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Ask HN: Why is there only one widely accepted syntax for mathematics?

• Someone 3y ago
  There isn’t. About every specialized corner has its own conventions. The derivative of a function, for example, can be written in at least 4 ways (https://en.wikipedia.org/wiki/Notation_for_differentiation https://en.wikipedia.org/wiki/Notation_for_differentiation) Also note that https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbols https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbo... lists only fairly common notation, but already says “may”, “in such-and-so field” or “it is also denoted as” lots of times, and mentions several notations with multiple meanings. The section on brackets, in particular, clearly shows the same notation can mean wildly different things. There is some notation that’s almost universal, such as using ‘+’ for edition, but that’s the case for programming languages, too.
  • quantified 3y ago
    Different flavors of set theory are slightly different. It doesn't take long to get into "specialization". Are there different notations for propositional logic and predicate logic?
    • Someone 3y ago
      There definitely were slight variations. https://en.wikipedia.org/wiki/Quantifier_(logic)#History https://en.wikipedia.org/wiki/Quantifier_(logic)#History: “Frege did not devise an explicit notation for existential quantification, instead employing his equivalent of ~∀x~, or contraposition” and “Peirce and Mitchell wrote Πx and Σx where we now write ∀x and ∃x.” (I like that notation, by the way) and “Peirce's approach to quantification also influenced William Ernest Johnson and Giuseppe Peano, who invented yet another notation, namely (x) for the universal quantification of x and (in 1897) ∃x for the existential quantification of x. Hence for decades, the canonical notation in philosophy and mathematical logic was (x)P* to express "all individuals in the domain of discourse have the property P," and "(∃x)P" for "there exists at least one individual in the domain of discourse having the property P." Peano, who was much better known than Peirce, in effect diffused the latter's thinking throughout Europe. Peano's notation was adopted by the Principia Mathematica of Whitehead and Russell, Quine, and Alonzo Church. In 1935, Gentzen introduced the ∀ symbol, by analogy with Peano's ∃ symbol. ∀ did not become canonical until the 1960s.”*
      • quantified 3y ago
        I like the notation of "product across X" for "for all". I don't intuit the "sum across x" for "there exists", that seems more distinct. Although I can see that applying OR of the predicate across the population would achieve the same result. It would be nice to express logical aggregations such as AND and OR across sets like sum and product can be. Since their symbols are not Greek letters, it's not as simple as choosing the capitalized letter. For that matter, a way of separating map-reduce from the reduction function, so that max, min, or other functions can be expressed uniformly, would also seem like an advancement. Probably it's been done and I'm simply ignorant.
• 1970-01-01 3y ago
  Because it's very much a language if you want to formally communicate your thoughts. The 'syntax' becomes flexible when you're into higher levels of math. Feynman inventing Feynman diagrams comes to mind. This was one way math became more intuitive and flexible for communication of novel thoughts.
• amadeuspagel 3y ago
  “It is a profoundly erroneous truism, repeated by all copy-books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.” ― Alfred North Whitehead, An Introduction to Mathematics