Under such a conception of the objective world and the world of thought and their relations the old dispute of realism versus nominalism would take a new aspect. _Universals in re_ even though they were admitted to exist would become universals no longer in the higher universe of thought. True universals would only subsist as universals of the world of mind. The laws of Form and Formal Thought would thus become of chief moment in philosophy and no one could be recognised as properly laying claim to the title of philosopher without proficiency therein. XI. FALLACY OF THE SPENCERIAN AXIOM. Concerning the “axiom of symmetry” only a few examples of its fallacy are needful. Mutual friendship is certainly a “symmetrical” relation, but _A_ and _C_ may be mutual friends and _B_ and _C_ mutual friends also, but it in no wise follows that _A_ and _B_ are friends. They may be decidedly unfriendly as we often see the case. Take a case of equilibrium the cases of which seem to be favorite ones with M. Mouret. We suppose that planets may be regarded as in a relation of equilibrium with the sun and yet these mere equilibrations with the sun do not make any equilibrium between them. They do not knock together it is true but this is due to their own direct relations and not their relations of equilibrium with the sun. The distances of points from each other is a “symmetrical” relation and yet point _A_ may be from point _C_ the very same distance that point _B_ is from _C_, but the distance of points _A_ and _B_ from one another may vary from coincidence to double the distance _A C_-_B C_. XII. NATURE OF ARITHMETICAL EQUALITY. Concerning the relation of “mathematical equality” there is no single relation that obtains throughout mathematics as such. There is numerical equality upon which the equality in service in numeric algebra is founded, and there is geometric equality, the equality of vectors, etc., all different from one another. M. Mouret seems to have only numeric equality in view. He claims this relation to be not only of a very simple nature but that it is the very foundation of the notions of magnitude and quantity. He even declares that mathematics could not exist without this relation. Did he lose sight of the usual proof of Fourier’s celebrated theorem?