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This topic may be more interesting than you thought!

What is it?

Theory of Knowledge (TOK) provides an opportunity for students to reflect on the nature of knowledge, and on how we know what we claim to know. It is a mandatory course for all students taking the IB Diploma Program, and this course will twist your mind upside down, back and forth 180° as you go along the 2-year program. Are the axioms in mathematics even true? Or does mathematics don’t even exist at all.

Theory of knowledge 1

What are the platonic solids and why are they an important part of the language of math?

PLatonic solids

Platonic solid, any of the five geometric solids whose faces are all identical , regular polygons meeting at the same three-dimensional angles. Tetrahedron (or triangular pyramid), cube, octahedron, dodecahedron, and icosahedron are all platonic solids, also known as five regular polyhedra.

The tetrahedron, cube, and dodecahedron first arise by Pythagoras, while the octahedron and icosahedron were first discussed by the Athenian mathematician Theaetetus. However, the entire group of regular polyhedra owes its popular name to the great Athenian philosopher Plato.

Cube

Dodecahedron

Icosahedron

Octahedron

Tetrahedron

According to Plato, he equated:

• The tetrahedron to the element ‘fire’ (the only element that gives heat, and is responsible for all transformative processes such as digestion).
• The cube to the element ‘earth’ (the most dense element, the stuff of rock and bone, giving solidity and form).
• The icosahedon to the element ‘water’ (the universal solvent and the ruling archetype for all fluids, including rivers and oceans of course, but also plasma and mucous).
• The octahedron to the element ‘air’ (the element of mobility, carrying ideas, inspiration and disease around the cosmos and our bodies).
• The dodecahedron that corresponds to ‘ether’, or ‘akash’, is commonly called “space,” and that is accurate insofar as it is everywhere around and within us. It is the vacuum; the matrix; the space through which we move; the three-dimensional field in which we arise, survive and transform. It is within the ethereal container that the other four elements dance and intertwine.

Theory of knowledge 2

To what extent do instinct and reason create knowledge?

In TOK, the use of instinct and reason are influenced by emotions, perception, culture, and experience on how people will make decisions in life and in every situation. For instance, people may act and make decisions differently when they act upon instinct or when explaining by using logical reasoning. An example for a real-life situation is when a stove catches on fire. You will instinctively splash water to the stove, while the right thing to do is to cover the fire with a wet cloth to put it out by using reasoning.

Emotions tell you how to immediately react to a situation, whether you are scared, nervous, or excited. This emotional reaction then tells you how to perceive a situation. Culture also plays a huge role in how your instincts cause you to react to a situation. Where you grow up and who you grow up with influence how you react. All three of these categories build up the experience of an individual. This is where instinct transitions into reasoned and logical reactions. The more experience you gain, the more likely you are to react out of reason and less out of instinct. This will let you be more knowledgeable about the particular situation, when you are more experienced that will lead you to use reasoning and instinct together to create the best possible outcome.

Do different geometries [Euclidean and non-Euclidean] refer to or describe different worlds?

Euclidean Geometry

Euclidean geometry, the study of plane and solid figures on the basis of axioms and theorems employed by the Greek mathematician Euclid. This type of geometry are taught in schools before non-Euclidean geometry attracted the attention of mathematicians on the second half of the 19th century.

Euclid proposed five unprovable but intuitive principles known variously as postulates or axioms. The axioms are as follows:

• Given two points, there is a straight line that joins them.
• A straight line segment can be prolonged indefinitely.
• A circle can be constructed when a point for its centre and a distance for its radius are given.
• All right angles are congruent.
• If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, will meet on that side on which the angles are less than the two right angles.

Non-Euclidean Geometry

A non-Euclidean geometry is a rethinking and redescription of the properties of things like points, lines, and other shapes in a non-flat world. Anything that opposes Euclidean geometry.

The essential difference between Euclidean geometry and these two non-Euclidean geometries is the nature of parallel lines: In Euclidean geometry, given a point and a line, there is exactly one line through the point that is in the same plane as the given line and never intersects it, whereas non-Euclidean geometry oppose it.

Theory of knowledge 3

Is it ethical that Pythagoras gave his name to a theorem that may not have been his own creation?

pythagoras theorem

The Pythagoras Theorem, named after the Greek mathematician Pythagoras states that: If we let c be the length of the hypotenuse (longest side of the right-angled triangle) and a and b be the lengths of the other two sides, the theorem can be expressed as the equation: a² + b² = c².

“Pythagoras wrote nothing, nor were there any detailed accounts of his thought written by contemporaries. By the first centuries BCE, moreover, it became fashionable to present Pythagoras in a largely unhistorical fashion as a semi-divine figure, who originated all that was true in the Greek philosophical tradition, including many of Plato’s and Aristotle’s mature ideas. A number of treatises were forged in the name of Pythagoras and other Pythagoreans in order to support this view.”

By the statement shown above, Pythagoras’ achievements were unclear and inaccurate. Some of his past works remains controversial on some of Pythagoras’ engagements in certain philosophies and theorems. His past works are stated to be doubtful that he had done it by himself, and it is not likely that it is he that proved the theorem. Therefore, it may have not been ethical to be named the “Pythagoras Theorem” as it may be proven by other philosophers and mathematicians other than himself.

USeful Quote

“Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.” 

– Galileo Galilei (1564 – 1642)