Robustness of Formalism

An interesting example is provided by the development of complex numbers. The fact that the square of any non-zero real number is positive had been generally accepted as implying that there could be no number whose square is negative. Sixteenth century algebra brought this into question. The quadratic formula, essentially known since antiquity, did seem to lead to solutions which did involve square roots of negative quantities. But those were simply regarded as impossible. But the analogous formula for cubic equations, discovered by Tartaglia and published in Cardano’s book of 1545, forced a rethinking of the matter. In the case of a cubic equation with real coefficients and three real roots, the formula led to square roots of negative numbers as intermediary steps in the computation. Bombelli discussed this in his book of 1572. In particular, he noted that although the equation x3 - 15x - 4 = 0 had the three roots 4, -2 + V3, -2 — V3, the Tartaglia formula forced one to consider V-109. Soon mathematicians were working freely with complex numbers without questioning whether they really exist in some “second plane of reality”. What this experience illustrates is the robustness of mathematical formalisms. These formalisms often point the way to expansions of the subject matter of mathematics before any kind of convincing justification can be supplied. This is again a case of induction in mathematical practice.

Leibniz referred to this very experience when asked to justify the use of infinitesimals. As Mancosu explains

...the problem for Leibniz was not, Do infinitely small articles exist? but, Is the use of

infinitely small quantities in calculus reliable?[1]

In justifying his use of infinitesimals in calculus, Leibniz compared this with the use of complex numbers which had become generally accepted although at the time, there was no rigorous justification.

In another example, the rules of algebra, including the manipulation of infinite series was applied to operators with scant justification. This can be seen in Boole’s [2] massive tract on differential equations in which marvelous manipulative dexterity is deployed with not a theorem in sight.

  • [1] "See [7] p. 172.
 
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