Term’s started again, and the combination of an incredibly light timetable (thanks, Imperial!) and the fact that I am loathe to make the 5 mile round trip twice in one day (it’s getting cold, and biking to college makes my ears hurt) means I have a delicious four hours to kill and not very much work, so I’m in the library, and I’m going to write a post that I’ve been planning for a while.
“When do you use equals, and when do you use threquals?”
One of my best friends is a theoretical physicist at Imperial, and we often have the stereotypical “Physics v Maths” arguments, where he tells me that “Maths is like a hammer, you just use it to do other, more useful things” and I shout back that “PHYSICS MATHS IS MESSY AND MAKES ME EXTREMELY ANXIOUS” and so on. One day we were both travelling back from Fulham to Cambridge quite late at night, and Tom asked me a question, which was, “When do you use equals, and when do you use threquals?” (‘Threquals‘, by the way, for anyone who’s unfamiliar with this little colloquialism, is another word for 3 parallel lines on top of each other, or, the identity symbol, ‘≡’). This was in the middle of August, and we have still yet to reach an agreement about who is right or not.
What tom said:
What Tom said (and I’m paraphrasing), is that if you set two functions equal to each other, say, f(x)=g(x), then that means that for some value(s) of x, the outputs f(x) and g(x) will be equal, for example, x² = x. If you say f(x) ≡ g(x), then it means for all input values in the domain, the functions are the same. He then went on to say that he uses threquals as a defining operator – that “equals does not have the power to define functions”. So if he was answering a question involving functions, he might start by saying “y ≡ x²“.
Why this makes me anxious:
This way of using threquals makes me incredibly nervous, and I told him that – I think I said something along the lines of “You’re bandying about with the identity like it’s nothing and it makes me very, very distressed”. The way that I see the identity(/threquals) is as follows. One can use the identity symbol when the expression that you write down is true for every value of x. No matter what you put in, the expression will remain the same. Hence, “cos² x + sin²x ≡ 1” is an identity, and so is “(x+y)² ≡ x² + 2xy + y²“, because whatever you stick in there, you’ll get out the same answer, but, the functions on either side are non-trivially different. (i.e. obviously 1≡1 but this helps nobody).
In addition, Tom being a physicist means he doesn’t know (or isn’t taught, certainly) about congruence classes – the ‘≡’ symbol also being used to denote two numbers being congruent modulo n, (for the non mathematicians: a ≡ b (mod n) simply means that n divides the difference (a-b), or, even more simply, if you divide a by n, you get the remainder b). While this doesn’t fit with my strict usage of identity, it does actually compliment it nicely. Usually, when you do modular arithmetic, you do all your calculations in a single framework – i.e, you start working in mod n, and you don’t switch halfway through. It is as if, for the purposes of your working, you enter a world where there are only n integers, and you forget about all the others until the end of the problem. Hence, when we do calculations of this kind, mathematicians often omit the mod n on each line, instead preferring to start with “working mod n…” and then just using ‘≡’ from then onwards. This is a very interesting thing to note. When we work in this world, a ≡ b means exactly that – “a is identical to b“. In mod n, there is no difference. So really, this is not a notation issue at all. It’s using ‘≡’ exactly as I think it should be used.
“≡, ∀, Let, :=“
The usage of the identity as a defining operator made me fret a lot in a sort of vague, worried way, where I couldn’t really explain why I hated the concept so much. Personally, if I were to define a function, I would either say “Let y = x“, “Let y = x, ∀x” or “y := x“. (I think this might be a difference between physicsy maths, which (I imagine) is quite calculation heavy, and pure maths, which is often very wordy, but I’m just hypothesising). ‘∀’, by the way, means “for all“, and ‘:=’ is the defining operator, though I’m not sure how widely its used. The distinction between all of these symbols is a bit of a grey area. I can’t imagine replacing the identity with any of these, for the simple reason that, in my opinion, an identity is something that transcends your current problem or line of reasoning. For example, you can start a question by saying “let x = 1“, and for the rest of the question, every x is also 1. But once you finish the question, x is no longer 1 – you cannot then start the next question taking x to be 1 as well. However, cos²x + sin²x is always 1, no matter what question you’re doing.
“0 ≡ 0“
I spoke with my other very good friend Richard about this (also on the London to Cambridge train, though going the other way). Richard is a mathematician as well, just about to go into his first year at Oxford (and also one of the smartest people I have ever met), and we had a nice chat about it, in which he sympathised completely with my irrational anxiousness about using identity in such a careless and irreverent way, and agreed that ‘≡’ was a powerful and elegant thing and deserved more respect. He then went on to say another very interesting thing – that, in his view, an identity is something that reduces down to “0 ≡ 0“. Think about this for a second. While it may seem unhelpful to work with statements that are essentially “0 ≡ 0“, this is exactly what the identity does. If you expanded out the Taylor series’ for cos²x and sin²x, and cancelled them, you would arrive at the statement “1 ≡ 1” (which directly implies “0 ≡ 0“, just subtract 1 from both sides!) with fairly little effort, and yet, the identity is widely considered to be a pretty nifty one. I think it’s an incredibly good way of explaining it, which also prevents you from using it as a defining operator, because “y ≡ x” doesn’t reduce down any further than that, so it’s an equation, not an identity.
Richard’s theory made me extremely happy, and also helped me justify my insistence that ‘≡’ is somehow grander and more royal than just a defining operator. So there we go. The threquals/equals debate rages on, but I feel like I’ve won.