If its radii be equal or approximately equal,the conclusions are true.We afterwards extend our reasoning to similar cases;but only one instance is demonstrated.The definition is merely a 'notice to ourselves and others,'stating what assumptions we think ourselves entitled to make;and in this way it resembles the major in the syllogism.The demonstration does not 'depend upon'it,though if we deny it,the demonstration fails.By this argument,Mill conceives that the case of mathematics is put on a level with other cases.We always argue from facts,and moreover from 'particular facts,'not from definitions.We start from an observation of this particular circle --a sensible 'thing'or object,as in arguing about natural history we start from observation of the camel.Hence we may lay down the general proposition,applicable to geometry as well as to all ordinary observation,that,all inference is from particulars to particulars.'(36)This is the 'foundation'both of Induction,which is 'popularly said'to reason from particulars to generals,and of Deduction,which is supposed to reason from generals to particulars.(37)This sums up Mill's characteristic position.
III.MATHEMATICAL TRUTHS
This attempt to bring all reasoning to the same type forces Mill to ignore what to others seems to be of the essence of the case.There are,he says,two statements:'There may exist a figure bounded by three straight lines';that is the fruitful statement of facts.'This figure is called a triangle';that is the merely nominal definition or explanation of words.Moreover,as he says,we may drop the definition by substituting the equivalent words or simply looking at the thing.It does not follow that we can dispense with the mode of apprehension implied by the definition.Whether we use the word triangle,or the words,'three lines enclosing a space,'or no words at all,we must equally have the conceptions or intuitions of lines and space.All demonstration in geometry consists in mentally rearranging a combination of lines and angles so as to show that one figure may be made to coincide absolutely with another figure.The original fact remains unaltered,but the ways of apprehending the fact are innumerable.Newton and his dog Diamond might both see the same circular thing;but to Diamond the circle was a simple round object;to Newton it was also a complex system of related lines,capable of being so regarded as to embody a vast variety of elaborate formulae.(38)Geometry,as Mill undeniably says,deals with facts.Newton and Diamond have precisely the same fact before them.It remains the same,whether we stop at the simplest stage or proceed to the most complex evolution of geometry.The difference between the observers is not that Newton has seen new facts,but that he sees more in the same fact.The change is not in the things but in the mind,which,by grouping the things in the way pointed out by the definitions,is able to discover countless new relations involved in the same perception.This again may suggest that even the fact revealed to simple perception is not a bare 'fact,'something,as Mill puts it,'external to the mind,'but is in some sense itself constituted by the faculty of perception.It contains already the germ of the whole intellectual evolution.The change is not in the thing perceived,but in the mode of perceiving.And,therefore,again,we do not acquire new knowledge,as we acquire it in the physical sciences,by observing new facts,discovering resemblances and differences,and generalising from the properties common to all;but by contemplating the same fact.All geometry is in any particular space --if only we can find it.We do not proceed by comparing a number of different regions of spaces,and inquire whether French triangles have the same properties as English triangles.To Mill,however,the statement that geometry deals with fact leads to another conclusion.We must deal with these facts as with other facts,and follow the method of other natural sciences.We really proceed in the same way whether we are investigating the properties of an ellipse or a camel.In either case we must discover truth by experience.
