Donald Hoffman's Eye Candy: Conscious Realism's Mathematical Models

Donald Hoffman is attempting to “come up with a mathematically precise theory of consciousness and [] space, time, and matter”. That project involves an essential commitment to idealism. But how substantive is the maths in his project?

[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) Hoffman’s Conscious Realism

iii) Why the Maths?

iv) Models as Idealisations

v) Examples of Hoffman’s Models

vi) Conclusion

Professor Donald Hoffman often uses phrases such as “precise mathematics” and “mathematical models” in reference to his own 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.”

As just stated, Hoffman often mentions “mathematical models”. However, he rarely says what he means by those two 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 any 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 with mathematics” here: he uses the word “maybe” 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 scientists like Hoffman

“can write down mathematical equations that actually describe the conscious experiences you will have”.

Does Hoffman may mean that there are mathematical models of the physical bases or correlations of what he calls “conscious experiences” or consciousness itself?(See ‘Neural correlates of consciousness’.) That is, is Hoffman simply taking about what goes on in third-person (physical) brains — along with observable physical and verbal behaviour — when such things are studied by scientists?

If so, then any commitment to correlations, brains and behaviour would make Hoffman’s position very much like that of those neuroscientists, psychophysicists and other scientists who’re also physicalists…

Yet Hoffman is strongly against physicalism.

More relevantly, Hoffman is an idealist who’s stated that “brains and neurons do not exist unperceived”.

That, at least in itself, isn’t a contradictory position on Hoffman’s part. It simply means that Hoffman’s position on what (in this case) brains and neurons actually are needs to be made explicit — at least to all those people who aren’t aware that he’s an out-and-out idealist.

But what about having mathematical models of conscious experiences themselves?

Of course many scientists of mind and consciousness reject this “binary opposition” in that they don’t even attempt to divorce consciousness and mind from the brain. And the primary reason why they don’t do so is that most of them are either explicitly or implicitly committed to physicalism. Indeed Hoffman himself is keen to stress the fact that most scientists (specifically when it comes to the brain, mind and conscious) are — if often tacit — physicalists (see here).

So, again, how does Hoffman fit into this debate?

The general point here, then, is that mathematical models exist in physics, biology, economics, etc. Yet can there also be mathematical models of experiences and conscious agents?

In terms of the latter, the answer is yes… in a sense. That is, only if the verbal and behavioural actions of “conscious agents”, along with what happens in their physical brains, are being modelled…

Yet all that isn’t only what Hoffman is attempting to do.

Conscious Realism

Donald Hoffman often 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 has a few postgraduate workers, and even a few fellow professors, working with him on his conscious realism. However, phrases such as “we have substantial progress on the mind-body problem” seem a bit too grand. However, it’s the passage which follows which is relevant to this essay. 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 (cognitive scientists) 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 been called “third-person science”. That is, in such a science, the researchers will rely primarily on two things:

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

Hoffman moves beyond all that. 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?

The 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?

More precisely, in what way are numbers and other mathematical tools used to explain how “conscious agents construct their physical worlds”?

This can easily be answered in one way.

Numbers or mathematics generally 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 precise mathematical description…

But why bother?

The other question is about how exactly maths makes sense of what goes on in minds or consciousnesses. 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 himself 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 “think[s] [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”). However, it’s often the case that these “idealisations” (or simplifications) go way too far.

So is this true of Hoffman’s models of consciousness, conscious agents and the rest?

Here it also needs to be stressed that real situations (or things) in the world are very complicated and thus models — especially Hoffman’s models — may be extremely approximate in nature. However, perhaps the problem is not even approximation when it comes to Hoffman’s supposed “modelling” of consciousness, experiences, “conscious agents”, etc.

Yet idealisations and simplifications are often very-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 that’s being modelled. And it’s precisely because models — all models (by definition) — go beyond that data 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 a model and what it models can be very vague, weak and even purely metaphorical/analogical — and that can even be the case when the model utilises much mathematics.

All this means that 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 also 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. Thus, 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. (The complex model may be hard to analyse and difficult to understand.)

Now how much of all the above also applies to Hoffman’s models and what he claims about them?

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 any given phenomena) accurately. In that sense, Hoffman’s models face the same problem which he stresses human “perceptions” face in his (part) evolutionary account of conscious realism.

**************************

Part Two 

Examples of Hoffman’s Models

The following are a few examples of Hoffman’s “mathematical models” — or mathematical charts (i.e., they’re hardly even mathematical graphs).

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

Hoffman uses the (supposedly) mathematical symbols of W, X and G in the above:

W = “a world”
X = an “experience”
G = a “conscious agent’s action”

Now once you have these symbols, you can 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 suchlike. Hoffman makes use of them too.

So what we have is a triadic set of relations between W, X 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 way?

For one, as mentioned earlier, this model is certainly an idealisation (or a simplification): all we have represented is a world (W), an experience (X) and an action of a conscious agent (G).

So why only these three (as it were) variables?

Why a single experience and a single action? (Unless X is meant to be a symbol for experiences or experience in general.) And why are X and G seemingly taken in separation of the rest of W? What’s more, how would an externalist or anti-individualist take this almost Cartesian position on a world, an experience and an action? And what about the agent (G) and his/her/its embeddedness in the world (see ‘Embodied embedded cognition’)?

It’s of course the case that Hoffman’s conscious realism may provide all the answers to these questions

So to recap.

Hoffman’s graph above is seemingly scientific. We have the symbols W, X 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?

And then Hoffman goes deeper — or at least his next graph is more complex than the first one.

Now we have this:

Here we have extra “mathematical symbols” and thus more (supposedly) scientific rigour.

In addition to the symbols W, X and G, we now also have the symbols P, A 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 perception which you have”.

According to Hoffman, “a conscious agent is just [yes, just] a sextuple” — that is, “(X, G, P, D, A, N)”.

This means that the connecting line from W (a world) to X (an experience) is symbolised by P (Hoffman’s “markovian kernel” — see ‘Markov kernel’). And X’s connecting line to G is symbolised by D (a “perception map” or a “decision map”). That is, an agent carries out an “action” in a (or the) world.

Again, how does that model help?

And is the model accurate?

What sort of world (if a conscious agent’s world ) can be summed up by a “sextuple” (X, G, P, D, A, N) — even if we acknowledge the importance of idealisation or simplification?

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”).

What does the image above really tell us?

If we didn’t get much meat out of the left-hand side of this image (as quoted above), then how can we get much more meat when we have both sides taken 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.

Three things are now worth mentioning here:

(1) Why the use of the mathematically-sounding title “combination theorem” (see mathematical combination)?

(2) Why is the above a theorem? (More mundanely, why use the word “theorem” at all?)

(3): What does Hoffman’s graph actually give us?

Conclusion

To offer a sceptical conclusion.

Perhaps all that Hoffman means by his frequent references to “using precise mathematics” (or, more often, to using “mathematical models”) is simply the use of what he calls “mathematical symbols”; which, in turn, are then placed in graphs (such as in those above).

Yet mathematical symbols can be used for anything and they can be used by anyone.

This raises the following question:

What does Hoffman mean by the words “mathematical symbol”?

Is Hoffman really doing something that’s very different to what Julia Kristeva did in the following passage? -

And what about this “equation” from Jacques Lacan? -

It’s not being said that Hoffman’s models are entirely in the same ballpark as the other two outré examples. However, they’re still largely gratuitous. More importantly and finally, it can be argued that Hoffman’s mathematical symbols are used to simply (as it were) tart up his speculative philosophical positions...

****************************

*) See my ‘A Contradiction in Donald Hoffman’s (Idealist) Fitness-Beats-Truth Theorem’, ‘Donald Hoffman’s Long Jump From Evolutionary Biology/Theory to Highly-Speculative Philosophy’ and ‘Donald Hoffman’s Case For An Idealist and Spiritual Reality’.

Wittgenstein and Heidegger vs. Science and Technology

i) Wittgenstein and Heidegger on Science and Religion

ii) Wittgenstein and Heidegger on Science and Philosophy

iii) Wittgenstein and Heidegger on Kierkegaard

“...the total world-view of modern man [has] let itself be determined by the positive sciences...[which has resulted in] an indifferent turning-away from the questions which are decisive for a genuine humanity.” - Edmund Husserl (in his 1936)

Wittgenstein and Heidegger on Science and Religion

Martin Heidegger once wrote of the “flight of the gods”. This was a reference to what he took to be the ascendancy of “scientific culture” in 20th century Western society and the concomitant rise of “instrumental rationality”. Heidegger – along with Edmund Husserl - also wrote of the scientific flight from “lifeworlds”.

This position can be compared to Ludwig Wittgenstein’s view that Western civilisation had taken a flight from God. Wittgenstein was also – or in parallel - strongly against scientism (to use a term not often used in his own day).

This scienceophobia (to use an equally rhetorical term) spread it wings and flew out of the domain of philosophy and into the world of literature. It can be seen in Iris Murdoch’s following words about the existentialists’ predicament:

“…the fearful solitude of the individual marooned upon a tiny island in the middle of the sea of scientific facts, and morality escaping from science only by a wild leap of will.”

It may be incorrect to say here that Wittgenstein’s position against scientism (unlike Heidegger’s) was apolitical and personal. Heidegger, on the other hand, believed that science can/does lead to tyranny. (This is very ironic when one considers his support for the Nazis and their highly-technological regime.) Yet, according to Rush Rhees, Wittgenstein once told him that “[t]yranny doesn’t make me feel indignant”. We can clutch at straws here and say that this was because Wittgenstein believed that although tyrannies can enslave the body, nevertheless they leave the soul free to do what it likes.

However, Wittgenstein did once say that “history had shown [him] that science and industry [have the power] to decide wars”. He thought, like Heidegger, that mankind had turned away from God (or “the gods”) and put its trust in “scientific progress” instead.

Wittgenstein also once said (to a friend):

“Just improve yourself, that is all you can do to improve the world.”

This was a good piece of Protestant theology (i.e., “faith, not works”) on Wittgenstein’s part.

Heidegger was generally more suspicious of religion than Wittgenstein, at least on the surface. Wittgenstein was much less concerned (that is, in his philosophical publications) with theology and religion than Heidegger. Heidegger also believed that religions should have a strong social aspect, which Wittgenstein didn’t believe. This may mean that Heidegger’s view of religion generally (if not of theology and metaphysics) might not have been as uncritical as Wittgenstein’s. Indeed Heidegger once said (quoting Nietzsche) that “Christianity is Platonism for the masses”.

Wittgenstein and Heidegger on Science and Philosophy

Wittgenstein argued (in his Blue Book and exactly like Heidegger) that what he believed to be the philosophical obsession with science could only lead us astray. Yet it wasn't only Wittgenstein's logical positivists who wanted to ask and answer questions in a scientific manner, someone like Husserl (as Heidegger argued) did so too. So just as in certain instances Heidegger saw religion as the source of metaphysics (though he didn't necessarily think that a bad thing), Wittgenstein believed (at least at one point) that our scientific yearnings were now the source of metaphysics. (He did think that is a bad thing.) In both cases, science is something beyond the rightful ken of philosophy. It is something that had a strong pull on the many philosophers both Heidegger and Wittgenstein criticised.

Wittgenstein on Søren Kierkegaard

“Man has the impulse to run up against the limits of language…This running-up-against Kierkegaard also recognised and even designated in a quite similar way (as running-up against Paradox). This running-up against the limits of language is Ethics.” - Wittgenstein’s remark about Heidegger.

The following is Søren Kierkegaard himself speaking about metaphysics and reason:

“[Do anti-religious philosophers] wish to monopolize the notion of ‘Reason’ for the philosophical project of epistemic self-sufficiency? Fine. We will call ourselves the Paradox…But when you say that the Paradox is in conflict with ‘Reason’ there is something of an…illusion. For this is but an echo of what the Paradox has been saying about its relation to that philosophical project since at least the time of the apostle Paul.” (1844/1985)

This Kierkegaardian and ambivalent attitude towards metaphysics and reason can also be detected in Wittgenstein.

On the one hand (as stated) Wittgenstein believed that science often initiates the metaphysical yearnings of many philosophers (i.e., in the early to middle 20th century). Though he also believed that there's indeed something about metaphysics that's deeply attractive (as did Heidegger). He might have also believed - as many other philosophers did and still do - that there's something deep and even magical about metaphysics. Though this deep and magical side of metaphysics is precisely what had turned philosophers (including Wittgenstein himself and Heidegger) against metaphysics. And here again we can see the influence of Kierkegaard (who was classed by the logical positivists as an “irrationalist”) on Wittgenstein.

Kierkegaard believed that traditional metaphysicians had deemphasised the differences between God and man. This “logocentric” position (to use a term often used by Jacques Derrida and others) also meant that at precisely the same time the distinctions between the logos (as it appeared in metaphysics and rationality) and ourselves were in certain senses being emphasised. So, according to Kierkegaard, the metaphysical tradition had fallen victim to the “forgetfulness of Being” (to use Heidegger’s later words). In so doing, reason and metaphysics were deified at the same time as “the subject” was being slowly obliterated.

Wittgenstein, on the other hand, recognised the deification of science; rather than than the deification of reason and metaphysics. Or as Richard Rorty wrote:

“…the source of realist…philosophy of science is the attempt…to make ‘Nature’ do duty for God – the attempt to make natural science a way of conforming to the will of a power not ourselves…” (1985)

In addition, just as Wittgenstein undervalued (or underemphasised) the importance of speech and language in religion, so too he conceded (if only implicitly and at certain times) that language (or words) can't tell us what is true or profound in metaphysics. This is strange considering the supposed “anthropocentrism” of the late Wittgenstein. Indeed Wittgenstein did say that the “expression of metaphysics [is a] fundamentally religious feeling”. A feeling partly inspired by the urge to bang on the doors that are the “limits of language”. Wittgenstein, like Kant before him, wanted to transcend the boundaries of reason (despite Kant’s point that this wouldn't give us knowledge) and also to take Kierkegaard’s leap of faith.

*****************************

Note: 

Perhaps it can be said in conclusion (if only for contrast) that there’s only one person who's more blinkered than a “scientistic philosopher” (if such there be): and that's a strongly anti-scientistic (or anti-science) philosopher. Wittgenstein shares his anti-scientistic (rather than anti-scientific - as in Heidegger)) trait with, among others, Friedrich Nietzsche, Thomas Nagel, Edmund Husserl, Colin McGinn, and many neo-Aristotelian/neo-Thomist philosophers.

Graham Priest's Dialetheism: Quantum Mechanics and... Nothing (3)

i) Nothing Both is and is Not an Object

ii) Ontologically Dependent on Nothing

iii) Possible Worlds and Dialetheism

iv) Is Dialetheism About Reality?

iv) Quantum Mechanics

**********************

One can stress the ontological nature of his dialetheism in that it has been claimed that isn't a formal logic. Instead, it's argued to be "a thesis about truth". Thus it's no surprise that Graham Priest ties it to nothing and therefore to ontology.

Nothing Both is and is Not an Object

Graham Priest hints at dialetheism in this passage:

“Nothing is not an object because you've taken away all objects to get it.”

Elsewhere, Priest says that nothing is an object. Thus nothing “both is and is not” an object. Hence the dialetheism.

So despite the work of Russell, Quine, Carnap, etc., Priest comes out with phrases such as

“we posit nothingness in advance as something that is such-and-such we posit it as a being a thing”.

No; we don't “posit” anything ontological when we use the word “nothing”. Some philosophers (very few) have done so; although “we”, generally, haven't.

Priest then offers us the contradiction.

The first horn of the contradiction is that “nothingness is not an object”. The second horn is that nothingness “is an object”. Dialetheists (or at least Priest) make sense of this by arguing that nothingness “depends like all objects on nothingness”. The argument is this:

i) All objects depend on nothingness.

ii) So if nothingness is an object,

iii) then it too must depend on nothingness.

iv) Therefore nothingness must depend on itself.

Priest poetically calls this “going from nothingness to nothingness”.

Then we get back to the other dialetheic horn – nothing is not an object. That is:

i) “Only objects depend on nothing/ness.”

ii) If nothing/ness is not an object,

iii) Then it doesn't depend on nothingness. (It doesn't depend on itself.)

Or in Priest's own words:

“[N]othing depends on itself. Though since it's not an object, it doesn't depend on itself.”

Here again we have a contradiction.

Or so the argument goes.

If Priest firstly accepts that “nothingness is an object”, then these contradictions are bound to follow. In other words, if we begin with the claim that nothingness is an object, then no wonder Priest can then say that

“nothing [] depends on itself; but since it's not an object, it doesn't depend on itself”.

Yes, that is indeed a statement of a contradiction. But do we have the reality of a contradiction?

(Readers will need to backtrack to the section on objects in order to see exactly why Priest believes that nothingness can be seen as an object.)

Here's some more dialetheism from Priest. He says:

“Since [nothing] is an object, it is something. But it is the absence of all things too; so nothing is nothing. It is no thing, no object. Here, Heidegger got it exactly right: What is the nothing?”

Then, in a note, Priest is more dialetheically explicit when he states the following:

“Nothing, then, is a most strange, contradictory, thing. It both is and is not an object; it both is and is not something.”

So Priest appears to be magicking dialetheisms (as it were) out of the air. Actually, he doesn't accept that all the contradictions he mentions (in various places) are real contradictions (i.e., he denies that logical “explosion” also applies to dialetheism).

For example, Priest rejects the posited contradiction found in Wittgenstein's Tractatus idea that the “form of the world” is ineffable. Why? Basically, because we can both speak about - and describe - it. On the other hand, Priest does seem to accept that “the ground of reality [] embodies another contradiction” because nothingness “both grounds itself and doesn't ground itself” and “it is and isn't an object”. There nothingness is “the contradictory ground of reality”.

Ontologically Dependent on Nothing

Priest then makes this massive claim:

“At the ground of reality you have nothing - this contradictory ground of nothingness. So at the ground of reality there is one enormous contradiction.”

Firstly, why is nothing a “ground”? Indeed what is it for “nothing to be the ground of reality”? And where, exactly, is the “contradiction”?

No wonder Priest then says that this “is good old-fashioned metaphysics”. (It is!) And it's no surprise, either, that Priest also ask: “What to make of the contradictory ground of reality?” And he concludes:

“I'm going to leave [that question] to theologians to make sense of that question.”

So why theologians and not philosophers?

Possible Worlds and Dialetheism

Priest uses possible-worlds speak to justify his dialetheism. Take this example:

“It might be thought that the fact that ¬(A ∧ ¬A) holds at a world entails that one or other A and ¬A fails; but this does not necessarily follow.”

Is Priest saying that ¬(A ∧ ¬A) holds at the actual world (i.e., our world); though not “necessarily” at all possible worlds? Or does Priest believe that only at other possible worlds A ∧ ¬A holds? In any case, the dialetheic position is that A ∧ ¬A is not necessarily false.

Priest offers us the following symbolisation of his position:

¬A is true at w iff is A false at w.

¬A is false at w iff A is true at w.

So what about Priest's A ∧ ¬A?

According to Priest, “it is possible for A to be both true and false at a world”. That is, of course, the dialetheic position. Yet does this position require possible-worlds theory when Priest (elsewhere) has said that it's also applicable at our world – the actual world?

Not surprisingly, Priest concedes that

“it is natural to ask whether there really are possible worlds at which something may be both true and false”.

He also believes that this is a “fair question”. Nonetheless, Priest also argues that

“the best reasons for thinking this to be possible are also reasons for thinking it to be actual”.

That seems to follow from the earlier possible-worlds logic. We can now argue the following:

i) If it's possible for A ∧ ¬A to be true at at least one possible world,

ii) then it's also likely to be - or possibly - true at our actual world.

Is Dialetheism About Reality?

In order to tackle Priest's later dialetheic views on quantum mechanics (see the next section), let's firstly take the words of Bryson Brown as an introduction.

Bryson Brown (in his paper 'On Paraconsistency') stresses “the world”, rather than words. That is, he's stresses ontology, not semantics.

More specifically, Bryson stresses the importance of inconsistency for dialetheism. He also says that dialetheists are “radical paraconsistentists”. He writes:

“[Dialetheists] hold that the world is inconsistent, and aim at a general logic that goes beyond all the consistency constraints of classical logic.”

Deriving the notion of an inconsistent world (or a world which contains contradictions) from our psychological and/or epistemological limitations (as well as from accepted notions in the philosophy of science and mathematics) is problematic. In other words, the epistemological position that we have inconsistent (or even contradictory) positions/systems can't also be applied to the world itself.

Another way to put this is in terms of set-theoretic paradoxes, as also mentioned by Bryson Brown.

Brown says that “the dialetheists take paradoxes such as the liar and the paradoxes of naïve set theory at face value”. That is, it may be the case that dialetheists choose - for logical and/or philosophical reasons - to accept paradoxes even though they also believe that, ultimately, they aren't true of the actual world. Then again, Brown continues by saying that dialetheists “view these paradoxes as proofs that certain inconsistencies are true”. Thus:

These inconsistencies are true of what?

True only of the paradoxes (in themselves, as it were)?

Or true of the world itself?

Again, this stress on the world may betray a naïve, crude and, perhaps, an old-fashioned view of logic. Nonetheless, Priest himself does mention “reality” on a few occasions. When discussing the virtue of simplicity, for example, he asks the following question:

“If there is some reason for supposing that reality is, quite generally, very consistent – say some sort of transcendental argument – then inconsistency is clearly a negative criterion. If not, then perhaps not.”

This again concerns reality. As it is, it's difficult to see how the world can be either inconsistent or consistent. This position is similar – or parallel – to Baruch Spinoza's philosophical point that the world can only, well, be. (Graham Priest is a Buddhist.) Thus:

“I would warn you that I do not attribute to nature either beauty or deformity, order or confusion. Only in relation to our imagination can things be called beautiful or ugly, well-ordered or confused.”

What we say about the world (whether in science, philosophy, mathematics, logic or everyday life) may well be consistent or inconsistent. However, the world itself can neither be consistent nor inconsistent. Thus, it seems to follow, that inconsistency is neither a “negative criterion” nor a positive criterion.

Quantum Mechanics

Priest says that

“those who worked on early quantum mechanical models of the atom regarded the [Neils] Bohr theory [as] certainly inconsistent”.

Priest then tells us:

“[Y]et its empirical predictions were spectacularly successful.”

Priest appears to be hinting at the following:

i) We had an inconsistent physical theory about the world (or about the atom).

ii) That theory led to “empirical predictions [which] were spectacularly successful”.

iii) Therefore it is possible that the world itself is inconsistent. (Or more strongly: The world is inconsistent.)

It must be stressed here that the meaning of the word “inconsistent” is very different to the meaning of the word “contradictory” (or “paradoxical”). Something can indeed be inconsistent because it contains contradictions. Though can't something be inconsistent without also containing (logical) contradictions?

Here's a passage from Priest on an aspect of quantum mechanics that he sees as being relevant to dialetheism:

“Unobservable realms, particularly the micro-realm, behave in a very strange way, events at one place instantaneously affecting events at others in remote locations.”

Priest gives another example of quantum happenings. This example is one of radioactive decay. He writes:

“[S]uppose that a radioactive atom instantaneously and spontaneously decays. At the instant of decay, is the atom integral or is it not?”

Now for the traditional logic of this situation. Priest continues:

“In both of these cases, and others like them, the law of excluded middle tells us that it is one or the other.”

Couldn't the atom be neither integral nor non-integral when it instantaneously and spontaneously decays? (Priest talks of either/or and “one or the other”; not neither/nor.) Or, alternatively, at time t, x may not be an atom at all!

So what of Priest's own (logical) conclusion when it comes to atomic decay? He claims that the aforementioned atom “at the point of decay is both integral and non-integral”. This isn't allowed – Priest says - if the law of excluded middle has its way. After all, the law of excluded middle tells us that the the atom must either be integral or nonintegral; not “both integral and non-integral”.

Note:

1) Some of the quoted words and passages from Graham Priest in the above are taken from the 'Everything and Nothing' seminar – a Robert Curtius Lecture of Excellence at Bonn University - which Priest gave. I relied on both the transcript and the video itself. However, I've edited a lot of what Priest says in that seminar to make it more comprehensible. For example, I removed many of the uses of the word “so”, added full stops, commas and suchlike. Hopefully, the philosophical content is kept intact. None of this applicable to the words and passages I quote which come from Priest's papers.


*) See my ‘Graham Priest — and Martin Heidegger! — on Nothing (1)’ and my 'Graham Priest & Martin Heidegger Take Language on Holiday: the Nothing (2)'.