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

https://news.ycombinator.com/item?id=39063293

by owenpalmer • 3 years ago

3 6 3 years ago

news.ycombinator.com

The obvious comparison is the large number of programming languages which use different syntaxes for different problems. I think this is a good thing. Imagine if we never moved beyond Fortran! Undoubtedly there's a more intuitive (and less ambiguous!) way to represent math.

Someone2• 3 years 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)

Also note that 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.

quantified2• 3 years 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?

Someone2• 3 years ago

There definitely were slight variations.

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.”*

1970-01-01• 3 years 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• 3 years 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

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

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