Saturday


Nicolaus Copernicus

Long ago I lost count of the number of times I have heard and read that we are now, in the present, living in a pivotal moment in history, that we are, in a sense, at the center of history, and that the present is a privileged moment in time. The idea that one is present at a pivotal moment in history, and that one’s actions in relation to the unfolding of events in the present will play a decisive role in the world that is eventually to emerge from accelerated apocalypticism, may be regarded as the Ptolemaic equivalent of historiography, i.e., an anti-Copernican idea.

Ptolemaic historiography, if there were such a thing, would insist upon the centrality of ourselves and our perspective in the history of the world, holding that we have a privileged perspective on history as a consequence of our position in time. There is a more conventional way to understand this kind of claim. In the introduction to his Lectures on the Philosophy of History, Hegel made a tripartite distinction between original history, reflective history, and philosophical history. The first of these three, original history, is characterized by histories written by individuals who have witnessed the events they are recounting, or who have heard about them first-hand. Such histories are witnesses to history in two senses: firstly in having directly observed history, and secondly in being a witness to the spirit of the time, which entails sharing the Weltanschauung of the participants in contemporaneous history. Ptolemaic history, then, is a form of original history, because it is predicated upon the centrality of contemporaneous historical actors within their own perspective of history.

Copernican historiography, on the other hand, would apply the Copernican principle in time as the Copernican principle already has been applied to space. We have a parallel to this in the cosmological principle and that Fred Hoyle called the perfect cosmological principle: the cosmological principle simpliciter was concerned with the spatial isotropy of the universe, and Hoyle’s perfect cosmological principle extended this isotropy to time as well. The “perfect cosmological principle” proposed by Hoyle, Bondi and Gold as a supplement to the cosmological principle as conventionally understood, intended to justify a steady-state model of the universe, has, like the familiar cosmological principle, been given many expositions, no two of which are precisely the same. For example, here is the formulation from the Encyclopedia Britannica:

“…the universe on average is not only homogeneous and isotropic in space but also constant in time…”

“Fred Hoyle” article in the Encyclopedia Britannica

…here is a formulation in a paper from 2015…

“…the universe should appear essentially the same to all observers in all places at all times.”

“A new perspective on steady-state cosmology: from Einstein to Hoyle” by Cormac O’Raifeartaigh and Simon Mitton

…and here is a formulation from the Routledge Encyclopedia of Philosophy

“…a homogeneous distribution of matter in an infinite space and throughout an infinite time.”

“Cosmology” article in the Routledge Encyclopedia of Philosophy, 1998

Hoyle’s perfect cosmological principle was not widely accepted. The stock answer as to why Hoyle’s perfect cosmological principle was rejected has been to refer to the observational pillars of the big bang cosmology (cf. The Four Pillars of the Standard Cosmology), and most especially the discovery of the CMBR as a confirmation of big bang cosmology. But big bang cosmology ought to be understood in this context as a natural history of the universe. The confirmation of any theory that postulates that the universe has a natural history would have been sufficient to overthrow the steady-state model of the universe. The big bang model of cosmological evolution is one among a class of possible natural histories for the universe.

If we must reject the perfect cosmological principle because the universe is evolving, and therefore appears differently at different times, must we also reject the possibility of Copernican historiography as a rejection of Ptolemaic historiography? I will come back to this, but I will first consider some formulations of the Copernican principle.

Like the many versions of the perfect cosmological principle cited above, there are many formulations of the Copernican principle. For example:

Principle 1.3 (The strong Copernican principle). There are no privileged observers in the universe.”

Hans Ringström, On the Topology and Future Stability of the Universe, Oxford University Press, 2013, p. 6

The generality of this formulation is equally applicable to space and time, unless “the universe” is construed to mean the universe only in its spatial extension and not its temporal extension.

…and another formulation…

“The Copernican principle has been a fundamental tenet of modern science since the 16th century and is also a cornerstone of modern cosmology. It states that we should not live in a special region of the universe.”

“Confirmation of the Copernican principle at Gpc radial scale and above from the kinetic Sunyaev Zel’dovich effect power spectrum” Pengjie Zhang and Albert Stebbins

The implicit distinction between privileged observers and privileged spatial locations appears in formulations of both the cosmological principle and the Copernican principle. An interesting distinction might be explicitly formulated on this basis, such that a privileged spatial region might exist, but that if no observer existed at this location then no privileged observations could be made, but we will set this possibility to one side for the nonce, except to say that a universe without an observer located at a privileged region of space is only a step away from a universe with no observers at all; on the possibility of unobserved universes, and the problems that follow from this idea, cf. my recent post, The Two Senses of “Observable Universe.”

The idea of a perfect cosmological principle and the idea of a Copernican principle, when taken together, imply the possibility of a perfect Copernican principle, generalizing the conventional Copernican principle so that it applies to time as well as to space. A perfect Copernican principle would assert that we do not (or, if you prefer, and in accordance with the formulation in the Zhang and Stebbins paper, we should not) live in a special region or era of the universe.

Given that the Copernican principle follows deductively from the cosmological principle — if the universe is spatially homogeneous and isotropic, it follows that there are no privileged observers, because there are no privileged positions in the universe from which an observer might observe — the perfect Copernican principle would follow from a perfect cosmological principle, and, given material implication, the falsification of any perfect cosmological principle could not entail the truth of a perfect Copernican principle following deductively from a perfect cosmological principle.

History undertaken in the Copernican spirit, i.e., Copernican historiography, would be history written with the perfect Copernican principle as a regulative principle. If the task of history is to write cosmological history, or human history set in the context of cosmological history (as is the case with big history), we cannot do this and remain true to the perfect Copernican principle. A history of the cosmos from from a human perspective (which is the only kind of cosmological history that we, as human beings, can write), is an anthropocentric history, and views the universe entire from the privileged moment in time occupied by human beings, which is a small slice of the evolutionary history of life on Earth, which is, in turn, a small slice in the evolutionary history of the Stelliferous Era, which is, in turn, a small slice in the history of the universe entire.

Big history, then, cannot be Copernican historiography, though one could plausibly argue that big history is the eventual result of viewing the world from a Copernican perspective. I think that this is case, and perhaps I will try to argue another day for a tension within the Copernican principle that leads, on the one hand, to big history, while on the other hand not being theoretically compatible with a strict interpretation of Copernicanism. It seems that not only does the universe evolve, and that human beings evolve, but also the perspective that human beings have of the universe they inhabit also evolves, and it evolves as the interface between human life and the universe.

On a human scale of history, however, I think that the perfect Copernican principle can be applicable. That is to say, if we restrict the scope of history to the human tenure on Earth, then something like the perfect Copernican principle obtains, as no one period of history can be judged to be privileged over any other era of history, and certainly not in terms of a perspective from within history to write history. Each era has the opportunity to write what Hegel called the “original history” of itself, and each era has the opportunity to write reflective histories of its own times taken together with all previous history. In this respect, later eras survey a greater portion of the human past, and so are “privileged” in respect to having more empirical content of human history at their disposal. However, on a purely theoretical level, the expanding empirical content of human history is irrelevant.

No doubt this assertion I have just made — that the expanding empirical content of human history is irrelevant — must sound very strange to the reader (except for those who have read me very closely, and these are few and far between). Let me try to explain. Copernican historiography is integral with what I have called history in an extended sense, i.e., extending distinctively historical modes of thought beyond a exclusive engagement with the past. History in an extended sense comprises both past and future, which are formally indistinguishable (or, better, formally complementary), however radically different they are empirically. I also made this point this in my paper A Manifesto for the Study of Civilization in which I first employed the phrase history in an extended sense:

“One form that the transcendence of an exclusively historical study of civilization can take is that of extrapolating historical modes of thought so that these modes of thought apply to the future as well as to the past (and this could be called history in an extended sense).”

In order to understand history from the perspective of the perfect Copernican principle (which is a little like understanding history sub specie aeternitatis), and thus to “de-provincialize” one’s conception of history (I take the word “de-provincialize” from Carl Sagan), it is sufficient to see that unprecedented events are always occurring, always have occurred, and will continue to occur for as long as any events whatsoever continue to occur and thus continue to supply a natural history to the universe. If our presence, or our location in time (regardless of our presence), were singularly unprecedented, we would be justified in asserting that we live at a special time in history, but even a casual survey of history will show that there is always something occurring that has never before happened in the history of the universe.

Unprecedented events occur with predictable regularity. At a temporal microscale, it could be argued that each and every new moment of time is unprecedented, as the structure of the universe in no way guarantees to us that time will continue to produce new moments. On the other hand, each new moment of time is a moment among moments, one of a class of moments, the totality of which makes up the totality of time, so that each new moment may be as unique as each snowflake, but all moments are alike in the way that all snowflakes are alike. Whether or not we see moments of time or snowflakes as unique or as all the same depends upon how fine-grained an account of identity we bring to the analysis. Thus, to fully develop the idea of a Copernican historiography it will be necessary (at some point, though not today) to analyze the conception of identity one brings to history, and the scope of history we are considering at any one time. This is already implicit above when I noted that restricting our scope from cosmological history to human history may yield a valid application of the perfect Copernican principle.

An extremely fine-grained account of history will yield the absolute novelty of every moment; a less detailed overview of history would perhaps eventually yield absolute repetition, as represented by Ecclesiastes’ famous line, “The thing that hath been, it is that which shall be; and that which is done is that which shall be done: and there is no new thing under the sun.” Or maybe not; this is something on which I will have to think further. Ecclesiastes’ principle implies a cyclical conception of history, which I reject, but more on this another time.

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Ecclesiates’ explicit denial of novelty in the world: The thing that hath been, it is that which shall be; and that which is done is that which shall be done: and there is no new thing under the sun.

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Wednesday


The Thomas Digges chart of a Copernican solar system from 1576.

The Thomas Digges chart of a Copernican solar system from 1576.

The full awareness of our sun being a star, and the stars being suns in their own right, was a development nearly coextensive with the entire history of science, from its earliest stirrings in ancient Greece to its modern form at the present time. During the Enlightenment there was already a growing realization of this, as can be seen in a number of scientific works of the period, but scientific proof had to wait for a few generations more until new technologies made available by the industrial revolution produced scientific instruments equal to the task.

bessel-and-parallax

The scientific confirmation of this understanding of cosmology, which is, in a sense, the affirmation of Copernicanism (as distinct from heliocentrism) came with two scientific discoveries of the nineteenth century: the parallax of 61 Cygni, measured by Friedrich Wilhelm Bessel and published in 1838, which was the first accurate distance measured to a star other than the sun, and the spectroscopy work of several scientists — Fraunhofer, Bunsen, Kirchhoff, Huggins, and Secchi, inter alia (cf. Spectroscopy and the Birth of Astrophysics) — which demonstrated the precise chemical composition of the stars, and therefore showed them to be made of the same chemical elements found on Earth. The stars were no longer immeasurable or unknowable; they were now open to scientific study.

Joseph von Fraunhofer invented the spectroscope, and first observed what are now called Fraunhofer lines.

Joseph von Fraunhofer invented the spectroscope, and first observed what are now called Fraunhofer lines.

The Ptolemaic conception of the universe that preceded this Copernican conception painted a very different picture of the universe, and of the place of human beings within that universe. According to the Ptolemaic cosmology, the heavens were made of a different material than the Earth and its denizens (viz. quintessence — the fifth element, i.e., the element other than earth, air, fire, and water). Everything below the sphere of the moon — sublunary — was ephemeral and subject to decay. Everything beyond the sphere of the moon — superlunary — was imperishable and perfect. Astronomical bodies were perfectly spherical, and moved in perfectly circular lines (except for the epicycles). Comets were a problem (i.e., an anomaly), because their elliptical orbits ought to send them crashing through the perfect celestial spheres.

Geocentric Ptolemaic cosmology by Orance Fine (1494-1555)

Geocentric Ptolemaic cosmology by Orance Fine (1494-1555)

This Ptolemaic cosmology largely satisfied the scientific, philosophical, moral, and spiritual needs of western thought from classical antiquity to the end of the Middle Ages, and this satisfaction presumably follows from a deep consonance between this conception of the cosmos and a metaphysical vision of what the world ought to be. Ptolemaic cosmology is the intellectual fulfillment of a certain kind of heart’s desire. But this was not the only metaphysical vision of the world having its origins (or, at least, its initial expression) in classical antiquity. Another intellectual tradition that pointed in a different direction was mathematics.

The imaginative background of Ptolemaic cosmology; an image of God as architect from a Moralized Bible, folio 1 verso, Österreichische Nationalbibliothek, Vienna.

The imaginative background of Ptolemaic cosmology; an image of God as architect from a Moralized Bible, folio 1 verso, Österreichische Nationalbibliothek, Vienna.

Mathematics was the first science to attain anything like the rigor that we demand of science today. It remains an open question to this day — an open philosophical question — whether mathematics is a science, one of the sciences (a science among sciences), or whether it is something else entirely, which happens to be useful in the sciences, as, for example, the formal propaedeutic to the empirical sciences, in need of formal structure in order to organize their empirical content. The sciences, in fact, get their rigor from mathematics, so that if there were no mathematical rigor, there would be no possibility of scientific rigor.

Euclid provided the model of formal thought with his axiomatization of geometry. Legend has it that there was a sign over the door of Plato's Academy stating, 'Let no one enter here who has not studied geometry.'

Euclid provided the model of formal thought with his axiomatization of geometry. Legend has it that there was a sign over the door of Plato’s Academy stating, ‘Let no one enter here who has not studied geometry.’

Mathematics has been known since antiquity as the paradigm of exact thought, of precision, the model for all sciences to follow (remembering what science meant to the ancients, which is not what it means today: a demonstrative science based on first principles), and this precision has been seen as a function of its formalism, which is to say its definiteness, it boundedness, its participation in the peras. Despite this there was yet a recognition of the infinite (apeiron) in mathematics. I would go further, and assert that, while mathematics as a rigorous science has its origins in the peras, it has its telos in the apeiron. This is a dialectical development, as we will see below in Proclus.

An early copy of Euclid's Elements, which axiomatically systematized geometry.

An early copy of Euclid’s Elements, which axiomatically systematized geometry.

Proclus expresses the negative character of the infinite in his commentary on Euclid’s Elements:

“…the infinite is altogether incomprehensible to knowledge; rather it takes it hypothetically and uses only the finite for demonstration; that is, it assumes the infinite not for the sake of the infinite, but for the sake the infinite.”

Proclus, A Commentary on the First Book of Euclid’s Elements, translated, with an introduction and notes, by Glenn R. Morrow, Princeton: Princeton University Press, 1992, Propositions: Part One, XII, p. 223. This whole section is relevant, but I have quoted only a brief portion.

There is no question that the apeiron appeared on the inferior side of the Pythagorean table of opposites, but it is also interesting to note what Proclus says earlier on:

“The objects of Nous, by virtue of their inherent simplicity, are the first partakers of the Limit (περας) and the Unlimited (ἄπειρον). Their unity, their identity, and their stable and abiding existence they derive from the Limit; but for their variety, their generative fertility, and their divine otherness and progression they draw upon the Unlimited. Mathematicals are the offspring of the Limit and the Unlimited…”

Proclus, Commentary on the First Book of Euclid, Prologue: Part One, Chap. II

Here the apeiron appears on an equal footing with the peras, both being necessary to mathematical being. “Mathematicals” are born of the dialectic of the finite and the infinite. Both of these elements are also found (hundreds of years earlier) in the foundations of geometry. As the philosophers produced proofs that there could be no infinite number or infinite space, Euclid spoke of lines and planes extended “indefinitely” (as “apeiron” is usually translated in Euclid). Even later when the Stoics held that the material world was surrounded by an infinite void, this void had special properties which distinguished it from the material world, and indeed which kept the material world from having any relation with the void. The use of infinities in geometry, however, even though in an abstract context, force one to maintain that space locally, directly before one, is essentially of the same kind as space anywhere else along the infinite extent of a line, and indeed the same as space infinitely distant. All spaces are of the same kind, and all are related to each other. This constitutes a purely formal conception of the uniformity and continuity of nature. One might interpret the subsequent history of science as redeeming, through empirical evidence, this formal insight.

proclus-on-euclid

The infinite is the “internal horizon” (to use a Husserlian phrase) and the telos of mathematical objects. Given this conception of mathematics, the question that I find myself asking is this: what was the mathematical horizon of the Greeks? Did the idea of a line or a plane immediately suggest to them an infinite extension, and did the idea of number immediately suggest the infinite progression of the series, or were the Greeks able to contain these conceptions within the peras, using them not unlike we use them, but allowing them to remain limited? Did ancient mathematical imagination encompass the infinite, or must such a conception of mathematical objects (as embedded in the infinite) wait for the infinite to be disassociated from the apeiron?

husserl-quote

The wait was not long. While the explicit formulation of the mathematical infinite had to wait until Cantor in the nineteenth century, Greek thought was dialectical, so regardless of the nature of mathematical concepts as initially conceived, these concepts inevitably passed into their opposite numbers and grew in depth and comprehensiveness as a result of the development of this dialectic. Greek thought may have begun with an intellectual commitment to the peras, and a desire to contain mathematics within the peras, consequently an almost ideological effort to avoid the mathematical infinite, but a commitment to dialectic confounds the demand for limitation. It is, then, this dialectical character of Greek thought that gives us the transition from purely local concepts to a formal concept of the uniformity of nature, and then the transition from a formal conception of uniformity to an empirical conception of uniformity, and this latter is the cosmological principle that is central to contemporary cosmology.

Geometry as represented by Raphael in The School of Athens.

Geometry as represented by Raphael in The School of Athens.

The cosmological principle brings us back to where we started: To say that the sun is a star, and every star a sun, is to say that the sun is a star among stars. Earth is a planet among planets. The Milky Way is a galaxy among galaxies. This is not only a Copernican idea, it is also a formal idea, like the formal conception of the uniformity of nature. (In A Being Among Beings I made a similar about biological beings.) To be one among others of the same kind is to be a member of a class, and to be a member of a class is to be the value of a variable. Quine, we recall, said that to be is to be the value of a variable. This is a highly abstract and formal conception of ontology, and that is precisely the importance of the formulation. This is the point beyond which we can begin to reason rigorously about our place in the universe.

The sun in its local stellar neighborhood as a star among stars.

The sun in its local stellar neighborhood as a star among stars.

We require a class of instances before we can draw inductive inferences, generalize from all members of this class, or formalize the concept represented by any individual member of that class. This is one of the formal presuppositions of scientific thought never made explicit in the methodology of science. We could not formulate the cosmological principle if we did not have a concept of “essentially the same,” because the “same” view that we see looking in any direction in the universe is not identically the same, but rather essentially the same. Of any two views of the universe, every detail is different, but the overview is the same. The cosmological principle is not a generalization, not an inductive inference from empirical evidence; it is a formal idea, a regulative idea that makes a certain kind of cosmological thought possible.

cosmological-principle

Formal principles like this are present throughout the sciences, though not often recognized for what they are. Bessel’s observations of 61 Cygni not only required industrialized technology to produce the appropriate scientific instruments, these observations also presupposed the mathematics originating in classical antiquity, so that the nineteenth century scientific work that proved the stars to be like our sun (and vice versa) was predicated upon parallel formal conceptions of universality structured into mathematical thought since its inception as a theoretical discipline (in contradistinction to the practical use of mathematics as a tool of engineering). Formal Copernicanism preceded empirical Copernicanism. Without that formal component of scientific knowledge, that scientific knowledge would never have come into being.

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