Prognostic practitioners and transient underdetermination

I wrote these notes on the problem of transient underdetermination in seventeenth century cosmology for a previous post on epistemic counter-closure, but the more I thought about it, the more I realised that it wasn't an appropriate example: ever since Pierre Duhem, the cosmological dispute between heliocentric and geocentric models been historically understood in philosophy to primarily be a case of underdetermination rather than epistemic counter-closure. Introducing any instances of apparent inferential knowledge predicated on false basis beliefs produces additional confusion rather than clarity.

Setting the historical stage

(Note: it is difficult to construct a label that accurately reflects the appropriate categories for these historical communities. It would be a-historical to label these individuals 'scientists': the relatively recent social community of scientists did not yet exist, since these individuals were, depending on where and when they lived, part-astrologer, astronomer, mathematician, engineer, cartographer and globe-maker. Furthermore, there is too much conceptual baggage carried over if these individuals were called 'astrologers', and 'astronomer' obscures just as much as the label 'astrologer'. However, their preferred titles of 'mathematicus' or 'astronomus' are more or less conceptually empty to the modern mind, carrying with them no baggage at all. Consequently, I have followed Robert Westman (2011) and decided to refer to these individuals as prognostic practitioners.)

Consider the development of heliocentric models from the early twelfth century up to the mid-sixteenth and early seventeenth centuries: the twelfth-century prognosticator Alpetragius proposed in his Kitāb fī l-hay’a an early heliocentric model, but insisted that it was at most a mathematical convenience, rather than a description of the physical world (Alpetragius’ work was relatively popular in Europe, yet bore little similarities to later theories of heliocentrism beyond the central claim that the Earth orbited the Sun. See: Goldstein, 1964).

At the time of the publication of Copernicus' De revolutionibus in 1543, planetary tables such as Erasmus Reinold's Prutenic Tables, based on a heliocentric model, were no more accurate than star tables based on geocentric models found in Claudius Ptolemy's Almagest, published more than a thousand years previously. 

Furthermore, Copernicus’ model suffered from the same problems that plagued Ptolemy’s geocentric model, namely the use of epicycles. That is to say, heliocentric models began as mathematical conveniences, and for a time continued to be little more than a mathematical convenience, for they were no more accurate in their predictions compared to geocentric and geoheliocentric models. Nor were they particularly simple in their construction, lacking the use of the ellipses to model planetary motion.

It was only at the time of Kepler's publication in 1627 of the Rudolphine Tables, based on Tycho Brahe's data, that Kepler's model of planetary motion was shown to better fit the astronomical evidence than planetary tables based on Ptolemy’s geocentric model. However, even Kepler’s elliptic heliocentric model provided only a rough approximation of the true orbit of planets.

To summarise the long and convoluted history of the development of heliocentrist models, over the course of a few hundred years, what began as a mathematical convenience soon developed into a realised theoretical system that, according to its adherents, described the physical structure of and natural laws governing the solar system. The model was smoothed out, simplified, and through its simplification became even more predicatively successful.

This is an often-seen trend in the creation of mathematical or geometrical models: the model begins as a convenience, and after the model is repeatedly corroborated, revised and refined, the convenience is no longer merely convenient, but, in the eyes of individuals that use the model, purportedly a mathematical reflection of the deeper structure of the world.

A problem with this attempt at historical reconstruction: underdetermination

This reconstruction is, obviously, a fairly positive or Whiggish view of the historical development of a progressive research programme: the prognostic practitioners did not realise the value inherent in their model at its birth, grew to appreciate it for its elegance and mathematical simplicity compared to its rivals during its teenage years, and as it matured soon preferred it for a number of its theoretical virtues when compared to its rivals. At this point, the prognostic practitioners were like Eugene Winger (1960) or Hilary Putnam in their diagnosis of the state of play: these mathematical models were so unreasonably effective that the best available explanation for their predictive success was that the model had captured some deeper structure of the world. Miracles don't come easily.

However, there is one issue that faces a reconstruction of the history of heliocentric models: there was a gradual increase in the confidence of prognostic practitioners in the predictive accuracy of Copernicus’ heliocentric models compared to the rival Ptolemaic and Tychonic models, and a growing commitment to realism over instrumentalism or conventionalism by prognosticators. This is not in dispute. But this is a sociological phenomenon. However, even if the competing forms of heliocentrism were, after Kepler’s model was developed, generally more accurate in their predictions than Ptolemaic and Tychonic models, prognosticators were at this very time unable to account for a debilitating objection to heliocentric models: the confidence in heliocentric models existed in despite of a defeating reason.

Consequently, we are faced with a puzzle: these prognosticators grew more confident in a research programme while at the same time the research programme faced a novel objection that had yet to be addressed, and could not be addressed with the available methods.

As Tycho Brahe, perhaps the leading astronomer of his era, noted, if the Earth orbited the Sun, there would have been a measure of stellar parallax: at different times of the year, the stars would have slightly different positions. However, no change in their positions was measured by Brahe or, for that matter, any other astronomer until the nineteenth century. 

In light of the failure to measure stellar parallax, it is worth reconstructing this motivation for the development of geoheliocentric models. In order for heliocentric models of the solar system to be viable, heliocentrists had to introduce an auxiliary hypothesis: the stars were so far away as to prevent the measurement of stellar parallax.

However, this auxiliary hypothesis that stars were necessarily remote to avoid any measurement of stellar parallax produced the star size problem: the diameters measured of stars were, due to the methods used in the production of lenses in telescopes available at the time, as well as measurements taken by Brahe with the naked eye and the astronomer's stave, did not reflect their true size. In fact, if heliocentrism were true and their available astronomical records were accepted, Procyon, one of Earth's nearest stellar neighbours, would consequently dwarf the size of the Sun, extending in diameter to the orbit of Saturn around the Sun. Sirius would, in turn, dwarf Procyon.

This exaggeration of their size meant that a heliocentrist was left with a conundrum: the local star, the Sun, was the only star in existence with a vastly smaller diameter when compared all other measured stars; all other stars were massive. The Sun would be a grain of sand in a universe full of pebbles and rocks. This conclusion was too absurd for Brahe, as well as other Tychonists and Ptolemacists: in order to fix the size of planets such that they were generally similar in size to the Sun, the stars would have to lie just beyond Saturn. With the stars fixed at this distance, the Sun was no longer an unexplained outlier; it was just one more grain of sand on a beach. As Albert van Helden says, 'Tycho's logic was impeccable; his measurements above reproach. A Copernican simply had to accept the results of this argument' (1985, 51).

Consequently, Tycho and Nicholaus Reimers Baer both independently developed their rival geoheliocentric models. Under both Tychonism and Reimerism, the planets orbited the Sun, while the Sun orbited the Earth; the stars were situated beyond Saturn, and were a comparable size to the Sun. Procyon, for example, would be larger than Saturn, although slightly smaller than the Sun; Sirius would be larger than Procyon but slightly smaller than the Sun; and so on.

Since geoheliocentric models better accounted for the lack of measured stellar parallax, geoheliocentric prognosticators would (presumably) have stronger grounds for accepting realism than their heliocentric counterparts. While geoheliocentric models sacrificed simplicity in order to account for the lack of measured stellar parallax, elsewhere there were significant gains: there were no longer the debilitating objection of the Sun's inexplicable deviance in size from the norm.

One possible response to Brahe's star-size problem would be to claim that this was merely an experimental monster that would be resolved with further developments of technology: the eye was simply not a reliable method for determining the relative size of distant stars: atmospheric refraction presumably distorted the size of each star, and astronomical prognosticators would have to wait for the development of more advanced methods, such as early telescopes. However, two generations after Brahe, Giovanni Battista Riccioli updated the star-size problem in his Almagestum Novum (1651) to account for telescopes, noting that Galileo and Simon Marius' measurements of the size of stars were in agreement with Brahe's (Graney & Grayson, 2011).

The size of stars observed with either the naked eye or telescopes were similar--that is, similarly inaccurate. It was only until a century after Riccioli that in 1838, Friedrich Wilhem Bessel measured stellar parallax, providing an explanation for the illusory size of stars seen with the naked eye and early telescopes (Gingerich, 2011, 136).

A brief summary of the historical problem-situation

The possibility space in the early seventeenth century was not limited to the previously mentioned models:  the French mathematician François Viète’s unpublished manuscript, Ad Harmonicum Coeleste Libri Quinque Priores, presents a model he dubbed ‘Francilinidean’. The model was ‘essentially Ptolemaic’, but departing ‘notably from Ptolemy’ by replacing a number of Ptolemaic epicycles with ‘epi-ellipses and elliptical deferents’ (Schofield 1965, 295 emphasis added).

The manuscript is dated to approximately 1600-1603 (Swerdlow 1975, 188), almost a full decade before the publication of Kepler's Astronomia nova (1609), making it one of the earliest examples of a model that proposed an elliptical orbit of planetary motion. However, Francilinideanism, although existent within accessible possibility space, was apparently unknown to early seventeenth century prognosticators.

There exist a number of other viable models, although they are more of historical curiosities: Martianus Capella proposed an early geoheliocentric model in the fourth century. Four centuries later, the Irish philosopher John Soctus Erigena proposed his own geoheliocentrist model that differed from Capella’s model in the number of planets that orbited the Sun. Kelallur Nilakantha Somayaji, an Indian prognosticator, may have developed another competing geoheliocentric theory that differed from both Reimers and Tycho (Ramasubramanian, 1994). These models existed within the accessible possibility space as well.

In sum, there were at least six viable models of the solar system during the early seventeenth century: Keplerism, Copernicanism, Tychoism, Reimerism, Francilinideanism and Ptolemacism. 

There are a number of theoretical virtues we can choose to rank each model:

Empirical fit: geoheliocentrists like Tychonicists and Reimerists, as well as geocentrists like Ptolemacists and Francilinideanists, could account for apparent (but ultimately erroneous) falsifiers for a theoretical system like the star-size problem, while heliocentrists like Copernicans and Keplerians could not.

Predictive accuracy: Kepleranism and Copernicanism were both far more predicatively accurate than their rivals, yet there was a trade-off for predictive accuracy of planetary motion: both models relied on rejecting the available empirical evidence as an experimental ‘monster’ that would be eventually resolved. Some early Copernicans, such as Christoph Rothmann and Philips Lansbergen, even answered the star-size problem by providing implausible theological arguments: they argued the size of all other stars was no problem, for God was not constrained in any way, and could choose to produce stars at any size he desired (Graney, 2013).

Simplicity of the model: geoheliocentric models like Tychonism and Reimerism were preferable to geocentric models like Ptolemacism and Francilinideanism, since they were often a better fit to the experimental evidence, yet geocentric models like Ptolemacism, while less predicatively accurate, were simpler than geoheliocentric models: all planets and the Sun orbited the Earth under geocentric models, while under geoheliocentric models all planets orbited the Sun, while the Sun orbited the Earth.

Lastly, it is even possible to compare heliocentric, geoheliocentric and geocentric models with one another: for geoheliocentric models, Tychonism was a heliostatic theory: it posited that the Earth was stationary; Reimerism posited that the Earth rotated around its axis. Thus Reimerism accounted for some experimental results that supported the rotation of the Earth better than Tychonism, although Reimerism now appeared ad hoc when compared to Tychonism, for it posited an absurdity that lead to Tycho developing his model as heliostatic: there was no available mathematical account for the movement of objects in space on a rotating Earth. As Ptolemy objected,

'All objects not actually standing on the earth would appear to have the same motion, opposite to that of the earth: neither clouds nor other flying or thrown objects would ever be seen moving towards the east, since the earth's motion towards the east would always outrun and overtake them, so that all other objects would seem to move in the direction of the west and the rear... Yet we quite plainly see that they do undergo all these kinds of motion, in such a way that they are not even slowed down or speeded up at all by any motion of the earth'. 

For geocentric models such as Francilinideanism, since it relied on elliptical orbits and eliminated epicycles, it was preferable to Ptolemacism on grounds of mathematical and conceptual simplicity, as well as predictive accuracy, for it was empirically equivalent to Kepler's model; for heliocentric models, Kepleranism was preferable to Copernicanism on these same grounds of predictive accuracy and simplicity. 

Depending on whichever theoretical virtue a prognostic practitioner preferred over its rivals, there were eminently plausible grounds for preferring and dispreferring any model. It was only with a theoretical backing for the mathematical laws that governed the movement of planets that would not be available until the publication of Newton’s Principia (1687) that an account for Kepler’s theory of elliptical heliocentrism had the requisite geometrical backing, as well as a viable alternative physics that unified planetary and terrestrial motion to rival Aristotelian physics. 

However, Newtonian mechanics as it is taught today, reflecting more of the formulations set out by Laplace and Langrange, bears little resemblance to Isaac Newton's mechanics. As the physicist Steven Weinberg said, 'Newton is pre-Newtonian'. Even the theoretical grounds for adopting heliocentricism were subject to centuries-long game of conceptual elimination, addition and reconstruction, for Newton's mechanical theory lacked any physical explanation for the movement of planets.

Examining the philosophical problem-situation

(We have, unsurprisingly, a case of Arrow's impossibility theorem in action: there were upwards of six distinct alternative models and at least three different desirable theoretical virtues for each model (simplicity, predictive accuracy, empirical fit). Ranking preferences for each model based on a Borda Count, for example, in which there are n models, and based on a criterion C, the best model is given n points, the second-best n-1 points, and so on, isn't available, since it violates a central assumption in Arrow's theorem, the assumption of binary independence. This leads to the result that the prognosticatory community necessarily could not rationally decide on a ranking of models. Davide Rizza (2014) shows that even weakening binary independence cannot salvage theory-choice.)

This is but a rediscovery of the problem that was first acknowledged in the fifth century B.C.E by Apollonius and Hipparchus: the models of the Sun's motion, the simple eccentric and epicycle-deferent, were geometrically equivalent. In 1543, a thousand years later, the very same problem was acknowledged by Copernicus: 'It makes no difference that what they [the ancients] explain by a resting Earth and a universe whirling round, we take up in the opposite way so that together with them we might rush to the same goal. For in such matters, those things that are thus mutually related agree, in turn, one with the other' (Westman 2011, 5). 

These differing models were themselves (more or less) empirically equivalent, and geometrically transformable from one to another, taking different equants at the point of orbit and introducing or eliminating epicycles as necessary to maintain equivalence. Thus there would have to be different grounds for theory-preference besides empirical fit, but no model was obviously preferable when compared to its rivals due to Arrow's theorem. Furthermore, even if one model had been preferable compared to all available rival models, the question arises whether the proposed criteria for theory-preference itself were suitable. 

While the standards adopted today pay far more attention to empirical and logical grounds (i.e. empirical adequacy, predictive success, coherence, amongst others), this papers over the previously accepted epistemic conventions of sixteenth and seventeenth-century prognosticators: this was a time that a criterion of knowledge was inexorably tied to coherence with Biblical exegesis and the writings of the Ancients. Rothmann and Lansbergen's appeal to the extra-natural power of God may be absurd to the modern-day way of thinking, but it was not outside the realm of acceptability to prognosticators. It is only in hindsight and several hundred years of cultural shift that the plausibility of a standard wanes. Consider the counterfactual: had history run a different course, Biblical authority may have been an overwhelmingly strong standard compared to other theoretical virtues, leading to an outcome in which prognosticators preferred geocentric models over their rivals, even if geocentric models were less predicatively successful. In sum, the very criteria for theory-preference may be historically contingent.

What would have been the most epistemically prudent action, given this case of transient underdetermination?

In this historical case-study, as with many examples from the history of science, heliocentric, geoheliocentric and geocentric prognosticators more often than not genuinely believed their theoretical system accurately reflected a deeper structure of the cosmos. However, this genuine belief was, in almost all cases, in error. 

Instead, for the modern-day anti-realist, the diagnosis is obvious: it would have been more appropriate to adopt a form of Alpetragius’s conventionalism. Some prognostic practitioners took this route:

Andreas Osiander’s claimed in his preface to Copernicus’ De revolutionibus that a heliocentric model was merely a useful method, and ‘need not be true nor even probable… if they provide a calculus consistent with the observations, that alone is enough’ (Copernicus, 1992; cf. Gingerich, 2004).

Giovanni Giovano Pontano concluded, ‘The circles, the epicycles, and all suppositions of this sort should, therefore, be regarded as imaginary; they have no real existence in the heavens. They have been invented and imagined so as to let the celestial motions be grasped and to exhibit them to our sight’ (Duhem, 1969, 55).

Cardinal Bellarmine claimed that Copernican theory produced a helpful and useful calculating device that can be used to ‘save the appearances’ and nothing more, for ‘[t]o demonstrate that the appearances are saved by assuming the sun is at the center and the earth is in the heavens is not the same thing as to demonstrate that in fact the sun is in the center and the earth is in the heavens’ (Drake 1957, 162-4).

In fact, as Gingerich notes, ‘nearly every sixteenth-century astronomer accepted Copernicus’s De revolutionibus as an up-to-date recipe book for computing positions of planets, but definitely not as a description of physical reality… [since] the idea of a spinning Earth was completely ridiculous’ (2011, 135-140). A conventionalist or instrumentalist would retroactively diagnose the problem as follows: past prognosticators did in fact or ought to have emulated Alpetragius, Osiander and Bellarmine, but spoke out of turn, for they were unreasonably optimistic. The conclusion (presumably) is inescapable: the historical data compels us to conclude that future scientists ought to be scientific anti-realists.

Bibliography

Blake, R. (1960). Theory of Hypothesis among Renaissance Astronomers, in Blake, Ducasse, and Madden, eds., Theories of Scientific Method: The Renaissance Through the Nineteenth Century. Seattle: University of Washington Press.

Bradley, J. (1727-1728). An account of a new-discovered motion of the fixed stars. Philosophical Transactions 32, 637-61.

Copernicus, N. (1992). On the Revolutions. Translation and Commentary by Edward Rosen. Baltimore, MD & London: The Johns Hopkins University Press.

Drake, S. (1957). Bellarmine to Foscarini, 12 April 1615, Opere, National Edition, ed. Antonio Favaro, 12, 171-2; from Discoveries and Opinions of Galileo, trans. Stillman Drake. Garden City: Doubleday.

Duhem, P. (1969). To Save the Phenomena: An Essay on the Idea of Physical Theory from Plato to Galileo, Chicago: University of Chicago Press, excerpted from Giovanni Gioviano Pontano, De rebus coelestibus, libri xiv.

Gingerich, O. (2004). The Book Nobody Read. Walker Books.

______. (2011). Galileo, the impact of the telescope, and the birth of modern astronomy. Proceedings of the American Philosophical Society. 155(2).

Goldstein, B.R. (1964). On the theory of Trepidation according to Thābit b.Qurra and al-Zarqāllu and Its Implications for Homocentric Planetary Theory, Centaurus, 10: 232–247.

Graney, C.M. & T.P. Grayson. (2011). 'On the telescopic discs of stars-- a review and analysis of stellar observations from the early 17th through the middle 19th centuries', Annals of science, lxviii: 351-73.

Graney, C.M. (2013). 'Stars as the armies of God: Lansbergen's incorporation of Tycho Brahe's star-size argument into the Copernican theory', Journal for the History of Astronomy, xliv.

______. (2015). Setting Aside All Authority: Giovanni Battisa Riccioli and the Science against Copernicus in the Age of Galileo, Notre Dame, Indiana: University of Notre Dame Press.

van Helden, A. (1985). Measuring the universe. Chicago: University of Chicago Press.

Ramasubramanian, K., M.D. Srinivas & M.S. Sriram (1994). Modification of an earlier Indian planetary theory by the Kerala astronomers (cf. 1500 AD) and the implied heliocentric picture of planetary motion. Current Science. 66(1).

Rizza, D. (2014). Arrow's theorem and theory choice. Synthese, 191(8): 1847-1856.

Schofield, C. (1965.) The Geoheliocentric mathematical hypothesis in sixteenth-century planetary theory, The British Journal for the History of Science. 2(8).

Swerdlow, N.M. (1975). The planetary theory of François Viète. Journal for the History of Astronomy. vi, 185-208.

Westman, R.S. (2011). The Copernican Question: Prognostication, Skepticism, and Celestial Order. Berkeley: University of California Press.

Winger, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications in Pure and Applied Mathematics 13(1).

7/717