Is there a mathematical theory of intelligence?

If we mean intelligence in general, or universal intelligence, and mathematical models as a mathematical representation of reality, the answer is not yet.

It is regardless the universality of mathematics and its models, as set theory, dynamic systems, statistical models, differential equations, game-theoretic models or logic models.

Anything in the physical or biological world is subject to analysis by mathematical models if it can be described in terms of mathematical expressions, concepts and language.

It all depends on how you declare or define intelligence, as its values, types and mechanisms, with methods, models and techniques to study it.

There are all sorts of models and approximations by means of mathematical logics, mathematical optimization, probability theory, statistic models, information theory, computer science. cognitive science or neuroscience.

Accordingly, an intelligence could be realized in many forms and modalities, as an information entity, an animal/human being, an advanced algorithmic system, a learning software application, complex data-processing software/hardware, a sophisticated computing device, statistical machines, or a goal-directed agent, which “intelligence measures an agent’s ability to achieve goals in a wide range of environments, situations, tasks and problems”.

The current situation is too divided, specific and fragmented. In many cases, intelligence reduced to all sorts of optimization algorithms, problem-solving methods or just neural networks.

The most popular is the problem-solving strategies, which includes as their methods and techniques:

• trial-and-error problem-solving from biological evolution to reinforcement learning intelligence, say, in some agent-environment framework of action, perception and reward spaces. ?The agent and the environment interact by sending action, observation and reward signals to each other.?
• inductive reasoning, concrete or empirical intelligence, "from a part to a whole, from particulars to generals, from the past to the future, or from the observed to the unobserved", subdivided into causal?inference, categorical inference of conceptualization and generalization, and analogical inference.
• deductive?inferences of abstract intelligence, which?guarantees the truth of its conclusions, if the premises are true, the conclusion is true as well, no probable, or likely to be true. It is a higher-order thinking about complex concepts, ideas and principles, formal, symbolic, hypothetical, abstracted from concrete experiences, objects, people, or situations. Such a cognitive ability develops throughout the childhood as children gain new abilities, knowledge, and experiences, going through four distinct stages of intellectual development, as to J. Piaget.
• realistic thinking intelligence, which is about the external environment, involving the decision-making process to make a judgment with accuracy and efficiency. It underlies the ability to discriminate discrete objects or items of information (e.g., distinguishing a cat from a dog, a favorite subject of Google-like image software applications).?It include the concept formation, an intelligence’s ability to classify specific items following some strict rules, identifying physical properties, developing hypotheses about which of the specific dimensions define a class, arriving at the rules of class membership, and testing various hypotheses to form real-world concepts.
• divergent (or creative) thinking leading to new information, or previously undiscovered solutions of problems demanding flexibility, originality, fluency, and inventiveness.?

As for the types or kinds or classes of possible intelligences, they could broadly include:

• natural intelligence (biological, animal and human intelligence)
• artificial intelligence, machine intelligence, cognitive intelligence, alien intelligence
• general/universal intelligence with a general intelligent mechanism transcending all the specific operations and functions of artificial or human intelligence, as perception, cognitive processing, learning, reasoning, decision-making, and action.

Natural Intelligence

Many intelligence algorithms mimic natural phenomena such as how animals organize their lives, how they use instincts to survive, how generations evolve, how the human brain works, and how humans learn.

Such nature-inspired optimization algorithms (NIOAs) are defined as a group of algorithms that are inspired by natural phenomena, including swarm intelligence, biological systems, physical and chemical systems, etc. NIOAs include bio-inspired algorithms and physics- and chemistry-based algorithms; the bio-inspired algorithms further include swarm intelligence-based and evolutionary algorithms.

NIOAs are an important branch of artificial intelligence (AI), and NIOAs have made significant progress in the last 30 years. Thus far, a large number of common NIOAs and their variants have been proposed, such as genetic algorithm (GA), particle swarm optimization (PSO) algorithm, differential evolution (DE) algorithm, artificial bee colony (ABC) algorithm, ant colony optimization (ACO) algorithm, cuckoo search (CS) algorithm, bat algorithm (BA), firefly algorithm (FA), immune algorithm (IA), grey wolf optimization (GWO), gravitational search algorithm (GSA), or harmony search (HS) algorithm.

Besides, computer scientists have designed many AI algorithms by imitating human intelligence with machines, going as artificial human intelligence or human AI. It could be exampled by reinforcement learning (RL) algorithms by trial and error with penalties and rewards and neural networks algorithms (ANNs, machine learning algorithms that mimic the way human neurons communicate with one another), as imaged below.

The idea is to apply the components of mathematical model,?variables or decision parameters; constants and calibration parameters; input parameters, data; phase parameters; output parameters; noise and random parameters. And you have to reduce all the richness of human intelligence to the "learning from data (experience)". “Intelligence is a very general mental capability that, among other things, involves the ability to reason, plan, solve problems, think abstractly, comprehend complex ideas, learn quickly and learn from experience.”

Real Intelligence

In all, human/animal Intelligence is identified with the causal power [ability, capacity or faculty] of thought and consciousness and reason and intellect, known as the mind or brains of a sentient organism that direct and influence mental and physical behavior.

As a matter of fact, intelligence is the major part of an intelligence that enables it to be aware of the world and its knowing, experiences and causal interactions.

So, to have a mathematical model of universal intelligence means to develop the conceptual model of reality/universe/nature/world, its causality and mentality and its science as world’s data and knowledge.

In other words, real intelligence measures an intelligence’s power to effectively interact with the world, including any complex environments.

Then artificial intelligence is real/natural/causal intelligence produced artificially, by humans and/or machines.

Today, the most popular idea of machine intelligence is mistakenly related with big data and statistics and data science and engineering. N. Chomsky argued that the field's heavy use of statistical techniques to pick regularities in masses of data is unlikely to yield the explanatory insight that science ought to offer. And, “the "new AI"- focused on using statistical learning techniques to better mine and predict data - is unlikely to yield general principles about the nature of intelligent beings or about cognition. ..."

Mathematical models of intelligence has no real power and validity without modeling reality and its causality, where a?model?is a simplifying image of reality or a simplified abstract view of a complex reality, as with a scientific model.?

As far as the scope and source of intelligence is the whole world of reality, in all its varieties and complexities, the mathematical model/representation of reality is a mathematical theory of intelligence. It needs to be completed with scientific modeling and conceptual models, all integrated by the causal model of interacting reality and mentality.

Real intelligence is not "what is measured by intelligence tests,” but what is causally interacting with the world, understanding, adjusting and manipulating it in the most efficient ways.

Resources

Trans-AI: How to Build True AI or Real Machine Intelligence and Learning

Reality > Causality > Superintelligence: the last invention of man, or how Trans-AI takes over the world

Real Superintelligence (RSI): Disrupting ML, DL, ANI, AGI, ASI