An Extra-Mathematical Program Explanation of Color Experience

ABSTRACT

In the debate over whether mathematical facts, properties, or entities explain physical events (in what philosophers call ‘extra-mathematical’ explanations), Aidan Lyon’s (2012) affirmative answer stands out for its employment of the program explanation (PE) methodology of Frank Jackson and Philip Pettit (1990). Juha Saatsi (2012; 2016) objects, however, that Lyon’s examples from the indispensabilist literature are (i) unsuitable for PE, (ii) nominalizable into non-mathematical terms, and (iii) mysterious about the explanatory relation alleged to obtain between the PEs’ mathematical explanantia and physical explananda. In this paper, I propose a counterexample to Saatsi’s objections. My counterexample is Frank Jackson’s (1998a) program explanation for color experience, which I argue needs recasting as an extra-mathematical PE due to its implicit reliance on reflectance, a property that suffers conceptual regress unless redefined with Fourier harmonics. Pace Saatsi, I argue that this recast example is an authoritative PE, non-nominalizable, and minimally esoteric. Important for the indispensability debate at large, moreover, is that my counterexample reifies Fourier harmonics without the Enhanced Indispensability Argument (an argument to which Lyon applies PE as a premise). Indispensabilists have long overlooked the conditionalization of a limited mathematical realism on property realism, and my counterexample to Saatsi exploits this conditionalization.

KEYWORDS:

• Extra-mathematical explanation
• Enhanced Indispensability Argument
• program explanation
• mathematical nominalism
• mathematical realism
• color objectivism
• reflectance physicalism
• Fourier analysis

Acknowledgments

I thank Michael Stoeltzner for feedback on previous drafts, and two anonymous reviewers for challenges that improved the paper. Significant portions of this work were completed during my 2020–2021 Russell J. and Dorothy S. Bilinski Dissertation Fellowship at the University of South Carolina.

Notes

1 I treat these three terms interchangeably, context permitting.

2 Some may object that PE can be construed as an epistemic explanation that advances human understanding rather than picking out worldly ontology (thanks to Martin Zach for raising this point during the Q&A of my presentation at the Ernst Mach Workshop IX: Non-causal explanation in physics, September 2020), but to engage that controversy in this paper would be a digression.

3 More recently, Shea (2018, §8.3) endorses program explanations of human behavior.

4 For biological and ecological details, see Lyon (2012, 561, 567, n. 10), Baker (2005, §2), or Berenstain (2017, §3.4).

5 ‘Extra-mathematical explanation is the … mathematical explanation of physical facts’ (Baker and Colyvan 2011, 326).

6 The following text of B1 and B2 is Lyon’s (2012, 572) paraphrase of Baker (2009), in my formatting.

7 I identified this conceptual coherence problem for reflectance, and its Fourier solution, in Danne (2020).

8 The probability-raising function of programming properties, which Jackson and Pettit (1990) sometimes describe with the verb ‘ensures’ (114), has been a controversial tenet of program explanation. Thalos (1998, 286-289ff) criticizes the notion that a probability-raising factor in an explanation could be non-causal, and Macdonald and Macdonald (2007, §3) reach a similar conclusion. I review Saatsi’s (2012; 2016) objections to the ‘ensuring’ relation in section 5, but cannot otherwise assess the merits of PE as a kind of explanation in the scope of this paper.

9 Jackson and Pettit (1990, 116) remark: ‘A program explanation of an event may provide information which the corresponding process explanation does not’ (emphasis mine). I take this ‘may’ qualifier to mean that program explanations do not necessarily provide the sought-after modally superior information.

10 Bliss and Fernández (2010) give an accessible and convincing heuristic for distinguishing impoverished from non-impoverished PEs, in part by diagnosing multi-ordered programming chains, but because those authors’ conclusions sometimes contradict Jackson and Pettit (1990, and elsewhere) about particular cases, I decline to elaborate Bliss and Fernández’s account, to avoid internal debates about PE proper. That some PEs are impoverished, all parties agree, and that point is what I need for my argument.

11 These conditions are spelled out by Hardin (1988: Chap. 2). One may note that by making objects the bearers of the disposition to look red, Jackson rules out various subjectivist ontologies of color, according to which color experience might be programmed or caused by observers, brains, or minds, etc.

12 Helpful on this topic is Byrne (2001), responding to McGinn (1996).

13 As useful as this quote is, it contains infelicitous wording inconsistent with the rest of Hilbert (1987). Namely, Hilbert appears in this quote to place incident flux in the numerator of the reflectance ratio; but no one does that; reflectance is always the ratio of reflected-to-incident flux per wavelength.

14 Notoriously, reflectance profiles and perceptual dispositions anticorrelate in many cases, and color objectivists must handle this problem in nuanced ways outside my purposes to describe (Byrne and Hilbert 2003).

15 See McFarland and Miller (1998; 2000) for objections to this assumption, which they claim goes undefended by Jackson.

16 On the distinction between categorical and dispositional properties, see Schrenk (2017, Chap. 2).

17 An anonymous reviewer asks why impoverished dispositions should be reified at all, and especially for the sake of ontic explanation. This question is a deep one about the fundamental motivations for program explanation, that I will be unable to answer authoritatively in this space. I defer to Bliss and Fernández’s (2010, §1) brief history of PE as a defense of the autonomy of special sciences like sociology, psychology, and economics.

18 Hilbert (1987, 2089-2092) and Byrne and Hilbert (2003, 53; cf. 20, n. 25) deny on mostly empirical grounds that color reduces ontologically to microstructure; I return to this point, although it does not affect my argument and I need not take sides in the dispute.

19 See McFarland and Miller (1998; 2000) and Wright (2003), for criticism.

20 I remind the reader that I am not defending color objectivism or providing any reason to believe it. The view is controversial, as Gert (2017, Chap. 3) recently argues on scientific and philosophical grounds.

21 And as Bob Mullen remarked to me at a reading group, scientific investigation outside the human-visible spectrum will have to rely on electromagnetic reflectance of some sort; for example to detect the quantum-physical or chemical behavior of microstructure (G).

22 Hilbert (1987, 1944 ff.) correlates color experiences to ‘triple[s] of integrated reflectances,’ whereby the SSR value corresponding to an objective color is the scaled sum of integrated reflectance values (measured from objects) at the three main wavelength bands of human retinal sensitivity. The classic resource on this procedure is Land (1977), who is intriguingly not a color objectivist.

23 An anonymous reviewer wonders if the program explanation for color experience is simply backward, since the ‘more fundamental’ categorical base (quoting my reviewer) seems to explain why a disposition obtains or manifests. Key to answering this objection, I think, is to point out that the singular event explained, such as the single momentous water molecule, or the single experience of seeing red, does not by itself reveal the state of a liquid as boiling, or the property of an object to look red in normal conditions to normal observers, respectively; an inference from multiple events may be needed to ascribe programming properties, or to draw modal inferences about what would have happened due to possible alternative events, but I do not know how Jackson and Pettit would answer the objection.

24 By ‘pulse,’ I mean a finite-duration sinusoid like Figure 1 (a) below, although my arguments apply to other-shaped pulses used in laboratories, such as Gaussian and hyperbolic-secant.

25 Formally, $E=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{x}^2\left(t\right)dt$for a continuous-time signal x(t) with total energy E (Haykin and Van Veen 1999, 20). The signals susceptible to average power ascription possess ‘infinite energy,’ therefore, because they tend to be periodic (Haykin and Van Veen 1999, 21), which means defined ‘for all t’ (Haykin and Van Veen 1999, 18), which in signal theory means for all of positive and negative temporal infinity; and integrating a squared periodic signal forever, by the equation just cited, yields infinite energy.

An anonymous reviewer helpfully asks, ‘Isn’t average power just the ratio of total power and total duration, and do finite signals not possess both quantities?’ I answer that the ‘total power’ of a signal is only well-defined per wavelength if its period is well-defined in the orthodox sense, which means defined for all of positive and negative temporal infinity. Otherwise, the average power of a continuous-time signal is $P_{avg}=\underset{T\to \mathrm{\infty }}{lim}{\displaystyle \frac{1}{T}\underset{-T/2}{\overset{T/2}{\int }}{x}^2\left(t\right)dt}$ (Haykin and Van Veen 1999, 20), the period T again being well-defined only throughout infinite time. Because finite-duration signals do not possess well-defined periods T, their period limit is taken to infinity by the equation just cited, and their average power is zero. Hence technologists who ascribe ‘average power’ to finite-duration signals are pretending that the period T of those signals is well-defined, when it is not.

26 Hirliman (2005, 31) denies that this inverse relationship reduces to a duality of quantum physics, and emphasizes instead that harmonic dispersion obtains squarely within classical electromagnetics. Hence my applying harmonic dispersion against Hilbert’s (1987) account of reflectance appears fair, since I am not foisting on his macroscopic account a remote quantum problem.

27 Visible light propagates in the hundred-terahertz (THz; 10¹² Hz) range, but harmonic dispersion occurs at all usable frequencies and pulse durations, including those of radio and radar.

28 ‘Frequency’ and ‘wavelength’ are interchangeable in physical optics. Frequency is speed of light divided by wavelength.

29 As the pulses of Figure 1 are centered at 300 Hz.

30 1 ps = 10⁻¹² seconds.

31 One may object that an asymptotic approach to zero never reaches zero, and because numerator and denominator regress at identical iterative rates (an arbitrary assumption), the pulse-SSR ratio retains its value of 1 like we assumed. The problem with this response is that a 5 W pulse incident to the mirror still reflects with ≈ 0 W at 650 nm, or never really propagated with anything close to 5 W in the first place; both absurd conclusions.

32 An anonymous reviewer suspects that the harmonic dispersion ‘becomes smaller and smaller’ through its regressive iterations, allowing the 80% bulk of the first iteration to reflect as expected (4 watts), but this construal of the VRR is mistaken. What is getting smaller and smaller is the power component of the central wavelength of interest (650 nm), and not just the sidebands, nor just the ‘noise’ or ‘error’ of the signal.

33 See additional objections to the VRR, with replies, in Danne (2020).

34 The Fourier transform also requires a per-wavelength phase shift and negative-frequency harmonics, which I omit for clarity.

35 Just as the portion of the pulse between 0.49 and 0.51 seconds in Figure 1 (a) is not cancelled-out by the dotted-trace harmonics in Figure 1 (c), but the regions of Figure 1 (a) outside 0.49 and 0.51 seconds are cancelled out by that superposition.

36 I am admittedly being vague about what kind of mathematical realism follows from the reality of harmonic-SSR (SSR_H). Are the harmonics real as Aristotelian immanent universals, transcendent Platonic universals, or as something else? I suspect that I need not answer that question in this paper (since Lyon 2012 does not in his paper), but I would allow that the Fourier harmonics are being instantiated as Berenstain (2017, §5.3) explains that term.

37 A number of objections to EMPEC must be bypassed in the interest of space. Some might wonder, for example, how the stimulus and manifestation of harmonic-SSR could really be infinitely long. I defer to Morganti (2013, 179) and Barrow (1998, Chap. 6) on the possibility of an infinitely-sized universe, through which infinite-duration light would have room to propagate. On whether a disposition with eternal manifestation would still be a ‘disposition,’ see Chakravartty (2013, 45) for the preliminaries of a positive answer.

Others might press me to explain why harmonic-SSR is not an unreal idealization of reflectance. Pincock (2014), for example, assumes an infinitely-deep ocean to calculate the phase speed of water waves, but he retracts that idealization at the end of his analysis as not ontologically referring; why cannot I do the same with the harmonics of harmonic-SSR? My simple answer is that Pincock faces no conceptual regress by withdrawing the infinitudes he posits, whereas I clearly do, in the form of the VRR.

Lastly, to those who object that I fixate on a Fourier reduction base when there are other mathematical methods for representing propagating light, I reply that the onus is on the objector to show which mathematics predicts harmonic dispersion but avoids the VRR; wavelet analysis, for example, fails to block the VRR, since wavelet bases are manifestly heterochromatic with spectra resembling the ‘envelopes’ of Figure 2. On the nominalizability of Fourier harmonics, see section 5.

38 Tollefsen (2013) summarizes one such pernicious saga.

39 An anonymous reviewer recommends that I add the references Morganti (2015; 2014), Bohn (2009), and Schaffer (2003) to my footnote 37 discussion on the plausibility of an infinitely spatial universe, but I think that these authors even better present reasons not to ‘fear’ the VRR, and to accept it as ontologically fundamental without dire need of ‘solution’, especially not solution by mathematical realism. Developing this tolerant and even warm response to the VRR would of course require a separate paper, but the thesis is intriguing.