In this lesson, Role of Mathematics in Artificial Intelligence Part I, we are going to study the importance of mathematics in the field of artificial intelligence. As you can see, this is part I of the story, how the ideas related to mathematics in the theory of AI need more space and words, of course!
Mathematics is the most prominent pillar of Artificial Intelligence (AI) learning. Mathematics gives language to express scientific thoughts and helps in deducing formal logic. The groundwork laid by philosophers provided important foundational concepts for AI, but the transition towards a formal scientific discipline necessitated the mathematical formalization of logic and probability, as well as the emergence of a novel field of mathematics known as computation. These mathematical tools and frameworks were crucial in enabling the development of AI as a rigorous and systematic scientific endeavor. In this part, we will learn about the role of formal logic and probability theory in development of the Artificial Intelligence.
The roots of formal logic can be traced back to the ancient philosophers of Greece, India, and China. However, it was the pioneering work of George Boole (1815-1864) that marked a significant advancement in the mathematical development of logic. Boole meticulously formulated propositional, or Boolean, logic, laying the groundwork for its formalization (Boole, 1847). Building upon Boole’s contributions, Gottlob Frege (1848-1925) further expanded the scope of logic by incorporating objects and relations, giving rise to first-order logic, which remains a fundamental tool in modern times. The efforts of Boole and Frege played a crucial role in shaping the mathematical foundations of logic and establishing the framework for logical reasoning and inference in various fields, including artificial intelligence.
In addition to its central role in the early period of AI research, first-order logic also had a profound impact on the broader field of computation itself. The development of first-order logic not only provided a formal language for representing and reasoning knowledge but also served as a catalyst for groundbreaking work by logicians such as Kurt Gödel and Alan Turing. Gödel’s incompleteness theorems demonstrated the inherent limitations of formal systems, while Turing’s concept of the universal Turing machine laid the foundation for the theory of computation. These seminal contributions, influenced by the principles of first-order logic, fundamentally shaped our understanding of the capabilities and limitations of computational systems, paving the way for the development of powerful algorithms and computational models that are at the core of artificial intelligence today.
Predicting the Future: The Role of Probability Theory in AI
The theory of probability plays a pivotal role in AI as it extends the principles of logic to address situations involving uncertain information, and consideration of utmost significance. The origins of probability can be traced back to the pioneering work of Gerolamo Cardano (1501–1576), who initially conceptualized it in the context of gambling events and their potential outcomes. Building upon this foundation, Blaise Pascal (1623–1662) further contributed to the development of probability by demonstrating how to predict the future of an ongoing gambling game and assign average payoffs to the participants.
Probability swiftly gained prominence as an indispensable tool in the quantitative sciences, enabling the handling of uncertain measurements and incomplete theories. Notable figures such as Jacob Bernoulli (1654–1705), Pierre Laplace (1749–1827), and others advanced the theory and introduced novel statistical methods to analyze and interpret probabilistic phenomena. In this progression, Thomas Bayes (1702–1761) played a significant role by proposing Bayes’ rule, a fundamental principle for updating probabilities in light of new evidence. This rule has become a crucial tool in the field of AI, facilitating the incorporation of new information and enabling more accurate and adaptive decision-making processes. Probability provides a mathematical framework to quantify and reason about uncertainty, enabling AI systems to make informed decisions and handle complex problems. And, Bayes’ rule allows for the updating of probabilities based on new evidence, enabling AI systems to iteratively refine their beliefs and predictions. This iterative learning process, combined with probabilistic models, forms the basis of Bayesian networks and probabilistic graphical models, powerful tools used in AI for representing and reasoning under uncertainty.
The fusion of logic and probability, exemplified by first-order logic and probabilistic reasoning, has given rise to a rich framework for AI. This integration allows AI systems to combine logical reasoning with probabilistic inference, enabling them to handle uncertain information, make rational decisions, and adapt their behavior based on changing circumstances.
Mathematics is one of the foundational pillars of Artificial Intelligence, but this is not the only pillar. If you’re interested in reading more about AI and its development. Read AI & Neuroscience, AI & Economics, and AI & Control Theory. If you want to read more about mathematics and artificial intelligence see this survey paper. Enjoyed reading this article. Consider sharing with your friends and loved ones. Happy Learning!