The belief that numbers are synonymous with objectivity has a long tradition in science. This belief lies at the heart of mathematical theories of prediction. It is also a belief that has become engrained in the public attitude toward scientific theories and knowledge: If something can be put in numbers, it is thought to be more likely to be ‘true.’ Numbers give things an air of respectability. How numbers came to be the benchmark for the perceived scientific value of a theory provides a useful, albeit brief, background to better understand mathematical theories of prediction, where they come from, and what they are meant to achieve.
13.1 Newton, and the Power of Equations
Isaac Newton is often considered to be the founding father of the view that the ultimate aim of scientific investigations is to yield general scientific laws. Crucially, he promoted the idea that such laws can, and should, be expressed in mathematical equations.
The world according to Newton is regular and its structure and patterns can therefore be described mathematically. Newton reached this conclusion, according to legend, when seeing an apple falling from an apple tree. He then established that the speed of a falling object increases every second. This observation called for a mathematical analysis but there wasn’t anything in the mathematics of pre-Newtonian times that could be used to describe such phenomena. Isolated in his country house during the bubonic plague of 1665 and 1666, Newton found the time to think deeply about mathematics and the relationship between movement and rate of change. Newton’s analysis of the rate of change led to his invention of the fundamentals of calculus, which he then used to formulate the laws of motion and universal gravitation.
The validity of the idea that there is a direct connection between mathematical equations and the science of physics and science more generally had been demonstrated in a dramatic fashion: Simple mathematical equations could be used to describe the natural world. Newton’s laws became the most fundamental laws of physics and are still widely used in engineering today. They were replaced by the theory of relativity and quantum mechanics only in the twentieth century.
One of Newton’s many insights was that if you can describe a physical system with equations and then solve said equations, then you can not only describe the past and present but also predict the future. Indeed, Newton mainly developed his new mathematics to achieve more accurate predictions in physics. Newton showed that physical phenomena could be expressed in mathematical terms. This enabled him to make precise numerical predictions which could then be compared with measured values.
The success of Newton’s approach was unprecedented. No scientific theories before had come even close to achieving a similar accuracy in the predictions that were confirmed by subsequent empirical work. A well-known case is the discovery of planet Neptune. Soon after Uranus was discovered in 1781, predictions about its orbit, based on Newton’s laws, were published. However, they were found to deviate strangely from the predicted motions. Some astronomers suspected Newton’s laws to not be accurate at large distances, but English astronomer John Couch Adams and French astronomer Urbain Le Verrier reasoned instead that an undiscovered massive planet caused Uranus to stray from its predicted motion. Adams and Leverrier put forward their mathematical predictions about Neptune’s orbit. It took a while until they were considered seriously, but once this happened, Neptune was found rapidly.
13.2 Scientific Determinism
Newton’s approach is one that many scientists to this day follow and is called scientific determinism. It is the belief that the world, including the mental world, can be described completely in all its glorious detail. Everything that happens or will happen in the future can be explained by what happened in the past; nothing is uncertain.
Scientific determinism culminated in the claim that came to be known as Laplace’s demon. Laplace claimed that
we ought then to regard the present state of the universe as the effect of its anterior state and as the cause of the one which is to follow. Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective situation of the beings who compose it – an intelligence sufficiently vast to submit these data to analysis – it would embrace in the same formula the movements of the greatest bodies of the universe and those of the lightest atom; for it, nothing would be uncertain and the future, as the past, would be present to its eyes.1
In other words, if you, or a demon, knew the position and momentum of every particle in the universe, and ran it forward, you could predict the future in all its detail.
Modern incarnations of this view include claims that soon it will be possible to replicate the human brain perfectly in a digital format and upload individual human minds, including the billions of neurons and their trillions of connections, to a computer.2 The assumption is that if you know all the detailed parts of the brain and how they work together then nothing will be unknown. Scientific determinism, in its strong version, has it that everything in the universe is fully determined and hence can, at least theoretically, ultimately be described. A perfectly predictable world is possible because there is no inherent randomness in its structure, or as Einstein put it: ‘God doesn’t play dice.’3
13.3 Probabilistic Predictions
The backlash to the idea of perfect scientific determinism came in the early twentieth century. At least two serious challenges to the claim that it may be possible to completely determine everything have arisen. As a result of these challenges, scientific determinism has fallen out of favor with a substantial number of scientists.
The first significant challenge to determinism comes from modern physics. Modern quantum physics suggests that the universe is inherently indeterministic. This is exemplified most famously in Heisenberg’s uncertainty principle, which states that it is not possible to measure accurately, at the same time, the momentum and position of a particle. According to the Copenhagen interpretation, quantum mechanics is intrinsically indeterministic; there is only a probability for the outcome of a possible measurement. This was an inacceptable state of affairs to Einstein who believed that there must be an objective physical world even at the tiny level of quantum phenomena.
Almost one hundred years after quantum mechanics was developed there is still no consensus about its fundamental assumptions, though the majority of physicists and mathematicians believe that randomness is a fundamental concept in nature.4 Interpretations of the uncertainty principle include that it may be simply, and ultimately, incomprehensible to the human mind, or that it may be caused by an unknown problem in the measurement process such as that measuring gives the particle a random kick, or that uncertainty is a property of the particle and its motion is inherently random.5 Others have argued that such an interpretation is an example of a mind projection fallacy and that probability distributions in quantum mechanics describe not a property of the world but simply a certain state of information about the world.6 Be that as it may, the assertion of quantum mechanics that outcomes of measurements can, even in principle, never be predicted with complete confidence, and that the best we can do is to calculate the probability of an outcome, has had a huge influence on the mind sciences and science as a whole. Prediction might come with a probability: Human minds might be hedging their bets.
A second challenge to strong scientific determinism has been the accumulated evidence that some events can be, at least in the long term, unpredictable. Weather forecasts are a case in point. It would be nice if you could plan your wedding day for the next year and choose the day with the perfect weather based on an accurate weather prediction. It would be nice, but it is not going to happen: There is no certainty for long-term weather predictions.
No certainty, however, does not mean random. Many people plan their wedding day to be around summer for a reason. Weather is not random, weather is chaotic. Randomness and chaos may seem like similar concepts, but they are not the same.
Chaos theory was developed in the early 1960s based on the results of weather prediction experiments. Its discoverer, Edward Lorenz, dissatisfied with the very limited success of weather predictions in the 1950s, set out to predict the weather more accurately. Lorenz constructed mathematical models using 12 differential equations representing temperature, wind velocity, temperature, etc., and was reasonably successful in predicting the weather a few days ahead using them.
On one memorable day for prediction research in 1961, he decided to save some time. Rather than starting the program at its initial conditions, he started it in the middle. One more change he made was to enter the data to three decimal places instead of the six decimal places he had used before. Then he went off to get a coffee, waiting for the number crunching to finish. His expectation was that he would get a close approximation to earlier results, reasoning that tiny initial perturbations would lead to tiny changes in the predicted outcome. But Lorenz’s computer did not return the expected result; using the same deterministic equations, the tiny change in initial weather conditions led to a completely different weather prediction.
This is the famous butterfly effect: Small variances in initial conditions lead to very different outcomes in a dynamical system. To get the idea across, Lorenz later used the analogy of a butterfly flapping its wings in Brazil causing a tornado in Texas.7 But the core insight of Lorenz’s chaos theory is the following: Outcomes in dynamical systems can be unpredictable, predictability is finite.8
Since Lorenz’s discovery, the computing power and data collection involved in weather forecasts have changed beyond what then could be imagined. The supercomputers used by the major weather agencies today complete several quadrillion (!) calculations every second. And, indeed, forecasts have improved with this onslaught of mathematics. Yet, you are probably still aware that, while weather forecasts for the next five or so days are reasonably accurate, forecasts beyond a week are nearly useless. Chaotic systems do not allow for accurate long-term predictions. Chaos is not randomness, but as Lorenz put it, it is “when the present determines the future, but the approximate present does not approximately determine the future.”9
The upshot of all this is that irregular and unpredictable behavior can arise from simple systems that can be described by simple mathematical laws. Additionally, simple systems can have complex solutions; there are limitations in predicting the future even if we understand everything else.
Situations with uncertain outcomes call for probability theory. Most of perceptual and cognitive processing requires “uncertain conjecture from partial or noisy information.”10 It is hence not surprising that the concept of probability has transformed explanations and reasoning in large sections of science and has had a huge influence on contemporary mathematical approaches to the predictive mind. The assumption is that assigning a probability to a future mental event is the best we can do.
13.4 What Are Mathematical Theories?
Deterministic and probabilistic mathematical theories have in common that they construct mathematical representations of real-world phenomena. On a basic level this can be regarded as a type of explicit problem-solving. This involves presenting the problem in ‘abstract form’ in symbols (often numbers, letters, or geometrical elements). These symbols are then manipulated in accordance with precise rules: Strings of symbols in sets of equations come to represent ideas.
The construction of mathematical theories involves testing whether experimental observations fit the postulated ‘mathematical rules.’ If they do not fit then the ‘mathematical rules’ may be refined, extended, or new ones may be formulated. Newly mathematically formalized ideas are validated by testing whether they align with observations but also by examining whether they are consistent with other, previously established, mathematical rules.
13.5 Pitfalls and Promises of Mathematical Theories
Mathematical theories of prediction thus describe how predictive phenomena manifest themselves on an abstract and quantitative level. The abstraction involved in mathematical formulizations however entails certain risks; it is important that, when abstracting, crucial aspects of the real-world phenomena described in the formulizations are not ignored. The abstract mathematical statements must mirror or closely correspond to the properties of the world they are supposed to describe.
It is important to note in this regard that there is no agreement among scientists on whether mathematics is a tool invented by humans to describe the world or actually exists prior to human cognition. Albert Einstein and David Hilbert, one of the most influential mathematicians and the founder of logic, believed it is a tool ‘only’; Kurt Gödel and Roger Penrose thought there are ‘mathematical truths’ in the fabric of the universe.
This is more than just a largely irrelevant philosophical discussion. The ‘mathematics has a reality independent of human cognition’ view implies that prediction is governed by mathematical propositions that exist in nature; an absolute mathematical truth reflecting the ‘hidden coding’ (a ‘mathematical language’) of the universe that can be discovered. The ‘mathematics is a man-made tool’ view on the other hand implies that prediction can be described mathematically but there is no ‘mathematical ground truth’ by itself that explains the predictive mind.
The ‘mathematics is a man-made tool’ view suggests that ‘mathematics alone’ may not tell us how the world works. ‘Pure mathematics’ for example did not lead to the discovery that Newton’s theories were wrong; this was revealed by experimental observations that did not fit the theories. New ideas about how the physical world may work were then proposed and mathematically formalized. In other words the new idea(s) arose from creative thinking and mathematical formalizations helped to refine them. Newton in fact invented calculus to formulize his new theories (and make them more precise).
Mathematical formulizations in short might reveal their biggest strengths when they are conceptualized as tools: mathematical redescriptions that lead to a better understanding of physical phenomena, for instance enabling more accurate experimental predictions.
Indeed, there is little doubt that mathematical formulization has served science well. Precision is arguably the most important strength of mathematical theories: Scientific ideas can be expressed unambiguously. Mathematical equations are precise statements of the presumed state of affairs.
Another strength is that new knowledge and understanding can be increased when modeling complex processes and systems mathematically: The manipulation of abstract statements might reveal some aspects of the predictive mind that may have remained obscure in nonmathematical approaches.
Each of the mathematical theories of prediction discussed in the next chapters comes with its own flavor. They all, for instance, differ in the degree to which they are probabilistic or can be made compatible with probability theory. Connectionist networks, information-theoretic accounts, Bayesian inference, and quantum cognition accounts, however, all have in common that mathematical formulizations are core aspects of the specific approaches to the predictive mind.
