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The first few times I saw this problem on social media I always chose nine. At some point I recall thinking about it potentially being ambiguous, but never gave it much thought. By the end of researching around I have now confidently landed on insufficient clarity.
This is a long read, but it seems to capture and respond to almost any nuance imaginable for the problem. Thanks to @Frogswap for providing it. To be clear I had landed on my answer before reading this, but it did solidify it for me. Not that there was anything wrong with persuading me in the comments as it was totally part of the purpose of this market, but I'm pretty sure this resolution would have ended up the same regardless.
If it was clear what the answer to 6 ÷ 2(1+2) was, this market wouldn't even exist in the first place. And even if it did exist, everyone would agree on the answer. What's really at issue here is whether 6 ÷ 2(1+2) means (6÷2)(1+2) or 6÷[2(1+2)]. But that's not a mathematical fact, it's a matter of convention. And since there is no agreed upon convention, there is no fact of the matter as to what the "correct" convention is, since either can be correct.
@JosephNoonan As has been mentioned already, by the order of operations you learn in grade school, the answer is 9, since PEMDAS says to first evaluate anything inside grouping symbols, then exponents, then multiplication and division from left to right, and then addition and subtraction from left to right. That means we evaluate it as 6 ÷ 2(1+2) = 6 ÷ 2(3) = 3*3 = 9. But that is not a universally used convention, even in the modern day. For example, if I want to write the Fermi-Dirac distribution, I will generally write it as 1/[exp((ε-μ)/kT)+1], or [exp((ε-μ)/kT)+1]^(-1). Either way, I omit the parentheses around kT because everyone knows that those are meant to be grouped together "under" the slash. Even the Wikipedia article uses this convention: https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics
So obviously, there are two distinct conventions in play, both of which are commonly used.
@JosephNoonan is there a formal competing convention to the formal PEMDAS convention taught in school, or is the competition simply non-rigorous adherence to the formal convention for the sake of brevity, much in the way that language and vocabulary is a living thing and words do not have fixed definitions but rather their meaning shifts as their usage shifts in the popular language.
I generally view math as rigorous and seeking to resist changes. Papers may introduce their own notation if they define it before hand, otherwise I would expect the formal convention to be assumed. If we cannot assume the formal convention then everything needs to be defined evetime and we can assume nothing.
edit:
The point of having a convention is so that if one does not explicitly define their annotation, then it should be assumed to be using the convention. If there are two competing conventions then this purpose is defeated and everyone should always define their notation, but then there is no longer a need for a convention because now everyone has to define their system of annotation.
@ShitakiIntaki Most of the time people just don't adhere strictly to any convention for the sake of brevity. But I think in practice this usually ends up with "multiplication by juxtaposition takes precedence over division", and some people have in fact explicitly adopted that as a convention.
@JosephNoonan Stating the Fermi-Dirac equation is effectively an argument to authority. All it is is a derived equation, the written form, whilst understood is still incorrectly written if you assume our mathematic system is well-defined. Everyone here arguing for 1/inefficient clarity is arguing our maths is not well-defined.
Stating the Fermi-Dirac equation is effectively an argument to authority.
The question being posed here is what convention is "correct". There is literally no possible argument except for an argument from authority or from popular usage. "Your grade school textbook teaches PEMDAS" is also an argument from authority.
Everyone here arguing for 1/inefficient clarity is arguing our maths is not well-defined.
The convention you use for order of operations isn't a mathematical fact. It's a fact about how we choose to write mathematical expressions. Some mathematical notations are ambiguous. This is completely trivial, and if your problem with "insufficient clarity" is that it implies this obvious fact, I don't see what the problem is.
The market is finally set up so that, whatever the resolution, two kinds of people will be very upset. Well done, everyone!
edit: I guess N/A or PROB(33, 33, 33) would make three kinds of people a little upset.
It has to be 1. We always percieve divide by symbols as slash's. 6 / 2(1+2). 6/6. 1. Unless it is interpreted as 6/2 x (1+2). 9. BUT, you should be doing multiplying the bracket by 2 first. https://extranet.education.unimelb.edu.au/SME/TNMY/Arithmetic/wholenumbers/operations/orderofops.htm
Some arguments from authority:
Oliver Knill, a preceptor at Harvard: https://people.math.harvard.edu/~knill/pedagogy/ambiguity/index.html
George Bergman, (now retired) Professor Emeritus at Berkeley: https://math.berkeley.edu/~gbergman/misc/numbers/ord_ops.html
Two Casio calculators:
https://i.stack.imgur.com/DlcVB.png
(taken from https://matheducators.stackexchange.com/questions/17033/a-pemdas-issue-request-for-explanation)
Note that these aren't cherry-picked (except the calculators), they're the first authoritative sources in a duckduckgo search for 'multiplication by juxtaposition priority', although that phrasing may be biasing results.
@JosephNoonan Intentionally not clarifying, just replying so people know I am paying attention. I’ll say I have thought about it, but I’m open to continue thinking about it as well. I guess at the end of this there’s no getting around my thoughts on the matter, but my reason for the lack of clarity is to encourage others to answer how they think over trying to predict how they think I will resolve. I’ll make sure to explain my resolution when I do though. Also, I will not be resolving this early.
@lieblius It sounds like you already have an idea of how you intend to answer this. Is there anyone betting with insider information on how you intend to answer?
By the way, our profits and losses will be determined by how you answer, so we are going to bet on how we think you will answer. If you just wanted to gather our opinions, then you should have made a poll.
@LukeHanks I have a resolution I could make if I was forced to but I do not have a clear cut answer. It can be assumed that the current likelihood in resolving any of these 3 answers is 33%. The less information I provide means the less my answer is inferable, a battle for correctness is much more fun than a poll. I have no ulterior motives regarding this market other than the mana gained from people making bets. This includes insider trades, I will not be voting on this question and I will also not bias my answer based on anybody that has or will vote on the question.
@lieblius Is it even possible to assume an equal likelihood of all three? That feels paradoxical at first glance; if it's equally likely to be 9 or 1, that would strongly suggest it is insufficiently clear. I'm probably off base here, I just can't wrap my head around it.
@Frogswap The equal likeliness for each are the initial conditions I have personally approached the problem with as well as the initial betting odds when creating the question. Sure there is probably a much more definitive answer in my mind but I don’t intend to get at it yet. And yeah if my intention was to solely use the odds to resolve, it would be unfortunate if 50% chose 9 and 50% chose 1.
@lieblius I'm not trying to be critical of you or add information to the market with that question, I just meant it as a tangential sort of logic puzzle. Like, shouldn't your initial belief have been that it was unclear until you had evidence to support one side or the other? And if you had enough evidence to push both answers to 33%, shouldn't that have been even more evidence of the lack of clarity? This is going to bother me until I find an intuitive answer, mostly because I suspect it does actually make sense to put 33% on each.
@Frogswap Hmm let me take a stab at it. I think the first thing we can work off of is that "6 ÷ 2(1+2) =" has numerical answer. "Insufficient Clarity" is not an answer to "6 ÷ 2(1+2) =", but an answer about the question "6 ÷ 2(1+2) =". Now is it possible that our references to "Insufficient Clarity" are actually talking about different things? If a reference to "Insufficient Clarity" is in regard to the market for the question "6 ÷ 2(1+2) =" then I see there would be a paradox, but if my reference is to the question "6 ÷ 2(1+2) =" itself then is there really a paradox? I think not. The market for the question "6 ÷ 2(1+2) =" and the the question "6 ÷ 2(1+2) =" itself should be independent. The betting odds can be ironic but I think it dodges the paradox. However, if we assume the betting odds are in any way informing my resolution, then yeah I might have walked myself into a paradox. I will have to decide whether it is worth wrestling with it to allow the odds to factor into my resolution or to disregard the odds entirely. In the spirit of my responses so far, I would suggest an equal likelihood of me choosing to do either haha. Hope I made some progress with it but I could just be waffling; as one usually does with conversations on paradoxes.
@lieblius I, for one, think "it'd make sense to put 33% on each, but if P(9) and P(1) are the same then that's indicative of insufficient clarity, therefore the formula is insufficiently clear" is a perfectly good argument for resolving "insufficiently clear"
@lieblius Is it resolved based on your opinion at close, then? I'm not asking you to reveal any information about your opinion, I just want to know what the actual resolution criteria are.
@JosephNoonan The resolution criteria will be my best logical attempt. I can exclude randomness for you, but there are universes where I trust my intuition over others and pick my opinion, as well as universes where I appeal to an external criteria that I deem fair and sound (even if I personally disagree). I wont make a response to it so the market doesn't instantly converge, but anyone is welcome to suggest a good idea as well as discourage a specific bad resolution. I'm open to considering things but I do have an idea so far. Apologies if this doesn't fully answer the question.
@lieblius Okay, so you are using the meta-criteria of "I will determine what criteria I think are best at market close"?