MOTION MOUNTAIN

266 8 thought and language
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Challenge 265 e
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Page 304
of yours!’ Rucker reports that mathematicians conjecture that there is no possible nor
anyconceivable end to these discussions.
For physicists, a simple question appears directly. Do infinite quantities exist in
nature? Or better, is it necessary to use infinite quantities to describe nature? You might
want to clarify your own opinion on the issue. It will be settled during the rest of our
adventure.
Functions and structures
Which relations are useful to describe patterns in nature? A typical example is ‘larger
stones are heavier’. Such a relation is of a specific type: it relates one specific value of
an observable ‘volume’ to one specific value of the observable ‘weight’. Such a one-toonerelationiscalleda(mathematical)functionormapping.Functionsarethemostspecific
types of relations; thus they convey a maximum of information. In the same way
as numbers are used for observables, functions allow easy and precise communication
of relations between observations. All physical rules and ‘laws’ are therefore expressed
with the help of functions and, since physical ‘laws’ are about measurements, functions
of numbers are their main building blocks.
A functionf, or mapping, is a thus binary relation, i.e., a setf={(x,y)} of ordered
pairs, where for every value of the first elementx, called theargument, there is onlyone
pair(x,y). The second elementyis called the value of the function at the argumentx.
The setXof all argumentsxis called thedomainofdefinition and the setYof all second
argumentsyis called therange of the function. Instead off={(x,y)} one writes
f:X→Y and f:x→y or y=f(x) , (98)
where the type of arrow – with initial bar or not – shows whether we are speaking about
sets or about elements.
We note that it is also possible to use the couple ‘set’ and ‘mapping’ to define all
mathematical concepts; in this case a relation is defined with the help of mappings. A
modern school of mathematical thought formalized this approach by the use of (mathematical)categories,aconceptthatincludesbothsetsandmappingsonanequalfooting
initsdefinition.*
Tothinkandtalkmoreclearlyaboutnature,weneedtodefinemorespecializedconcepts
than sets, relations and functions, because these basic terms are too general. The
most important concepts derived from them are operations, algebraic structures and
numbers.
A (binary)operation is a function that maps the Cartesian product of two copies of a
setXinto itself. In other words, an operationwtakes an ordered couple of arguments
*Acategory isdefinedasacollection ofobjectsandacollection of ‘morphisms’,or mappings. Morphisms
can be composed; the composition is associative and there is an identity morphism. More details can be
foundinthe literature.
Notethateverycategorycontainsaset;sinceitisunclearwhethernaturecontainssets,aswewilldiscuss
below, ,itisquestionablewhethercategorieswillbeusefulintheunificationofphysics,despitetheirintense
andabstractcharm.
Motion Mountain – The Adventure of Physics copyright © Christoph Schiller June 1990–November 2015 free pdf file available at www.motionmountain.net