jamest wrote:You don't consider it significant that it's potentially impossible to objectify or universalise the explanatory framework of mathematics?

No I do not consider it significant at all. Languages evolve in response to a society's communication needs. A society that does not have a need to communicate certain ideas would likely not have a vocabulary to communicate those ideas. Vocabulary evolves to accommodate needs. Can you show me a medieval english equivalent of the term "path integral"? The english language evolved the vocabulary to communicate the idea of a "path integral". It did not have it right from English version 1.0 (whenever that might have been).

jamest wrote:Maths requires a particular form of subjective expression, then?

No, not any particular form. A large variety of subjective expressions can communicate mathematical ideas, as is evidenced by the fact that the Chinese, Japanese and most other people are able to do math in their own language.

jamest wrote:
I've suddenly realised that many/most mathematical concepts (plus; minus; equals; velocity; time; space; sets; infinity; nothing; etc. etc.) have a definite meaning within the English language which might not translate well, if at all, in all languages. Therefore, I wondered whether mathematics made sense from the perspective of all languages?

Not a few of these concepts also have several mutually exclusive meanings in English as well, depending on what field of maths one is looking at. To many speakers of English, it's not obvious either what you're talking about if you go on about power series, semigroups, ... most of maths is a specialist jargon not part of the language as used by the overwhelming majority of the speaker community. Nor are they things one can expect people to learn just by being part of the wider speaker community.

Could Old English or Latin have been used to develop such a specialist jargon? Sure. Did they? No, although descendant languages did. Some of this requires using terms in ways that kind of doesn't really fit with how English is used in other contexts, and the same goes on occasion in other languages as well (I recall a lot of the maths I've learned in Swedish is phrased in ways that don't sound very natural to Swedish.

Here, really, the wittgensteinian notion of language games is very important: maths is a family of language games. Language works different when you're doing maths than when you're teaching someone to swim or telling someone how to slice fish into fillets. Of course, in some predominantly non-mathematical language games, sudden 'subgames' may emerge where some maths (most often relatively simple maths - arithmetics and such; even a restricted part of arithmetics is probably very common: counting by using some successor function) pop up.

However, if you take a short text of this specialist jargon and present it to a random (heck, even relatively well-educated) native anglophone, it's rather likely they will not grasp the contents.

In the case of tiny tribes in Brazil or New Guinea, specialist jargons have not been developed yet, in part because no one in these tribes have had the need for abstract algebra or such. It may also be possible there's subfields of maths for which no English jargon has been developed, although there may be jargons in, say, Chinese or Russian for those subfields.

But the fact that maths don't translate into every language is easiest verified by noticing that some parts of math don't even translate into everyday English (which is a language).

Gödel's incompleteness theorem even basically kind of says that some maths will always evade being entirely translatable into languages.

I've always assumed that mathematics was the most objective of languages. I'm wondering whether this is in fact the case.

It is objective, but it's not very useful in day-to-day communication as you can't say much that is humanly relevant about things with it.

We also find that not only is maths a cultural thing as far as notation goes, there's also subcultural ideas within maths that at times have influenced notation and even how to think of what a solution is in subtle ways: for quite some time, mathematicians avoided having negative numbers standing by themselves.
x+1 = 0 was preferred over x = -1. This in part is the reason why we normally write eiπ + 1 = 0, rather than eiπ = -1.
Such things may even to this day influence us, it's just it's difficult to spot one's own biases!

Thanks for that informative and interesting post, Zwaarddijk.