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To What Extent Do We See With Mathematics?


Variable XWhen I first became fascinated with mathematics’ tightly knit abstract structures, its prominence in physics and engineering reassured me. Mathematics’ indisputable value in science made it clear that my preoccupation with its intangible expressions was not pathological. The captivating creative activity of doing mathematics has real consequences.

During my graduate school years, I began to consider that the appearance of reality actually depended on the kind of mathematics we use to see it. Was it possible that the use of mathematical ideas, like a lens, could bring some aspects of the world into sharp focus while blurring others to the point of invisibility? A new mathematics, whose development is being led by author and theoretical physicist David Deutsch, may actually highlight what mathematics can do to help us “see” our reality, and maybe even tell us something about how the process works. Deutsch is best known for his pioneering work on the quantum theory of computation, where some of the more mysterious quantum phenomena are harnessed to dramatically enhance computation. While his new mathematics is related to the quantum theory of computation, it is also distinct from it. He calls the new mathematics constructor theory: a theory designed to tell us, in the most general sense, what is and is not possible in the physical world.

Deutsch has discussed constructor theory before. An unexpected early success of the theory, he has said, has provided a new foundation for information theory. Information theory involves the quantification of information. But perhaps most relevant to this discussion is that information theory equates abstract things such as words, coded data and algorithms with physical things such as like electric signals, chemical exchanges and molecular coding. Since they are all information, the employment of information theory is transdisciplinary. Just a few of the disciplines included in its range of application are physics, electrical engineering, linguistics and neurobiology. The processing of information, expressed in the formalism of mathematics, captures the action of many kinds of systems.

The physical nature of information is one of the keys to constructor theory. “There is no such thing as an abstract computer,” Deutsch explains. Information is “instantiated in radically different physical objects that obey different laws of physics.” In other words, information becomes represented by an instance, or an occurrence, like the attribute of a creature determined by the information in its DNA. In Deutsch’s October 2012 presentation for an online discussion group, he makes the point more personally.

I'm speaking to you now: Information starts as some kind of electrochemical signals in my brain, and then it gets converted into other signals in my nerves and then into sound waves and then into the vibrations of a microphone, mechanical vibrations, then into electricity and so on, and presumably will eventually go on the Internet.

The ‘information’ is the only thing that remains unchanged in each of these transformations and this is the focus of constructor theory. Constructor theory is meant to get at what Deutsch calls this “substrate independence of information,” which necessarily involves a more fundamental level of physics than particles, waves and space-time. And he suspects that this ‘more fundamental level’ may be shared by all physical systems.

Information and knowledge are, for Deutsch, part of the stuff of physical life. In his book, The Beginning of Infinity, Deutsch compares and contrasts human brains and DNA molecules. “Among other things,” he says, they are each “general-purpose information-storage media…” This information he calls knowledge, and he aligns biological information with explanatory information when he says that knowledge “is very unlikely to come into existence other than through the error-correcting process of evolution or thought.” The non-explanatory nature of biological knowledge is distinguished only by its limits, affecting largely only the physical attributes of creatures and their immediate experience. The explanatory nature of human knowledge, however, allows us access to things far beyond our immediate experience.

Constructor theory begins with the understanding that every physical system can be understood in terms of transformations, where one physical system is changed by the action of another. The object causing the transformation is called the constructor. But a key-defining characteristic of a constructor, as Deutsch’s colleague Chiara Marletto explained to me, is that in the process the constructor “remains unchanged in its ability to cause the transformation again.” In a philosophical paper on the subject,

Deutsch gives the heat engine in thermodynamics as an example of a constructor. Continually converting thermal energy into mechanical energy, or heat into work, it operates cyclically, returning to its initial state at the end of each cycle, able to repeat the process over and over again. In this configuration, the constructor together with its input and output states jointly describe an isolated system.

But, Marletto also explained, the constructor itself, the thing that causes a transformation, is abstracted away in constructor theory, leaving only the input/output states.

These input/output states are expressed as “ordered pairs of states” and are called construction tasks. The idea is no doubt a distant cousin of the ordered pairs of numbers we learned about in algebra. The composition of tasks, or networks of tasks, are also defined to account for the interaction of more than one system. One of the challenges of building the theory is defining the relationships among tasks (or the algebra of tasks), so that they make sense with respect to one another and are able to accurately express known physical laws. Constructor theory’s own laws, which are actually “laws about laws,” have been given the name principles. Principles describe the constraints on other laws rather than the behavior of physical objects directly. “We guess principles,” Marletto told me, “and then build a mathematical structure to be consistent with that conjecture.” The algebra must make sense within itself, be able to express known physical laws, and be able to define the laws (or principles) of constructor theory. Describing something of their thought process Deutsch tells us:

We try and think what it means, find contradictions between different strands of thought about what it means, realize that the algebra, and the expressions that we write in the algebra, don't quite make sense, change the algebra, see what that means and so on.

The basic principle of the theory is that subsidiary theories, or all physical theories compatible with constructor theory, must be expressible entirely in terms of statements about which tasks are possible, which are not, and why. “If you have this theory of what is possible and what is impossible,” Deutsch says, “it implicitly tells you what all the laws of physics are.” In a constructor-theoretic description of the physical world “what actually happens is seen as an emergent consequence of what could happen, rather than vice-versa.”

Although it resembles other abstract algebras, the algebra of constructor theory is new. But Marletto and I agreed that it rests on very familiar, fundamental notions: ordered pairs, one-to-oneness, inputs and outputs, compositions and the very idea of a mathematical transformation. There is a thread that leads from constructor theory algebra back to the algebra that most of us learn in high school or college. But the hopes for this algebra are far-reaching. Constructor theory principles, together with the laws of subsidiary theories that are compatible with these principles, are expected to produce new laws which have no equivalent in existing theories.

Translation into Constructor Theory Graph

Image provided by Chiara Marletto

I am intrigued by the fact that, after centuries of observation and skillful experiment, and after centuries of analysis and mathematical insight, that this algebraic, information-driven mathematical structure may be able to reach the depth necessary to produce new physical theory, and one that is not expressible within the framework of current theories. The algebra is not designed to systematize current theories, but rather to find their foundation and then open a window onto things that we have not yet seen.

So where does the relationship between physical reality and mathematical reality actually lie? Perhaps it lies in mathematics’ relationship to cognition itself. Cognitive processes are themselves transformations—of sensory data into the perceived attributes of the things in our experience, or of associated experiences into meaningful narratives or into symbol. Fundamental cognitive processes work in mathematical ways. Neural cells in the visual brain, for example, are specialized to respond to abstractions like verticality. Distance is discerned by the measurement of discrepancies between the view of each of our two eyes, and probabilities are calculated when the brain brings meaning to new sensory data using past experience. Perhaps mathematics, as a thoughtful activity, is a continuation of what the body is already doing—what it accomplishes outside of our awareness. Then mathematics works to shape our experience of deeper realities, as if the body is working to see more, or to see differently. And this would support the confidence that Deutsch expresses near the end of his presentation, that the work of science can bring us closer to what is really there.

Since the early part of the 19th century, mathematicians and physicists alike have pondered the relationship between mathematics and physics. Over the course of the intervening years questions about their relationship have broadened and given rise to new questions. Mathematician and computer scientist Gregory Chaitin, for example, has recently introduced a mathematical or, more precisely, a software account of biology in his book, Proving Darwin: Making Biology Mathematical. Once we discovered software, Chaitin has said, we were able to see that it is everywhere around us. DNA, he explains, is “a universal programming language found in every cell,” and, consequently, biology is about algorithmic information. Chaitin has begun the construction of a mathematical biology, where genetic history can be described algorithmically. In his working models, there is no difference between mathematical creativity and the biological creativity of evolution.

The proposals from both Deutsch and Chaitin share the view that mathematics is integral to biological processes. Perhaps they each represent an evolution of thought in science, where the distinction between what was once considered mechanical action and thoughtful action becomes less clear and where thoughtful action is understood as part of the life of the universe.

The views expressed are those of the author and are not necessarily those of Scientific American.

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