Hence we must suppose that geometry deals either with 'non-entities'or with 'natural objects.'(45)Arithmetic fares little better.When we say that two and one make three,we assert that the same pebbles may,'by an alteration of place and arrangement'--that is,by being formed into one parcel or two --be made to produce either set of sensations.(46)Each of the numbers,2,3,4,etc.he says elsewhere,'denotes physical phenomena and connotes a physical property of those phenomena.'(47)Arithmetic owes its position to the 'fortunate applicability'to it of the 'inductive truth'that the sums of equals 'are equal.'(48)It is obvious to remark that this is only true of certain applications of arithmetic.When we speak of the numbers of a population,we imply,as Mill admits,no equality except that each person is a unit.(49)We may speak with equal propriety of a number of syllogisms or of metaphors,in which we have nothing to do with 'equality'or 'physical properties'at all.Further,as he observes,(50)it is the peculiarity of the case that counting one thing is to count all things.When I see that four pebbles are two pairs of pebbles,Isee the same truth for all cases,including,for example,syllogisms.Mill admits,accordingly,that 'in questions of pure number'--though only in such questions --the assumptions are 'exactly true,'and apparently holds that we may deduce exactly true conclusions.That ought to have been enough for him.He had really no sufficient reason for depriving us of our arithmetical faith.He can himself point out its harmlessness.As he truly says,'from laws of spare and number alone nothing can be deduced but laws of spare and number.'(51)We ran never get outside of the world of experience and observation by applying them.If we count,we do not say that there must be four things,but that wherever there are four things there are also two pairs of things.The unlucky 'pebble'argument illustrates one confusion.
'Two and two are four'is changed into 'two and two make four.'
The statement of a constant relation is made into a statement of an event.Two pebbles added to two might produce a fifth,but the original two pairs would still be four.The space-problem suggests greater difficulties.Space,he argues,must either be a property of things or an idea in our minds,and therefore a 'non-entity.'If we consider it,however,to be a form of perception,the disjunction ceases to be valid.The space-perceptions mark the border-line between 'object'and 'subject,'and we cannot place its product in either sphere exclusively.The space-relations are 'subjective,'because they imply perception by the mind,but objective because they imply the action of the mind as mind,and do not vary from one person or 'subject'to another.To say whether they were objective or subjective absolutely we should have to get outside of our minds altogether --which is an impossible feat.Therefore,again,it is not really to the purpose to allege that such a 'thing'as a straight line or a perfect circle never exists.Whether we say that a curve deviates from or conforms to perfect circularity,we equally admit the existence of a perfect circle.We may be unable to mark it with finger or micrometer,but it is there.If no two lines are exactly equal,that must be because one has more spare than the other.Mill's argument seems to involve the confusion between the statement that things differ in space and the statement,which would be surely nonsense,that the spare itself differs.It is to transfer the difference from the things measured to the measure itself.It is just the peculiarity of space that it can only be measured by space;and that to say one space is greater than another,is simply to say,'there is more space.'As in the case of number,he is really making an illegitimate transfer from one sphere to another.A straight line is a symmetrical division of space,which must be taken to exist,though we cannot make a perfectly straight line.Our inability does not tend to prove that the 'space'itself is variable.In applying a measure we necessarily assume its constancy;and it is difficult even to understand what 'variability'means,unless it is variability in reference to some assumed standard.If,as Mill seems to think,space is a property of things,varying like other properties,we have to ask,In what,then,does it vary?All other properties vary in respect of their space-relations;but,if space itself be variable,we seem to be reduced to hopeless incoherence.
Thus,to ascribe necessity to geometry as well as to arithmetic is not to ascribe 'necessity'to propositions (to use Hume's language again)about 'matters of fact.'The 'necessity'is implied in a peculiarity which Mill himself puts very forcibly,(52)and which seems to be all that is wanted.An arithmetical formula of the simplest or most complex kind is an assertion that two ways of considering a fact are identical.When I say that two and two make four,or lay down some algebraical formula,such as Taylor's theorem,I am asserting the precise equivalence of two processes.I do not even say that two and two must make four,but that,if they make four,they cannot also or ever make five.The number is the same in whatever order we count,so long as we count all the units,and count them correctly.So much is implied in Mill's observation that counting one set of things is counting all things.The concrete circumstances make no difference.The same is true of geometry.
