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Mathematics can be shown to be a logical development of certain basic ideas ;
mathematics can be reduced to logic.
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The number of a class is the class of all those classes which are similar to it,.
(Classes are similar when their members can be put into a one-to-one relation with each other.)
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A relation is symmetrical when if one thing has the relation to another,
the other has the same relation to it ;
a relation is transitive when if one thing has the relation to a second,
and the second has the same relation to a third,
the first has the relation to the third.
(Other relations are defined.)
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An infinite cardinal number satisfies the equation,n equals n plus l.
(An infinite collection has parts which have as many terms as the infinite collection itself.)
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By distinguishing between types of entities it is possible to avoid paradoxes which have perplexed philosophers for centuries.
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Mathematical truths are a priori and have nothing to do with facts about the world ;
they are logical tautologies.
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