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All future symbols are symbolically explained by means of these three. Even these three can be explained by means of the notions of relation and class; but this requires the Logic of Relations, which Professor Peano has never taken up. It must be admitted that what a mathematician has to know to begin with is not much.
Those who wish to know the nature of these things need only read the works of such men as Peano or Georg Cantor; they will there find exact and indubitable expositions of all these quondam mysteries. In this capricious world, nothing is more capricious than posthumous fame. One of the most notable examples of posterity's lack of judgment is the Eleatic Zeno.
Professor Peano, who is assisted by a very able school of young Italian disciples, has shown how this may be done; and although the method which he has invented is capable of being carried a good deal further than he has carried it, the honour of the pioneer must belong to him. Two hundred years ago, Leibniz foresaw the science which Peano has perfected, and endeavoured to create it.
All these, however, are a concession, and are swept away by Professor Peano. One symbol stands for zero, another for number, and a third for next after. What these ideas mean, it is necessary to know if you wish to become an arithmetician. But after symbols have been invented for these three ideas, not another word is required in the whole development.
The great master of the art of formal reasoning, among the men of our own day, is an Italian, Professor Peano, of the University of Turin. In the ordinary mathematical books, there are no doubt fewer words than most readers would wish. Still, little phrases occur, such as therefore, let us assume, consider, or hence it follows.
Thus the individuals whose susceptibilities reach the rarefied atmosphere of those peaks where there exists an exquisitive difference between Frege and Peano, or between Sassetta's earlier and later periods, may be good stanch Republicans at another level of appeal, and when they are starving and afraid, indistinguishable from any other starving and frightened person.
Then we bring in the straight line joining two points, and begin again with , namely, on the straight line joining a and b, there is at least one other point besides a and b. There is at least one point not on the line ab. And so we go on, till we have the means of obtaining as many points as we require. But the word space, as Peano humorously remarks, is one for which Geometry has no use at all.
This method is due partly to Peano, partly to another Italian named Fano. To those unaccustomed to it, it has an air of somewhat wilful pedantry. In this way, we begin with the following axioms: There is a class of entities called points. There is at least one point. If a be a point, there is at least one other point besides a.
In Geometry, as in other parts of mathematics, Peano and his disciples have done work of the very greatest merit as regards principles. Formerly, it was held by philosophers and mathematicians alike that the proofs in Geometry depended on the figure; nowadays, this is known to be false. In the best books there are no figures at all.
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