11.5 Mathematics and geometry

Ancient Greek mathematicians revolutionized the field, shifting from practical applications to abstract thinking and deductive reasoning. They developed axiomatic systems and proofs, setting the foundation for modern mathematics. Their work influenced philosophy, architecture, and astronomy.

Greek mathematical traditions, including Pythagoras' theorem and Euclidean geometry, had far-reaching impacts. These concepts found practical applications in architecture, engineering, and navigation. The emphasis on logical reasoning and proof became the standard for establishing mathematical truth, shaping future scientific inquiry.

Origins of mathematics

• Mathematics originated from the practical needs of ancient civilizations to count, measure, and calculate
• Early mathematical concepts developed independently in various cultures (Mesopotamia, Egypt, India, China) before spreading and influencing each other
• The development of number systems and arithmetic operations laid the foundation for more advanced mathematical thinking

Early counting systems

Top images from around the web for Early counting systems

• 

  Early Counting Systems | Mathematics for the Liberal Arts View original

  Is this image relevant?

• 

  Early Counting Systems | MA 124 Contemporary Mathematics View original

  Is this image relevant?

• 

  Early Counting Systems | Mathematics for the Liberal Arts View original

  Is this image relevant?

• 

  Early Counting Systems | Mathematics for the Liberal Arts View original

  Is this image relevant?

• 

  Early Counting Systems | MA 124 Contemporary Mathematics View original

  Is this image relevant?

1 of 3

Top images from around the web for Early counting systems

• 

  Early Counting Systems | Mathematics for the Liberal Arts View original

  Is this image relevant?

• 

  Early Counting Systems | MA 124 Contemporary Mathematics View original

  Is this image relevant?

• 

  Early Counting Systems | Mathematics for the Liberal Arts View original

  Is this image relevant?

• 

  Early Counting Systems | Mathematics for the Liberal Arts View original

  Is this image relevant?

• 

  Early Counting Systems | MA 124 Contemporary Mathematics View original

  Is this image relevant?

1 of 3

• Ancient cultures used various objects (pebbles, shells, knots on a string) to keep track of quantities
• Tally marks and notches on bones or sticks served as early forms of recording numbers
• The development of verbal counting systems and number words facilitated the communication and manipulation of quantities

Development of number systems

• Mesopotamian cuneiform numerals represented numbers using a base-60 system (sexagesimal)
• Egyptian hieroglyphic numerals used a base-10 system with distinct symbols for powers of ten
• Greek and Roman numerals adapted and refined earlier number systems, with the Greeks using a modified base-10 system (Ionian numerals) and the Romans using a combination of letters and symbols (I, V, X, L, C, D, M)

Emergence of arithmetic and algebra

• Basic arithmetic operations (addition, subtraction, multiplication, division) developed to solve practical problems (distributing goods, calculating areas)
• Egyptians and Babylonians used tables and algorithms to perform complex calculations (multiplication, division, square roots)
• Algebraic concepts (solving linear equations, quadratic equations) emerged in response to more advanced mathematical problems (land surveying, tax collection)

Greek mathematical traditions

• Greek mathematics built upon the knowledge of earlier civilizations but placed a greater emphasis on abstract thinking and deductive reasoning
• Greek mathematicians sought to understand the underlying principles and relationships between mathematical concepts
• The development of axiomatic systems and mathematical proofs distinguished Greek mathematics from the more practical approaches of earlier cultures

Pythagoras and his followers

• Pythagoras (c. 570-495 BCE) founded a philosophical school that combined mathematical study with mystical and religious beliefs
• Pythagoreans believed that numbers and their relationships held the key to understanding the universe
• Pythagorean theorem ($a^2 + b^2 = c^2$) for right triangles was a major discovery, with far-reaching applications in geometry and trigonometry

Euclidean geometry

• Euclid (c. 300 BCE) compiled and systematized earlier geometric knowledge in his influential work "Elements"
• Euclidean geometry is based on a set of axioms and postulates from which all other geometric propositions can be logically derived
• Euclid's five postulates (parallel postulate, equality of right angles, equality of all right angles, congruence of triangles, and the whole is greater than the part) served as the foundation for geometric reasoning

Archimedes' contributions

• Archimedes (c. 287-212 BCE) made significant advances in geometry, mechanics, and hydrostatics
• Developed formulas for the volume and surface area of a sphere and cylinder
• Used the method of exhaustion to approximate the value of pi ($\pi$) and calculate the area under a parabola

Apollonius and conic sections

• Apollonius of Perga (c. 262-190 BCE) wrote a treatise on conic sections (circles, ellipses, parabolas, hyperbolas)
• Defined conic sections as the intersections of a plane with a double cone
• Developed a system of naming and classifying conic sections based on their properties

Practical applications

• Greek mathematics found practical applications in various fields, demonstrating the utility of abstract mathematical concepts
• Mathematical knowledge facilitated the design and construction of complex structures, the understanding of celestial phenomena, and the exchange of goods and services

Architecture and engineering

• Geometric principles informed the design of temples, public buildings, and infrastructure (Parthenon, aqueducts)
• Proportion and symmetry, derived from mathematical ratios, were key elements of Greek architecture
• Engineers used mathematical calculations to determine the strength and stability of materials and structures

Astronomy and navigation

• Greek astronomers used geometry to model and predict the motions of celestial bodies
• Eratosthenes (c. 276-194 BCE) calculated the circumference of the Earth using geometric methods
• Hipparchus (c. 190-120 BCE) developed trigonometric tables and a system of latitude and longitude for navigation

Commerce and trade

• Mathematical concepts (fractions, percentages, interest) were essential for financial transactions and record-keeping
• Merchants used arithmetic to calculate prices, exchange rates, and profits
• Governments employed mathematicians to manage tax collection, land distribution, and public finances

Mathematical tools and techniques

• Greek mathematicians developed and refined various tools and techniques to facilitate their work
• These tools and techniques allowed for greater precision, consistency, and rigor in mathematical investigations

Greek numeral system

• Greeks used a modified base-10 system (Ionian numerals) with symbols for units, tens, and hundreds
• Alphabetic numerals, assigning numerical values to letters of the Greek alphabet, were also used
• The Greek numeral system facilitated arithmetic calculations and the representation of large numbers

Use of compass and straightedge

• Compass and straightedge were the primary tools used in geometric constructions
• Straightedge used to draw straight lines between points
• Compass used to draw circles and arcs, and to transfer distances
• Geometric constructions (bisecting angles, constructing perpendicular lines) relied on the use of these tools

Mathematical proof and deduction

• Greeks emphasized the importance of logical reasoning and deductive proof in mathematics
• Mathematical statements were derived from a set of axioms and postulates using logical arguments
• Indirect proof (reductio ad absurdum) used to demonstrate the truth of a statement by showing that its negation leads to a contradiction
• Deductive reasoning became the standard for establishing mathematical truth and distinguishing it from empirical observation

Influence on philosophy

• Greek mathematics had a profound influence on philosophical thought, shaping ideas about knowledge, reality, and reasoning
• Philosophers drew upon mathematical concepts and methods to explore questions of truth, beauty, and the nature of the universe

Mathematics as ideal of knowledge

• Plato (c. 428-348 BCE) viewed mathematics as the highest form of knowledge, leading to understanding of eternal truths
• Mathematical objects (numbers, geometric figures) seen as perfect, unchanging forms that exist independently of the physical world
• Engaging in mathematics considered a means of training the mind to contemplate abstract ideas and reason logically

Plato's theory of forms

• Plato's theory of forms held that the physical world is an imperfect reflection of a realm of perfect, eternal forms
• Mathematical objects (circle, triangle) seen as examples of perfect forms that exist in this higher realm
• Studying mathematics considered a way to access and understand the true nature of reality

Aristotelian logic and reasoning

• Aristotle (384-322 BCE) developed a system of formal logic based on deductive reasoning
• Aristotelian logic used syllogisms (arguments consisting of a major premise, a minor premise, and a conclusion) to derive new knowledge from established truths
• Mathematical proofs served as a model for the rigorous application of logic and reasoning in other fields of inquiry

Legacy and transmission

• Greek mathematical knowledge was preserved, translated, and built upon by subsequent civilizations
• The transmission of Greek mathematics to the Islamic world and beyond had a lasting impact on the development of science and philosophy

Preservation in Islamic world

• During the Islamic Golden Age (8th-14th centuries), scholars translated and studied Greek mathematical texts
• Islamic mathematicians (Al-Khwarizmi, Omar Khayyam) made significant contributions to algebra, trigonometry, and numerical analysis
• Preservation of Greek mathematical knowledge in the Islamic world ensured its survival and continued influence

Translation and dissemination

• Greek mathematical texts were translated into Arabic, Persian, and Latin, making them accessible to scholars across the Mediterranean world
• Translations of Euclid's "Elements" and other works spread Greek mathematical ideas to Europe, North Africa, and Asia
• Transmission of Greek mathematics facilitated the exchange of knowledge between different cultures and intellectual traditions

Impact on Renaissance and beyond

• Rediscovery of Greek mathematical texts during the Renaissance (14th-17th centuries) fueled advances in science and philosophy
• Renaissance mathematicians (Fibonacci, Cardano, Descartes) built upon Greek ideas to develop new concepts and techniques (decimal fractions, probability theory, analytic geometry)
• Greek emphasis on deductive reasoning and mathematical proof influenced the development of the scientific method and modern mathematics
• Legacy of Greek mathematics continues to shape the way we understand and apply mathematical knowledge in various fields (physics, engineering, economics)