The Mathematical Universe [Book Review]

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.

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