Dear Jeff,
You begin by noting just what a strange and limited language is mathematics, with examples 'pass the salt' and 'watch out', etc. Your examples of Roman versus Arabic numerals somewhat parallels Donald Palmer's essay, where he points out that modern science and technology would not be feasible (possible?) without the decimal number system. Typically the actual numeric system in use is glossed over.
I fully agree that counting (the physical implementation of Peano Axioms) leads to mathematics, producing the natural numbers and, per Kronecker, allowing man to build 'all the rest' of math. In my essay I elaborate on 'counters as key'. In one comment you ask "could counting exist without logic?" You seem to think yes, but in my essay I tend to think no. I view 'logic' not as mathematical argument, but as the basic fact of nature that allows AND and NOT gates to exist at every scale as fundamental physical structure, whether implemented at the level of protein, neuron, or silicon.
What I really enjoyed was your speculation about "language" in terms of other species or senses, from echolocation to dog's sense of smell. You make a fascinating point, re dogs use of 'smell' to communicate 'Moon', that "the Moon does not have a smell, but Democrat and Republican states are not actually red and blue."
Also like your statement "science cannot find truth, but can find understanding, and failure is an important tool in understanding."
I invite you to read my essay and welcome your comments.
Best regards,
Edwin Eugene Klingman