🦖 History of Mathematics & Logic

🦕 Chronological Development

• 🦕 Who discovered zero? Can you replicate ancient geometric proofs?

🦤 Biographical Anecdotes

• 🦤 “Eratosthenes measuring Earth’s circumference,” solving ancient math challenges.

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┼╌ ≈ 500 BCE: The first pure mathematician began to appear in Greece, India, and China.
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┼╌ 1501-1576 Gerolamo Cardano
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┼╌ 1550-1617 John Napier (Logarithms)
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┼╌ 1596-1650 René Descartes
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┼╌ 1601-1665 Pierre De Fermat
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┼╌ 1707-1783 Leonard Euler
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┼╌ 1811-1832 Evariste Galois
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┼╌ 1912-1954 Alan Turing
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In prehistoric times humans were already able to quantify. This was very useful for survival (More food, good. Bigger tribe, avoid. Tiger bigger than rabbit.).

They also were able to identify different shapes and sizes and had some numeric concepts. Counting, addition, subtraction...

From then, the history of mathematics is a story of discovering. As principles were established (by nature or thought), the emergence of properties was followed by those explorers we call mathematicians.

These laws emerged from the axioms into universal, eternal, and unchanging laws that transcend specific realms of existence, as only an equanimity on the premises produces an equanimous result. 2+2 = 4 is true for berries, mammoths, and independent of human minds to count them.

It quickly evolved along to commerce, stock and storage, dimension valuing such as weight and length measurements, area of land (geometria).

Societies also evolved to be benefited by these discoveries, that got more complex, adding multiplication and division to the basis of mathematics, and roads, canals, and pyramids to these societies.

New applications got discovered, such as astronomy, navigation, engineering, bookkeeping... New patterns and ideas emerged.

🍎 Counting is the most primitive example of mathematics. And it was clearly something humans discovered about the world, rather than invented. The number of apples in a basket obeys the rules of arithmetic. So, if I add one apple, the number in the basket increases by one. This has always been the case, whether people were aware of the fact or not.

🦴 Early societies had a limited vocabulary for counting: ‘one, two, three, many’ is associated with some primitive societies, and indicates primitive computation and scientific ability.

🫱🏼 The early societies would typically have employed their fingers for counting. As such, societies that counted fingers have a 5 "influence". Others counted the phalanxes in the fingers (the lines in the fingers) have a tendency for 12. It is remarkable that no society is known to count also fingertips, which could have been a 16 base (so related to binary.

🥖 Oddly enough, the French (who else) use a completely off base, the base 20. It is to suppose early gales would count with the toes too.

👨🏻‍🎨 This can only be shown how ridicule it is with a joke:

The math teacher asked the french teacher to help with maths for an exam. For a simplification exercise 4*20 + 10, a student left it blank. The French teacher marked it as correct!

Lebombo bone

🦴 The Lebombo bone, named after the Lebombo Mountains in Africa where it was discovered by archaeologists in the 1970s, is a lower leg bone from a baboon. It’s special because, deliberately cut into its length, are twenty-nine notches, believed to be tally marks.

🦣 The bone has been radiocarbon dated to around the year 40,000 BC, making it the world’s oldest mathematical artefact.

🕵🏻‍♀️ The most plausible reason for these markings is counting the days of a menstrual cycle or moon phases.

🦴 The Ishango Bone shines further light into the understanding of early humans of basic arithmetic, such as multiplication and division patterns, as well as some number properties and some primes.

The Levante

First numeric systems in recorded history belong to the Mesopotamian Empire. It related to grain storage and taxation.

Babylonians

🌾 The Babylonian civilization flourished in the crescent valley of the Tigris and Euphrates rivers. This civilization lasted from 2000Bc to 300 BC.

𒀱 Various cuneiform tablets containing mathematical texts have been discovered. Including tables of multiplications, divisions, squares, cubes, square roots, Pythagorean triples...

🕰 The base 60 (sexagesimal, still in common use for time measures) was common.

🧙🏼‍♂️ The Babylonians were able to represent arbitrarily large numbers or fractions with just one marking.𒐕

𒐕 𒀸 𒐑𒐄𒐞𒐂𒐥

🌙 They later developed works on astronomy too. They could predict eclipses and some other astronomical events. They associated some bodies with de divine and mapped the zodiac and other night sky areas with mystical undertones. The priests were called Magi, where the word Magic comes from.

Egipcian

𓀬 They used symbols for powers of tens as quantity, and repetition for additional units.

𓆐𓆐𓂮𓆾𓍦𓎊𓐀

𓀙𓀗 The Egyptians were familiar with geometry, arithmetic and elementary algebra. They had techniques to find solutions to problems with one or two unknowns.

𓁟 They also represented fractions

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The Greeks

🏛 The Greeks made major contributions to western civilization including mathematics, logic, astronomy, philosophy, politics, drama and architecture.

Thales and Pythagoras applied logical arguments to mathematics.

Socrates and Plato applied logical types of reasoning to philosophical questions.

Pythagoras

🧔🏽‍♂️ Pythagoras was a philosopher and mathematician who had spent time in Egypt becoming familiar with Egyptian mathematics. ç

🛐 He lived on the island of Samos, and formed a secret society known as the Pythagoreans. They included men and women and believed in the transmigration of souls, and that number was the essence of all things. They discovered the mathematics for harmony in music with the relationship between musical notes being expressed in numerical ratios of small whole numbers.

📜 Pythagoras' obsession with formality conducted him to develop the first mathematical proof: The Pythagorean Theorem.

🔳 It is believed that the first person to prove the existence of irrational numbers was a member of the Sect. Specifically, he built a proof for the square root of 2 not to be rational. He disappeared after going for a walk with other sect members.

Euclides

🏺 Provided a systematic foundation for geometry, in around 300 BC.

📐 His work is known as ‘The Elements’ and consists of 13 books. The early books are concerned with the construction of geometric figures, number theory and solid geometry. He set out the fundamental postulates describing the nature of circles, straight lines, and angles. These were constructed by the rules of logic and inference.

Aristoteles

Logic as a discipline was founded by Aristotle and developed by philosophers for centuries.

• Aristotle invents syllogistic logic in Organon, which means tool
• Using geometry as his model, he saw that science consisted of proofs, proofs of syllogisms, syllogisms of statements, and statements of terms.
• • Categories: terms
• • Interpretation: statements
• • Prior Analytics: syllogisms (argument structures that, by their very design, appear to be indisputably valid.)
• • Posterior Analytics: proofs.

In a syllogism, the premises and conclusions fit together in such a way that, once you accept the premises as true, you must accept that the conclusion is true as well — regardless of the content of the actual argument being made.

• Premises:
  • All men are mortal.
  • Socrates is a man.
• Conclusion:
  • Socrates is mortal.

Classical China

🧮 The earliest known example of what we today recognize as the hand-held abacus was invented in China approximately 5,000 years ago. Consisting of wood and moveable beads, this counting tool assisted its human operator by keeping a running total of items added.

Classical India

The oldest known ancestors of the modern symbolic system were found in caves and on coins around Bombay dating back to the 1st century AD

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㆓ 2
㆔ 3
＋ 4
Ի 5
५ 6
੭ 7
ᔕ 8 
𞋶 9

Medieval Europe

🗡 During the Middle Ages, arithmetic was taught in European universities as one of the seven liberal arts, alongside grammar, logic, rhetoric, geometry, music, and astronomy

Al-Khwarizmi, father of Algebra

👳🏽‍♂️ Abu Abdullah Muhammad ibn Musa al-Khwarizmi contributed greatly to the language and concepts of algebra. Around A.D. 820.

🌘 In addition to his resounding developments in various fields of mathematics, he also contributed greatly to astronomy, geography, the inner workings of clocks, and the degree measurements of angles.

🌒 Khawarizmi developed the basis for modern arithmetical notation, based on the Hindu numerals. He was the first to spread the decimal number system, now commonly referred to as Arabic numerals. He also introduced the idea of zero to Arabs and Europeans.

؎ Kitab al-Jabr w’al- Muqabala, was the first book on Algebra. The word algebra is derived from the term al-jabr, which can be taken to mean “reunion of broken parts,” “reduction,” “connection,” or “completion.” The rest of the title loosely translates to “to set equal to” or “to balance.” 𓍝

Fibonacci

Leonardo da Pisa (aka Fibonacci)(c. 1170–1250), an Italian mathematician, traveller, and tradesman, discovered that the potential of algebraic computations using the Hindu (Arabic) notation for numbers far exceeded the capacities of the Roman numeral system that was standard in Europe at that time.

Descartes

Descartes was the discovery of relationships between geometric measurements and algebraic methods, now referred to as analytic geometry.

Pascal

⚙️ In 1642, a French mathematician Blaise Pascal (1623–1662) invented the first mechanical adding machine.

Renaissance

Newton

Leibnitz

Complex Numbers

Pi

Modern mathematics

About 150 years ago, mathematicians found that logic was an indispensable tool for grounding their work as it became more and more abstract.

As a basis for understanding logic, philosopher Bertrand Russell set down three laws of thought. These laws all have their basis in ideas dating back to Aristotle, who founded classical logic more than 2,300 years ago.

Computer Era