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?

  • 3
    No. The hard problem is about explaining why/how there is experience, your F is about what: what sorts of physical systems produce experience and what kinds of it. There is a habitual substitution for gravity, etc., where "what" answers are given to "why" questions because the "why" part proper is trivial. One physical third person thing gives rise to another, no extra explanation needed. This substitution in general, and gravity analogy in particular, do not work for experience, hence the explanatory gap. In the end, your F is the sum total of the "easy" problems, on the list and off.
    – Conifold
    Aug 4, 2023 at 21:24
  • 1
    @Conifold Then what kind of "why" would you deem sufficient, and how do you explain Chalmers' own comments that a solution to the hard problem would be like a theory in physics, and that it would "specify basic principles telling us how experience depends on physical features of the world"?
    – causative
    Aug 4, 2023 at 21:28
  • 1
    It is as Chalmers says: tell us why/how third person its give rise to/are accompanied by first person what-it-is-likes, not just what of the former correlate to what of the latter. An auxiliary problem is access to whats: it is hard to see how what-it-is-likes can even be converted into third person representational form to become the relata your F proposes to correlate. A big part of the hard part is in your pre-produced "mathematical representation of subjective experiencing", which sounds almost oxymoronic without a why/how first-to-third person psychophysical bridge that Chalmers asks for.
    – Conifold
    Aug 4, 2023 at 21:45
  • 1
    @Conifold But you didn't answer either of the questions I just asked you. What kind of "why" are you talking about, if you aren't talking about the kind of "why" used in physics? And how do you reconcile your exclusion of physics-like explanations, with Chalmers' comments that the solution would be a physics-like principle telling us how experience depends on physical features of the world? (You are right that finding the right representation for the output of F would be among the many challenges of determining F.)
    – causative
    Aug 4, 2023 at 21:56
  • 1
    Chalmers' "physics-like" is not the like you have in mind. He himself argues for the explanatory gap on the grounds that any satisfactory explanation would have to be unlike objectified explanations of physics. The likeness here is purely structural, of a 'fundamental' physical/metaphysical sort rather than particularized biological sort.
    – Conifold
    Aug 4, 2023 at 22:15

6 Answers 6


Seems like you are totally on a wrong track.

  1. You cannot expect that a mathematical formula could be applied in a context with no parameters to put in said formula.

  2. Subjective experiences are not direct consequences of "physical system states". Subjective experiences are based on what we know and feel about those "physical system states", what they mean to us.

  • Regarding "Subjective experiences are based on what we know and feel about those "physical system states", what they mean to us.": What is "we" in this context, if not the physical system states themselves? The sentence doesn't seem to make much sense, without defining it.
    – user000001
    Aug 5, 2023 at 13:31
  • Begs the question.
    – Corbin
    Aug 5, 2023 at 13:35
  • @user000001 Naturally "we" refers to all and any of us human beings. Physical systems refer to all physical systems our own bodies included. Only our minds can experience anything, assign any meanings to "physical system states". Aug 5, 2023 at 15:35
  • 1
    @PerttiRuismäki: But this assumes that the mind exists as an entity separate from the body, and not just as a property of the brain. If the "hard problem" is solved, which is what the Q is about, then your position will have been proven false. This answer then is just asserting that the hard problem cannot be solved, but not really giving any evidence for it.
    – user000001
    Aug 5, 2023 at 15:48
  • 1
    @user000001 I am not assuming anything. The mind is not separate from the body. It is only doing different things. The mind thinks and experiences, the body does only physical things. Aug 5, 2023 at 16:22

I agree with Conifold and Pertti. There are two points you might like to consider. Firstly, take, for example, a picture of Donald Trump, say, on a computer screen. It is possible, albeit not as a single formula, to equate the frequencies and intensity of the light from each pixel on the screen with underlying energy states in the LED, but what is not possible is to equate the energy states with the fact that the colours and intensities of all of the pixels amount to a representation of Donald Trump.

More generally, physical formulae equate quantities both in terms of their numeric values and their dimensions. You cannot, for example, have a physical formula that equates kilograms with metres. Given that, for your idea to work in principle it would have to be possible to quantify all aspects of consciousness both in terms of a numeric value and a set of dimensions. I defy you to suggest, even in broad brush terms, how that might be done.

  • "I defy you to suggest, even in broad terms, how that might be done" - for that I suggest you read Chalmers' paper thoroughly, particularly sections 6 and 7. I think a lot of dualist thinkers have latched on to Chalmers' ideas as a weapon against physicalism, without realizing how minimalist Chalmers' dualism really is. He says, "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."
    – causative
    Aug 5, 2023 at 15:24
  • Of course, you're free to disagree with Chalmers, or to accept some of his ideas while rejecting others. But it's very clear from the latter half of his paper that he intended his "hard problem" to be answered in a way similar to how physics answers physics questions.
    – causative
    Aug 5, 2023 at 15:27
  • @causative many thanks. When I get a moment I will examine the sections you mentioned and report back. Aug 5, 2023 at 15:41
  • I think his principle of organisational invariance, in which he states that a network of neutrons in the brain when modelled on a like for like basis in silicon should give rise to the same experience in the silicon as in the brain, raises some puzzles. Suppose there were enough humans to model a brain, with each human acting as a single neuron, so that instead of replicating a brain in silicon we replicate it by a collection of billions of humans acting in concert, would the collection of humans, as a single entity, have the same experience as the brain the humans were modelling? Aug 5, 2023 at 15:55
  • I'd say yes, and I bet Chalmers would too.
    – causative
    Aug 5, 2023 at 16:07

Re: explanations and formulae

If all you needed for an explanation was a formula describing the observations, we wouldn't need a theory of gravity, a theory of elliptical planetary motion, or even heliocentrism. Ptolemy's 2nd century model of circular orbits around the Earth, with circular epicycles of the planets around the orbits, is sufficient to describe all planetary motion. Given enough epicycles, you can describe any trajectory at all, such as this youtube video illustration of Homer Simpson using an epicyclic orbit. The math won't be as easy as Kepler's, but it'll be much easier and neater than setting up a computer to brute force a numerical solution in General Relativity. Ptolemaic epicycles therefore probably have more mathematical simplicity than the modern physics explanation, too.

To explain a phenomenon is to describe in terms of more fundamental principles, not to find an equation that describes it and call it a day. For instance, Newtonian gravity answers the following:

Suppose the heavens and the earth behave according to the same principles. What phenomena observed on Earth might proceed according to principles which also account for the motion of the planets described by Kepler's equations? What would those principles have to be to match both sets of observations, and how could we test them?

Re: Hard Problems

I would suggest that a problem is a Hard Problem when the question may not mean anything - but which seems like it must mean something, for some unknown modification of our definition of truth or some unknown rephrasing of the question, if we could only uncover a deeper knowledge of the universe. Physics has lots of these. I would suggest, "What is happening on Mars right now?"

  • Newtonian gravity ultimately comes down to a single formula: that the force between two masses m1, m2 separated by distance r is given by G m1 m2 / r^2. Why should this force be present? Newtonian gravity has no answer. It just is. Every theory in physics is like this. You can ask "why" a thing happens and receive more and more detailed explanations, but in the end there is a formula that you just have to accept as the way things work. (Also, GR is more correct than Newtonian gravity; a theory is first evaluated by how well it matches observations, and only then by its simplicity.)
    – causative
    Aug 5, 2023 at 15:16

Regarding pure apperception, Immanuel Kant at one point writes:

Apperception is something real, and the simplicity of its nature is given in the very fact of its possibility. Now in space there is nothing real that is at the same time simple; for points, which are the only simple things in space, are merely limits, but not constituent parts of space. From this follows the impossibility of a definition on the basis of materialism of the constitution of my Ego as a merely thinking subject. But, because my existence is considered in the first proposition as given, for it does not mean, "Every thinking being exists" (for this would be predicating of them absolute necessity), but only, "I exist thinking"; the proposition is quite empirical, and contains the determinability of my existence merely in relation to my representations in time. But as I require for this purpose something that is permanent, such as is not given in internal intuition; the mode of my existence, whether as substance or as accident, cannot be determined by means of this simple self-consciousness. Thus, if materialism is inadequate to explain the mode in which I exist, spiritualism is likewise as insufficient; and the conclusion is that we are utterly unable to attain to any knowledge of the constitution of the soul, in so far as relates to the possibility of its existence apart from external objects.

Now, then, are there discursive qualia? When are things like certainty or clarity tantamount to felt states? Regarding clarity, one recent analyst says:

Some philosophy – Wittgenstein's would be an example – is written in clear sentences, yet most people find it obscure at a first reading. This is because the prime location of clarity in philosophy is not sentences but structures. Only if a reader can relate what he is currently reading to a wider framework does he know where he is. Coherent utterance in all discursive media – not only language but mathematics, for example, or music – possesses two kinds of structure at the same time. In this article these are distinguished, and their radically different relationships with language shown. In the process, the commonest causes of unclarity are also identified.

If there are discursive qualia, there might (presumably) be mathematical qualia (c.f. references to spatial qualia in some analyses). Perhaps there is "something that it is like" to count objects: many of us count one-by-one but some of us are adept at subitizing, then.

Putting the above together: if a mathematical formula existed to explain qualia reductively, would such a formula be able to handle the relativity and subjectivity of mathematical representations themselves? Consider Benacerraf's identification problem, then: a formula that could explain qualia would be tantamount to an explanation as to why some identifications were closer to correct than others (in explanation-theoretic terms). This might be helplessly circular, then, however (does the formula explain its own qualia-theoretic manifestation?).

All that being said, one might wonder if pure qualia are even a kind of thing to be explained (or: we need a specific concept of explanation and we can ask about the qualia of explanations themselves). If there were a divine being in the classical sense, with divine qualia, it would be hard to say that those were explicable any more than the divine nature moreover: there seems to be something possible to say like "qualia precede questions of physical/mathematical explanations," and then the hard problem of their existence is like asking for an explanation of the concept of explanations (asking for this such that the definition of the concept is applied from the outside). I myself dare not say that it would be absolutely impossible to come up with a representation that would socially count as a mathematical formula, and which socially counted as explaining qualia, depending on the vagaries of history. (For the time being, a scientist recently lost a bet with Chalmers that consciousness would by now be explained, which goes to show that Chalmers was, and perhaps still is, willing to entertain the possibility of losing such bets himself.)


  1. If "can in principle" means per Chalmers' own sense of the problem, then since Chalmers was willing to make a decades-spanning bet over the issue, then he himself must have thought that there was some (at least epistemic) possibility that the bet was not trivially made and so that there might be some hitherto-undiscovered conception of the problem by which it might be resolved in favor of physical reductionism. Or maybe he did make the bet flippantly, I don't know (I don't know much about his work, much less his personal character, but offhand he doesn't strike me as a bad-faith reasoner).
  2. If "can in principle" means according to perhaps unclear or vague standards of metaphysical possibility/principles that go beyond Chalmers' representation of the problem, then we'd have to draw on theories of metaphysical possibility, and hope to specify them nontrivially (or else we might say "for all we know, there could be a principle such that..."). We'd also have to draw on theories of explanation and of mathematical objects (if these objects in their purest state are uninterpreted symbols/strings or whatever, for example, then their use in explaining qualia would be so as to explain themselves too, viz. to account for semiotic qualia, which might seem impossible or pointless).

So: in terms of sheer possibility and the mere concept of something being "explained in principle," then it does seem as if we might, one day, come up with something that is mathematically formulaic that explains consciousness modulo the hard problem, or else we would find that the hard problem is, if not illusory, yet not something fit for a solution-by-explanation.

  • Yes as you quote from CPR he does say: and the conclusion is that we are utterly unable to attain to any knowledge of the constitution of the soul. But Dreams of a Spirit seer is much more ambiguous — whether the mystic (Swedenborg) was simply delusional or came closer to seeing (aperceiving) the noumena behind the phenomena is a question that Kant scholars dispute about
    – Rushi
    Aug 5, 2023 at 16:29
  • 1
    @Rushi I suppose Kant could have changed his mind or contradicted himself, wouldn't be the only time he did one of those things (seemingly). As far as using that quote here, I was mainly trying to give an introductory example of how philosophy-of-math and -of-space intersect the OP question, not as a refutation of the possible explanation-type that the OP is asking for. Unfortunately, I don't understand the hard problem of consciousness very well so I shouldn't be too confident in any such direction, though. Aug 5, 2023 at 16:34

Interestingly, I just listened to this interview with a neuro-mathematician who is proposing exactly what you are calling for -- a math formula explaining consciousness. Here is the interview with Ogi Ogas, https://www.youtube.com/watch?v=TmNqg1vssoo and here is a link to his book, Journey of the Mind: https://www.amazon.com/Journey-Mind-Thinking-Emerged-Chaos/dp/1324050578/ref=sr_1_1?keywords=ogio+ogas+books&qid=1691260499&sr=8-1

Note this math is "process" math, and is emergent and wholistic, not reductionist.

I believe that Chalmers, who generally rejects emergentism and wholism, might not endorse this approach to such a function F. Chalmers is a dualist because he is convinced that consciousness does not reduce to matter, and reduction is the right way to do explanations. BUT -- you don't have to convince Chalmers; all you need to do to answer your question is show such math is possible. And Ogi asserts that this demonstration has been successfully accomplished.

I do not think Ogi's claim is true. My belief is that Ogi's assertion of identity between processes and basic experience -- is false, and we can easily show this with falsification tests by observing how most of our processing is done without awareness.

However, Ogi's claim basically shows that your thesis is possible. One can create a math model for the emergence of consciousness, and if it matches all our data about consciousness, then we COULD treat it as the answer to the hard problem. The issue that makes the hard problem "hard" is that any such model -- violates the principles of physicalism.

Ogi's math model is based on strong emergent dualism. Awareness is an emergent phenomenon, that appears as processes develop properties. Consciousness is a higher tier emergent phenomenon, based on processes of processes. Ogi's math model is dualist, and explicitly breaks the causal closure of physics.

This tells you something that I consider to be a general truth about such a math model. Consciousness appears to be causal, and if one takes this data about consciousness seriously, then it will be causal in the math model. Such model would then not be a "physicalist" model. The "hard problem" is not hard for dualists, and any math model F we build for consciousness today, will be a dualist one.


I strongly suspect that consciousness is completely determined by the physical state of the universe. (Lots of evidence points to this, and I know of no evidence that suggests it is not the case.)

For this reason, I believe that such a formula exists.

On the other hand, because physics per se does not contain any information relating to consciousness per se, I believe that such a formula could never be deduced.

(This conclusion is also supported by the fact that we cannot measure consciousness. Furthermore, any attempt to measure the physical state of someone's consciousness apparatus must apparently interfere with the consciousness itself — a kind of uncertainty principle.)

  • A person can, however, directly know what they personally are conscious of. This non-empirical information might be combined with empirical data to help find the right formula.
    – causative
    Aug 9, 2023 at 4:10
  • I agree that there is the obvious direct knowledge of consciousness that each being experiences. But there is still no way to make details physical measurements about the being's state at the time, or any way to translate the experience of consciousness into precise physical terms. Aug 9, 2023 at 5:38
  • It looks like you have a lot of unjustified presuppositions here. Physics is indeterministic, so there is no way it can determine anything. Science is pluralist, so physics does not encompass consciousness (which is within psychology). And math models need not be physicalist, one can have a math model that is idealist, dualist, emergent, or whatever. So a math model of consciousness is not equivalent to a physicalist model describing consciousness.
    – Dcleve
    Aug 9, 2023 at 6:51
  • There is no evidence that the universe is indeterministic. Only that we cannot predict the future thanks to the Heisenberg uncertainty principle. Not the same. Aug 9, 2023 at 15:15
  • " Science is pluralist, so physics does not encompass consciousness": I do not understand this claim. Regardless of the higher sciences, to the best of our knowledge, everything is subject to the laws of physics and there are no other laws that govern how events evolve in time. Sep 5, 2023 at 18:46

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