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To invert a and b is to put b in the place of a and a in the place of b. In the event that a was in the first place and b in the second, b must be in the first place and a in the second. Moreover, in the event that a is numeral 1 and b is numeral 2, 2 must go in the place of 1 and 1 in the place of 2. Substitution theory is thus one that combines numerals and numbers. This paper shows that many misunderstandings in mathematics can be compared to confusion between numerals and numbers. On a positive side, it shows how important are all the numberings or parameterizations that turn out to be pervasive throughout mathematics, as devices to “set ideas.” A mathematical structure cannot, in general, be reached without the medium of such points of reference. The paper shows in particular that any consistent structuralist explanation of mathematical objectivity cannot be sustained without full attention.