A Brief History of Mathematical Logic
Logic formalises the concepts we often take for granted in mathematics: proofs, truth, and statements are some of these core foundations, upon which we build all mathematical research. This article explores the history of mathematical logic ― we trace a path from the work of ancient scholars to our present, diverse branches of study.
The Ancient East
Developments in Ancient Eastern mathematical logic were largely clustered in just two countries, China and India.
“Who really knows?
Who will here proclaim it?
Whence was it produced? Whence is this creation?
The gods came afterwards, with the creation of this universe.
Who then knows whence it has arisen?”
― Nasadiya Sukta, on the origins of the universe
India
Indian logicians were generally motivated by religion. Vaisheshika and Nyaya were two of six contemporary Hindu schools of philosophy; Vaisheshika proposed that all objects in the universe were reducible to a finite number of atoms, much like how complex plane Euclidean geometry theorems are constructible from just five fundamental axioms, while the Nyaya school distinguished itself by accepting testimonies and analogies as valid in logical argument. Catuṣkoṭi, also known as the tetralemma, was a system refined by the Buddhist philosopher Nagarjuna and was a predecessor of our modern Boolean logic. Jain and Buddhist logic tended to be concerned with the nature of knowledge ― how is knowledge derived, and when is it reliable?
China
In China, the Mohist school of thought reigned supreme. Founded by the philosopher Mozi, it dealt with issues relating to valid inference and the conditions of correct conclusions. Mohism was special in its preference for rhetorical analogies over mathematical reasoning; persuasive arguments and speech were preferred over drawing conclusions from the rigorous examination of evidence, which may sound familiar to the politically astute reader of the present. Later on, the Logicians grew out of Mohism ― they’re often credited with discovering formal logic, which abstracted away finnicky particulars and allowed a mathematical study of reasoning for the first time.
The Ancient West
“Valid reasoning has been employed in all periods of human history.” ― Wikipedia
Egypt and Babylon
The origins of mathematical logic in the West can be reasonably traced back to geometry. The word stems from the ancient Greek γεωμέτρης, meaning ‘land measurers’, which speaks to the practical testing of geometric formulas which were being discovered at the time ― logic may have developed as a more abstract version of this demonstration to show the validity of conclusions.
Equally, skilled mathematicians in ancient Egypt and Babylon may have developed precursors to modern mathematical logic. Esagil-kin-apli, the chief scholar of a Babylonian king in the 11th century BCE, produced a medical diagnostic handbook based on assumptions that we would now generally refer to as axioms. At the same time, Babylonian astronomers were using internal logic ― collections of consistent rules and reasoning ― to predict where planets would be in the night sky.
Mathematical Logic in Greece
Ancient Greece heralded the start of empirical methods being replaced with actual proof. Thales and Pythagoras were Greek philosophers who emphasised the connection between proving concepts and mathematical development: once you could prove abstract mathematical truths, it was acceptable to use those theorems to prove more complex ideas. This idea was very similar to the concept of deductive systems, in which one starts with given truths and considers what can be worked out.
Fragments of early proofs are still preserved in the works of Euclid:
Euclid of Alexandria came up with three basic principles of geometry, the abstraction of which forms a basis for mathematical logic:
- Certain propositions (statements) must be accepted as true without demonstration; such a proposition is known as an axiom of geometry.
- Every proposition that is not an axiom of geometry must be demonstrated as following from the axioms of geometry; such a demonstration is known as a proof or a “derivation” of the proposition.
- The proof must be formal; that is, the derivation of the proposition must be independent of the particular example in question.
“Let no one ignorant of geometry enter here.”
— Inscribed over the entrance to Plato’s Academy.
Plato was a fourth-century philosopher raised three questions about philosophical logic:
- What is it that can properly be called true or false?
- What is the nature of the connection between the assumptions of a valid argument and its conclusion?
- What is the nature of definition?
Plato suggested that the necessary connection between the assumptions of a valid argument and its conclusion corresponds to a necessary connection between “forms”, expanded upon in a philosophical concept aptly named the Theory of Forms. His works generally concern the search for a definition of some important concept — justice, peace, goodness, for example — and it is likely that he was impressed by how important definitions were in mathematics. Plato considered every definition to be a Platonic Form: the common nature present in different particulars. In mathematical logic, we understand this idea to be linked to abstraction; a theorem, once proved, can be applied to any appropriate context.
Aristotle
The work of Aristotle have had enormous influence in contemporary Western thought, being translated into Latin in the Middle Ages and employed as standard texts in many of Europe’s universities. He was the first formal logician, demonstrated the principles of reasoning by employing variables to show the underlying logical form of an argument.
“If it is rainy, then I will get wet if I’m outdoors” becomes “A implies B”; A and B are the variables corresponding to these statements, allowing a further abstraction that allows us to consider the logical argument context-free. He was also the first to distinguish the validity of these relations from the truth of the premises themselves, which could be false — as long as the reasoning is consistent, we consider the statements to be sound.
In addition, he produced a collection of logical works called the Organon, the earliest formal study of logic, discussed and analysed what makes a syllogism (a valid argument), and considered non-formal logic — the study of constructs including fallacies, which are invalid deductive arguments.
Medieval Europe and the Middle East
The works of Al-Kindi, Al-Farabi, Avicenna, Al-Ghazali, Averroes and other Muslim logicians were based on Aristotelian logic and were important in communicating the ideas of the ancient world to the medieval West. They explored conditional syllogisms, which were forms of valid argument based on conditions.
If I don’t wake up, then I can’t go to work. If I can’t go to work, then I won’t get paid. Therefore, not waking up means that I won’t get paid.
In Medieval Europe, Aristotle’s work was also the foundation of new insights. Christian philosopher Boethius contributed heavily to the development of Scholastic logic, a form of Aristotelian logic that was developed between 1200 and 1600 CE. Because Christianity was widespread at the time, these developments led to the application of logical techniques to attempt to prove theological concepts such as the existence of God.
The Rise of Modern Logic
In the mid-19th century, inspired by the methods of proof used in mathematics, there was a renewed interest in logic. Rigor was at the heart of this renaissance, leading to the names “symbolic” or “mathematical” logic that we employ today.
Modern logic’s rules of operation are determined only by the axioms and rules of valid inference we apply to them; we don’t start with ordinary language and try to find an equation which matches it — we start with abstract mathematics, draw conclusions, and then try to express our conclusions in ordinary language.
World War II led to an explosion of mathematical logic. It was too large and burgeoning to contain within just one field; logic diversified into model theory, proof theory, computability theory, and set theory, leading to a picture of logic today that is the composite of various intersecting disciplines.
Further exploration
- Stanford’s Introduction to Logic course on Coursera
- Notes on Mathematical Logic from the University of Illinois
