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Most written languages have a direction to be written, for example, most european languages are read from left to right, and arabic based languages are generally written right-to-left.

In mathematics, there is not this syntactic formation of words, letters etc, so is there any method of 'reading' them, can they be read from the left or the right. Is this a feature of truly 'logically' formal langauges?

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  • Formal expressions are "read" recursively, in reverse to how they are formed, and the rules for that are much more precise and explicit than in any natural language, see well-formed formulas. Indeed, they were programmed into computers long before any text recognition software was designed. As a matter of fact, the same recursive structure underlies interpretation of sentences of natural languages too, over and above superficial conventions like left to right, see generative grammar.
    – Conifold
    Jan 5, 2023 at 22:38
  • In usual mathematical jargon, that used by mathematical textbooks, you have yo read them according to the language used: from left to right for e.g. europea languages. Jan 6, 2023 at 11:48

4 Answers 4

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mathematical expressions have agreed-upon "grammatical conventions" for how they are to be interpreted, so that anyone solving the same equation will obtain the same result. In their simplest form, these are the rules of algebra that we learn in high school.

In some cases, there may be ambiguities in the equation which are resolved by including guides (in the form of, for example, nested parentheses) which specify the order in which different operations in the equation are to be performed.

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Asymmetry is a mathematical condition, and already with exponential notation, we have cases of mathematical assertions that can't be read arbitrarily in one direction or the other: besides 24 = 42, there is no solution from natural numbers for ab = ba (I mean, aside from when a = b, of course). (There are solutions among transfinite numbers, e.g. ℵ0ω "probably" = ℵω0, but those instances are still "rare" in the sense that most lists of alephs will not, in written speech, be lists of singular cardinals; but this is trivial.) In fact, there are even relations, not equivalence relations strictly but in the same general family (generic symbol: "~", although in logic we more often use "~" for "not") where A ~ B might be true but B ~ A would not be (e.g. A < B might be true but B < A would not be).

In category theory, much can be made of the distinction between left- and right-adjoint functors. But I doubt this is due to some essential property of leftwards vs. rightwards directionality in the zone of practical human experience. (That most people are right-handed does not seem metaphysically necessary.)

Going back to the exponentiation context, here's a discussion on MathSE that's relevant.

The gist of all this is that in one sense, there is a correct way to read a given piece of mathematical prose, but it is not second-order correct that this is how the piece is read (i.e., setting the local standard of correctness does not depend on meeting a broader standard of correctness for setting local standards; at least, not so far as I know). You can see this in the Benacerraf identification problem, where translating back and forth from {}-notation into ∈-notation (and then into Arabic numerals, as "implementing" natural numbers in the form of sets) is fairly open-ended when it comes to what kinds of translations are (or seem) ultimately permissible.

EDIT: in sec. 2.3 of the SEP article on substructural logic, they go over an "order question" in the display of premises in arguments, a question that opens up the possibility of a (mathematical) logic where X, Y has a different force of implication than Y, X. Again, this is not to say that the higher-order context of this difference mandates right-to-left or left-to-right reading, just that it mandates some local decision about reading order.

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Formulae follow grammatical conventions as @niels nielsen said. But there is more:

  • Formal proofs are completely structured by convention: See e.g. the Proof Explorer. Note that for different such proof libraries, the logical basis (type theory or not, ZFC or ZF, ...) might differ, as well as the display of consequences, but the subject Mathematical Logic does definitely set many constraints as convention. (e.g. beginning with axioms and proving by specified inference rules) Nevertheless note that formal proofs are not something which is the average mathematician's daily bread.
  • Real mathematical texts are normally a mixture of formulae (precisely constrained by rules), formal language ("Let ... a natural number", note that special grammar conventions apply like whether "Let" is used in conjunction with "be".) and less formal parts, which summarize, name the target in simple words, explain a concept, refer to a source and so on.

As @Conifold mentioned, formulae are often defined recursively. In Mathematical logic, this leads so far as to prove something "by induction over the structure", i.e. by seeing the formula as trivial, in finitely many steps recursively defined string. Despite this, there is some informal use (at least in papers in Western languages) of the reading direction: for example, a = b may stand for "o. k., a shall mean b".

In addition, I would like to bring in the ideas of a term rewriting system and of the Chomsky hierarchy.

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  • Note that formal maths writing distinguishes between the signs '=' and ':='. The first 'a=b' means the two quantities a and b are the same and could be read from left to right or right to left. Rewriting it as 'b=a' would have the same meaning. On the other hand 'a:=b' means quantity a is defined by quantity b and this relation is not symmetric. You need to know b first and then can get a from it.
    – quarague
    Jan 7, 2023 at 7:17
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A good question. Unfortunately, LaTex is disabled for Philosophy SE.

You of course know that some mathematical operations are commutative (a + b = b + a). Given so, how would you deal with the following exercise?

12 + 1 + 18 + 19 = x. What is x?

You could ...

  1. Read from left to right: 12 + 1 is 13, 13 + 18 is 31, 31 + 19 is 50, x = 50

  2. Read from righ to left: 19 + 18 is 37, 37 + 1 is 38, 38 + 12 is 50, x = 50

  3. Read from the anywhere: Do speed math and choose made-for-each other pairs of numbers. 12 + 18 is 30, 19 + 1 is 20, 30 + 20 is 50, x = 50

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