Suppose an alien life comes to Earth, and challenges us to answer a question that will allow them do determine if we can communicate without ambiguities and solve controversies in a rational way. The question is:

Dear Humans:

We know that 48÷2×(9+3) is 288, and we know that 48÷2y is 2 when y is (9+3).

Given that, what is the result of expression 48÷2(9+3)? Is it 2 or 288?

They warn us that there is only one right answer, based on math and linguistics theorems, which is either 2, or 288. In other words, we cannot give answers such as "ambiguity", "depends", "neither", "you are cheating", "2 or 288", etc. We must pick 2 or 288, and give one of these numbers as the answer. If we answer correctly, they will proceed with friendly relations. Otherwise, they cannot forward us about the outcome of a wrong answer.

They give us 10 hours to come with an answer. The government quickly assembles a committee with experts in both math and linguistics. If you were a member of such committee, which answer would you defend, and why?

  • I would answer 288, using linear order to disambiguate the competition between the application of the two operations, division vs. multiplication. In other words, given equi-level operations like division and mulitiplication, I would rely on the linear order to decide which is applied first. In this case, it is division because division appear to the left of multiplication. The conundrum is mentioned in Wikipedia: en.wikipedia.org/wiki/Order_of_operations. – Tim Osborne Dec 1 '13 at 7:11
  • Thanks, @TimOsborne, but your criteria conflicts with the universally accepted answer for 48÷2y, which is 2. The implied multiplication 2y is prioritized over the division. Why wouldn't that hold for the implied multiplication of 2(9+3)? – fernacolo Dec 1 '13 at 7:34
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    @TimOsborne, ehm... so in your book "linear"="left-to-right"? – dainichi Dec 2 '13 at 0:57
  • @fernacolo - I can't remember the exact phrasing for it right now (and it's too late to go find out) - but 2y expressed like that is a "term" (I think) and is evaluated like (2 x Y), ie earlier than 2 x Y would be without the brackets. 2(9+3) isn't "well defined" mathematically, that kind of expression only applies to variables. – Ryno Dec 2 '13 at 6:17
  • @Ryno That's why we must resort to what is defined in the input. And it's not true that the kind of expression only applies to variables. You can find many real world examples where numbers and parenthesis are concatenated without an explicit multiplication symbol, like for instance, the series definition of cosecant (en.wikipedia.org/wiki/…). – fernacolo Dec 2 '13 at 7:44

From a linguistic perspective, there is no such thing as a theorem. There are some recognized heuristics for determining the difference of meaning/function of linguistic units. @fernacolo demonstrated one of these above. But there are more interesting things linguistics (the study of all of language - not just words, clauses and sentences) has to say about this.

This example shows the difference between axiomatic and conventional aspects of the practice of mathematics. How notation is used to denote the order of operations is a matter of convention. The explicit conventions in the language of maths as used on Earth is that x and simple concatenation both mean multiplication and that multiplication has an order priority over other operations. Therefore, using these two conventions the answer is 2. However, there is also a human tendency to group things together so when polled, a group of people will be about evenly split between the two answers, quite possibly giving different answers at different times. However, the alien letter seems to imply that there is a difference in interpretation of notating multiplication by concatenation or by 'x'. But this is completely arbitrary. There is no such convention among mathematicians. We can guess that the aliens are using these two notations to indicate the prioritization of operations but that is on a very small sample and it is conceivable that this holds some other meaning to them.

An alien lifeform with any level of linguistic insight, should be asking, not whether we have a way of communicating without ambiguity - because we don't, but rather whether we have procedures that help us disambiguate. And the answer to that is yes. In math, to resolve potential ambiguities about the order of operations, we use bracketing, which would remove any ambiguities about the answer in this problem. We have another way of resolving ambiguities which asking about each other's assumption. Therefore, removing that possibility from the interaction is misunderstanding the totality of linguistic tools at our disposal. The 'human' way of answering the question would be: "It seems that you make a distinction between 'x' and concatenation to indicate the order of operations. Is that correct?"

  • First, in math there's no priority of multiplication over any other operation. Multiplication and division have same priorities, and both are used in input. Second, it's actually possible to communicate without ambiguity, and software developers do it all the time with programming languages. While math is not a programming language, it's supposed to be independent of culture, as long as we agree on conventions. Third, this is not about using existing conventions, but about identifying conventions and relations rationally in a given domain. The input represents domain, so it can be arbitrary. – fernacolo Dec 2 '13 at 1:37
  • Sorry, but this sentence is a contradiction: "it's supposed to be independent of culture, as long as we agree on conventions". Culture is a set of agreed conventions. So if we have to agree on them (even implicitly), it is culture-dependent. Also, it's true that there is no priority of multiplication over division underlying mathematics, but it is a standard convention in mathematical notation (which again is culture-dependent). – Dominik Lukes Dec 23 '13 at 19:40

Using only the alien's input (i.e., ignoring cultural differences among scholars, experts, tools and literature), we must observe that the aliens are informing us about what they believe to be universally accepted conventions:

  • 48÷2×(9+3) = 288
  • 48÷2y = 2, given y = (9+3)

From that, we can identify two clearly distinct semantic elements:

  • The explicit multiplication happens after division, hence 48÷2×(9+3) = 288.
  • The implicit multiplication happens before division, hence 48÷2y = 2 when y = (9+3).

All common elements (numbers, symbols, etc) are behaving equally in both sentences, and we can assume that they must behave equally in the question expression.

Given that, the expression 48÷2(9+3) has an implicit multiplication. Hence it must happen before division, and the correct answer is 2.

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    This is a good example of semantic analysis done by a linguist. However, I would argue that the corpus is too small for saying that there are two 'clearly' distinct semantic elements. – Dominik Lukes Dec 1 '13 at 8:33
  • I agree with you about the corpus being small, which can cast doubts in similar, real-world analysis. However in this scenario, the aliens have restricted the domain. So I believe we can rely on this restriction to establish semantic relations with confidence. – fernacolo Dec 1 '13 at 8:37
  • It seems to me that there is more than one way to interpret the "data" given by the aliens, since the "corpus" is so small, as @DominikLukes notes. Why is the implicit/explicit distinction more valid than the variable/constant distinction mentioned in the comments above or the presence/absence of parentheses? Both in the original question and in this response, it's not clear whether the aliens are testing us on our awareness of our own notational conventions or if they just happen to use a system that looks a lot like ours but may or may not follow the same conventions to a tee. – musicallinguist Dec 2 '13 at 16:02
  • @musicallinguist Among competing interpretations, we should use the one with fewest assumptions (en.wikipedia.org/wiki/Occam's_razor). We can assume that 2y and 2(9+3) has the same meaning, given y "is" (9+3), as stated in input. In order for these expressions to have different meanings, with need to make more assumptions, possibly arbitrary ones, and that would violate the rule of Occam's Razor. – fernacolo Dec 2 '13 at 17:04
  • What do you mean by 'meaning'? If you mean that they both denote 'multiply the quantity 9+3 by 2', then we can assume 2×(9+3) and 2(9+3) to have the same 'meaning', as well. But according to the implicit/explicit theory, they don't have the same status in the order of operations. Likewise, 2y and 2(9+3) might both denote 'multiply the quantity 9+3 by 2', but they might be treated differently in their order-of-operations status. In natural languages, words that denote the same thing may pattern differently morphologically or syntactically; I don't see how Occam's Razor is relevant. – musicallinguist Dec 2 '13 at 17:37

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