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In teaching, we sometimes necessarily oversimplify concepts. Terry Pratchett famously referred to this as Lies to children:

A lie-to-children is a statement that is false, but which nevertheless leads the child’s mind towards a more accurate explanation, one that the child will only be able to appreciate if it has been primed with the lie.

In online discussions about lies to children, most examples I have come across are in the natural sciences, most frequently physics and chemistry. There are some examples on the Wikipedia page as well. Does anyone know of any good examples in mathematics and statistics, particularly at the undergraduate level or higher?

I don’t mean examples where we as teachers simplify and give hand-waving arguments because the level required for a rigorous proof is far too high. What I have in mind are explanations that are completely and utterly incorrect, but somehow still helpful.

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    $\begingroup$ Some people saying that it's not correct to say "infinitely close". But I disagree (with reference to non-standard analysis). I'll keep trying to think of something. Closest is saying square root of a negative is undefined, maybe? $\endgroup$
    – Sue VanHattum
    Jan 26, 2023 at 22:13
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    $\begingroup$ I think you would have more luck targeting the question at a younger age range than undergrads. You might get omissions in the intro calc sequence, and the applications problems will have a lot of gross simplifications, but those simplifications are not of the math. $\endgroup$
    – Adam
    Jan 27, 2023 at 1:56
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    $\begingroup$ While searching for good examples myself, I predict that there will be many answers suggesting things that are only lies, i.e., bad didactics or plain wrong, such as “learning long division is directly important for real-life application” or “1.5 is closer to 1 than to 2 and that’s why we round down”. $\endgroup$
    – Wrzlprmft
    Jan 27, 2023 at 8:23
  • $\begingroup$ What about the idea of imagining an infinitely tall triangle, such that the sides are functionally parallel when calculating interference patterns in the double-slit experiment? $\endgroup$
    – Dugan
    Jan 27, 2023 at 16:05
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    $\begingroup$ Would you include ‘lies’ that are never stated, but merely implied by usage and example? I'm thinking particularly of the misapprehension that almost all real numbers are rational (which covers the answers to most problems, most polynomial coefficients, and most numbers you meet — especially for younger students). $\endgroup$
    – gidds
    Jan 27, 2023 at 18:46

20 Answers 20

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Young children 5-8 years old, are taught to subtract the smaller number from the bigger number. They are told that you can't subtract a bigger number from a smaller number. This lie has its advantages and helps cement the order of the numbers in a subtraction sentence. Then when students learn about negative numbers they can easily unlearn the previous "lie" that you can't subtract a smaller number from a bigger number.

Of course, there is often a bright 7 year old who has been taught about negative numbers at home who objects to not being able subtract bigger numbers from smaller numbers.

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    $\begingroup$ A similar thing with division, before fractions and/or decimals are introduced. $\endgroup$
    – OrangeDog
    Jan 27, 2023 at 11:39
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    $\begingroup$ Not a lie, just a statement in which not all terms are fully defined. It is certainly true that there is no natural number that you can add to a given natural number to obtain one smaller than that. $\endgroup$
    – Carsten S
    Jan 27, 2023 at 12:17
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    $\begingroup$ @OrangeDog At least with division, teaching that tends to come along with the concept of "and 3 remainder". So teaching fractions adds the extra layer of what else you can do with the remainder. $\endgroup$
    – Graham
    Jan 27, 2023 at 13:15
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    $\begingroup$ "This lie has its advantages and helps cement the order of the numbers in a subtraction sentence." This sounds to me like "this completely misleads the children but it's good because it helps them get the right answers on standardized tests without having to think too hard about making those standardized tests any good" $\endgroup$ Jan 27, 2023 at 18:19
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    $\begingroup$ @leftaroundabout: And this is of course precisely what we would like children in elementary school to learn... :-) Reminds me a bit of Arnold who once complained: "To the question 'what is 2 + 3' a French primary school pupil replied: '3 + 2, since addition is commutative'." $\endgroup$ Jan 27, 2023 at 19:07
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We usually teach:

$$\int\frac1xdx=\ln{|x|}+c$$

Whereas it should be:

$$\int\frac1xdx = \begin{cases} \ln{x}+c_1 & x>0 \\ \ln{(-x)}+c_2 & x<0 \end{cases}$$

Why don't we teach the correct version? Using the "wrong" version usually leads to the right final answer, whereas using the correct version would probably overburden struggling calculus students.

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    $\begingroup$ I got through four years of mathematics at university without noticing this! To be fair, once we got to complex integrals, I realised pretty quickly that the "log |x| + c" solution isn't exactly right; but it's funny that if we never leave the real numbers then it's wrong in a different and incompatible way. (I say "incompatible" because in the complex numbers you don't get to choose c1 and c2 totally independently.) $\endgroup$
    – kaya3
    Jan 27, 2023 at 12:27
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    $\begingroup$ Even on the reals, the second version isn't exactly "the correct one". The correct version says that the function $x\mapsto 1/x$ is not integrable on any domain that includes $0$. What you actually can do is obtain one antiderivative that works restricted to the positive and one that works restricted to the negative reals, but writing them both in a single equation is more sloppy than the simple version that excludes the negative numbers entirely. $\endgroup$ Jan 27, 2023 at 19:04
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    $\begingroup$ @leftaroundabout The second version isn't sloppy: if a function $f \colon U \to \mathbb{R}$ happens to be defined on an open subset $U \subseteq \mathbb{R}$ (possibly disconnected), then $\int f(x) dx$ can (should?) be interpreted as the set of all functions $F \colon U \to \mathbb{R}$ satisfying $F'=f$ in $U$, and this is what we get. $\endgroup$ Jan 27, 2023 at 19:16
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    $\begingroup$ The fancy way of saying this is that the 0th de Rham cohomology group of $\mathbb{R} - \{0\}$ is given by the locally constant functions. The "+C" should really stand for a locally constant function in each instance, not just a constant. $\endgroup$ Jan 27, 2023 at 20:23
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    $\begingroup$ @Stef It avoids the need to introduce an exception to the rule that the integral of a function is "some other function plus c", and in all of the exercises in that textbook that require an integral of 1/x, the integral is over an interval that doesn't contain the pole anyway, so there would be no need to write a solution with two different c variables just to ignore one of them. At that level, the goal of a problem is often not "describe the family of all functions whose derivative is this", but rather "find the area under this curve". $\endgroup$
    – kaya3
    Jan 29, 2023 at 11:37
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The idea that “a number” means “this decimal expansion”, rather than the expansion being a way of representing a number that has some more set-theoretic definition. It's the de facto truth for everyone in the world who isn't going into rigorous mathematics, but thinking of the decimal as “the number” and other properties as things that are true about it is the root of the classic confusion around $0.999... = 1$.

When I was introduced to the concepts at the undergraduate level, we couldn’t even get through the basic definitions without acknowledging that $0.999... = 1.000...$ as an assumption of how the representation is constructed to allow us to prove that numbers map (almost) uniquely in both directions.

This relates closely to Patrick Stevens' answer – a straightforward understanding of rational and real numbers works intuitively, but the formalism has to go the other way from the intuition.

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    $\begingroup$ One could also formalize real numbers as infinite decimal expansions, with the equivalence with 9s imposed. I think I actually saw a writeup of this a few years ago, and proving that you have a totally ordered complete field based on this definition was not as horrible as you might imagine. $\endgroup$ Jan 27, 2023 at 12:32
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    $\begingroup$ That makes sense, although if we were going to do that I'd definitely want to see an argument that the equivalence was a natural thing to impose. (I suppose the algebraic argument based on 9.999... = 10 * 0.999... does supply that.) $\endgroup$
    – LizWeir
    Jan 27, 2023 at 13:13
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    $\begingroup$ @LizWeir non-integral quantities are taught primarily as fractions initially. This has a number of advantages: it allows the introduction of concepts like factors, etc and matches natural language more closely. (Children will already be eating "three quarters" of a cake, not "0.75 of a cake"). At this point other representations for integers like tallymarks and roman numerals are often taught. This is much maligned but they are culturally marginally important, and from a mathematical perspective help break the "a number is merely this sign" mistake. $\endgroup$
    – Dannie
    Jan 27, 2023 at 14:04
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    $\begingroup$ Decimals are initially treated as merely the awkward language spoken by calculators and computers, and usually alongside ten times-tables and basic SI prefixes so that "moving the dot" helps the kids understand the structure of them and their practical use in trades, crafts, etc. $\endgroup$
    – Dannie
    Jan 27, 2023 at 14:05
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    $\begingroup$ I'll note that 0.999...=1 is not true under all number systems. Some students are (intuitively) working in a different mathematical system (which allows infinitesemals) than the standard one. Mathematical systems with infinitesimals can have similar power/rigor/coherency to the "standard" system. Nonstandard Student Conceptions About Infinitesimals discusses this from the perspective of educators. It's approachable, despite being over 100 pages. $\endgroup$
    – Brian
    Jan 27, 2023 at 14:16
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The average

where the lie-to-children is the word "the".

Ask anyone what "the average" of a set of values is, and immediately you'll be told the arithmetic mean. That's how it's taught initially, and that's what everyone falls back to by default.

A little further on in primary school, you're taught that actually there are three kinds of averages - mean, median and mode. But only three.

Study actual statistics though, and you get onto least-squares, standard deviation and other ways to know how confident you are about your "average", fitting to polynomials or other functions, Bayesian statistics, and so on.

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    $\begingroup$ "Most people have an above-average number of arms." $\endgroup$ Jan 27, 2023 at 12:51
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    $\begingroup$ I disagree. I have never heard anyone say "the average" to mean anything other than the mean. $\endgroup$
    – Daron
    Jan 27, 2023 at 13:23
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    $\begingroup$ @Daron But which mean? Arithmetic, geometric, harmonic, something else? The term ‘mean’ has the same issue to a large extent, it almost always means the arithmetic mean, but it’s still technically ambiguous. $\endgroup$ Jan 27, 2023 at 16:00
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    $\begingroup$ When you use standard spreadsheet software, you need to be comfortable with the fact that the AVERAGE() function returns the arithmetic mean and not anything else. $\endgroup$ Jan 27, 2023 at 16:30
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    $\begingroup$ @Graham If no other context is given, "The average" almost always means the mean. This is how language works. I would not call the existence of the median and mode a lie to children. It is just a change of convention. $\endgroup$
    – Daron
    Jan 27, 2023 at 17:53
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Whether or not the derivative $\frac{dy}{dx}$ is a fraction. Similarly, what, exactly, are $dy$ and $dx$?

This actually goes through several iterations of lies:

  1. We first hammer it into Calc I students that $\frac{dy}{dx}$ is not a fraction, but instead the limit of fractions and that we write it as a fraction for intuition. We teach that $dx$ should be thought of as "an infinitesimal change in $x$" and sometimes even mention that this isn't strictly accurate, but does mirror the intuition of Newton, Leibniz, and others during the historical development of infinitesimal calculus.

  2. Somewhere in a differential equations or physics course, we teach students to manipulate $\frac{dy}{dx}$ as a fraction. But only in special circumstances and that it's not really a fraction; it just plays one on TV. All of these fraction manipulations are simply algebraic shorthand for things like change of variables in integration when solving a separable differential equation.

  3. Somewhere in a differential geometry/calculus on manifolds course we introduce the idea that $\frac{dy}{dx}$ is, in fact, a fraction relating vectors in tangent spaces. And simultaneously introduce the idea that a derivative is a linear mapping between tangent spaces and not a fraction. And that $dx^i$ is simply a map that returns the $i$-th coordinate of a vector in a tangent space.

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"Random variable."

...because, as we all know, a random variable is neither random nor a variable. It is a real-valued function. But if we tried to introduce the concept, the feeling, of a stochastic, uncertain, incompletely known environment using such a deterministic terminology as "real-valued function", we would certainly fail. Hence, random variable it is.

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    $\begingroup$ Is a random variable really a fiction? It may be that they don't exist in the world of digital computers that we currently normally use, but that's not to say they don't exist. $\endgroup$ Jan 27, 2023 at 15:28
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    $\begingroup$ I've never been fond of this somewhat-snarky take. Random variables are the mathematical model for things that appear random in the world, and are used as such in applications all the time. $\endgroup$ Jan 27, 2023 at 16:35
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    $\begingroup$ -1 for two reasons: (i) This is not an example of a lie-to-children (or students), since it's not something that is "corrected" later on. On the contrary, random variable is standard terminology which is used all the time by mathematicians working in probability theory. (ii) I feel obliged to say that I don't find the reasoning in your second paragraph convincing. As @DanielR.Collins pointed out, random variables are commonly interpreted as quantities that are "variables" in a model and that are random. $\endgroup$ Jan 27, 2023 at 18:12
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    $\begingroup$ Random variables are subsets of "real-valued functions", assuming you're only meaning that the codomain is real (the domain need not be). But not all "real-valued functions" are random variables, so it's incorrect to equate the two. And variables can hold functions! It is a variable (holding a function), and it is formalizing some notion of "randomness". "Random variable" is more correct. $\endgroup$ Jan 27, 2023 at 18:27
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I was introduced to the real numbers as "all the points on a [two-sided infinitely long] line".

  • At best this is a circular definition. It's certainly very sloppy, and those words could be used to describe many different objects.
  • The real-world intuition is incorrect. By the time you're looking at a small enough scale for it to matter whether you've got $\mathbb{R}$ or $\mathbb{Q}$, space has stopped behaving like $\mathbb{R}^3$.

But it's intuitively obvious what it means. Eventually you'll be introduced to "Dedekind- or Cauchy-completion of the rationals" to formalise what it means for something to be a "point on a line" in the intended sense.

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  • $\begingroup$ Well, but for actually writing down 5 or 6 geometric definitions that come immediately from intuition this is completely rigorous...see Bell's Smooth Infinitesimal Analysis, for example, where he builds out almost all of calculus and the basics of differential geometry for an (inequivalent!) axiomatization of the reals using Euclidean geometry alone. Every function is continuous, and infinitely differentiable---made possible by a constructive ambient logic preventing definition of point discontinuities. The theory has topos models, and so is as consistent as ZF(C?). $\endgroup$
    – Duncan W
    Jan 28, 2023 at 3:19
  • $\begingroup$ Cute! Although to convince me that this is a counterexample, you'll have to show me a high school teacher who avoids using excluded middle in their arguments involving the reals (or else proves that the instances of LEM they use are intuitionistically true). $\endgroup$ Jan 28, 2023 at 8:10
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    $\begingroup$ "I was introduced to the real numbers as "all the points on a [two-sided infinitely long] line". " What do you mean by "two-sided line"? $\endgroup$ Jan 31, 2023 at 20:40
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    $\begingroup$ A line rather than a ray. $\endgroup$ Feb 1, 2023 at 18:33
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$ $

$\LARGE\mathbb R$

Students are introduced to real numbers long before they are ready for the formal definition. At second level they are primed for dealing with fractions and not fractions and told there are numbers like $\pi$ where the decimal expansion goes on forever.

The formal definition is in terms of completion of a metric space or Dedekind cuts. I was in my early twenties when I first encountered it.

Interesting fact: The metric space definition is already circular if you insist on defining a metric as a function $d: X \times X \to \mathbb R$. This assumes $\mathbb R$ is defined and cannot be used to create the definition. Your homework is to repair this problem.

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    $\begingroup$ Define $\mathbb R$ using something other than Cauchy sequences, and then feel free to use Cauchy completion to define other complete metric spaces? $\endgroup$
    – Dave
    Jan 27, 2023 at 16:06
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    $\begingroup$ Initially the (usual) metric $|\cdot| : \mathbb{Q}\times\mathbb{Q} \to \mathbb{Q}$ can be used to complete $\mathbb{Q}$ into $\mathbb{R}$. $\endgroup$
    – Aeryk
    Jan 27, 2023 at 17:39
  • $\begingroup$ @Aeryk Ten points! $\endgroup$
    – Daron
    Jan 27, 2023 at 17:51
  • $\begingroup$ @Dave That also works but requires back-and-forthing. Nine points! $\endgroup$
    – Daron
    Jan 27, 2023 at 17:52
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    $\begingroup$ You can define the real numbers using Cauchy sequences without circularity - just swap out "for all real epsilon > 0" with "for all rational epsilon > 0". The only reason the definition of Cauchy sequences depends on the real numbers is because you're using a definition of Cauchy sequences of real numbers. $\endgroup$
    – kaya3
    Jan 28, 2023 at 9:10
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I was told in high school that Euclidean geometry can be derived from the five postulates written by Euclid, but this is not the case. Several of Euclid's proofs have holes in them and one can create models of the five axioms that do not satisfy those results (e.g., in $\mathbb{Q}^2$ a line can be closer to the center of a circumference than the length of its radius and still not inersect the circle).

A rigorous foundation of Euclidean geometry was done by Hilbert in 1899 and it requires 20 axioms, which exclude the above models and make all classical theorems valid.

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    $\begingroup$ Are these lies told to children specifically? I feel like these are just inherited lies that had been told to adults for a very long time. (And when I put it that way, I feel uncomfortable calling them lies, because certainly Euclid did not set out to deceive anyone about the various models of his axioms.) $\endgroup$ Jan 27, 2023 at 17:54
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    $\begingroup$ Euclid made a honest mistake, assuming that the text was not corrupted by copying it over the centuries. That his 5 axioms are good in taught in high schools, at least in Italy a few years ago. $\endgroup$
    – Rad80
    Jan 27, 2023 at 21:51
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A lot of things around the limitation and construction of number spaces come to mind such as:

  • You cannot divide 5 by 2.
    You cannot take the square root of a negative number.
    You must not divide by zero.

  • Real numbers hold more reality than imaginary numbers.

  • Imaginary numbers hold some reality because they have applications in electrical engineering, quantum physics, …

  • Mathematicians invented complex numbers because they wanted to take the square root of negative numbers.

    (There may be some truth to this one. It’s not a good motivation though.)

  • We use complex numbers to be able to solve more equations.

  • Infinity is not a valid result and you cannot perform computations with it.

Whereas the truth is more like:

  • All number spaces (and infinity) are artificial constructs.
  • “Advanced” number spaces (and infinity) allow us to treat certain problems more efficiently or avoid irrelevant pathologies.
  • Taking the square root of a real number (and similar) is most often not something we do for its own sake, but something that comes up in application and is either a way to an useful answer or not.
  • The restrictions of number spaces reflect restrictions of real-life or inner-mathematical applications and it’s our duty to apply them as appropriate.
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  • $\begingroup$ "The restrictions of number spaces reflect restrictions of real-life applications and it’s our duty to apply them as appropriate." Maybe if you're an applied mathematician… $\endgroup$
    – wizzwizz4
    Jan 27, 2023 at 17:04
  • $\begingroup$ @wizzwizz4: Now that you are saying this, I might as well add inner-mathematical applications to the list. That only leaves those few people who investigate the particular consequences of certain restrictions for their own sake. $\endgroup$
    – Wrzlprmft
    Jan 27, 2023 at 18:16
  • $\begingroup$ Well, I still cannot take the square root of a negative number. As soon as I extend square roots to more than just nonnegative real numbers, suddenly every number except zero has two square roots, so it's no longer possible to take the square root of a number. $\endgroup$
    – Stef
    Jan 27, 2023 at 20:34
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    $\begingroup$ @Stef: That’s a problem you also have when taking the square root of positive numbers. For example, it is only by convention that $\sqrt{4} = 2$ and not $\sqrt{4} = -2$. We can introduce similar conventions to make the square roots of negative numbers (and complex numbers) unique. $\endgroup$
    – Wrzlprmft
    Jan 27, 2023 at 20:57
  • $\begingroup$ Sure, you can come up with conventions for anything and everything if you want to. But that doesn't make "lies" all statements made by other teachers who are not using your conventions. $\endgroup$
    – Stef
    Jan 27, 2023 at 21:07
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My biggest pet peeve:

"A vector is a quantity with both magnitude and direction"

One is required to say this to pick up marks on on A-level physics exam, say, despite it being very, very wrong and arguably placing emphasis on the wrong concepts. I am not yet at university, but I imagine there is a large culture shock when students are first exposed to linear algebra. That said, introducing any other notion seems to be very counter productive to the high school education, and with regards to one actually needs to be able to do, the "magnitude-and-direction" lie is successful.

Many examples of this can be found in high school mathematics curricula. Unrigorous notions or incorrect definitions of the derivative, even of "increasing/decreasing function", integral, etc. are all lies that tend to cause students some difficulty unlearning in the first years of university but are essential for teaching the applications of high school maths to the typical high schooler.

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    $\begingroup$ What about "A vector is a quantity with both magnitude and direction" is not true? Maybe not all vectors have magnitude and direction, but the ones being discussed at A Level do, so in the context it is presented it makes sense to me... $\endgroup$ Jan 31, 2023 at 20:53
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    $\begingroup$ @AdamRubinson It's not at all the definition of a vector. It's a really annoying pseudo-definition... but it is a reasonable thing to teach at A level, as you say, which is precisely why I deem it a suitable response to this post. $\endgroup$
    – FShrike
    Jan 31, 2023 at 20:56
  • $\begingroup$ Yeah, that's true: definitionally, a vector is a more abstract object. $\endgroup$ Jan 31, 2023 at 20:57
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I think sometimes when introducing $\pi$ to children they are told that it's exactly $3.1415$ or some other arbitrary number of decimals, or even that it equals $22/7$

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    $\begingroup$ I wouldn’t classify this as a lie to children as specified in the question. Of course it’s said to children and it’s a lie, but I am very skeptical that this is a didactic benefit to this as in “leads the child's mind towards a more accurate explanation, one that the child will only be able to appreciate if it has been primed with the lie”. $\endgroup$
    – Wrzlprmft
    Jan 27, 2023 at 8:18
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    $\begingroup$ Never heard of this $\endgroup$
    – OrangeDog
    Jan 27, 2023 at 11:39
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    $\begingroup$ I don't recall ever being told this is the exact value, it was always explained as an approximation. $\endgroup$
    – Barmar
    Jan 27, 2023 at 15:40
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    $\begingroup$ Even if this qualifies, it’s not exactly something that matters at all. The margin of error involved in just using 3.1415 instead of 3.1416 (correctly rounded), or 3.14159, or 3.141592654, or some higher number of digits is so small that it just doesn’t matter even in many scientific applications. Even 22/7 is not all that horrible when dealing with small circles and low precision (it’s only about 0.04% larger than π, so it’s still useful in real life because it’s predictably a tiny bit high). $\endgroup$ Jan 27, 2023 at 16:18
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    $\begingroup$ Some teachers teach badly – but, unfortunately, I am aware of this being taught. @AustinHemmelgarn It matters when you're working with (e.g.) Taylor series representations of trigonometric functions, since thinking pi is rational will lead to contradictions. $\endgroup$
    – wizzwizz4
    Jan 27, 2023 at 17:07
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We can define irrational numbers to be those numbers which are not rational, and then define the real numbers to be the union of the rationals and irrationals.

The problem with this definition is that it first requires use to have defined the set of real numbers.

Here is a related discussion:

Why does the widespread erroneous definition of "irrational number" persist without being taught?

If we try to define irrational numbers as "numbers that are not rational" then we unwittingly capture numbers like $\sqrt{-1}$, quaternions, $\aleph_0$, surreal numbers, etc.

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Anyone who was taught about Venn Diagrams was probably told that a set is a collection of objects.

This is, of course false. But, it is a handy way to think about sets, and even when you get around to getting to a more formal notion of sets- you'll still think of them as collections of objects, while reasoning about them.

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  • $\begingroup$ In what way is this false? As far as I understand the axioms of set theory, a set is more or less a formalization of a "collection of objects" (but my knowledge of set theory is quite limited). $\endgroup$ Jan 30, 2023 at 5:29
  • $\begingroup$ Sets are not defined objects in set theory.Sets can relate to each other, in two ways.Equality and elementhood. While the Axioms do define equality between sets , the axioms don't define elementhood.Sets being collections of objects is just a particular way to think/talk about the elementhood relation. But, that's not in any way related to what sets are. Formally, sets are variables which satisfy some formulas in the language of set theory. There are various set existence axioms, If a formula can be derived from one of those axioms , the Variables which satisfy it are sets. $\endgroup$ Jan 30, 2023 at 7:01
  • $\begingroup$ There are many interpretations of "elementhood" that are consistent with the axioms. Instead of collection of objects, you can think of it as less-than, or subset, or, stronger than, etc... the formula you build which are satisfied by variables, assign intuitive meaning to elementhood. $\endgroup$ Jan 30, 2023 at 7:17
  • $\begingroup$ Sets aren't at their core, a formalization of collections of objects. They are variables which satisfy properties which can be defined with well formed formula ( the technical term for formula in the language of set theory). - So, to be an element of a set is to satisfy some property. As sets are defined by what elements they have, that's what sets are: a formalization of what it means to define something in terms of the properties that it has. $\endgroup$ Jan 30, 2023 at 7:45
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    $\begingroup$ @MichałMiśkiewicz Thinking of any collection of sets as a set leads to contradictions, like Russel's paradox (Consider the collection of all the sets that do not contain themselves. If this is a set, called R, then it is either in or not in R, Both of these lead to a contradiction). An illustration of this is the barber paradox (Define the barber as a person who shaves those and only those who do not shave themselves. Does a barber shave himself?). $\endgroup$ Mar 4, 2023 at 9:12
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As OP asks for examples "particularly at the undergraduate level or higher", here is a meta-mathematical "lie-to-children" that undergraduates tend to learn (often probably implicitly) when they major in mathematics:

Validating a mathematical result means checking how every single step in the argument follows logically from the previous steps.

The skill to check how every single step of a proof follows from the previous steps is obivously something that one needs to learn well. So I don't think one can avoid that the students get, at some point, the impression that this consecutive validation of steps is the essence of checking a mathematical result. However, when more experienced mathematicians read a result and its proof, many of them approach it quite differently then reading every single step in the proof in detail.

Consequences of this (maybe unavoidable) "lie-to-children" can be observed on many occasions (for instance, on several Stackexchange sites):

  • Some (many?) people seem to be under the impression that a mathematical theory were essentially a logical house of cards that collapses once one removes a single piece. Experience shows that this is not the case, though: mistakes in research papers do occur on a regular base and at varying degrees of severity, yet mathematics (or subfields of it) is far from collapsing. Several reasons for this are discussed in the answers to this MathOverflow post.

  • Many PhD students believe that the "canonical" way to referee a mathematical research paper is to check every single step in all the proofs. However, this is hardly what one does in practice. In this post I described how checking proofs during peer review tends to work in practice.

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Independence in probability theory is not independence in usual sense because it does not take into account causation.

Addition1: The notion of independence is well-known en.wikipedia.org/wiki/Independence_(probability_theory). For example, events $A$ and $B$ are called independent if $\mathbb{P}(AB)=\mathbb{P}(A)\mathbb{P}(B)$. As we can see, this is not at all the same as causal independence or smth.like that, because there is not a word about causes in this definition. And if smb. speaks about independence, for example "Student John's grade in physical education does not depend on the size of Napoleon's army", it doesn't look like he means that $\mathbf{P}(AB) = \mathbf{P}(A)\mathbf{P}(B)$ for the corresponding events.

Addition2: This is not the first time someone put a dislike, failing to formulate a single objection against the obviously correct thesis)

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    $\begingroup$ What do you consider independence in the usual sense? I wouldn’t consider two causally connected events independent in any sense that I am aware of. $\endgroup$
    – Wrzlprmft
    Jan 27, 2023 at 14:15
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    $\begingroup$ You call this an "obviously correct thesis", but I don't see it. Can you explain this in the Bayesian formalism? $\endgroup$
    – wizzwizz4
    Jan 27, 2023 at 17:05
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    $\begingroup$ @wizzwizz4 Independence is determined within the Kolmogorov axioms. And the Bayesian approach is defined within the framework of Kolmogorov's axioms. The notion of independence is well-known en.wikipedia.org/wiki/Independence_(probability_theory). For example, events $A$ and $B$ are called independent if $\mathbf{P}(AB) = \mathbf{P}(A)\mathbf{P}(B)$. As you can see, this is not at all the same as causal independence, because there is not a word about causes in this definition. That's why I call the obvious thesis obvious. $\endgroup$ Jan 27, 2023 at 17:15
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    $\begingroup$ @BotnakovN. If you included that comment in your answer, I would still disagree with it (because I don't think that independence "in the usual sense" is causal independence) but I would think it's a fine answer because I would know what you mean. Right now, your answer is incomprehensible without that comment. (I did not downvote, though, to be clear.) $\endgroup$ Jan 27, 2023 at 17:51
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    $\begingroup$ The definition via P(A and B) = P(A) * P(B) is not a simplification - it is the full definition of "independence" used in probability theory. There is no later stage of education where a student will learn that independence in probability theory is not really defined that way. It is really defined that way. You just dispute that "independence" is a suitable word for the concept, so your dispute is with the terminology used in the field, not with how it is taught. $\endgroup$
    – kaya3
    Jan 28, 2023 at 12:18
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1-2-3-4-5-6-7-8-9-10... Well, not if you are using binary...

The angles on a triangle add up to 180 degrees... Only if the triangle is on a flat surface.

Multiplication is repeated addition. Have you ever multiplied, say, (-3)*(-4) and wondered about how you were adding up a negative number of times? No?

Most of elementary mathematics when you get to abstract algebra, really. Adding two positive numbers always gives a positive, larger number? Nope, sorry, we are in Z7 today so 4+4=1. (Or we are actually adding time--if it is currently 10pm and your homework is due in 11 hours, 10+11=9am.)

Subtraction is adding a negative number. Unless you are subtracting a negative number, oops.

IMO math is absolutely crammed full of lies to children, to the point where any elementary math teacher who uses the words "always", "never", "every", etc., is probably automatically fibbing.

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    $\begingroup$ I claim these aren't lies, but either unusual notations or generalisations. "Well, not if you are using binary" - that is, if the symbols "10" do not in fact mean the natural number ten. "We are in Z7 today", so the symbol "4" does not refer to the natural number four. "The angles on a triangle add up to 180 degrees", unless the triangle isn't drawn on a plane. "Multiplication of natural numbers is repeated addition". None of your statements are false, they're just misleadingly or sloppily phrased. $\endgroup$ Jan 27, 2023 at 23:48
  • $\begingroup$ I also thought about "Multiplication is repeated addition" for this question, but there's one big problem : it's not a lie at all. If you post a link to the corresponding question, you might as well post a link to the accepted answer, which basically refutes Devlin's rant. Math is full of examples which are correct for a class of numbers, and need to be extended for superset. It doesn't mean it's a lie to describe the example with a simpler version for subsets. $\endgroup$ Jan 28, 2023 at 21:20
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$$\frac{1}{0} = \infty$$

I was told this by my Math teacher in my 8 or 9 grade. I only knew at that time about infinity was that it is a very large number; so large that no can ever write it.

I don't expect it to be very common though.

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Students are often taught the chain-rule as a trivial cancellation law. In reality, there are intricacies within the chain-rule.

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    $\begingroup$ Welcome to the site. Can you be more specific? Showing an example of how the chain rule is taught and how it should be taught would clarify details for the reader. $\endgroup$
    – Amy B
    Jan 30, 2023 at 9:58
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In the US, UK, and former UK colonies, at certain primary-school grades, you can't leave your final answer as an improper fraction. You'll lose marks for answering

$$\frac{3}{4}+\frac{1}{2}=\frac{5}{4}.$$

Students are taught that there is something "improper" and wrong about leaving the above as your final answer.

For full marks, you must answer

$$\frac{3}{4}+\frac{1}{2}=\frac{5}{4}=1\frac{1}{4}.$$

Some grade levels later, students discover this "rule" was nonsense.


This seems to be mainly a US/UK thing: See What is the rationale for distinguishing between proper and improper fractions?.

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  • $\begingroup$ This feels like just a lie (or rather bad convention) to me rather than a “lie to children”. Most of the answers to the linked question go along the lines of: It is good to know this exists or to avoid weird representations (like 22/7), but that’s no reason to enforce this. $\endgroup$
    – Wrzlprmft
    May 25, 2023 at 6:43
  • $\begingroup$ @Wrzlprmft: See the linked discussion: It seems the main reason for teaching students how to convert improper fractions into mixed numbers is that the latter commonly appear in everyday life (at least in the US and UK). $\endgroup$
    – user18187
    May 25, 2023 at 6:54
  • $\begingroup$ Not only the US/UK thing. $\endgroup$
    – Rusty Core
    May 30, 2023 at 0:42

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