Blue Book: Difference between revisions

Let us use the following example to clear this up. We say that the equation x² = ‒ 1 has the solution ± √<span style="border-top:1px solid; padding:0 0.1em;">‒1</span>. There was a time when one said that this equation had no solution. Now this statement, whether agreeing or disagreeing with the one which told us the solutions, certainly hasn't its multiplicity. But we can easily give it that multiplicity by saying that an equation x² + ax + b = 0 hasn't got a solution but comes α near to the nearest solution which is β. Analogously we can say either “A straight line always intersects a circle; sometimes in real, sometimes in complex points”, or, “A straight line either intersects a circle, or it doesn't and is α far from doing so. These two statements mean exactly the same. They will be more or less satisfactory according to the way a man wishes to look at it. He may wish to make the difference between intersecting and not intersecting as inconspicuous as possible. Or on the other hand he may wish to stress it; and either tendency may be justified, say, by his particular practical purposes. But this may not be the reason at all why he prefers one form of expression to the other. Which form he prefers, and whether he has a preference at all, often depends on general, deeply rooted {{BBB TS reference|Ts-309,48}} tendencies of his thinking.