Conscious Realism’s Mathematical Models

Donald Hoffman’s Own “Eye Candy”

By Paul Austin Murphy

Illustration by GeorgePeters/iStock/Standard License
[The title above is an ironic take on Donald Hoffman’s ‘Reality is Eye Candy’, which was a presentation he gave in 2017 at a SAND conference.]

i) Introduction
ii) Conscious Realism
iii) Why the Maths?
iv) Models as Idealisations
v) Examples of Hoffman’s Models
vi) Conclusion

Donald Hoffman. Photo by: Themindoftheuniverse/Wikimedia Commons/CC BY-SA 4.0

Professor Donald Hoffman often (very often) uses phrases such as “precise mathematics” and “mathematical models” in reference to his philosophical position — conscious realism. Hoffman explains why he does so in the following words:

Part of my background is in psychophysics. This is the science of studying conscious experiences and building mathematical models. Your conscious experiences are not random things. We do careful experiments and can write down mathematical equations that actually describe the conscious experiences you will have. They’re mathematical, so conscious experiences can be described by mathematics.

Donald Hoffman

As just stated, Hoffman often mentions “mathematical models” though he rarely says what he means by those words. And he rarely offers us any examples of these models. (The ones I’ve found will be discussed later.) There may be a good reason as to why Hoffman doesn’t give us many examples. For example, he says that we “must admit that maybe consciousness can be described with mathematics.”

Hoffman doesn’t say that consciousness has been described by mathematics here: he uses the words “maybe consciousness can be described with mathematics” instead. Yet elsewhere Hoffman keeps on talking about his mathematical models of consciousness (as well as of experiences).

Models of “conscious experiences”?

What form do they take? And what does it mean to claim that psychophysicists like Hoffman “can write down mathematical equations that actually describe the conscious experiences you will have”?

Does Hoffman mean that he has mathematical models of the physical bases or physical correlations of what he calls “conscious experiences” (or consciousness itself)? That would be fine. But to have mathematical models of conscious experiences themselves seems like a category mistake.

The point here, then, is that mathematical models exist in physics, biology, economics, etc. However, can there really be mathematical models of experiences and conscious agents? In terms of the latter, the answer is ‘yes’; though only if the physical and behavioural nature and actions of conscious agents are being modelled. However, that’s not what Hoffman is attempting to do.

Conscious Realism

Donald Hoffman uses the word “we” when he should really use the word “I”. Take this eulogy to his own conscious realism. Hoffman writes:

Here there is good news. We have substantial progress on the mind-body problem under conscious realism, and there are real scientific theories.

It can be conceded that Hoffman may have a few postgraduates, and even a few fellow professors, working with him on his conscious realism. So the phrase “we have substantial progress on the mind-body problem” seems a bit too grand for anyone’s liking. However, it’s the passage which follows which is relevant to this piece. Hoffman continues:

We now have mathematically precise theories about how one type of conscious agent, namely human observers, might construct the visual shapes, colors, textures, and motions of objects.

Now that’s fair enough. It can easily be seen how scientists (i.e., cognitivescientists) can construct “mathematically precise theories” about how “human observers might construct the visual shapes, colors, textures, and motions of objects.” The thing is that Hoffman goes much further than this. He has done so by entering the domain of speculative philosophy. Not only that: the reference to constructing shapes, colours, and the motions of objects can all be placed under what’s often called “third-person science.” That is, in such a science the researchers will primarily rely on two things:

  1. The “reports” of the subjects in experiments.
  2. The neuroscience, etc. of vision.

Hoffman moves beyond 1 and 2. He claims to have constructed a “mathematically precise” theory or “model” of consciousness, experiences, cognitive agents, etc. too. In addition, Hoffman also uses such mathematical models to defend (or simply describe) his philosophical position of conscious realism. Now what we have here is a huge jump from the neuroscience/cognitive science (mentioned in the quote above) to Hoffman’s speculative philosophical position.

Why the Maths?

My question is simple. When Hoffman says that his theory “gives mathematically precise theories about how certain conscious agents construct their physical worlds,” what does he mean by that?

More precisely, in what way are numbers and other mathematical tools/functions used to explain how “conscious agents construct their physical worlds”? This can easily be answered. Numbers or mathematics can be used to describe or explain just about anything. For example, if I randomly throw a deck of cards on the floor, the positions of all the individual cards can be given a mathematical description. But why bother?

The other question is about how precisely maths makes sense of what goes on in minds or consciousness. Here again maths can be used (perhaps arbitrarily or pointlessly) to do so. More to the point, what work is the maths doing in Hoffman’s philosophical position of conscious realism?

Hoffman compares what he’s doing to what Alan Turing did. In Hoffman’s own words:

[T]he tip from Turing is that Alan Turing decided to give a theory of what is computation and he came up with this really simple formalism. A little machine that has a finite set of states finite set of symbols some simple transition rules and it turned out he could prove that any computation could be done by this simple little device called the Turing machine and that was what launched the theory of computation computer science.

Consequently, Hoffman continues by asking us this question: “[C]an we do the same thing for consciousness?” That is:

Can we come up with a simple formalism which will handle all aspects of consciousness?

And again in the following we may have a category mistake when Hoffman asks this question:

Can we come up with a mathematically precise theory of consciousness and, from that, boot up space, time, and matter?

What’s more, Hoffman tells us that he thinks that “a precise mathematical science of consciousness is possible.”

Models as Idealisations

No one will have a problem with the fact that mathematical models can — or always do — idealise what it is they’re modelling. For example, this is the case with ideal gases, point particles, massless ropes, and lots of stuff in boxes (see Lee Smolin’s “physics in a box”). What’s more, it’s often the case that these “idealisations” (or simplifications) go too far. Is this also true of Hoffman’s models of consciousness, agents and the rest?

Here we’ll also need to stress the fact that real situations (or things) in the world are very complicated and thus models — especially Hoffman’s models — may be extremely approximate in nature. Despite that, perhaps the problem isn’t even approximation when it comes to “modelling” consciousness, experiences, etc.

Yet idealisations and simplifications are often good things.

For a start, a model must provide us with more than “empirical data.” Put simply, models serve a purpose that’s beyond any painstaking description of every aspect of what it is that’s being modelled. And it’s precisely because models — all models (by definition) — go beyond that data (or beyond description) that there can be the following problems:

  1. Models can oversimplify.
  2. Models can bear little relation to what it is they model.
  3. The relations between models and what they model can be very vague, weak and metaphorical/analogical in nature — and that can even be the case when the model utilises mathematics.

So how much of all the above applies to not only Hoffman’s models themselves; but also what Hoffman says and claims about them?

Each mathematical model also has to take into account the to and thro between accuracy and simplicity. These and other scientific criteria are always being played against each other. This means that other factors must come in, such as the “predictive power” of the model. In addition, simplicity is supposed be cherished in the theories of physics and when it comes to mathematical modelling. In addition, if a model is complex, then it will more faithfully reflect the thing that is modelled. If it’s too complex, on the other hand, then it won’t serve the purpose of being a model very well. (That is, the complex model may be hard to analyse and difficult to understand.)

All this means that Hoffman’s mathematical models (if they are mathematical models) need to account for the question as to whether or not they really do describe systems (or reality) accurately. In that sense, Hoffman’s models face precisely the same problem which he often stresses human “perceptions” face in his (partly) evolutionary account of conscious realism.

Examples of Hoffman’s Models

The following are a few examples of Hoffman’s “mathematical models” — or mathematical graphs.

Firstly, we have this mathematical model of what Hoffman calls a “conscious agent”:

Hoffman uses the (supposedly) mathematical symbols of WX and in the above:

W = the world
X = an experience
G = an agent’s action

Now once you have these symbols you can of course play with them. In Hoffman’s own words:

We can translate this into some mathematical symbols. We have a world W, experience X and action G… and then we have a map [see next image], a Markovian kernel… and an integer counter [n] which is going to account for the number of perceptions you have.

And so on. And where you have mathematical symbols, you often also have maps, graphs, grids and such like. Hoffman makes use of them too (in the following).

So what we have in the above is a triadic set of relations between WX and G. Does it tell us anything? Is it gratuitous? And even if it’s not actually mathematical in nature, does it still help us in some other way?

As mentioned earlier, this model is certainly an idealisation (or a simplification): all we have represented is the/a world (W), an experience (X) and an action of a conscious agent (G). So why only these three phenomena? Why a single experience and a single action? (Unless X is meant to be a symbol for experiences or experience in general.) And how are X and G taken in separation from the rest of W? How would an externalist and/or anti-individualist take this seemingly Cartesian position on a world, an experience and an action? (Of course Hoffman isn’t a Cartesian from either a philosophy of mind or an ontological point of view — he’s vocally against “dualism.”) And what about the agent (G) and his/her/its embeddedness (as in embodied cognition) in the world (W)?

It’s of course the case that Hoffman’s conscious realism will provide all the answers to these questions. That is, if one accepts conscious realism, then these problems become non-problems. In simple terms, externalism, anti-individualism and embeddedness have no purchase in Hoffman’s system. That’s simply because his philosophical system is literally all about conscious agents interacting with other conscious agents.

So to recap.

Hoffman’s graph above is very sexy and seemingly scientific. We have the symbols WX and G for a start. Not only that: the letters are connected in a geometric graph. But so what? How does this graphic and symbolic representation help matters? More importantly, what does it really say? And is this really a mathematical model?

Then Hoffman goes deeper — or at least his next graph is more complex. Now we have this:

Here we have extra “mathematical symbols”. In addition to the symbols WX and G, we now also have the symbols PA and D. Thus:

A = “action map”
P = “a Markovian kernel
D = “a perception map” or a “decision map”
N = “an integer counter” which “counts the number of perceptions which you have”

According to Hoffman, “a conscious agent is just[sic] a sextuple” — that is, a conscious agent is just this set: (XGPDAN).

Thus, the connecting line from W (a world) to X (an experience) is symbolised by P (a “markovian kernel”). And X’s connecting line to G is symbolised by D (a “perception map” or a “decision map”). That is, when an conscious agent carries out an “action” (or a “decision”) in that world, this is symbolised by D.

Again, how does the model help? And is the model accurate? What sort of world (W) — if a conscious agent’s world — can be summed up by a “sextuple” (XGPDAN) — even if we acknowledge the importance of idealisation or simplification?

(I’m willing to concede that I may have partly misread Hoffman’s symbolisations or models. Nonetheless, I don’t believe that would have even a slight effect on my criticisms.)

Things get even deeper here:

Here we have a symbolic and graphic representation of “two conscious agents,” not one. In addition, we have N1 and N2 (both “integer counters”). But what does the image above really tell us? And if we didn’t get much meat out of the left-hand side of this image (as quoted above), then how can we get more meat simply because we’ve put both sides together?

Finally, we have this:

In the above, “each dot is a conscious agent” and “each link is a connection between conscious agents where they are communicating with each other”. Even Hoffman must admit that the placings of the agents (the pink dots) and the resultant shapes of these agential interrelations are completely arbitrary. (There are symbolisations of triadic interrelations and quadratic relations; which, in turn, are related to other geometric relations.) This, however, may not matter to the philosophical point that Hoffman is attempting to get across.

Two things are worth mentioning here. 1)The use of the mathematically-sounding title “combination theorem” (see mathematical combination). 2) What does that graph actually give us? Indeed why is the above a theorem? (Or, more mundanely, why use the word “theorem” at all?) In any case (as already stated), I could randomly throw a deck of cards onto the floor and then represent them graphically (with symbols ’n’ all). What would that show? Well, it would show us where all the cards landed. But what more would it show? Probably not much more.


To offer a final sceptical conclusion.

Perhaps all that Hoffman means by his frequent references to “using precise mathematics” (or, more often, his references to using “mathematical models”) is simply that he uses what he calls “mathematical symbols”; which are then placed in graphs(such as in those above). But mathematical symbols alone can be used for anything and they can be used by anyone.

This also raises the question: What does Hoffman mean by the words “mathematical symbol”?

Finally, is Hoffman doing something that’s really that different to what Julia Kristeva did? Take this passage (which is replete with mathematical symbols and references) from Kristeva:

And here’s an “equation” from Jacques Lacan:

Finally, I’m not saying that Hoffman’s models are completely in the same ballpark as the other two examples above. Nonetheless, they are still, I believe, gratuitous. And they’re also used to tart up (as it were) his extremely speculative philosophical positions.

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