What,then,is really implied in the doctrine that all knowledge rests upon experience?One of Mill's intellectual ancestors lays down the fundamental principle.It is absurd,says Hume,(39)to try to demonstrate matter of fact by a priori arguments.'Nothing is demonstrable unless the contrary implies a contradiction.Nothing that is distinctly conceivable implies a contradiction.Whatever we conceive as existent we can also conceive as non-existent.There is no being,therefore,whoa non-existence implies a contradiction.''Matter of fact,'then,must be proved by experience;but,given a 'fact'we may deduce necessary consequences.All necessity may be hypothetical;there is an 'if'to every 'must,'but remembering the 'if'the 'must'will be harmless.It can never take us beyond experience.The existence of space itself cannot be called necessary;but space once given,all geometry may 'necessarily'follow,and imply relations running through the whole fabric of scientific knowledge.Mill agrees that a 'hypothetical'necessity of this kind belongs to geometry;and adds,that in any science whatever,we might,by making hypotheses,arrive at an equal necessity.(40)But then,he goes on to urge,the hypotheses of geometry are not 'absolute truths,'but 'generalisations from observation,'or 'inductions from the evidence of our senses,'(41)which,therefore,are not necessarily true.This led to his keenest controversies,and,in my opinion,to his least successful answers.He especially claims credit in his Autobiography for having attacked the 'stronghold'of the intuitionists by upsetting belief in the a priori certainty of mathematical aphorisms.In fact,his opponents constantly appealed to the case of mathematics,and Mill assumes that they can be met only by reducing such truths to the case of purely empirical truths.He argues boldly that the 'character of necessity ascribed to the truths of mathematics'is 'an illusion.'(42)Geometry and arithmetic are both founded upon experience or observation.He goes indeed still further at times.
At one place he even holds that the principle of contradiction itself is simply,one of our first and most familiar generalisations from experience.'We know,'by the simplest observation of our own minds,'that belief and disbelief exclude each other,and that when light is present darkness is absent.(43)Mill thought himself bound,we see,to refer to experience not only our knowledge of facts,but even the capacities,which are said by another school to be the conditions of perceiving and thus acquiring experience.If he had studied Kant,he might have reached a better version of his own view.As it was,he was led to accepting paradoxes which he was not really concerned to maintain.He had to choose between a theory of 'intuitions'--so understood as to entitle us to assert matter of fact independently of experience --and a theory which seems to make even the primary intellectual operations mere statements of empirical fact.Since necessary statements about matters of fact must be impossible,he argues that we cannot even draw necessary inferences from observed fact.Not content with saying that all necessity is hypothetical,he argues that all necessity,even the logical necessity of contradiction,is a figment.If he does not carry out a theory which would seem to make all reasoning unsatisfactory,he maintains,at least,that the hypotheses or assumptions involved in geometry,and even in arithmetic,are generalised from experience,and 'seldom,if ever,exactly true.'If the assumptions are inaccurate or uncertain,the whole superstructure of science must also be uncertain.
The nature of his argument follows from his previous positions.He treats space and number as somehow qualities of the 'things,'or as attributes which we observe without in any sense supplying them.His argument upon geometry begins by asserting that there are no such 'real things'as points or lines or circles.Nay,they are not even possible,so far as we can see,consistently with the actual constitution of the universe.It is 'customary'to answer that such lines only exist in our minds,and have therefore nothing to do with outward experience.(44)This,however,is incorrect psychologically,because our ideas are copies of the realities.A line without breadth is 'inconceivable,'and therefore does not exist even in the mind.
