About Philosophy
In this page I just want to talk about my mathematical philosophy thinkings. Actually many of them can be found in my Zhihu page in Chinese.
Wittgenstein
My favourite philosopher is Wittgenstein. His book Tractatus Logico-Philosophicus is really worth reading! In the book, he says
- 1.1: The world is the totality of facts, not of things.
- 2: What is the case – a fact – is the existence of states of affairs.
- 2.01: A state of affairs (a state of things) is a combibnation of objects (things).
So what’s the difference between facts and things? A fact does not only consist of many objects (things) but also their complicated relations. This philosophy is really like that of category theory where we think morphisms between objects are more important than onjects themselves. It seems the world is a very complicated category. But note that this book was completed in 1918 much earlier than the work of Eilenberg and Mac Lane.
The most famous sentence in this book is
7: What we cannot speak about we must pass over in silence.
But for me the following proposition seems more important.
4.116: Everything that can be thought at all can be thought clearly. Everything that can be put into words can be put clearly.
As a corollary we can say words are the boundary of thoughts. I think the history of higher categories and higher stacks has proven this proposition true. It also motivates us to think about mathematics and put some really complicated concepts in words.
Nietzsche
We can also consider how etymology can be applied to mathematics which I learn from Nietzsche. In his book On the Genealogy of Morality Nietzsche analyzes the meaning of words ‘good’ and ‘bad’, and tries to prove moral concepts are resulted from social differences. For the word ‘bad’ he finds how words ‘common’, ‘plebeian’ and ‘low’ finally change into it.
The best example for the latter is the German word ‘schlecht’ (bad) itself: which is identical with ‘schlicht’ (plain, simple) – compare ‘schlechtweg’ (plainly), ‘schlechterdings’ (simply) – and originally referred to the simple, the common man with no derogatory implication, but simply in contrast to the nobility. Round about the time of the Thirty Years War, late enough, then, this meaning shifted into its current usage. (On the Genealogy of Morality, Chapter 1 Section 4)
On the other hand in the next section, he connects the Germann word ‘gut’ (good) with ‘den Göttlichen’ (‘the godlike man’) and den Mann ‘göttlichen Geschlechts’ (the man of ‘godlike race’).
There is another example due to Jean-Paul Sartre. In his book L’Être et le Néant (Being and Nothingness) he tells us the word ‘person’ originates from the Latin word ‘persona’ which means masks worn by actors on the stage. Therefore he thinks this can show that people are originally divorced from themselves. Your real life is the same as an actor acting in a play. The masked you is different from the real you, but you take the masked you as real because you are good at seeing yourself through the eyes of others.
Now how can we do etymological analysis in mathematics? For example here we try to understand the mathematical concept ‘homomorphism’. We have ‘homo → homos (Latin) → same (English)’ and ‘morphism → morphine (Latin) → form (English)’. Then from this point it’s clear that ‘homomorphism’ means ‘structure-preserving’ which should preserve the same structures for mathematical objects. As for ‘isomorphism’, ‘iso (Latin) = equal’ and since the word ‘equal’ is stricter than ‘same’, isomorphisms should preserve structures in the stricter sense.
There is another example ‘homology’. ‘-logy’ originates from ancient Grace which just means ‘class’. Hence ‘homology’ should mean ‘similar or corresponding in some way’. In addition, this word is not unique to mathematics and it first appeared in biology in which it means similarity of the structure, physiology, or development of different species of organisms based upon their descent from a common evolutionary ancestor. A common example of homologous structures is the forelimbs of vertebrates, where the wings of bats and birds, the arms of primates, the front flippers of whales and the forelegs of four-legged vertebrates like dogs and crocodiles are all derived from the same ancestral tetrapod structure.
Now we all know in mathematics ‘homology’ is an abelian group but this was discovered by Noether in 1925. Before this, it’s used to describe some conditions in the path integeral. For example given two points in a space, we have two paths connecting them. For integerable funtions P and Q, we consider the differential form Pdx+Qdy. If its integral along the two paths are equal, then these paths are homologous. In this early history, Riemann did a lot of works. You can find details in Weibel’s survey A History of Homological Algebra.
As we have talked before that in biology ‘homology’ means similarity of the structure, physiology, or development of different species of organisms based upon their descent from a common evolutionary ancestor. Different (co)homology theories may have the same ancestor. From Grothendieck’s axiomation in Tohoku, we know how we can obtain a (sheaf) cohomology theory in a really general context which is also valid in a topos. Moreover nearly all cohomology theories such as de Rham cohomology and singular (co)homology can be represented by the sheaf cohomology. In arithmetic geometry there are also many (co)homology theories with deep connections. So what’s the ancestor of them? I think it’s an interesting question.
Spengler
Another inspiring philosophy I think is due to Spengler in his book Der Untergang des Abendlandes (The Decline of the West) where he talks about civilizations and thinks they are akin to biological entities, each with a limited, predictable, and deterministic lifespan. When Toynbee studies different civilizations in A Study of History, the philosophy of history he uses is also physiognomy. This kind of physiognomy is called ‘comparative cultural morpholog’ as well. It studies history and culture from the perspective of morphology. Actually in ancient China there was also a subject of physiognomy. At that time, experienced people could judge a person’s information such as his future status, etc just based on his appearance. Looking at the photo of Shiing-Shen Chern, in the past we believed those who had big ears that were higher than their eyebrows would have great achievements.
The philosophy of comparative cultural morpholog is that as a living existence, conscious life always expresses itself and culture is the container of life expression. The essence and destiny of each culture are closely related to the activities of life living in that culture. Life presents itself as various symbols, so different parts of the same cultural whole, such as literature and mathematics, will have morphological relationships, or similarities. As a kind of collective life, culture itself has cyclical characteristics. Although different cultures are different life forms, just as the age stratification of different people is roughly the same, the life cycle of different cultures is similar. So cultures in the same life cycle will show similarities.
If we limit our attention to the field of mathematics, then according to this point of view, the field of mathematics will actually show ‘characteristics of life’. Different concepts or areas born from the same womb have similarities in morphology, and their historical development trajectories and goals have similar characteristics. The most typical examples are ‘homology’ and ‘homotopy’, both of which come from the investigation of the nature of space itself. Their properties and developments are very similar. For example for ‘homology’ we have long exact sequences induced by derived functors but for ‘homotopy’ we also have fiber sequences and cofiber sequences (Puppe sequences). We have the subject homological algebra to develop algebraic structures of homology in the abelian snese but we also have the subject homotopical algebra (model categories) to develop algebraic structures of homotopy in the non-abelian sense.