What a great word 
I’m going to struggle to drop it into casual conversation mind you
That’s not true. In Ohio, all unincorporated land (that is, not in a village or a city), is in a township. Townships do have elected officials, but usually they offer very few public services and might not even have their own law enforcement agencies, relying on the county sheriff’s office for policing.
Technically speaking, all land is in a township in Ohio, but usually incorporated municipalities push out or replace the township’s jurisdiction. When a township has been completely eaten up by incorporated municipalities, however, it is still considered to exist on paper and is thus called a “paper township.”
Re: boustrophedontically
Actually, I think the word is boustrophedonically.
(ETA: If you’re casting about for opportunities to drop this into casual conversation, consider that desk-jet style printers usually do this. If there is a print head that moves across the paper as it prints (as opposed to a laser type of printer), then it probably prints alternate lines as it moves in alternate directions.)
Mathematics columnist John Allen Paulos once wondered if the plural of “spouse” is “spice”.
Maybe we say “math” instead of “maths” because the combination of “th” followed by “s” is simply kind of hard to pronounce.
Why would arithmetic not be considered a part of mathematics? (ETA: Saying that someone isn’t doing math if one isn’t doing proofs is just sort of a “No True Mathematician” argument, isn’t it?)
What better definition would you propose?
The OED thinks it means “the science of space, number, quantity, and arrangement, whose methods involve logical reasoning and usually the use of symbolic notation, and which includes geometry, arithmetic, algebra, and analysis; mathematical operations or calculations”. That looks like a rational defensible definition to me, and it corresponds with the way I observe the word to be used. Arithmatic is definitely an aspect of mathematics.
In what way do you establish that 3+3=6 without a proof? The proofs involved may be trivial, by computation, but they’re still proofs, and remembering back to lessons in arithmetic and counting in high school, what I’d recognise now as distributivity and commutativity properties of addition and multiplication were being discussed with hand-wavy “proofs” of why they should hold.
In my kid’s primary school (in the UK), they now refer to maths class as “numeracy”, and English as “literacy”.
But why are space, number, quantity, and arrangement all lumped together into the same science?
Why are paediatrics, obstetrics, diagnostics and psychiatry all lumped together under medicine? Why are ethics, aesthetics, semantics and epistemology all lumped together under philosophy?
They just are; that’s how the words are used; that’s what they mean.
I have often wondered this, and I was hapoy to see this OP, as I was thinking of asking it myself. I guess I missed the original OP, and like it often happens, when I come out here to ask a question, I start reading and forget starting the OP I wanted to start in the first place.
With that said, this is one of those words that truly drives a nail in my brain.
I thought the word “maths” was something that came from Australian English, as without exception the only people that I have ever heard say “maths” on a regular basis were Aussies. (Actually, it often sounds like Matt’s, not Maths, when my aussie friends say it.)
Anyway, i am interested in the uptick of the usage of “maths”. I have a BS in math… Not maths, but math. I hate when someone says “maths”. It sounds like a piece of chalk on a blackboard to my ears. Which brings me to another question… Why?
Why does “Maths” bug me so much? Anyone else feel this way, about math-maths or any other similar word differences?
I am not going to argue which is right… Obviously, “math” is the correct abbreviation to mathematics. 
but more than any single word, “maths” sets me on edge in a way that I can’t understand. “Maths” sounds so… uneducated to me. And yet, when I stop to think of the logical shortening of mathematics, it is hard not to understand why we Americans dropped the trailing ‘s’ in the first place.
I know is is just one of those quarky things, but I am now wondering… Does any other country that has in its history a colonization from the British Empire say “math” and NOT “maths”, or is the US the only outlier?
Why, in particular, is geometry lumped in as math?
I read a suggestion once somewhere that arithmetic and higher math, as well as geometry, can all be lumped together as applications of group theory or ring theory.
Like most words in ordinary language, “math” doesn’t have a formal, fixed scope, but a number of senses related by a web of family resemblances. Things that are like other things that people call “math” tend to be things that people call “math” (and that’s fine!). Certainly, arithmetic calculation and geometric proof are both among the archetypes of the concept, as well as whatever happens in university mathematics departments and elementary school math classes, and myriad other things besides.
Which isn’t to say I have no sympathy for the spirit of your remarks (though I find even an emphasis on proof as the sine qua non a bit parochial; for example, much of what I’ve considered great mathematics has been less about proof and more about finding a new perspective on some concept)! For different purposes, in different contexts, one may use the term “mathematics” with more emphasis on, or exclusion of, various of these aspects.
If I were forced at gunpoint to give as good a rigid definition of math as I could manage, after listing all the above objections to my patient assailant, I suppose I might say “Mathematics is generally the study of abstract rules/patterns and their consequences”. And to the extent that space, number, quantity, and arrangement are all considered in terms of such abstract properties, it thus makes some sense to lump them together into this field.
By that definition, it would seem that spelling is a subfield of mathematics (albeit one with a great many rules, most of which are just of the form “If the word is _____ it is spelled _____”).
Expanding on a previous point:
I share the view that what most students spend most of their “math educations” doing leaves them with a rather perverted idea of mathematics. But, as I said, I don’t think proofs are the fundamental thing either, and let me illustrate by way of example.
At some point in my youth, I was intrigued by an older friend’s graphing calculator, and he spent some time explaining and demonstrating to me how an arithmetic function can be thought of as describing a geometric figure. I had never been exposed to that idea before, and found it fascinating. It would have to be considered a landmark in my mathematical development (just as it was a paradigm shift for the mathematical world centuries ago). Whatever doing math is, moments like that must be part of it. Yet it would be a stretch to describe what this example was about as “doing proofs”. Rather, I would say the essence of math for me is largely this process of discovering new ways to see and understand things, which sometimes involves proofs of propositions, but covers a lot of other ground as well.
[For what it’s worth, I believe most students are first taught the idea of graphing functions prior to high school geometry. The standard math curriculum isn’t completely devoid of valuable content! (Conversely, I’m under the impression the way in which “proof” is dealt with in the typical high school geometry class is about as stilted, bureaucratic, and rote as any calculational ritual…)]
Well, you can see now why I pleaded even at gunpoint not to have to fix a definition… The fuzziness of definition is my point; there isn’t really a hard line between “The ‘geometry axioms’ I’m interested in are so-and-so, with consequence such and such” and “The ‘spelling rules’ I’m interested in are so-and-so, with consequence such and such”. But the former tend to be simpler or more ubiquitously arising rules, and so “mathematicians” tend to study them more than the latter.
I’m sure if you wanted you could find some sort of NLP paper which treats orthography as a mathematical entity.