It seems like Tim likes to further classify things into "reasonable-sounding" and "unreasonable-sounding" implication statements, but I just don't follow the lines of that distinction at all.
Consider. Suppose that in a tennis match each point is won or lost independently of all other points, and suppose further that the server has a 95% chance of winning each point. It seems reasonable to assume that the probability of winning the game from a winning position is always greater than the probability of winning the game from the beginning.
But this is not true! The probability of winning the game from 40:30 up is less than the probability of winning from Love All (99.986% vs 99.991%).
Some students, even those who are quite competent at probability and can follow the calculations, find that this offends their sensibilities and decide that it's all rubbish.
People get very upset if you contradict their common sense. To claim that the sentence:
If n is an integer and both even and odd, then n=17.
(edited to add the actual probabilities)
Pretty sure no one cares, but I thought I'd add it for completeness.
So mathematicians can only prove theorems by working with the definition of "implies" that makes no causality requirement.
However, there is a sense (albeit informal) in which mathematicians recognise causality, and he alludes to that when he says that probably no one would say that the Riemann hypothesis implies Fermat's last theorem. I think that all mathematicians (or at least those who learned from Herstein!) would agree that Sylow's theorem on the existence of p-subgroups of all possible orders is equivalent to the weaker statement about the existence of p-Sylow subgroups; but no mathematician would mean by that merely that both are true.
I am however interested in this guy's next article, which will hopefully be a 8-page document regarding how, unlike in plain English, "x or y" in logic is still true if x and y.
Well it would sound pretty stupid to say "x XOR y" in regular conversation, which is what is usually meant. I'm trying to imagine how it would sound -- "ok, you can get a pony zhor an xbox?", "you can get ice cream ecks-or health insurance?" Also, not many people would probably talk to you.
(Comment: I'm fairly sure I know the answer, but I don't think it's so trivial as to not make for an interesting discussion on HN. In particular, I'm not sure that it's really less or more confusing to the sufficiently uninitiated.)
This article turned out to be a fascinating read, for example, the fact that zero is even leads to 1 having an even number of prime numbers, a slightly counterintuitive proposition, which turns out to be important in the definition of the Mobius function and ... there goes 15 minutes.
Where in the world do people get this idea? Every time that I teach a class where the number-theory content is elementary enough for this to come up, someone thinks either that it's both even and odd, or neither even nor odd.
It's hard for me as a mathematician: a number is even if it's divisible by 2; 0 is divisible by 2; so 0 is even.
(EDIT: Err, sorry, I was rebutting those who claim that it's neither. I should say instead: a number `n` is odd only if `n + 1` is even; 0 + 1 = 1 is not even; so 0 is not odd. Maybe this is a bit less convincing?)
I honestly can't see the confusion—except to attribute it to a memory of a flustered grade-school teacher who was unprepared for a question and so had to recall his or her own flustered grade-school teacher ad infinitum.
(Oh, or is it maybe a confusion with infinity, which, if we were to try to classify it, would have some claim to being both?)
I think it is based on mistaken beliefs like these :
* 0 is not a number, it's the absence of a number. Therefore, standard number properties like parity does not apply.
* All even numbers are twice another number, which is not true of 0.
Also, very early on, people learn that 0 is "weird", e.g. you cannot divide by it. Also there's something that irks the common psyche about treating the statement "this set has zero elements" similarly to "this set has 5 elements".
This doesn't show the stupidity of the general public (or some of your students), it's just that 0 is among those abstract mathematical concepts (like infinity) that you just have to get used to. Remember that it was invented much later than natural numbers.
P.S. People also see other weird conventions (e.g. 1 is not a prime, 0!=1) and may think that this is another one.
Once we are confident that infinity is not a number, why would we expect it to have parity?
> Oh, or is it maybe a confusion with infinity, which, if we were to try to classify it, would have some claim to being both?
… although the old maxim “What's a number?” “Whatever you treat as a number” (which I encountered in Linderholm's wonderful book (http://www.amazon.com/Mathematics-made-difficult-Carl-Linder...) ) means that the various flavours of infinity, considered as ordinals or cardinals, have some claim to being numbers; and it is often helpful in measure theory, and my own work, to treat even plain old infinity as being a number on at least as good a footing as any other (positive real).
> Once we are confident that infinity is not a number, why would we expect it to have parity?
Why? Seriously, though: what do you mean by parity (EDIT: I mean even-ness) of a set? Almost assuredly, something like “it can be partitioned into two disjoint subsets that can be put into bijection”; but then you've just defined even-ness of cardinals (which are sets after all), and all infinite cardinals are even (in fact, satisfy 2κ = κ; EDIT: and odd; they also satisfy 2κ + 1 = κ).
It's a property of _integers_, or any system with a natural homorphism into the integers mod 2 under addition. It's certainly not a property of numbers in general. Is pi even or odd? How about sqrt(2) + i*e?
With that said, as I have argued elsethread, I certainly do think that there are occasions when infinity can be considered a number; the surreal numbers are an elegant example.
We could just throw away these short hands and just integers, positive integers, negative integers.
Which actually led me to another potential reason for the confusion. Is zero positive or negative? OK, now is zero even or odd? I can imagine some just see zero with weird properties.
We could just throw away these short hands and just [use]
integers, positive integers, negative integers.
On the other hand, from the perspective of mathematics, the natural numbers are very different to the integers. For one thing, they're wellordered whereas the integers are not. Obviously a set-theoretic definition of the natural numbers (as the finite ordinals) will start at 0 since it's just identified with ø. Set-theoretic definitions of the integers are typically in terms of equivalence classes of pairs of natural numbers (or arbitrary elements from these equivalence classes if you want to work in ACA0) .
 See SoSOA p.10 for details. http://www.math.psu.edu/simpson/sosoa/
I think that's the best explanation I've heard. Thanks! (Also you can't divide by it (although (http://www.bbc.co.uk/berkshire/content/articles/2006/12/06/d...) …), so who knows what trickery it's up to?)
Hmm, I've never encountered that—which is not to say you're wrong. I'll bet you'd have a hard time finding someone post-secondary school who wouldn't answer instantly that -2 was even, though.
(A disclaimer: The only people with whom I've had occasion to have this conversation have been students in (my) college mathematics classes, so I guess I should clarify that these are the people of whom I'm really thinking.)
EDIT: Incidentally, this reminds me of one of my favourite pedagogical techniques, which I was heartbroken to discover recently is not original to me. After discussing even and odd numbers, as a very basic introduction to modular arithmetic, I like to introduce and discuss ‘threeven’ and ‘throdd’ numbers, and discover that they don't behave quite as nicely—what to do, what to do ….
Me: OK, let's get started. Is 5 even or odd?
Me: And 4?
Me: How about 3?
Me: And 2?
Them: OK, OK, I get it!
For instance consider If unicorns exist, then mice do not exist. Well in fact unicorns have been created by transplanting the horn buds of a goat so that they would grow into a single straight horn instead of being two horns. So in some sense unicorns exist, but yet mice still exist. Obviously the statement is false. Sure, you can argue, that they aren't really unicorns. Fine. Suppose that with genetic engineering we make it happen without the operation. Mice would still exist.