ToK Essay 2 Nov 23: Beautiful Patterns

PT#2 N23  If “the mathematician’s patterns, like the painter’s and the poet’s, must be beautiful” - what would be the impact on the production of mathematic and artistic knowledge?”

ToK Essay 2 Nov 23 beautiful patterns:

This question could be approached as a consideration of the ways in which knowledge is made (produced, constructed, discovered) rather than about the knowledge itself. The PT does direct us to consider the “impact on the production of knowledge”.

Beautiful subject rather than object.

If it’s about the production of knowledge, rather than the knowledge produced, its about subject rather than object. The word “beauty” could lead a lot of people to think that it’s about the object of knowledge production, however we’d advise students to focus on subject rather than object.

Obviously, subject and object are linked, and the former may lead to the latter. Maybe a good place to start is to think about the reasons why mathematicians make knowledge, and compare it with the possible reasons for artists making knowledge. 

Purposes of knowledge production.

There’s potentially a nice contrast in the purposes for knowledge production in The Arts and Mathematics. We could compare the debate in Mathematics between Pure and Applied Maths with the debate in The Arts over whether artists make knowledge for themselves or for their audience.

That debate in Maths would be that Pure Maths is made purely to extend mathematical knowledge, whilst Applied Maths is made to solve real world problems. Conversely, the debate in the Arts would be that some art is made just for the artist to express their inner world, whilst other art is made to engage the audience  (“the knower”).

Link "beautiful patterns" and interpretation. 

Beauty is often thought of as a relativist concept (“beauty is in the eye of the holder”).

We’d advise students to be wary of relativist arguments as they are rarely sufficient to attract high marks in the ToK Essay. 

Potentially, a more substantial argument would be that true beauty in the production of knowledge should be free from / independent of external constraints such as audience preferences in the arts, and real world problems in Mathematics. 

A further argument could be developed around the role of interpretation in the production of knowledge. Processes of interpretation of knowledge in the arts could change both how patterns are identified and how they are used in the production of knowledge. 

As such the role of interpretation in the production of artistic knowledge could change the explanations, justifications and perspectives pertaining to that knowledge - which in itself could change the definition and attribution of beauty.

In Maths the interpretation of pre-existing knowledge (e.g. axioms, theorems and models) could affect both the justifications and evidence used by mathematicians. This could be further applied to the discussion of the relationship between forms, objects and theorems in Maths. 

The role of context in the production of (beautiful) mathematical and artistic knowledge.

There’s a really interesting debate to be had about whether the patterns are merely context bound, or are they universal. This debate could be developed to consider whether beauty is also context bound or universal. This could be further developed to consider whether the methodologies of knowledge production are context bound or universal.

If the patterns, beauty and even the methodology are context bound then there may not be an underlying near aesthetic structure to mathematical and artistic knowledge (as implied by Mr Hardy) because that definition of beauty is ever changing. 

These arguments, and many more developed in far more detail in the notes: 10 Arguments for ToK Essay #2 available from TokToday - those notes contain

• Over 9000 words of ToK content.

• knowledge arguments

• evaluation points

• Implications of knowledge arguments

• suggested real life situations (with references)

We also have a list of 25 questions that you can ask Artificial Intelligence (such as ChatGPT) about ToK Essay #2. These questions are designed to get relevant content which is appropriate for this ToK Essay.