Russell, "the greatest achievement of which our own age has to boast.... It was assumed as self-evident, until Cantor and Dedekind established the opposite, that if, from any collection of things, some were taken away, the number of things left must always be less than the original number of things.
From him to our own day, the finest intellects of each generation in turn attacked the problems, but achieved, broadly speaking, nothing. In our own time, however, three men Weierstrass, Dedekind, and Cantor have not merely advanced the three problems, but have completely solved them.
Russell was telling us, it was only the other day that Cantor and Dedekind observed that although time continually eats up the days and years, the possible future always remains as long as it was before. This happens to be a fact interesting to mankind.
At last Georg Cantor has found the answer, and has conquered for the intellect a new and vast province which had been given over to Chaos and old Night. It was assumed as self-evident, until Cantor and Dedekind established the opposite, that if, from any collection of things, some were taken away, the number of things left must always be less than the original number of things.
Twenty years ago, roughly speaking, Dedekind and Cantor asked this question, and, what is more remarkable, they answered it. They found, that is to say, a perfectly precise definition of an infinite number or an infinite collection of things. This was the first and perhaps the greatest step. It then remained to examine the supposed contradictions in this notion.
At the door, rapt apparently in the contemplation of the mountain peaks which glowed rich orange in the last lingering sun-rays, but really watching which way the sheep on the moor were taking, stood the innkeeper, a brawny, sodden-visaged, blear-eyed six feet of brutishness, holding up his hose with one hand, for want of points, and clawing with the other his elf-locks, on which a fair sprinkling of feathers might denote: first, that he was just out of bed, having been out sheep-stealing all the night before; and secondly, that by natural genius he had anticipated the opinion of that great apostle of sluttishness, Fridericus Dedekind, and his faithful disciple Dekker, which last speaks thus to all gulls and grobians: "Consider that as those trees of cobweb lawn, woven by spinners in the fresh May mornings, do dress the curled heads of the mountains, and adorn the swelling bosoms of the valleys; or as those snowy fleeces, which the naked briar steals from the innocent sheep to make himself a warm winter livery, are, to either of them both, an excellent ornament; so make thou account, that to have feathers sticking here and there on thy head will embellish thee, and set thy crown out rarely.
The solutions, for those acquainted with mathematics, are so clear as to leave no longer the slightest doubt or difficulty. Of the three problems, that of the infinitesimal was solved by Weierstrass; the solution of the other two was begun by Dedekind, and definitively accomplished by Cantor. The infinitesimal played formerly a great part in mathematics.
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