Dr. Natalia Janson, senior lecturer in applied mathematics, discusses: Can We Resolve the Mind-Body Problem with Mathematics? at TEDxLoughboroughU Conference (Transcript)
Listen to the MP3 Audio here: Can We Resolve the Mind-Body Problem with Mathematics by Natalia Janson at TEDxLoughboroughU
Dr. Natalia Janson – Senior Lecturer in Applied Mathematics
In the 1600s, a philosopher and mathematician, René Descartes, suggested that the mind was an immaterial entity, isolated from the body and — notably from the body and from the brain.
On the contrary, in the middle of the 20th century, a philosopher, Gilbert Ryle, proposed that the mind is simply the physical processes occurring in the brain, and there is no separate entity which needs to be called the mind. So, in fact, he proposed that the mind was material.
In between these two extreme standpoints, there was a theory of a 19th century philosopher, Arthur Schopenhauer, who admitted that the mind and the body were distinct entities, but the properties of the mind were largely determined by the properties of the body to which it belonged.
There has been a lot of philosophical dispute about whether the mind is totally unobservable, or perhaps observable. In the 20th century, it has been widely recognized that we need to start handling the mind with the methods of natural sciences. The ancient mind-body problem about the relationship between mental and physical processes has been reformulated with account of the achievements of modern science.
Nowadays, the scientific hypothesis is that the mind is the product of the brain, and it emerges from interactions between the brain components. So the mind-body problem now reads: What is the exact relationship between the brain and cognitive functions, between the brain and behavior? Several people hypothesized that the mind could be some field produced by the brain, possibly an electrical field.
A psychologist, Benjamin Libet, proposed that the mind is not a physical force field, not like an electrical or a gravitational one, and is rather some immaterial field which is not directly observable.
What else do we assume about the mind? It develops throughout our life. Indeed, we are not the kids we used to be: we learn, we change every day, we develop. It is largely shaped by the environment, by nurture, for example. It governs our behavior. And, importantly, it is assumed that all minds are unique.
There seems to be some contradiction between what different theories attribute to the mind. If only one of these theories is correct, does it mean that all others are wrong? Or is there any chance that maybe all of these theories are correct? Well, in science, it happens from time to time, when conflicting theories are reconciled by the appearance of a more general theory. And a very famous example is the wave-particle duality of light, which was resolved when quantum mechanics appeared.
To study the mind with the methods of natural sciences, we first of all need its rigorous definition. But it should be such a definition, which connects the mind with brain data. And there are two kinds of brain data: Brain activity data, which tell us what the brain does; and the brain connectivity data, which inform us how the brain is made.
There is one standard way to define an object: to list all of its features. Can’t we simply define the mind as an object having all the features listed here? No, because it will not connect the mind with the brain, and it will not explain how features which seem to contradict each other, to exclude each other, can coexist. So, basically, we need to resolve the mind-body problem first. In mathematics, there happens to be an entity, which is called into being by the brain, and which combines all of the mind features listed here in a non-conflicting manner.
So let us start, from the brain. Thousands of papers are published on neuroscience every year, and our knowledge of brain mechanisms is vast and growing. However, it did not resolve the mind-body problem so far. To get to the core of this problem, I am going to ignore most of this knowledge and I will focus only on the most important, significant brain facts.
The brain consists of a large number of neuron cells, which are connected to each other. These neurons produce repetitive splashes of electrical activity, called firings, which never stop. And importantly, the neural activity occurs spontaneously, in the sense that nobody tells the neuron when to fire. The neuron analyzes signals at its input and decides what to do next entirely on its own. So, overall the brain is a large system which demonstrates spontaneous activity.
To describe spontaneously behaving devices of any origin, from a mechanical pendulum clock to a beating heart, in mathematics there is a special theory, called theory of dynamical systems. ‘Dynamics’ means ‘change of time’. To construct a model for a spontaneously behaving device, we make two steps. First of all, we identify all quantities which change in time and call them variables. In neurons, such variables are usually ionic currents, the membrane voltages, concentrations of various chemicals inside the cell, and so on. The full collection of such variables is called a state. We assume that these variables are coordinates of a certain point in a special space, called state space.
State space is not like the physical 3D space in which we live. It is a non-physical space. And going from the physical 3D space to a non-physical, multi-dimensional state space was a huge mathematical breakthrough. When the neuron demonstrates activity, its variables change in time and the state point moves along a certain path.
To describe the whole brain, we need to introduce state variables for all of its neurons — and there are about 100 billions of them — and to form the joint state space, because the neurons are connected. So the dimension of the state space of the brain is really large, much larger than 3.
So the second step in constructing a model is to specify the rules which govern the behavior of the system. And in our terms these are the rules which govern the motion of the state point in the state space. We do it as follows: At every point in the state space we put an arrow which has a certain length and a certain direction. This arrow is called the velocity vector. The full collection of such arrows is called the velocity vector field. It works as follows: Imagine that our state point is represented as a football moving in the state space. And imagine that at every point in the state space we put a footballer who does two things: He accepts the ball from where it arrives and kicks it in a given direction with a given speed. In between the two kicks, the football travels for just a tiny amount of time before it is caught by the next footballer. And the next footballer kicks it in a different direction with a different speed. That is why the football moves.
So this metaphor explains why our field of arrows is called the velocity vector field: it is because speed with direction is called ‘velocity’, and in mathematics all quantities with direction are called ‘vectors’. So we know that if we consider a car moving along a curved village road, the velocity of the car tells us how quickly and in what direction the car moves in the physical 3D space. But here the velocity vectors tell us how quickly and in what direction the state point moves in the non-physical state space.
Interestingly, when all these arrows are permanently fixed in time, do not change, they still prompt the system to evolve, to behave, and they fully determine all properties of this behavior. So the velocity vector field of the device plays the role of the ruling force which dictates the system what to do in every possible situation.
You can see that this object is much more sophisticated than a straight line or a surface. And, in fact, it was introduced by Henri Poincaré, a mathematician — largely thanks to him — especially to handle complex systems. The actual model has this form, and it is called a dynamical system. These equations mean that, at every point in the state space, we have a velocity vector whose components are rates of change of individual variables, or velocities. The values of these components depend on the location in the space, as we say in mathematics, they are functions of this location.
How do we know what these values are? Well, in this Universe all devices obey the same laws of physics and chemistry, and that includes living systems and the brain. In brain models, these components are simply the combinations of physics and chemistry laws which we apply to various parts of the brain with account of the brain’s architecture, such as the connectivity map. So essentially, the velocity field of a device is simply a mathematical representation of the physical structure of this device. Living systems have been modelled as dynamical systems, and perhaps the most famous neuron model in this form is due to Hodgkin and Huxley and is now over 60 years old.
More recently, there were several attempts to model the whole brain as a dynamical system. So, ultimately, to build the model of the brain means to determine the brain’s velocity field. It is important to appreciate that when we observe the brain activity in an experiment during finite time, we only observe just one trajectory in the state space of the brain, which corresponds to only one behavioral pattern of the brain. But the velocity field holds information about all possible behavioral patterns the given brain can exhibit in principle.
To resolve the mind-body problem, I propose to identify the mind with the velocity field of the brain. Also, to the question: How is the brain connected to behavior?, my answer is: through the velocity vector field which the brain creates. Let me explain how. Go back to the list of features of the mind, and now — shuffle them a bit. And now compare these features against the features of the velocity field of the brain. Is our velocity field a field? Yes. Is it called into being by the brain? Yes, there is no velocity field of the brain without the brain. Does it emerge from interactions between the brain components? Are its properties determined by the physical properties of the brain? Yes and yes, most certainly.
Does it govern our behavior? Let us see. Suppose your body moves. What makes it move? Muscle contraction. What makes muscles contract? The appropriate coordinated firing of special neurons, called motor-neurons, which connect the brain with the muscles.
What makes motor-neurons fire appropriately? The firings of the neurons in the brain to which they are connected. And these are governed by the velocity field of the brain. So, ultimately, the velocity field of the brain governs the behavior of our body. Is it distinct from the brain? Yes, if only because, unlike the brain, it exists in the non-physical state space. Does it coincide with the brain? In a certain sense, yes, because the velocity field of the brain is only a mathematical representation of the brain’s physical structure. It is a bit like saying that what you see on this slide is a straight line, but all you see here is a collection of ink particles deposited on paper.
Is our field material and observable? Yes, because you can measure its components in an experiment as the rates of change of neural variables, or velocities. Is it immaterial and unobservable? Yes, because it exists in a multi-dimensional space, which we cannot experience with any of our senses, and which we cannot observe directly with any measuring instruments. Is it directly connected with both kinds of brain data? Yes, as we just discussed. Does it develop? Let us see.
By now, you might be wondering: If for any spontaneously evolving device or organ, such as the mechanical clock or a heart, one can introduce a velocity field and call it a mind, why do the clock and the heart not think and why does the brain think? This concerns with the physical architecture of the brain. In all devices or organs, the architecture is rigidly fixed. Any changes are slow and lead to less efficient functioning. The brain is fundamentally different because it is plastic. It is the only device in this Universe, known to us, which functions by means of modifying its own architecture in real time. The connections in the brain appear, disappear, weaken and strengthen non-stop.
When the brain modifies its connections, it simultaneously modifies its velocity field. So the brain’s velocity field is unique because it is plastic, it develops. How exactly it develops depends on the signals, on the stimuli, which we receive at our sensors, which means on the environment.
Finally, it is well appreciated that all brains are different, and in different brains there are different velocity fields. So, the velocity field of every brain is unique. So we have ticked all the boxes here.
To summarize: To resolve the mind-body problem, I propose to identify the mind with the plastic velocity field of the brain. But somebody might say: These arrows in a multi-billion-dimensional space cannot represent the mind, because they are not real, they are invented by mathematicians. But so are any laws of physics and chemistry represented as mathematical equations. I do not know if these laws are real or not, but the very real physical world functions as if these laws were real. The velocity field of the brain is only these laws combined in a special manner. So if we agree to regard the laws of nature real, we should be consistent and accept the reality of the velocity field of the brain.