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I am not sure if this is the right stackexchange, but I wanted to ask if all group of mathematical notations are a mathematical statement. I want to ask this, because it seems to me that it's not the case, but I don't know what are the other classes a group of mathematical notations can fall under. It seems it's not the case, because a statement seeks to claim something as the truth, and not all group of mathematical notations seek to do that. By mathematical notations, I mean mathematical symbols.

Sayaman
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    No. Not all expressions are grammatical. e.g., $x3++^y$ is meaningless. But what expressions are acceptable depends on your convention. The set of strings of symbols that are acceptable can be defined by a formal language – Jair Taylor Mar 28 '19 at 22:49
  • Just as not all strings of English letters are English statements, not all strings of mathematical symbols are mathematical statements. For a string to be a statement it has to follow the rules of syntax in its language. In some parts of math those rules are very definite. In English and many areas of math there is some flexibility, but there are still far more meaningless strings than meaningful strings. The statements are a subset of the meaningful strings. – Ross Millikan Mar 28 '19 at 22:53
  • Math isn't a programming languages. Math notation is designed to make it easy for humans to read math, not to turn everything into a formal system. – anomaly Mar 28 '19 at 23:24

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I may be misreading your question, but I'd say no. For example, $\pi$ is used to denote the ratio of a circle's circumference to its diameter and so simply denotes a (irrational) real number as opposed to any sort of statement.

Additionally, mathematical symbols can be grouped in a way that renders them meaningless; for example, $$a^\varnothing-\sum_{\text{David Hilbert}}\prod_!\iint^=.$$

As an aside, there are also meaningful mathematical statements which are impossible to prove (at least with the current, standard axioms, and, I suppose, assuming their consistency). For example, $$2^{\aleph_0} = \aleph_1.$$

Caveat: Of course, someone can come along and define some new notation. This may render a once meaningless string meaningful.

Gary Moon
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  • For example, an integral outside an equation, what is it? It's not a statement since it's not an equation, so what would you call it? – Sayaman Mar 28 '19 at 23:03
  • I would call it poppycock. But, you could also call it flapdoodle, gibberish, nonsense or any number of other things based upon your linguistic preference. On a more technical level, one could call it a string. – Gary Moon Mar 28 '19 at 23:08
  • It's a symbol, used out of context. Just like if I random,ly put a punctuation mark somewhere it doesn't make sense. – ConMan Mar 28 '19 at 23:12
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    It's a good post, but especially +1 for the idea of summing over all David Hilbert. – anomaly Mar 28 '19 at 23:28
  • @anomaly Thanks. – Gary Moon Mar 29 '19 at 01:06
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Just like any language, mathematics has grammar and vocabulary. There are meaningful ways to put symbols together, and meaningful ways to arrange groups of symbols, and these are essentially defined by a combination of assertion and consensus - by which I mean, anyone can say what something means but then everyone needs to agree on that for it to ultimately have meaning, at least in that context.

For example, I can't just go around talking about the mathfulness of something and expect you to understand that. Similarly, I can't just write $3++4$ and expect you to know what that might represent. However, I can say that "mathfulness" means "the ability to express something in a mathematical formulation", and then when I talk about the mathfulness of circular motion you have an idea of what I'm getting at. I can also say that $++$ is defined as an operator on two numbers such that $a++b := a + b + a \times b$, and now you know that $3++4 = 19$.

But someone else can say that "mathfulness" means "density of mathematical notation", and talk about the trends in mathfulness of journal articles. And they can also define $a++b := a^b + b^a$ and say that $3++4 = 145$. And then we need to have a way to distinguish those different meanings, which will depend on the context in which they're used and whether one meaning is used more often than the other and hence recognised more often.

ConMan
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