# Imagination and reason in Mathematics

Philosophical Remarks

When did we first fall in love with mathematics? For me, it was in class 6.

My father exposed me to a problem from Euclidean geometry. We were traveling in Kausani.

After days of frustration and failed attempts, I could put together the ‘reason’ that made ‘everything fit together perfectly’. The problem was solved and beauty of ‘pure reason’ revealed itself. It was breathtaking. I fell in love!

This has been the guiding principle in my teaching efforts. At the core of mathematics is ‘reason’. We definitely draw inspiration from real world observations. However, one does mathematics because he or she adores ‘reason’ itself and not the observations.

Years later, I was exposed to Kant’s ‘Critique of Pure Reason’ (thanks to my wife). I was tempted to switch to Philosophy. The promise to go beyond reason was alluring.

Methodology

The philosophical foundation of this eight - week course in beautiful mathematics is therefore well-grounded in these personal experiences. I hope to expose the students to the enchanting beauty of ‘reason’. This is planned in the following manner:

• Begin with some observations of objects
• After repeated observations, find a pattern. A pattern is, roughly speaking, Plato’s ‘form’. It is something that reveals itself when you threw away all the ‘unimportant elements’ from your observations.
• Once observation and pattern recognition are accomplished, one employs ‘reasoning’ to see why the pattern could be relevant in broader contexts.

Hence we have the following recipe:

Observation —> Pattern Recognition —> Generalization

Here is a concrete example:

• Observe platonic solids (try drawing them, draw 2-dimensional projections, etc.)
• Recognize pattern by counting vertices, faces, and edges (Euler’s number)
• Generalize that Euler number is, in essence, an ‘invariance phenomena’. Invariance is omnipresent. Employ that to understand golden ratio (that is see the application of invariance principle in a completely different context)

There are two other things, that I would love to try in this course

• Dialectical investigation in the line of Tarasov. Roughly speaking, this appeals to two steps
• deconstruction of a big idea into fundamental pieces
• reconstruction of the big idea from those pieces.
• Rabindranath’s experiments with pedagogy, especially relating to the objects of observation. Rabindranath recognized that if one stays close to nature and social fabric at the observation stage, then the pattern recognition and generalizations are fundamentally altered. Though I have not experimented or studied this claim in detail, it seems plausible.

Cheenta - Filix Level 1 Math Olympiad Starter module.

Day 0 - Warm up with beautiful problems and drawings.

Day 1 - Platonic Solids (Cube, Tetrahedron, Octahedron, projections)

Day 2 - Platonic Solids (Icosahedron, Dodecahedron, projections)

Day 3 - Counting the simplexes

Day 4 - Invariance principle (Euler number)

Day 5 - Invariance principle (Golden ratio)

Day 6 - Algorithms (Fibonacci number generator)

Day 7 - Algorithms (Fibonacci number generator)

Day 8 - General problems from invariance principle

How the sessions are designed?

• Each session begins with a ‘Motivation problem sheet’. Students are expected to try these problems on their, possibly even before attending the class. They are allowed and encouraged to discuss amongst themselves.
• The discussion kickstarts with a big problem or big idea. Lectures are limited to 15 minute slots. Students will need to ‘do’ mathematics after each such 15 minute slot.
• The session ends with a ‘Follow up problem sheet’. These problems are ‘collaborative homework’. They are most effective when students discuss them in groups.

Key Points

#### Start here

• Faculty: Ashani Dasgupta, founder-faculty at Cheenta, pursuing Ph.D. at University of Wisconsin Milwaukee, USA, in Geometric Group Theory
• Who may attend the course: Students of class 3 to 7 or anyone who wants to fall in love with mathematics
• When are the sessions: Friday 6:30 PM I.S.T. to 8 PM I.S.T.
• When is this starting: November 22, 2019
• What is the cost: Rs. 4899 for two-month enrollment (free for existing students at Cheenta)
• How are the classes conducted: Online, live, interactive
• How to enroll: Email us at support@cheenta.com to apply or click on the following link to join.

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### Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.