math-pedagogy

[tawny slate]

what is yoneda lemma

also im about to lose connection for a bit

will reply later

[tight star]

Well, in the category I’m thinking of

It just says that if m and n are nonnegative integers, then $m | n \iff \forall d \in \mathbb{Z}_{\geq 0}, d | m \implies d | n$

[burnt vessel]

Pseudonium

[tight star]

We’ve applied this with $m = \text{gcd}(a + 2b, 2a + b)$ and $n = 3$

[burnt vessel]

Pseudonium

[vagrant meadow]

can you explain what the lemma is in general and how you're applying it to this problem? just for my own edification

[tight star]

um…

it’s gonna be a little difficult to explain what it is in general

unless you already know what categories and functors and natural transformations are?

[tight star]

If you’re comfortable with category-theoretic terminology I can say

[long pelican]

Using $\to$ for divides for the moment, #math-pedagogy message says
[(m\to n)=\prod_{d:\bN}\prod_{d\to m}d\to n.]
More generally for any contravariant functor $G$
[G(m)=\prod_{d:\bN}\prod_{d\to m}G(d).]
This formulation is sometimes called the Ninja yoneda lemma using ends (which I wrote as products)
( https://en.wikipedia.org/wiki/Yoneda_lemma#In_terms_of_(co)end_calculus )

[burnt vessel]

Icy0
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[tawny slate]

ok im back

and lets pretend i have no clue what a category is

is there an eli5 to help me understand why thinking about it in terms of categories is useful or generalizes to other applications?

[tight star]

hmm so

im not sure how else I would’ve done the problem by the proof method I presented

without using universal properties?

because we had to convert a single divisibility statement into 2

and also convert 2 into 1

which is exactly what the universal property talks about

though - I didn’t explicitly mention categories, did I?

[tawny slate]

nope

[tight star]

the idea of a universal property is the takeaway, I’d say

[tawny slate]

also the universal property sounded like you just pulled it from thin air, but to me it feels like a common sense statement about gcd that is "obviously true"

ahhh ok so this problem motivates the idea of a universal property?

[tight star]

it sounded like I pulled it from thin air?

[tight star]

yep

[tawny slate]

yeah like you just started with it

[tight star]

right

but you said it seems clear why it’s true?

[tawny slate]

in the sense that given the eli5 definition of gcd, it follows logically almost trivially, intuitively

yes

[tight star]

ok that’s good

it turns out like

loooots of things in maths have a universal property

and you can use it to help you in problems

e.g. the universal property lets you define gcd for multiple integers

say for 3 integers

given m, n, k, the gcd is the unique integer satisfying

$\forall d \in \mathbb{Z}_{\geq 0}, d | m \wedge d | n \wedge d | k \iff d | \text{gcd}(m, n, k)$

[burnt vessel]

Pseudonium

[tight star]

and in fact, using the universal property, you can show this exists

gcd(m, n, k) = gcd(gcd(m, n), k) = gcd(m, gcd(n, k))

and you can extend this to arbitrarily many integers too

[tawny slate]

ok but what other kinds of objects have this universal property thing

all i have right now is gcd

[tight star]

so, lowest common multiple does as well!

given integers m and n

[tawny slate]

that is quite an underwhelming example

[tight star]

the lowest common multiple is the unique nonnegative integer satisfying

[tawny slate]

i would be surprised if it didnt have a universal property if gcd has one

[tight star]

$\text{lcm}(m, n) | k \iff m | k \wedge n | k$

[burnt vessel]

Pseudonium

[tight star]

You also have union and intersection of sets

Also the product of sets

And groups

And topological spaces

As well as the disjoint union of sets

The free product of groups

[tawny slate]

ok, that one is slightly more interesting, and seems to draw a more clear picture of what a universal property is

[tight star]

So for union

[tawny slate]

i got that one

can you do groups

[tight star]

The universal property is $A \cup B \subseteq C \iff A \subseteq C \wedge B \subseteq C$

[burnt vessel]

Pseudonium

[tight star]

So, given groups G and H

You can form a group called the “direct product”, G x H

it satisfies the following universal property

take some other group Z

then group homs Z -> G x H correspond bijectively to pairs of group homs Z -> G and Z -> H

what G x H “does” is let you collapse 2 group Homs into one

much like how gcd lets you collapse 2 divisibility statements into 1

[tight star]

and also go back

[tawny slate]

so lemme see if i got this

whats intersect in latex again

cap, right

[tight star]

$\cap$

[burnt vessel]

Pseudonium

[tawny slate]

the universal property of intersect is $A \subseteq B \cap C \iff A \subseteq B \wedge A \subseteq C$

[burnt vessel]

CosmoVibe

[tight star]

Exactly!

[tawny slate]

and the reason these "universal properties" are cool

isnt just the generalization of them

but also that they can help syntactically attack proof problems

[tight star]

Yep

[tawny slate]

by combining or splitting statements

ok

[tight star]

Mhm

[tawny slate]

seems reasonable

is there a more like elementary math example of them besides gcd/lcm

and union/intersect

[tight star]

Hm…

What counts as elementary math?

[tawny slate]

lets say pre univ or at least undergrad

[tardy ember]

cartesian product of sets maybe?

[tight star]

Yep

[tardy ember]

i have no idea which topics are "pre-univ or undergrad"

[tight star]

Also the universal property of subsets and quotients

[tawny slate]

ok sureeeee but now im wondering what kind of problem involving cartesian products would benefit from this

[tight star]

Well I have a classic one from topology

[tawny slate]

quotients?

[tight star]

For Cartesian products of topological spaces

[tight star]

Yep these have a universal property too

[tardy ember]

how about ``$(A \times B) \times C \cong A \times (B \times C)$''?

[burnt vessel]

bee [it/its]

[tawny slate]

oh! interesting

[tight star]

In fact like

Proving this is basically the same as

[tardy ember]

they're isomorphic because a map from a set $Z$ into either of them is just maps $Z \to A$, $Z \to B$, $Z \to C$

[burnt vessel]

bee [it/its]

[tight star]

Proving gcd(m, n, k) = gcd(gcd(m, n), k) = gcd(m, gcd(n, k))

It’s basically the same proof

gcd and Cartesian products of sets

Are very different things

But it turns out they “do” similar things

Let you interconvert between 2 arrows and 1

[tardy ember]

yeah the associativity of gcd/lcm, union/intersection, cartesian product of sets, even disjoint union of sets, are all actually special cases of the same theorem

[tight star]

Mhm

[tawny slate]

i have an intuition for this, but the form of the proof is not something im used to, would you be able to write up a very clear explanation of this?

ah wait im confusing myself

i need to first figure out the univ prop of cartesian products

[tight star]

Right

It’s the following

Given sets A and B

You can form the Cartesian product A x B

It has the following universal property

Given some other set Z

Functions from Z -> A x B correspond bijectively to a pair of functions Z -> A, Z -> B

[tawny slate]

ahhhhhhhhhh ok

[tight star]

You can use this to show preimage preserves every set operation you could think of

Union

Intersection

Complement

Set difference

Symmetric difference

There’s one proof that shows these are all stable under preimage

In fact, the same proof shows that any collection of Venn diagram regions is stable under preimage

[tardy ember]

...well given that that's not true i'm now rather curious what proof you're imagining

[tight star]

Wait it’s not true?

[tawny slate]

i was gonna say that it sounds fishy

[tardy ember]

wait no hang on maybe i'm wrong

[tight star]

I’m almost certain it is true

Because I have a nice categorical proof in mind

[tawny slate]

surely there are nonreversible set operations

[tight star]

Give me an example?

[tawny slate]

hmmmmmm

is there really none

[tight star]

Lol

[tardy ember]

ok the counterexample i had in mind doesn't make sense (i was taking the preimage of something that's not actually a function)

[tight star]

Mhm, right

You use the universal property of Cartesian product in the proof

[tawny slate]

ok i can see it now but

still seems pretty abstract

and i definitely dont see any point of bringing in yoneda for that problem

we solved it without it

[tight star]

solved what without it?

Oh, preimage also satisfies its own universal property

Which connects it to direct image

namely, that $A \subseteq f^{-1}(B) \iff f(A) \subseteq B$

[burnt vessel]

Pseudonium

[tawny slate]

why is that a universal property

that doesnt look like combining or splitting of statements

[tight star]

A universal property in general means

“Arrows into your object have a nice alternative description”

Or “arrows out of your object have a nice alternative description”

At least the way I use it

[tawny slate]

ok i guess thats the category theory talking

[tight star]

So, arrows into the preimage have a nice alternative description

As arrows out of the direct image

It’s a kind of “mutual universal property”

You can use this to show direct image preserves unions

[tight star]

Let me know if you do find a counterexample

But uh given that I think I have a proof

I don’t think you will..

im also happy to give the proof if you want

[daring gulch]

Voluntarily removed
Oh THANKS

Why indeed

[daring gulch]

-_-

[tight star]

Is it fair to say that one of the main reasons fractions are hard is that it’s the first time you might meet an equivalence relation?

[modern trench]

https://youtu.be/EMNqJQaf08E?feature=shared

[minor turtle]

not really

i do agree with you that this is one of the first times that an equivalence relation is taught in school, but I think that fractions are just hard because students don't understand that finding sums and differences and products of fractions are extremely connected to that equivalence relation

I am just speaking from my anecdotal evidence here^ I used to tutor students who were behind in algebra 1, and some of them really had a hard time adding fractions, and I found that showing them a picture and having them do practice adding them with the visual aids often worked

[tawdry venture]

what kind of picture? pies? or number line

[minor turtle]

boxes with dividing lines, you can make this as interactive as you want, but I find fractions as literal objects to be nice as well, get scissors out and cut them up with the student yk

or split up chocolate etc

[tawdry venture]

oh right ok so sorta like number blocks

[minor turtle]

yeah i think if one had disposable number blocks that you could cut up or found an appropriate way to separate them to think about fractions that could work. I am partial to chocolate because I really like chocolate, and students find it fun

[tawdry venture]

🍫

[tight star]

Hmm, so without the picture, what problems arise when adding fractions?

[tawny slate]

i posted an answer that you responded to previously, and based on what i wrote i disagree, because i think its due to the fact that even say adding/subtracting fractions requires a huge number of technical steps that require both conceptual understanding and muscle memory (memorizing times tables)

but even if your answer does get to the heart of labeling the most difficult aspect of it as an "equivalence relation", i think especially when referring to a more basic concept like fractions, it helps more to describe what is happening in the brain than the math itself. none of the students learning fractions will understand what an "equivalence relation" is, so it is not helpful in describing what is hard about them. we dont say that students struggle with counting because it's the first time they encounter recursion on omega

trying to describe whats going on in the students' heads helps you more clearly understand what is it about an equivalence relation that you think is so difficult for students to grasp

i also dont think it is the first time they encounter equivalence relations, pretty sure equivalence relations start as early as addition, as (a,b) -> a+b, dont see what makes division special here

[tight star]

Well it’s like

Addition can also be viewed as a function

Fractions feel more like an actual equivalence relation

[tight star]

Oh yeah, I wouldn’t start by saying “ok everyone you’ve gotta be careful about the equivalence relation we’re using”

It’s more for discussing amongst people who teach fractions

[cosmic ibex]

At the elementary school level, the rationals should not be presented as something we create with equivalence relations and whatnot. They're just points on the number line that are already there, which we find a useful (?) way to speak about as the result of certain division operations. In that view it shouldn't be more mysterious that 4/6 = 6/9 than it is that 2+3 is also 4+1.

[tight star]

Right, so you’d recommend thinking of fractions in terms of lengths?

[cosmic ibex]

Yeah, I think that's the primary intuition.

[tight star]

Got it

[vocal phoenix]

Yeah, and I don't think most people teach fractions in school in terms of equivalence relations, but I think it's a valid point the equivalence relation is still there even if we don't talk about it explicitly.

So different symbols will represent the same fraction, such as 3/6 and 1/2

I'm not exactly familiar with the order of elementary math education, but this might be the first time students encounter this kind of situation.

[tight star]

Yeah whereas with integers, there’s always a canonical base-10 way to write them down

[modern trench]

Do they usually teach division before fractions?

[tight star]

And you don’t really need to convert to a different representation before adding, subtracting or multiplying

Whereas for fractions, sure there’s a canonical representation where the denominator is positive and coprime to the numerator

But you can’t just use this representation for doing fraction arithmetic

[tawdry venture]

yeah, and then it's division with remainder.

[cosmic ibex]

Hmm I think I dimly remember when I was in primary school there was a sequence in the textbook entitled "number names" which tried to teach us that "2+3" could itself be a name for the number that "5" is also a name for.
I suppose that was an attempt to prepare us for the situation with fractions where the only way to speak about some numbers is to state an arithmetic operation that produces it.

[vocal phoenix]

Sounds clever, because the general thing that's needed here is being prepared to express the same thing in various ways.

Whether or not that's how it's done will probably hugely depend on country and generation

The difficulty in distinguishing between a number and its representation is very common, as evidenced by the recurring "is 0.999.... the same as 1" discussions.

Also it only just occurred to me, after all those years, that the Polish word for "fraction" has exactly the same etymology as in English ("ułamek" from "łamać", which is "to fracture")

Neat

[cosmic ibex]

It helps that even among the naturals there are usually numbers that you can only speak about by describing a canonical computation that computes them. E.g. if you add nine to twelve and want to speak aloud the number you get in German, you'll find yourself saying, quite literally, one+twenty.

[tawny slate]

really think this should be emphasized, because i think this is not only useful for fractions, but also expressions in general

i cant even begin to describe how many times i got students who were able to distribute

c(a+b)

but not

(a+b)(a+b)

just because they couldn't yet understand to treat (a+b) as a "name" for c

really really excellent tip

[tight star]

This is fascinating to me

I think there was a convo here a while ago about how “substitution” seems to be a difficult concept to grasp

It’s actually related to a few things

Difficulties with substitution make function evaluation hard

It makes what you said about distributivity hard

And it’s also related to that lemma i talked about recently where all you needed to do was substitute d = m, but the student was unsure whether this was allowed

I think maybe I’ve heard this come up with the injectivity definition?

$\forall x, y, f(x) = f(y) \implies x = y$

[burnt vessel]

Pseudonium

[tight star]

Where students are confused because “if x and y are the same, why did we use different letters for them”

So perhaps lots of this comes back to issues to do with variables and substitution?

[noble hare]

Yeah I find that students really struggle with "dummy variables" from theorems. E.g. if you had a theorem involving two arbitrary functions f,g, and then you were given a problem to solve using that theorem but the functions in that problem are also called f,g and perhaps not playing the same roles as the f,g in the theorem. I've seen many students implode when faced with this

[tight star]

I see i see

Yeah i had an instance recnelet where a student here was confused about dummy variables in limits

Say in $\lim_{\Delta x \to 0}$

[burnt vessel]

Pseudonium

[tight star]

Where they thought that this meant delta x was “infinitely small”

I said no, it’s always finite

[modern trench]

Petition to teach first order sentence manipulation in high school

[tight star]

But their confusion was “if it’s finite, you should be able to quantify it. So how small is it?”

I’ve found that explicitly expressing things in terms of functions often helps eliminate confusions like these

To a quite surprising degree

Which comes back to my general suggestion to just use functions more as a tool for expressing ideas when teaching

[long pelican]

I’ve always thought working with quantification (universal and existential) was thrown under the bus in math education for no good reason

Well I know the reason now; it's partly because it's unprecedented for appearing on a standard and partly because it would take creativity and original thought to figure out how to test it, whereas it doesn't take creativity and original thought to test a mechanical skill

And partly because, being precedented, there would likely not be consensus on the proposed ways to test it

[tight star]

What do you mean by this?

[long pelican]

Universal quantification = for all [dummy variable]
Existential quantification = there exists [dummy variable]

[tight star]

oh I know what universal and existential are lol

i mean, in what sense do you think they’re thrown under the bus?

and what would you prefer?

[long pelican]

It's never taught, only left implicit and for the (top x% of) students to infer, if lucky

[tight star]

it’s never taught?

[long pelican]

Nope

Depends on country maybe?

[tight star]

hm, quite possibly

how would you teach it?

[long pelican]

Implementing a change like that would take some thought but

teach first order sentence manipulation in high school
might be a place to start

[tight star]

what is that?

[long pelican]

I won't try to speak for @modern trench

[tight star]

mhm

[long pelican]

So pinging him

[tight star]

but - say you were tutoring someone

how would you explain universal and existential to them?

[long pelican]

Through examples, probably

[tight star]

right..

[long pelican]

Can't really say more, the path depends on how quickly they pick it up and where they get confused

Depending on path you'd move onto dissecting an example, showing more complex examples, playing games, or whatnot

[tight star]

i wonder if viewing it as a function could help too

[long pelican]

Since the central new idea for the student to grasp here is dummy variables in language, what are you proposing to view as a function?

[tight star]

So

For both existentials and universals, you have to start with the idea of “proposition with a free variable”, right?

[long pelican]

Yeah

[tight star]

Once you have that, you can then bind the variable to a quantifier to make a new proposition

[long pelican]

So analogous to union and intersection, or product and sum

Over a variable

[tight star]

So the idea is - a proposition with a free variable is a function $X \to {0, 1}$

[burnt vessel]

Pseudonium

[tight star]

Where X is the collection that the proposition ranges over

You can substitute an input, and you get an output of “true” or “false”

Then, just type-wise, the idea of both existentials and universals is

A function $(X \to {0, 1}) \to {0, 1}$

[burnt vessel]

Pseudonium

[tight star]

The input you have to give is a proposition with a free variable

And the output is a truth value

I’ve found a few times that replacing dummy variables with “take a function as input” helps more than you’d think

What universal does is look at the function and “AND” all the outputs

Whereas existentials “OR”s all the outputs

[long pelican]

It's a nice proposal. Textbook writers should run A/B tests for it instead of doing nothing

[tight star]

For example, this is how i fixed the student’s confusion with limits

Instead of using dummy variables, i phrased it in terms of taking a function as input

This is the kind of thing i mean with using functions as a tool to express ideas

[tawny slate]

curious, what do you by it requires creativity and original thought to figure out how to test it? isnt it pretty easy to test?

[modern trench]

(your opinion is probably more valid than mine since I have not taught)

I think first order sentence manipulation is pretty much mechanical though

[tight star]

What is this

[long pelican]

Maybe it's easy for us but at the very least you can't use old test items as a template, for example

[modern trench]

Like,

Given that [1st order sentence], [1st order sentence], and [1st order sentence] are true, which of the following must be true?

[cosmic ibex]

I'm doubtful it would do much good to teach first-order manipulations as a mechanical skill without also stressing the meanings of the sentences.

[tight star]

Ok what is a 1st order sentence

[modern trench]

∀∃∧∨~φ()

[tight star]

Huh..

[tardy ember]

so for instance $\forall x (P(x) \to Q(x))$ is a first-order sentence

[burnt vessel]

bee [it/its]

[tight star]

What makes it first order

[modern trench]

Possibly with basic number theory or sets mixed in

[tardy ember]

the fact that there's no quantifiers over predicates or functions

that would be second-order

[tight star]

What’s a predicate..

[modern trench]

φ

[tardy ember]

for instance $\forall P \exists x : P(x)$ is second-order because of the $\forall P$

[burnt vessel]

bee [it/its]

[tight star]

Ok

I kinda get it..

Though it’s difficult to see how this is relevant

[tardy ember]

well because, uh, $\forall$ and $\exists$...?

[burnt vessel]

bee [it/its]

[tight star]

But why does first order matter

[modern trench]

Because it helps a lot with logical thinking

[cosmic ibex]

I don't think anyone is proposing to teach the term "first-order", since students are not going to see higher orders anyway.

[tight star]

Isn’t this not first order by your definition

[modern trench]

Yes this is one problem that teaching first order logic will solve

[tight star]

Huh…

[tardy ember]

...well almost all actual mathematical reasoning is done in first-order logic, given that afaik there isn't any sensible proof system for second-order logic that isn't actually first-order logic

[tight star]

It’s literally quantifying over functions

How is it first order

[modern trench]

Functions are sets not predicates

[tardy ember]

the functions are the objects in the domain

[tight star]

I’m just getting more and more confused..

[tardy ember]

ZFC is a first-order theory where the objects happen to be sets

[tight star]

And you say this is supposed to help with logical thinking…?

[cosmic ibex]

Nobody is saying that knowing the meaning of the term "first order" will help anyone with logical thinking.

[tight star]

Then why are we discussing first order at all

Why not just like

Logic

[modern trench]

Because logic is much much broader

[tight star]

I guess I just don’t get why first order is special

[modern trench]

I'm specifying first order logic as the logic you need to do math

[cosmic ibex]

Because someone thought they were speaking to an audience who also knows higher-order logic and they wanted to reassure them that the higher order logic was not what they were proposing to teach in school.

[tight star]

So what about this is not math

[modern trench]

The logic taught in any first year proofs and logic course is called first order logic

[tight star]

I don’t get that

[modern trench]

I'm just trying to refer to that by name

[tight star]

You quantify over functions all the time

[noble hare]

am i being adressed or are you making a point to someone

[tight star]

Oh sorry i may have accidentally pinged you

[tardy ember]

that is maths, being done in a first-order theory like ZFC, where there are objects that we happen to sometimes call "functions"

what's actually happening there is just quantifying over objects

[tight star]

“For all continuous functions f and g, f + g is continuous”

[tardy ember]

and then the theory also has a predicate named \in

[modern trench]

Yes, and I'm saying we should teach quantifiers properly

[tight star]

That’s quantifying over functions

[cosmic ibex]

You. Are. Misundestanding. Their. Purpose. With. Saying. "First-order".

[tight star]

To be clear this is the definition of first-order I am using

Because it’s the one you gave me

[cosmic ibex]

They were just trying to say they were not proposing to teach the other hairy stuff that appears in techincal formal logic courses.

[tardy ember]

from the perspective of the logic, these things aren't functions, they're just objects

[tight star]

So functions aren’t functions???

[cosmic ibex]

You're not even listening, are you?

[modern trench]

Functions in the usual sense is not predicate in the logic sense

[tardy ember]

ok let me give a different definition

[tight star]

It seems like you’re trying to tell me that functions aren’t functions

[tardy ember]

first-order logic is logic where the only type of quantifier is over objects

[tight star]

So I don’t know what you mean by function

[tight star]

Ok

So what counts as an object

A predicate isn’t an object…?

[modern trench]

In ZFC (which is all of math), the objects are sets

So functions are objects

[tight star]

Ok..

[tardy ember]

first-order logic doesn't have a notion of "a thing not being an object", the objects are all that exists

[tight star]

Then why is it not all logic

[modern trench]

This system is called first order logic

[cosmic ibex]

Because there's also a formalization of logic that treats functions and sets as distinct from obejcts.

[tight star]

Ok

[modern trench]

There are many other logics

[cosmic ibex]

That is the formalization Arki was explicitly proposing not to teach.

[tight star]

This is a little difficult for me to grasp…

[tight star]

Then why’d they bring it up

[cosmic ibex]

Sigh.

[tardy ember]

they didn't, they brought up first-order logic

as the thing they were proposing to teach

[tight star]

But like why make the distinction that it’s first-order

[cosmic ibex]

Sigh.

[tight star]

Sigh?

[modern trench]

It's the name of the system with ∀∃∧∨~φ, with excluded middle, modus ponens, etc

[noble hare]

to indicate they are not proposing teaching higher order logic?

[tight star]

Uh

Right

[tardy ember]

also "first-order logic" is a useful technical term even if you're not trying to distinguish it from higher-order logic

[tight star]

I thought that would be obvious

[modern trench]

There are many different logics, and first order logic is one of them

[tight star]

Why would you try to teach higher-order logic

…whatever that is

[modern trench]

I'm just referring to first order logic by name

Nothing else

[cosmic ibex]

Nobody is proposing teaching higher-order logic in school.

[tight star]

I guess I still don’t quite understand what the adjective “first-order” is doing there

[tardy ember]

like something like modal logic isn't exactly "higher-order", but it also isn't what the name "first-order logic" refers to

[cosmic ibex]

SIGH

[tight star]

But that’s fine im not a logician

[tardy ember]

it isn't

the entire name "first-order logic" is doing something

that's not the adjective "first-order" applied to the word "logic", that is a single morpheme, that sequence of words is itself a technical term

[tight star]

Huh

Oh

[cosmic ibex]

Why do you keep complaing about that word after you have been told a dozen times now what it means in the context?

[tight star]

That’s so weird..

[noble hare]

just like someone might say we should teach more "naive set theory" in schools, it might be obvious to some that they dont mean ZFC set theory if they dropped the naive, but its a weird point to be stuck on that just because its "obvious" to some the adjective should be left out

[tight star]

Cause I thought it was an adjective

Whereas what they were describing just sounded like

Logic

I’m not even really sure what you mean by “a logic”

[tardy ember]

"logic" is used to refer to so many different things, and "first-order logic" is a useful way to refer to actually a very tiny slice of the things that people call "logic"

[cosmic ibex]

They were specifying which of the many kinds of logic they were proposing to teach!

[tardy ember]

nobody is, that's why we didn't just say "logic"

[tight star]

There are many kinds of logic?

[modern trench]

The average proofs and logic class teaches first order logic and just calls it logic

[tardy ember]

there are so many things that get called logic and there isn't really any formalised common feature between them afaik

it's as vague as a word like "number"

[tight star]

I think maybe you were assuming I knew a lot more than I actually did

And your use of technical terms just confused me

[modern trench]

Ok anyways, I'm proposing we teach manipulation of symbols ∀∃∧∨~φ in high school

[tight star]

Right

Didn’t we say that already…?

[tardy ember]

how about $\to$ as well

[burnt vessel]

bee [it/its]

[tight star]

Or hmm, could you be more specific with what you mean by “manipulation”?

And - maybe try to avoid first-order..

[cosmic ibex]

Yes, but you have been spending an inordinate amount of energy complaining about the words it was said with.
If not even rephrasing it to not use the words you object to will satisfy you then what on earth will?

[modern trench]

Know what the sentence intuitively means, and whether or not there is an implication between two sentences

[tight star]

Lol why are you so annoyed

[tight star]

An example…?

Sorry what you’re saying is just

A little abstract for me to grasp

[cosmic ibex]

You're the one who has been spending half an hour on throwing a tantrum about "first-order", and you call me annoyed?

[tight star]

What counts as a “manipulation”?

I’m not throwing a tantrum?

I’m just asking what first-order meant

[modern trench]

.

[noble hare]

would you propose simple deductions?

[tardy ember]

yes and you spent half an hour asking it

[tight star]

Ok could you give an example of this?

[tight star]

Cause I was confused for half an hour?

[cosmic ibex]

And ignoring all the explanations you get?

[tight star]

I wasn’t ignoring?

The issue was that you said you’re not allowed to quantify over functions

[noble hare]

ah i just saw your own msg you replied to

[tight star]

But your use of function was different to my understanding of the term

And to be honest im still not sure what you mean by the term function

[modern trench]

"Does ∀x∃y.φ(x,y) imply ~∀x∃y.~φ(x,y)?"

[tight star]

Ok

So questions like that, you’d want to teach in school?

[modern trench]

Yes, perhaps at an appropriate level of difficulty

[tight star]

I see

Sure that makes sense

[modern trench]

I don't want people memorizing names like modus tollens or disjunctive syllogism though

[tight star]

I… hmm

Back to words I don’t know

But that’s ok

[tardy ember]

(i don't
i don't have a definition that classifies all things into "function" or "not function", there is a type of thing named "a quantifier over functions", which obeys different logical rules to "a quantifier over objects", and which does not exist in first-order logic)

[tight star]

I think I remember modus ponens

[modern trench]

Otherwise it's just circle geometry all over again

[tight star]

Ok then maybe you can see why I was confused

If you can’t even tell me what the words you’re using mean

[cosmic ibex]

...

[tardy ember]

modus tollens: given "if p then q" and "not q" infer "not p"
disjunctive syllogism: given "p or q" and "not p" infer "q"

[tight star]

Oh ok cool

[tardy ember]

it doesn't really matter what exactly they are though, the point is that they're technical words for extremely obvious principles and memorising them is not useful at all for actual reasoning

[tight star]

Right

Sure, that tracks

[tardy ember]

in fact i had to look up what the second one was, which really illustrates how pointless memorising it would be

[tight star]

So I guess… what you’re saying is that in order to teach quantifiers, we have to make questions involving quantifiers…?

[modern trench]

We have to teach how to manipulate them at least

[cosmic ibex]

In order to each anything you need to have practice questions involving it?

[tight star]

Right, sure

Yeah

But what im asking is whether you mean anything specific

[tight star]

Cause to me this seems obvious

But it seemed like maybe you had something more specific in mind

[modern trench]

Yes, that is to teach quantifiers at all

[tight star]

Right

[modern trench]

In high school

[tardy ember]

given that this took us half an hour i'm not convinced it was obvious actually? but alright sure

[tight star]

I think we can agree on that..

[tight star]

It seems more like a failure of communication between us

I think the statement “in order to teach something you should ask questions about it” is obvious

But it wasn’t clear to me that’s what you were saying

Mostly cause I didn’t (and probably still don’t) understand what first-order logic is

So I got hung up on that

Anyway I guess I should not look into logic lol

Seems like it’d just confuse me

[noble hare]

Too category theory pilled

[modern trench]

Probably the opposite

[tight star]

Does category theory make you worse at logic?

[tight star]

I mean, based on your explanations at least

Perhaps there are others which would work better for me

[noble hare]

Nah was in jest

[tight star]

I see

clearly I have some communication issues with members of this server..

Anyway

So perhaps we should teach quantifiers and their manipulation in school

Seems to be the upshot

[noble hare]

Yeah I mean the current system in the UK (atleast from my uni experience) was to bombard students with naive set theory in every module in first semester and prefix the first analysis course with baby first order logic (truth tables and whatnot)

[cosmic ibex]

I'm not sure we really ought to teach symbolic quantifiers in school, but perhaps more focus on navigating their natural-language equivalents might help some.

[tight star]

Hmm…

I remember going to the logic and set theory course at my uni in 3rd year

And they mentioned truth tables there

Otherwise I’m not actually sure they mentioned truth tables at uni explicitly

It was kinda just assumed knowledge?

[noble hare]

Assumed from where?

[tight star]

Like school

I also remember that course did

Propositional logic vs predicate logic

But tbh I don’t remember the difference..

Though maybe that’s an example of what you mean of like

Different kinds of logic

[cosmic ibex]

Predicate logic has quantifiers, propositional logic doesn't.

[tight star]

See the issue is now like

I have an idea of what quantifier means

But im not sure if the way you’re using it is the same

So maybe im not sure what you mean by quantifier

[cosmic ibex]

Quantifiers are the \forall and \exists symbols.

[tight star]

After all I thought I knew what functions were but apparently there’s a different definition

[tight star]

Ok

And… ok so I guess predicate logic is the only kind that could be

Uh, first or higher order

[cosmic ibex]

Correct.

[tight star]

Since the other one doesn’t even involve quantifiers…

I don’t remember what order we did

To me first order just means like

First term in Taylor series

[cosmic ibex]

If you didn't learn a distinction, then without a doubt the predicate logic you learned was first-order.

[tight star]

How do you know that

Maybe I’d have to see an example of what higher-order logic even means..

[cosmic ibex]

Because I know higher-order logic, and it makes very little sense to teach it without explicitly contrasting it to the usual first-order logic.

[tight star]

Right

I guess I’ll probably never know what higher-order logic is

But apparently I don’t need to…?

[austere delta]

First order = use quantifiers on variables
Second order = use quantifiers for sets
Third order = use quantifiers for sets of sets
Etc

[cosmic ibex]

The same way I can say "if you remember you learned some geometry, but you didn't learn of any distinction between solid geometry and plane geometry, then it was without a doubt plane geometry you learned".

[austere delta]

So for example the least upper bound property is a second order property because it is a statement about "all sets of real numbers"

[cosmic ibex]

It's confusing Pseudonium that ZFC can quantify over sets and sets of sets just fine using just first-order logic.

(Which is not an unreasonable confusion, by the way).

[tight star]

Yeah

[austere delta]

Yeah, ZFC has its own little internal idea of what a set is

[tight star]

There’s an idea of sets outside ZFC…?

[tight star]

I see I see

[austere delta]

I mean you have a model for ZFC.

And if you want to reason about that model you kinda have to be outside it

[tight star]

Maybe we should move this to foundations

[austere delta]

Probably. Tropos would probably also do a way better job at explaining than me

[cosmic ibex]

(moved to #foundations)

[tight star]

Ohhhhh

I remember hearing about second-order peano stuff!

[tawny slate]

could you elaborate? i dont follow

[long pelican]

Since there are no old test items on quantification they can base new questions off of, they will have to create some questions from scratch

[tardy ember]

we don't have any existing test questions about it

so you have to do at least something original, which is harder than copying what you were doing last year

[tawny slate]

ah i see

in that case though, is it really that difficult to make test questions for that?

[long pelican]

Like I said, not for us

[tawny slate]

ahhhh

i actually had a massive pet project i was going to work on where i would create through both manual curation and custom tailored scripts and meta-scripts to collect and/or generate all kinds of math questions of all kinds

gonna at least add this topic to my notes in there

[jovial sinew]

So what are your opinions on skipping pre-alg? Just learned that someone I know is planning to do so

[tawny slate]

imo that is a highly contextual question

it really depends on an individual basis

math, imo, is more of a "chain of prerequisites" than most other school subjects, so falling behind is pretty detrimental

so as long as they know the basics, should be fine

in general, however, i think pre-alg is more acceptable to skip than say geometry or pre-calc

[jovial sinew]

Ok.

[spice matrix]

quite the opposite

categorical logic is really interesting. Moreover there’s curry howard lambek.

[tight star]

Oh wait yeah I remember hearing about categorical logic

[wispy slate]

why to not teach abstract definitions first (for example, some abstract algebra definitions, or more general theories) ?
and then go for examples by teaching the basic courses
namely to teach first real analisis, and then , with the foundation of real analysis doing more calculus ?(problem solving)

(assuming students have a good foundation in mathematics , and mathematical maturity , like math olympiad students)

[tight star]

Uh well i definitely know I would’ve struggled if it was done this way

I much prefer going from examples to abstraction than the reverse

[vocal phoenix]

Because it tends not to work that well

[reef nacelle]

Why would you? I think learning things mostly as it has been historically evolved is alot more rewarding

Than going the other direction

Also the same problem I have for learning proof writing or reading them (like embrace the mistakes instead of hiding them)

[tight star]

Hmm is this really to do with history tho

[reef nacelle]

No not at all, but I mean mostly how a certain concept evolved is better viewed sometimes historically imo

You get a sense of what motivated the people to study it

And sometimes even a better perspective too

[tight star]

Right that’s fair

[wispy slate]

I am thinking in this analogy,
suppose you are going to a safari and the definition of a fierce is given, then , when you are on safari, you see a tiger , and you are able to deduce is a fierce, then if you want to know more about a tiger you study the tigers,
then you continue going and you find a lion, know you know a lion is also a fierce, and so on and so on.
so , at the end , it doesn't mean not meeting lions or not meeting tigers, but once you meet, you dont memorize them as lions, or tigers, you memorize them as fierces, then if needed you make the distinction between a tiger , lion ,etc

[tight star]

Hm

Do you have a concrete example of how you’d do this for math?

[wispy slate]

I has been studying abstract algebra before university , so when I encountered function composition in a class, I realize in certain cases it is just a binary operation, so in this analogy fierce equals binary operation, lion equals composition

[tight star]

Well…

It’s only a partial binary operation

[wispy slate]

it is for some cases

[tight star]

Monoids!

Well I would not recommend teaching binary operations before function composition

[wispy slate]

why not?

[tight star]

Cause the former is too abstract

[wispy slate]

but what's the problem of being abstract?

[reef nacelle]

In the same sense learning the generalized stokes theorem is too abstract if you haven’t learnt the prior stuff

If anything you’re losing stuff by doing this

[wispy slate]

why losing?

[reef nacelle]

You might miss subtleties that for most people is only realized with small steps rather than taking big generalizing steps so to speak

As you may know, often when solving a problem, it’s easier to handle a smaller but easier problem

[tight star]

It’s hard

[reef nacelle]

I think this is analogous to this

[tardy ember]

yeah abstraction can be helpful but i think you do also need to see concrete examples

[wispy slate]

yeah , but I am don't mean to skip less general theories, such like skiping what is being generalized , but , instead a proccess of
definition --> examples --> generalization

[tardy ember]

like you can often get intuition from playing with actual objects that you wouldn't come up with just by staring at a definition

[tight star]

Seeing definitions before examples often makes them feel unmotivated

[wispy slate]

instead of
examples --> generalization --> definition

[tight star]

At least for me

Like “why should I care about this definition”

So in your case, id be “ok why should I care about binary operations”

It just feels plucked from thin air

[wispy slate]

but if the students does know the teaching strategy, I mean, trusts that the work al ready done may be important

[tight star]

Welllll

[reef nacelle]

Ironically I would say this process of dealing with it more concretely first actually helps one to later on start becoming more comfortable with not having intuition of abstract objects, I guess this is common for grad students

[tight star]

Didn’t work for me

[tardy ember]

i think it makes sense to put the definition "first" in the sense of like, "higher up the page", just so you have available the language you need when you then immediately start giving examples

[tight star]

In part this is why I switched to physics

Like I totally trust my professors knew what they were talking about

Just didn’t mean I knew what they were talking about

[wispy slate]

and also , despite being abstract, its still concrete, more concrete than intuition for example

[tight star]

Uh

No

Intuition feels very concrete to me

[tardy ember]

if you try to give examples of binary operations without even having said what a binary operation is then it's going to be less clear what exactly the commonality between them you're trying to highlight actually is

[tardy ember]

i think "how concrete is intuition" is a type error

[tight star]

Sure, I get this

[tardy ember]

you can have intuitions about anything

[tight star]

But I feel like you would’ve seen lots of examples of binary operations beforehand

And then it’s just about giving a name to a pre-existing concept

Like you’ll at least have seen addition and multiplication

And subtraction

Sometimes the issue is not parsing the definition but understanding the definition

I am very familiar with this from trying to learn category theory

[tardy ember]

well that's what the examples are for

you parse the definition enough to know what type of object the examples are going to be, then look at examples to understand what the definition actually means

[tight star]

Right…

[tardy ember]

or, well, depends on the area

[tight star]

I guess I just prefer examples first

[tardy ember]

sometimes the thing you want to do with a definition is actually just prove some properties about it

[tight star]

But I know there are some people who genuinely prefer abstraction first

I met a bunch in my category theory class

[wispy slate]

imagine if I tell you a function is a machine , for example , or thinking it like a metaphor, like some physicist do,
vs telling you that a function is a type of set.
one is more intuitive and the other is more abstract, but functions as sets are more concrete , at least i think they are

[tight star]

Lol I prefer the first

I am a physicist after all

[reef nacelle]

I think they’ve secretly grown to have this attitude thanks to being very concrete and so fourth early on, and built the comfort to think like this(?)

[tardy ember]

some objects are extremely difficult or impossible to give an example of in any useful sense

[tight star]

I have no idea how their brains work

[reef nacelle]

Damn

[tight star]

Maybe it’s just too different to how mine works

[wispy slate]

why ? I mean which object?

[tardy ember]

well for instance

models of ZFC

[tight star]

Lol

[reef nacelle]

Well I guess it’s just one aspect of mathematical maturity tbh

[tight star]

I guess I never reached this then

[tardy ember]

inaccessible cardinals (if any exist then you can define a canonical one, the first one, but this does nothing to help with building intuition)

free ultrafilters on N

[reef nacelle]

Hm what if you’re overestimating this ability?

[tardy ember]

in general a lot of set theory is objects that exist but that you can't exactly write down

[tight star]

Wdym

[reef nacelle]

Like, I don’t think there’s a level you reach so to speak, but only grow to get better at it if I can put it that way

[wispy slate]

but you can define them?

[tight star]

Well I always prefer concrete to abstract

[wispy slate]

sorry still havent studied set theory in so deep

[tight star]

So in that sense I’ve made no progress

[tardy ember]

yes

well in the sense that you can define what "an ultrafilter on N" is

[wispy slate]

then you have something concrete

[tardy ember]

you can't necessarily give a definition of any particular one

[wispy slate]

the definition

[tardy ember]

that's a very strange definition of "concrete"

[wispy slate]

that object is exactly what the definition is talking about

[tardy ember]

has any mathematician ever done anything that wasn't concrete, according to you?

[tight star]

Definitions don’t really feel concrete to me

[wispy slate]

I would say metaphors aren't concrete, but i dont konw if metaphors are really made by mathematicians , (instead of standard definitions)

[tardy ember]

also like, being able to give one particular example of a thing doesn't mean that the way it's specified is useful for anything whatsoever

[reef nacelle]

Maybe our ideas of concrete and abstract are different. Casually using function terminology for people not used to them or even category theory lemmas on problems not meant for them certainly seems to highlight your skills; unless your overestimating the abilities of the ones you’re helping. Since to me atleast that’s a flavour of being abstract

[tardy ember]

for instance in L, there is a well-ordering of the entire universe, so for any type of object that exists, there is a first object of that type, by any particular choice of global well-ordering (there are several possible choices that behave quite similarly but are technically different)

[tight star]

I don’t think that counts as abstract though

[tardy ember]

but afaik there isn't anything useful you can say about, like, the $<_L$-first bijection between $\mathbb{R}$ and $\aleph_1$

[burnt vessel]

bee [it/its]

[tardy ember]

it's a bijection between R and aleph 1, ...and that's about it

or that's not everything you can know but like, if you want to work out what value this bijection has at 0, i would be surprised if the answer is actually provable in ZFC, and also it would depend on how you defined the reals

this is not a concrete example in any sense that's useful for human intuition

[wispy slate]

isn't that a little extreme example?

[tardy ember]

this is pretty normal for set theory

there are a lot of types of object that, if you're in L, you can technically "give an example of" but in a way that leaves it completely unclear what any of its properties are, and in some other universes there are just no definable examples ("definable" in the formal logic sense of "specified by a formula" - so essentially you can't describe them)

[wispy slate]

but that's the reason why you can't just rely on definitions

but I was trying to say , that if you know what a possible definition could be.
then when you see it (in action) , maybe you are able to recognize it. (find examples)
so in terms of time, you can understand better a definition , just for knowing it before

[tardy ember]

it's the reason why you have to, sometimes, because sometimes examples don't usefully exist

[tight star]

Like I think I’m a little uncomfortable with the assertion that mathematical maturity = more abstraction

I don’t think abstraction should just be pursued for its own sake

Or that abstract = better

[reef nacelle]

It’s just one aspect, it’s a lot more than that of course

[vocal phoenix]

I agree, mathematical maturity should mean you're more comfortable with abstraction and see where it's valuable, but abstraction isn't a goal I consider intrinsically worthy.

[tight star]

Right

I guess… I am a little more comfortable with abstraction than before

But it’s still easy for me to get overwhelmed by it

See, uh, people trying to explain higher-order logic to me..

[tardy ember]

like some things technically arguably "construct" the object that they claim to be proving the existence of, but with the resulting object being so complicated it's not actually useful for understanding the definition, except in the sense that any use of a definition helps with understanding it

[tight star]

Adjoint functor theorem comes to mind

[reef nacelle]

I think human minds seem abstract in nature, the fact that we have something such as mathematics is amazing, and I think it’s due to the idea of abstraction in some sense

[tight star]

For me, it’s much more about the ability to choose when you want to be abstract and when you want to be concrete

And the ability to translate between them

Rather than saying “X perspective is the correct one”

[wispy slate]

but in the example you give, why to define something you dont know anything about , and you are not even sure if you are going to find it?

its like to define an animal which you are not going to see in the safari? right?

[modern trench]

Surely part of the goal is to investigate an interesting definition

[tight star]

But how do you know the definition is interesting

[tardy ember]

well using things can still be useful even if you can never write down an example of one

[modern trench]

Nonprincipal ultrafilters are very interesting

[tight star]

Because…?

[tardy ember]

it's like defining an animal that you will never see in the safari but that is still there and interacts with other animals

[reef nacelle]

Historically the definitions usually come at the very end of some interesting particular concept or idea

[modern trench]

Ultraproducts, stone cech, etc

[wispy slate]

then why you say it isnt concrete , for example that definition

[tardy ember]

even if you're only interested in the other animals, it's still useful to be able to predict how they respond to an animal that happens to be invisible

[tight star]

It is?

[modern trench]

Definition and explicit construction are two different things

[tardy ember]

because i'm using the normal definition of "concrete" that's actually useful, instead of the definition by which all of mathematics is concrete

[wispy slate]

ohh

[modern trench]

One can define the reals as "the unique complete ordered field"

[tight star]

Right

I never really liked that definition..

[modern trench]

But constructing (via Cauchy sequences / Dedekind cuts) it is another matter

[tight star]

Without a construction, it’s not clear to me such a thing exists

[modern trench]

Another example might be real numbers that can't be described

[tight star]

Yeah though then you have to be careful

I keep hearing about these models where every real is describable

[modern trench]

[wispy slate]

then , if there are objects which is hard to find examples, what different makes to begin with abstraction or not?

[tight star]

Beginning with abstraction feels unmotivated

[wispy slate]

(without skipping examples , same content , different order)

[tight star]

Is a big thing for me

[modern trench]

We can prove objects exist and deduce properties about them without having to construct them explicitly

[tardy ember]

...well you have to begin with abstraction, because, what else is there? you can't give examples if that's impossible??

[tight star]

I mean in a sense this is already how it’s done in a math degree

Definition theorem proof definition theorem proof definition…

I don’t particularly like this way

But it’s the standard way

[modern trench]

That's another matter though

[wispy slate]

but you doesn't beggin by mathematical analysis , then calculus, rather you do calculus then analysis

[tight star]

Otherwise why would you care about analysis

[wispy slate]

cause you priori know that it gives a better structure for calculus (or at least you trust)

[tight star]

How do I know that

If I don’t even know calculus

I think… maybe I have issue just trusting things blindly

I thought that’s not what you have to do in math

In math you can actually see why things are true

[tardy ember]

well sometimes you have to trust something for some time

like on a smaller scale

[tight star]

The scale of a whole course in analysis?

[tardy ember]

like on a smaller scale
sometimes when you read a proof it just starts doing something and you have no idea how it's relevant to the statement being proven

but if you just trust that it's going somewhere, you read through the proof and then you can see what it was all for

[tight star]

I guess..

[spice matrix]

I think sometimes it’s reasonable or possible to motivate an abstract definition through pure theory.

But in general this is not possible. Often definitions require elaborate constructions that do not build on previous theory motivations in any obvious way. Instead they are the abstraction because they work to describe all the examples we have in mind and not really for any other obvious reason.

Sometimes we can rationalize an “obvious” theoretical reason after the fact in an elegant manner. But the effectiveness of this is debatable, and by sheer fact of being afterwards is often ad hoc.

So very often, it is really good to start from the examples in mind then abstract away later.

[tight star]

I don’t tend to like those proofs much though

[wispy slate]

no , but i dont mean a big change or something like that, but instead, for example knowing from the beggining of calculus , that lebesgue integration is a thing, idk

[tight star]

Oh gosh

Constructing the lebesgue measure is awful

Riemann integral is so much simpler

[wind ginkgo]

I think both are hard.

[tight star]

Yeah I agree but

I think it’s clear lebesgue is harder

[vocal phoenix]

It's simpler in principle, but so much more tedious to work with.

[tight star]

Yeah I don’t disagree

I just know that I would not have understood integration

If lebesgue was done first

[spice matrix]

In hindsight, I like the lebesgue measure better tbh. I’d start with riemann integral first since it’s simpler.

But I’d only do it for singlevariable then move on to lebesgue later.

[modern trench]

Definition of Lebesgue measure:
The completion of the measure generated by μ([a,b]) = b-a

[tight star]

Yeah I don’t know

What that means

[modern trench]

The idea is "simple" in the same sense as this

[tight star]

Yeah I don’t like that definition of the reals either

[spice matrix]

Lebesgue measure permits a very satisfying theoretical description once you have the concepts in place.

The unique translation invariant measure on R^n which evaluates to 1 on the unit cube.

[tight star]

Sure but I don’t know such a thing exists

[spice matrix]

Yeah that’s why we have to construct it.

[modern trench]

The completion of that probably?

[tight star]

Exactly

And the construction is awful

At least from what I’ve heard

[modern trench]

Idk who you heard it from

[tight star]

Some friends who did measure theory

[spice matrix]

I don’t think the outer measure construction is that awful. It’s just a particular variant of filling up a space with a bunch of little boxes.

[wispy slate]

but I dont mean to begin directly with abstraction , but instead, beginning with the purpose of abstraction, but intentionally, I mean warning students about the more general definitions first , so as the go in their studies , they can find the generalizations by themselfs

[vocal phoenix]

Lebesgue integral is very intuitive and pleasant to define and work with, the Lebesgue measure is a bit of a pain.

[tight star]

Right but the measure is necessary for the integral

[vocal phoenix]

Specifically, the outer measure is a very nice definition, but showing that it's a measure on a certain sigma-algebra, that's where the effort is

[spice matrix]

I can agree with that

[vocal phoenix]

But taht's the only tedious part really

[tight star]

Like I don’t want to have to take 5 lectures just to integrate polynomials

[spice matrix]

Outer measure is satisfying imo. Proving it has the properties we want can be difficult.

[vocal phoenix]

And you only need to do it once

[spice matrix]

But that’s just rigorous mathing lol.

[vocal phoenix]

Then you can just bask in the joys of Lebesgue integration

[spice matrix]

Proofs aren’t always easy

[cosmic ibex]

Does that description really force us to use the entire Lebesgue measure?

[modern trench]

Is there any situation where you actually need to interpret reals as "Dedekind cuts" / "Cauchy sequences" rather than just "complete ordered field"?
Other than proving existence

[spice matrix]

I’m not sure what you’re asking.

[vocal phoenix]

Also you still need to prove that such a measure exists

[spice matrix]

~~ Proving existence. ~~

you said other oops

[cosmic ibex]

E.g. the restriction of the Lebesgue measure to the Borel sigma-algebra is also translation invariant and maps the unit cube to 1.

[modern trench]

One could define reals as "complete ordered field" and shove the proof of existence into the appendix

[tight star]

I wouldn’t like that though

[vocal phoenix]

Yeah, If I was using the uniqueness characterization I'd say it's the unique Borel measure that's translation invariant and has the prescribed value on the cube.

If you're talking abstract measures on a topological space you're unlikely to not be talking Borel anyway

Taking completion afterwards if you really need it

[spice matrix]

Yeah to be really pedantic we would take completion lol

[cosmic ibex]

But isn't the Lebesgue measure defined on a strictly larger sigma-algebra than the Borel one?

[modern trench]

Yea it's the completion

[vocal phoenix]

I'd say it's a bit unclear, I wouldn't have problem with the term "Lebesgue measure" being used even for the restriction to Borel sets.

On a pedantic level it's not the same measure as the complete Lebesgue measure because the domain is different, but it's a level of pedantry that I can't imagine being impactful

[modern trench]

But the "standard Borel measure on R^n" has a name

[vocal phoenix]

Especially because Lebesgue-measurable sets are "almost" Borel anyway

[spice matrix]

it’s a harmless abuse that even my favorite analysis book does.

Like it’ll define lebesgue measurable sets. But then it just proves every theorem about borel measurable instead lol.

[vocal phoenix]

Is that name not "Lebesgue measure"?

[modern trench]

Lebesgue measure is the completion of that

[vocal phoenix]

Unless you want to be cute and talk of Haar measure

[cosmic ibex]

It's been a looong time since I took measure theory, but as far as I recall, the major complication in defining the Lebesgue measure was that it needed to be extended to that larger sigma-algebra (and I got the impression that this was necessary for the Lebesgue integral to be as well-behaved as we wanted it to).

[tight star]

To be fair I don’t know much measure theory

But from what I’ve heard, I know I wouldn’t have coped with it in first year

[vocal phoenix]

Certainly not, completion is generally a simple operation.

[tight star]

Which is when they introduce the Riemann integral

[vocal phoenix]

It's proving that the Borel sets are Lebesgue-measurable, that's the hard bit

[modern trench]

The completion of a σ-finite measure is the minimum extension to a measure such that subsets of measure zero sets are measure zero

[cosmic ibex]

Huh. I thought "Lebesgue measurable" means "in the Lebesgue sigma-algebra" which by definition was an extension of the Borel sigma-algebra.

[spice matrix]

The way Tao does it is singlevariable does Riemann.

Multivariable does lebesgue. Which imo is the right way to do it. Multi is a pain to do rigorously regardless of if it’s riemann or lebesgue. So may as well put in the work for the more useful one imo.

[tight star]

Oh yeah I think for multivariable it’s fair

[vocal phoenix]

And you literally do it by taking your original sigma-algebra, adding all subsets of sets of measure zero, and generating a sigma-algebra based on that

[tight star]

But not single-variable

[vocal phoenix]

It turns out that what you get is sets that are unions of sets from your original algebra and subsets of measure zero sets.

[modern trench]

Yea so the Lebesgue measure is the completion of the standard Borel measure

[vocal phoenix]

And this lets you straightforwardly define the extended measure on them

[vocal phoenix]

Sure, but this is genuinely the first time I'm seeing someone quibble with using the term "Lebesgue measure" even when the domain is the Borel sigma-algebra

[cosmic ibex]

(There was a discussion a few days ago saying that the extension could be in a simple way or a complex way that includes Caratheodory something).

[vocal phoenix]

The hard and significant part of Lebesgue measure really isn't the completeness

It's being defined on Borel sets, it's the measure of cubes being equal to their volume, and it's the translation invariance.

[cosmic ibex]

Perhaps I'm misremembering. ¯_(ツ)_/¯

[spice matrix]

Since this is a pedagogy channel. I’m just gonna put in my vote that the best sequence for Real Analysis is Tao’s Analysis I and II books. Followed by Axler’s Measure, Integration and Real Analysis

[cosmic ibex]

But I still think I remember a fairly easy first part that defined a measure on the Borel sets, and then a weird and hairy second part that extended it to some non-Borel sets, which (in a way I absolutely don't recall how) was imporant -- and only the final result got to be called Lebesgue.

[vocal phoenix]

What you remember really doesn't sound like Lebesgue measure.

Or at least if you can recall that "easy part that defines a measure on the Borel sets", I'll be very interested to see it.

[cosmic ibex]

It's 20+ years ago.

[reef nacelle]

And was this the way you were taught at first or some opinion after the fact of some particular book you had before?

[vocal phoenix]

The easy part is the definition of outer measure using infima of total lengts of countable covers by intervals.

Then the gnarly part is showing that this is in fact a measure on a certain sigma-algebra, which is at least as large as the Borel sets.

Then the final easy part is showing that this measure is complete.

[spice matrix]

well that’s anti-climatic

[vocal phoenix]

With a gnarly additional bit of showing that the measure of an interval is equal to its length

[vocal phoenix]

Yep, as I said, all Lebesgue-measurable sets are essentially Borel

The difference being null

[spice matrix]

I’ve only taken a single basic undergraduate level analysis course that used Lay’s book.

I learned analysis from these books on my own before taking that course and I’ve taken a peek at many other books including Rudin. And this is my opinion after that.

[cosmic ibex]

I'm not even sure by now what "complete" means in this context.

[modern trench]

Subsets of measure zero sets are measure zero (in particular measurable)

[spice matrix]

https://en.wikipedia.org/wiki/Complete_measure

[vocal phoenix]

I don't have an opinion on Tao (haven't read him), but Axler is my default recommendation for first contact with measure theory.

Folland for the more ambitious reader, the kind of person who enjoyed Rudin's PMA

[reef nacelle]

Would a probability route be wise for a first contact with measure theory?

[vocal phoenix]

A rigorous course in probability basically is measure theory with a focus on a specific class of measure spaces, at least early on.

[reef nacelle]

So in some sense, this gives practical motivation?

[vocal phoenix]

But you'll be learning the probabilistic terminology so you might eventually need to learn to translate it into standard measure-theoretic terminology

"Event" vs "measurable set", "random variable" vs "measurable function", "expected value" vs "integral" etc.

[reef nacelle]

Ah i see

i was recommended the Jacod and Protter book Probability Essentials

Which is supposed to lean into the more measure theoretic aspect

[hollow musk]

I have not read the rest of this conversation, so forgive me if I'm saying something very redundant, but I'm much in favour of using examples to motivate concepts. I think that dropping a high level of abstraction and then going onto more concrete examples often misses the point: it tells the reader nothing about why this definition is even needed, nor why this particular definition was chosen. I also feel like authors should more frequently set the stage before introducing higher abstraction, and one of the best ways to do that is with concrete, but useful, examples.

[vocal phoenix]

Yep, I've already said it but I'll say it again that I agree with that point of view

There are occasional exceptions where too much specificity can obfuscate the essential ideas, but they're exceptions

[hollow musk]

[vocal phoenix]

One such exception is the (extensively discussed above) Lebesgue integral; where I'm very much against starting with Lebesgue integration specifically (as in, integration of functions from R to R with respect to the Lebesgue measure), but rather with general theory of measure and integration (albeit with the Lebesgue measure and integral as an important example that receives a lot of focus)

The jump from Lebesgue integral specifically to general measure and integration is comparatively small, and gives you incredible scope and flexibility for this small inconvenience.

[tight star]

Lollllll i managed to namedrop monads in #prealg-and-algebra

But also it was again genuinely surprising how much phrasing things in terms of functions helped clarify things, even to someone completely new to them

For interest the convo starts here #prealg-and-algebra message

[tight star]

it seems that functions are an easy enough concept for people to just pick up on-the-fly

even at very early levels of math

I mean from that particular convo, I remember being confused during school whether “squareRoots” would even count as a function

[grim spindle]

There's some cases in which I don't see how this is feasible, like with the definition of Topology. I don't see how you can motivate it well via examples without at least going first through metric spaces, but even then

[gritty spade]

I remember that the fact that addition is a function shook me...

[tight star]

ooh really?

that’s fascinating to hear

[gritty spade]

I exaggerated a little, but knowing that everything in math could be united within the boundaries of set theory (category of sets with functions as morphisms) was fascinating

[tight star]

right right

i think maybe this comes from the emphasis on graphing functions?

which kinda restricts you to functions which take in a number, and spit out a number

and addition isn’t that

[gritty spade]

Yes, it disturbed me back in middle school

We didn't give formal definitions

[tight star]

that’s also probably why squareRoots wouldn’t count as a function in school

cause it doesn’t give back a single number

[tight star]

so what are you suggesting?

[gritty spade]

I'm not a teacher, but I suggest students be given a proper definition, or work with diagrams (arrows)

Work with different representations of functions

The graph is only one of them

[tight star]

so for a proper definition, you’d mean defining relations between sets, and then a function as a special type of relation?

[gritty spade]

Yes

[tight star]

right…

im not sure how well that’d go over

[gritty spade]

but it might be slightly ambitious

[tight star]

i usually just do “a function is an input-output machine” for the first time

[gritty spade]

i was going to reply that

It is good enough

Let me give an example

$f : x \mapsto \begin{cases} 0 \quad &\text{if} \quad x \not\in \mathbb Q \ \frac 1 q \quad &\text{if} \quad x = \frac p q, p \wedge q = 1\end{cases}$

[burnt vessel]

Valentin

[gritty spade]

I had this one for an oral interrogation

Needless to say it is hard to graph on a piece of paper...

[tight star]

ah, the popcorn function!

[gritty spade]

it has a name ?

[tardy ember]

https://en.wikipedia.org/wiki/Thomae's_function

[gritty spade]

Are you teachers ? What level are you teaching ? just out of curiosity

[tight star]

i just graduated

from doing a math degree

[gritty spade]

And you want to teach ?

[tight star]

hmm

im going on to do a phd in condensed matter physics atm

so not explicitly teaching

although as a grad student i’ll be a TA for some stuff ofc

[gritty spade]

So you also graduated in physics ?

[tight star]

nope!

i did a math degree

though i did do a lot of applied math courses

[gritty spade]

Anyways what year did you start doing "real math" (I mean with formal definitions) ?

[tight star]

uh

year 1?

first year

of university

[gritty spade]

Me too but I heard that calculus was introduced without being formalized in the US system (although I don't quite understand it well)

[tight star]

im not in the US system!

[gritty spade]

ok that explains a lot! where are you from ?

[tight star]

well, i did my degree at a European uni

[gritty spade]

Europe is pretty large you know

[tight star]

mhm

[gritty spade]

I'm from France, in the equivalent of a 1 year of uni but in a high school (it's very specific)

[tight star]

ok

[vernal raptor]

idk if i only feel this way because my mathematical maturity isn't high enough

but i often feel that the contents of textbooks have very little to do with the exercises at the end of each chapter

namely, chapters will dedicate dozens of pages to proving some standard result

but when it comes to the problems, all that matters is the result itself

the proof technique of the result presented in the chapter is completely irrelevant

for example:

Hatcher (and likely most other standard texts in AT) dedicates many many pages toward proving excision via barycentric subdivision

however, when it comes to the problems, all that matters is the excision property itself, and the concept of barycentric subdivision itself is completely unneeded

[austere delta]

I guess most authors would feel unsatisfied with giving a long list of results without proof.

But yeah, usually the results, examples and they're applications are more important than the exact details of each proof.

[tight star]

though in my experience it’s often useful to know the proofs very well

because the strategies that get used in proving them also appear in other problems

[austere delta]

Oftentimes they do, but I think the results are chosen more on their relevance than the relevance of its proof

[tight star]

I guess I wouldn’t really know

Since the start of university I haven’t really learnt math from textbooks

[midnight scarab]

That's very true for example in analysis

[tight star]

Oh right

Yeah analysis is probably my favourite part of math

So maybe that’s where I got that from

[austere delta]

At least for early courses, I feel the proof techniques needed are often a little more advanced than the ones expected of students to reproduce

I guess it varies

[midnight scarab]

Less true for like proving that the Eilenberg-Steenrod axioms apply to a given homology theory

[tight star]

yeah way over my head lol

[vernal raptor]

right this is exactly my point

having to read so many pages proving that simpicial homology is indeed a valid homology theory

by sorting out the intricacies of simplicial complexes

but then every problem is just like

[tardy ember]

i think the other thing that proofs can be useful for is intuition? both about the result itself and about the area in general

not always true but like

if you just look at a result sometimes you just go "what. how"

whereas with a proof, sometimes you get lost because it's too long and can't really string it all together into what feels like a single coherent reason, but it's still something

and you can do things like feed in a particular confusing example and see how the proof handles that specific case

you can try to construct a counterexample and then have the proof point out exactly what you did wrong that makes it not actually a valid counterexample

etc.

some things are just unhelpful technical details sometimes (although which things varies between people - when someone excludes things that they think of as technical details and you don't, that's how you get proofs where you just can't follow what's going on, that seem to make big leaps between steps)

but also in a sense a proof of a statement is a reason that the statement is true

[tight star]

though even then there’s like

the reason something is true and the moral reason something is true

[tardy ember]

but a very strong form of reason, it's not some fuzzy concept that if you try to drill down you see it is just a heuristic ("this arithmetic progression probably has infinitely many primes in it? the primes are basically distributed randomly right?"), you can keep poking it forever and it will keep explaining itself in more detail and you can go all the way down to the axioms if you want to

[tardy ember]

sometimes, but... also sometimes not

this might just be that my intuition is really well-calibrated on how proofs work, but i can't really think of any reasoning that i would regard as actually trustworthy (and not just heuristics) and that doesn't have an obvious translation into a proof

maybe i'm just failing to think of one...?

[tight star]

no what I mean is

[tardy ember]

("obvious" as in, it's allowed to be annoying, but it shouldn't require introducing genuinely new mathematical ideas)

[tight star]

you can have a fully rigorous proof of a statement

but it might not be the “moral” reason why it’s true

[tardy ember]

yeah i can't really think of any examples of that

[tardy ember]

i guess that's fair, but i'd also say like

[tight star]

taken from https://eugeniacheng.com/wp-content/uploads/2017/02/cheng-morality.pdf

[tardy ember]

what this is establishing is more, "p -> p" is provable from those axioms

[tight star]

I think she discusses it more coherently than I can

[tardy ember]

this is a part of demonstrating that these axioms actually capture anything resembling what we think -> should mean

[tight star]

id also say stuff like

Commutativity of multiplication

[tardy ember]

and i think this just is a pretty good explanation of the reason that "p -> p" is provable from these axioms

[tight star]

bruh

well maybe you’re just very good at logic

cause I can’t parse that proof at all

[tardy ember]

i didn't say it was easy to read