CHAPTER 13: Deductionism, or the Proof Shall Make you Free


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Both Greek and religious influences thus favored deduction over induction…

In effect, the Christian faith under Constantine was formalized more or less axiomatically. The Church could thereby provide a unified social order and system of justice, according to divine law, of which it was the sole guardian and interpreter. Steven Weinberg describes the religious version of deductionism as “wishful thinking”: “Unlike science, religious experience can suggest a meaning for our lives, a part for us to play in a great cosmic drama of sin and redemption… For just these reasons, the lessons of religious experience seem to me indelibly marked with the stamp of wishful thinking.” [Weinberg Dreams of a Final Theory, p255] While I quite agree that religion involves wishful thinking, the thesis of this book is that science does too! The deductionist program, and the dream of a final theory in particular, are examples of wishful thinking.


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This modeling of language served as a paradigm for the modeling of nature…

Perhaps because they were written only, classical Latin and Greek had been grammatically formalized, and served as models in the 16th century to formalize and teach vernacular tongues. The transition, from a language that was simply occupied from within to an outside view of it as an object to dissect, reflects the growth of subjectivity. It represents a transition from a naïve to a self-conscious perspective.

In the watershed year of 1492, Antonia de Nebrija published the first grammar of the Castilian tongue, which he called “artificial Castilian,” proposing that a standardized Spanish would be the necessary basis of empire. Just as Euclid’s Elements later served as the paradigm of formalization, so Greek and Latin grammar inspired the formalization of Spanish and other languages.

The very term ‘language’ superseded ‘tongue’, to underline the constructed aspect of language. This led to consideration of the vernacular itself as having been constructed (informally and unconsciously) in the first place. Hence, speech was not divinely conferred at birth but socially constructed. Modern studies, on the other hand, reveal the extent to which language capacity is genetically innate.


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And so is the… belief that physical reality essentially is mathematics.

In the medieval hierarchy of the sciences, physics and natural philosophy had been considered superior to mathematics. The new central roles of mathematics entered natural philosophy through Galileo, who replaced dogmatic adherence to the philosophy of Aristotle with equally dogmatic faith in a mathematical interpretation of nature: “…Aristotle looked at nature as a process by which things fulfill their potential… This concern with teleology was allied with the belief that natural philosophy could be built directly on perception and that mathematics could not explain the colorful and qualitatively determined facts of common experience. Galileo considered such an approach naïve and misleading, and he sought to transcend the limitations of Aristotelian empiricism by claiming that reality is mathematical in form and that mathematical theory should determine the very structure of experimental research… Plato had held the physical world was a copy or likeness of a transcendent, ideal world of mathematical forms; it was an inexact copy, and therefore physics could never yield absolute truth but only likely stories. Galileo, by contrast, held that the world actually consisted of the mathematical primary and secondary qualities and their laws and that these laws were discoverable in detail with absolute certainty.” [William R. Shea, “Galileo and the Church” in Lindberg & Numbers God and Nature, p123-4]


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Hence, knowledge of man-made systems represents the most reliable knowledge.

Cf. Danilo Marcondes de Souza Filho “Skepticism and the Philosophy of Language in Early Modern Thought” Pontifícia Universidade Católica do Rio de Janeiro, Brazil: “For this reason, he gives philology, as a science of signs and of interpretation, a capital role in the constitution of human knowledge. Language, as a system of signs, is now given importance precisely as a human product, in the opposite direction of rationalism which thought it imperfect for this very motive. If signs and meaning are human creations, then the meanings of these signs are plainly accessible to human beings.”


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On the other hand, we can at least hope to understand human institutions.

The notion of maker’s knowledge is usually credited to Vico. Yet, Bacon had at least implied it, recognizing that knowledge of the natural world is necessarily open-ended, in contrast to analytic truths. Similarly, Hobbes argues that we only know the cause of something with certainty when we are in fact the cause of it. Spinoza and Locke held related ideas. Hobbes assumes a “state of nature” to be chaos, in which the condition of uncertainty would be unbearable. This is the prime reason for the social contract—that is, the need for the State to enforce order.


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…it is knowledge of the made as opposed to the found.

Cf. Bertrand Russell’s “knowledge by acquaintance,” and the privileged “direct” access to one’s own perceptions, in contrast to inferred knowledge of the external world. As a privileged knowledge of cultural artifacts, maker’s knowledge parallels the certainty Descartes had found in subjective experience as a mental artifact. In that context, cf. also Karl Popper’s “Third World.”


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The principle of maker’s knowledge expresses the allure of deductive systems.

According to Spinoza, only the certainties of reason can make us happy; we are bound to remain unsettled and unsatisfied by fallible empirical knowledge. For, such knowledge depends on things beyond our control; only reason affords absolute confidence, because it dwells within us and is independent of the external world. [Eric Schliesser “Spinoza and the Philosophy of Science: Mathematics, Motion, and Being” Philpapers/SCHSAT-23]


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Vico’s distinction had to do with the human versus the divine creator.

Aristotle distinguished between natural things, which contained within themselves the power of their material realization, and artifacts, which did not. He regarded objects in general, whether natural or artificial, as form imposed upon matter, or matter realized in form. However, he also distinguished between ‘maker’s art’ and ‘user’s art’, giving the example of the helmsman, who naturally knows what the helm is for, whereas its manufacturer knows best the properties of the materials of which it is made. (Compare this to our modern concept of design, which incorporates both.) Moreover, Aristotle’s concept of form includes what we would today call function. He takes a teleological view of both artifice and nature. It’s unclear what he might say about today’s synthetic materials. [Joachim Schummer “Aristotle on Technology and Nature” Philosophia Naturalis 38 (2001) pp105-120]


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While nature may not be perfectly knowable, science as a humanly defined construction ought to be.

Scientific knowledge includes the cumulative results of repeatable experiments and observations, which are records of human interactions with nature. Theories come and go, and though future research may invalidate specific theoretical conclusions, the very requirement of repeatability insures that empirical results should stand as a cumulative record of human actions.


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…resides in the power to invoke clear and precise existence by decree or fiat.

Animal experiments show that numeric symbols can be recognized as indictors of intuitively cognized quantities. [Cf. I. Diester and A. Nieder “Numerical values leave a semantic imprint on associated signs in monkeys” J. of Cognitive Neuroscience Jan 13, 2009] For example, a monkey can learn to associate an arabic numeral with approximately the quantity it represents for humans. Of course, people make use of the same kind of perceptual estimation. In contrast to animals, however, they have also defined the arabic numeral to represent an exact quantity. This is an enormous conceptual leap—not only the first step in arithmetic, but also in establishing deductive as opposed to inductive thought.


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Even in familiar realms, idealization is often misleading.

For instance, as a mathematical operation, the baker’s transformation is reversible; but as a physical process, there are minute deviations from the idealization, so that mixing is actually irreversible. Similarly, reversing a film of billiard balls is not the same as reversing their real motion. To reverse a film treats actual interactions as though perfectly defined at each frame. The change of frame amounts to a logical operation, and so is reversible. Nature is not a sequence of such operations. The universe is not a projector running a film, nor a computer running a program. Cf. Ilya Prigogine and Isabelle Stengers Order Out of Chaos: Man’s new dialogue with nature Bantam 1984, p269-70: “The baker’s transformation transforms each point into a well-defined new point. Although the series of points obtained this way is ‘deterministic’, the system displays in addition irreducibly statistical aspects… Although the evolution of a point is reversible and deterministic, the description of a region, however small, is basically statistical.”


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Like a board game, a deductive system (or formal axiomatic system) has playing pieces and rules defining possible moves…

Formally, a deductive (axiomatic) system consists of symbols, formation rules, axioms, and transformation rules—together with the theorems they may generate. Formation rules are the basic conventions governing how the symbols may be strung together. Axioms are basic assumptions—the input of the system. Transformation rules specify how expressions may be handled in order to derive conclusions (theorems). Theorems are the conclusions or output of the system. The axioms are the equivalent of algorithmically incompressible sequences. The rules of the system serve to expanded sequences from those axioms—which is the opposite of compression.

My intention here is to introduce a general notion rather than a precise mathematical definition of deductive or formal system. The formal elements of a deductive system correspond to the ingredients of a game, such as a parlor board game like Monopoly, which has playing pieces, rules defining possible moves, a starting position and a goal, and a playing space or field-of-action that is conceptual in essence but may be embodied physically, as it is in the Monopoly or chess board. Similarly, such formal elements also correspond to a machine or mechanical system, as well as to a program for running it—which is, in effect, a text. All these related notions are primarily conceptual, but may have physical embodiments. Deductive system, axiomatic system, game, machine, program, algorithm, blueprint, script, and text are more or less equivalent expressions of the same general idea.


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Then its premises may appear to coincide with truths about the world…

A machine, or other technological invention, is an embodied deductive system. Yet, its physical existence must be distinguished from its conceptual existence. As a real object, it is both made and found—a product of human definition that is composed of real materials. These aspects of its being do not perfectly coincide; the physical machine can wear out or malfunction, whereas the information defining its blueprint is timeless.


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On the other hand, to formalize mechanics, for example, meant first regarding its fundamental concepts as fictions…

Ernst Cassirer The Philosophy of Symbolic Forms Vol 1: Language Yale UP 1955/1980, p76: “In this sense, Hertz came to look upon the fundamental concepts of mechanics, particularly the concepts of mass and force, as ‘fictions’ which, since they are created by the logic of natural science, are subordinate to the universal requirements of this logic…” In other words, Hertz articulated a 19th century shift toward formalism, dominated by requirements of logical consistency, which played a key historical role in the evolution of the modern place of mathematical models.

The “best system” approach is to view natural laws as “consequences of those propositions which we should take as axioms if we knew everything and organized it as simply as possible in a deductive system.” [Frank Ramsey Foundations of Mathematics Humanities Press 1978, p138] The best systems approach consciously substitutes deductive for real systems. The fact that it is “best” should alert us to the fact that the substitution can never be perfect: “What we value in a deductive system is a properly balanced combination of simplicity and strength…” [David Lewis Counterfactuals Harvard UP 1973, p73] Lewis admits, of course, that “the virtues of simplicity and strength tend to conflict,” so that a proper balance is a matter of preference. In other words, deductionism is an ideal that can serve us if we don’t take it too literally. While no deductive system can exhaust reality, it may give the misleading impression of being able to do so simply because it is self-contained and complete within itself as an axiomatic system.


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“…the rationality is ours, to which nature has no obligation to conform.

Nature appears reasonable to us, within limited domains of experience, because we have evolved adaptively to it in those domains, not because it was designed to fit human reason, nor because there is some pre-established harmony between reason and reality other than evolutionary adaptation. While deductionism enshrines the expectation that nature is rational, outside these domains there is no guarantee that reality is commensurate with rational schemes at all.


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…when one thinks of nature as literally consisting of idealizations of various sorts.

For example: the engineering student’s apocryphal cow is a roundish object with negligible protuberances, which may be mathematically approximated by a perfect sphere. In the Platonic tradition, it is the sphere that is real and true; the physical object is but a crude imitation! This inverts the natural order; the human mind idealizes experience with real objects, redefining them as elements of a constructed system, while disowning responsibility for doing so.

Cf. Norbert Elias “Knowledge and Power”, in Society and Knowledge: contemporary perspectives in the sociology of knowledge and science 2nd Edition, ed by Nico Stehr and Volker Meja, p221: “The rational deductionist is a secularized relative of the man who did not need to look through Galileo’s telescope… He does not allow for a world existing independently of the knower’s almighty intellect and his axiomatic belief.” And p216-17: “There are in our time strong tendencies toward the production of a kind of academic knowledge which cannot be tested by means of experiments, case studies, statistical measurements, or in any other way… Perhaps one could speak of a new deductionism or of neoscholasticism.”


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…acknowledges that scientific models and laws of nature are matters of convention.

Perhaps with false modesty, some postmodern deconstructionists have disclaimed any pretension to objectivity, deliberately characterizing their philosophical and cultural writings as fiction. This is in keeping with the self-consciousness of modern thought—which distinguishes convention from truth. Most scientists, on the other hand, have not given up a pretension to objective truth, and would certainly never characterize their work as fiction. At most, they might underline its provisional or pragmatic value. A theory might prove to be factually wrong, but is certainly never intended merely to entertain.


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…or else conversion factors between different accounting systems.

Hence, Weinberg Dreams of a Final Theory, footnote p139: “Plank’s constant simply provides the conversion factor between… older systems of units and the natural quantum-mechanical unit of energy… It can be shown that energy defined in this way has all the properties that we normally associate with energy, including conservation; indeed, the invariance of the laws of nature under the symmetry transformation of resetting our watches is why there is such a thing as energy.” Similarly, c is often considered a conversion factor between time and space.

There are some old arguments against deductionism, or against the commensurability of reality with pre-conceived schemes. Oresme had argued that any two celestial motions were probably incommensurable—for example, a siderial day and a solar year. (This was an argument against astrology, by way of showing that the cosmos is not a predictable, repetitive system—a deductive system such as astrology was.) [James Franklin The Science of Conjecture: evidence and probability before Pascal John Hopkins U Press 2001, p143] A similar argument might be made against the likelihood that fundamental constants can be reduced to one another, or to some common theoretical basis.


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…into the deductive paradigm that has become the scientific ideal.

Euclid’s influence is literally seen in Newton’s geometric solutions in the Principia, of problems that ironically would be solved today with the calculus he invented. Groundwork for this transformation is apparent in Plato’s theory of the elements, identified with the regular geometric solids, of which there are logically only five. The corpuscles of each element are not just shaped like their geometrical solid; rather they are that form and nothing more. This belief can be compared with modern assertions that physical reality, at the bottom level, is nothing other than mathematics, geometry, or information. On a deeper level, the deductive paradigm is grounded in the basic fiat of definition, which transcribes common meanings of words into precise definitions.


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“…Kepler tried to interpret the planetary orbits in terms of the five regular solids, and Galileo insisted on exactly circular orbits.

Today, Kepler’s three laws are fundamental, whereas his geometric planetary scheme is forgotten as an unscientific fantasy. Nevertheless, Kepler’s scheme worked within five percent accuracy to account for the planets known at the time. Its cleverness gained him employment with Tycho Brahe. Kepler also thought that the incommensurate number of days in a year was due to a perturbing influence of the sun, and should properly be a tidy 360 days! Similarly, Galileo’s laws of motion are now basic physics, while his Aristotelian view of planetary orbits is ignored as metaphysical as well as incorrect.


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…we have hardly the need to assimilate physical reality to idealized mathematical schemes.

For example, many theorists are persuaded by symmetry arguments—the modern equivalent of Galileo’s conviction regarding circular orbits. Principles of symmetry or invariance reflect a search for factors that do not depend on the state of the observer—that is, a search for objective rules of nature in the most general sense, independent of local “accidental” realities that can impose differing conditions on observers. It is thus a search for generality more than reality per se. For example, far from any gravitating mass that might bias the state of the observer, theoretically there is no preferred direction in space, which therefore has symmetry with respect to direction. But this is a special circumstance, certainly not normal for earthbound observers, for whom “down” is a preferred direction. Outer space is not empty, but full of gravitational influences, removal from which can only be approximate and ideal.


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…imperfect reflections of a deeper reality expressed by equations.

Weinberg’s Platonism gives symmetry principles causal power; particles and forces exist because symmetry breaking makes them necessary: “It is the local symmetry between different frames of reference in space and time that makes gravitation necessary, and in much the same way it is a second local symmetry between electrons and neutrons (and hence between up quarks and down quarks and so on) that makes the existence of the photon, W, and Z fields necessary.” [Weinberg Dreams of a Final Theory, p146]

At the outset of the twentieth century, following a different bias, symmetry arguments were uncommon, considered to involve aesthetic judgments that were not particularly scientific. [Holton TO, p282] Cf. also Weinberg, p159: “It seems that in the 1930s it was simply not good form to write papers based on symmetry principles. What was good form was to write papers about nuclear forces. If the forces turned out to have a certain symmetry, so much the better… But the symmetry principle itself was not regarded… as a feature that would legitimize a theory… Symmetry principles were regarded as mathematical tricks; the real business of physicists was to work out the dynamical details of the forces we observe.”


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…Einstein, who was one of the first to introduce modern symmetry arguments.

Just as Einstein reasoned from the apparent asymmetry of forces involved in Maxwell’s theory (induction) to arrive at SR, so he reasoned about the asymmetry of induced effects due to motion in gravitational fields to arrive at GR. Weinberg concludes that the existence of a field or force can be deduced from symmetry principles. Yet, Einstein began with the force (gravitation), together with symmetry (equivalence) of reference frames.


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…indeterminacy…is just what makes nature real rather than ideal.

The debate between Einstein’s realism and Bohr’s positivism reflects the general philosophical question of whether physics describes reality or human knowledge. A description can be complete if it refers to the state of knowledge, yet incomplete if it purports to describe reality. In that sense, Bohr and Einstein were talking at cross-purposes. In terms of a black box, a macroscopic object is either in the box or not, at a given time; a probabilistic description of the box’s content is incomplete from a realist perspective. The state of the box (before opening) is understood differently in the two perspectives. Einstein expressed common sense in pointing out that there can be no intermediate state between an exploded and an unexploded bomb.

While causality is traditionally considered a metaphysical (rather than logical) relationship, the states of a deterministic system are actually logically entailed rather than “caused.” Such systems are logically closed. The appearance of causal power, or determinism, rests on the power of reason, not on the power of observation. [See: Carl Hoefer “Causality and Determinism: tension or outright conflict?” Oct 24 2004 web archive]



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If nature is real, however, no theory can be complete in this sense.

It is interesting to speculate that Einstein’s campaign against the “incompleteness” of the existing quantum theory might have been influenced by his friendship with Kurt Gödel. [See Rebecca Goldstein Incompleteness: the proof and paradox of Kurt Gödel Atlas Books (Norton) 2005] The gist of Gödel’s theorems is that not all of mathematical truth can be formalized. It seems Einstein stopped short of concluding that, in a parallel way, physical reality must transcend any given theory.

Einstein first expressed his deductionism by raising the constancy of the velocity of light to the status of an axiom in Special Relativity. If the constancy of c was not to remain a mystery to be explained, it had to be treated as something given, i.e. a postulate. Similarly, Newton’s absolute time was less a metaphysical speculation than a postulate necessary to proceed with his science. The Uncertainty Principle may also be regarded as a postulate. [cf. Smolin The Life of the Cosmos, p245]


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Determinism… reflects the…hope that nature can be reduced to a deductive system.

Determinism is “the claim that the full specification of the state of the world at one time is sufficient, along with the laws of nature, to fix the full state of the world at any other time.” [Cambridge Dictionary of Philosophy, p613] This definition does not necessarily imply predictability, since even if determinism were true one might not be in a position to know this full specification. In truth, one is never in this position, because the world is not a finitely specifiable system. One cannot, except arbitrarily, assume a fixed finite number of factors involved, or that all the “correct” ones can be definitively identified.


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…(as in hidden-variable theories)…

‘Hidden variables’ might represent properties of the measuring device or system, as well as properties of the system measured. The point of hidden variable theories is to restore causality (predictability regarding individual objects), but the determinism involved might concern any or all parts of the interacting system. The “hidden” information might reside in some distant part of the universe, well outside an isolated system. [Cf. Smolin Time Reborn, p155-6]


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…an exhaustive account is not.

EPR continues: “If, without in any way disturbing a system, we can predict with certainty (i.e., with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity.”[A. Einstein, B. Podolsky, and N. Rosen “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” Physical Review Vol 47 May 1935 (italics theirs)] However, a ‘probability equal to unity’ characterizes deductive systems, not natural ones, which always demonstrate a probability less than one. In fairness, Einstein apparently later regretted the inclusion of a definition of physical reality in EPR, since he considered that the main thrust of the paper did not hinge on it.


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Some commentators…had recognized the limits of the deductive program.

Sir John Herschel Preliminary Discourse on the Study of Natural Philosophy (1830), quoted in Enrico Bellone A World on Paper: studies on the second scientific revolution MIT Press 1982, p71: “It was the radical error of the Greek philosophy to imagine that the same method which proved so eminently successful in mathematical, would be equally so in physical enquiries, and that, by setting out from a few simple and almost self-evident notions, or axioms, everything could be reasoned out.”


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Eddington expressed similar views.

Sir Arthur Eddington, quoted in Peter Johnson The Constants of Nature: a realist account Ashgate 1997, p3: “My conclusion is that not only laws of nature but the constants of nature can be deduced from epistemological considerations, so that we can have a priori knowledge of them.” However, not all of Einstein’s contemporaries shared such optimism for the deductive approach, nor believed that constants could be derived from an ultimate theory. Planck, for example, viewed science as fundamentally inductive. [John D. Barrow Theories of Everything: the quest for ultimate explanation Fawcett/Balantine 1991, p122] Yet, Johnson comments: “While at first sight Eddington’s views may seem curious, in fact modern unified field theorists are working along similar lines…”

Cf. also David Layzer Cosmogenesis: the growth of order in the universe Oxford UP, p6: “The universe of modern physics is an enormously expanded and elaborated version of the perfectly ordered but static and lifeless world we encounter in Euclid’s Elements, of which it is indeed a descendant.” Similarly, Stephen Toulmin The Return to Cosmology: postmodern science and the theology of nature U. of Calif. Press 1982, p238: “The philosophers and scientists of post-Renaissance Europe, it seemed, were at last able to live the life of pure reflection (or theoria) which Aristotle had acclaimed as the highest good.” Cf. Aristotle himself: “…the proper object of unqualified scientific knowledge is something which cannot be other than it is.” [Posterior Analytics I, 2 (71 b 15)]


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It permeated the rationalism of Christian Europe…which is alive and well today.

Weinberg, for example, excuses Kepler’s Platonic theory of the planets as a justifiable approach akin to modern symmetry arguments: “The kind of speculative reasoning [Kepler] applied to the solar system is very similar to the sort of theorizing that elementary particle physicists do today; we do not associate anything with the Platonic solids, but we do believe for instance in a correspondence between different possible kinds of force and different members of the Cartan catalogue of all possible symmetries. Where Kepler went wrong was not in using this sort of guesswork, but in supposing… the planets… incorporated into the laws of nature at any fundamental level.” [Weinberg Dreams of a Final Theory, p164]