math-pedagogy

[sharp hawk]

So the attempt is to define numbers in terms of sets. Sets and their elements, as the constitutive blocks for the concept of number

[tardy ember]

(a set is transitive if it contains every set that is an element of one of its elements)

[tardy ember]

yes

more specifically from ZF

[sharp hawk]

So basically element 0 and elements succ(0), succ(succ(0)), etc .

[tardy ember]

i mean the equivalence between that and "element of all limit ordinals" is somewhat nontrivial but yes, the natural number are 0, S(0), S(S(0)), etc.

more precisely they are the initial algebra of the endofunctor F(X) = 1 + X of the category of sets
which if you unpack the definition means that for any set X, and an element z : X and a function s : X -> X, there is a unique function f from the set of natural numbers to X such that f(0) = 0 and f(n+1) = s(f(n))

(although of course just saying that doesn't prove that any set with that property actually exists)

[sharp hawk]

Now the issue is, that these are labels, they don't describe the actual concept of number.

It's saying ' numbers are 0 and s(0), s(s(0)), etc such that they satisfy certain axioms like s(s(x))+s(0)=s(s(s(x)))'

This is an abstract definition, that describes a certain mechanism (the addition), as observed from the outside by looking at the outer clothes (the labels, 0, s(0) etc)

The axiom (the rule written above) for addition came from where ?

[tardy ember]

i don't see why this is a problem...?

well firstly, you need to have some axioms or you're not going to get anywhere

[sharp hawk]

Providing labels and how they act formally, does not give any insight of the inner mechanism of that concept

[tardy ember]

(with just first-order logic you can't even prove that there are two distinct things)

[sharp hawk]

It's basically evading the concept itself

[tardy ember]

ok well insight about the inner mechanism of numbers... is just not really the point of maths?

[sharp hawk]

Another definition:

A number means measuring a certain magnitude using a chosen unit or equal parts of that unit;

The name of that measure is what we call number

[tardy ember]

like we don't care how the numbers are constructed, we consider two different complete ordered fields to be basically the same thing because there's a structure-preserving bijection between them so anything you can say about one of them that's in terms of that structure will be true about all of them

[sharp hawk]

Well we can't call any random thing a 'number' just because it satisfies some formal axioms, without us specifying what we mean by 'number'

[tardy ember]

and what mathematicians generally mean by "number" is anything that satisfies a particular set of axioms

[sharp hawk]

This leads to problems, it is not workable

[tardy ember]

what problems?

[sharp hawk]

trying to define natural numbers in terms of elements of sets, and then 'rational numbers' in terms of natural numbers, and then real numbers in terms of rational numbers, does not work.

In the first place from where did the idea of rational and real numbers appear? From axioms? 🙂 Or from the definition I showed above.

[tardy ember]

well the real-world reason we're studying them is... well idk, does it matter? maybe we looked at the universe and it seemed to have real numbers in it, maybe we're just playing around, you don't need a reason to decide to look at a particular object

from a mathematical perspective, it doesn't matter, you can throw together any collection of concepts and as long as each individual step makes sense the overall construction is a valid object to study

[sharp hawk]

It turns out it's valid only if the concept was valid.
We can give axioms AFTERWARDS. They are about the outer form, how we work with them . But the way we work with them (these mathematical objects) is not random, it is exactly based on what they represent intrinsically

Real numbers were not just invented axiomatically

They came from the fact that we realised them when attempting to measure the diagonal of a square

It's not that some dude thought of axioms and this is how the concept came into existence.
No.

[tardy ember]

yes, that's true
actual mathematicians in the real world did not just decide to consider a complete ordered field completely at random, they thought about it because spacetime in the real world is (or at least seems to be, at a human scale) continuous

does it matter?

[sharp hawk]

It matters if we want to be clear what we work with.
Otherwise let's call any random thing a 'number'. Would this make mathematics better? Or just make it a non-mathematical mess

[tardy ember]

you can be clear what you're working with by just... making definitions

the real numbers are the complete ordered field

and in fact, if you try to define numbers in terms of the real world, that makes it less clear

what exactly can you do with a measure of a certain magnitude using a chosen unit or equal parts of the unit?

[sharp hawk]

Now if someone wants to consider any symbol a number, and invent axioms upon axioms , that's all right but what he is working with is just artificial constructions (labels), he is doing linguistic play, not mathematics

Certainly such play works sometimes, but it's only when it is not separated from the actual original meaning of the concept it formalized.

[sharp hawk]

By real world we mean geometry

[tardy ember]

if you have a set of measures of a certain magnitude using a chosen unit or equal parts of the unit, is there a largest one? and how would you know that?
how many measures of a certain magnitude using a chosen unit or equal parts of the unit are there? how would you know?

[tardy ember]

alright we just disagree about what mathematics is then

i think any kind of study of purely formal objects is mathematics

[sharp hawk]

A largest magnitude?

[quiet jackal]

a priori it is not

it happens to work out in this case

[tardy ember]

even if you define something completely at random and not because you saw it in the real world, like the hyperreal numbers
if each step is formally valid, it is valid and it is mathematics

[sharp hawk]

Even if it leads to contradictions?

[tardy ember]

well if you get a contradiction, you've just reached the mathematical conclusion that the axioms you were using are inconsistent

[sharp hawk]

And it indeed does lead. But we ignore them so we think it doesn't

[tardy ember]

...?

are you saying that there's a contradiction that's provable from ZF?

[sharp hawk]

Explain this question please

[tardy ember]

well, suppose there's a set, and each element of the set is a measure of a certain magnitude using a chosen unit or equal parts of the unit
is there some element of this set that is larger than all of the other elements of the set?

and aside from the actual question itself, i'm more interested in how you would write code to recognise a valid argument that answers this question

[sharp hawk]

5 means give units for example.

Since by the 2nd requirement/postulate of Euclid any segment can be extended, we can get a magnitude of 6 units, 7 units, and so on. No biggest magnitude

(Meant to reply to the other message about biggest measure )

[tardy ember]

5 means give units for example.
?

i don't get what that's in response to

are you saying that's a contradiction in ZF?

[sharp hawk]

Yes

[tardy ember]

...ok, yeah, that's another terminology difference then

i use "contradiction" to refer to a proof that some statement is both true and false

ZF does not prove that the number 5 does have a unit, so the fact that it doesn't is not a contradiction

[sharp hawk]

By design ZF is self contradictory. Now when the contradiction is the norm, it has become difficult to spot other contradictions it leads to because we don't see them as contradictions

[tardy ember]

ZF does not prove that a statement is both true and false, so it's not inconsistent by my definition

[quiet jackal]

this conversation is obviously inappropriate for this channel

[sharp hawk]

Specifically I am talking about the axiom of infinity that is self contradictory

[tardy ember]

i have no idea what you mean by "contradiction", but i don't think the existence of contradictions by your definition makes something not mathematically valid or interesting

[sharp hawk]

This reply was meant for the other message about biggest measure

Not for zf, my bad.

[sharp hawk]

I will give an example

I come up with an axiom that there is a number bigger than 5 but smaller than 4

This is already contradictory. But we can ignore this fact, as it is done in zf, and surprise, we will think it leads to no contradictions

[tardy ember]

ok well if "number" means real number, then yes, a reasonable set of axioms about the real numbers and the assertion that some number is > 5 and < 4 is a contradiction, in the sense that you can use it to prove both 0 = 1 and 0 != 1

[sharp hawk]

If we can measure a magnitude by a unit or equal parts of it, we call this a number. So rational numbers are numbers

If we can't measure it, we call it an irrational or incommsurable magnitude

And check this out

Both numbers and irrational magnitudes, are magnitudes.

Adding one magnitude to another magnitude gives a magnitude

Adding one number to a number gives a number

5+6=11

Adding one number to an irrational magnitude gives an irrational magnitude

That's why 1+sqrt(2) remains as it is , does not simpify into a single number

Adding two commensurable magnitudes

Sqrt(2)+sqrt (2), simplifies 🙂 to 2*sqrt(2) and sqrt (2) is the unit that measures both terms and also the result of their addition

[tardy ember]

...yep, all of that is also true in the complete ordered field, you're just using somewhat nonstandard terminology for it

[sharp hawk]

It certainly comes out of this 🙂

Which is, geometry

We made the structure more abstract, more algebraic , which is cool

But it only works because it is more than pure algebraic form. It is a solid concept behind

[tardy ember]

...i don't think that's true?

like, what would be the "solid concept" behind the hyperreal numbers

[sharp hawk]

That one is wrong man

Just imaginary contradictory stuff....

[quiet jackal]

contradictory with what

[sharp hawk]

Not rooted in a valid concept. And it just fools us

[tardy ember]

ok so what's the contradiction
how do you prove that some statement is both true and false

[sharp hawk]

With what I just said

[quiet jackal]

what?

[sharp hawk]

There is no magnitude highest than all other magnitudes

[quiet jackal]

lol

[sharp hawk]

In geometry.

[tardy ember]

it doesn't matter that the hyperreal numbers are "imaginary"

you can still get useful results from them

[quiet jackal]

ok let's change the subject slightly

what do you think about hyperbolic geometry?

[sharp hawk]

Certainly not 🙂

[tardy ember]

yes you can

why wouldn't you be able to?

[sharp hawk]

I want to learn it, don't have much experience with it

[quiet jackal]

obviously he is not going to engage with you on foundations lol

[sharp hawk]

Because it's just nonsense 🙂

[quiet jackal]

ok but surely you think it is "contradictory"?

[tardy ember]

it isn't nonsense, it's the ultrapower of the real numbers

[quiet jackal]

it has to reject the parallel postulate

[sharp hawk]

Empty words and invalid concepts

It contradicts the second postulate of Euclid

[quiet jackal]

ok

[tardy ember]

no it doesn't

[quiet jackal]

so you do think hyperbolic geometry is invalid

[tardy ember]

none of the postulates of Euclid are about ultrapowers

[quiet jackal]

because euclid's postulates are to be upheld at all costs

?

[sharp hawk]

No because it dies not contradict any postulate:)

[quiet jackal]

it does

[tardy ember]

they're about geometry, and the hyperreal numbers are not a geometric object

[sharp hawk]

No

[quiet jackal]

parallel postulate does not hold in hyperbolic geometry

[sharp hawk]

This is a misconception, it does not

[quiet jackal]

what?

[tardy ember]

consider the following postulate: that I am on Earth
but the Sun is not on Earth, and so the Sun is invalid

[sharp hawk]

The so called non-euclidean geometries are not contradictory to euclidian geometry, at all

They are really not non-euclidean, they just work with different mathematical objects, in a different context. That's all

https://en.m.wikipedia.org/wiki/Poincaré_disk_model#:~:text=In geometry%2C the Poincaré disk,diameters of the unit circle.

In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk that are orthogonal to the unit circle or diameters of the unit circle.

See... straight lines are not straight lines any more

[tardy ember]

yep, and that also applies to the hyperreal numbers

the hyperreal numbers are not points or straight lines, they're objects in a context that is not geometry

[sharp hawk]

It would be cool, but that's an open question. This is what I am asking you to ponder upon, whether it is so or not.

I'd say not, because it still works with the same magnitudes

[quiet jackal]

lol

just lol

[tardy ember]

they don't, they work with the ultrapower of that set of magnitudes

each "magnitude" in the context of the hyperreal numbers is an equivalence class of infinite sequences of magnitudes

straight lines aren't straight lines any more, they're infinite collections of lines

[sharp hawk]

'bigger than all other magnitudes' it is still working with the original mathematical objects, claiming to extend them (but in a contradictory manner)

The so called non-euclidean geometries are not doing that. They are simply not working with straight lines any more that's why the postulate about parallelism can change.

[tardy ember]

where did you get the idea that the hyperreal numbers have something "bigger than all other magnitudes"

there is no largest hyperreal number

[sharp hawk]

Correct me sure.

[tardy ember]

and even if there was, the hyperreal numbers aren't inherently claiming to be magnitudes at all

[sharp hawk]

Is there a number bigger than all natural numbers? That's what I meant

[tardy ember]

there is no hyperreal number bigger than all hypernatural numbers

there are hyperreal numbers bigger than all natural numbers, but in that context the hyperreal numbers aren't numbers, they just act sort of like numbers

[sharp hawk]

Bigger than all natural numbers .

Well there is a supernumber bigger than 5 and smaller than 4. But it's not a number, it's a supernumber

[tardy ember]

to be clear, all that this actually means is that we have defined a set of pairs of hyperreal numbers, and that there is some hyperreal x such that (f(n),x) is in the set for every natural number n, and f the embedding of the natural numbers into the hyperreal numbers

[tardy ember]

yep, and that's consistent

[sharp hawk]

In that case what does it even mean to compare a non-number (a hypperreal), to a number?

[tardy ember]

it necessarily implies that "supernumbers" don't follow all of the rules that numbers do, but there's nothing inherently wrong with defining something that doesn't follow all the rules that numbers do

[tardy ember]

that is a good question

[sharp hawk]

That's a question for us to ponder upon

[sharp hawk]

.

[quiet jackal]

no the point is that straight lines in hyperbolic geometry are not the same as straight lines in euclidean geometry. yet hyperbolic geometry captures physical, observable phenomena

a more correct example of the latter is general relativity

[tardy ember]

it turns out that there's an obvious embedding of numbers in hyperreal numbers

[sharp hawk]

Sure

[tardy ember]

so then the question is just how do you compare two hyperreal numbers

and it turns out there's also a fairly natural way of doing that

[quiet jackal]

ok so then why would euclid's system be fundamental? we know since einstein that our realities are not flat

[sharp hawk]

Please show a case that is a counter-example

[tardy ember]

all of this is just definitions, we could have chosen some random other relation and named that "<" and got that 2 < 0 in the hyperreal numbers, but there's one particular way that has nice properties and that one particular way is what people almost always mean when talking about comparison of hyperreals

[quiet jackal]

f(x) = 0 for x nonzero, f(0) = 1

[sharp hawk]

I did not say it is fundamental, just that if we extend upon it we (our extension) should not be contradictory

Non-euclidean geometries do not extend upon it, do they?

[quiet jackal]

i think you are claiming that a lot of things are "extensions" of euclidean geometry

[tardy ember]

...pretty much nothing in maths is extending upon euclid's postulates

[quiet jackal]

without any evidence

[tardy ember]

usually what people do is just define some other type of object

[sharp hawk]

Claiming that 'there is a hypper real bigger than all natural numbers' is an extension upon natural numbers , upon Euclid:)

[tardy ember]

no it isn't

[quiet jackal]

no it does not become true just because you say it for the 5th time

[tardy ember]

the hyperreal numbers don't extend the real numbers, they're just a definition of another type of thing

[sharp hawk]

So Euclid was not working with natural numbers then ?

[quiet jackal]

actually i think it is widely agreed that a historical emphasis on axiom-based geometry significantly hampered the growth of mathematics

[tardy ember]

euclid probably worked with natural numbers at some point

but the important thing is that hyperreal numbers... aren't natural numbers

the existence of a hyperreal number bigger than all natural numbers does not imply that there is a natural number bigger than all natural numbers, because it was never claimed that that hyperreal number is a natural number

euclid's postulates do not imply that there cannot be a hyperreal bigger than all natural numbers because euclid's postulates were not about hyperreal numbers

[sharp hawk]

Euclid's postulates do not imply there cannot be a non-magnitude bigger than all other magnitudes.

Do you agree that the word bigger 's meaning has changed? So better use a different word, not 'bigger' which is reserved for comparing magnitudes

[tardy ember]

i guess that's reasonable
...so what should we call it

[quiet jackal]

no it is not reasonable

why would you accept this goalpost moving

[sharp hawk]

Otherwise we overload the word with different meanings and the whole theory becomes ambiguous

[tardy ember]

it... does actually make sense? like, yes it's a silly convention to only use "bigger" on real numbers, but it makes far more sense than everything else that's happened so far

let's just say that the "<" relation on the hyperreal numbers is denoted *<

[quiet jackal]

no it is only ambiguous when one party in the conversation interprets everything in a maximally uncharitable way

[tardy ember]

the notation doesn't matter, whether you use the same word for *< on hyperreals and < on reals doesn't actually affect anything about the mathematical content

it only really becomes "ambiguous" if you also don't write the canonical embedding of reals into hyperreals and just do it implicitly, but even then, it turns out that *a *< *b iff a < b (where a is a real number and *a is the corresponding hyperreal), so the ambiguity isn't important

[quiet jackal]

you have been told over and over again that hyperreal numbers are not natural numbers

so why would the corresponding order

be the same as for naturals

[sharp hawk]

I meant it is not right to say that a hypperreal is 'bigger' than some number number. Because bigger is reserved for ordinary numbers that have a clear geometric meaning

[quiet jackal]

that is a fiction

that you imposed

and just revealed

[tardy ember]

is that your only objection to the hyperreals? if we put * in front of every word that has a geometric meaning, to clarify that we might be using it to mean something else, is the result consistent?

[sharp hawk]

So you see, what happens. We cancel the actual meaning of what 5 centimeters > 3 centimeters means

And give it various ambiguous meanings. Axiomatic meanings inconsistent with the original meaning.

[quiet jackal]

if X is not a natural number then anything i say about X should not be interpreted as a statement about natural numbers

no

again

there is no ambiguity

bee has been remarkably patient trying to explain this example to you

[sharp hawk]

Just 1 meaning, you're saying ?

[quiet jackal]

in excruciating detail

you are either not reading the messages

or not asking for clarification when you do not understand

[sharp hawk]

If there's just 1 meaning for the word 'bigger' then a hypperreal is bigger than a natural in the same way a natural is bigger than another natural .

[quiet jackal]

and finally you start nitpicking the use of english because "the meaning of centimeters is canceled"

[quiet jackal]

no obviously there are multiple meanings

[sharp hawk]

Which is not possible because a hypperreal is not a natural.

Thus 'bigger' now has more than 1 meaning, and this is called ambiguity

[quiet jackal]

but there is only one

in the context of what bee is saying

[sharp hawk]

He agreed to use *>

[tardy ember]

it*

[sharp hawk]

It. Sorry 🙂

[tardy ember]

so... if we use *> instead of "bigger", does that make the hyperreals consistent, according to you?

[sharp hawk]

A bit yes, it's a step towards consistency

[quiet jackal]

lol

[tardy ember]

...what other problems are there?

[quiet jackal]

just wait 20 minutes and another will be gradually revealed

[sharp hawk]

It is a great step, because we ordinarily mix up different mathematical objects or operators, we think they're the same because they are designated by the same word or symbol

[tardy ember]

do you think it's still inconsistent, or is this one step enough?

[sharp hawk]

Well it is not clear what *> means geometrically or in some other practical way

I am curious to learn what practical or theoretical applications does it have

Is this theory more useful than my theory of a supernumber which is ^> than 5 and ^< than 4 ?

(^> means just ^> , and ^< means ^< )

[tardy ember]

well the main important thing about the hyperreal numbers is the "transfer principle", which is that for a certain class of statements about real numbers, if the statement is true, you can basically put *s in front of everything and the resulting statement is still true

so since 4 < 5 is true, *4 *< *5 is true

and it also goes the other way: if *3 *< *6 is true (which it is), then 3 < 6 is true

the other thing is that there are hyperreal numbers that are *> all real numbers

in the sense that for some hyperreal x, for any real number r, x *> *r

(i'm putting *s in front of real numbers to represent a particular map from the real numbers to the hyperreal numbers)

...hm
ok all of the actual applications that i'm immediately thinking of are... somewhat complicated

but it does turn out to be useful, to have the transfer principle but also these numbers that are *> all real numbers

(by the way if you wanted to actually use these, you should probably look up exactly what the transfer principle is, the actual definition has some important details, and if you don't know the details it's very easy to mess up)

[sharp hawk]

Works the same way, just calculating the derivative of f(x),x nonzero, at 0, would do,
No need then for left and right limits of the derivative as that teacher has done in the pdf

[tardy ember]

...but then you get the wrong answer, the derivative of f at 0 doesn't actually exist

and its derivative everywhere else is 0

[sharp hawk]

I agree it does not exist

I thought it exists in mainstream calculus.

[quiet jackal]

????

[tardy ember]

$\lim_{h\to 0} \frac{f(h)-f(0)}h = \lim_{h\to 0} \frac{-1}h$

[burnt vessel]

bee [it/its]

[tardy ember]

and that doesn't exist, because it goes to either +inf or -inf depending on which side you approach it from

[quiet jackal]

or f(x) = x for x>1, and f(x)= x-1 for x<= 1

or f(x) = |x|

or any number of other examples

is “mainstream calculus” a subset of “rigorous mathematics”

or is it something else

[sharp hawk]

It is not...

[quiet jackal]

no idea what that is then

never heard of it

[sharp hawk]

It's the usual calculus, which is not rigorous; only believed to be rigorous.

[quiet jackal]

lol

fascinating

[tardy ember]

i feel like we have different enough backgrounds that "the usual calculus" is ambiguous

like is this, define limits in terms of epsilon-delta, then define $f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}h$

[burnt vessel]

bee [it/its]

[tardy ember]

or is it something else

[quiet jackal]

its whatever euclid would have wanted

obviously

draw a secant line

using a straightedge

[sharp hawk]

I have a question

[quiet jackal]

ok

[sharp hawk]

Since f is not continuous at 0, do we need to bother about the derivative at 0?

[tardy ember]

well a function can't be differentiable at a point where it isn't continuous

[sharp hawk]

Yeah...

[tardy ember]

beyond that i'm not really sure what you mean by "do we need to bother"

i guess it is true that there isn't much point explicitly computing the derivative there, if we know that it's not going to have one because it's not continuous

[sharp hawk]

So no need to calculate the left hand and right hand limit of the derivative

These exercises are pointless, discussing about them would be useful if they really made sense but they're just artificial playing

[sharp hawk]

The Newton leibnitz calculus

Turns out there's a better way, friends.
Using a parallel secant rather than a non-parallel secant

[quiet jackal]

wow

[sharp hawk]

This, along with another geometrical discovery, makes it possible to obtain the derivatives and integrals without limit theory.

Limit theory remains useful in calculating series , but no need for it for derivatives and integrals

[quiet jackal]

wow

[sharp hawk]

Yeah it's new stuff, makes everything easier. It's called New Calculus

[quiet jackal]

wow

i havent heard of it, is it new?

[sharp hawk]

Yes

[tardy ember]

...is this a different definition of a derivative, or just a different way of computing the same values as the standard definition?

[sharp hawk]

Gets to the same formulas for the derivatives and integrals obviously

Using a different definition, that does not rely on limits, just on parallelism

[quiet jackal]

https://thenewcalculus.weebly.com

[tardy ember]

...how do you define parallelism

[sharp hawk]

I have my own articles on it.
Yes, John Gabriel is the discoverer

I learned much from him

[quiet jackal]

i can tell

are you his disciple

or just an admirer

[sharp hawk]

Admirer but since I have spent a lot of time and finally understood his stuff (also found some small mistakes in some of his proofs in some other secondary topics, which he acknowledged and corrected )

I can also be called a disciple, I consider the new calculus very nice and easier to learn and at least as powerful

[long pelican]

John Gabriel is a household name in this discord

but not in a good way

[quiet jackal]

i wonder why people never try to revolutionize the foundations of more modern topics in mathematics

[sharp hawk]

The secant parallel to the tangent to the graph of the differentiable function.

We define differentiable as having such a tangent which is unique, for every point in the domain

[long pelican]

Emily Riehl, Jacob Lurie, Adeel Khan?

[quiet jackal]

its always calculus, distribution of prime numbers, or euclidean geometry

[quiet jackal]

sorry i should say

[long pelican]

Mike Shulman

[quiet jackal]

outsiders to the field

[long pelican]

oooo

[tardy ember]

what's a tangent to a graph?

[sharp hawk]

Because there are issues bro. For example dx and dy are still infinitesimals

[quiet jackal]

could it be that they dont have enough understanding of mathematics to engage critically with anything developed after 1700?

[sharp hawk]

A straight segment not crossing the graph at the point of tangency

[long pelican]

Non-cynical answer is fewer people understand it, especially outsiders, and given a probability p of being a crank the chance that 0 of a small set is a crank is much higher than 0 of a larger set

[quiet jackal]

yes of course

also like

the more math you learn

the harder it is to fall down these kinds of rabbit holes

[long pelican]

Speak for yourself, I'm in 3 rabbit holes myself but I'm exploring them privately 🤣

[quiet jackal]

yeah i mean

[sharp hawk]

https://m.youtube.com/watch?v=TJ6Y6LpfJhU&list=PLhe-vn_Mq9Y4y6i5FZ5ihJffmBTGfYesH&index=3

[quiet jackal]

we need people to go down rabbit holes to make discoveries

but also the more you know, the better your heuristics are

and similarly, the more you engage with the community/literature

so i dont mean it becomes harder to get absorbed in math problems or whatever

but rather that you develop a sense of what things are likely to pan out

and a sense of when to give up on things

[tardy ember]

$$3x^2 + \frac{3x(n^2-m^2)+n^3+m^3}{n+m}=f'(x)$$ $$3x^2 + 1 + \frac{3x(n^2-m^2)+n^3+m^3-n-m}{n+m} = f'(x)$$ \ therefore the derivative of $x^3$ is $3x^2+1$?

[burnt vessel]

bee [it/its]

[tardy ember]

(using the example from around 1:30)

[sharp hawk]

No, you cannot set m=n=0 in THAT one; yours is different 🙂 than the one in the video

[tardy ember]

so what makes the one in the video work and my one not?

[sharp hawk]

Btw it has such a simple mechanism, you won't believe. After you understand it,

Your question is FAQ1 from my article .

[tardy ember]

...which article

[sharp hawk]

Your LHS represents the slope of a parallel secant line.

Distances m and n can't be 0 since it's a secant.

[sharp hawk]

Lemme turn on the computer

[tardy ember]

...that objection applies just as well to the computation they did in the video though?

[sharp hawk]

No

[tardy ember]

the first equation here is literally taken from the video, assuming i didn't write it wrong

[sharp hawk]

I will explain step by step

1sec.

The New Calculus Derivative

dx²/dx
=[(x+n)²-(x-m)²]/(m+n) [A] slope of the (parallel) secant line
=2x+n-m [B] slope of the tangent line+the difference in slopes
In [B] we set n=m=0 (but not in [A]. See the explanations below to understand why this is valid), and we are done, we have got the general derivative dx²/dx=2x, without using limits.

In general for any smooth function f:
f'(x)
=[f(x+n)-f(x-m)]/(m+n) [A] slope of the (parallel) secant line
=f'(x)+Q(x,m,n) [B] slope of the tangent line+the difference in slopes

Explanation how it works:
Expression [A] is defined only when m≠0 or n≠0, because it encodes only the calculation for the slope of a secant line (when m=n=0 the line is not a secant any more, it becomes the tangent and [A] specifically -by design- describes the secant, not the tangent.), but expression [B] is always defined (even when m=n=0) because it encodes both f'(x) (the slope of the tangent) and the Q(x,m,n) (the tangent and secant slope difference) which when added together give the slope of the secant line (aka the same result as [A]).
So in equation [B], Q(x,m,n) can be 0 (which simply means that the secant is parallel to the tangent)
So we never set m=n=0 in [A], just in [B]. (what is happening behind the scenes is geometry)

FAQ 1. When we use [A] to calculate the slope of a parallel secant line, how do we know in the resulted [B] which part is f'(x) and which part is Q(x,m,n)?
Answer: Simply by setting n=m=0, what remains is f'(x) and that's it. It really is easy and beautiful. Q(x,m,n)=0 is called "the auxiliary equation".

Definition [D1]
"A function is smooth over a given interval if it is continuous over that interval AND only one tangent line is possible at any point in the interval. Inflection points are excluded because no tangent line is possible at points of inflection, only half-tangent lines."

[tardy ember]

ok so what if you instead simplify ((x+n)^2-(x-m)^2)/(m+n) = 2x + n - m + L(n,m)
where L(n,m) is 1 when m + n = 0, and 0 otherwise

this is valid because it gives the same results for all m,n where A is defined

and then you set n = m = 0 and get that the derivative is 2x + 1

[sharp hawk]

If you do that,

it basically means that you changed the [B] from the geometrical theorem
"In general for any smooth function f:
f'(x)
=[f(x+n)-f(x-m)]/(m+n) [A] slope of the (parallel) secant line
=f'(x)+Q(x,m,n) [B] slope of the tangent line+the difference in slopes"

you artificially added a 1 in [B].

[tardy ember]

what's "artificial" about it?

maybe the way they did it was actually artificial

[sharp hawk]

it is not true that if you add 1 to [B] the equality holds.

[tardy ember]

how do you know which way isn't artificial, in order to compute a derivative?

[sharp hawk]

just like adding 1 million to [B] also makes the equality not hold. The geometric theorem is saying that the equality holds when you don't artificially add numbers just to change the value as you like 🙂

[tardy ember]

so how exactly do you work out which value of [B] doesn't have extra numbers added to it?

[sharp hawk]

you basically modified [B]

[tardy ember]

how would you write a program that looks at a computation of [B] and works out whether it was artificially added to or not

[sharp hawk]

if you compute [A], you precisely get [B] 🙂

and here's a theorem about that

[tardy ember]

i computed [A] and i precisely got 2x + n - m + L(n,m)

that agrees with A at every point where it's defined

[sharp hawk]

NO because that L(n,m) is just 1 , written with letters. you ADDED it

you added 1 to it without any reason.

[sharp hawk]

FAQ 2. But this requires that the difference in slopes Q(x,m,n) which we get when we calculate [A]), never has m+n as a denominator. How do we know this for sure?
Answer: Theorem [T1]. Q(x,m,n) never has m+n as a denominator.
Proof
For any parallel secant we have:
[A]=[B], that is
secant slope=tangent slope+ slope difference
,in mathematical symbols:
[f(x+n)-f(x-m)]/(m+n)=f'(x)+Q(x,m,n), where Q(x,m,n)=0
⇔[f(x+n)-f(x-m)]/(m+n)=f'(x)
Since RHS=f'(x) does not depend on m+n, it must be that m+n is a "real" factor of every term in [f(x+n)-f(x-m)]. QED. (from the book An Introduction to the Single Variable New Calculus, page 121)

[tardy ember]

ok so to compute a derivative with New Calculus, you need to have a reason for each step, it's not sufficient that each step is correct?

[sharp hawk]

what you did is not correct.

[tardy ember]

the value of A is always equal to 2x + n - m + L(n,m)

if A has a value, then n + m is not 0, and so L(n,m) is 0

[sharp hawk]

but it is 1 in [B], when, m=n=0, simple as that. And you can't just add 1 as you please.

[tardy ember]

so there's some constraint on B other than that it's equal to A at every point where it's defined, and what is that constraint?

so if any subterm of B is equal to 1, the computation is incorrect?

[sharp hawk]

secant slope=tangent slope+ slope difference

NOT

secant slope=tangent slope+ slope difference+1

[tardy ember]

i'm saying the tangent slope is 2x+1

and you haven't explained why i'm wrong

the expression for B i gave you is equal to A

[sharp hawk]

you have to explain WHY you add 1 , or 1 million, in [B] where there was no 1 or 1 million.

[tardy ember]

so i was right earlier

it's not sufficient for the step to be correct

the step has to have a reason for it

even if A is equal to B, there's this other step where you explain what each subterm of B is there for

[sharp hawk]

what you did is not correct. You cannot randomly add 1 to [B] and claim this is correct

[tardy ember]

A is still equal to B

[sharp hawk]

[A] doesn't exist when m=n=0

m=n=0 implies it is not a secant. [A] is about the secant.

[tardy ember]

ok well

in 2x + n - m

A doesn't exist when m = n = 0

m = n = 0 implies it is not a secant. A is about the secant

therefore 2x isn't the right answer, apparently

[sharp hawk]

how so? when it is not a secant, it is the tangent 🙂 you still have [B] when m=n=0

[B] is about the tangent 🙂 plus slope difference. which when 0, gives us exactly the slope of the tangent

[tardy ember]

ok well we still have 2x + n - m + L(n,m) when m=n=0

[sharp hawk]

you randomly added an L(n,m) which you say equals 1 when m=n=0. This is invalid. you cannot just add stuff (in this case 1) to [B]

obfuscating the 1 with letters doesn't make this valid.

[tardy ember]

you cannot just add stuff (in this case 1) to [B]
ok fine then, B is (2x + n - m) * (1 + L(n,m))

i can't just add stuff, but there's no rule against just multiplying stuff, so this is fine, right?

[sharp hawk]

you can add 0 sure

[tardy ember]

yes, and i did

L(n,m) is 0

[sharp hawk]

no because you said it equals 1 when m=n=0. so you added 1. not zero

[tardy ember]

i didn't add 1

it was already there

the value of A at m=n=0 is actually 1

[sharp hawk]

?

explain please.

[sharp hawk]

dx²/dx
=[(x+n)²-(x-m)²]/(m+n) [A] slope of the (parallel) secant line
=2x+n-m [B] slope of the tangent line+the difference in slopes

[tardy ember]

if you simplify ((x+n)^2-(x-m)^2))/(m+n), you get 2x + n - m + L(n,m)

that is an expression that is always equal to ((x+n)^2-(x-m)^2))/(m+n)

[sharp hawk]

you get this 2x+n-m

[tardy ember]

that's because you randomly decided to subtract L(n,m)

[sharp hawk]

no L(n,m) there.

that L(n,m) is added by you and then you make it equal 1. this is invalid.

[tardy ember]

what exactly makes your choice to subtract L(n,m) correct, and my choice to leave it there wrong?

they're both valid simplifications

they're both always equal to A

[sharp hawk]

AND

[tardy ember]

what is a simplification, what extra condition is there beyond not changing the value?

[sharp hawk]

AND, that's all?
No, it's not all.

AND it suddenly becomes 1 in [B] because you make it so. This is invalid.

[tardy ember]

it suddenly becomes 0 in [B] because you make it so

this is invalid

[sharp hawk]

well, the simplification is 2x+n-m .

[tardy ember]

you randomly decided to not add L(m,n)

what is a simplification?

[sharp hawk]

you can add ONLY ZERO

NOT A ZERO THAT BECOMES 1 at random

that's what made your attempt invalid, I hope it is clear

[tardy ember]

what is a simplification?

what does it mean for one expression to be a simplification of another? how would you tell?

if you simplify 1/x, what is its value at x = 0?

[sharp hawk]

now that's ...

🙂

❤️ nice discussions with you by the way

[tardy ember]

let's define J(x) to be 1 at x=0, and 0 otherwise

now consider these two expressions:
[A] x/x
[B] x/x + J(x)

[sharp hawk]

then J(x) is not ALWAYS 0

so it is not 0. because zero is always zero, not just sometimes.

[tardy ember]

these expressions define the same function: no value at 0, and 1 everywhere else

[tardy ember]

if you simplify A and evaluate at 0, you get 1
if you simplify B and evaluate at 0, you get 2

so what you have here isn't actually a way to take a derivative of a function

it's a way to take a derivative of an expression

for some functions there are multiple distinct expressions that give that function as their values, which, according to your rules for simplifying that you haven't explained, simplify to expressions with different values

[sharp hawk]

what does adding a NON-ZERO thing such as J(x) has to do with anything?!!

if you want to reason validly, add 0.

[tardy ember]

if it was a function, you wouldn't be able to tell

the only reason it matters is that this is not a method that works to differentiate functions

[sharp hawk]

if that function is NOT ALWAYS 0, it is not valid to add it.

[tardy ember]

yes

exactly

[sharp hawk]

but you attempted to add it!

so what you did is invalid

[tardy ember]

this means that what we're dealing with isn't functions

if these were functions then it would be valid

because if two functions have the same value at any point then they are equal

x/x + J(x) and x/x have the same value at every point, but they are not equal

so they aren't functions

so you can't use this method to differentiate a function

[sharp hawk]

no it is not valid in ANY way to add NON-ZERO in just the RHS or in just the LHS

and that is what you did

it's invalid.

[tardy ember]

are you even listening to me

i know it's invalid

that is my entire point

what i'm saying is that it IS invalid in this system to do that

which implies that your system, in which it IS invalid to do this, does not satisfy function extensionality

because x/x is DIFFERENT from x/x + J(x), because you added something that isn't zero

they have the same value at every point AND THEY ARE DIFFERENT

and this is not how functions as they are normally defined behave

[sharp hawk]

certainly . And what relevance does a system have in which you add non-zero in just the RHS . I don't get that

[tardy ember]

the relevance is that that's the system that mathematicians normally use

[sharp hawk]

huh?

adding non-zero in just the RHS is what is normally used?

[tardy ember]

well it isn't done that often because it isn't actually useful

but it's a valid operation in the normal view of functions

which is function extensionality

a function IS a mapping of inputs to outputs

if two functions map the same inputs to the same outputs, they are the same function

[sharp hawk]

then please let's get back to the earth

[tardy ember]

but in your system, x/x is the same mapping as x/x + J(x), but not the same expression

so the things you're dealing with aren't mappings

[sharp hawk]

we area dealing with geometry here, simple

[tardy ember]

so what you've defined here is not the same as the standard derivative

there are functions for which it doesn't even make sense to ask whether it has a NC derivative, because the NC derivative is not defined on functions

[sharp hawk]

what we have here is a geometric theorem that allows us to calculate the slope of the tangent line of a differentiable function at any point

that's all

[tardy ember]

not a function

[sharp hawk]

YES

only SMOOTH functions are dealt with in NC

[tardy ember]

...no

i feel like you're not even reading what i'm saying

[sharp hawk]

esoteric functions are not considered because it turns out they're not useful in practice

[tardy ember]

you're saying that x/x and x/x + J(x) are different

this implies that you're not dealing with functions

because they contain the exact same input-output pairs

if they were functions, that would imply that they're the same

but according to you they're not the same

so they're not functions

[sharp hawk]

I would love to understand what you mean so please give an example of what you call functions

[tardy ember]

$f = {(x,x) : x \in \mathbb{R} \land x \neq 0}$

[sharp hawk]

and how adding a non-zero J(x) just on the RHS, is valid, with your functions

[burnt vessel]

bee [it/its]

[tardy ember]

here's an example of a function

it is a set of ordered pairs

[sharp hawk]

a non-smooth function

ok

[tardy ember]

i don't actually know what "smooth" means in this system so i have no idea

by standard definitions it is infinitely differentiable, except at 0 where it's not defined

[sharp hawk]

Definition [D1]
"A function is smooth over a given interval if it is continuous over that interval AND only one tangent line is possible at any point in the interval. Inflection points are excluded because no tangent line is possible at points of inflection, only half-tangent lines."

NC **only **deals with smooth functions

no other functions are taken into consideration, by design, because it is not useful to do that.

[tardy ember]

a set of ordered pairs is a function if for any x, if (x,y) and (x,z) are both in the function, then y = z

[tardy ember]

can there be two smooth functions that are different, but that are the same function?

[sharp hawk]

well, the smooth function is the one for which we calculate the slope of the tangent at some point

and [A] [B] are just the LHS and RHS of a GEOMETRIC theorem

so in order to address it, you have to address the geometry because it's just geometry written algebraically nothing else

[tardy ember]

now if we take this and add J to it, the resulting function is \ $$g = {(x,y) : x \in \mathbb{R} \land x \neq 0 \land y = f(x) + J(x)}$$

[burnt vessel]

bee [it/its]

[tardy ember]

since x is not equal to 0, J(x) is 0

so this is the same as \ $$g = {(x,y) : x \in \mathbb{R} \land x \neq 0 \land y = f(x)}$$

[burnt vessel]

bee [it/its]

[tardy ember]

which by extensionality is equal to f

so $g = f$

[burnt vessel]

bee [it/its]

[sharp hawk]

the smooth functions are not [A] or [B] and they don't need to 🙂 we don't calculate the slopes of [A] or of [B] 🙂

We USE [A] and [B] (the lhs and the rhs of the geometric theorem) to calculate the slope of the tangent of a smooth function f such as f(x)=x²

[tardy ember]

what are [A] and [B] then, if they're not smooth functions?

[sharp hawk]

so no need to address "two smooth functions" because it's not what we're doing

[tardy ember]

since they'e also not functions

[sharp hawk]

f(x)=x² is the smooth function

and we want its derivative that's all

[tardy ember]

so would $f = {(x,y) : x \in \mathbb{R} \land y = x^2}$, the function that maps any real number $r$ to $r^2$, be different and not a smooth function?

[sharp hawk]

and it's all geometry. so it's not possible to address it in any other way than geometric because that's what I am presenting you: a geometric theorem

[burnt vessel]

bee [it/its]

[tardy ember]

...right, ok, that's the issue here

you're using a definition of function that's... geometric, apparently

[sharp hawk]

this f(x)=x² is a smooth function yes

[tardy ember]

...ok now i'm just confused

[long pelican]

Pre-Cauchy/Weierstrass/others (responsible for building math on set theory) definition of function

[sharp hawk]

but I regard ℝ just as geometric magnitudes that's all

rational and irrational lengths

[tardy ember]

ok, are $f(x) = \frac xx$ and $f = {(x,y) : x \in \mathbb{R} \land x \neq 0 \land y = 1}$ the same function?

[burnt vessel]

bee [it/its]

[tardy ember]

the second one is the function that is not defined at 0, and maps any nonzero real number to 1

[sharp hawk]

the first one equals 1 when x is non-zero , and the second one does the same thing?
If so, they are the same function

[tardy ember]

alright \ are $f(x) = \frac xx + J(x)$ and $f = {(x,y) : x \in \mathbb{R} \land x \neq 0 \land y = 1}$ the same function?

[burnt vessel]

bee [it/its]

[tardy ember]

they're both equal to 1 when x is non-zero, and not defined when x = 0

[sharp hawk]

so J(x) is 0 for any non-zero x

[tardy ember]

yes

[sharp hawk]

yes

and then you will want to make it 1, for x=0 ?

[tardy ember]

yes, J(0) = 1

[sharp hawk]

LHS and RHS (that [A] and [B]) are not the same function, certainly

[A] is the slope of the SECANT
[B] the slope of the TANGENT + the slope difference

[A] and [B] are about two different geometrical objects. Which have the same slope (that's how parallelism comes into play and proves useful)

[tardy ember]

are $f(x) = \frac xx$ and $f(x) = \frac xx + J(x)$ the same function?

[burnt vessel]

bee [it/its]

[sharp hawk]

yes, if J(x) is 0 for any non-zero x

[tardy ember]

yep, it is

so now if we "simplify" both of them...

$f(x) = 1$, and $f(x) = 1 + J(x)$

[burnt vessel]

bee [it/its]

[sharp hawk]

but simplification works only when x is non-zero

[tardy ember]

for one of these, f(0) is 1, and for one of them, f(0) is 2

so we "simplified" the same function into two different functions

[sharp hawk]

because we attempted to simplify for the case x=0 too

[tardy ember]

and yes, you're right, it doesn't actually make sense to do this for x = 0

alright so looking back at trying to... calculate the derivative of x^2? or whatever it was we were doing?

we have ((x+n)^2-(x+m)^2)/(n+m), i think

[sharp hawk]

we can simplify [A] because m and n are never both 0 in [A] since it is the slope of the parallel secant

[tardy ember]

alright here it is

so we simplify it to 2x + n - m

but for the exact same reasons, this doesn't make sense if n + m = 0

[sharp hawk]

certainly

[sharp hawk]

.

in [A] we simplify it only when m and n are not both 0

[tardy ember]

...and then we plug in n = m = 0 and claim the derivative is 2x

but if n = m = 0, then n + m = 0 and the simplification was invalid

[sharp hawk]

here is the subtlety : we plug in [B] not in [A], and [B] is about a different geometrical object

[B] is about the tangent

where m and n CAN be 0

it's parallelism that makes the equality possible

I know exactly what you mean

[tardy ember]

ok but you can't just produce sense out of nowhere by talking about geometry

[sharp hawk]

if you analyze it carefully you will see it is valid

[tardy ember]

for the exact same reasons as before, ((x+n)^2-(x+m)^2))/(m+n) and ((x+n)^2-(x+m)^2))/(m+n) + J(m+n) are equal functions

but if you "simplify" the second one, you get 2x + n - m + J(n+m), which is a different function that we're claiming is B

[sharp hawk]

2x + n - m the simplification was valid because m and n were not both 0, do you agree ?

and 2x + n - m represents the slope of the parallel secant

[tardy ember]

if m + n isn't 0, then simplifying it to 2x + n - m is valid

if m + n isn't 0, then simplifying it to 2x + n - m + J(m+n) is valid

[sharp hawk]

ok. this is the first step towards understanding this mechanism.

and what [B] tells us is that the slope of ANY (parallel or non-parallel) secant, i.e. [A] , which is 2x + n - m

is the same as the slope of the TANGENT, plus the difference in slopes Q(x,m,n)

2x + n - m for non-zero m,n EQUALS the slope of the TANGENT, plus the difference in slopes Q(x,m,n)

[tardy ember]

...ok, yes, that's definitely true for some definition of Q(x,m,n) (you can just define it as "2x + n - m - the slope of the tangent")

[sharp hawk]

Q(x,m,n) means the difference between the tangent's slope and a (parallel or non-parallel) secant's slope

[tardy ember]

...so how does it help to know that the slope of a secant is equal to the slope of that secant plus the slope of the tangent minus the slope of the tangent

[sharp hawk]

yeah, good question.
the slope of the TANGENT, does not depend on m and n, do you agree with this?

[tardy ember]

yes

[sharp hawk]

so the resulted portion in [B] which does not depend on m and n, is precisely the tangent's slope

[tardy ember]

...ok but how do you tell what the portion that "doesn't depend on m and n" is

like, you can write it as 2x + n - m and then it looks like it's 2x

or you can write it as 2x + 1 + n - m - m/m and then it looks like it's 2x + 1

[sharp hawk]

very valid question , and we will address it now

[sharp hawk]

FAQ 2. But this requires that the difference in slopes Q(x,m,n) which we get when we calculate [A]), never has m+n as a denominator. How do we know this for sure?
Answer: Theorem [T1]. Q(x,m,n) never has m+n as a denominator.
Proof
For any parallel secant we have:
[A]=[B], that is
secant slope=tangent slope+ slope difference
,in mathematical symbols:
[f(x+n)-f(x-m)]/(m+n)=f'(x)+Q(x,m,n), where Q(x,m,n)=0
⇔[f(x+n)-f(x-m)]/(m+n)=f'(x)
Since RHS=f'(x) does not depend on m+n, it must be that m+n is a "real" factor of every term in [f(x+n)-f(x-m)]. QED.

[sharp hawk]

let me think about the way you wrote it

the way you wrote it, makes [B] undefined for m=0 , but [B] (which represents tangent slope+ slope difference) is always valid, even for m=n=0

so it is impossible actually to get something in that form 🙂

[tardy ember]

alright what if we write it as 2x + 1 + n - m - sin^2(m) - cos^2(m)

[sharp hawk]

it's correct and it works 🙂

set m=n=0 and you'll see that we get the derivative 🙂

[tardy ember]

...ok wait, so we're not taking the part that doesn't depend on m and n, we're taking the value when m = n = 0?

[sharp hawk]

indeed, it seems in this case not. We just take the whole of it.

[tardy ember]

but doesn't that have the exact same issues as earlier?

we got this expression from simplifying something that isn't defined at m = n = 0

[sharp hawk]

so that would be a better description. that we set m=n=0 and all that remains, is the derivative.

[tardy ember]

if, in the process of simplifying A, we had added J(m+n), which doesn't change what function it is (because either way it's not defined at m + n = 0)

[tardy ember]

then now we'd have 2x + n - m + J(m+n) and we'd get 2x+1 as the derivative

[sharp hawk]

(and also note that in practice, it won't happen that you add a constant 1 which you then subtract as sin^2(m=0) )
so it was for the sake of that example, that we had to change the description; but basically it's still that m=n=0 and what we obtain is the derivative

Because look,

in that example, setting m=n=0 rendered a non-zero Q(x,m,n), which is impossible

if you see that, you will understand why I claimed it's actually natural and easy and direct.

[tardy ember]

but if we're setting m=n=0, then we run into all the issues from earlier

because we got this expression in the first place by simplifying A

and A isn't defined at m=n=0

[sharp hawk]

[A] is the secant line

the magic is in the geometry: a parallel secant line HAS the same slope as the tangent it is parallel to

that's all it means.

that's all that [A]=[B] means, and there is no issue

[tardy ember]

there... is... though...?

like ignoring the geometry and looking at the actual computation we're doing

[sharp hawk]

well, look deeply into it

[tardy ember]

we take A, which is not defined at m=n=0

we simplify it

we then evaluate it at m=n=0

[sharp hawk]

no, we don't evaluate [A]

for m=n=0

[tardy ember]

we evaluate the result of simplifying A, at m=n=0

you can call that B if you want but it won't change the fact that we computed it by simplifying A

and didn't we agree earlier that that kind of thing doesn't work?

$f(x) = \frac xx$ and $f(x) = \frac xx + J(x)$ are the same function, but if you ``simplify'' both of them and evaluate at $x=0$, you get different answers

[burnt vessel]

bee [it/its]

[tardy ember]

so if you take two different expressions for A that are the same function, one with an extra J(m+n)

you simplify both of them

you get two different expressions for B

you evaluate at m=n=0

you get two different answers

[sharp hawk]

the slope of the secant 2x + n - m for non-zero m,n EQUALS the slope of the TANGENT, plus the difference in slopes Q(x,m,n)

geometry tells us this. And we don't need to set m=n=0 in [A] , it would be absurd since a secant exists only when m and n are not both 0

[sharp hawk]

in your example it happens because you made J(x) have a non-zero value

[tardy ember]

...do you agree that the actual computation we're doing, when you ignore all the geometry, is to take A, an expression that is not defined at m=n=0, simplify it, call the result of simplifying it B, and evaluate B at m=n=0?

[sharp hawk]

WE CANNOT IGNORE the geometry.

geometry and only geometry makes [A]=[B] make sense , a very precise sense

[tardy ember]

...hm

ok let's look in more detail then

[sharp hawk]

[A] and [B] encode calculations for the slopes of the geometrical objects called tangent and secant

[tardy ember]

i'm going to reply to this message again for reference

[sharp hawk]

sure

[tardy ember]

we start with $A=\frac{(x+n)^2-(x-m)^2}{m+n}$

[burnt vessel]

bee [it/its]

[tardy ember]

this is the same function as $A = \frac{(x+n)^2-(x-m)^2}{m+n} + J(m+n)$, right?

[burnt vessel]

bee [it/its]

[tardy ember]

because whenever either expression is defined (so m + n is not 0), they're the same

[sharp hawk]

yes, given that J(m+n) is 0 when m+n is nonzero

[tardy ember]

so we simplify

$A = 2x + n - m$

[burnt vessel]

bee [it/its]

[tardy ember]

$A = 2x + n - m + J(m + n)$

[burnt vessel]

bee [it/its]

[sharp hawk]

simplification works ONLY when m+n is non-zero.

[tardy ember]

yep

[sharp hawk]

agree,good.,

[tardy ember]

and now we call these expressions B

so we have either $B = 2x + n - m$, or $B = 2x + n - m + J(m + n)$

[burnt vessel]

bee [it/its]

[tardy ember]

and now we evaluate at m = n = 0

$B = 2x$ \ $B = 2x + J(0)$

[burnt vessel]

bee [it/its]

[sharp hawk]

but this is not what [B] is, [B] is NOT the result of [A] , that's the subtle key

[B] just happens to EQUAL to [A] always

[tardy ember]

...so how do we actually find B, if simplifying A to get it doesn't always work?

[sharp hawk]

you do find [B], but [B] is a different geometrical object 🙂 it is not a consequence of [A], it just equals [A] always .

[tardy ember]

but... how

how do we find B

[sharp hawk]

we simplify [A] WHEN m+n is NOT 0

[tardy ember]

yeah we tried that, and we got 2x + n - m + J(m+n)

and that's not correct

even though it is a valid simplification of A, when m + n isn't 0

[sharp hawk]

what is not correct is that you make J(m+n) be NON-ZERO at some point. 🙂 obviously it's invalid

you are adding a non-zero value.......... and thus you are changing the equality from the theorem [A]=[B]

[tardy ember]

??

ok so what procedure would i follow to avoid doing that

[sharp hawk]

[A]=[B] IS NOT about adding a non-zero value as you please, only on RHS

well there is no reason to even THINK of J(n,m).

you are adding it out of nowhere, it is not needed, it does not appear in our theorem

so no need for us to think about it. let's think about the theorem itself, what IT says

[tardy ember]

ok but like, in more complicated situations

[sharp hawk]

the thing is simple: you are not saying that J(m+n) IS ALWAYS ZERO . just sometimes. well, that makes it NOT ZERO

[tardy ember]

if i was reading a proof written by the devil how would i make sure that they didn't slip something like this in somewhere

something more complicated than J, that only works when m + n isn't 0

[sharp hawk]

if you want to add something just on LHS or just on RHS, it has to be ZERO ALWAYS , otherwise adding it is not valid.

[tardy ember]

ok, slight problem with that, this proof written by the devil never actually adds anything to anything, they just claim a value for B out of nowhere, and then prove that it gives the same value as A when n + m != 0

so how am i supposed to check it now

[sharp hawk]

how to prove that [A]=[B] when m+n≠0 ?

[tardy ember]

no they did prove that

just, clearly that isn't sufficient for the value of B at m=n=0 to be the derivative

so how do i check that this expression that they're claiming is always equal to B (and which is always equal to A when A is defined) actually is

[sharp hawk]

so they did prove that

slope of secant=slope of tangent_not_depending_on_m_plus_n + some difference which is a sum of factors of x,n,m

and that

slope of parallel secant=slope of tangent_not_depending_on_m_plus_n + 0

so if we simplify LHS in this last equation above
we get
part_not_depending_on_m_plus_n + a sum of factors of x,n,m, EQUALS slope of tangent_not_depending_on_m_plus_n + 0

[tardy ember]

...no they didn't mention anything about parallel secants actually

they just gave an expression, claimed that it's always equal to B, and proved that it's always equal to A when m+n isn't 0

(i don't know how you'd check whether a secant is parallel without just, already knowing what the derivative is?)

[tardy ember]

and this isn't enough, because their "B" might have some weird J(m+n) thing in it

but then what would be enough?

[sharp hawk]

⇒ part_not_depending_on_m_plus_n=slope of tangent_not_depending_on_m_plus_n

[sharp hawk]

In other words:
We actually NEVER get to m=n=0.

When I said 'we set m=n=0 in [B]' it was only to help one visualize which part can vanish (in order to identify the derivative that remains). But that part depending on m and n in [B] is already ZERO for all NON-zero pairs (m;n) corresponding to parallel secants.
It is already vanished, all the time. And we don't set m=n=0, we just identify which part is the derivative

Really simple.
'And how do you identify?' by using the fact that all the terms which depend on n or m are part of the expression which is 0 (vanished); computationally setting m=n=0 also would vanish it but it's already vanished anyway for all NON-zero m,n, so 'setting m=n=0 in [B]' is just a metaphor to help one see which part is to be discarded so that the derivative remains

[sharp hawk]

This should solve all the issues your rightfully pointed out about my description of the equation [A]=[B].
My description earlier was not accurate enough since I said we can set m=n=0 in [B], which is not the case as it would invalidate the equation.

I acknowledge that I should describe more carefully that we never set m=n=0 in the equation and there is no need for that. It just happens that computationally if one sets m=n=0 in [B] this results in the part depending on m and n vanishing and what remains is the derivative. But that part in the equation actually is already always vanished (zero), it's just symbolically written in letters (which sum up to 0 always).

@tardy ember thanks for pointing out the sloppiness in my earlier description, you're great I have rarely discussed like that with anyone

It's the first time I feel like falling in love, on a math server?! Wdf

[tardy ember]

...but wouldn't you need to have a parallel secant in order to work out which terms vanish for a parallel secant? and if you have that then you already know what the derivative is anyway

also there might not be any parallel secants

x^3 is strictly increasing, so all of its secants have positive slope, but its derivative at x=0 is 0, its tangent line there is horizontal

[sharp hawk]

Yes, and you don't know the derivative

It's from that (the facts that

• the secant is parallel and that - the tangent slope does not depend on m and n)
  that the derivative is found

[sharp hawk]

FAQ 3. How to prove there is a secant line whose slope is the same as the tangent?
Answer: It's quite trivial to visualize it: it's a proof without words (geometric) that even a child can understand.

One can also provide a proof in words
Definition [D2.1]
"A tangent line is a line such that no other straight line can fall between it and the curve." - Apollonius work on Conics, around 225 BC)
Definition [D2.2]
"A tangent line is a (straight) line that touches a curve at a point without crossing it at that point, and extends to both sides"
The above two definitions refer to the same thing (they are equivalent).

Note: only non-straight lines have tangents; a straight line cannot be tangent to itself.

Theorem [T2]
Any tangent to the graph of a smooth function has a parallel secant.
Proof:
By definition of a tangent line [D2.2], the curve has one and only one point A in common with the tangent. [TG]
By definition of smoothness, there is one and only one tangent line at point A. [SM]

[SM] implies we cannot rotate the tangent about point A and obtain a new tangent. This implies that if we rotate it x radians about point A, we obtain a secant AB. Now we can rotate it -x radians about point B and we get a new secant BC that is parallel to the original tangent through A. QED.

We can observe it the other way around too:
Theorem [T2.2]
Any secant s is parallel to some tangent
Proof:Let s=AB, where A,B∈graph of the function.
A and B can be dragged along the graph to become A'B' such that A'B'∥AB and A'B'<AB. We can go on dragging until the distance A'B' becomes 0, that's when A'=B' is the point of tangency and the parallel to AB through that point is the tangent. QED.

[tardy ember]

Now we can rotate it -x radians about point B and we get a new secant BC
what if there isn't a C?

that definitely gets you a line through B that's parallel to the tangent at A, but that line might not be a secant

[sharp hawk]

impossible, when I wrote this proof I thought well about this. But let me think how to formulate in words an answer to this question

[tardy ember]

and in the example of x^3
what exactly is the horizontal secant of x^3?

[sharp hawk]

at point 0? there is no issue with the cubic because there is no tangent at 0 that's why there is no parallel secant

this is consistent with the definition of the tangent line

[tardy ember]

...oh, huh
ok yeah i guess by that definition there is no tangent there

although that does imply that you can't compute the derivative of x^3 at x=0 which is a bit weird

[sharp hawk]

yes

it feels weird because in the mainstream calculus the definition has been changed; but the fact is there is no tangent line there so there is no issue and it is consistent

[sharp hawk]

this would imply that the point B is where the graph ends, so it is not smooth at B so indeed no derivative there. But in NC we work with smooth curves always
In case we run into this situation it means we are at a point where it is not smooth and there is no derivative there so there is no issue, of course the theorem doesn't apply, because the theorem applies ONLY where the graph is smooth

just as |x| (often given as an example in mainstream calculus) is not smooth at x=0

[tardy ember]

(...well it does also imply that the product rule isn't true by this definition, and also just that this is in fact computing something that's different from the standard definition of derivative, but yes it is consistent to define it this way)

...wait does it imply that the product rule isn't true?

[sharp hawk]

please expand on the product rule, how is it made untrue by this

[tardy ember]

ok actually the product rule might still be true? because x isn't differentiable, by this definition

[sharp hawk]

I have proved the product rule too

and the chain rule 🙂 and more...

The Product Rule in New Calculus

If function h is the product between functions f and g.
h(x)=f(x)⋅g(x)
then
h'(x)
={h(x+n)-h(x-m)}/(m+n)
where as usual m and n are horizontal distances from the point of tangency (x,f(x)) to the endpoints of a secant line that is parallel to the tangent
={f(x+n)g(x+n)-f(x-m)g(x-m)}/(m+n)
={f(x+n)g(x+n)-f(x-m)g(x-m)+0}/(m+n)
={f(x+n)g(x+n)-f(x-m)g(x-m)+f(x+n)g(x-m)-f(x+n)g(x-m)}/(m+n)
swap term 2 and term 4
={f(x+n)g(x+n)-f(x+n)g(x-m)+f(x+n)g(x-m)-f(x-m)g(x-m)}/(m+n)
take out the common factors f(x+n) and g(x-m)
={f(x+n)⋅[g(x+n)-g(x-m)]+[f(x+n)-f(x-m)]⋅g(x-m)}/(m+n)
split into two fractions
=f(x+n)⋅[g(x+n)-g(x-m)]/(m+n)+[f(x+n)-f(x-m)]⋅g(x-m)/(m+n)
=f(x+n)⋅g'(x)+f'(x)⋅g(x-m)
by an earlier proof we can set m=n=0
And we're done:
f(x)⋅g(x)=f(x)⋅g'(x)+f'(x)⋅g(x)
no esoteric limits stuff involved

now let me check that part by an earlier proof we can set m=n=0 again.

[tardy ember]

(tbh i feel like this stuff about "set m=n=0" probably actually turns out to just be limits, if you interpret it as a limit then you get almost exactly the standard definition of a derivative, and i don't see how else you can formalise it)

[sharp hawk]

it gets to the same results as limits, but it is not limits

the area of a triangle could be calculated by limits and integrals, but it can also be calculated simply base*height/2 which is not using limits any more

[tardy ember]

i... don't get what it is then

like, how do you formalise it

how would you explicitly write out a statement involving the variables f, x, y, that's true if and only if f'(x) = y

[sharp hawk]

the expressions are always exact, in terms of fixed lengths, no need of limits

dx and dy are not infinitesimals any more, they are fixed lenghts. dx is that m+n

the mean value of f'(x), which we'll call f'(µ), times dx gives the area under the graph of f

[tardy ember]

with the standard definition that looks like \ $\forall \varepsilon > 0 , \exists \delta > 0 , \forall h , (0 < h < \delta \Rightarrow \left|\frac{f(x+h)-f(x)}h - y\right| < \varepsilon)$

[burnt vessel]

bee [it/its]

[tardy ember]

(or at least i think that's right)

[sharp hawk]

yes

[tardy ember]

assuming i wrote that correctly, that statement is true if and only if f'(x) = y, and that is the standard definition of a derivative

what's the analogous statement here? what is the definition of "f'(x) = y" that you're using?

[sharp hawk]

[A]=[B]
⇔slope of the secant=f'(x)+differences in slopes
⇔f'(x)=slope of the secant-differences in slopes
in other words
f'(x)=slope of the parallel secant <-

[tardy ember]

...is that supposed to be an answer to this or is it something else

[sharp hawk]

yes, f'(x) is equal to the slope of the parallel secant which is {f(x+n)-f(x-m)}/(m+n) for some m;n (not any m,n)
No other definition for the derivative

[tardy ember]

ok well uh

what are n and m

and what is a parallel secant

if the definition you're proposing is $\exists n,m : y = \frac{f(x+n)-f(x-m)}{m+n}$ then that implies most functions have multiple derivatives at any one point \ if it's something else then you need to be more clear about what it is

[burnt vessel]

bee [it/its]

[sharp hawk]

why do you say it can have multiple derivatives?

[tardy ember]

well like, if you put in m=1 n=1, you might get a different value from if you put in m=1 n=2

[sharp hawk]

because there are many possible (m;n) pairs?

[modern trench]

???

[sharp hawk]

ah  🙂 no , you always get the same value for ANY secant that is parallel to the tangent. lemme find a drawing

[tardy ember]

ok so what does it mean for a secant to be "parallel to the tangent"

what statement in terms of f,x,m,n is only true if the secant defined by m,n is parallel to the tangent

[modern trench]

i.e. How do you define the tangent line without using definition of derivative circularly

[tardy ember]

that doesn't explain what a "parallel secant line" is

[sharp hawk]

a secant that is parallel to the tangent

[tardy ember]

and what does it mean for a secant to be parallel to the tangent

[sharp hawk]

secant mean straight line that intersects the graph of the function at two distinct points

so the question is what does it mean for two lines to be parallel, right?

which is a very good question

[modern trench]

More of "what is the definition of a tangent line (the red line)"

[sharp hawk]

orange line is parallel to the green line
⇔ the two red arcs are of equal lengths

[sharp hawk]

Definition [D2.3]
"A tangent line is a straight segment not crossing the graph at the point of tangency"

[modern trench]

What if the graph is like y = |x|

Or y = x³

[sharp hawk]

what about it...

[modern trench]

Multiple line through (0, 0) that don't cross the graph

[sharp hawk]

no issues, and we have discussed these few hours ago

[tardy ember]

the blue line doesn't cross the red line

[sharp hawk]

yeah there is no derivative there

[modern trench]

All lines through y = x³ cross (0, 0)

[sharp hawk]

so no derivative there.

[modern trench]

Or what does cross mean

[sharp hawk]

in NC.

no tangent ⇔ no derivative, simple and consistent

[tardy ember]

...hm
ok yeah i guess this... does work...? the question now is just what "cross" means

[modern trench]

Does cross mean going from above the graph to below the graph

[sharp hawk]

I think it's easy to understand what cross means and what a tangent line is

[sharp hawk]

yes, no tangent there

[tardy ember]

he's declared that that isn't a tangent line

[sharp hawk]

since it crosses

[modern trench]

I think your approach to calculus is very similar to what people originally did

[sharp hawk]

certainly

[modern trench]

Some centuries ago

[sharp hawk]

certainly the original definitions were more consistent than what was done afterwards

[tardy ember]

well i have no idea how to write it entirely in terms of just arithmetic operators and comparison and quantifiers and stuff like that, so it doesn't actually seem particularly easy...?

[sharp hawk]

that's not an issue because geometry is maths without relying ONLY on math operators

[modern trench]

For nice enough functions (piecewise C1) this certainly suffices to do a lot of calculus

But the point of more recent developments is to make stuff rigorous

And much more general

[tardy ember]

the closest i can think of is that "crossing" a line means that in a sufficiently small open set that includes the point of intersection you're going to a different connected component, but that kind of requires a notion of continuity

[sharp hawk]

do you need math operators to prove that x² is continuous ?

[tardy ember]

ok well the issue with that is

if you have an argument entirely in terms of the axioms of ZF, you can check it

you can write a program to check it

you can mathematically analyse the space of all possible arguments and do things like find statements that provably can't be proven

[sharp hawk]

that's a good topic I am also interested in: can geometry be checked automatically ?

[tardy ember]

if you have an argument in terms of geometry... how do you do that stuff

i suspect that what you'd have to do is write into the program a list of known-valid geometric inferences, and require an argument to be a sequence of those

but at that point that's just axiom-based reasoning where the thing the axioms describe happens to be geometry, and i don't see a benefit of doing that but axiomising geometry instead of set theory

[sharp hawk]

so, one question is whether automatic theorem provers would be able to understand what a tangent line is or not

[modern trench]

One way this is advantageous over basing math in geometry is this you can do analysis ("calculus") on spaces with arbitrarily high dimensions (even infinite-dimensional, say function spaces)

[tardy ember]

and while theorem provers are the most blatant example, there's also like, humans might not know what's a valid geometric argument

[modern trench]

Also a lot of issues with rigor in calculus eventually had to be resolved by stating everything using limits

[sharp hawk]

humans understand geometry without much difficulty

both have their merits

I am not against using formalizations

[modern trench]

Yes, and in fact what is taught in many calculus classes is more or less just geometric intuition

But for doing modern math beyond that you need rigor

[sharp hawk]

geometry is rigorous too

[modern trench]

Well, yes, but the geometric intuition that is taught is not very rigorous

Since the actual definition of limit isn't even usually taught in calculus classes

[tardy ember]

...defining "crossing" as "i think humans can easily understand this" isn't rigorous

[sharp hawk]

it is not good for machines

[tardy ember]

like imagine if you had two humans who disagreed on some weird edge case of whether two weird curves cross or not

[sharp hawk]

you are used more to formulas than to simple words, but in fact they are both just symbols which express a certain meaning

[tardy ember]

what would they do about that

the definition wouldn't help at all, they could just end up disagreeing forever

that's not how maths is supposed to work

[sharp hawk]

if one gets the meaning, that's rigorousness

[tardy ember]

actually i am pretty used to words, a lot of things aren't usually written with entirely formulas

but like, usually the words all have definitions

[sharp hawk]

well that's what happening in the mainstream calculus .............. Look:
"When called upon to teach real analysis some years ago, I was amazed to find that the definition of continuity in our textbook (Bartle and Sherbert 1992) [1] was significantly different from the one I had learnt as a student from Siddons et al. (1952) and Hardy (1952), even though that definition is at the root of the entire subject."
"it is not only undergraduates who find continuity difficult."
"Courant (1937) had two definitions that appear to conflict with each other."
"The definitions in all three of Jordan (1882), Whittaker (1902), and Hardy (1908) must have been found unsatisfactory, because they were changed in the second editions of those books: Jordan (1893), Whittaker and Watson (1915), and Hardy (1914)"
"That least acceptable case has now become the most acceptable."
"The conflict between definitions of pointwise continuity seems not to be well
known."

• J F Harper, Victoria University of Wellington, New Zealand "Defining continuity of real functions of real variables" doi:10.1080/17498430.2015.1116053 https://sci-hub.se/10.1080/17498430.2015.1116053

"All three of those nonequivalent definitions of pointwise continuity are still in use." - Wikipedia https://en.wikipedia.org/wiki/Continuous_function

[tardy ember]

no that's completely different

that's the same word being given multiple distinct definitions that are all entirely precise

the people there don't disagree on any statement of mathematical fact, they're just using slightly different conventions

for any one of those definitions of continuity, it is impossible to get stuck on the definition and therefore be unable to resolve a disagreement about whether a function meets that definition or not

[sharp hawk]

and the dilemma was what to do if different people give distinct definitions for "crossing"

[tardy ember]

no

the dilemma was what to do if different people have different intuitions about "crossing"

[sharp hawk]

can that be the case?

[tardy ember]

with the entire problem being caused by a lack of definition

[sharp hawk]

can you understand crossing differently than I ?

then it's an issue, yeah

[tardy ember]

probably?

[sharp hawk]

but is it really the case?

[tardy ember]

i mean in simple cases we'd probably agree, but like

surely you can have some weird function where it's difficult

[modern trench]

Well I think "crossing" is sufficiently well-defined if you use this defn

[tardy ember]

the same way that there are functions that are continuous everywhere and differentiable nowhere, or functions which have a derivative but the derivative is not continuous

[sharp hawk]

'This story got me scared. Starting from 1993, multiple groups of mathematicians studied my paper at seminars and used it in their work and none of them noticed the mistake.... A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail. - Vladimir Voevodsky

"the fact that he was able to prove a "weaker and more complicated lemma which turned out to be sufficient for all applications" matches my own experience. For instance, while working on a recent project I discovered no fewer than nine mistaken theorem statements (not just mistakes in proofs of correct theorems) in published or almost-published literature, including several by well-known experts (and two by myself). However, in all nine cases it was simple to strengthen the hypothesis or weaken the conclusion in such a way as to make the theorem true, in a way that sufficed for all the applications I know of.

I would argue that this is because the mistaken statements were based on correct ideas, and the mistakes were simply in making those ideas precise. Or to put it differently, mathematicians get our intuitions from "well-behaved" objects: sometimes that intuition can be wrong for "pathological" objects we didn't know about, but in such cases we simply alter the definitions to exclude the pathological ones from consideration.'

[modern trench]

There exists some open interval around p such that there is a line segment with domain the open interval around p that stays above or below the function (nonstrictly)

The problem is, this places lots of limitations

Since you can't even differentiate say y = x³ at 0

[sharp hawk]

what's the limitation? in what way affects us? it's consistent with the fact that there is no tangent there

[sharp hawk]

"On the other hand, people do sometimes get mistaken ideas. For instance, here's another quote from Voevodsky's article:"

    I can see two factors that contributed to this outrageous situation: Simpson claimed to have constructed a counterexample, but he was not able to show where the mistake was in our paper. Because of this, it was not clear whether we made a mistake somewhere in our paper or he made a mistake somewhere in his counterexample. Mathematical research currently relies on a complex system of mutual trust based on reputations. By the time Simpson’s paper appeared, both Kapranov and I had strong reputations. Simpson’s paper created doubts in our result, which led to it being unused by other researchers, but no one came forward and challenged us on it.```

"So what do you do as a student? In addition to the other good advice that's been given, I think one of your primary goals should be to train your own intuition. That way you will be better-able to evaluate whether a given result, or something like it, is probably true, before you decide whether to read and check the proof in detail.

Of course, there is also the position that Voevodsky was led to:"
```And I now do my mathematics with a proof assistant. I have a lot of wishes in terms of getting this proof assistant to work better, but at least I don’t have to go home and worry about having made a mistake in my work.```

[modern trench]

It is consistent, but usually it would be nice if very well-behaved functions like all polynomials could be differentiated everywhere

Also you lose nice properties like "f(x) and g(x) differentiable at p implies f(x)g(x) differentiable at p"

[modern trench]

So what's the definition of the derivative of x³ at 0

[tardy ember]

(that's not a counterexample because by this definition x isn't differentiable)

[tardy ember]

ok actually it is true that this turns out to not be true

x^2 and 1/x both have derivatives at x=1, but their product doesn't

[tawdry venture]

christ what the hell went on in here

[tawdry venture]

...

1. "christ" was an interjection, and not actually referring to any religious figure
2. why instantly act so hostile and call me names?

<@&268886789983436800>

[toxic talon]

yeah that's a good expectation

[plain valve]

Indeed, especially since I just muted lol

[tawdry venture]

ok at least i have time to block this fucker lmao

[toxic talon]

,prune 100 --from 595406671107325983 --force

[burnt vessel]

Couldn't find a member matching 595406671107325983!

Feb 4, 2024, 3:03 PM

Couldn't find the requested user, cancelling purge.

Feb 4, 2024, 3:03 PM

[plain valve]

lol did people unironically ask about new calculus above

[toxic talon]

idk either way it doesn't fit at all in this channel

[fallen needle]

god complex on discord is actually wild.

[cloud zealot]

holy shit, the real john gabriel showed up and i missed it

bump

[wheat swift]

speak of the devil and he shall appear moment

[tawny slate]

What.... just happened

lol I found a Google groups link that says he died last year

[long pelican]

You sure that's not the actor?

[tawny slate]

https://groups.google.com/g/sci.math/c/EFV3KFBMaak

[long pelican]

ok there's 2021 for the actor and 2023 for the non-actor

[tawny slate]

I have no clue anymore

[long pelican]

yep I browsed a bit and saw conflicting information

[cloud zealot]

https://www.reddit.com/r/matheducation/comments/1ahhn6c/no_more_simplifying/

[winged urchin]

Overall I believe most of us here would likely scream YES to something like this

We don't simplify to just make it look nicer, we 'simplify' to do something.

I do find it amusing that they claim that 3/sqrt(2) is 'clearly' simpler than 3sqrt(2)/2. It seems the author is committing the same sin they are speaking out against ahah

Though I do wonder, what would we say to a student whom we may take a mark off for say leaving an answer as pi^2 + sin(pi) + 1 vs. showing they know what sin(pi) is and then I would be tempted to say pi^2 + 1 is 'simpler' but what can we say to avoid that nasty word?

I think simplify is one of those dirty words though that show up far too much. Like 'cancel' or 'moving terms'

[tawny slate]

Pretty much no disagreements there. I always tried to be mindful of the usage of "simplify" to things that are more clearly "simpler", but also giving explicit examples where "un-simplifying" is preferred in order to highlight that the point isn't simplification itself but just picking the right tool for the right situation, which is a much more nuanced and difficult problem

[vocal phoenix]

I shall work to make the term "complicate" more popular.

[lethal leaf]

I mean sometimes it is just simplification: consider combining like terms

But also yes in some cases a factored form is simplified, or a fully expanded form

Context is key and that's hard to teach at times

[winged urchin]

Actually I just thought... there are occasions where even combining like terms doesn't make things 'simpler'

Like when you are trying to find a pattern in a series or sequence

If you add up all the numbers into one then you can lose the overall pattern

[lethal leaf]

yea

For example solving for a closed form of a recurrence by "unrolling"

[lethal leaf]

hence this message

[winged urchin]

Math has taught me that we need to be very VERY careful saying completely general things. "This is like this ALWAYS"

If we think hard enough there's some weird example that defies our expectations

[wise onyx]

Yep

[tall bolt]

This isn’t really the place to discuss this, maybe try #discussion or #math-discussion this is for people to discuss teaching techniques

[hazy crag]

Okay, thanks for telling me. I just thought math anxiety to be part of learning mathematics

[nova turtle]

meow

[plain valve]

Yikes

[maiden swan]

so true

[spring ocean]

easy ban

[blissful carbon]

.reopen

[tight star]

How do you get people more comfortable with abstraction?

In terms of a specific case where this popped up - i was reading through the intro and elim rules for “forall” and “there exists” and feel like I understand them a lot better now, but I’ve also been told that they can be a bit too abstract for people starting out with proofs…

[cloud zealot]

"Here are so-and-so concrete examples. Observe so-and-so pattern(s) or that they are similar in a crucial, to-be-defined way." State precisely the unifying idea(s). That's one way to do it.

[cloud zealot]

what are these introduction and elimination rules exactly

[tight star]

So for there exists, it’s

In order to introduce/make “there exists x P(x)”, you have to have previously proved P(a) for some a

And to eliminate/use “there exists x P(x)” to prove Q, you have to prove that “forall x, P(x) implies Q”

For forall I treat it more like a function, so proving “forall x P(x)” is kind of like making a function that takes an arbitrary object x and provides a proof of P(x), though I know formally it’s not actually this ofc

And then you can use it like a function too

[cloud zealot]

so...i don't see a need to say this kind of stuff. it's okay to interpret quantifiers in an intuitive way in an introduction to proof.

[tight star]

Hmm I think my issue was

I interpreted “there exists” much more as “there’s a specific object a for which P(a) holds”

But that’s not really what it actually means

Which messed with my intuition until I saw the rules for it

I mean, I guess a big thing here is that I don’t really know how to teach people proofs

[cloud zealot]

are you okay with following a textbook

[tight star]

What do you mean?

[cloud zealot]

are you trying to teach people proofs for a class?

maybe you can use the textbook as a guide

[tight star]

Not at the moment, no

But it might be something I have to do in the future

I’m not actually sure what good “proof” textbooks there are…

[cloud zealot]

#book-recommendations message

[tight star]

Oh, thanks!

[tawdry venture]

ok so

i have a set of tests to grade

the test consists of 7 problems of moderate difficulty, but all the problems are approximately equally hard

the teacher did not assign point values to the problems

i, as a TA, might want to assign them all a uniform point value

and i am deciding between 3 pts or 4 pts per problem

the problems themselves are radical inequalities

i am kind of leaning towards 4 so that i can give half credit (2 pts) for a correct answer without working

what do y'all think

gonna ask a colleague in the meantime

[next relic]

(1) correct answer without working would be 0 in my book unless I made it explicit that I don't want any explanations.
(2) you may want to have all the questions adding up to 25 or 30, just for roundness :p

[tawdry venture]

25 and 30 are both non-multiples of 7.

[next relic]

That means some questions are worth 3 and some are worth 4.

Or some 4 and some 5.

The number of marks should commensurate the amount of work anticipated.

Without knowing the exact questions on the test, I can't really go deeper. But I think 2-5 marks per question, depending on difficulty, is appropriate.

If you want have a high average, you can also give more marks on easier questions and fewer marks on harder ones.

If you want a low-headache, simple to use rubric:
Method 3 + Accuracy 1

Method
1 for some valid attempt, even if it doesn't go anywhere
2 for substantial progress
3 for totally correct procedure

Accuracy
0.5 if there's a minor error
1 if fully correct

Answer only, no working will only earn accuracy marks.

@tawdry venture (sorry for ping again!)

[tawdry venture]

i wish i'd taken a photo

i left them at school

but like, to tell you what i remember, they were all inequalities of the form $\sqrt{f(x)} < g(x)$ (or $\leq, >, \geq$ in place of $<$), where $f$ and $g$ were each polynomials of degree 1 or 2

[burnt vessel]

Ann

[tawdry venture]

so all 7 questions had equal difficulty in my eyes

[next relic]

then the real question is why did you need to give 7 diff questions on the same stuff

Yeah then I agree, they prolly have equal difficulty.

[next relic]

@tawdry venture this rubric would prolly work. You can also remove 0.5 in accuracy if you don't want any hassle. :D

[tawdry venture]

i was going to ask

the options i saw for not dealing with half points are to double the whole thing, or to do M3 + A2

[next relic]

M3 + A2 means 40% on accuracy and 60% method.

How much do you value accuracy over procedure?

(You mind if I keep pinging you or it's good?)

Most exams I've seen usually do 50/50 or more generous on method marks.

[tawdry venture]

i don't mind pings.

well, from here anyway.

as in this channel

[next relic]

Good, cause I feel bad pinging you every 3 seconds :p

3/2 also works. What's the passing mark at your uni?

[tawdry venture]

i don't think i can give a numerical answer to how much i value accuracy over procedure

i work at a school

[next relic]

70 is your passing grade?

@tawdry venture

[tawdry venture]

70 as in 70%..?

[next relic]

Yes, sorry.

[tawdry venture]

no, i think it's lower than that.

plus, i don't convert score into a grade

the teacher who gave me these tests to grade does

i only tally up the scores

[next relic]

Right, do they scale the marks on the tests?

Or it's just a straight percentage calculation?

[tawdry venture]

i don't actually know.

i couldn't catch the teacher earlier today.

so i will have to ask her tomorrow how she wants it done.

[next relic]

But anyway I think M3/A2 is a decent model, enough weight on accuracy to make students take it seriously.

[tawdry venture]

ok

[next relic]

Dw about it it's just me being a nerd. :p

Bc your test grades don't have the same meaning if she scales it at the end of the term.

[tawdry venture]

this is all a low-scales, non-urgent test btw

sogjhgld

low-stakes* sorry

hands autocorrected

[next relic]

Oh then it doesn't matter then :D

Just do 3/2 and keep your life simple!

Even 2/2 if you feel like it!

[tawdry venture]

ok

[tall bolt]

70 passing is wild, 70 is an A here

[long pelican]

I like 70 being an A better

[next relic]

55 is passing at my uni. 60 was passing in hs in my province.

And anything above 85 is an A for us.

Honestly I prefer 70 being an A, kinda like the UK system.

[long pelican]

90 being an A means I know without having to look, that the exams are all going to be rote

[next relic]

Unfortunately that's how it works in the states, unless the prof curves heavily.

[long pelican]

I am in the states haha