The Parity of Zero
solipsys.co.uk
solipsys.co.uk
I think the people claiming that the right answer is obvious are making a mistake much more severe than the mistake made by people who guessed that zero was exempt. They are confusing the direction of reasoning by trying to prove axioms from conjectures by first assuming that their conjectures are theorems.
You are right that the definition of even-ness is how we determine whether zero is even or not (by that definition).
But there are reasons to choose one definition over another. Yes, you can choose to define zero to be exempt. But you can look at the patterns of numbers that are "obviously even" and observe that adding 2 to an even number results in an even number. So it's reasonable that subtracting 2 from an even number should result in an even number.
That's not a definition, but it's a pattern to observe, and leads one to think that maybe defining zero to be even makes sense, and fits with existing patterns.
You can define it how you like, but some definitions are more useful than others. That's why 1 is no longer considered to be prime, -1 is considered by some to be prime, and why even-ness is defined in such a way as to include zero. It creates fewer exceptions to general properties.
It's more convenient, more useful, and simpler, than any of the alternatives. Patterns provide a reason to create the definition we use.
When we are trying to figure out what can be encompassed in a term, the term's plain, non-mathematical meaning should be considered-- at the very least as a tiebreaker to equally valid definitions.
Consider the question about earth rotating around the sun and how long it takes. The answer to the first part is readily available BECAUSE ITS PART OF OUR LANGUAGE AND CULTURE. We periodically hear about Copernicus' heretical discovery. We hear the phrase 'one more trip around the sun', we see models of the solar system, etc. So the right answer is just instinctively top of mind.
In other words you can get this right by being completely ignorant of celestial mechanics because you almost can't avoid "knowing" it by osmosis.
The question about the duration of the trip is a much harder question. You need to either recall a much less frequently referenced definition, or to model it out in your head. And guess what, people are not likely to do that work while answering a stupid poll. I bet if the question included "take your time and if you get this right you will win $100", the correctness rate would shoot up.
On the parity of zero, it probably goes like this "it seems like it's even but they are asking a poll about it so there's probably a trick. There could be some rule that makes zero special" and because it's an irrelevant quiz they say "neither" and move on with their lives. Again, if you phrase it to slow them down and offer a reward, more people would take time to think it out and answer correctly.
It feels like what is really frustrating the author is that people don't totally nerd out thinking through twitter polls.
I, in common with most people who do maths, consider 0 to be even, but I also remember that maths is just a mental game we play, where we give all kinds of entirely abstract, unusual meanings to a variety of terms in order to get something useful out of them. It's like saying that there are infinities bigger than other infinities (because we define bigger to mean 'not able to be put into a correspondence'), or that the sum of natural numbers is -1/12 or when we say that x^0 is 1 because that makes all kinds of things work so much better. All this terminology is useful, but the way the article talks make it sound as if it is ordained by a higher power that it must be this way rather than simply contingent on what we find most useful.
To complain that someone is stupid because they're using less useful or standard terminology is unfair, especially if they're not playing the same game you're playing - dividing cookies evenly is a totally different game to maths, and it's reasonable to have different definitions.
As someone who has occasionally been able to make strong arguments that the 'wrong' answer on a multiple choice question is more correct than the 'right' answer, there's also something pretty elitist about giving people so little room to explain themselves. You can tell almost nothing about a person by whether they fit into one of four boxes you've chosen for them.
I am genuinely stunned and baffled as to how you have come to the conclusions you have, and I'm going to take some time to try to work out how that's happened. I'm utterly dismayed that it comes across as being elitist.
I think it's great that you're trying to analyse why people are giving such an answer, and you may be largely right in what you've come up with, but I think a more charitable explanation that you don't seem to have considered at all is that their answer makes logical sense in their framework, and that the educational mission you are on should be considering frameworks rather than rightness and wrongness and 'educating' them out of their current view.
To help me, when you say:
> I think a more charitable explanation that you don't seem to have considered at all is that their answer makes logical sense in their framework ...
I can't find a framework in which it makes any logical sense. Can you suggest one?
There have been three suggested already
1. the ancient greek view (in which one was not fully even or odd).
2. the view where the definition of even requires having 2 as a factor. This is not as good a definition of 'even' as having no remainder when divided by two for all the reasons the wikipedia article will tell you, but it's logical and consistent, and quite probably tracks with common (non-mathematical) usage best.
3. The other view is about how we use the concept of evenness in normal language. In normal language, and for good reason, 0 is a special case. I don't tell people that a Zoo has an even number of unicorns. If I'm talking about sharing cookies between my two children, I might say that 2 cookies are even, or that 4 cookies are even, I will not say that an empty cookie jar has an even number of cookies in it.
There may be others, I'd be the wrong person to ask, since I consider 0 to be even.
But the reason the article came across as elitist to me is because while repeatedly stressing that you believed in taking a charitable interpretation of the responses and the reasons for them, your starting point seemed to be the wrongness of the wrong answers, and your charitableness seemed limited to trying to find reasons for why those people were so wrong.
An actually charitable treatment that followed your described principles might be very interesting.
I didn't read it as elitist, but that's because I share your interest in the topics being discussed. Many poeple don't. For someone who doesn't care whether the Earth goes around the Sun or the Sun around the Earth, or what the correct mathematical definition of "even" and "odd" is, your article absolutely could come across as elitist, because you're implying that they should care, and that your job is to explain to them why they should care.
But as a matter of simple pragmatic fact, there is no reason why they should care. As a matter of simple pragmatic fact, it is perfectly possible for someone to live an entire human life without ever having to know whether the Sun goes around the Earth or the Earth around the Sun, or what the definition of parity is. So people might not be interested for the simple pragmatic reason that they see no reason why they need to be interested.
One can make a case that there are benefits to be gained from being interested in such things even if they don't appear to have any pragmatic relevance; but making that case takes a lot longer than one blog post.
He didn't read it, though, did he? You specifically wrote they are "not stupid", and had a greek chorous thing going with "they don't care".
You're writing isn't elitist here, but I think it moves very quickly up the ladder of stasis in an unjustified way (https://owl.purdue.edu/owl/general_writing/the_writing_proce...).
At my school we've been trying to sort out this phenomenon for about five years, and we're still wrangling at the stasis of definition, with understanding the nature of what we see in our students. An example:
After I teach imaginary numbers, I give my students an (ungraded) assessment with these questions:
1. Please circle the imaginary numbers in this list. (about half imaginary)
2. Please perform the following multiplications. (mixed; about half imaginary x imaginary)
3. What is the definition of an imaginary number?
4. In your answers to question two, please circle all the products that are themselves imaginary numbers.
5. Is the product of two imaginary numbers imaginary? Why or why not?
We've never explicitly discussed #5, but its answers flows necessarily from the previous answers. What I'm really interested in is whether the answer to #5 is coherent with the answers to the previous four questions. I'd say for about a third of students it is not. For example, a student correctly wrote that i * i = -1, and correctly identified -1 as not imaginary, but then said for #5, "Yes. Because when you multiply two imaginary numbers it doesn't become unimaginary."
We see that play out in essentially every subject, and it's gotten much more prevalent in the last, say, ten years. It's especially noticeable in our 8th grade formal logic class, where a meaningful fraction of students appear unable to reason propositionally or syllogistically at all.
We have some ideas about what's driving the change, but are still trying to figure out which of them are correct and what we can do about it. My personal theory is it's something to do with syntax and semantics, with some students not associating meaning with words in the way that one would expect. It's so hard to extract signal from the noise about that though.
In my case, the problem is understanding proper street and transportation network design, and then having to live in the United States where the vast majority of the landscape is literally designed to create long driving trips with everyone stuck in traffic. The professionals upholding the anti-functional design sometimes understand this, but generally shrug as they get a few extra shrubs planted around an intersection feeling they’ve done the best they can. “It’s the whole system, what can I do?”
It’s deeply frustrating and discouraging to experience hundred-year mistakes freshly poured in concrete on a regular basis, which I believe are just as this author suggests, mostly attributable to the large majority of people not caring about how roads and streets (or anything else, for that matter) actually work. Combined with the strong tendency of humans to prefer the familiar and status quo over the novel and new... and once things become shitty they will tend to stay that way forever.
Even within the United States, are you unable to find an urban geography to your liking, say in Manhattan or San Francisco?
Upon further analysis, you might actually find that the street and transportation network design in any given area is largely optimal given the relative costs of time, land, labor, transportation and energy. That doesn't mean it's enjoyable to be stuck in traffic - that sucks, and there are alternatives (like living in Manhattan), you just need to pay for them - they're not absent due to widespread ignorance.
https://en.m.wikipedia.org/wiki/Commissioners%27_Plan_of_181...
Analogous questions at slightly more sophisticated levels:
"Is zero a natural number?" "Is 1 a prime?" "Does an algebraic ring contain the multiplicative identity?"
(At least two of those rely on definitions that have been ambiguous at some point.)
As with the PEMDAS "trick" questions that show up sometimes on the internet, this is a trick question to what I say: it is what you want to be.
Integers/Positive/Rational etc numbers have an exact definition. "Natural numbers" are "numbers that appear in nature" or some finicky definition like that.
If you want 0 to be there it's fine. If you don't you're wrong but it's fine as well ;)
If you want "positive integers > 0" say that.
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Lexicon / Oxford Dictionary
https://www.lexico.com/en/definition/natural_numbers
> The positive integers (whole numbers) 1, 2, 3, etc., and sometimes zero as well.
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Wikipedia
https://en.wikipedia.org/wiki/Natural_number
> Some definitions, including the standard ISO 80000-2,begin the natural numbers with 0, corresponding to the non-negative integers 0, 1, 2, 3, ...
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Math World / Wolfram
https://mathworld.wolfram.com/NaturalNumber.html
> The term "natural number" refers either to a member of the set of positive integers 1, 2, 3, ... (OEIS A000027) or to the set of nonnegative integers 0, 1, 2, 3, ...
Is zero positive?
Back when I was in grad school we informally polled the mathematicians in the department on whether 0 should be a natural number, and got about an even split. The key factor being what area of math you specialized in. So, for example, people studying logic and combinatorics thought it should be because the empty set comes up a lot. But people studying number theory and combinatorics thought it shouldn't because they are always dealing with things enumerated and starting at 1.
When I went through school, I was expected to memorize that natural numbers start with 1, and distinguished them from whole numbers that include 0. But this is an entirely arbitrary distinction. As https://en.wikipedia.org/wiki/Natural_number says, there is no general consensus on whether 0 should be included, and there are a variety of unambiguous terminologies that are preferred for cases where it matters.
There are also plenty of correct answers, but it's interesting to see the reasoning behind the other answers. This suggests that the problem isn't rote learning but due to people reasoning in the wrong way.
This this. That's why that BS about "the sum of all positive integers "is" -1/12" gets down on my nerves so deeply. Or elementary teachers finding out that students came up with solutions with "a new method" and immediately dismissing it as wrong.
Yes, 80% of math teachers are bad and 80% of students are disinterested. But we can do better.
On to the subject, if 0 is neither even/odd then it is "something else" (SE for short) that needs a definition. And we need rules for how numbers go from Even/Odd to SE. Thankfully these are not needed because 0 is Even
There is no reference to the actual poll, so it's not clear what the actual structure of the poll is.
That said the reason to prefer a model where the Earth is stationary but rotating over a model where the Earth is both stationary and non-rotating (but everything else is) is somewhat technical and not something I would expect everyone to get correct (though it's technically high-school level physics).
Perhaps the main reason people would answer questions like that wrong is that frankly the most useful model for day to day activities is one where the Earth is completely immobile.
We just had this conversation last night at dinner with my middle schooler. She thought the Earth's orbit was closer and farther during the year. Asked her just now and she hadn't been aware the seasons are reversed north and south.
Now she's taking her cell phone flashlight and shining it at the ceiling at different angles :)
Then again, not knowing the cause has almost no practical consequences for most people. I doubt that knowing the answer would make them less susceptible to most false claims about climate change, for example.
http://dianetfalcon.weebly.com/knowing--learning/harvard-gra...
I had forgotten that it also had a 'phases of the moon' question.
It makes me think of people complaining about some of the high school maths. While, most of it isn't needed for the average person on most days, it happens in everyday life that you can save yourself some time and effort by using a little math.
True (in even bases).
> Since 30 is even so must be 0
This is essentially the converse of the previously statement. Fortunately, the statement is based off an equivalence so it is also True.
https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-t...
Hell, I've noticed that even a number of mathematicians will exclude zero or the empty set (or other such trivial objects) from definitions even when there is no reason to, even when there's no edge case there. Sometimes zero or the empty set really is an edge case an acts differently, but most of the time it doesn't, and yet a number of mathematicians will exclude it anyway just because, IDK, they never really got comfortable worthing with the empty set, I guess.
If I subtract one even number from another, is the result always even?
If they say yes, then they just proved the even-ness of 0. If they say no, then they must allow for some exception. Like, the result is even only for positive results.
Then I would ask them what purpose is served by having that exception.
I've seen with my own eyes people claiming that 0 is odd.
I can sort of understand why people is confused and say that 0 is not even, but I can't imagine why someone would say that 0 is odd.
(Also, the sample in your your twitter poll is probably biased and includes too many people that cares about math. A poll in the general population would get worse results.)
Then again, most any number can be odd -- in the appropriate context. Behold:
"Alexander was a great general. Great generals are forewarned. Forewarned is forearmed. Now, four is an odd number of arms for a general to have. Four is also an even number. And the only number that is both odd and even is infinity. Therefore, Alexander, the great general, had an infinite number of arms."
“Uhm... uh... it’s definitely more even than odd” she answered in full agreement with the result of the Twitter poll.
When I was a kid I carefully recorded the position of the sun for a year, and plotted the sun's "motion around the earth". Then I flipped it around and had the earth's orbit. (I also plotted Venus and Mars, as much as I could see them, and they plotted fine, once I switched to heliocentric.
I seem to remember that this was how Kepler came up with equal angles in equal time; I am sure I read it in a book and followed what it said, but that was many decades ago.
So the reasons are readily available without any tool but a pen and paper, but I don't think they are even meaningful to most people. I can't really blame them.
How did you measure how far away the Sun was? Without that information you can't reconstruct the Earth's orbit as you describe.
I also remember looking up a method involving looking at the moon at the right phase to calculate the distance to the sun but I do remember I never got that to work.
There are two possible reasons for people not caring:
1. Their childhoods were fucked up to the point that they never had the time to think about math.
2. They're stupid
So, maybe the initial reaction from the responders ("that they are stupid, ignorant, or even worse") wasn't all that wrong.
Aaron Swartz put it in more positive words: "What people call intelligence just boils down to curiosity."
The bigger question is about why one frame of reference is considered to be correct and others incorrect. You would have to prove that it is impossible to come up with a set of mechanical conventions to suit the model that doesn’t contain a contradiction. It seems unlikely that this has been attempted independently by most professional astrophysicists. So ultimately even the most learned rely on consensus more than principle, and their own depth of reasoning may have easily put themselves on the other side of Copernicus’ heresy trial. It’s pretty ridiculous, right? Like, how do you explain why ghosts only come out at night if the sun isn’t chasing them around the earth? Obviously wrong. Stupid heliocentrists.
Some people conceptualize the prime numbers as a mathematical sequence definitionally — akin to the Fibonacci sequence. And mathematical sequences have both inductive-case members — that is, members created by the inductive-step “algorithm” for the sequence — and also base-case members — members by fiat, that bootstrap the sequence into being.
In the mathematical inductive sequence of the prime numbers, you clearly need 1 there in order to “get the sequence going.” Any parsimonious formula you could define for a prime-sieve induction step, needs to start off with a 1 already “in the sieve” in order to generate the sequence you want it to generate. Any special-case that excludes 1 (or rather, unit values) from “being prime”, can be reformulated into a syllogism for 1 (/unit values) being base-case members of the primes sequence.
And yet, it turns out that the definition of “prime” isn’t actually “a member of the generative sequence of the prime numbers”, but rather more specifically “an output of the generative process of the prime numbers.”
This is unusual for a mathematical property — usually we use an adjective “Foo x” to refer to an x that shows up as a member in a mathematical object Foo. But prime-ness and membership-in-the-primes-sequence aren’t equivalent. Primality as a property, was created as a mathematical tool to reason about perfect numbers; and you don’t get perfect numbers (or at least, not the same set of perfect numbers) if you label 1 as prime. But this, obviously, has nothing to do with the primes sequence!
It’s as if we had a concept of a “Fibonacci number” that didn’t include the base-case members of 0 and 1. It would feels kind of strange and arbitrary to laymen, who probably know about the Fibonacci sequence, but who wouldn’t realize that there might be something interesting about the Fibonacci-ness property of the inductive members, that doesn’t apply to the base-case members.
I was specifically taught as a child that zero is neither even nor odd, because zero has no value. I don't think it was confusion or misunderstanding.
Understanding the value of "nothing" or "the opposite of something" is what separates us from most other brained-animals, correct?
I understand there are patterns that can be applied. However, I never once was misserved by that understanding, either through academics or physical life.
Ultimately, it's more upsetting today to try to reset that thought process in my mind. So to me, zero is neither even or odd, and I'm happy with that.
There were no eclipses: the moon was just not full.
I wonder how they arrived at this conclusion.
Is this a strange coincidence or is this a common mistake?
1. Of course there are stupid people, almost regardless of the definition of stupidity. Not enough to explain the outcome, though.
2. Under the hypothesis that (most) respondents simply regurgitate, I can sort of understand why they would respond "neither": they have never been taught or can't remember (BTW, this is a classical error in a survey). But why would they say it's odd or both odd and even? Who taught them that?
IMHO this isn't a thing that needs to be/should be remembered. It isn't about being taught if zero is odd or even. This is more about people understanding the world around them, people integrating math and science into their world view, instead of as a separate thing they had to do back in school.
Stupid != ignorant
Many folks who don't know math often have a significant level of skill in other areas.
There are many folks who just didn't bother to learn math, and either just regurgitated the material to pass tests and promptly forgot it, or flat out cheated.
It's sad, as math is extremely useful in many situations. I'm not even talking about algebra, trig or calculus. Just basic arithmetic, fractions and exponents.
Algebra can also be quite useful in everyday life too, and not just WRT numerical problems.
Algebra uses clearly defined rules to perform operations. If you can master that, more general problem solving becomes much easier.
That many people don't have such skills is sad. But our education system seems to favor moving people up and out rather than ensuring competency in the elements of the curriculum. And more's the pity.
And stupid people exist, really. If you take IQ, a considerable portion, almost one in six, of the population scores below 85, classified as intellectual disability, borderline functioning, or worse. These people do not know what's odd and even. It's not sad. Many can lead normal lives, and find joy in other things.
I didn't say "daily life" I said "everyday[0] life."
The difference is that there are times (certainly not daily) in normal (which is the sense I used "everyday") life where not algebra specifically, but the idea that structuring a problem properly makes it easier to solve is extremely useful.
>And stupid people exist, really.
Not sure where I said there aren't stupid people. I said that being ignorant of something (in this case, math) didn't necessarily mean one is stupid.
I'm getting the sense that I didn't communicate my points very well to you, or something was lost along the way. Sorry about that.
If I subtract one even number from another, is the result always even?
(maybe after first asking about addition)
If they say yes, then they just proclaimed the even-ness of 2 - 2 = 0.
If they say no, then they must have made a weird arbitrary exception. Like, the result is even only for positive results. Then I would ask them if having an exception serves any purpose.
People who think of a subject as random facts don't even try to build a mental model. And therefore they don't know the most obvious things that would come out of a mental model.
Even after all those years, it probably feels a little odd to people (pun intended). For example, few public transport authorities have a line (tram, bus, whatever) numbered 0. They usually start with 1.
In the UK there are 8 train/railway stations with a platform 0, and another is being built.