Most mathematical ideas are built by logic and language alone. We need language and symbols to name the attrubute (number) that is common to •••• and MMMM and we just "agree" on that language -- four or vier or cuatro, for example. We also agree on a name for |||||||| and BBBBBBBB. These names can differ from culture to culture. But after that, we have no choice about what we call a combination of •••• and ••••. It must be what, in English, is called eight.
But there are a few mathematical conventions -- things we just agree on without being forced to agree by logic -- beyond language alone.
Order of operations, mathematical symbols...
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