In a previous blog post, we discussed the relationship between Artificial Intelligence and Philosophy. We argued that Artificial Intelligence constitutes a disciplinary field where efforts have been made to create intelligent entities like humans from various perspectives. Still, at the same time, it involves the tools and applications that have gained so much popularity today.
We emphasized that the development of Artificial Intelligence is a matter that should be taken with utmost seriousness due to the potential risks involved. Furthermore, we maintained that a proper understanding would result in better regulation, ensuring the safety and well-being of humanity while fostering the development of various AI applications.
Next, we will discuss the mathematical foundations of Artificial Intelligence according to Stuart RUSSELL and Peter NORVIG. Our authors explain that mathematics is another foundational disciplinary field of AI in their book Artificial Intelligence: A Modern Approach.[i] In this regard, we will analyze the contributions made from mathematics to logic since the 19th century. These mathematical developments facilitated its application in computer science and, over time, contributed to AI.
While RUSSELL and NORVIG primarily discuss the relationship between mathematics and logic and its influence on the development of Artificial Intelligence, it is also important to note that statistics, probability, linear algebra, and calculus form the foundations of AI. This time, we will continue with the exposition from the book Artificial Intelligence: A Modern Approach for the purposes outlined in the first post in this series.
Algorithm: A Fundamental Concept
The stream of logical positivism contributed to the intellectual landscape that allowed the development of Artificial Intelligence during the 20th century. The idea of a theory of the mind as a computational process was championed by Rudolf CARNAP (1891-1970). The German philosopher argued that knowledge comprises a set of logical theories linked to sensory observation sentences, as outlined in the previous blog post.
In the year 825, the Persian polymath Muhammad ibn Musa al-Khwarizm (780-850) wrote a manual on algebra aimed at solving linear and quadratic equations. This book was translated into Latin as Liber Alghoarismi de practica arismetrice in the early 12th century. It is important to note that the term ‘alghoarismi‘ or ‘algorismi,’ which evolved into ‘algoritmo’ in Spanish, is derived from the Latinized version of the name al-Khwarizm.[ii]
Then, algorithms are instructions to solve problems, perform calculations, or process data. At the same time, the concept of an algorithm concerning artificial intelligence is prominent in mathematics; RUSSELL and NORVIG present other ideas that shaped logic during the 19th and 20th centuries. Later, in connection with the development of logic, we will discuss some mathematical concepts that play a significant role in AI.
Logic and Mathematics
The English mathematician George BOOLE (1815-1864), alongside other thinkers, discussed the concept of an algorithm within the framework of deductive logic. Meanwhile, the German mathematician Gottlob FREGE (1848-1925) incorporated objects and relations into BOOLE‘s discussions on logic.
Additionally, RUSSELL and NORVIG point out that mathematical ideas from the 20th century also contributed to the development of Artificial Intelligence. In this context, the Polish-American Alfred TARSKI (1901-1983) demonstrated “how to relate the objects in logic to objects in the real world” through his theory of reference.[iii]
Moreover, the Austro-Hungarian Kurt GÖDEL (1906-1978) formulated in 1930 ‘an effective procedure to prove any true statement in the first-order logic of FREGE and [Bertrand] RUSSELL (1872-1970), but that this first-order logic could not capture the principle of mathematical induction needed to characterize the natural numbers.’[iv]
Later, GÖDEL demonstrated the limits of deduction. Then, through his incompleteness theorem, this mathematician confirmed that “in any formal theory […], there are true statements that are undecidable in the sense that they have no proof within the theory.[v] According to the incompleteness theorem, it can be proven that ‘some functions on the integers cannot be represented by algorithms -meaning- they cannot be computed.’[vi]
Turing Machine and Gödel’s Theorem
In 1936, the English mathematician Alan TURING (1912-1954) invented a model that allowed the analysis of the functioning of a computational system. The term ‘Turing Machine‘ was coined by the American mathematician Alonzo CHURCH (1903-1995), who served as TURING‘s doctoral thesis advisor.[vii] This model aimed to demonstrate the existence of fundamental limits on the power of mechanical computation.[viii]
In this sense, TURING sought to characterize computable functions based on Gödel‘s incompleteness theorem. While the meaning of this concept cannot be easily defined, the Turing Machine – according to the CHURCH-TURING thesis – can compute computable functions of natural numbers or, at least, provide a sufficient definition.[ix] However, as RUSSELL and NORVIG explain, TURING also demonstrated the existence of functions that could not be computed.
A fundamental problem: tractability
According to the authors of Artificial Intelligence: A Modern Approach, in computer science, there are three critical problems related to this field: 1) decidability, 2) computability, and 3) tractability. Decidability means that “a decision problem is decidable if there is an effective method to derive a correct answer.”[x] Meanwhile, computability is “the ability to solve a problem effectively.” In computer science, this latter problem is studied with mathematical logic and the theory of computation.[xi]
Following RUSSELL and NORVIG, intractability is crucial in Artificial Intelligence, as it has a more significant impact than decidability or computability. According to our authors, a problem is intractable “if the time required to solve instances of the problem grows exponentially with the size of the instances.”[xii] Then, exponential growth implies “that not even moderately large instances can be solved in any reasonable time.” Therefore, it is better to “divide the general problem of generating intelligent behavior into tractable subproblems rather than intractable ones.”[xiii]
On the other hand, our authors explain that the NP-completeness theory solves the problem of intractability. The term NP-completeness derives from the concept of Non-deterministic Polynomial-time completeness, where the term non-deterministic involves the concept of a Non-deterministic Turing Machine. The latter, inherent to the computer science theory, signifies “a theoretical computational model whose governing rules specify more than one possible action in a given situation.”[xiv]
Similarly, RUSSELL and NORVIG point out that the theory of NP-completeness was developed by Stephen Arthur COOK (1939) and Richard Manning KARP (1935), who demonstrated “the existence of large classes of canonical search and combinatorial reasoning problems that are NP-complete.” Finally, according to our authors, any class of problems that can be reduced to NP-complete problems is intractable.[xv]
Moreover, probability is another branch of mathematics that impacts the field of Artificial Intelligence. Regarding this discipline, RUSSELL and NORVING only mention contributors to its development, such as Gerolamo CARDANO (1501-1576), Blaise PASCAL (1623-1662), James BERNOULLI (1654-1705), Pierre LAPLACE (1749-1827), and Thomas BAYES (1702-1761). While they present some basic elements of probability, they do not explicitly elaborate on how they have influenced Artificial Intelligence.[xvi]
Some Conjectures
Artificial Intelligence is a complex field where diverse disciplines such as mathematics, computer science, linguistics, and neuroscience intersect. This disciplinary field’s scientific and engineering framework is justified by its foundational objective: creating intelligent entities. Humans are currently (supposedly) the most intellectually advanced living beings, which explains the need for various disciplines to converge in attempting to develop entities similar to us.
In the previous blog post, when analyzing the relationship between Philosophy and Artificial Intelligence, we identified – following RUSSELL and NORVIG – an intellectual trend in the philosophical landscape that contributed to this disciplinary field. In this regard, philosophical ideas are significant as they allow us to understand the complex intellectual and social landscape within which Artificial Intelligence develops.
From the perspective of intellectual history, one must first comprehend its linguistic context to understand the meaning of an idea. According to Quentin SKINNER, it implies that it is necessary to capture the intentional force with which the expression is uttered to grasp the definition of a specific term. Therefore, it is essential to know what the author was doing by using that term in addition to the meaning of what was said.[x]
On the other hand, in this instance, the influence of mathematics on Artificial Intelligence can be discerned. The best example of this is Gödel‘s incompleteness theorem and how, based on it, Alan TURING sought to formulate a meaning for computable functions through his machine. This example from Stuart RUSSELL and Alan NORVIG allows us to understand the impact of certain theses emerging in one disciplinary field on others.
We reiterate that the goal of these blog posts by IP Consultores is to contribute to understanding Artificial Intelligence‘s state of the art. Ours are mere conjectures intended to encourage more people to delve into the intricate intellectual landscape of this disciplinary field. Our approach is significant as it promotes the idea that the debate on the implications of AI is a crucial matter in the early 21st century.
Ultimately, the more elements we must comprehend its scope, limitations, risks, and benefits, the more beneficial the public discussion will be for everyone. Therefore, despite my intellectual limitations, I only aim to contribute another perspective in this vast field called Artificial Intelligence.
[i] RUSSELL, Stuart, and NORVIG, Peter, Artificial Intelligence. A Modern Approach, 3rd ed., Pearson Education, New Jersey, 2010, pp. 7-8.
[ii] KATZ, Victor J., Stages in the History of Algebra with Implications for Teaching, Educational Studies in Mathematics, 2007, p. 190.
[iii] RUSSELL, Stuart, and NORVIG, Peter, Artificial Intelligence. A Modern Approach, 3rd ed., Pearson Education, New Jersey, 2010, pp. 7-8.
[iv] ibíd, p. 8
[v] ibídem.
[vi] ibídem.
[vii] Wikipedia, Turing machine, https://cutt.ly/MwZNLEI9, website consulted on January 31, 2024
[viii] ibídem.
[ix] RUSSELL, Stuart, and NORVIG, Peter, Artificial Intelligence. A Modern Approach, 3rd ed., Pearson Education, New Jersey, 2010, p. 8.
[x] Wikipedia, Decidability, https://cutt.ly/6wX6Glv8, website consulted on February 6, 2024.
[xi] Wikipedia, Computability, https://cutt.ly/EwX6DLLW, website consulted on February 6, 2024.
[xii] RUSSELL, Stuart and NORVIG, Peter, op. cit., note 1, p. 8.
[xiii] ibídem.
[xiv] Wikipedia, Non-deterministic Turing machine, https://cutt.ly/xwX6FnPE, website consulted on February 6, 2024.
[xv] RUSSELL, Stuart and NORVIG, Peter, op. cit., note 1, pp. 8-9.
[xvi] ibíd., p. 9.
[xvii] SKINNER, Quentin, Visions of Politics. Volume I: Regarding Method, Cambridge University Press, 2002, p. 82.
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