Can a proof be just words?

Exactly as thorough as you would have to be using any other kinds of symbols. It's just that vast messes of symbols are hellish for humans to read, but sentences aren't. Adding symbols to something doesn't make it more rigorous, less likely to be wrong, or really anything else. Symbols are useful for abbreviating in situations where this adds clarity, and making complex arguments easier to follow, but shouldn't be used where they do not help in this regard.


Yes they can and I'm of the opinion that symbolism and notation should be avoided unless it serves to simply the presentation of the material or to perform calculations. For example you want to cut a cube so that each face has a three by three grid of smaller cubes similar to the Rubix cube and with a little thought and experimentation once might conjecture that six is the minimal number of cuts. The best proof of this that I know of is simply "Consider the faces of the center cube." They require six cuts because there are six faces and it follows immediately. No symbols or calculation but still logical and mathematically sound.


Natural language for expressing mathematical statements can be indeed vague and ambigous. However, when you study mathematics, one thing you will usually learn at the beginning is how to use mathematical terminology in a rigid, unambigous way (at least for communication with other people trained in mathematical terminology). This process takes usually some time if you are not a genius (I guess it took me about two years at the university until I became reasonable fluent), so unfortunately I fear I cannot tell you a small set of rules which kind of language is "right" for mathematical proofs, and which is "wrong". This is something you can only learn by practicing.

Hence, the answer is IMHO "yes, words are fine, when used correctly by a trained expert". (Amazingly, one could say the same about more formal proofs using symbols.)

Note that historically, before the 18th century, proofs using natural language was the de facto standard in mathematics. Most of the symbolic notation we usually use today was developed in the 18th and 19th century.

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