> Lying is a core part of communicating mathematics. We lie to kindergarteners when explaining fractions. We lie to fourth graders when approaching limits...
Hard disagree.
Lying is with intention to deceive.
Teaching is simplifying with the intention that they understand and get the correct intuition.
Math is not about lying, that's just silly.
Terry Pratchett, The Science of Discworld:
> As humans, we have invented lots of useful kinds of lie. As well as lies-to-children ('as much as they can understand') there are lies-to-bosses ('as much as they need to know') lies-to-patients ('they won't worry about what they don't know') and, for all sorts of reasons, lies-to-ourselves.
> Lies-to-children is simply a prevalent and necessary kind of lie. Universities are very familiar with bright, qualified school-leavers who arrive and then go into shock on finding that biology or physics isn't quite what they've been taught so far. 'Yes, but you needed to understand that,' they are told, 'so that now we can tell you why it isn't exactly true.'
> Discworld teachers know this, and use it to demonstrate why universities are truly storehouses of knowledge: students arrive from school confident that they know very nearly everything, and they leave years later certain that they know practically nothing. Where did the knowledge go in the meantime? Into the university, of course, where it is carefully dried and stored.
I don't get it?
Nevertheless you don't have to lie to kids in any field, science, art or otherwise.
Brilliant
I don’t remember us getting to fractions in kindergarten, but maybe the curriculum has radically changed since the early 70s.
What exactly is the lie? 1/4 and 3/8 equals 5/8. Is there’s something more to that? Is that fundamentally wrong?
> Is there’s something more to that?
Yes: this is about building the quotient field (field of fractions) [1] for some integral domain, or more generally, building the localization ([2], [3]) of a commutative ring with respect to some given set that is closed under multiplication (the special case of the quotient field for a ring R is obtained when one chooses R\{0} as such a set).
--
[1] https://en.wikipedia.org/w/index.php?title=Field_of_fraction...
[2] https://en.wikipedia.org/w/index.php?title=Field_of_fraction...
[3] https://en.wikipedia.org/w/index.php?title=Localization_(com...
Don't we still teach kids that e.g. 3/4=6/8, that they need to make a common denominator to add, and that they should cross multiply to check equality? I suppose we don't teach zero divisors, but otherwise, jargon aside, I'd be hard pressed to explain how we don't teach kids that fractions are members of ZxZ* mod (ad-bc).
Lies to children are like... time-reversal symmetry.
It's mighty pretentious to say that one needs all that theory to simply answer the question lol. For many questions, only the most rudimentary theory is plenty to get an answer, that is exactly the same answer as a more elaborate theory would yield.
If you just want to do some stupid computations: sure.
But this is not what mathematics is centrally about. The central point is the kind of thinking about the respective topics (and understanding it) which these more abstract definitions encode.
Understanding the topic just enough to do some elementary computations does not give you the kind of thinking that is often near a transcendental experience.
Just to give one example: the reason why the localization of a commutative ring (a generalization of the field of fractions) is introduced is that many properties of ring hold if and only if they hold for all of its local rings; see for example [1]. This means to understand some property of a commutative ring R, we "just" have to understand its (simpler) local rings.
This is an example why one wants to study such ideas; on the other hand, I can imagine sooo many more exciting things to do with my life than dividing numbers by each others to form fractions. :-)
[1] https://en.wikipedia.org/w/index.php?title=Localization_(com...
Now chemistry on the other hand…
OK there’s still no intent to deceive but almost all of the “rules” you learn have giant exceptions