Is it a general truth that when two equations can be described with the same sentence, only varying intonation they are equivalent?

Question

My question is the effect of something that I have noticed during some of the courses I am taking as a first year maths bachelor, that in tutorials we will be asked to proof that two statements are equal but that when I try to describe the left and right hand side I end up with one and the same sentence, I will give three examples: \begin{equation} (A^m)^{-1} = (A^{-1})^m \end{equation} Both the left and right hand side are described by the sentence "the inverse of matrix A to the power of m" \begin{equation} f^{-1}(A\cap B) = f^{-1}(A) \cap f^{-1}(B) \end{equation} With both A and B being supsets of the real numbers, both the left and right hand side are described by the sentence "the values that when the function f is applied to them end up inside set A and also inside set B", something similar can be done replacing the intersect with a union and the 'and' with an 'or'. \begin{equation} (AB)C = A(BC) \end{equation} This example is discussed in three blue one browns series on linear algebra, both the left and right hand side are described as "first apply transformation C, then transformation B, and finally transformation A"

Of course, the fact that two sides are described the same is not a rigorous proof since it could be that sneaky business is applied in words with double meanings - although often that should be easy enough to spot - but I do think this is a rather interesting patern and I would absolutely love to know if anyone can come up with an example where both the left and right hand side are described the same and still not equal.

Answer 1

The sentences you provide lack punctuation. Hence they are (potentially) ambiguous. Whereas "the inverse of matrix A to the power of m" uses correct syntax, semantically it is ambiguous. It does not adequately describe either $(A^m)^{-1}$ or $(A^{-1})^m$.

Intonation is a form of verbal punctuation which is applied by someone who interprets a sentence. Verbally there are homophones, homonyms and homographs - so it depends on whether the sentence is spoken or written. Context can also be necessary as a clue to the intended meaning. Furthermore, the structure of sentences (syntax) can differ between languages.

So I think there is no general answer to your question, and the question itself is not mathematically interesting. It merely highlights the inadequacies of spoken languages for mathematics.

In mathematics 'punctuation' is provided by brackets (and other conventions such as BODMAS) rather than by intonation. Sometimes also the context is crucial in interpreting a mathematical statement. In mathematics it is essential to avoid all ambiguity. Mathematics is a written language rather than a spoken language. Verbal intonation has no role.

Answer 2

Mathematica has a Speak function, where (for instance)

Speak[x^2]

will produce the aural utterance "x squared."

Speak[(x y) z] and Speak[x (y z)] each give the identical utterance "x times y times z," where of course these need not be equal in non-associative algebras.