DEMONSTRATION, AND NECESSARY TRUTHS. 1G5 



one into another language : though how, after such a series of transla- 

 tions, the fact itself comes out 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 arc peculiarities in the processes 

 of arithmetic and algebra which render tho abine theory very plausi- 

 ble, 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 difficulty, 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 reconcilable 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, inter- 

 secting other lines, forming an angle with one another, and the like ; 

 but not so a and h. These may represent lines or any other mao-ni- 

 tudes, but those magnitudes are never thought of; nothing is realized 

 in our imagination but a and h. 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 beginning, when 

 the premisses are translated from things into signs, and the end, when 

 the conclusion is translated 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 seem to have come to one of Bacon's Pre- 

 rogative Instances ; an experimcntuvi crucis on the nature of reasoning 

 itself 



Nevertheless it will appear on consideration, that this apparently so 

 decisive instance is no instance at all ; that there is in every step of 

 an arithmetical or algebraical calculation a real induction, a real infer- 

 ence of facts from facts ; and that what disguises the induction 

 is simply its comprehensive nature, and the consequent extreme 

 generality of the language. All numbers must be numbers of some- 

 thing : there are no such things as numbers in the abstract. Ten must 

 mean ten bodies, or ten sounds, or ten beatings of the pulse. But 

 though numbers must be numbers of something, they may be numbers 

 <if anything. Propositions, therefore, concerning numbers, have the 

 remarkable peculiarity that they are propositions concerning all things 

 whatever ; all objects, all existences of every kind, known to our 

 experience. All things possess quantity ; consist of parts which can 

 be numbered ; and in that character possess all the properties which 

 are called properties of numbers. That half of four is two must be 

 ti-ue whatever the word four represents, whether four men, four miles, 

 or four pounds weight. We need only conceive a thing divided into 

 four equal pai'ts, (and all things may be conceived as so divided,) to be 

 able to predicate of it every property of the immber four, tliat is, 

 every arithmetical proposition in wliich the number four stands on one 

 side of the equation. Algebra extends the generalization still further: 

 every number represents that particular number of all things without 



