear and undisputed title as the `father of modern science',”[<cit.>.] since this is what gives them the one point of connection they are able to establish between medieval and modern science. Historians have been able to match up some aspects of Renaissance Aristotelian tradition with the scientific methodology and practice of Galileo. From this they are inclined to infer that Galileo was influenced by this tradition. But there is little direct evidence to this effect. Galileo is said to have found Aristotelian teachings "boring."[Vincenzo Viviani, Racconto istorico della vita del sig.r Galileo Galileo (1654), OGG.XIX.602, <cit.>.] Insofar as there are similarities between him and earlier thinkers who tried to tackle similar questions, this can very well be due largely to common sense rather than direct philosophical influence.

It would be much harder to find substantive links between medieval authors and mathematically competent people such as Kepler or Newton. Since Galileo mostly writes philosophical prose and rarely if ever substantive mathematics, it is much easier to try to construe his works as related to the preceding philosophical tradition. The strength of those alleged continuities in philosophical thought between Galileo and preceding Aristotelian tradition is highly debatable. But it is a moot point in any case unless Galileo is accepted as a founder of modern science. The entire argument stands and falls with this false premiss. Therefore, if one proves, as we have done above, that Galileo was a mediocre scientists of negligible importance to the mathematically competent people who actually achieved the scientific revolution, then the continuity thesis collapses like a house of cards.

The defenders of the continuity thesis are equally ineffectual in establishing the second false premiss of their argument, namely the alleged absence of these “new” ideas in Greek thought. In fact, even continuity thesis advocates make no secret of the fact that the medieval tradition was built on “remnants of Alexandrian science.”[<cit.>.] For example, “although we are left with few monuments from the profound research of the Ancients into the laws of equilibrium, those few are worthy of eternal admiration.”[<cit.>.] Obviously, “masterpieces of Greek science … [such as the works of] Pappus, and especially Archimedes, … are proof that the deductive method can be applied with as much rigor to the field of mechanics as to the demonstrations of geometry.”[<cit.>.] Galileo himself embraced Archimedes as his role model in no uncertain terms. “As far as genius is concerned, [Galileo] claimed Archimedes had exceeded everybody else, and called him his master.” “Galileo claimed it was possible to walk safely, without stumbling, on earth as well as in heaven, as long as we remain in Archimedes's footsteps.”[Niccolo Gherardini, Vita di Galileo Galilei (1654), OGG.XIX.645, 637, <cit.>.]

How, then, can continuity thesis advocates acknowledge these “masterpieces” “worthy of eternal admiration” from antiquity, yet at the same time attribute the scientific revolution to medieval or renaissance philosophers? By writing off those ancient works as minor technical footnotes to an otherwise thoroughly Aristotelian paradigm. Only if this picture is accepted can any kind of greatness be ascribed to the pre-Galileans, as is evident from passages such as these:
Some philosophers in medieval universities were teaching ideas about motion and mechanics that were totally non-Aristotelian [and] were consciously based on criticisms of Aristotle's own pronouncements.[<cit.>.]
Admittedly, most of these … significant medieval mechanical doctrines … were formed within the Aristotelian framework of mechanics. But these medieval doctrines contained within them the seeds of a critical refutation of that mechanics.[<cit.>.]
The medieval mechanics occupied an important middle position between … Aristotelian and Newtonian mechanics. … [Hence it was] an important link in man's efforts to represent the laws that concern bodies at rest and in movement.[<cit.>.]
The impressive set of departures from Aristotelianism achieved by medieval science nevertheless failed to produce genuine efforts to reconstruct, or replace, the Aristotelian world picture.[<cit.>.]
If Aristotle is taken as the baseline, this looks quite impressive indeed. But why should Aristotle be accepted as the default opinion? Aristotle was one particular philosopher who was a nobody in mathematics and lived well before the golden age of Greek science. Medieval and renaissance thinkers indeed mustered up the courage to challenge isolated claims of his teachings almost two thousand years later, while mostly retaining his overall outlook. This does not constitute great open-mindedness and progress. Rather it is a sign of small-mindedness that these people paid so much attention to Aristotle at all in the first place. In my view, it is not so much impressive that they deviated a bit from Aristotle as it is deplorable that they framed so much of what they did relative to Aristotle, even when they disagreed with him. This is very different from post-Aristotelian thought in Greek times, where there is no evidence that any mathematician paid any attention to Aristotle's mechanics.

In any case, “extravagant claims for the modernity of medieval concepts” suffer from “serious defects.”[<cit.>.]
There was no
such thing as a fourteenth-century science of mechanics
in the sense of a general theory of local motion applicable
throughout nature, and based on a few unified principles.
By searching the literature of late medieval physics for just
those ideas and those pieces of quantitative analysis that
turned out, three centuries later, to be important in
seventeenth-century mechanics, one can find them; and
one can construct … a “medieval science of
mechanics” that appears to form a coherent whole and to
be built on new foundations replacing those of Aristotle's
physics. But this is an illusion, and an anachronistic fiction,
which we are able to construct only because Galileo and
Newton gave us the pattern by which to select the right
pieces and put them together.[<cit.>.]
The main piece of such precursorism is the so-called “mean speed theorem.” This trivial theorem states that, in terms of distance covered, a uniformly accelerated motion is equivalent to a constant-speed motion with the same average speed (Figure <ref>). Some people praise this as an “impressive” achievement[<cit.>.]—“probably the most outstanding single medieval contribution to the history of physics,” derived by “admirable and ingenious” reasoning[<cit.>.]—even though these authors did nothing with this trivial theorem and only deduced it to illustrate the notion of uniform change abstractly within Aristotelian philosophy. Later the theorem became central in “Galilean” mechanics since free fall is uniformly accelerated.[<ref>.] But it “was, in fact, never applied to motion in fall from rest during the 14th, or even in the 15th century” (only in the mid-16th century there is a passing remark to this effect within the Aristotelian tradition, “without any accompanying evidence”).[<cit.>.] Let us not radically inflate our esteem for the Middle Ages by anachronistically praising them for pointing out a trivial thing that centuries later took on a significance of which they had no inkling.[For example, it played no significant role in the thought of the first person to prove it, Oresme. <cit.>.] Let us instead recognise the theorem for the trifle that it is. Then we shall also not have any need to be surprised when it turns out that Babylonian astronomers assumed it without fanfare thousands of years earlier still.[[Ossendrijver , 2016].]

In a similar vein we are told that there are “unmistakeable Jesuit influences in Galileo's work”[<cit.>.]: “Above all Galileo was intent in following out Clavius's program of applying mathematics to the study of nature and to generating a mathematical physics.”[<cit.>.] The preposterous notion that this was “Clavius's” program can only enter one's mind if one only reads philosophy. It was obviously Archimedes's program, except, unlike Clavius, he proved his point by actually carrying it out instead of sermonising about what one ought to do in philosophical prose. Philosophers (ancient and modern alike) have a tendency to place disproportionate value on explaining something conceptually as opposed to actually doing it. After all, that is virtually the definition of philosophy. Hence they praise certain Aristotelians for explaining some supposedly profound principles of scientific method even when “it is quite clear that [none of them] ever applied his advocated methods to actual scientific problems.”[<cit.>.] Descartes—a mathematically creative person—knew better: “we ought not to believe an alchemist who boasts he has the technique of making gold, unless he is extremely wealthy; and by the same token we should not believe the learned writer who promises new sciences, unless he demonstrates that he has discovered many things that have been unknown up till now.”[Descartes to Van Hogelande, 1639/1640, <cit.>.] Unfortunately, such basic common sense is often lacking among historians and philosophers assigning credit for basic principles of the scientific method.

There is a contradiction in the way modern historians try to trace many aspects of the scientific revolution to roots in the middle ages. On the one hand these historians like to claim that the traditional view of the scientific revolution is ahistorical and based on an anachronistic mindset, whereas their own account that sees continuity with the middle ages is more sensitive to how people actually thought at the time itself. Ironically, however, their view, which is supposed to be more true to the historical actors' way of thinking, is actually all the more blatantly at odds with how virtually all leaders of the scientific revolution thought of the middle ages. “The scientific achievement of the Middle Ages was held in unanimous contempt from Galileo's time onward by those who adhered to the new science. Leibniz' scathing verdict `barbarismus physicus' neatly encapsulates the reigning sentiment.”[<cit.>.] This was not for nothing. Leibniz was an erudite scholar well versed in the philosophy of the schools. But he was also an excellent mathematician. The latter enabled him to pass a sound judgement on medieval science.

§.§ Epilogue

Galileo could not have asked for better co-conspirators than those modern academia have provided for him. He desperately needs his audience to view Aristotle as the default baseline against which all his works should be evaluated. He desperately needs his audience to be ignorant of mathematical authors. And he's in luck. The default training of historians of science is not higher mathematics and physics, but seminars based on non-mathematical authors such as Aristotle. So the people tasked with being Galileo experts are by design the people most inclined to accept Galileo’s deceit. Pretending that Archimedes doesn’t exist serves both their purposes and Galileo’s, since they share Galileo’s aversion to proper mathematics. For example, the annual bibliography of works on the history of science published by the flagship journal of the History of Science Society consistently lists well over ten times as many works on Aristotle as on Archimedes, and almost as many more on “Aristotelianism.”[Number of entries in the subject index of the annual Isis comprehensive bibliography for the years 2003–2017: Archimedes 42, Aristotle 482, Aristotelianism 339. There has never been an entry on “Archimedeanism,” though I for one would welcome it.]

Such are the inclinations and predispositions of the philosophy-trained humanists who dominate the field today. But people steeped in mathematics see the world differently—a fundamental schism in the outlooks of modern historians of science and the historical figures they are trying to understand. If we read authors like Copernicus, Kepler, Galileo, Descartes, and virtually everyone else who made a contribution to the mathematical sciences, we find endless praise for Archimedes and bottomless contempt for Aristotle. With this as our baseline our view of Galileo is radically transformed.

Following Galileo's sentencing by the church, an author of a book to be printed at Florence was told by the Inquisition to change the phrase “most distinguished Galileo” into “Galileo, man of noted name.”[<cit.>.] Though I am not generally on the side of the Inquisition, I have come to the conclusion that this particular decree is sound. Instead of “Galileo, father of modern science,” we would do better to make it “Galileo, man of noted name.”

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