By "answering the hard problem with a formula," what I mean is to give a formula F that takes as input a mathematical representation of a physical system, and produces as output a mathematical representation of what the physical system is subjectively experiencing, if anything.
For this it is important to be clear on the definition of the hard problem. I'll refer to Chalmers' paper on the subject.
About the easy problems, he writes:
The easy problems of consciousness include those of explaining the following phenomena:
- the ability to discriminate, categorize, and react to environmental stimuli;
- the integration of information by a cognitive system;
- the reportability of mental states;
- the ability of a system to access its own internal states;
- the focus of attention;
- the deliberate control of behavior;
- the difference between wakefulness and sleep.
Now, none of the above match the description of F; F is not trivially "easy" by virtue of being on the list. It is not on the list.
Now, about the hard problem, Chalmers first dances around the subject a little bit:
The really hard problem of consciousness is the problem of experience... As Nagel (1974) has put it, there is something it is like to be a conscious organism. This subjective aspect is experience. ... What unites all of these states is that there is something it is like to be in them. All of them are states of experience.
This is too vague to take it as a firm definition of the hard problem. What is the problem of experience? Fortunately, Chalmers starts to give more authoritative definitions of the hard problem, by asking questions clearly intended to be part of the hard problem:
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?
The answer given by the formula F is, "we experience the quality of deep blue because our subjective experience is determined by F, and the system is in a state that, according to F, experiences the quality of deep blue."
We may compare such an answer to the theory of gravity, which answers questions such as, "Why does this planet move in an ellipse?" The answer given by the theory of gravity is a formula, we might denote by H: "this planet moves in an ellipse because its motion about its star is determined by H, and H says the planet moves in an ellipse." In both cases, the question asks why a behavior is seen - a subjective experience in one case, and a certain motion in the second case - and the answer given is, "because it obeys the formula, and the formula says that will happen."
In the case of gravity, this is considered an adequate explanation - provided that the formula is simple enough and agrees with all the other laws (which, sadly, physicists have not achieved yet.) Then, could an explanation of this kind also be considered an adequate explanation in the case of subjective experience? Is there a "hard problem of gravity" unanswerable by any formula? (This is not another question, just rhetorical musing.)
Continuing with Chalmers:
How can we explain why there is something it is like to entertain a mental image, or to experience an emotion?
We can explain it like this: because subjective experience follows F, and F says that when the brain entertains a mental image or experiences an emotion, there is a corresponding subjective experience.
It is widely agreed that experience arises from a physical basis, but we have no good explanation of why and how it so arises. ... Why should physical processing give rise to a rich inner life at all?
F would be an explanation of why and how it arises. The only unanswered question would be why subjective experience follows F - but, just as with laws of gravity, our explanatory regress must terminate somewhere, and as long as that terminal point is sufficiently simple and correct, we must be satisfied with that.
Chalmers then goes on to talk about what he thinks are the problems with current scientific approaches to consciousness. Then he gets to what he thinks is needed to solve the hard problem:
In particular, a nonreductive theory of experience will specify basic principles telling us how experience depends on physical features of the world. These psychophysical principles will not interfere with physical laws, as it seems that physical laws already form a closed system. Rather, they will be a supplement to a physical theory. A physical theory gives a theory of physical processes, and a psychophysical theory tells us how those processes give rise to experience.
... If this view is right, then in some ways a theory of consciousness will have more in common with a theory in physics than a theory in biology
These passages seem to be perfectly satisfied by our hypothetical F; perhaps Chalmers himself would agree that our mathematical formula F could be a solution to the hard problem.
Some more quotes by Chalmers supportive of physics-like explanations:
There is nothing particularly spiritual or mystical about this theory—its overall shape is like that of a physical theory, with a few fundamental entities connected by fundamental laws. It expands the ontology slightly, to be sure, but Maxwell did the same thing. Indeed, the overall structure of this position is entirely naturalistic, allowing that ultimately the universe comes down to a network of basic entities obeying simple laws, and allowing that there may ultimately be a theory of consciousness cast in terms of such laws.
The fundamental laws of nature are part of the basic furniture of the world, and physical theories are telling us that this basic furniture is remarkably simple. If a theory of consciousness also involves fundamental principles, then we should expect the same. The principles of simplicity, elegance, and even beauty that drive physicists’ search for a fundamental theory will also apply to a theory of consciousness.
A nonreductive theory of consciousness will consist in a number of psychophysical principles, principles connecting the properties of physical processes to the properties of experience. We can think of these principles as encapsulating the way in which experience arises from the physical. Ultimately, these principles should tell us what sort of physical systems will have associated experiences, and for the systems that do, they should tell us what sort of physical properties are relevant to the emergence of experience, and just what sort of experience we should expect any given physical system to yield.
That last quote in particular sounds exactly like a specification for F.
Related: this post.
Anyway, to repeat my question: is there anything about the hard problem of consciousness that means that it cannot be answered by a formula giving the dependence of subjective experiences on physical system states?