11 ms·

The Parity of Zero

79 points by ColinWright 6y ago

tgv 6y ago
Interesting take, but 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?
- 0xTJ 6y ago
  I disagree with what you say in the second point, unless they've forgotten what odd and even mean in that sense. You aren't taught that 56,437 is odd, it just is. 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.
- nobody9999 6y ago
  >1. Of course there are stupid people, almost regardless of the definition of stupidity. Not enough to explain the outcome, though. 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.
  - tgv 6y ago
    I agree our education system is wrongly geared. I do not agree on the usefulness of algebra in daily life, nor on that of maths beyond primary school. I've been trying to use this as a motivation too, but it falls flat. Calling it extremely useful is almost taking the piss. 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.
    - nobody9999 6y ago
      >I do not agree on the usefulness of algebra in daily life, 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. [0] https://www.merriam-webster.com/dictionary/everyday https://www.merriam-webster.com/dictionary/everyday
0xTJ 6y ago
This seems more like a failing of education systems/parents encouragement (and recursively on that second one, I'm not just saying blame parents). Not everyone is going to be interested in STEM, but building intuition about the world is important. Seeing the pattern in odd/even, thinking about stuff orbiting in space instead of the simply the everyday world where the sun appears on one side and sets on the other. It reminds me of that US thing where a tweet said that $500M distributed amongst the entire US population would be over a million dollars per person, instead of over one per person. I think it got picked up by the news, without anyone noticing that $500M/328M != $1.xM. 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.
jjbinx007 6y ago
I think the problem with whether zero is even or not lies partly with the fact that if I have one apple and you take one way I have no apples. Most people wouldn't say I had an even number of apples when I didn't have any at all.
- recursive 6y ago
  We say -2 is even, despite the fact that -2 apples makes even less sense.
dade_ 6y ago
The illustrative tangent is far more disturbing. It suggest that ~80% of people have no clue what causes the seasons to change on Earth. Small wonder most people believe anything on social media.
- kortex 6y ago
  I disagree. The questions are worded in a way that leads the reader to think the second question is about the first question's second clause. The issue here is people not reading carefully - if the blog is reporting the poll correctly. There is no reference to the actual poll, so it's not clear what the actual structure of the poll is.
  - contravariant 6y ago
    Though even if they understood it as 'using the geocentric model, how long does it take for the sun to go around the earth?' their answer would be wrong. 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.
- ignoranceprior 6y ago
  I wouldn't be surprised if a large proportion of people were not aware that Earth's axial tilt is responsible for the seasons.
  - rrauenza 6y ago
    ....and that it's the angle of incidence not that the tilted part is "closer" 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 :)
aesthesia 6y ago
Another possibility is that people do remember that evenness has to do with divisibility, but they have too many concerns about divisions involving zero. Perhaps they inadvertently exchange the divisor and the dividend and end up thinking that the parity calculation involves division by zero. Or maybe they remember the rule against division by zero too broadly and think it applies to zero as a dividend as well.
gus_massa 6y ago
You should wait a few years and make the twitter poll again, but give three options: even/none/odd 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.)
- nobody9999 6y ago
  >I've seen with my own eyes people claiming that 0 is odd. 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."
- jonsen 6y ago
  Make it four options. See I passed the question to the kitchen: “Wyfe! Is zero even or odd?” “Uhm... uh... it’s definitely more even than odd” she answered in full agreement with the result of the Twitter poll.
burlesona 6y ago
This is a remarkably optimistic take on things that I see and agree with, but struggle not to feel deeply cynical about. 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.
- tacostakohashi 6y ago
  Why must you live in the United States? Are you not free to leave at any time to a place more in line with your preferences? 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.
  - navait 6y ago
    Manhattan and the NYC metro area was absolutely built with cars in mind, mostly under the vision of Robert Moses.
    - mortehu 6y ago
      I don't think the Commissioner's plan of 1811 was made with automobiles in mind. Although there's a highway next to each river, they don't really define traffic in Manhattan. https://en.m.wikipedia.org/wiki/Commissioners%27_Plan_of_1811 https://en.m.wikipedia.org/wiki/Commissioners%27_Plan_of_181...
vmilner 6y ago
The even number question and the Earth/Sun question seem subtly different in that I can imagine people knowing that 0/2 = 0, but being unsure if the definition of "even" includes that. (I can imagine they might hazily think "even" requires being twice a positive integer). This isn't the case with the Earth/Sun question. 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.)
- capitainenemo 6y ago
  Don't you mean definition of even is X%2=0 ? 0%2=0
  - CJefferson 6y ago
    Many people think even means "can be divided by two (by which they mean, when divided by 2 the result is an integer). Yes, that is logically equivalent to "x%2=0", but that isn't how (I think) most people imagine it.
    - capitainenemo 6y ago
      I understood what he meant to say (divisible by zero with no remainder - that is, modulus of zero) it was just the equation was surprising. It was technically correct but not useful in context. Otherwise 16/2=0 would fit as well. So figured it was worth clarifying in case anyone else was surprised too. :)
  - vmilner 6y ago
    I was (clumsily) trying to say: "know that 0/2 is an integer (i.e. zero), but are not sure if an integer result is enough to qualify 0 as even"
    - capitainenemo 6y ago
      Ah. Gotcha. That makes sense.
raverbashing 6y ago
> We need to find a way to show that maths is coherent, and not just a jumble of "stuff." We need to find a way to get people to engage with wondering why things are as they are, to see that it's not arbitrary and unmotivated. 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
ignoranceprior 6y ago
This reminds me of an essay by Elizer Yudkowsky: https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-the-teacher-s-password https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-t...
stevefan1999 6y ago
0 is even because 0 = 0 (mod 2)
- 0xTJ 6y ago
  No one is debating whether or not 0 is even here. The article mentions that it's well-settled. This is about why the results show so many "normal" people not seeing it as even. Framing it as modular arithmetic isn't going to help those people.
deleted 6y ago
[deleted]
scotty79 6y ago
My mom just gave me most curious answer. Even numers end with even digit. Since 30 is even so must be 0.
- justincredible 6y ago
  > Even numbers end with even digit 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.
  - scotty79 6y ago
    I phrased the statement to exactly express equivalence not just implication.
- mannykannot 6y ago
  That is an interesting way of looking at the question. The pattern works if and only if your radix is even.
kens 6y ago
It's interesting to look at the question "Is zero even or odd" on Quora because people give their reasoning for why zero is odd. e.g. "Zero is neither an odd nor an even number because it is divisible by both 2 and 3 and also because it is divisible by any nonzero number." "Zero is neither even nor odd, neither prime nor composite, neither positive nor negative.It is far more special than that, its a unique number." "Answer is no, because zero does not have the elemental properties to qualify as being a number." "Now, anything without value cannot be termed either even or odd." 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. Link: https://www.quora.com/Is-zero-even-or-odd https://www.quora.com/Is-zero-even-or-odd
xyzelement 6y ago
This article is painfully oblivious to how people function! It reaches the correct (and I assume apriori) conclusion that people don't care to connect things in their head, but it gets it from very silly data. 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.
- SoSoRoCoCo 6y ago
  That's a good summary. There's something elitist about this paper that I couldn't quite put my finger on, and I think you got it. Perhaps obsessing too much about twitter polls and what constitutes "stupidity" is what chafed. When someone starts attempting to categories who is "stupid" (even though OP states "people are not stupid" at the end of the article), my BS detector starts going off. Especially when it is focused on minutia (parity of zero) and "good enough" knowledge (earth/sun). The earth and sun orbit each other, but the focal point is inside the sun (and every atom in the universe orbits each other in an N-dimensional space), why is that dismissed as "nitpicking" by the author, but parity of zero is somehow an important indicator of stupidity? smh.
  - 6y ago
tromp 6y ago
I'd ask any person who doubts that 0 is even: 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.
tromp 6y ago
I'd ask any person who doubts that 0 is even: 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.
- wcarey 6y ago
  A counter argument to you approach: If I multiply two positive integers by one another is the result always greater than either integer? Turns out there is one exception to that rule. Does it serve a purpose? How do you know?
  - tromp 6y ago
    Note that your rule already has a condition of positive inputs. It's not surprising that for inputs satisfying an inequality, the result also satisfies one. A simpler example is: If I subtract 1 from a positive integer, is the result always positive? That also has an exception. That's all due to 1 being an exceptional (the smallest possible) positive integer. Similar to how 0 is an exceptional natural number.
    - wcarey 6y ago
      Right - I was aiming for something analogous to your rule, which has the condition of even inputs. I do prefer your simpler example. Either way, I'm not sure your rhetorical strategy would be persuasive to someone who was unsure of the parity of zero (because, as you say, zero is an exceptional number).
btilly 6y ago
This reminds me of http://www.phys.ufl.edu/~det/phy2060/heavyboots.html http://www.phys.ufl.edu/~det/phy2060/heavyboots.html. (TL;DR it is common for people to think that if you dropped a pen on the Moon, it would float away. When those people are asked how the APOLLO astronauts didn't float away the most common response is that they had heavy boots.) 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.
ajarmst 6y ago
My guess is part of the issue is people remembering that 0 is neither positive nor negative and neither prime nor composite and naturally extending it to even vs. odd. But even here, one has to be careful about what definition of "even" you're employing. = 0 (mod 2)? Has two as a factor? One less than an odd number? !(x % 2)? I remember my confusion when I found out that the answer to "is 0 a natural number" was "it depends" and I still run into people who insist 1 is prime, which I think is insane. A lot of these things depend on the norms of the mathematical community discussing them.
- derefr 6y ago
  To me, the “is 1 prime” question gets at a very subtle distinction people don’t realize they’re making. 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.
whatshisface 6y ago
Something's wrong with all of the "deduce from patterns" answers in the article's comments. None of the patterns are deduced! It's definitely not an axiom that an even number minus and even number is even; that would have to be proven from the definition of even-ness. The definition of even-ness is what decides whether or not zero is even. It would be equally coherent to define 0 to be even, or exempt from even-ness. An example of a definition where exemption would make sense would be to say that even numbers are those whose unique prime factorizations contain a factor of two. 0 is exempted from having a factorization, so exemption from even and odd would follow naturally. Whether or not 0 is even is a random accident of math history with at most a little justification from convenience. Nobody can be blamed for failing to deduce it because it cannot be deduced. 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.
- wcarey 6y ago
  Indeed. And Euclid’s definition of evenness (Def VII.6) applies only to numbers, which for Euclid excludes both 0 and 1. So there’s at least one vaguely axiomatic formulation of number theory where the correct answer is probably “neither because it’s not the sort of object to which you can predicate evenness or oddness”.
- ColinWright 6y ago
  Yes and no. 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.
Sniffnoy 6y ago
I'm a bit doubtful of Wright's claim here that people only think zero is neither even nor odd only because they've been told so. I think a lot of people just don't like edge cases and want to exclude them. Now zero isn't really an edge case for parity, but someone who's not too used to thinking about zero might just consider it as an edge case generally (possibly even thinking of it as only sort of a number -- remember, people took a long time to accept zero as a number!), especially if they're thinking more in terms of whole numbers than in terms of integers. 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.
- shadowofneptune 6y ago
  Imagine explaining this question to someone without a concept of zero. Can nothing be cut into two nothings? Does that make it something?
inglor_cz 6y ago
Zero was adapted into Western mathematics fairly late and we still see the effects of its previous absence in the calendar. 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.
- ColinWright 6y ago
  Trivia: In the UK there are 8 train/railway stations with a platform 0, and another is being built. https://en.wikipedia.org/wiki/Platform_0 https://en.wikipedia.org/wiki/Platform_0
  - inakarmacoma 6y ago
    And in several countries, floor #13 literally does not exist. No button. They just skip the inauspicious floor entirely.
- jey 6y ago
  Is there any calendar system that numbers days from zero? I'm sure that the Gregorian, Tamil, Chinese, and Discordian calendars don't. The closest thing I can think of is strftime's %w and %W flags for numbering days and weeks starting from zero.
md224 6y ago
The belief that zero is neither even nor odd makes sense once you consider a statement like “An even number of Unicorns exist” or “I possess an even number of magic lamps.” It’s technically correct, but it also sounds kind of strange, simply because we’re not accustomed to assigning parity to an empty collection.
- deleted 6y ago
  [deleted]
mannykannot 6y ago
The solar system example has some epistemic implications. Epistemology convetionally begins with a definition of knowledge as a justified belief in a proposition that is true, and the people who gave the 1-day answer did indeed believe the true proposition that the Earth revolves around the sun (subject to the caveats in the article). Was that belief justified? Well, they they probably received it from a knowledgable source, and if that is not regarded as justification, then by that standard, no-one knows much at all. It is also clear, however, that they do not have much understanding of the situation to which the proposition pertains. Epistemology, IMHO, should spend less time fussing over knowledge being a collection of propositions, and focus more on what it means to understand something.
gumby 6y ago
I think few even look at or understand the "moving stars" (planets). Also it feels few understand how seasons work. Every solstice and equinox prompts a news article or two explaining what the terms mean. Four times a year. So for them "Sun moves around the earth" is actually just as good as "Earth goes around the Sun." 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.
- pdonis 6y ago
  > 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". How did you measure how far away the Sun was? Without that information you can't reconstruct the Earth's orbit as you describe.
  - 6nf 6y ago
    Maybe they just looked it up?
  - gumby 6y ago
    I remember measuring the angle from a stick in the schoolyard (which wasn't far from home). A science teacher helped me. 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.
deleted 6y ago
[deleted]
namenotrequired 6y ago
Similarly, two (unrelated) smart people told me, isn't it cool that we can see the earth's shadow on the moon. 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?
young_unixer 6y ago
> They don't care. 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."
jl2718 6y ago
If you fix the reference frame to one in which the rotation is fixed to the earth but the center is fixed to the sun then you have the earth revolving around the sun in about a day. 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.
rwmj 6y ago
I quite like this article and it reminds me of when I first saw a mathematical proof. I was blown away that you could do this, and I went on to study maths and logic at university. But the most remarkable thing is that I only saw that proof when I was 15 or 16. In other words I'd gone through a decade of maths classes at a decent school and no one ever taught me about proofs! It was all simply memorizing facts and methods up til then.

79 points by ColinWright 6y ago

tgv 6y ago

Interesting take, but 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?

0xTJ 6y ago
I disagree with what you say in the second point, unless they've forgotten what odd and even mean in that sense. You aren't taught that 56,437 is odd, it just is. 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.
nobody9999 6y ago
>1. Of course there are stupid people, almost regardless of the definition of stupidity. Not enough to explain the outcome, though. 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.
- tgv 6y ago
  I agree our education system is wrongly geared. I do not agree on the usefulness of algebra in daily life, nor on that of maths beyond primary school. I've been trying to use this as a motivation too, but it falls flat. Calling it extremely useful is almost taking the piss. 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.
  - nobody9999 6y ago
    >I do not agree on the usefulness of algebra in daily life, 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. [0] https://www.merriam-webster.com/dictionary/everyday https://www.merriam-webster.com/dictionary/everyday

0xTJ 6y ago

This seems more like a failing of education systems/parents encouragement (and recursively on that second one, I'm not just saying blame parents). Not everyone is going to be interested in STEM, but building intuition about the world is important. Seeing the pattern in odd/even, thinking about stuff orbiting in space instead of the simply the everyday world where the sun appears on one side and sets on the other. It reminds me of that US thing where a tweet said that $500M distributed amongst the entire US population would be over a million dollars per person, instead of over one per person. I think it got picked up by the news, without anyone noticing that $500M/328M != $1.xM. 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.

jjbinx007 6y ago

I think the problem with whether zero is even or not lies partly with the fact that if I have one apple and you take one way I have no apples. Most people wouldn't say I had an even number of apples when I didn't have any at all.

recursive 6y ago
We say -2 is even, despite the fact that -2 apples makes even less sense.

dade_ 6y ago

The illustrative tangent is far more disturbing. It suggest that ~80% of people have no clue what causes the seasons to change on Earth. Small wonder most people believe anything on social media.

kortex 6y ago
I disagree. The questions are worded in a way that leads the reader to think the second question is about the first question's second clause. The issue here is people not reading carefully - if the blog is reporting the poll correctly. There is no reference to the actual poll, so it's not clear what the actual structure of the poll is.
- contravariant 6y ago
  Though even if they understood it as 'using the geocentric model, how long does it take for the sun to go around the earth?' their answer would be wrong. 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.
ignoranceprior 6y ago
I wouldn't be surprised if a large proportion of people were not aware that Earth's axial tilt is responsible for the seasons.
- rrauenza 6y ago
  ....and that it's the angle of incidence not that the tilted part is "closer" 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 :)

aesthesia 6y ago

Another possibility is that people do remember that evenness has to do with divisibility, but they have too many concerns about divisions involving zero. Perhaps they inadvertently exchange the divisor and the dividend and end up thinking that the parity calculation involves division by zero. Or maybe they remember the rule against division by zero too broadly and think it applies to zero as a dividend as well.

gus_massa 6y ago

You should wait a few years and make the twitter poll again, but give three options: even/none/odd 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.)

nobody9999 6y ago
>I've seen with my own eyes people claiming that 0 is odd. 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."
jonsen 6y ago
Make it four options. See I passed the question to the kitchen: “Wyfe! Is zero even or odd?” “Uhm... uh... it’s definitely more even than odd” she answered in full agreement with the result of the Twitter poll.

burlesona 6y ago

This is a remarkably optimistic take on things that I see and agree with, but struggle not to feel deeply cynical about. 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.

tacostakohashi 6y ago
Why must you live in the United States? Are you not free to leave at any time to a place more in line with your preferences? 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.
- navait 6y ago
  Manhattan and the NYC metro area was absolutely built with cars in mind, mostly under the vision of Robert Moses.
  - mortehu 6y ago
    I don't think the Commissioner's plan of 1811 was made with automobiles in mind. Although there's a highway next to each river, they don't really define traffic in Manhattan. https://en.m.wikipedia.org/wiki/Commissioners%27_Plan_of_1811 https://en.m.wikipedia.org/wiki/Commissioners%27_Plan_of_181...

vmilner 6y ago

The even number question and the Earth/Sun question seem subtly different in that I can imagine people knowing that 0/2 = 0, but being unsure if the definition of "even" includes that. (I can imagine they might hazily think "even" requires being twice a positive integer). This isn't the case with the Earth/Sun question. 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.)

capitainenemo 6y ago
Don't you mean definition of even is X%2=0 ? 0%2=0
- CJefferson 6y ago
  Many people think even means "can be divided by two (by which they mean, when divided by 2 the result is an integer). Yes, that is logically equivalent to "x%2=0", but that isn't how (I think) most people imagine it.
  - capitainenemo 6y ago
    I understood what he meant to say (divisible by zero with no remainder - that is, modulus of zero) it was just the equation was surprising. It was technically correct but not useful in context. Otherwise 16/2=0 would fit as well. So figured it was worth clarifying in case anyone else was surprised too. :)
- vmilner 6y ago
  I was (clumsily) trying to say: "know that 0/2 is an integer (i.e. zero), but are not sure if an integer result is enough to qualify 0 as even"
  - capitainenemo 6y ago
    Ah. Gotcha. That makes sense.

raverbashing 6y ago

> We need to find a way to show that maths is coherent, and not just a jumble of "stuff." We need to find a way to get people to engage with wondering why things are as they are, to see that it's not arbitrary and unmotivated. 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

ignoranceprior 6y ago

This reminds me of an essay by Elizer Yudkowsky: https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-the-teacher-s-password https://www.lesswrong.com/posts/NMoLJuDJEms7Ku9XS/guessing-t...

stevefan1999 6y ago

0 is even because 0 = 0 (mod 2)

0xTJ 6y ago
No one is debating whether or not 0 is even here. The article mentions that it's well-settled. This is about why the results show so many "normal" people not seeing it as even. Framing it as modular arithmetic isn't going to help those people.

deleted 6y ago

[deleted]

scotty79 6y ago

My mom just gave me most curious answer. Even numers end with even digit. Since 30 is even so must be 0.

justincredible 6y ago
> Even numbers end with even digit 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.
- scotty79 6y ago
  I phrased the statement to exactly express equivalence not just implication.
mannykannot 6y ago
That is an interesting way of looking at the question. The pattern works if and only if your radix is even.

kens 6y ago

It's interesting to look at the question "Is zero even or odd" on Quora because people give their reasoning for why zero is odd. e.g. "Zero is neither an odd nor an even number because it is divisible by both 2 and 3 and also because it is divisible by any nonzero number." "Zero is neither even nor odd, neither prime nor composite, neither positive nor negative.It is far more special than that, its a unique number." "Answer is no, because zero does not have the elemental properties to qualify as being a number." "Now, anything without value cannot be termed either even or odd." 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. Link: https://www.quora.com/Is-zero-even-or-odd https://www.quora.com/Is-zero-even-or-odd

xyzelement 6y ago

This article is painfully oblivious to how people function! It reaches the correct (and I assume apriori) conclusion that people don't care to connect things in their head, but it gets it from very silly data. 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.

SoSoRoCoCo 6y ago
That's a good summary. There's something elitist about this paper that I couldn't quite put my finger on, and I think you got it. Perhaps obsessing too much about twitter polls and what constitutes "stupidity" is what chafed. When someone starts attempting to categories who is "stupid" (even though OP states "people are not stupid" at the end of the article), my BS detector starts going off. Especially when it is focused on minutia (parity of zero) and "good enough" knowledge (earth/sun). The earth and sun orbit each other, but the focal point is inside the sun (and every atom in the universe orbits each other in an N-dimensional space), why is that dismissed as "nitpicking" by the author, but parity of zero is somehow an important indicator of stupidity? smh.
- 6y ago

tromp 6y ago

I'd ask any person who doubts that 0 is even: 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.

tromp 6y ago

I'd ask any person who doubts that 0 is even: 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.

wcarey 6y ago
A counter argument to you approach: If I multiply two positive integers by one another is the result always greater than either integer? Turns out there is one exception to that rule. Does it serve a purpose? How do you know?
- tromp 6y ago
  Note that your rule already has a condition of positive inputs. It's not surprising that for inputs satisfying an inequality, the result also satisfies one. A simpler example is: If I subtract 1 from a positive integer, is the result always positive? That also has an exception. That's all due to 1 being an exceptional (the smallest possible) positive integer. Similar to how 0 is an exceptional natural number.
  - wcarey 6y ago
    Right - I was aiming for something analogous to your rule, which has the condition of even inputs. I do prefer your simpler example. Either way, I'm not sure your rhetorical strategy would be persuasive to someone who was unsure of the parity of zero (because, as you say, zero is an exceptional number).

btilly 6y ago

This reminds me of http://www.phys.ufl.edu/~det/phy2060/heavyboots.html http://www.phys.ufl.edu/~det/phy2060/heavyboots.html. (TL;DR it is common for people to think that if you dropped a pen on the Moon, it would float away. When those people are asked how the APOLLO astronauts didn't float away the most common response is that they had heavy boots.) 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.

ajarmst 6y ago

My guess is part of the issue is people remembering that 0 is neither positive nor negative and neither prime nor composite and naturally extending it to even vs. odd. But even here, one has to be careful about what definition of "even" you're employing. = 0 (mod 2)? Has two as a factor? One less than an odd number? !(x % 2)? I remember my confusion when I found out that the answer to "is 0 a natural number" was "it depends" and I still run into people who insist 1 is prime, which I think is insane. A lot of these things depend on the norms of the mathematical community discussing them.

derefr 6y ago
To me, the “is 1 prime” question gets at a very subtle distinction people don’t realize they’re making. 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.

whatshisface 6y ago

Something's wrong with all of the "deduce from patterns" answers in the article's comments. None of the patterns are deduced! It's definitely not an axiom that an even number minus and even number is even; that would have to be proven from the definition of even-ness. The definition of even-ness is what decides whether or not zero is even. It would be equally coherent to define 0 to be even, or exempt from even-ness. An example of a definition where exemption would make sense would be to say that even numbers are those whose unique prime factorizations contain a factor of two. 0 is exempted from having a factorization, so exemption from even and odd would follow naturally. Whether or not 0 is even is a random accident of math history with at most a little justification from convenience. Nobody can be blamed for failing to deduce it because it cannot be deduced. 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.

wcarey 6y ago
Indeed. And Euclid’s definition of evenness (Def VII.6) applies only to numbers, which for Euclid excludes both 0 and 1. So there’s at least one vaguely axiomatic formulation of number theory where the correct answer is probably “neither because it’s not the sort of object to which you can predicate evenness or oddness”.
ColinWright 6y ago
Yes and no. 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.

Sniffnoy 6y ago

I'm a bit doubtful of Wright's claim here that people only think zero is neither even nor odd only because they've been told so. I think a lot of people just don't like edge cases and want to exclude them. Now zero isn't really an edge case for parity, but someone who's not too used to thinking about zero might just consider it as an edge case generally (possibly even thinking of it as only sort of a number -- remember, people took a long time to accept zero as a number!), especially if they're thinking more in terms of whole numbers than in terms of integers. 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.

shadowofneptune 6y ago
Imagine explaining this question to someone without a concept of zero. Can nothing be cut into two nothings? Does that make it something?

inglor_cz 6y ago

Zero was adapted into Western mathematics fairly late and we still see the effects of its previous absence in the calendar. 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.

ColinWright 6y ago
Trivia: In the UK there are 8 train/railway stations with a platform 0, and another is being built. https://en.wikipedia.org/wiki/Platform_0 https://en.wikipedia.org/wiki/Platform_0
- inakarmacoma 6y ago
  And in several countries, floor #13 literally does not exist. No button. They just skip the inauspicious floor entirely.
jey 6y ago
Is there any calendar system that numbers days from zero? I'm sure that the Gregorian, Tamil, Chinese, and Discordian calendars don't. The closest thing I can think of is strftime's %w and %W flags for numbering days and weeks starting from zero.

md224 6y ago

The belief that zero is neither even nor odd makes sense once you consider a statement like “An even number of Unicorns exist” or “I possess an even number of magic lamps.” It’s technically correct, but it also sounds kind of strange, simply because we’re not accustomed to assigning parity to an empty collection.

deleted 6y ago
[deleted]

mannykannot 6y ago

The solar system example has some epistemic implications. Epistemology convetionally begins with a definition of knowledge as a justified belief in a proposition that is true, and the people who gave the 1-day answer did indeed believe the true proposition that the Earth revolves around the sun (subject to the caveats in the article). Was that belief justified? Well, they they probably received it from a knowledgable source, and if that is not regarded as justification, then by that standard, no-one knows much at all. It is also clear, however, that they do not have much understanding of the situation to which the proposition pertains. Epistemology, IMHO, should spend less time fussing over knowledge being a collection of propositions, and focus more on what it means to understand something.

gumby 6y ago

I think few even look at or understand the "moving stars" (planets). Also it feels few understand how seasons work. Every solstice and equinox prompts a news article or two explaining what the terms mean. Four times a year. So for them "Sun moves around the earth" is actually just as good as "Earth goes around the Sun." 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.

pdonis 6y ago
> 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". How did you measure how far away the Sun was? Without that information you can't reconstruct the Earth's orbit as you describe.
- 6nf 6y ago
  Maybe they just looked it up?
- gumby 6y ago
  I remember measuring the angle from a stick in the schoolyard (which wasn't far from home). A science teacher helped me. 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.

deleted 6y ago

[deleted]

namenotrequired 6y ago

Similarly, two (unrelated) smart people told me, isn't it cool that we can see the earth's shadow on the moon. 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?

young_unixer 6y ago

> They don't care. 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."

jl2718 6y ago

If you fix the reference frame to one in which the rotation is fixed to the earth but the center is fixed to the sun then you have the earth revolving around the sun in about a day. 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.

rwmj 6y ago

I quite like this article and it reminds me of when I first saw a mathematical proof. I was blown away that you could do this, and I went on to study maths and logic at university. But the most remarkable thing is that I only saw that proof when I was 15 or 16. In other words I'd gone through a decade of maths classes at a decent school and no one ever taught me about proofs! It was all simply memorizing facts and methods up til then.