Dr. WHEWELL, ON THE FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 175 



in our conceptions, it must always be possible to derive one of these elements from the other, if we 

 are satisfied to accept, as proof of such derivation, that one always co-exists with and implies the 

 other. Thus an opponent may say, that our ideas of space, time, and number, are derived from 

 our sensations or perceptions, because we never were in a condition in which we had the ideas of 

 space and time, and had not sensations or perceptions. But then, we may reply to this, that we no 

 sooner perceive objects than we perceive them as existing in space and time, and therefore the ideas 

 of space and time are not derived from the perceptions. In the same manner, an opponent may say, 

 that all knowledge which is involved in our reasonings is the result of experience ; for instance, our 

 knowledge of geometry. For every geometrical principle is presented to us by experience as true ; 

 beginning with the simplest, from which all others are derived by processes of exact reasoning. 

 But to this we reply, that experience cannot be the origin of such knowledge ; for though experience 

 shows that such principles are true, it cannot show that they must be true, which we also know. We 

 never have seen, as a matter of observation, two straight lines inclosing a space ; but we venture 

 to say further, without the smallest hesitation, that we never shall see it ; and if any one were to 

 tell us that, according to his experience, such a form was often seen, we should only suppose that he 

 did not know what he was talking of. No number of acts of experience can add to the certainty of 

 our knowledge in this respect; which shows that our knowledge is not made up of acts of experience. 

 We cannot test such knowledge by experience; for if we were to try to do so, we must first know 

 that the lines with which we make the trial are straight ; and we have no test of straightness 

 better than this, that two such lines cannot inclose a space. Since then, experience can neither 

 destroy, add to, nor test our axiomatic knowledge, such knowledge cannot be derived from expe- 

 rience. Since no one act of experience can affect our knowledge, no numbers of acts of experience 

 can make it. 



15. To this a reply has been oiFered, that it is a characteristic property of geometric forms that 

 the ideas of them exactly resemble the sensations ; so that these ideas are as fit subjects of experi- 

 mentation as the realities themselves ; and that by such experimentation we learn the truth of the 

 axioms of geometry. I might very reasonably ask those who use this language to explain how a 

 particular class of ideas can be said to resemble sensations ; how, if they do, we can know it to be 

 so ; how we can prove this resemblance to belong to geometi'ical ideas and sensations ; and how 

 it comes to be an especial characteristic of those. But I will put the argument in another way. 

 Experiment can only show what is, not what must be. If experimentation on ideas shows what 

 must be, it is different from wliat is commonly called experience. 



I may add, that not only the mere use of our senses cannot show that the axioms of geometry 

 must be true, but that, without the light of our ideas, it cannot even show that they are true. If we 

 had a segment of a circle a mile long and an inch wide, we should have two lines inclosing a space; 

 but we could not, by seeing or touching any part of either of them, discover that it was a bent line. 



16. That mathematical truths are not derived from experience is perhaps still more evident, 

 if greater evidence be possible, in the case of numbers. We assert that 7 and 8 are 15. We find it 

 so, if we try with counters, or in any other way. But we do not, on that account, say that the 

 knowledge is derived from experience. We refer to our conceptions of seven, of eight, and of addi- 

 tion, and as soon as we possess these conceptions distinctly, we see that the sum must be fifteen. 

 We cannot be said to make a trial, for we should not believe the apparent result of the trial if it 

 were different. If any one were to say that the multiplication table is a table of the results of experi- 

 ence, we should know that he could not be able to go along witli us in our researches into the founda- 

 tions of human knowledge ; nor, indeed, to pursue with success any speculations on the subject. 



17. Attempts have also been made to explain the origin of axiomatic truths by referring 

 them to the association of ideas. But tliis is one of the cases in which the word association has 

 been applied so widely and loosely, that no sense can be attached to it. Those who have written 

 with any degree of distinctness on the subject, have truly taught, tluit the habitual association of tlie 



