Mathematics has a more limited domain than language, since language can be used to consider everything, whereas mathematics is restricted to the formal or quantitative aspects of things. Mathematical notation is like language in that its symbols typically have little natural connection to what an expression is about; for example, in 3 + 4 = 7, the symbols for the numbers are conventional. But mathematical symbols have a clear and determinate subject matter, unlike language, which can treat of everything. So, language can describe mathematics and calculation, but not vice versa.
Language, mathematical symbols, diagrams, and drawings all put our thought into a form that we can perceive with our senses. For example, we may try to solve an equation in our head and then realize that we need to write it down or say it aloud, and doing so makes it easier to work through.
But compared with other aids such as diagrams or mathematical symbols, language has three advantages for representing our thought. First, language depends entirely on what is natural to every normal human being, namely the mouth, tongue, ears, and intellect, and so it is always possible and near at hand for everyone. Second, the sounds produced by the mouth can be very varied, and almost none of them has an intrinsic connection, as sound, to the things we describe in our words. Third, language has a universal scope and can be used to represent anything, far beyond the expressive power of even the most advanced mathematics.
Diagrams and drawings can, of course, be very helpful. But even the diagrams that we make are almost always things that we make sense of with language, since we put names and labels on their parts and often describe in words what the parts mean. In the same way, when a mathematical problem cannot be solved in its given formulation, we can always state the problem in language and consider it at length in more general terms. And at the deepest level of mathematical inquiry, it is always valuable to ask what the expressions and the results really mean, even if that may not be something we can express to someone who is not already a mathematician.
This also has an important and philosophical implication for mathematics: mathematics can never be a complete and closed system, because it relies on words and concepts beyond itself, that is, on terms and principles common to other disciplines.
None of this takes away from mathematics, which is extremely powerful for understanding the world. The point is only that language is more general than mathematics and that it grounds mathematical thinking. That is why the classical tradition and this course following it begin with the arts of language, that is, grammar, logic, and rhetoric, rather than with mathematics and mathematical representations.
Next, we consider why we must begin with words.
Continue to Why the Classical Tradition Begins with Words.
Or go back to Where to Begin for Higher Studies in General to explore another starting point.