MOTION MOUNTAIN

262 8 thought and language
F I G U R E 165 Devices for the definition of sets (left)
and of relations (right).
In conclusion, all concepts are based on the idea that nature is made of related
parts. This idea leads to complementing couples such as ‘noun–verb’ in linguistics,
‘set–relation’ or ‘definition–theorem’ in mathematics, and ‘aspect of nature–pattern of
nature’ in physics.These couples constantly guide human thinking, from childhood onwards,asdevelopmental
psychology can testify.
What are sets? What are relations?
“Alles,waswir sehen,könnte auchanderssein.
Alles,waswir überhauptbeschreibenkönnen,
könnte auchanderssein.Esgibt keineOrdnung
derDinge apriori.*
LudwigWittgenstein, Tractatus, 5.634”
Defining sets and defining relations are the two fundamental acts of our thinking. This
can be seen most clearly in any book about mathematics; such a book is usually divided
into paragraphs labelled ‘definition’, ‘theorem’, ‘lemma’ and ‘corollary’. The first type
of paragraph defines concepts, i.e., defines sets, and the other three types of paragraphs
express relations, i.e., connections between these sets. Mathematics is thus the explorationofthepossiblesymbolicconceptsandtheirrelations.Mathematicsisthescienceof
symbolicnecessities.
Sets andrelations are tools of classification; thatis whytheyare also thetools of any
bureaucrat. (See Figure 165.) This class of humans is characterized by heavy use of paper
clips, files, metal closets, archives – which all define various types of sets – and by
the extensive use of numbers, such as reference numbers, customer numbers, passport
numbers, account numbers, law article numbers – which define various types of relations
between the items, i.e., between the elements of the sets.
Both the concepts of set and of relation express, in different ways, the fact that nature
can be described, i.e., that it can be classified into parts that form a whole. The act of
grouping together aspects of experience, i.e., the act of classifying them, is expressed in
formal language by saying that a set is defined. In other words, a set is a collection of
elements of our thinking. Every set distinguishes the elements from each other and from
the set itself.This definition of ‘set’ is called thenaive definition. For physics, thedefinitionissufficient,butyouwon’tfindmanywhowill
admitthis.Infact,mathematicians
have refined the definition of the concept ‘set’ several times, because the naive definition
does not work well for infinite sets. A famous example is the story about sets which do
* ‘Everything we see could alsobe otherwise. Everything we describeat all could alsobe otherwise. There
isnoorderofthings apriori.’
Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–November 2015 free pdf file available at www.motionmountain.net