弗雷格 算术基础 读书笔记 5

Reading notes 5

Views of certain writers on the concept of Number

Individual numbers are different from the generalconcept of Number. Frege has decided in favor of the view that the individualnumbers are best derived from the number one together with increase by one. Butas long as the number one and increase by one are themselves undefined, thesedefinitions remain incomplete. And also because of their generality, we can’tdefine them from the definitions of the individual numbers, but only from thegeneral concept of Number.

Frege opposes the attempt to think of numbergeometrically, as a ratio between lengths or surfaces. He agrees with Newton’sunderstanding that number is not a set of units as the relation in the abstractbetween any given magnitude and another magnitude of the same kind which istaken as unity, because this is an apt description of number in the widersense. But it is incomplete also, for it presupposes the concepts of magnitudeand of relation in respect of magnitude, and Newton defines number as arelation between magnitudes, was not geometrical magnitudes only, but alsosets, this expression “relation between a set and the unit of the set” tells usno more than the expression “number by which a set is determined”.

Is number definable? The general inclination is, onthe whole, to hold that Number is indefinable, that is more because attempts todefine it have failed than because anything has been discovered in the natureof the case to show that it must be so.

Is Number a propertyof external things?

What concept of Number is? Are they the same to thewords hard or heavy or red, which have for their meanings properties of externalthings?

For M. Cantor, number originates only by abstractionfrom objects. And for E. Schroder, number is modelled on actuality, derivedfrom it by a process of copying the actual units with ones. So for M. cantorand E. Schroder, number is on a level with colour and shape, and treats numberas property of things.

Frege agrees with Baumann the idea that numbers aren’tconcepts extracted from external things: the reason being that external thingsdo not present us with any strict units. For colour and hardness are theproperties of external things, but numbers are not. To say the leaf is green,we are saying something belongs to a surface independently of any choice ofours, green is the objective property of the leaf. but to say the leaf is onewill mean different to different person, because the number of the leaf is notthe objective property of it. So long as we are looking at the leafs in differentperspectives, we will find different numbers of it.

What is it that the number belongs to as a property? Mill

replies as follows: the name of a number connotes, “of course, some property

belonging to the agglomeration of things which we call by the name；and that property is the characteristicmanner in which the agglomeration is made up of, and may be separated into,parts. Frege disagrees with Mill. Firstly, the characteristic manner is not theonly manner the agglomeration separated into parts. There are many manners wecan use to separate agglomeration, so there is no such a property to belong tothings as a characteristic manner. Secondly, if the numbers belong to things,then what does the number 0 belong to?

It does not make sense that what is by nature sensibleshould occur in what is non-sensible. We cannot see numbers from objectsdirectly, what we can see are just things or parts that constructed them.

For Mill the number is something physical,for Locke and Leibniz it exists only as a notion, and Berkeley thinks thenumber is entirely the creature of the mind. Mill was wrong, however, there mayhave a difference in number to which no physical difference corresponds.

Is number something subjective?

After discussing that numbers are not theproperty of physical things, Frege said this line of thought may easily lead usto regard number as something subjective. So we need to make some psychologicalenquiry. But the description of the psychological processes will not precedethe forming of a judgement of number. Whatever accurate a description of themental processes is, it will not take the place of a genuine definition of theconcept. Number is something objective, which is independent of our ideas andeverything of the sort.

What is objective for Frege? Something objectiveare distinguished by Frege from what is handleable or spatial or actual. They arenot things that we can see or touch or something physical, nor something imaginedor created by our mind. We may interpret geometrical axioms or the word “point”differently in terms of our respective intuitions. But we can still say thatthey have objective meanings. So Frege understand objective to mean what isindependent of our sensation, intuition and imagination, and of all constructionof mental pictures out of memories of earlier sensation, but no what isindependent of the reason.

Schloemilch calls number the idea of theposition of an item in a series. If number were an idea, then arithmetic would bepsychology, since psychology are subjective, everyone will have differentideas, that is what Frege will not agree. So objectivity cannot be based on anysense impression, which as an affection of our mind is entirely subjective, but only on the reason.