You are wrong because your explanation fits perfectly
what an Engineer does, which is to solve problems
And since programming is a form of expression mathmatically.
the following defination of Engineering from Websters proves that programing is an Engineering discipline.
The application of scientific and mathematical principles to practical ends such as the design, manufacture, and operation of efficient and economical structures, machines, processes, and systems.
Art Tatum: Isnt technical drawing a form of communication?, How about an Equation that describes a system, all are the answers of the Problems which Engineers try to creatively solve. The Product of Engineering is communication, the communication of the solutions to the questions which were orignally posed to them.