DEMONSTRATION AND NECESSARY TRUTHS. 



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process in algebra is but a succession 

 of changes in terminology, by which 

 equivalent expressions are substituted 

 one for another ; a series of transla- 

 tions of the same fact, from one into 

 another language ; though how, after 

 such a series of translations, the fact 

 itself comes out changed, (as when we 

 demonstrate a new geometrical theo- 

 rem by algebra,) they have not ex- 

 plained ; and it is a difficulty which 

 is fatal to their theory. 



It must be acknowledged that there 

 are peculiarities in the processes of 

 arithmetic and algebra which render 

 the theory in question 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 difl&culty, 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 demonstra- 

 tion we have a mental diagram, if not 

 one on 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 realised in our imagination 

 but a and 6. The ideas which, on the 

 particular occasion, they happen to re- 

 present, are banished from the mind 

 during every intermediate part of the 

 process, between the beginning, when 

 the premises 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 contend that the reasoning process 

 hastodowithanjrthingmore? Weseem 

 to have come to one of Bacon's Pre- 

 rogative Instances ; an experimentum 

 cruets on the nature of reasoning itself. 

 Nevertheless, it will appear on con- 

 sideration, that this apparently so de- 

 cisive instance is no instance at all ; 

 that there is in every step of an arith- 

 metical or algebraical calculation a 

 real induction, a real inference of facts 

 from facts ; and that what disguises 

 the induction is sifnply its compre- 

 hensive nature and the consequent 

 extreme generality of the language. 

 All numbers must be numbers of 

 something ; 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 some- 

 thing, they may be numbers of any- 

 thing. Propositions, therefore, con- 

 cerning numbers have the remarkable 

 peculiarity that they are propositions 

 concerning all things whatever ; all 

 objects, ail 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 true 

 whatever the word four represents, 

 whether four hours, four miles, or four 

 pounds weight. We need only con- 

 ceive a thing divided into four equal 

 parts (and all things may be conceived 

 as so divided) to be able to predicate 

 of it every property of the number four, 

 that is, every arithmetical proposition 

 in which the number four stands on 

 one side of the equation. Algebra 

 extends the generalisation still farther: 

 every number represents that par- 

 ticular number of all things without 

 distinction, but every algebraical 

 s5mabol does more, it represents all 

 numbers without distinction. As soon 

 as we conceive a thing divided into 

 equal parts, without knowing into 

 what number of parts, we may call 

 it a or X, and apply to it, without 



