What are some fundamental differences between mathematics as a language and language spoken by humans? I heard mathematics is more restricted, but aside the fact that it's restricted I can't think of any fundamental differences between mathematics and spoken language.
Natural languages are by definition evolved naturally, mostly from spontaneous verbal interactions between humans, over large groups of people and many generations for the main natural languages. Sometimes, some authority legislates over various grammatical rules and how certain words should be defined, but the structure and logic of the language is nobody's invention.
Mathematical language is indeed more "restricted" in the sense that it evolved from interactions taking place in a much smaller group of people, namely, mathematicians, and it turns out that this had bad consequences.
Natural languages are mostly evolved from spoken utterances, rather than written ones, at least initially, although it could be argued that this is changing fast with the generalisation of printed books and more recently of computers and the internet. Mathematics, on the contrary, from the start mostly evolved from written material, although speech must have had some impact.
There is no fundamental difference in terms of the objects considered in natural language expressions and mathematical language ones. Mathematical objects are mostly abstract concepts, and increasingly so. But geometric figures are directly inspired by concrete objects and most concrete objects inspire an abstract generalisation based on them. So the difference is one of degree rather than anything fundamental. Mathematics is closer in this respect to metaphysics and to the more abstract corners of philosophy than to the language used on the streets of big cities.
The sort of logical reasoning used in mathematics is also identical to logical reasoning as used outside mathematics. The differences in what comes out of each comes entirely from the fact that mathematics is a symbolic language, which greatly reduces ambiguity and help improve the performance. This is an important difference, not a fundamental one.
One consequence of the restricted demographics of mathematics is that it is more susceptible to expediency. Natural languages are open systems. In spite of the Queen and of the Académie française, no one owns English or French. Not even Putin owns Russian. No Chinese emperor ever owned Chinese. Mathematics is in this respect much more vulnerable to fads, whims, ideology and wrong ideas.
First, the empty set. Try to talk about the empty forest as the forest that has no trees, and you will be laughed out of the room. Yet, mathematicians found nothing to object to the notion of empty set, even after Frege in person took the pain to spell out why it was a nonsensical concept and a silly idea. Why was the notion of empty set ever introduced into mathematics? The notion of empty set is a recent invention. It only became "necessary" after mathematicians had adopted mathematical logic as a foundation to mathematics. Thus, it is the vagaries of the very small group of mathematicians and philosophers who developed mathematical logic which succeeded in imposing a nonsensical concept on the millions of mathematicians active all over the world.
Second, the material implication. Contrary to what its name suggests, the material implication is not a sort of logical implication. It is not an implication at all. That is, ϕ ⊃ ψ is not equivalent to the conditional "if ϕ is true, then ψ is true". And this even if we restrict ϕ and ψ to mathematics. Yet, most mathematicians today take the material implication to mean exactly "if ϕ is true, then ψ is true", at least if ϕ and ψ are mathematical expressions. This is obviously also a direct consequence of the adoption of mathematical logic.
This is arguably a gigantic blunder, so big that it is even impossible to fathom the possible consequences in the long term, never mind the potential safety risks in the short term. The main cause of this situation was that the approach to mathematical logic taken by 19th and 20th century philosophers and mathematicians has become hegemonic.
Somewhat similar events are possible with natural languages, but never so extreme and always limited to small groups of people. Natural languages are driven by meaningfulness and the principle of reality. The mathematical language is more susceptible to the vagaries of human imagination and the arbitrariness of what seems expedient on the moment.
I heard mathematics is more restricted, but aside the fact that it's restricted I can't think of any fundamental differences between mathematics and spoken language.
Mathematics isn't just restricted compared to human language, it is fundamentally limited to its own abstract universe. Maths knows nothing about humans, animals, rocks, music, religions, activities, or time. We can apply maths to all of those things, and the physical and social sciences heavily depend on maths, but pure maths knows nothing of the universe we live in. And the universe humans live in is the universe of human language.
The main difference between the language of mathematics and human spoken language is that the former can never carry feelings. In other words, IT CAN NEVER CONVEY FEELINGS; but the latter may not be so. But this does not imply that mathematics is the only such language.
Because human spoken language can influence the auditory sensory system, different pitches or tones (in the same terms, phrases or in sentences) can produce different emotions in the listener. This also is not possible in the language of mathematics.
And since it deals only with abstract ideas IT CAN NEVER SPECIFY AN OBJECT.
Language of mathematics can be changed into spoken language. Therefore we can say that the language of mathematics is a subset of human spoken language (if the idea transacted only is considered).
To understand a difficult idea we usually need the help of our eyes also. Although it will be difficult to understand, everything that can be done with the language of mathematics can be done with spoken language. So the fundamental differences can only be assessed from that point of view.
Since verbal communication includes both spoken and written form, that convenience is here in the language of mathematics also. So language of mathematics is easier to comprehend when compared to spoken form.