Studium InstitutePreface to the Trivium

Path · mathematics and symbols

Mathematics Needs To Be Interpreted

One might object to the claim of the classical trivium that words must come first in our studies by arguing that the highest truths are scientific theories, that these are stated in mathematical form, and that mathematics is therefore the greatest instrument of the human mind for stating important truths. But on careful consideration it is clear that mathematical expressions do not have meaning in themselves, except in so far as that meaning is given by language; and mathematical expressions certainly cannot interpret themselves. For example, one of the laws of physics may be given as a formula, but the formula is incomplete until we have been told what its terms mean: if F is force, m is mass, and a is acceleration, then Newton’s Second Law of motion, which relates together the force applied, the mass of the body, and its resulting acceleration, can be expressed as F = ma.

It is true that a notation such as F = ma is convenient because it is easy to use and easy to manipulate. Such equations can even be taught to students without requiring that they fully understand them. But that should not obscure the fact that a notation is useless without the correct interpretation, and any such interpretation is and must be given in language.

The same holds for mathematics as a whole. The fundamental definitions and results of mathematics are stated primarily in language, as definitions and axioms; and even where they are stated primarily in symbols, the symbols are always given meaning by definitions in language. This is why mathematical formulas can always be read out loud (and why new notation is usually accompanied by a phrase that tells the reader how to speak the notation). Likewise, advanced symbolic systems are typically learned both through instruction given in a language and through examples of the use of the system, and learners often find it necessary to say to themselves in words what the symbols mean.

This is true even of the most general part of modern mathematics, mathematical logic. Mathematical logic is built on a formal language in which terms and rules are precisely defined; but those terms and rules can only be stated and explained in a natural language, in what is called the metalanguage of the system (usually English). So, even the most general and powerful tools of mathematics must be expressed and defined in language.

Language, therefore, is necessary, even if it is not sufficient, for stating mathematics and for ordering mathematical thought. And, thus, language is more general and expressive than mathematics.

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