Are thoughts more fundamental to our reality than particles? “Well, how can you talk if you haven’t got a brain?” Dorothy asked the scarecrow. And after a moment’s glance toward the sky, he replied honestly, “I don’t know.”

Gottfried Wilhelm Leibniz, an influential mathematician and philosopher, whose work spanned the late seventeenth and early eighteenth centuries, argued methodically that the building blocks of reality are actually dimensionless units that have more the character of a mathematical point than an elementary particle. Leibniz imagined a universe of matter built up from non-material substances. These fundamental substances resemble thoughts more than the atomic building blocks of matter conceived in 19th century physics. While modern quantum conceptions have shattered the character of these tiny material elements, they continue to influence the popular understanding of fundamental particles such as the recently discovered Higgs boson.

Another influential pioneer, this time in the study of learning and perception, Johann Friedrich Herbart, was of the opinion that ideas are not passively pushed around by experience, but rather that they struggle to gain expression in consciousness. He even used the term ‘self-preservation’ to describe their action. Leibniz proposed a similar notion. Very recently, MIT cosmologist Max Tegmark suggested that the reason mathematics is so effective in describing reality may simply be that reality is a mathematical thing. In a recent interview for Science he describes mathematical structure as “abstract entities with relations between them.” “They don’t exist in space and time,” he explains, rather “space and time exist in them.”

What most people learn about mathematics in high school, and even in college, provides a very dull image of the discipline. Mathematics looks like a difficult and tedious set of rules, a painfully learned toolkit, required by scientists to do the work of science. But mathematics may very well be the human activity that best reflects the working relationship between mind and matter, or between thought and material. It has already facilitated a striking reconsideration of the fundamental nature of our physical reality in modern physics. But its completely symbolic worlds, which are only discovered with very careful introspection, may also have much to say about the significance of our thoughts.

Two very recent efforts demonstrate novel extensions of the use of mathematics. They each reveal unexpected parallels - one between physical processes and language processes, and the other between mathematics and organic activity. The first is from Bob Coecke, Professor of Quantum Foundations at Oxford University, who has observed that the way words interact with each other to bring meaning to a sentence, is very similar to the way particles interact in quantum mechanical processes. The other is the work of Gregory Chaitin, mathematician and prolific author, who has invented a mathematical life form that can maintain itself, and that can evolve. This is Chaitin’s first step in what he hopes will be a new approach to biology and the study of evolution.

Let’s look at Coecke’s work first. He has pioneered the use of a graphical mathematics to simplify the calculations of quantum physics. The work is based on category theory, a branch of mathematics that focuses less on mathematical objects themselves and more on the maps that transform them – on the things you do to mathematical objects. In the mathematics familiar to most, category theory concerns actions like summing, multiplying, projecting or translating, rather than objects like numbers, spaces or vectors on which these actions operate. One of the consequences of this shift is that category theory can find relationships even among the different branches of mathematics.

Coecke uses the structure of category theory to give meaning to the graphics he has designed. The figures involved are not complicated. They are lines and boxes. Lines represent systems, or the things one has chosen to look at. These may be quantum mechanical systems or classical physics systems. Boxes represent processes, or operations that can take a system of type A, for example, to a system of type B (figure 1). The processes represented by the boxes could be, Coecke says, either cosmological or ones produced by experiment. It is the arrangement of boxes and lines (which Coecke calls wires) that tell the story. For example, a cup-shaped line indicates a system with two outputs. When a pair of cups with inverted orientations is combined, a single output system remains.

Once these abstractions are given particular definitions (whether mathematical or physical) their behavior, which is read in a very intuitive fashion, accurately reflects what is known within that setting. While the statements are diagrammatic, by virtue of their relation to category theory, they capture quantitative content, as well as qualitative content. The computing is in the movement of boxes along the lines (or wires) of the diagram. It is a calculus, Coecke says, that can express quantum action more simply than the one line symbolic statements of equations.

Figure 2 shows a diagrammatic expression of quantum teleportation - the process by which quantum information can be transmitted, exactly, from one location to another without moving through the intervening space. Alice and Bob are common names given to the bearers of quantum information.

With generalities as broad as the ones in category theory, Coecke’s diagrammatic calculus has application to biological processes as well. But it is its application to linguistics that is perhaps the most unexpected. In a podcast produced in April 2012 by the Foundational Questions Institute, Coecke characterized both quantum mechanical systems and language systems as “things that flow in wires.” The state of a physical system flows in a wire and the meaning of a word flows in a wire. A transitive verb, for example is thought of as a process with three wires, one requiring an object, one a subject, and one producing the sentence.

A sentence

Coecke makes the argument that his methods are more effective than current linguistic models because, while current models use either the meaning of individual words or the grammatical rules that govern their combination, they cannot combine the effects of word meaning and word order. It should not be overlooked that the success of this approach to language relies on the observation that the words are interacting.

Lets now look at Chaitin’s mathematical organisms as he describes them in his most recent book, Proving Darwin: Making Biology Mathematical. His work begins with the observation that “After we invented software we could see that we were surrounded by software. DNA is a universal programming language and biology can be thought of as software archeology – looking at very old, very complicated software.”

Chaitin starts with the simplest case – a single software organism that has no body, no population, no environment and no competition – what he calls a toy model of evolution. The organism is assigned the task of naming extremely large positive whole numbers. To do this effectively, this mathematical life form needs to invent addition, multiplication, and exponentiation. If you have the large number N, for example, and you want to find a larger one, it will be necessary to consider N + N, or N times N, or N to the nth power, or N to the N to the nth power, and so on.

By design, successfully finding a number larger than the last one found increases the fitness of the organism/program. A mutation is an algorithmic modification, a computer program. The original organism produces the mutated organism as its output. Because this mathematical problem can never be solved perfectly, the ongoing evolution of these organisms is assured.

These toy models, as Chaitin calls them, provide a way to measure evolutionary progress and biological creativity. It is not surprising, therefore, that Chaitin claims that biological creativity equals math creativity. And this leads to a potentially important shift in perspective. He refutes social Darwinism with social metabiology according to which, “the purpose of life is creativity, it is not preserving one’s genes. Nothing survives, everything is in flux…everything flows.”

Chaitin admits that it’s too early to tell if the ideas in metabiology will bear the fruit for which he hopes. But this conceptual shift in evolutionary theory will no doubt stimulate questions about biological creativity, evolution, and mathematics. As he discusses what mathematics might yet achieve, Chaitin says this:

Math itself evolves, math is completely organic. I am not talking about what Newtonian math might ultimately be able to achieve, nor what modern Hilbertian formal axiomatics might ultimately be able to achieve… and not even what our current postmodern math might ultimately be able to achieve. Each time it faces a significant new challenge, mathematics transforms itself. (Emphasis added)

As a living language of science, that can also be an effective model of language, mathematics may yet help uncover the way that thought resides in nature.

When the scarecrow finally meets the wizard, the straw in his head remains, but he gets a diploma. The representative of thought that emerges is mathematics. He tries to state the Pythagorean theorem, as if the diploma alone can produce this kind of thought. But he doesn’t get it right.