Burning off some anger at well meaning but objectionable FB people

Another one of these things is floating around facebook and watching the commentary about it made me angry. SO here we are. The original looks something this: 

9 = 90
8 = 72
7 = 56
6 = 42
 
3 = ?

And as typical, there are a variety of answers, and as usual, most of them seem to make 'sense' to the commentator, but are untenable in any respectable arithmetic system, which is one in which the following hold

1. we're working in base 10, and not hexadecimal or something (I can't do bases in my head fast enough to know if, given the equations we know, the answer would be different in hexadecimal or something)
2. arithmetic operations are vaguely consistent
3. that the operator/symbol = means 'is equivalent to'
4. arithmetic operations respect RST, that is Reflexivity (that α must be equivalent  to α [i.e. itself]), Symmetry (that α+β must be equivalent to b+α), and Transitivity (that if α is equivalent to β and β is equivalent to γ then α must be equivalent to γ)

In such a system the answer is (and pretty much can only be) 12. 

Now I'm happy to discuss possible arithmetic systems until the cows come home. I'm happy to discuss possible operations. I'm happy to disagree about the right answer. What made me angry was the people who were trying to be 'reasonable'. "Well," they suggest, "the answer is 12, but I suppose the answer could be ... if ..." But it couldn't be. Here's why.

First, a word on notation. There is no sense in which the expression 9 = 90 makes any sense, in a system where the = operator means 'is equivalent to '. If 9 = 90, then what does = mean? If = means 'is equivalent to' we can assume that there is some unexpressed function being applied to the numbers on the left, resulting in the numbers on the right. Let's use the notation f(x) for this operation, so the list should read f(9)=90, f(8)=72, etc. So let's assume the best of intent, that = means =, that RST hold, and so on. 

Given the four examples we know the right answers for, it would appear that f(x) is represents the operation x*(x+1). In which case f(3) is clearly 12. f(x) could only yield the equations it does, i.e. f(9)=90, f(7)=56, etc. and f(3)=18 if the f(x) operation was actually x*(some number above it in the list). This is not what I would accept as a coherent function in any reasonable arithmetic system.

But arithmetic operations cannot be defined in terms of what's around them, or any circumstantioal information.  If f(x) were anything like the above, then changing the order or spatial arrangement would yield idiotic results:

f(9) = 90
f(6) = 54
f(7) = 24
f(8) = 56

But if that were so, we'd have violated RST because f(6) cannot be dependend on to equal f(6). It makes no sense to refer to circumstances in the definition of arithmetic functions. You can't define + in terms of what's around it, or where it is on the page, or even whether you're adding apples or oranges. α+β cannot equal one thing at the top of a page and something else in the middle, any more than it can (reasonably) equal one thing on Tuesday, and something else on Thursday, unless it's a leap year, unless the leap year ends in 00.*

*I suppose there is a case to be made that operations could refer to intrinsic properties of its operands, i.e. f(x) is a different operation if x is even vs odd, or positive vs negative. But that involves universal properties of numbers (in the system in question), i.e. something which are deterministic and 'available' by virtue of the number system you're using, and totally independent of the context in which the number appears**

**I'm not sure this would actually make sense, but I asked someone earlier how you multiply or divide Roman numerals.  I mean, I get that your average Roman kid probably didn't have arithmetic homework in which they regularly had to multiple 46 by 331 or something.  But someone must have had to work out how many denarii to charge someone buying XLII eggs and III denarii a pop. The vest answer I could find needed to refer to whether or not the operands (or their derivatives in the procedure) were even or odd (in the sense that the procedure described halving some numbers and if the number was odd, they had to remember to deal with the remainder.  But I digress.

And even if we did allow that kind of variability, there's still no way to interpret both f(9) at the top of the list and f(8) in the middle of the list, since there is no definable object to use with the f(9). 

If we decide to allow that like the blank space above the top of the list allows for some other, unknown, object to be part of f(x) at the top of the list, then I want to know whether than exception to f(x) also applies in the case of f(3), since there is equally a blank space above the f(3) line and the f(6) line. Which would still be inconsistent with f(3)=18 (f(3) would equal 30, I guess).  In fact f(3) would have no determinable answer, and the arithmetic system being described would fall apart; RST would not and could not apply.***

***I haven't thought seriously about RST and they means in an arithmetic system since high school. Long before my present understanding of number theory set in. This has actually been kind of fun, in the way that potentially learning how to multiply and divide Roman numerals was fun. But I'm still angry at these "well, I guess it could be 18..." people.

So the answer has to be f(3)=12 give my assumptions, and anything else results in an inconsistent and untenable arithmetic. Thus spaketh Hagiwara. Go in peace.