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Andy SmithAndrew P. Smith, who has a background in molecular biology, neuroscience and pharmacology, is author of e-books Worlds within Worlds and the novel Noosphere II, which are both available online. He has recently self-published "The Dimensions of Experience: A Natural History of Consciousness" (Xlibris, 2008).



A Review of Mike Hockney's
The Mathematical Universe

Andrew P. Smith

According to an old, fairly well-known joke, a clock that is stopped is better than one that loses time—say, five seconds every hour—because the stopped clock is right twice every day, whereas the slow one is right only once every year. Mike Hockney, I take it, would not find this funny at all. On the contrary, he would think it obvious that the stopped clock was better.

“The scientific method,” Hockney argues in his book The Mathematical Universe (2014), “produces successful theories; it does not produce true theories.”(650-652)[1]. And success, in the absence of truth, Hockney seems to believe is little better than no success at all:

Scientific materialism is absurdly incoherent. (653)

Scientific materialism is a joke. It's now much closer to Berkeleyan empiricist idealism than it is to any kind of materialism.(676-677).

Scientists are living in a fantasy world if they think they will ever explain reality.(911)

Science doesn't know what anything is. It's just a set of measurements with labels attached.(915-916).

Scientists – wedded to empiricism rather than rationalism – have evolved beyond the Abrahamists but are still seriously retarded.(1043-1044).

All scientists are inherently irrational.(1062)

Scientists, locked into their sensory worldview…lack the imaginationand intuition to be truly intelligent. It's up to the elite – ontological mathematicians – to put these scientific apes in their place.(1306-1308).

Science is the quasi-religious worship of sensory experience, and the rejection of rational unobservables.(1802-1803).

Science's understanding of the fundamental ground of reality is insane.(4856).

Science is anti-rationalism. Science is as bad and deluded as the other great enemy of rationalism: religious faith.(5187-5188)

The tragedy is that science is now run by fools.(5399-5400)

To many scientists and their supporters, statements like these are sure to inflame. The successful theories of science, after all, have provided us with all of our modern technology, including medicine, communications, transportation, and so on. Hockney's alternative view, which as we shall see is based on reason and particularly mathematics, could not by itself have given us the computers and the internet that allow him to disseminate this view. It would not have allowed us to eradicate many diseases and to extend the average lifespan of members of our species. It would not have given us rapid access to almost any place on earth, as well as allowed us to explore outer space.

Nor is science just about utility, helping us survive and prosper. It has also delivered a fairly coherent picture of our world. As E.O. Wilson described in Consilience (1998), the different areas of science—physics, chemistry, biology, behavior—fit together to describe how primordial atoms evolved into molecules; molecules evolved into cells; cells evolved into organisms; and organisms evolved behavior, including our own. The result is a history, beginning billions of years ago and culminating in modern human societies. For the most part, this story hangs together remarkably well.

So what is Hockney's problem? He is concerned with ultimate questions, and particularly one that science not only has not yet solved, but which many thinkers, probably including some scientists, think science may never solve: how did the universe begin, i.e., how did something arise from nothing? The mainstream scientific view is that the universe came into existence with the Big Bang, about fourteen billion years ago, which created its fundamental constituents, matter, space and time.

Here is the problem with that view, according to Hockney:

What and where were the laws of physics before the physical universe came into being? What laws were controlling the Big Bang as it happened? What laws caused it in the first place? Whatever caused and controlled the Big Bang must, of necessity, have preceded the Big Bang, yet scientists openly say that space, time and matter did not exist prior to the Big Bang. In which case, there's nothing left within the empiricist, materialist paradigm to account for how the Big Bang happened…
No scientist has ever plausibly explained where scientific laws come from, how they interact with mutable, material things and how they exist at all.(565-573).

I think this is a fair criticism. Certainly scientists themselves are aware of this problem. They are also aware of other rather embarrassing deficits if not down-right contradictions in their theories, also pointed out by Hockney, such as the paradoxes and multiple interpretations associated with quantum mechanics, the inability to reconcile this theory with other theories such as relativity into a single grand theory, the lack of any way to test the claims of some ideas such as string theory, and so on. Science is clearly an ongoing, incompleted project that features not just gaps in its knowledge, but implications that sometimes make no sense to any of us.

The question is, is there a better alternative? For example, is there a solution to the problem of something from nothing? Hockney and his fellow Illuminati, as they call themselves, think there is, and that it lies in making reason, particularly mathematics, the primary source of our knowledge, rather than sensory observations:

only mathematics delivers seamless truth since it's based on analytic a priori truths and not on contingent, ad hoc hypotheses subjected to unreliable experimental verification.(653-654).

Human reason – within the strict context of mathematics – is always right and always superior to empiricism.(682-683).

Only mathematics allows absolute knowledge of existence to be attained. (3969-3970)

You simply cannot rely on your senses if you want definitive answers to existence. If you want answers, you must use your reason, not your senses. Mathematics is the ultimate subject of reason. It has no connection whatsoever with the senses. The whole of mathematics can be worked out without ever looking at the world.(4515-4518)

Science, to be plausible when it comes to the ultimate questions, must be based on rationalist mathematics and not on the empiricist scientific method that is incapable of addressing infallible, absolute truths.(4640-4642).

In The Mathematical Universe, Hockney lays out his case in support of this. I found this treatise creative, interesting, provocative and certainly worth the read. It definitely stimulated a lot of thinking on my part, and I wouldn't hesitate to recommend it to scientists, philosophers and anyone else deeply interested in the fundamental questions of our existence. Nevertheless, I think it has some fairly serious flaws, which I will discuss here.

A Total Zero

According to Illuminism, as Hockney's view is called, the world begins with monads, zero-dimensional points that have always existed and are so structured that they contain, in effect, all of mathematics:

Illuminism begins with the simplest possible thing – a single mathematical point. This is the monad, the basic unit of existence. Being unextended, it conforms with Descartes' definition of a thinking mind…it contains all the numbers between zero and infinity in all directions, signs and orientations. These numbers exist in anextremely precise way, guaranteed to produce a net result of nothing so that the point is ultimately defined by the number zero, the inverse of which is infinity.(415-422).

Those with some background in philosophy will immediately see in the concept of monads the influence of Leibniz, whom Hockney greatly respects. In fact, Hockney traces the Illuminist movement back to Pythagoras, the great Greek philosopher and mathematician, through Liebniz and on to the present day, describing it as a mostly ignored view that has been handed down from generation to generation by a small group of adherents. Though the central underlying principle has remained constant—mathematically-based reason is the key to understanding ourselves and our world—the details of the view have been modified somewhat over the generations. Thus Hockney's version of monads is somewhat different from Leibniz's, and in particular, it is that last descriptive statement that Hockney claims resolves the something from nothing problem. As he goes on to expound:

In order for zero to be the inevitable and inescapable net result of the combination of infinite numbers, all of the numbers must conform with the most powerful analytic formula in the whole of mathematics – Euler's Formula, the great jewel of mathematics: eix = cos x + i sin x…

What's so remarkable about Euler's Formula is that it produces perfect balance between negative and positive numbers, between real and imaginary numbers and between zero and infinity. No element is privileged over any other. The net ontological effect of theformula is zero (since the circle's negative half perfectly cancels its positive half), yet this is an “infinite” zero, a structured “nothing” that goes on forever!...

In order to include all possible ontological numbers, it's necessary to introduce a more generalized form of Euler's Formula:

A e i( fx + ö) = A cos (fx + ö) + i A sin (fx + ö)

where A is amplitude, f is frequency and ö = the phase angle (phase shift). In the frequency domain, the three elements necessary to specify a wave are amplitude, frequency and phase, so this generalized formula allows all possible waves to be accommodated. (426-450)

Bringing in waves is essential to Hockney's view of how monads create the phenomenal world, which I will discuss later. But for now, let's return to the something from nothing issue.

A “simple” point is therefore nothing of the kind. It's an infinite information system, based on a superposition of infinite waves of every conceivable permutation, all of which put together produce a sum of zero (total and infallible balancing to zero).

Simply by defining a wave as the mathematical basis and definition of energy, a point is transformed into a repository of infinite, balanced energy. And bear in mind that this energy is necessarily eternal. Euler's circle never stops spinning. Nothing can ever halt it – because ultimately there's nothing there! An Euler circle is simply an ingeniously ordered and structured nothingness that can never perish. It's always rotating and can never stop. Energy is just eternal motion.(451-458)

The foundational, uncaused causes of reality must be mathematical and they must have the property of being “nothing” since nothing is the compulsory rational ground state of existence. Mathematics has the ultimate rational trick up its sleeve – because “nothing” can also be something. It's precisely because something and nothing can be equated (via an equation as simple as, for example, 2 - 1 - 1 = 0) that we are all here at all; that anything is here.(579-582).

What is “nothing”? It's categorically not “non-existence”. Nothing is actually something. Something is mathematically structured nothingness. The generalised Euler Formula is exactly the miraculous mathematical instrument that allows nothing to be structured.(996-998)

So the Illuminist solution to the problem of “something from nothing” is to equate the two. Nothing, properly understood, is “ingeniously ordered and structured”, creating “an infinite information system, based on a superposition of infinite waves of every conceivable permutation, all of which put together produce a sum of zero”.

It's not Sufficient to be Sufficient

In assessing this view, the first thing to emphasize is that this is simply a postulate, no different in principle from the assumption of universal scientific laws. How does Hockney know that this Euler-based system exists? The short answer is that he believes this is the only reasonable scenario:

Since it has only one element, the Euler universe is in best accord with Occam'sRazor. There's nothing outside it. It explains everything. It leaves no gaps, hence obeys Leibniz's principle of continuity. It's the basis of Leibniz's principle of sufficient reason, and of all causality. It enshrines all of Plato's immutable, eternal laws (considered mathematically).(4556-4560)

Leibniz's principle of sufficient reason, which says everything happens for a reason, is especially important to Hockney, who uses it in two ways: 1) to demonstrate that the scientific worldview does not follow this principle, in that no sufficient reason can be given for why there are particular scientific laws rather than others, why that which is observed through the senses should be regarded as real, and so on; and 2) to show that a sufficient reason can be given for the existence of the monads:

Via [Euler's] formula, existence can be maintained at its necessary groundstate of zero (nothing), while always being something. (Any non-zero resultant cosmic energy is forbidden. There is no sufficient reason why the cosmos should have any arbitrary energy, and why such an energy should be above the ground state.)(441-444)

If one monad can exist with no net energy, what sufficient reason could prevent the existence of others, also with zero net energy? In fact, what could prevent the existence of infinite such “nothings”?(465-467)

The first law of thermodynamics (stating that energy can be neither created nor destroyed) is, rationally, a statement that the energy of the universe is always zero (because there could never be a sufficient reason for the energy to be greater than zero, and if the energy of the universe is always zero then it automatically follows that there can't be any more or less of it).(588-591)

I think Hockney is at his best here, arguing against the apparent arbitrariness of the conventional scientific view of energy. Another, even better, example of this arbitrariness is found in the fundamental constants or parameters of the universe, which have to be set within very narrow limits for the universe to exist. No one understands why this is the case, and the fact that this is so upsetting to most scientists and other theorists is basically a validation of Leibniz's principle. We believe there should be a reason for why things are the way they are. Einstein famously objected to the notion of a universe based on a roll of the dice.

Nevertheless, just as there is no empirical evidence that supports the notion that the empirical approach is a valid means of understanding the truth, there is no rational argument that supports the principle of sufficient reason. This is basically an intuitive notion.[2] The universe could in fact be to some extent random and arbitrary, and certainly there is a lot of empirical evidence for the role of contingency in evolution, in general, and in everyday events, in particular. The philosopher Quentin Meillassoux (2008) has even argued that natural laws may result from a contingent process.

Moreover, I want to point out several features in the above passages that open Hockney up to more specific criticism. First, though as I noted earlier, he disparages scientific empiricism, he has no problem making use of its products when this suits his purposes, e.g., the first law of thermodynamics.[3] This is critical to his arguments supporting the existence of monads, even though he argues all along that the insights of science, while useful, are not true. If the first law were ever proved false—and Hockney seems to believe that's possible for any scientific law—it could be quite damaging to his case.

Second, as I noted in passing earlier, Leibniz himself admitted we can't always know the reason underlying some phenomenon. But Hockney, in these statements and elsewhere, presumes that we can. For example, he assumes because no reason can currently be given for the total or net energy being at a particular non-zero value, then it must be that there is no reason. Many scientists, of course, would dispute this. Science has a long history of overcoming apparently conflicting or nonsensical implications of its theories, and many would regard it as premature to jettison some notion just because it currently appears arbitrary or unexplainable.

Finally—and I think this goes to the heart of the limitations of the rationalist approach-- Hockney generally uses the principle of sufficient reason in a negative sense. He does not argue that there is a reason for something so much as that there is no reason for the alternatives. Thus not only must the net energy of the universe be zero because there is no reason for a non-zero value, but there must be an infinity of monads, because there is no reason why there should not be. He might argue that this is not inconsistent with Leibniz, but logically, saying that there is a reason for everything that exists is not quite the same as saying anything exists if there is no reason for it's not existing. I understand that adopting both versions makes for a more elegant, internally consistent view—everything is what it is for a reason, and nothing is excluded without a reason--but in practical terms, it raises problems.

One problem, as I just pointed out, is that it depends on knowing for certain there is no reason for the alternatives, which is generally not the case. But even more, this use of sufficient reason presents a particular difficulty in that it results in an asymmetry in the argument, a different standard of proof depending on which side one is on. The classic example of this is the argument over whether God exists. A skeptic can point out that no one can provide a reason, i.e., proof, of this. This is using the principle of sufficient reason in the positive sense. But a believer can counter that no one can provide proof that God doesn't exist. This is the using the principle in the negative sense, and here and generally elsewhere, it's far more difficult to counter—the famously impossible “proving a negative”. Hockney, I think, believes the two go hand in hand, that the one always implies the other, which in his mind justifies using the weaker negative standard to establish the stronger positive standard.

Science, of course, avoids these problems precisely because it demands empirical proof, or at least evidence. As the quotes I provided earlier imply, Hockney is very critical of the empirical method, contrasting it unfavorably with truth, which he asserts is to be found only in mathematics. He believes sensory evidence to be fallible—as shown by the failure of so many scientific theories in the past. So he would no doubt dismiss any requests to provide evidence in the scientific sense for his theory of dimensionless monads. But it would seem quite…well, reasonable, to ask him to provide mathematical proof of this, and he clearly can't. He can't provide some mathematical equation or series of equations that proves—in a way that the principle of sufficient reason can't--that these monads exist (that, presumably, is why he has written this book instead). So it seems to me that he is in much the same boat as the scientists he so savagely criticizes. Just as scientists presume certain laws that can't be confirmed by sensory observation, so Hockney presumes mathematical monads that can't be confirmed by a rigorous mathematical proof.[4]

The Reason for Reason

In other words, even Hockney implicitly acknowledges that pure mathematics can take us only so far. Not all reasoning involves, or only involves, mathematics, and he must make use of such non-mathematical reasoning in order to build and support his theory. He does claim that “any reason that cannot be linked to mathematics and grounded in mathematics, is rejected.” (3355-3356) But obviously the truth of that statement depends on how one defines “links”. I have just pointed that he can't provide a strictly mathematical proof for the existence of monads. He does, he must, rely on reasoning that consists of more than mathematical tautologies.

This raises an obvious question: how reliable is such reasoning? Is it really superior to sensory observations, as Hockney claims?

In recent years, a view has arisen that provides a serious challenge to this notion. One of its pioneers and most influential proponents, George Lakoff (1999), has argued that reason is embodied, meaning it's not something independent of the sensory world, but derived from it:

Reason is not disembodied, as the tradition has largely held, but arises from the nature of ourbrains, bodies, and bodily experience. This is not just the innocuous and obvious claim that we need a body to reason; rather, it is the striking claim that the very structure of reason itself comes from the details of our embodiment…

Reason is evolutionary, in that abstract reason builds on and makes use of forms of perceptual and motor inference present in "lower" animals…Reason is thus not an essence that separates us from other animals; rather, it places us on a continuum with them…

Reason, arising from the body, doesn't transcend the body. What universal aspects of reason there are arise from the commonalities of our bodies and brains and the environments we inhabit...

Real human beings are not, for the most part, in conscious control of--or even consciously aware of--their reasoning. Most of their reason, besides, is based on various kinds of prototypes, framings, and metaphors.(69-91)[1]

In his book Philosophy of the Flesh, Lakoff provides many examples of how our reasoning makes use of concepts that are derived directly from our bodily experiences. The point is not so much whether reasoning is right or wrong as that it is a way we evolved to interact with the world, just as the use of our sense organs is. In other words, our reason developed not as a tool to find what Hockney regards as the truth of our existence, but simply to help us survive. Hockney wants to put reason on a pedestal, towering above the senses, and uniquely able to penetrate to the core of all the ultimate questions. But that wasn't why our reason evolved, and there is no guarantee that it's capable of doing this.

Nor is reason ever capable of accomplishing anything on its own. Reason, associated with the frontal and associative areas of the cerebral cortex, was a relatively recent evolutionary addition to a brain driven, in lower vertebrates, much more by emotion. Our own brains, of course, have retained these emotional areas, such as the limbic system, and as neurologist Antonio Damasio has argued (2005), these and some newer ones are critical to the exercise of reason. Without emotions, there would not be a reason for reason—i.e., there would not be a goal which reason might help us attain.

Hockney is clearly enormously passionate about his view of monads. This passion is what drives him to write books about it, and to fill those books with pleas to the reader to understand what he's saying. But passion is not reason, it's emotion. Reason does not care what the reader does or believes. Reason can help us evaluate the consequences of different courses of action, but emotion is critical to the actual choice we make. Reason can tell us that 1 + 1 = 2 is true and that 1 + 1 = 3 is false, but only emotion can tell us that truth is preferable to falsehood.

So reason, like sensory observations, is based in the brain, and potentially fallible. But Hockney nevertheless regards it as something we can potentially achieve a universal consensus about, and argues that this is not possible for sensory observations:

The sensory “fact” that the sky is blue is not proof that the sky is blue. A blind or colour blind person would have no ideawhat you meant if you referred to “blue”. (1795-1796)

The use of this argument is ironic. Hockney wants to show that empirical observation is fallible, but to do it, he has to use…another empirical observation. What he doesn't mention here is that the very fact that we can tell that some people are color blind is a triumph of the empirical approach. We can do this because this approach is not based on the evidence of one individual, which may be fallible, but on the evidence of many individuals. It's through this corrective factor that we can not only meaningfully describe the sky as blue, but explain why it may not appear that way to some people.

And reasoning works in much the same way. We don't accept the “reasonable” or “logical” conclusions of one person. Their thought processes have to be replicated. Just as there are people who can't see the color blue, there are also people who can't reason logically. Many studies have demonstrated, in fact, that the majority of people perform quite poorly on some rather simple tests of logic and reasoning (Kahneman 2011). Hockney is well aware of this; in fact, he despairs at the fact that most people's minds are mired in mythical or otherwise flawed worldviews just because they lack the ability to reason well. But what he doesn't seem to grasp is that the standards that tell us this don't come primarily from one or a handful of geniuses like Leibniz. They come from a broader social process that is very much like the one used to validate sensory impressions.

Hockney continues:

Moreover, you can have no idea whether your perception of blue is the same as anyone's perception of blue.(1796-1797)

True, but so what? What counts is consistency. If everyone can agree to call certain shades of color blue, the actual experience they are having is irrelevant (not to mention incapable of being determined by mathematics, either). And mathematics is not really so different. If someone agrees with me that 1 + 1 = 25, it doesn't matter if his experience of that equation is very different from mine. In particular, the concept of “equal” could, for all we know, be experienced in profoundly different ways by different people. But as long as we use it in a consistent manner, it doesn't matter.

So I think Hockney fails to appreciate that the distinction between reason and empiricism is not clear-cut. Both are dependent on a process of achieving consensus among individuals. This process frequently involves taking something on faith or trust, something that Hockney again ascribes only to empiricism:

Does the fact that the Higgs boson has purportedly been discovered make you think you are one jot closer to the ultimate truth of existence? You certainly can't directly sense the Higgs boson. You have to take its existence on trust – trust in CERN's Large Hadron Collider and the scientists and engineers operating it and interpreting its output.(775-777)

Has Fermat's last theorem been proven? I think so, but I have to take the proof on trust, as I doubt I'm capable of understanding it. Knowledge of all kinds has become so sophisticated that very few people can directly verify any new finding of any importance. Certainly demonstrating that Euler's formula could actually create the world as we see it would require an exceedingly powerful and complex mathematics that very few people would be capable of understanding.

The bottom line here is that most of the knowledge we have about the world beyond us is social: it depends on individuals agreeing on certain things, and in trusting in many agreements in which we don't directly participate. This knowledge may exist on a spectrum, with mathematics the most certain of all. But there are many sensory-derived facts that are as universally acknowledged as being true as any mathematical equation: that the sun rises every twenty-four hours; that humans and other organisms require certain substances external to their bodies to stay alive; that we become unconscious and dream at night; that everyone eventually dies.

Even these facts are not eternal. They could change some day. But so can reason. Hockney expresses glee in pointing out the long history of failed scientific theories, as evidence (empirical, of course) that empiricism is flawed. But how about the even larger graveyard of failed philosophical theories, all based on the supposedly infallible reason?[6] Philosophy is actually worse than science in this regard, because while failed scientific theories are generally abandoned for good, with the entire scientific community eventually moving on, philosophers continue to argue even today over ideas expounded centuries or thousands of years ago. If reason is so superior to empirical observations, why is it impossible to come to a consensus on so many of its conclusions?

To conclude, reason, like sensory observations, is a product of the human mind, and the human mind is not only fallible, but subject to change. Hockney claims he can ground all his reasoning in mathematics, but he manifestly cannot. This point will become even clearer later, when we examine his ideas about the space-time world. Moreover, as I will also discuss later, even pure mathematics is not necessarily immune in this regard.

Design Flaws

So one problem with Hockney's Illuminist view is that it's based on an unproven—perhaps even, as Hockney himself triumphantly emphasizes, from an empirical point of view, unprovable—assumption: that such monads as he describes actually exist. He believes their existence can be proven purely by reason. They uniquely explain how something can arise from nothing. Science does not accept such reasoning as a proof. It might accept a rigorous mathematical proof, but such is clearly not possible, in much the same way that no sensory observations can prove the metaphysical assumption that only phenomena that can be observed with the senses are real.

However, just because the existence of these monads can't be proven in a scientific sense doesn't mean we should ignore the possibility. As I conceded earlier, their existence would seem to avoid many of the arbitrary elements in the current scientific worldview. So let's examine the idea a little more closely. Is it coherent? Can we actually imagine monads of the kind Hockney postulates?

Hockney likens these monads to computers, unconscious information processors. Maybe this is a reflection of my impoverished intellect, but I find it very hard to understand how anything remotely like a computer could be said to exist in a dimensionless point of balanced nothingness. We can certainly conceive of a computer that contains the Euler equation, and is programmed to convert various numbers, or values of x, into various output waves. But would we regard this computer program as a balanced nothingness? No. Such a program reflects a complex organization of information. In the world we are familiar with, such a program only comes to exist through material processes that have work, expenditure of energy, done on them.

At first glance, Euler's formula is relatively simple, considering what it can accomplish. After all, it can be written in a single line, and contains only a few terms. But that doesn't mean it's simple. If it were relatively simple, like 1 + 1 = 2, every individual on earth of a certain age, from the beginning of recorded history, would be familiar with it. In fact, it required one of the greatest mathematical minds of all time to formulate it. When we consider how complex the human brain is, the kind of ideas it's capable of creating, then realize that only one out of hundreds of millions of such brains ever grasped this equation on its own, and then only after long, hard work—we have a crude but I would say highly relevant measure of how fantastically complex it actually is. Hockney himself describes it as “the most complex object in the universe”. (4430) I myself would not go that far, but everyone should be able to agree it's not as simple as it appears.

In other words, Hockney's view of the monad smacks of intelligent design. Maybe it's not in the same league as a God who waves a magic wand and creates the earth and every species of life on it, but neither is it an obvious improvement over the exquisitely balanced parameters or constants that have to exist in order, according to the scientific worldview, for the universe to be possible. Hockney himself in fact describes it as “unconscious intelligent design.” (3429)

Hockney is no fan of the traditional view of God, or of what he refers to as conscious intelligent design. The standard argument against God, or an intelligent designer, which Hockney himself repeats in this book, is to ask, where did God come from? We might ask of him, where did Euler's equation come from? His reply is that as a mathematical truth, it's eternal. This response can't be dismissed as easily as the response that God is eternal. Unlike a conventional superintelligence, this equation does not seem to require a long evolutionary process. To many intelligent minds, this equation does seem to be something that is just there, which has always been there.

But this is not something that can be said with any certainty. Euler's equation, as far as we know, can only be understood or visualized by a human brain, which certainly is not eternal. It applies to circles, idealizations of real world phenomena, and would make no sense in a world in which such phenomena didn't exist. The fact that it has been true for all of our existence does not establish that this truth is eternal and independent of our existence. This is a point I will return to later.

Moreover, the principle of sufficient reason, which Hockney uses to rationalize the existence of his monads, is not necessarily his good friend here. This principle says everything exists for a reason, which today we would interpret as a cause. Yet Hockney views these monads as eternal, hence uncaused. Leibniz himself would not view this as a problem, because by reason he meant anything accessible or potentially accessible to the human intellect. And Hockney is saying, if I understand him correctly, that the reason for the existence of monads in this sense is that they are the only thing that can reconcile the issue of something from nothing. Using reason, we can conclude that monads must exist.

But if the monads satisfy this requirement, of existing because we, someone, can figure out a reason for their existing, doesn't this reason imply a cause? If the monads exist because they uniquely balance something to zero, isn't this a cause? If they didn't achieve this balance, they wouldn't exist.

This might seem like stretching the definition of a cause, but consider the alternative. If the need to achieve a balance is not regarded as a cause, then the monads must have come into existence through pure chance—which, to Hockney, is the devil incarnate. All we can say is that it just happened that way. In this case, there is no more sense to the scheme than to the scientific view of precisely tuned fundamental constants.

In other words, we have two choices here. On the one hand, we can regard reasons as real things, substances, which are capable of causing the monads to come into existence. Then we have to ask what causes reasons, or if they are eternal, how we reconcile this with nothingness.

Alternatively, we can regard reasons as just thoughts that pass through our mind, which point to but don't themselves participate in reality. In this case, they don't cause the existence of the monads, but by the same token, neither can they be taken as infallible indicators of the existence of anything. That being the case, the existence of monads is, as far as we can tell, either the result of chance or some cause which is currently beyond our understanding.

Hockney, as one would expect, subscribes to the former view:

The world did not arise through chance. It did not randomly leap out of non-existence. It arose out of reasons that are eternal.(3430-3432)

How, then, are these reasons—which in Hockney's system are much like scientific laws—reconciled with nothingness? The point of hypothesizing a zero-dimensional monad was that it could balance all numbers to zero. But now, it seems, that in addition to Euler's equation and all these numbers, certain reasons also have always existed. Don't they affect the balance?

What Counts?

To summarize the discussion so far, one problem with Hockney's conception of monads is that they seem too good to be true. The world begins with a very complex equation that is capable of balancing numbers to a net zero. This seems like intelligent design, and Hockney freely admits that it is. It appears to be an improvement over the traditional view of God, in that the monads reconcile something with nothing, but at the price of postulating something, the Euler equation, that is exceedingly complex and which therefore seems highly improbable.

Moreover, Hockney's monads don't contain just this equation. They also contain all possible numbers, real, irrational and imaginary, which function as the inputs into the equation, which in fact is not just an equation, but a sort of computer program. In a real computer, such inputs don't just float around freely. They have to be separated from the program, in a distinct storage compartment, with another process responsible for inputting them into the program in an orderly manner. This requires further organization. Except that all this has to occur in a dimensionless point, where there is no space, and therefore not even the possibility of separation.

This raises further problems. Recall that all the numbers have to balance to zero for the entire ensemble to manifest as zero or dimensionless. But this is only the case if the Euler program is operating on all the numbers simultaneously. If it were to operate on only one number at a time, or even some numbers, there would be a net, non-zero outcome. How does the program operate on all numbers at once? In a conventional computer, this would require multiple programs, in fact an infinite number of them.

Perhaps this is what Hockney envisions. Or perhaps, since there is no time as well as no space in the monad, the program by definition is operating on all numbers simultaneously. However, according to Hockney, the phenomenal, dimensional world of space-time results when some of these outputs are “emitted” from the dimensional point. Why wouldn't this disrupt the balance, since some values are now being treated differently from others? This is a point I will return to later.

Finally, since the Euler equation itself is not a number, wouldn't that upset the balance? After all the numbers have been converted into outputs, which sum up to zero, we still have the equation itself. I have already pointed out that its simplicity is deceptive, that it's highly complex. But from the point of view of balancing to zero, even the simplest equation in the world, even 1 + 1 = 2, would create a problem. It is not a number, so it can't balance out with the other numbers. Hockney seems to treat the equation itself as zero or invisible, making no contribution to the overall balance other than through its processing of numbers, but what allows him to do that?

All of these points might be regarded as technical or practical issues associated with the monad-as-computer analogy. But there is also a deeper, conceptual problem with this view. Information carries knowledge or meaning, but to do this implies an observer, apart from the information processor, that interprets its output in a certain way. Computers can be said to process information because their output—for example, numbers printed on a tape, or pixels flashed on a screen—is meaningful to human beings. In the absence of humans, what a computer does can be described simply as electrons moving through transistors. There is no information whatsoever in this movement apart from the human interpreter.[7]

In the case of the human mind, information is understood to be processed through the activity of large ensembles of densely interconnected neurons. The interpretation process must be built into the system, that is, in the scientific view, it must consist of other neuronal processes, and how this is possible is a vexing problem. But we can say fairly confidently that numbers, for example, as well as various mathematical manipulations involving numbers, must be represented by activity in various neural networks.

This understanding, however, applies only to a conscious human mind. What happens in an unconscious mind or nervous system? An interesting possible example, which I have discussed (though not in connection to the problem of information) elsewhere[8], is provided by the desert ant. This species lives in a mostly featureless terrain, and when it leaves its nest to forage for food, often traveling long distances (relative to its size), it must have a way of finding its way back home. Studies have shown that it uses a process called dead reckoning. From its position with respect to the sun, the ant determines the direction it's moving at any moment, and apparently from the movement of its legs, it also estimates the distance it moves in that direction. Though it frequently changes direction as it forages for food, in a sort of random walk, this information is constantly updated, and stored as a series of vectors (this distance in this direction, another distance in that direction, and so on) in the retina of its eyes. It then processes this information into a single vector sum, which, when it's ready to return home, can be used to provide a straight line route back to the nest.

We don't know for sure, of course, that the ant is unconscious, has no experience of itself and its surroundings, but most scientists would presume it to be, and even if it isn't, we can for the sake of argument imagine it being so. If it is unconscious in this sense, can we still speak of this information as meaningful? After all, an unconscious ant, unlike a conscious human being, is not interpreting this information. It is not saying to itself, I am here, I am here, I want to go there, and so on.

The answer is yes, the information is still meaningful, but only because the ant is distinguished from its environment, with which it's interacting. This interaction forms a framework within which particular kinds of activity in its nervous system correlate with particular positions in the environment, and in turn with particular behavior by the ant. One might describe this as unconscious meaning. The meaning is not experienced, but it is implied in the process of sensing, or picking out, certain patterns from the environment. In the same way, we can say that other insects, such as honeybees, that can distinguish different colors, may not experience these colors consciously—they may not have the qualia of red or blue, for example--but still are engaged in a process that makes the colors meaningful. And more generally, we can say that there are many important examples of unconscious meaning or information—e.g., DNA—that follow the same principle. The nucleotide bases in DNA, in triplets, pick out specific amino acid-transfer RNA complexes. This is what makes the information in DNA meaningful.

So we can speak of information processing occurring in an unconscious system, but only if that system is interacting with some other system. But a zero-dimensional monad of the kind Hockney describes is not interacting with another system. Everything is contained—if even that is the right word for a zero-dimensional point—within the monad. So it becomes highly problematic to speak of information in such a system. Information relative to who or what?

When Hockney says that the monad contains every possible number, he is assuming that numbers have an independent existence, that they have meaning in the absence of any relationship to anything beyond them. In other words, that their meaning is inherent in them. But this is not how we understand numbers in the familiar world. They only become numbers in relation to other things. Numbers (much like words, in fact) are defined, in the first place, in relationship to each other; the concept of two only becomes coherent if we have a concept of one, and so on. These relationships are, as Hockney notes, tautological. But more important, and less apparent, numbers are also defined in relationship to other processes.

Return to the computer again. Any number can be represented by a computer process, but the key word here is represented. What we call the number four, for example, is just some movement of electrons in transistors, generally coupled to some more user-friendly display, to which we have arbitrarily given this definition. And this definition is only possible if there is a conscious observer who defines the process as meaning the number four. In the case of an unconscious computer, such as an insect, the number four is represented by some pattern of neural activity, which becomes meaningful by correlating with some feature or process in the environment. In a world where there were only such unconscious organisms, it might be difficult to speak of the number four, but as human beings considering such a world, we can identify some process that in effect does function to represent that number. There is a sense in which we can say that the number four exists in that world.

Of course, the number four can be represented much more directly, such as by four objects, such as four pebbles, as our pre-literate ancestors might have done. One might argue that in the absence of conscious observers, we can still speak of the number four existing in that world. Most scientists accept that in the early universe, there were atoms that existed in some kind of finite number; at some particular location at some particular time there could have been four atoms of some kind. But this is still a representation process. The number four is embodied in material objects, which are doing the representing.

In the zero-dimensional monad, there is nothing, no object and no process, that can represent the different numbers. Hockney seems to think these numbers can just float around—sort of like those cut-outs found in some cartoons--made fully meaningful by some unconscious mental process. But as we have just seen, mental processes, conscious or unconscious, involve representation, and if there are only numbers, there is nothing to represent them.

Hockney might reply that I am assuming that mental processes result from physical ones. In his view, everything is mental, so information processing is not embodied in anything. His theory is a form of idealism, and I will discuss it from this point of view later. But even a disembodied mind must function through some kind of representation, that is, meaning or information must emerge from a relationship of one thing to something else. “4” is not a thought; a thought is about “4”. This is how our minds—the only possible model for these monads that an idealist can use—work.[9]

To summarize this section, the monads as Hockney describes them present several problems, both practical and conceptual. Practically, it's difficult to see how they could process numbers in a way maintaining a necessary balance of zero. Conceptually, they are supposed to contain numbers, but there is no process within the monad that can represent these numbers.

An Illusionary Illusion

The monads as Hockney envisions them are, initially, unconscious minds existing in a zero-dimensional point. In Kantian terms, they are noumena, the reality underlying the phenomenal world of appearances. How do we go from them to this familiar phenomenal world? According to Hockney, this happens through Fourier transformations, producing an infinite number of sine and cosine waves, which can combine in myriad ways to form any conceivable pattern:

Joseph Fourier asserted that it was possible to expand any arbitrary function in the form of a trigonometric series. Any ontological pattern can be made by adding, or superposing, sinusoidal basis waves. That is, any complex pattern you will encounter in the real world can be broken down into a collection of simple sinusoidal waves…
The whole mystery of existence is contained within Fourier mathematics because it's none other than the means by which unextended Cartesian minds (frequency domains) communicate withextended Cartesian bodies (spacetime entities)…
Monads are composed of nothing but eternal, immutable sine and cosine waves of every conceivable type (via the generalized Euler Formula)…
Via inverse Fourier transforms, thismonadic frequency information can be combined to create any spacetime representation. In other words, mental “ideas” can be converted into physical “bodies” via Fourier mathematics. Mind is the basis of matter, not the other way around (as scientific materialists have always claimed). The phenomenal world is simply a mathematical way of presenting noumenal frequency (mental) data.(819-844).

Fourier mathematics (as recognised by the Illuminati alone and not by the general mathematical community, and not even by Fourier himself), turned the Cartesian unextended, thinking domain into an eternal monadic frequency domain (the Soul Domain), and the Cartesian extended, material domain into a spacetime domain (the World)…The miracle of existence is the miracle that Fourier mathematics can present the same information in two radically different ways.(4041-4049)

So with the right input, Euler's formula can produce any particular sine or cosine wave, and through various combinations of these waves, any pattern at all can be created. For example, even something as complex in shape as the human body could be created, at least visually, through an exceedingly complex combination of such waves.

How, though, does this Fourier transformation, converting mental frequencies to space-time patterns, actually occur?

In physics, there's a crucial distinction between bosons and fermions, the two fundamental classes of particles. The bosonic wavefunction is said to be symmetric with regard to particle exchange (meaning that infinite bosons can occupy the same quantum state). The fermionic wavefunction on the other hand is antisymmetric with regard to particle exchange (meaning that no two fermions can occupy the same state). If there are infinite monads all occupying the same Singularity, they can all be regarded as mental bosons. However, the application of a simple antisymmetry operation converts them into mental fermions. This has the most astounding consequence: it confers unique coordinates on each of the monads and instantly creates an extended Cartesian coordinate grid.
The monads haven't actually moved anywhere – they are still inside the Singularity – but they now have unique identifiers (coordinates) and this produces the effect (illusion) of all monads now being separated from each other. To put it another way, they now have extended (fermionic) relations with each other. A Cartesian extended world has come into being. Yet the remarkable thing is that it's entirely constructed from points (from minds) and their mathematical relations...

The infinite Cartesian grid is still wholly contained within the Singularity and is thus a mathematical illusion. Existence can only ever take place within a single point (everything that exists is contained within the Singularity: we live inside the Big Bang Singularity, and the Big Bang itself took place within that Singularity, and remains contained within it). At no point does anything ever leave. The Big Bang was nothing but an internal mathematical restructuring of the Singularity via an antisymmetry operation andFourier mathematics. (477-508)

So according to Hockney, the phenomenal world is an illusion. Nothing actually exists but minds, which are dimensionless monads. Our familiar world of bodies moving about in space and time are created by these minds.

But this presents an obvious problem. In order for an illusion to exist, or occur, there has to be a perceiver, an experiencer. But Hockney has previously described the monads as initially unconscious, and compared them to computers. How can unconscious forms of existence experience an illusion? How can they experience anything at all? And if they can't, who or what is experiencing this mathematical illusion?

Hockney, then, seems to run into the same problem he argues is a fatal flaw of science. According to him, quantum mechanics is nonsensical, with its implication that an observer is necessary to collapse the wave function. There were no observers before humans evolved, certainly none before life evolved, so how can we conceive of the early universe of atoms coalescing into galaxies, stars and planets?[10] But conscious monads, it would seem, are necessary to bring the phenomenal world into existence, and initially they aren't conscious.

Hockney, I think, is claiming that the transformation can occur unconsciously, which is why he appeals to the boson-fermion relationship. Bosons and fermions are certainly presumed to be unconscious. But Hockney's use of this relationship is just a metaphor, and hardly capable of doing any heavy lifting in his theory. Other than the fact that bosons in the same quantum state can occupy the same space, and fermions can't, the analogy fails. According to our scientific understanding of them, bosons and fermions have no mental properties; certainly bosons are nothing like the monads that Hockney describes. It's true that some kinds of bosons, in theory, can act like dimensionless points, but they generally don't behave this way under experimental conditions. Nor can bosons of this kind transform themselves into fermions in the way that Hockney describes unextended monads forming extended space time. Moreover, bosons, in the view of physicists, are not privileged as real in contrast to unreal fermions.

If Hockney is using this analogy just for illustrative purposes, to provide an idea of what his transformation entails, I have no problem with it. But if he is arguing that this provides evidence or support for his view, I have to disagree. And even if he wants to use it in this way, it does not provide a reason to believe that the space-time world is any less real than the zero-dimensional world of the monad.

I want to add that his scheme also does not answer the question of why the particular space-time world that we experience is created. Beginning with an infinite number of waves, any type of form can be created. Why or how, then, did a world with stars and galaxies and planets, and a planet with particular lifeforms, come into existence? For someone who strongly objects to what he considers the arbitrary or contingent nature of some of science's conclusions, because they don't conform to the principle of sufficient reason, he is remarkably silent about this.

Thinking is Faster than You Think

Hockney's claim, then, is that unconscious monads create an illusionary world of dimensions. In order to experience this world, they must become conscious. How do they do this? What mathematical formula or transformation allows or ensures that this happens? Hockney—who, recall, claims to reject all reasoning not grounded in mathematics--provides none. Here is what he says:

Standard consciousness is mind filtered through the prism of space and time. Space and time can have the effect of slowing down and localising thought, eventually to the extent that it can reflect upon itself. ..Consciousness arises when thought is slowed down, allowing choices to be pondered and made. Unconscious animals can never ponder choices. They always act instantly – by programmed instinct.(3712-3716).

This is a really strange view, bordering on the nonsensical. He has emphasized earlier that only mind, in dimensionless monads, is real, and that space-time is an illusion, created by a Fourier transformation. He never explains how monads, initially unconscious, can experience an illusion. But now, he claims they become conscious by “filtering” through the illusionary world? So consciousness is an illusion? The only thing that is real is the unconscious monad? The perfectly evolved human being is the one who is most deeply immersed or “filtered” in the illusion?

Even setting these contradictions or inconsistencies aside, Hockney is conflating consciousness, as a general phenomenon, with one type of consciousness. Human reflective consciousness is relatively slow, but as appreciated through the work of Daniel Kahneman (2011), it's generally accepted that there is a faster type of thinking as well. The experience of emotions, which also occurs consciously, is likewise very fast, as is the experience of simple sensory impressions, or qualia. The last could be said to be the most basic form of consciousness, probably experienced by many other animals than ourselves, as well as by babies and young children, who have not yet developed the ability to reflect.

So clearly the experience of consciousness by humans is not always a slow process. While Hockney contrasts it with the instinctive, unconscious behavior of animals, which he regards as much faster, humans—as is typical of evolution of the brain—have preserved the earlier fast forms while adding a slower form. For example, our emotional response to certain patterns or triggers in nature is very much like that of lower vertebrates, though the latter may not experience conscious emotions at all.

Nor does consciousness necessarily depend on the experience of time and space. As I have discussed in detail (Smith 2009), there is a form of consciousness, that I refer to as zero-dimensional, that does not involve the experience of time, space or separate objects. Evidence for this is provided not only by the reports of mystics, which many scientists regard with considerable skepticism, but also by the behavior of certain very primitive organisms, as well as by newborn infants. While consciousness of this sort may never be purely zero-dimensional, that is, completely free of the experience of space, time and distinct objects, its existence shows that conscious experience does not require a well-developed sense of these parameters. It certainly strongly suggests that some kind of consciousness may be completely independent of them.

Finally, conversely, the perception of time and space may not necessarily involve conscious experience. A large body of evidence demonstrates that a great variety of organisms, including some fairly simple invertebrates, can sense, distinguish and respond to events that occur in time or space (Smith 2009). But they don't necessarily do so consciously. Indeed, most of our own behavior is unconscious, and yet responds to events in time and space. There are also neurological patients, for example, those exhibiting blindsight, who can respond to events in time and space without any conscious awareness.[11] So the distinction between consciousness and unconsciousness can't be made on the basis of sensitivity to the dimensional world of time and space.

Thus Hockney's proposed explanation for how monads become conscious fails on a number of levels, in large part because it's based--not surprisingly, given his acknowledged contempt for empiricism--on a poor knowledge of relevant empirical facts. Frankly, it sounds ad hoc, something he made up on the fly in an attempt to fill in critical details of the theory. There are yet other problems with it that I will discuss shortly.

But in concluding this section, I want to emphasize that even if one accepts this view that consciousness results from this “filtering” process, it still would not solve the mind-body problem, as he claims. Hockney views the mind-body problem as one of reconciling unextended or dimensionless mind with extended, dimensional matter. It's true that it was framed in this manner by Descartes, and this is why, for example, Hockney thinks the boson-fermion analogy is a good model for understanding how mind and matter could interact. But the mind-body problem today is understood as reconciling qualia, the raw experience of consciousness, with insentient matter. And his “filtering” scheme does not even address this issue, let alone propose a solution to it.[12] Nowhere does he explain how the slowing down of mind would result in qualia—and as I just pointed out, it can't, since qualia can be associated with much faster forms of consciousness. Granted, the problem of consciousness is perhaps the most difficult of all problems in science, and no one else has an answer, either. But most philosophers and scientists at least have the perception to understand that they don't, and the humility to acknowledge this.

Interactions, not Inner Actions

So far, I have criticized Hockney's Illuminist treatise based on aspects of the view that are specific to it. But Illuminism is an example of a more general class of theories, known as idealism, and must deal with criticisms that apply to all members or examples of that class. Most theories of mind are either materialist or idealist.[13] Materialist theories, of which the current scientific worldview is the outstanding example, regard the world around us as real, and existing independently of our existence. Idealism, in contrast, views this world as in some manner a creation of our minds.

Support for idealism has traditionally come from the basic fact that we can know nothing about the world except through our minds. The entire world that we experience outside of our minds could be an illusion, idealists suggest, created by the mind. Hockney follows in this tradition when he argues that reason, and particularly mathematics, is the key to understanding existence. Again and again and again in this book, he drives home the point that empirical, sensory evidence, which is used to construct scientific knowledge of the world, is fallible, has frequently been found to be wrong.

I have already suggested one flaw with this view. Hockney's position presupposes that we have a way of understanding the world, through mathematics-based reason, that is independent of our sensory impressions, indeed independent of the world of time and space. But he has no way to prove this through purely mathematical reasoning; he must appeal to a form of reasoning that does not involve mathematics. As I discussed earlier, a growing body of evidence supports the conclusion that reason in this sense, as much as sensory observations, is a product of the body and material brain. Reason did not evolve so that human beings could understand the ultimate truth of existence. It evolved because it helped us, along with our senses, survive in the world. That being the case, the same arguments that are used to support skepticism with regard to the truth of sensory observations apply equally to reason. Both are products of a fallible mind.

Even the notion that pure mathematics is eternal and exists apart from our sensory existence is highly debatable. Hockney makes much of the fact that mathematical equations are tautological, true by definition. From a purely practical point of view, of course, this tautology greatly limits their usefulness. We could not have gotten anywhere in the world if the only knowledge we relied on was of this kind. Hockney's worship of mathematics reaches absurd levels when he says “We have no need of experiments, except as a secondary check.” ( 5378-5379).

But an earlier discussion highlights a deeper issue. All knowledge or information, including that of mathematics, presupposes an interaction of the phenomenon to be known with either a conscious observer or with an environment, in order to provide meaning. That being the case, we can question whether even mathematics can be understood as completely independent of time and space. Hockney, observing that mathematical equations and proofs have stood unchanged over the course of history, argues that they are eternal. But how are we, mortal beings whose species has existed for only a tiny sliver of time on earth, to judge that anything is eternal? Just because mathematical proofs that were developed thousands of years ago are still valid today does not establish that they are eternal. They may simply reflect something about the structure of the human brain that cognizes them, something that hasn't changed in all that time, but which could change in the future.

So one problem with idealism in general is that it presupposes an infallibility of the human mind that simply isn't supported by the evidence. In addition to this point, though, there is another serious problem, one that is more commonly raised. If everything is created by the mind, why does every mind experience more or less the same world? How is it that I see the same things you do, or at least see a world that is consistent in the same way that yours is? Not only can two human beings agree to a very large extent on what they see in each other's presence, but they can agree that what they see continues to exist when they are no longer in its presence. To reconcile this with idealism, it's necessary to postulate that every individual mind is not just creating the same world—that's difficult enough to imagine—but creating it in a way that allows it to persist even when they aren't experiencing it. In other words, a world that supposedly requires mentation to come into existence continues to exist even when no one is thinking about or experiencing it.

What is Hockney's solution to this? Again, he fails to provide a transparent answer:

all minds collectively create a collective, objective material world – the one we all inhabit!

Matter is nothing but a collective rather than individual mental output. Matter is what we get when we all dream collectively rather than separately. We all have two dream states: private dreaming (when we are asleep) and public dreaming (when we are “awake”).(2593-2596)

This is obviously nothing but hand-waving. How do the collective minds produce the same world? Where is the math, or even a modicum of reasoning, to provide the details of the process?

Hockney uses dreaming as an example of the private world of monads, but this is a faulty analogy. The difference between the dream world and the so-called waking world is not simply private vs. public. Dreaming represents an entirely different state of awareness, in which one mind does not interact with other minds. A collection of dreaming monads can't produce a shared world.

So Hockney would need to modify this account, to explain why a single monad—which remember, contains Euler's equation, along with every possible real and imaginary number—produces by itself a world in which time and space sometimes exist, but are frequently warped. Then he needs to add another account explaining how the same individual monads, can, in effect waken, so that they are in a state where they can interact with other awakened monads, and create a shared world of time and space. What mathematical equations or transformations distinguish these two kinds of individual monads?

But this is not the most serious problem with his view. As just noted, for a shared world of time and space to appear, the individual monads must interact. In Leibniz's philosophy, the monads were described as “windowless”, and incapable of such interaction. Hockney, however, describes them as “windowed”, meaning they can interact. According to him, here is how they do this:

All individual monads are able to release a small amount of their energy (a low energy band) into the shared space – the Cartesian grid formed by all active monads – and the whole material universe we experience is simply the interaction of all of that mind energy in an arena of mathematical extension.(508-511).

Again, we see that he does not provide a coherent explanation of just how this interaction results in a shared world. But here is the additional problem. Hockney also emphasizes that these monads are uncaused:

All monads are uncaused causes. They do not depend on anything else for their existence (they have no Creator), hence they are not determined by any causal chains. That means that they are causal initiators (causal agents), but themselves are inherently uncaused. That is what it means to be free, to be capable of exhibiting free will and free choice.(519-522)

How can monads interact with each other if they are uncaused? The word “interaction” presupposes cause and effect.

I think what Hockney means is that the monads cause certain events that result in the space-time world and all phenomena occurring within it, but are not themselves affected by these phenomena. But they clearly are affected by them. If the public, shared world of space-time is experienced by monads differently from their private experience, how are the monads said to be not affected? And what does “affected by” mean if not “the effect of”, i.e., the result of some cause?

Recall that monads are initially unconscious, becoming conscious only through “filtering” through space-time. How can one understand such filtering except as a cause of consciousness? Before the production of space-time, there was no consciousness; now there is. How is space-time not a cause of consciousness? Remember that space-time results from a collective effort of all the monads, their “public dream”. So when an individual monad experiences this public domain, how is that experience not caused in large part by the other monads? Since monads are described as causal agents, the space-time world must be an effect caused by them. How then can its effects on each individual monad not also be the result of causes?

Out of Touch with Unreality

This problem of causality becomes acute when Hockney discusses evolution of the monads, a process that involves reincarnation:

Our journey to divinity actually relies on reincarnation – on continual physical death followed by rebirth. Our bodies die, our minds do not. If you stopped your body from dying you would stop the natural progression and evolution of the mind and remain fixed forever in a low-efficiency state while those who accepted bodily death would keep advancing and transmuting until their minds were capable of divine expression. Nothing could be more foolish – and show less understanding of true reality – than atheistic transhumanism. Bodily death and reincarnation is nature's greatest gift to us because it's the sole means by which we can become Gods and fulfill our destiny.(882-887)

If I understand this correctly, individual monads become increasingly conscious and rational by means of their “filtering” through the world of space-time. When an individual dies, the monad, in effect, loses contact with the space-time world, yet it retains memories of this world, making it more rational than it was before. When it's reborn into a new individual, therefore, it starts out more rational than it did in its last reincarnation. So through a process of repeating births, deaths, and reincarnations, the individual, though trading one body for another, becomes increasingly more rational, more evolved.

How in the world can this process possibly be understood except as monads being subjected to cause? They start out unconscious. Together they create a world of space and time. Somehow, through their interaction in this world, they become more conscious, and this consciousness is a permanent change, one that persists even after the monad loses touch with this world. By any other name, this is cause and effect.

So while I can understand regarding the initial existence of the monads as uncaused, I don't see how their subsequent history is consistent with this. If they are truly beyond space and time, then they should be independent of the causal world, but their interactions with it that Hockney describes simply are not consistent with this. He seems to want to have his cake and eat it, too. He wants them to change and evolve, but to do this without responding to any causes.

One reason Hockney wants to view monads as uncaused, clearly, is so that free will is possible. He correctly argues that the scientific view that all phenomena involve cause and effect (or quantum randomness, which is also not free) is inconsistent with this. But why not bite the bullet, and admit that there is no free will? While a number of scientists and philosophers have recently defended the notion of free will (Dennett 2004; Churchland 2008; Murphey et al. 2009; Gazzaniga 2011), the more perceptive and honest of them realize that any brain process they may postulate is not in fact free from cause and effect. As Churchland in particular understands very well, the notion of free will as behavior that is uncaused is dead; the project is to understand how we behave in a way that is somehow consistent with some measure of individual responsibility.

Free will is not simply inconsistent with cause and effect. The concept is incoherent. Why would we act except for reasons—in the broad sense of the term, including in response to irrational desires? Why do we choose one behavior over another, except because we prefer it, and how do we prefer it in the absence of justifying reasons (not all of which are necessarily conscious). The only alternative is to act purely randomly and spontaneously, but beyond the fact that there would be no point—not to mention no survival value—in doing so, that kind of behavior is not free, either.

Any intelligent person should be able to understand this, but many people have what I can only regard as deep bias against accepting it. Such individuals, apparently, lack enough insight into their own behavior to see that it's always the result of some cause, but for rationalists, who believe in reason rather than their lying eyes, this shouldn't be a problem. Yet consider this quote from Leibniz that Hockney provides:

There is always a prevailing reason which prompts the will to its choice, and for the maintenance of freedom for the will it suffices that this reason should incline without necessitating.(2321-2322)

It should be obvious to anyone that “incline” is a weasel word, a fudge factor designed to ensure a desired result. Notice that Leibniz doesn't say reason inclines, therefore we have free will; he says we have free will, therefore reason must incline. We have seen earlier that Leibniz, whom Hockney considers one of the greatest geniuses of all time, was the same man who formulated the principle of sufficient reason, which basically states that everything has a cause. He almost certainly understood that all of our behavior results in response to causes. But unable to embrace this, he hedges, saying that reason only “inclines”, as if there could be any other factor in our actions except some other reason.

So Hockney's entire view of the evolution of monads is loaded with contradictions—and on top of all that, these “uncaused initiators” are initially unconscious! They apparently exhibit free will without even being aware of it. They cause the world of space-time by emitting energy, which somehow does not disrupt the exquisite balance that is necessary to maintain them as a net zero, or dimensionless. Through “filtering” in this space-time world, they become conscious, but this, too, is an uncaused change, one that also does not disrupt their dynamic balance. They change further over time, becoming more rational, which certainly must involve changes in the wave frequencies, yet even this does not disrupt the balance.

Hockney seems to be struggling with this problem when he says:

The difference between mind and matter is subtle and originates in the immediacy with which a net dimensionless effect is achieved. Mind depicts the situation where a net dimensionless effect is immediately achieved. This corresponds to the frequency domain – which is crucially outside space and time. Matter, on the other hand, depicts the situation where a net dimensionless effect is not immediately achieved. This corresponds to the spacetime domain, where the space and time elements conspire to obstruct the immediate balancing of positive and negative elements, or make this balancing much more complex.(3509-3516)

But again, this is hand-waving; it doesn't solve the problem. Here is the problem: for mind to change—to have conscious experiences in space-time, to think about these experiences, to have memories of them, and so on—it must undergo a change in the relative amounts of various frequencies. But this means the delicate balance is upset, and zero-dimensionality is lost. Conversely, if mind always maintains a balance, as Hockney is saying here, then it doesn't change.

His view of reincarnation, moreover, is itself rife with problems. I will pass over the dearth of real evidence for the process[14], and just take Hockney's account at face value. Reincarnation involves the mind, or soul, of a deceased person reinserting itself into a newly born person: “we keep acquiring, via reincarnation, new mathematical spacetime bodies to which our mathematical frequency minds can link”.(4045) Why? If monads collectively create space-time, why can't a monad, newly freed from a dying body, reincarnate in the form of a fully mature mind? The material brain, after all, is supposed to be created by the mind, and each mind has access to the entire shared world of space-time, so why does the new mind have to start over from square one each time?

As indicated in an earlier quote, Hockney sees reincarnation as necessary so that the constantly evolving mind is not stuck in an aging body; it can start out in a new one. But why must the brain in this body also be undeveloped? Why does it require two decades or so of development before it can make use of its superior rationality (assuming, again, that there is any evidence for this in the first place)?

For that matter, why is there aging and death at all? “Death,” he says, “is a permanent break from the collective dream, and only reincarnation can then restore the link.”(2598-2599) If monads create the space-time network which includes human bodies and brains, why don't these bodies and brains persist? Why are they born, then mature and die? Why do individual monads, which are supposed to be eternal, and which are supposed to contain an infinity of numbers, and an infinity of energy, in the form of sine and cosine waves, lose touch with the space-time world? That world, says Hockney is created by emission of energy. How can an individual monad, which he claims contains an infinite amount of energy, ever run out of it?[15]

The answer to all these questions, of course, is that empirical science has backed Hockney into a corner. The process has to work as I just described because the evidence of our senses reveals that we are born, mature, age and die; that people in the middle of life do not suddenly become far more rational; that sudden apparently uncaused changes do not abruptly occur in space-time; and so on. The evolution process that Hockney refers to, in which minds become more rational, is one that very well could be happening. Like him, I think it's happening, and I certainly hope it's happening. But to the extent that it is happening, it's a result of social, not individual, evolution, which means all the changes are occurring in the space-time world. Hockney himself effectively admits this, by describing the growth of consciousness and rationality as dependent on interaction with this world.

What he doesn't seem to see is that one can easily and completely account for social evolution without reincarnation at all, since each new generation of individuals is exposed to the social changes that persist beyond the life of any one individual. All that reincarnation adds to the story is the notion that individual minds are immortal, that the more evolved minds found in one generation are the same minds that were less evolved in a previous generation. But absent compelling evidence for this, which he doesn't have, Occam's razor, which he claims to have great respect for, suggests that he should drop this notion entirely.

So What?

I have discussed several of what I consider to be serious flaws or deficiencies in the Illuminist view of existence. Perhaps the most serious criticism that can be made of Illuminism, though, is to ask: So what? What if this view were basically correct? How much difference would it make?

For those interested in how the universe began, of course, it would represent a huge breakthrough in our understanding, but what impact would it have on our everyday lives? As I noted earlier, Hockney scornfully dismisses a profound physical discovery like the Higgs boson as having very little meaning for most people. But would Illuminism fare any better?

Would it change the basic facts of how we evolved? For the most part, no. Illuminism says that the phenomenal, extended world is created by Fourier transformation, but as far as I can tell it accepts almost wholesale, as it must, the scientific view of that world, including galaxies, stars, planets, the earth, and all forms of life on earth, including ourselves. At least in this book, Hockney—wisely, I would say—does not attempt to claim that his wave theory can explain either how or why these various objects or lifeforms arose, by challenging basic physics, chemistry, biochemistry, genetics, Darwinism, neuroscience, and so on. He does claim that he has solved the wave-particle problem--with the unoriginal and experimentally unsubstantiated idea that particles travel in waves—as well as that he and other Illuminati are the only ones who understand the nature of time. But I don't know any scientists who would accept these claims.

Does Illuminism suggest new forms of technology? No. Technology is the development of what is successful or useful, and Hockney has already yielded that domain, quite willingly, to science. Of course, technology is ultimately based on findings in basic research, but that research, if it's to lead to practical applications for phenomenal beings, has to be empirical. As I pointed out before, tautology may be the most certain kind of knowledge, but by itself it's also the most useless. Unless you are a first grade teacher, repeating 1 + 1 = 2 does not get you dinner. Even far more sophisticated mathematical equations generally have no known practical value, and those that do are found to do so only through empirical breakthroughs.

Would Illuminism provide new insights into human psychology or behavior, lead to new predictions in social processes or economics? No. As I discussed earlier, while it claims to reconcile the unextended and extended worlds in a unitary monism, it does not address the central issue of qualia. The aspect of the theory with the most profound potential implications, and which is also the most amenable to empirical tests, is probably its view of reincarnation. But as I discussed earlier, this is also one of the weakest parts of the theory. Not only does it lack empirical support, but the social consequences of the process are much more easily explained in terms of the scientific worldview.

As far as I can see, all the Illuminist theory does, at best, is graft a different beginning onto a story that is fourteen billion years old. This theory might be bold and elegant, aesthetically pleasing to those who grasp it, and might avoid some of the difficulties associated with the current scientific view. But as long it depends purely on reason for its validation, I don't see how it can ever be taken very seriously by the wider community. As I pointed out earlier, Hockney is passionate, which is to say, highly emotional, about his theory, and thinks it's essential to humanity that they come to see it in the same way he does. But the best chance for that to happen, in my view, is for him to overcome his distaste for empiricism and focus his intellect on devising ways of empirically testing it, even if this can only be done indirectly, on its implications. Empirical support for it would help validate not just the concept of monads, but indeed, the rational approach used to formulate them.

Beyond that, what I think Hockney and other Illuminati simply don't get is that science has solved most of the important pieces of the puzzle already, the parts that actually matter in our everyday lives. By this, I don't mean that science has discovered everything worth discovering, that it has brought us complete understanding of existence—that is a mistake that is made perennially--but that it has established that the empirical method has proved more capable of exploring our everyday world than any other. This method does not by any means dismiss mathematics and reason; on the contrary, they are essential to it. But neither does science rely exclusively on them, as Hockney seems to think it should.

Nor is the scientific method set in stone. It's constantly changing in response to new data and new ideas. Thus today we have scientific, or attempted scientific, investigations into phenomena that traditionally would have been considered outside the domain of empiricism, such as the development of moral behavior; the basis of political and economic beliefs; the nature of altered states of consciousness; and as I discussed earlier, the structure of reason. There is even, contra Hockney, an entire field known as experimental mathematics, which simply recognizes the fact that even a genius like Leibniz does not come to his insights in a flash, but through a long sequence of thoughts, which can be codified; and with the aid of computers, new theorems can sometimes be found before a formal proof is established. These and other explorations represent a great expansion of science not simply in terms of its subject matter, but necessarily also in terms of its methodology. In particular, the term empirical is not necessarily restricted any more to observations made through the senses, but can apply to any kind of knowledge that can potentially be shared between independent observers.

Even Hockney, as I have discussed earlier, makes use of empirical findings in his theory. It would be impossible for him not to, and in fact, the modifications he has made in Illuminism as it was understood by his predecessors have been shaped by these findings. No matter how brilliant the rational mind, it's always limited by its empirical knowledge. Hockney regards Leibniz, Pythagoras and some other philosophers of the past as the greatest minds in history, surpassing any modern or more recent scientists and philosophers. But they did not know things that most educated people know today, and which often have enormous implications for philosophy: for example, that matter is composed of atoms that are identical for any particular substance; that all sensory stimuli undergo extensive processing before we become conscious of them; and that most of our thoughts and other mental processes, including those underlying so-called voluntary behavior as well as language, occur unconsciously. Even Leibniz respected empiricism, evident in one of his most famous quotes: “the mark of a genuine idea is that its possibility can be proved, either a priori by conceiving its cause or reason, or a posteriori when experience teaches us that it is in fact in nature.”16 A little study of this unquestionably brilliant mind reveals, in fact, that he believed knowledge resulted, or could result, not primarily from the work of a few isolated geniuses, but from the collective efforts of large numbers of people—foretelling today's scientific community that Hockney is so critical of.

The other strong recommendation I make to Mike Hockney is to tone down the rhetoric and cut out some of the long, repetitious rants against science and other “anti-rationalists” that I think mar the exposition. “Enlightened people,” he says, “see cooperation with others as the best way to increase their mutual power.(4133) But by enlightened, he apparently means people who share his views. No religious zealot expresses more certainty in the truth and the righteousness of his belief in God than Hockney does in his mathematical ontology--an approach, he boasts, that “can rationally rubbish all of the rival approaches.”(3360). Einstein's theory of relativity is “lunatic” (4378); Bertrand Russell is a “fool.” (5504) He even suggests that he and his fellow Illuminati constitute a “master race” (1044), the only ones who can save the rest of us clowns from our stupidity. Whatever the validity of its ideas, this book is also a testament to the fact that hyperrationality (his own description of his views, not mine) can become a blind, emotionally-driven obsession with reason that threatens the existence of respect, humility, compassion, and uncertainty--all essential tools, I would say, to anyone who wants to exert influence in the marketplace of ideas.


  1. Numbers in parentheses refer to the location in the kindle version.
  2. Though Hockney describes his approach at times as intuitive, he doesn't equate intuition with reason, because he also describes (somewhat incorrectly) the mystical or spiritual approach as intuitive. But he obviously believes that intuition is the way to truth. I don't disagree that it frequently is, but intuition, unlike reason, can't be easily shared or replicated by independent minds. Hence, intuitive insights are not considered validated unless they can be supported by reason.
  3. Another example is his assertion that there are four basic types of people, and that “it's an astonishing truth that how we understand the world is dependent on our psychological type.”(692-693) So it seems that sometimes, after all, empiricism can lead to truth. Even more ironically, he sometimes seems unaware of how poor the empirical evidence he appeals to is, as when he uncritically repeats the old, and clearly exaggerated, differences between the right and left halves of the brain, virtually equating the former with the dimensionless monads and the latter with scientific materialism. (1130-1133) Below, I will provide other examples that show that he frequently has a poor understanding of empirical evidence that is critical to some of his arguments.
  4. To be fair, Hockney does indicate that this book was written for a general audience, and that he therefore has avoided as much as possible the use of mathematical terms and equations. In fact, Euler's equation is about the only formula in the entire book. But given the audacity and implications of his claims, it would be reasonable for the reader to expect, if Hockney could provide significant more support for them mathematically, an appendix in which this case was provided in more detail. After all, if some of his claims could be proven mathematically, this would resolve several major long-standing disputes in science and/or philosophy. In any case, in most instances where I argue that Hockney has not provided mathematical proof of some claim, such as the existence of monads, it's because this is basically impossible by definition, in the same way that sensory observations can't validate the empirical approach. I expect that any detailed mathematics he does have involves further elaboration of what goes on in these monads, not proof that these events actually occur.
  5. The equation might be better written as “one plus one equals two”, because just as the sky is not always blue, another point Hockney uses against empiricism, neither does 1 + 1 always equal 2. In the binary system, for example, 1 + 1 = 10. And there are still other systems, such as those composed of vectors, where even one plus one does not equal two.
  6. One of my favorite jokes has a university president complaining to the chair of the health and sciences department that they are spending too much money on research. The chair points out that the money is necessary for sophisticated machines and other technology used to perform experiments. The president counters that theoretical physicists get by without all this equipment. “All they need is pencils, paper and wastebaskets!” he exclaims. Then after a moment, he adds, “And our philosophy professors are even better, because they don't use wastebaskets.”
  7. Some physicists, such as Seth Lloyd (2006), define information purely in terms of bits, and argue that information was present in the universe from the very beginning. However, even Lloyd's definition of information presupposes interacting systems. One particle obtains information about another particle when it collides with it.
  8. Smith (2009), pp. 121-124.
  9. This is the property of intentionality, which Hockney mentions as an argument in support of dualism (2802-2803). However, no materialist philosopher I'm aware of regards it in this way (Daniel Dennett, the king of reductionists, practically made his name with his work on intentionality). More to the point, Hockney doesn't seem to understand that while intentionality might be a feature of a disembodied mind, it could not be a feature of the monads as he describes them. The monads are supposed to consist of nothing but Euler's equation and all kinds of numbers. Anything else added to this mix would destroy the balance to zero. Intentionality might be a property of an unextended mind in the classic Cartesian sense, but then the zero-dimensionality has to be explained in some other manner (Or not; it's not entirely clear that unextended has to mean zero-dimensional, that mind can't exist in time and space. Mind, in the admittedly confusing way that Descartes described it, does seem to exist in this way). Since Descartes was not addressing the something from nothing issue, his version of mind does not have to operate within the strict rules that Hockney has set himself.
  10. Not all physicists argue that a conscious observer is necessary to collapse the wave function. For example, Lloyd (2006) says: “All that is required to destroy the [wave function] is for some system, no matter how small, to get information about the position of the particle. If the particle knocks a passing electron or molecule of air, for example, that, too, will destroy the [wave function]. (p. 108)
  11. Hockney himself points this out in his book (2800-02), but in addition to missing this obvious lesson, he fallaciously argues that blindsight proves that consciousness requires more than just physical processing. He doesn't seem to understand—again, perhaps because of his antipathy for empirical-based reasoning—that the brains of such patients are different from those of ordinary people, and that these differences are undoubtedly correlated with the lack of consciousness.
  12. Hockney could address this problem by postulating that the original monads were actually conscious in the most fundamental, or what I call zero-dimensional, sense: simply aware, without any of the classical distinctions of consciousness present, such as self vs. other or time, space and objects. Their interaction with space-time would then result not in consciousness itself, but a more complex form of it sensitive to these dimensions, and eventually capable of reflection, and so on. He would then be advocating a form of panpsychism, something akin to what I have discussed (Smith 2009). The difference, though, is that his version would still be idealistic, with nothing existing in reality except consciousness. In most versions of panpsychism, including mine, both mind and matter are equally real.
  13. Panpsychism, the notion that all forms of matter are to some extent conscious, can be considered in some respects a third alternative. But panpsychist theories are generally materialist, in that they regard the material world as just as real as mind. And idealist versions are also possible. See note 10, above.
  14. I will also mention only here that much of his “evidence” for reincarnation amounts to gross misinterpretations, e.g., his point that memory is not localized in any one place in the brain is not evidence for a non-physical storage. Again, one can't expect someone who dismisses empirical evidence to be very good at interpreting it.
  15. For that matter, why does a monad emit energy in the form of sine waves, forming this space-time illusion, in the first place? If there is no cause, this is tantamount to saying, don't ask, you just have to accept that it happens. This is basically the same kind of excluding of certain questions as off limits that Hockney criticizes science for doing, when it claims that only sensory phenomena are real.
  16. Ariew and Garber, p. 26.


Ariew, R. and Garber, D. (1989) Leibniz: Philosophical Essays Indianapolis: Hackett Publishing.

Churchland, P.S. (2008) Do We Have Free Will?

Damasio, A. (2005) Descartes' Error: Reason, Emotion and the Human Brain. New York: Penguin

Dennett, D.S. (2004) Freedom Evolves. NewYork: Penguin

Gazzaniga, M.S. (2011) Who's in Charge? Free Will and the Science of the Brain. NewYork: HarperCollins

Hockney, M. (2014) The Mathematical Universe. Hyperreality Books

Kahneman, D. (2011) Thinking, Fast and Slow. New York: Farrar, Straus and Giroux

Lakoff, G. (1999) Philosophy in the Flesh. New York: Basic

Meillassoux, Q. (2008) After Finitude. An Essay on the Necessity of Contingency. New York: Continuum

Murphey, N., Ellis, G.F.R., and O'Connor, T., eds. (2009) Downward Causation and the Neurobiology of Free Will (Understanding Complex Systems). NewYork: Springer.

Wilson E.O. (1998) Consilience. New York: Knopf.

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