The Hard Problem of Consciousness: A Mathematical Approach T.R. LEFFLER [email protected] We can potentially make progress on the hard problem of consciousness (Chalmers, 1995) – the seemingly intractable problem of explaining how qualia arise from certain physical systems – via the mathematically tractable problem of explaining how objectively unmeasurable aspects (identified with qualia) could arise from certain physical systems that are otherwise objectively measurable. In short, it is proposed that qualia may correspond to non-divergent singularities in the mathematical descriptions of certain aspects of certain complex physical systems. This proposal may have been foreshadowed by Srinivasa Ramanujan. It could have significant implications for the prospects of the experimental verification of consciousness in physical systems and for the prospects of the generation of artificial consciousness (AC). 1 The hard problem of consciousness and the explanatory gap At the core of the mind-body problem is the hard problem of consciousness (Chalmers, 1995): The really hard problem of consciousness is the problem of experience ... Why is it that when our cognitive systems engage in visual and auditory information- processing, we have visual or auditory experience: the quality of deep blue, the sensation of middle C? ... It is widely agreed that experience arises from a physical basis, but we have no good explanation of why and how it so arises. Our lack of an explanation of how conscious experiences (i.e. qualia) arise from certain physical systems represents an explanatory gap (Levine, 1983) in our understanding of the world. The explanatory gap is central to the hard problem (Majeed, 2015). To bridge the explanatory gap – and thereby make progress on the hard problem – an explanation of how qualia arise from certain physical systems is needed. For example, to bridge the explanatory gap, we need an explanation of how the conscious experience of pain arises from the physical system consisting of “the firing of C fibers” (Levine, 1983). Leading theories of consciousness – including physicalism (e.g. Churchland, 1983; Dennett, 1991), property dualism (e.g. Chalmers, 1996), neutral monism (e.g. Russell, 1927; Strawson, 1994), and integrated information theory (e.g. Tononi, 2004; Koch, 2012) – each attempt to bridge (or circumvent) the explanatory gap and thereby make progress on the hard problem. However, at present, there is no satisfactory explanation 2 of how qualia arise from certain physical systems, and we are thus left with the hard problem and the explanatory gap. This paper will propose an explanation of how qualia arise from certain physical systems. In the spirit of Galileo’s intuition that the book of nature is written in the language of mathematics, and in light of the “unreasonable effectiveness” (Wigner, 1960) of mathematics in explaining other aspects of the world, the explanation to be proposed is mathematical. 2 The foundation for a mathematical explanation of how qualia arise from certain physical systems Although it does not do so yet, there is reason to suspect that physics will eventually provide an accurate description of the world (Weinberg, 1992). However, as the existence of the explanatory gap makes clear, any such description of the world will not be a complete description of the world unless it explains how qualia arise from certain physical systems. At present, physics is silent on this point. A fundamental reason for this silence – and the foundation for the mathematical explanation to be proposed – is a fundamental dichotomy in the nature of ordinary physical systems, such as jet engines, on the one hand, and the special physical systems that give rise to qualia, such as brains, on the other hand. One the one hand, a physical system such as a jet engine is a system that is objectively measurable in full. On the other hand, a physical system such as a brain is a system that, at least in certain states, is objectively measurable in part and objectively unmeasurable in part. More precisely, a jet engine is a physical system for which all aspects of the system can be fully described by a set of fields with objectively measurable numerical magnitudes at all coordinates and by a set of properties (e.g. mass, temperature, etc.) with likewise objectively measurable numerical magnitudes. However, the brain of, for example, a person looking at a deep blue light, is a physical system for which most aspects of the system can be described by such a set of fields and properties with objectively measurable numerical magnitudes, but for which certain aspects of the system (namely, the conscious experience of the quality of deep blue) cannot be so- described. The deep blue qualia that arise from such physical system (as opposed to the neural correlates of such qualia) are rather aspects of such system that are objectively unmeasurable, in principle.1 A description of such aspects of such physical system is thus nowhere to be found in the sort of description that would otherwise fully describe a physical system such a jet engine (namely, a set of objectively measurable numerical 1 Recognition of this fundamental dichotomy in the objective measurability of the physical aspects of physical systems, on the one hand, and qualia, on the other hand dates to at least Leibniz’s (1714) mill argument: “[S]upposing that there were a mechanism so constructed as to think, feel and have perception, we might enter it as into a mill. And this granted, we should only find on visiting it, pieces which push one against another, but never anything by which to explain a perception.” 3 magnitudes of fields and properties). Any description of a physical system such as the brain in question that consists only of a set of objectively measurable numerical magnitudes of fields and properties is thereby fundamentally incomplete. Since qualia are irreducible to description in such physical terms, they have unsurprisingly been “left out of the physicalist story” (Jackson, 1982) and so the explanatory gap and the hard problem persist. 3 A proposed mathematical explanation of how qualia arise from certain physical systems Notwithstanding the omission of qualia from the physical picture of the world, we know that the world in fact includes certain physical systems (e.g. brains) from which, at least in certain states, qualia arise. Thus, we can ask the question: What would a complete mathematical description of a physical system from which qualia arise look like? In order to account for both the objectively measurable aspects of such a system and the objectively unmeasurable aspects of such a system, we might expect that such a complete mathematical description would include both (a) numerical values representing the magnitudes of the objectively measurable aspects of such system (i.e. its ordinary physical aspects) and (b) non-numerical values representing the objectively unmeasurable aspects of such system (i.e. its qualia). In fact, there are equations that generate just such mathematical descriptions. These are equations featuring mathematical singularities – points at which the value returned by such an equation is numerically undefined. To illustrate, suppose that the firing of a particular type of neuron (N) is determined to be the neural correlate of the experience of the quality of deep blue. A firing of N is thus a physical system from which arises, seemingly inexplicably, an objectively unmeasurable aspect in the form of deep blue qualia. Suppose further that this physical system has a particular aspect (Ω) that is mathematically described by the following equation over the course of a given firing of N: Ω 1 where v is a variable that flips sign from negative, through zero, to positive around the moment of the firing’s peak intensity. This equation features a non-divergent singularity at v = 0, which corresponds to the moment of the firing’s peak intensity. At that moment, Ω’s magnitude is numerically undefined (not infinite). 4 The dynamics of Ω over the course of a given firing of N would thus correspond to the following graph: Fig. 12 For a given firing of N, at all times before the moment of the firing’s peak intensity, Ω is an aspect of the physical system that has an objectively measurable numerical magnitude (e.g. when v = -0.5, the magnitude of Ω is 0.004) and can thus be identified with physical aspect the system. However, at the moment of the firing’s peak intensity at v = 0, an anomaly occurs. At this moment, Ω is an aspect of the physical system that has no objectively measurable numerical magnitude (due to division by zero), but rather has a magnitude that is non-numerical, and thus it cannot be identified with a physical aspect of the system. At this moment (and for so long as v = 0), Ω is rather an aspect of the physical system that has a non-numerical magnitude and is thus objectively unmeasurable. Finally, at all times after the moment of the firing’s peak intensity, Ω is once again an aspect of the physical system that has an objectively measurable numerical magnitude (e.g. when v = 0.5, the magnitude of Ω is 0.004) and can thus once again be identified with a physical aspect of the system.3 2 Fig. 1, Fig. 2 and Fig. 3.b were created with Desmos. 3 These dynamics can likewise be illustrated where Ω is the magnitude of a field that is part of the physical system in question, where, for example: Ω 1 Fig. 2 At all coordinates (v, w) other than (0, 0), this field has an objectively measurable numerical magnitude and can thus be identified with a physical aspect the system. However, at (0, 0), such field’s magnitude is non-numerical, and thus cannot be identified with a physical aspect of such system.