Learning Peano arithmetic from examples, for instance learning to count over the natural numbers, is a classic machine learning task. Modern machine learning systems can only solve this task approximately, that is they can learn to count over a certain interval but not, say, to arbitary values. Same goes for other arithmetic. This, despite the progress in apparently more complex tasks such as classifying objects in images etc. Of course, if you don't know what Peano arithmetic is, you can't imagine why it may be a useful test of learning ability.

But I think your question is more along the lines of "why do I need Peano arithmetic to build an image classifier?" (or any machine learning application). To be clear, you don't. But without knowledge of some fundamental concepts in mathematics and the theory of computation, you won't understand the power and limitations of any system you may be capable of buidling. As a lay person, that's by the by. As a machine learning researcher, that'd be disastrous.

Unfortunately, as the OP points out, such lack of deep background in AI research is common today, and probably responsible for the absolute quagmire in which machine learning research finds itself.