DEMONSTRATION, AND NECESSARY TRUTHS. 331 



changed, (as when we demonstrate a new geometrical 

 theorem by algebra,) they have not explained ; and it 

 is a difficulty which is fatal to their theory. 



It must be acknowledged that there are peculiari- 

 ties in the processes 6f arithmetic and algebra which 

 render the above theory very plausible, and have not 

 unnaturally made those sciences the stronghold of 

 Nominalism. The doctrine that we can discover facts, 

 detect the hidden processes of nature, by an artful 

 manipulation of language, is so contrary to common 

 sense, that a person must have made some advances 

 in philosophy to believe it ; men fly to so paradoxical 

 a belief to avoid, as they think, some even greater dif- 

 ficulty, which the vulgar do not see. What has led 

 many to believe that reasoning is a mere verbal 

 process, is, that no other theory seemed reconcileable 

 with the nature of the Science of Numbers. For we 

 do not carry any ideas along with us when we use the 

 symbols of arithmetic or of algebra. In a geometrical 

 demonstration we have a mental diagram, if not one 

 upon paper ; AB, AC, are present to our imagination 

 as lines, intersecting other lines, forming an angle 

 with one another, and the like ; but not so a and b. 

 These may represent lines or any other magnitudes, 

 but those magnitudes are never thought of; nothing 

 is realized in our imagination but a and b. The ideas 

 which, on the particular occasion, they happen to 

 represent, are banished from the mind during every 

 intermediate part of the process between the begin- 

 ning, when the premisses are translated from things 

 into signs, and the end, when the conclusion is trans- 

 lated back from signs into things. Nothing, then, 

 being in the reasoner's mind but the symbols, what 

 can seem more inadmissible than to pretend that the 

 reasoning process has to do with anything more ? We 



