Abstract
A permissivist framework is developed to include images in the reconstruction of the evidential base and of the theoretical content. The paper uses Newton’s optical theory as a case study to discuss mathematical idealizations and depictions of experiments (‘equidiametric’ and ‘harmonic’ spectral images, ray-paths in drawings of prism-experiments), together with textual correlates of diagrams. Instead of assuming some specific type of theoretical content, focus is on novel traits that are delineable when studying the carriers of a theory. The framework is developed to trace elliptic and ambiguous message design (polysemy), and utilizes variegated acceptance (heterogeneous uptake and polyphony) as an asset. Newton’s resources (novel diagrammatic carriers, mathematical idealizations, and neologisms) allowed for various framing modes and reconstructions, entailing various judgements concerning the theoretical content, the evidence base, and Newton’s use (and/or abuse) of mathematics. Elliptic presentation of the theory’s proof-structure and ambiguities (also pertaining to the function and material setup of the crucial experiment) influenced uptake, contributed to the process of opinion-polarization, and the acceptance/rejection of the theory. The study suggests that the analysed carriers of theoretical content (including visuals) have an argumentative function, and one of their uses is to adjust the burden of proof.
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Notes
The details of why the Opticks could not become a fully blown physico-mechanical theory and stayed in the realm of mixed mathematics is discussed in the informed analysis of Ducheyne (2012, Ch. 4, esp. pp. 220–222).
As the trait corresponds to some act of making distinctions between carriers, the account can in principle give both an extensional and an intensional account of properties and relations, both “unity in multiplicity” and “multiplicity in unity” can be addressed. See e.g. the non-numerical arithmetic developed by Brown (1969).
The reconstruction develops an artefact-human knowledge-mobilization process, building on an externalized reading of positions and argumentative utterances. As opposed to internalized ‘mental state’ reconstruction of a position, this approach proceeds via studying commitments. The perspective is discussed in more detail by the standard pragmadialectic theory, a widely used discourse analytic approach to argumentation utilizing a speech-act theory (Eemeren and Grootendorst 2004). I have earlier used the model to study argumentative aspects of the debates surrounding Newton’s NT (Zemplén 2008), and potential use for constructivist approaches in science studies has been discussed in Kutrovátz (2008).
The analysis follows Frederik Stjernfelt’s lucid account of Peirce’s notions of diagrammatical reasoning in the sciences and focuses on simple and double signs, not triple signs (Arguments), although some propositional content seems easily determinable (e.g. light here passes through these prisms), in accord with Peirce’s picture theory of propositions. For Peirce “... a diagram with a label—say, a geometrical figure with legend—may express a Dicent symbol—a full-fledged proposition, and the manipulation of that diagram, in turn, may express an Argument” where “[t]he triple structure of the Argument refers to the idea that it not only is a sign for its object by means of the Rheme and the Dicisign presented in the premiss, but also involves the same object a third time, now appearing as that to which the conclusion pertains.” (Stjernfelt 2015, p. 1045).
A recurring feature of the text is the use of conceptual chiasmus, to give an example (Fig. I, III, 13.) the triangle ABC is “a Triangular Imaginary Plane whereby the Prism is feigned to be cut transversely through the middle of the Light. Or if you please, let ABC represent the Prism it self, looking directly towards the Spectator’s Eye with its nearer end...” (O4: 31).
In a different, less geometrized idealization by Mariotte, it was drop-like and cusped at one end (Lohne 1968, p. 171). During the controversy the length/breadth ratio, the shape, and the conditions of observability became debated.
Some of the core assumptions characterizing contemporary debates make it rather difficult to study normative constraints on vocabulary extension, as the restriction in Willard-Kyle’s recent contribution illustrates a general trend: “... cognitive interests cannot be a part of the evidence since they do nothing to affect the confidence one is rationally permitted to invest in a hypothesis” (Willard-Kyle 2016).
In the Optical Lectures the parts of the spectral image occupied by the colors are proportional to “a string divided so it would cause the individual degrees of the octave to sound” (Newton 1984, p. 545). The historical development of the ‘harmonic’ spectrum is discussed in Topper (1990). This carrier is connected to a more extensive heuristic programme utilizing musical analogies in Newton’s optical and cosmological work (Guicciardini 2013).
As only early reproductions depicted five circles, later editions included six circles, this connection was severed. The addition to the equidiametric circle-spectrum correlated with ‘the advertized precision’, the normative constraint on the image length/breadth ratio, specifying the ‘experimental’ desiderata for admissible reconstruction of the experiment.
The seven colors of the rainbow is even today a widely used trope, and as long as Newton’s theory was seen as a theory of primary colors (the early nineteenth century, before the rise of physiological optics) this reading was not uncommon. For example, Coleridge wrote to Tieck (4 July 1817): “To me, I confess, Newton’s positions, first of a Ray of light, as a physical synodical Individuum, secondly that 7 specific individua are co-existent (by what copula?) in this complex yet divisible Ray ... have always appeared ... monstruous FICTIONS!—and in this I became perfectly indifferent, as to the forms of their geometrical Picturability.” (Burwick 1986: p. 54). Today a widespread reading assumes “the existence of innumerable colours in white light ... How could there not be an infinity of coloured rays forming the spectrum when each part of it is illuminated by a ray following a uniquely refracted path?” (Hall 1993: p. 64).
Geometrically optically some diagrams can be considered agrammatical, like Fig. I, IV, 18. On Fig. II, II, 8. (not reproduced here) the virtual position of the Sun (when the spectral image is watched through a prism) is defined by the angular size, the dicisign S and circle appear in the mapping, but the source is only represented as a single incoming ray, with no angular size.
Newton in some experiments describes the penumbra, measuring it, but not discussing the issue theoretically.
Newton’s early notebook drawing of a subjective prismatic setup shows (MS Add. 3996, 122r) the prism aligned to cause minimum deviation of the ray, which falls perpendicularly on the eye. In later note-books ‘single line’ drawings depicted prismatic experiments with light-sources, and at times Newton multiplied these to depict larger quantities of rays with the use of pencils or bundles (various examples in MS Add. 3975). The theoretical foundations were provided by Barrow’s work on pencils of rays.
Theoretical developments before Newton utilized the slight discrepancy of the obliqueness of the rays due to angular size of the source to account for different colors, e.g. Marci’s Thaumantias, (1648), see (Garber 2005). Angular sizes of the Sun were generally exaggerated in camera obscura drawings.
Huygens further developed the spherical model of light-propagation, conforming to traditional geometrical optics.
The ‘inverted image’ convention was used both by early opponents (Pardies) and late critics. Rizzetti (2010 (1727) Plate II. Fig. 7), for example, combines the mapping of the inverted Sun (and a penumbra), the mapping of the emergence of the two colored regions after the refraction (and the disappearance of the white region) with spectral images in his geometrical optical disambiguation. After the Newtonian revolution the parallel-ray diagrammatic convention was utilized ever more often, and also appeared in attacks on Newton’s theory. Castel (1740: p. 414), for example uses a parallel incident incident beam to explicate the observation that the colors arise from the two symmetrical edges of the refracted beam.
It is not certain that this was the same prism as the one Newton originally used when describing the experimentum crucis (Mills 1981).
Morphing is discussed in Domenico Bertoloni Meli’s account of the transformation of mechanics as a series of operations leading from object to another, a process of creative transformation. Morphings and techniques of ‘dematerialization’ (where some relations valid for a constrained case is supposed to remain valid in an unconstrained one) characterized many Early Modern discovery-procedures (2010, p. 581), and I suggest they also played a role in extending the boundaries of mixed mathematics to include more and more of natural philosophy via a merger with physics.
As one reviewer of the manuscript noted, the “Newtonian attempt to separate certain theory from physical speculation led Newton into tensions when he passed from words to images”. A century ago Neurath studied some of the tensions (1914/5, 1915) and used the blurred edges of the refracted image to develop a notion of vagueness present in the theory. These edge-phenomena are depicted and described in different Parts of O., testifying to the complex relationship between text and image. Later Neurath developed the notion of Ballungen (Cartwright et al. 1996).
For a minimal theory of frames see an elegant approach delineating ways of handling heterogeneity and overcoming frame differences (Wohlrapp 2014). Frame criticism raises an objection presupposing an external perspective, frame hierarchization subsumes a divergent frame to another, frame harmonization makes two frames compatible, and frame synthesis provides a superseding higher frame.
“And therefore, that we may not be wanting to them, or our selves, in a matter of so much importance, as the full Confirmation of it by nervous and apodictical reasons; especially when the Determination of that eminent and long-lived Controversie concerning QUIDDITY or Entity of Light, Whether it be an Accident, or Substance, a meer Quality, or a perfect Body? seems the most proper and desiderated subject of our present speculations, as the whole Theory of all other sensible Qualities (as Vulgar Philosophy calls them) is dependent on that one cardinal pin, since Light is the nearest allied to spiritual natures of all others, so the most likely to be incorporeal: we must devote this short Section to the perspicuous Eviction of the C O R P O R I E T Y of Light” In §1 of Sect II. Book III (Charleton 1654 (1966), p. 204).
A typical example of the ‘geometrization’ of natural phenomena is Huygens’ solution to the problem of anomalous refraction, involving less simple mathematics (spheroidal light propagation) than expected (spherical propagation), but still ‘contained’ the phenomena within the realms of geometry (an oblique ellipse, as the ray and wave intersect obliquely), see (Dijksterhuis 2004, p. 168). Such geometrization continued to be productive for the coming centuries in the discipline.
,,Nor is reflection necessary, for there is no reflection here, nor finally do we need many refractions, for there is only one refraction here. But I reasoned that there must be at least one refraction—and, in fact, one whose effect was not destroyed by another—for experience shows that, if the surfaces MN and NP are parallel, the rays, being straightened as much in one as they were able to be bent in the other, would not produce these colours. I did not doubt that light was also needed, for without it we see nothing. And Moreover, I observed that shadow, or some limitation on this light, was necessary;” translated by Stephen Gaukroger (AT VI 330-331).
Hooke’s experimentum crucis showed that conditions simpler than Descartes’s suffice for the emergence of colors from light (Hooke 1665).
Burden of proof was further decreased by Oldenburg, and editorial omissions from the printed version of the NT influenced both the topical potential available to Newton during the controversy, and the theoretical content that was published as his position (Zemplén 2016).
“Ontological commitment to a natural kind amounts to the belief that some entities are entities ‘of the same kind”’ (Contessa 2006, p. 461). The ‘harmonic-spectrum’ diagram suggests that there are seven kinds of entities (sorts of Rays), and the ‘equidiametric circle’ diagram implies ontological commitment to an innumerable number of theoretical kinds, entities that have the properties (sorts of refrangibilities) attributed to them by the theory. If rays of light differ in refrangibility and the refrangibility of light rays is strictly connected to their color, then there are as many types of light-rays as simple colors (seven; harmonic spectrum), and there are as many types of simple colors as refrangibilities (indefinite; equidiametric spectrum).
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The work was supported by the “Integrative Argumentation Studies” NKFI-OTKA K 109456 Grant.
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Zemplén, G.Á. Diagrammatic carriers and the acceptance of Newton’s optical theory. Synthese 196, 3577–3593 (2019). https://doi.org/10.1007/s11229-017-1356-5
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DOI: https://doi.org/10.1007/s11229-017-1356-5