#incompleteness

Wabi-sabi is a traditional Japanese aesthetic and worldview centered on the acceptance and appreciation of imperfection, impermanence, and the incomplete. It is deeply rooted in Buddhist teachings, particularly those relating to the transience of life. The philosophy of wabi-sabi celebrates the beauty found in simplicity, humility, and natural processes. Here’s an exploration of its core principles and concepts:

1. Impermanence (Mujō)

Transience of Life: Central to wabi-sabi is the Buddhist concept of impermanence (mujō), which acknowledges that all things are in a constant state of flux and decay. This perspective encourages an appreciation for the present moment and the beauty of fleeting experiences.

Natural Aging: Wabi-sabi finds beauty in the natural aging process. The patina of wear and the signs of use on objects are celebrated as they reveal the passage of time and the story of their existence.

2. Imperfection (Wabi)

Beauty in Flaws: Wabi-sabi embraces the idea that nothing is perfect, and it is in the imperfections that true beauty resides. This could be seen in asymmetry, roughness, and the uniqueness of handmade objects.

Simplicity and Humility: The wabi aspect emphasizes simplicity and humility. It values modest, rustic beauty over ostentatious or overly elaborate designs. This often translates to an aesthetic of minimalism and restraint.

3. Incompleteness (Sabi)

Unfinished and Evolving: The sabi aspect of wabi-sabi appreciates the incomplete and the evolving nature of things. It suggests that beauty is a process, not a fixed state, and that objects and experiences are always in a state of becoming.

Quietness and Serenity: Sabi also conveys a sense of quietness and serenity. It reflects a meditative quality, an appreciation for solitude and the tranquility found in understated beauty.

4. Connection to Nature

Natural Materials: Wabi-sabi often involves the use of natural materials that age gracefully over time, such as wood, stone, and clay. These materials reflect the organic processes of growth and decay inherent in nature.

Organic Forms: The aesthetic favors organic, irregular forms that mimic the irregularities found in nature, as opposed to geometric perfection.

5. Mindfulness and Presence

Living Mindfully: Embracing wabi-sabi encourages living mindfully and being present in the moment. It’s about appreciating the here and now, and finding contentment in the current state of things, despite (or because of) their imperfections.

Acceptance: Wabi-sabi involves accepting the natural cycle of growth and decay, and finding peace and contentment in this acceptance. It’s about letting go of the pursuit of perfection and embracing the reality of impermanence.

Examples in Practice

Tea Ceremony: The Japanese tea ceremony is a quintessential example of wabi-sabi in practice. The ceremony emphasizes simplicity, natural materials, and the appreciation of imperfections in the tea utensils, such as a crack in a teacup that adds character and history.

Kintsugi: The art of kintsugi, where broken pottery is repaired with gold or silver lacquer, highlights the philosophy of wabi-sabi. It transforms damage into beauty, celebrating the history and imperfection of the object.

Gardens: Traditional Japanese gardens often embody wabi-sabi principles with their asymmetrical layouts, natural elements, and deliberate incorporation of aged or weathered features.

This is either going to be the first installment of a long running series or something I will never do again. (We'll see, don't know yet.)

Like the name suggests each iteration will showcase a theorem with its proof, all in one picture. I will provide preliminaries and definitions, as well as some execises so you can test your understanding. (Answers will be provided below the break.)

The goal is to ease people with some basic knowledge in mathematics into set theory, and its categorical approach specifically. While many of the theorems in this series will apply to topos theory in general, our main interest will be the topos Set. I will assume you are aware of the notations of commutative diagrams and some terminology. You will find each post to be very information dense, don't feel discouraged if you need some time on each diagram. When you have internalized everything mentioned in this post you have completed weeks worth of study from a variety of undergrad and grad courses. Try to work through the proof arrow by arrow, try out specific examples and it will become clear in retrospect.

Please feel free to submit your solutions and ask questions, I will try to clear up missunderstandings and it will help me designing further illustrations. (Of course you can just cheat, but where's the fun in that. Noone's here to judge you!)

Preliminaries and Definitions:

B^A is the exponential object, which contains all morphisms A→B. I comes equipped with the morphism eval. : A×(B^A)→B which can be thought of as evaluating an input-morphism pair (a,f)↦f(a).

The natural isomorphism curry sends a morphism X×A→B to the morphism X→B^A that partially evaluates it. (1×A≃A)

φ is just some morphism A→B^A.

Δ is the diagonal, which maps a↦(a,a).

1 is the terminal object, you can think of it as a single-point set.

We will start out with some introductory theorem, which many of you may already be familiar with. Here it is again, so you don't have to scroll all the way up:

Exercises:

What is the statement of the theorem?

Work through the proof, follow the arrows in the diagram, understand how it is composed.

What is the more popular name for this technique?

What are some applications of it? Work through those corollaries in the diagram.

Can the theorem be modified for epimorphisms? Why or why not?

For the advanced: What is the precise requirement on the category, such that we can perform this proof?

For the advanced: Can you alter the proof to lessen this requirement?

Bonus question: Can you see the Sicko face? Can you unsee it now?

Expand to see the solutions:

  A foreigner, imbued with Christian ideas, naturally feels inclined to substitute for Tao the term by which he is accustomed to denote the Supreme Being--God. But this is only admissible if he is prepared to use the term 'God' in a much broader sense than we find in either the Old or the New Testament. That which chiefly impresses the Taoist in the operations of Nature is their absolute impersonality. The inexorable law of cause and effect seems to him equally removed from active goodness or benevolence on the one hand, and from active, or malevolence on the other. This is a fact which will hardly be disputed by any intelligent observer. It is when he begins to draw inferences from it that the Taoist parts company from the average Christian. Believing, as he does, that the visible Universe is but a manifestation of the invisible Power behind It, he feels justified in arguing from the known to the unknown, and concluding that, whatever Tao may be in itself (which is unknowable), it is certainly not what we understand by a personal God--not a God endowed with the specific attributes of humanity, not even (and here we find a remarkable anticipation of Hegel) a conscious God. In other words, Tao transcends the illusory and unreal distinctions on which all human systems of morality depend, for in it all virtues and vices coalesce into One.

The Christian takes a different view altogether. He prefers to ignore the facts which Nature shows him, or else he reads them in an arbitrary and one-sided manner. His God, if no longer anthropomorphic, is undeniably anthropopathic. He is a personal Deity, now loving and merciful, now irascible and jealous, a Deity who is open to prayer and entreaty. With qualities such as these, it is difficult to see how he can be regarded as anything but a glorified Man. Which of these two views--the Taoist or the Christian--it is best for mankind to hold, may be a matter of dispute. There can be no doubt which is the more logical.