# 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