弗雷格 算术基础 读书笔记 6

Reading notes 6

Views on unity and one.

Does the number word “one”stand for a property of objects?

According to Schroder, each of the thingsto be numbered is called a unit. But why must we first bring the things underthe concept of unity, instead of simply defining number right away as a set ofthings? Maybe we are thinking to call the things units, we can add ourdescription at them, to regard numbers as the property of the things. In thiscase a unit would be an object characterized by the property of numbers. But reasonhave already been given as conclusive against the view that number is aproperty of things.

Grammatically, to use numbers as predicatemeans nothing. Because numbers are not the property of things. It is difficultto give a definition of the property“one”. Leibniz and Baumann have been given two wrong definitions ofthe number “one”.

Baumann bases the concept of one on certaincriteria for being one, namely being undivided and being isolated. If this werecorrect, then we should have to expect animals, too, to be capable of havingsome sort of idea of unity. But becomes known to us through the exercise ofthose higher intellectual powers which distinguish men from brutes.Consequently, such properties of things as being undivided or being isolated,which animals perceive quite as well as we do, cannot be what is essential inour concept.

It seems that we indicate unity as aderivative from one, but it is not true, because unity is connected not so muchwith one as with united or unitary. When we say something is one is differentto say something is a unity. Unity means there are objects in it, and theseobjects are united as unitary. But when we say something is one, we mean thenumber of is one. When we say the Earth has one moon, we are saying the numberof satellite of the Earth is one. The word of satellite is not only subject tothe Earth, we can use it to Jupiter or Mars as well.

G. Kopp calls whatever is thought of asself-contained and incapable of dissection, whether we become acquainted withit through the senses or otherwise, an individual; and individuals which are tobe numbered he calls ones, and one here is in the sense of unit. But theattempt to find something strictly called a unit and to be numbered has failed.So if there are no such thing that can be undivided or indivisible, we shouldnot to define numbers in this way.