We are used to thinking that mathematics uses symbolic language spontaneously and that this is its “natural” language. Probably most of us is not even able to imagine how could one solve algebraic equations without using the expressions we are used to, using letters and symbols to represent unknowns, data and mathematical operations.
In reality it was not always like this. The idea of representing symbolically unknowns and known values using letters was introduced by Fibonacci (Leonardo Pisano, 1170-1250), even if Diophantus (III century AD) had already foreshadowed the matter, without arriving at a true symbolic representation. We have to wait though François Viete who in his In artem analyticam isagogé (1591) demonstrates the usefulness of symbolic language.
To give an idea how things were going on before the end of the 16th century, let us consider the example of Tartaglia (Niccolò Fontana, Brescia 1499 – Venezia 1557), the discoverer of the formula of cubic equations (although the formula had been discovered, at least in part, before him by Scipione del Ferro, who, however, had not made it public. This happened on the part of Gerolamo Cardano, to whom Tartaglia himself had revealed it). Tartaglia, in the absence of the algebraic language we are used to today, to remember the procedure (today we would say the algorithm) to solve cubic equations like x3 + 2x = 5 or similar used verses, which of course he had memorized (note that what he calls the “thing” is the unknown, what today we define x):
Look at verses in the Italian version. A translation of a poem in 1500 Italian language is outside my skills.
From this we understand that algebra would not have made much progress without the use of the symbolic representation. The lack of symbolism is also the main reason why the ancient Greeks have developed only the geometrical aspects of algebra.
A leap of this kind has not occurred in the other sciences, the only exception being logic, which over time has passed from the colloquial language of Aristotelian logic to symbolic language, where the classical syllogisms (1. Socrates is a man, 2. Men are mortal, 3. Socrates is mortal) can be reduced to symbolic expressions.
Experimental sciences, especially physics, but not only, make great use of the mathematical language and therefore of its symbolism, but they do not use a specific symbolism to describe their reasoning or the inferences (or inductions) based on their data. This can be a big limitation in the age of computers and artificial intelligence: everything that goes into a computer needs a symbolic representation.
Perhaps the time has come to try to overcome this obstacle and adopt different forms of representation and description of scientific experiments?
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