Anonymous at Fri, 7 Jun 2024 02:02:03 UTC No. 16218642

>>16218636
you have to think of everything as a limit even tho it's not explicitly shown. the dx is just a short hand for the limits. then it will make sense for you.

Anonymous at Fri, 7 Jun 2024 02:15:49 UTC No. 16218655

>>16218636
It's confusing because the author has glossed over the fact that [math]\lim_{\Delta x\to 0} \frac{\Delta y}{\Delta x} = \frac{dy}{dx}[/math] (where [math]\Delta[/math] represents a small finite change in a quantity). When we take limits at the end any quantity involving [math]\Delta x[/math] will tend to zero and disappear.

Anonymous at Fri, 7 Jun 2024 02:25:15 UTC No. 16218679

>>16218636
That author sucks. It's a horrid explanation about derivatives and a confusing misuse of notation.

Anonymous at Fri, 7 Jun 2024 02:28:07 UTC No. 16218682

>>16218655
No, he mentions limits further down. He makes a clear distinction between the notion of an infinitesimal and that of a limit. And I agree with you, limits are clearer. I'm just trying to understand why infinitesimals make sense but this is clearly not the resource that will illuminate me. Non-standard analysis is a bit too advanced for me though.

Anonymous at Fri, 7 Jun 2024 02:35:35 UTC No. 16218686

>>16218679
To be fair, the beginning lacks but the text picks up nicely after if you get the shoddy infinitesimal explanation out of the way. My favourite part is how he derives the Taylor series in section 12.6, never seen it done like that and it seems it's the historical approach.

Anonymous at Fri, 7 Jun 2024 02:35:40 UTC No. 16218687

>>16218682
It would have been a lot simpler if he had instead just said (dx)^2 is so small it can be ignored. So dy = 2x dx, and then do the last step.

Anonymous at Fri, 7 Jun 2024 02:43:28 UTC No. 16218701

>>16218682
Oh right, I was just going based on what it said in your image. I've just downloaded the pdf and had a quick look. It seems like it's an introductory calculus text which skips over the introductory groundwork. It appears the author interprets the word "intuitive" to mean "hand-wavy".

>this is clearly not the resource that will illuminate me
Yeah this is not a book that will give you a solid understanding of the hyperreals. You need a text on set theory or model theory which deals with ultrafilters and ultraproducts.

Anonymous at Fri, 7 Jun 2024 02:53:58 UTC No. 16218719

>>16218636
>If you discard the dx term, why is it still an equality? Shouldn't that read:
Because dy/dx is not infintesimal, and 2x is not infinitesimal, but dx is infinitesimal. It's basically the same as writing 1+(1/infinity)=1. It's not true in the realm of infinitesimals, but outside the realm of infinitesimals, it is true, because outside the realm of infinitesimals, infinitesimals are so small that they are negligible and can therefore be treated as being equal to 0. The time they CAN'T be treated as being equal to 0 is when we are looking at the ratio of one infinitesimal to another infinitesimal. But simply adding an infinitesimal to a normal number can be disregarded; i.e. 1+1/infinity=1.

And in the case of derivatives, it's basically just a limit, so if you want to not discard infinitesimals and instead look at it as a limit, then as >>16218655 said, if dy/dx=2x+dx, then the limit of dy/dx as dx approaches 0 is simply 2x+0.
>>16218682
Anon you replied to is not saying that an infinitesimal is a limit. He is saying that a derivative is a limit involving infinitesimals; that is, dy/dx is the limit of (delta x)/(delta y) as (delta x) approaches 0.

Anonymous at Fri, 7 Jun 2024 02:55:26 UTC No. 16218725

If you want a long in-depth book on infinitesimal calculus, read "Calculus: An Infinitesimal Approach" by Keisler.
May be easier or more difficult than the text you have now depending whether you learn better when given more info, or if the extra info just confuses and distracts you. I have not read the book you linked but I did read most of Keisler's book and I enjoyed it,.

Anonymous at Fri, 7 Jun 2024 02:57:31 UTC No. 16218727

>>16218719
>(delta x)/(delta y)
Got that backwards, obviously meant to say (delta y)/(delta x)

Anonymous at Fri, 7 Jun 2024 09:30:21 UTC No. 16219156

>>16218719
Thanks, this made sense. Now this explains why author says to divide by dx first instead of discarding infinitesimal quantities. >>16218687

Anonymous at Fri, 7 Jun 2024 10:21:23 UTC No. 16219218

>>16218636
dy = dx = 0

Vard at Fri, 7 Jun 2024 10:23:08 UTC No. 16219222

Math in the modern era is full of fallacy and conspiracy, it probably doesn't work that way OP, you're just taught it does.

Vard at Fri, 7 Jun 2024 10:24:59 UTC No. 16219226

>>16218725
You can't parse this, thus, they direct you to a book which affirms it's true that you must trust like a Bible to enter their circle.

Anonymous at Fri, 7 Jun 2024 10:27:32 UTC No. 16219231

>>16218636
Has anyone here tried non-standard Analysis? Is it less schizo? dx dy etc. always bothered me deeply. In Lebesgue integration it's a measure function, in Riemann Integration it's an interval/limit-of-an-interval, in diff geom it's a differential form, for Leibiniz it was an algebraic infinitesimal etc.

Anonymous at Fri, 7 Jun 2024 10:42:07 UTC No. 16219264

>>16218636
It's just a way of keeping an accurate accounting of vanishingly small numbers. It can blow things up if it is in a denominator, but if it's additive like that you can take the limit to 0 and kill the term because it's... infinitesimal. It's approximately equal to in the same sense that 0.999... is approximately equal to 1.0.

Anonymous at Fri, 7 Jun 2024 11:52:24 UTC No. 16219401

>>16219231
I tried Henle and Kleinberg and the construction of the hyperreals seems very laborious to me. To me real analysis is more approachable. After I’m done with book in OP I will probably go over something like Understanding Analysis by Abbott

Anonymous at Fri, 7 Jun 2024 16:23:42 UTC No. 16220020

>>16218636
Most handwavy arguments in calculus and diffeq have actual formal proofs for showing that they work... they're just overly laborious and you end up derailing the discussion of the bigger issues to focus on the minutia of why the methods and approximations work.

ex. I ran through a full, detailed proof showing why the separation of variables method works for my intermediate mechanics course this semester, and even the truncated proof is still like four or five pages long, and the students don't really get that much out of the experience. Telling them that you can split dy/dx into a dy and dx like it's a fraction isn't technically accurate and it obscures a lot of the things going on under the surface to get to that point, but it lets you get to the actual meat and potatoes of working through separation of variables problems more quickly.

In this case your author is glossing over some of the deeper discussion of limits to get to the fucking point.

Anonymous at Fri, 7 Jun 2024 16:50:28 UTC No. 16220058

>>16219264
.999... isn't approximately equal to 1. It's strictly equal to 1.

Using certain notations and hyperreals it can be approximately equal to 1, but that .999... and the normal .999... aren't the same .999...

Anonymous at Fri, 7 Jun 2024 17:06:29 UTC No. 16220101

>>16219231
>Is it less schizo?
Limits aren't schizo. They're just handwavey and it's really easy to write something technically wrong even if everyone would understand what you meant.

I find nonstandard analysis more straightforward and easy to comprehend at lower levels, but your mileage may very. I know some people hate it.

Frankly I think all of calculus is just different people grappling with the fact that d, Δ, δ, and ∂ are different Ds.

Anonymous at Fri, 7 Jun 2024 17:10:06 UTC No. 16220116

>>16220101
working with well defined notions isn’t hand wavy (eg. delta-epsilon def of limits)

Anonymous at Fri, 7 Jun 2024 17:20:22 UTC No. 16220141

>>16220020
>I ran through a full, detailed proof showing why the separation of variables method works for my intermediate mechanics course...........the students don't really get that much out of the experience
I agree with this perspective for engineering or other sciences, where application is more important than theory. However, OP's image came from a site called "intellectual mathematics". I think it's fair to expect a little more rigour from a mathematics text. Also, I would argue that OP's question indicates that glossing over the details has actually led to more confusion in this case, not less.

>and even the truncated proof is still like four or five pages long
How did it run to 5 pages? Even if you're proving the chain rule before getting to the separation of variables itself, it shouldn't be that much work.

Anonymous at Fri, 7 Jun 2024 19:18:38 UTC No. 16220352

>>16219231
>Is it less schizo?
>schizo
It is now mandatory to channel Errett Bishop.

Our educational system contains many interesting paradoxes. We tell the students to get involved in the world, and the curriculum becomes increasingly abstract. Courses in sociology, anthropology, economics, et cetera are introduced, in which mathematical models are discussed. We describe the advantages of having a liberally educated citizenry, to students whom we make increasingly anxious and decreasingly able to think for themselves. We ask the students to understand, and we examine them on facts and technique.

We go to great lengths to see that students are given equal opportunities, and competition becomes more and more intense. The student who is able to achieve superlative grades will go to medical school, if he wishes, and perhaps elect to become a surgeon making hundreds of dollars an hour. Another student who does not take his courses quite so seriously may end up with a menial job that pays less than a hundred dollars a week, and be lucky to get it.

Mathematics now plays an important role in the process of determining who will get what, in part because it is considered to be especially difficult, especially objective, and especially useful to contemporary man. Tests of mathematical proficiency are regarded as a fair and efficient means of eliminating large numbers of superfluous aspirants to choice degrees. Unfortunately, in the process of testing larger and larger numbers of terrified candidates for success, we are telling our students that performance is the name of the game. If they emerge from their courses with any interest in mathematics at all, it will not be a thoughtful interest. It is bad form to ask what it all means. Since reality is so elusive, models are the order of the day, and truth is relative to the model, a kind of super-chess.

Anonymous at Fri, 7 Jun 2024 19:19:39 UTC No. 16220354

>>16220352
Contrary to the expressed intentions of some of its founders, the new math has contributed much to the mystification of students. From a primary concern with numbers and geometrical objects, the pre-college curriculum has moved on to open sentences, sets of sets, distinctions between numbers and numerals, and the like. The students quickly get the idea that they are not supposed to take it seriously: the teachers do not, do they?

Now the colleges have been more conservative, but a new book, Elementary calculus by H. Jerome Keisler, could change all that. To quote from the book: "In 1960 Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the twentieth century." Again: "Recently, infinitesimals have had exciting applications outside mathematics, notably in the fields of economics and physics. Since it is quite natural to use infinitesimals in modelling physical and social processes, such applications seem certain to grow in variety and importance. This is a unique opportunity to find new uses for mathematics, but at present few people are prepared by training to take advantage of this opportunity."

No evidence of these claims is given in Keisler's book, but the students will not notice that. Those students who think that mathematics is about something will be disabused. To quote Keisler: "Do not be fooled by the name 'real number'. The real number system is a purely mathematical creation which may or may not give an accurate picture of a straight line in physical space." Again: "In discussing the real line we remarked that we have no way of knowing what a line in physical space is really like. It might be like the hyperreal line, the real line, or neither. However, in applications of the calculus it is helpful to imagine a line in physical space as a hyperreal line."

Anonymous at Fri, 7 Jun 2024 19:20:41 UTC No. 16220355

>>16220354
What are we to make of these statements? Is Keisler describing mathematics as we know it, and the world as we have come to perceive it? The answer would appear to be "no", but perhaps we have not kept pace. If not, and his statements are true, the evidence should be somewhere in the book. So let us examine the book.

Keisler gets down to business on p. 25 by defining the average slope between two points on a curve in the usual way. Then he computes the average slope [math]2x_0 + \Delta x[/math] between two points on the parabola [math]y = x^2[/math]. Reasoning nonrigorously, as he calls it, he then neglects the [math]\Delta x[/math] (because it is very small) and gets the value [math]2x_0[/math] for the slope at [math](x_0,y_0)[/math]. The argument is repeated, this time for velocities. The trouble with these intuitive arguments, he says, is that it is not clear when something is to be neglected. "What is needed is a sharp distinction between numbers which are small enough to be neglected and numbers which aren't. Actually, no real number except zero is small enough to be neglected."

Since Keisler wants to "neglect" the [math]\Delta x[/math] (and gives the students the impression that we need to neglect the [math]\Delta x[/math]), he would seem to have reached an impasse: On the one hand [math]\Delta x[/math] represents a nonzero real number, and on the other he has told us that no real number except zero is small enough to be neglected. The impasse is broken by forgetting that [math]\Delta x[/math] is a real number, calling it something else (an infinitesimal), and telling us that it is all right to neglect it.

Actually the presentation is much more complicated than that. We are not told what an infinitesimal [math]\Delta x[/math] is, or what [math]f(x + \Delta x)[/math] means. Instead, the matter is treated axiomatically. Perhaps the intuitive content is intended to be supplied by our imagining a line in physical space as a hyperreal line.

Anonymous at Fri, 7 Jun 2024 19:21:43 UTC No. 16220358

>>16220355
It is not until p. 298 that Keisler relates his development of calculus to the usual one, and puts everything in what to him is its proper place. The conventional definition of limit is grudgingly given. He tells the student that "Indeed, the whole point of our infinitesimal approach to calculus is that it is easier to define and explain limits using infinitesimals".

This claim deserves examination. Of course, it is all in the axioms. I sometimes tell mathematicians whose only concern is to deduce theorems from axioms to add the axiom "0 = 1". They are outraged, ostensibly because that axiom would be inconsistent. What really bothers them is that it would make mathematics too easy, and put them out of business.

In the sense that Keisler has developed limits from a supposedly consistent system of axioms, they have been explained. But he has not explained the axioms. They are mere conveniences for generating proofs, whose intuitive content will certainly excape the students. If you do not believe this then read them, axioms V* and VI* in particular.

Of course, the usual notions all get defined, sooner or later, in the usual way, because calculus is about the real numbers. The book offers no evidence that the hyperreal numbers are anything except a device for proving theorems about the real numbers. They are not even an efficient device, depending as they do on axioms V* and VI*, among other things.

The technical complications introduced by Keisler's approach are of minor importance. The real damage lies in his obfuscation and devitalization of those wonderful ideas. No invocation of Newton and Leibniz is going to justify developing calculus using axioms V* and VI* -- on the grounds that the usual definition of a limit is too complicated!

Anonymous at Fri, 7 Jun 2024 19:22:44 UTC No. 16220361

>>16220358
Although it seems to be futile, I always tell my calculus students that mathematics is not esoteric: It is common sense. (Even the notorious [math]\varepsilon[/math], [math]\delta[/math] definition of limit is common sense, and moreover is central to the important practical problems of approximation and estimation.) They do not believe me. In fact the idea makes them uncomfortable because it contradicts their previous experience. Now we have a calculus text that can be used to confirm their experience of mathematics as an esoteric and meaningless exercise in technique.

Anonymous at Fri, 7 Jun 2024 21:01:34 UTC No. 16220581

>>16220058
.999... is only a valid concept if we are dealing with hyperreals. So no, .999... is not exactly equal to 1. It is infinitely close to 1, but there is a non-zero infinitesimal difference.
If we disregard the concept of hyperreals, then we disregard the concept of .999... itself.

Anonymous at Fri, 7 Jun 2024 22:40:36 UTC No. 16220780

>>16220361
Bishop is criticizing apples for not being oranges.

Anonymous at Fri, 7 Jun 2024 22:45:27 UTC No. 16220790

>>16220581
>.999... is only a valid concept if we are dealing with hyperreals.
Or geometric series. Or fractions.

Whether you define it as 3/3, 9/9, or the sum of 9/10^n for n=1 to infinity, it's the same shit. Literally fuck all about repeating decimals requires hyperreals.

Anonymous at Fri, 7 Jun 2024 22:52:26 UTC No. 16220812

>>16220581
>>16220790
Ooh, you can also drive the point home by moving from base-10 to base-9. I fucking defy you to tell me .999... in base-10 is anything but 1 in base-9 and further defy you to tell me that 1 in any base is anything but 1.

Anonymous at Fri, 7 Jun 2024 23:16:10 UTC No. 16220864

>>16220790
3/3=1
9/9=1
sum of 9/10^n on (1,infinity)=.999...
.999...=/=1
and in fact .999...=/=.999... more often than not
In fractions .999... is sometimes used as an approximation but it is just that.
>>16220812
There is no '9' symbol in base nine. In base nine, the number nine is written as '10'.

Anonymous at Fri, 7 Jun 2024 23:50:57 UTC No. 16220913

>>16220864
>There is no '9' symbol in base nine. In base nine, the number nine is written as '10'.
Yes. Thank you, Sherlock. So how would you write .999... in base 9? You understand decimal decimals can be written in other bases, right? As an example, .5 in binary is .1.

Fucking idiot.

>3/3=1
3*1/3=3*.333...
3/3=.999...
>9/9=1
9*1/9=9*.111...
9/9=.999...

>sum of 9/10^n on (1,infinity)=.999...
No. It's 1. The same way that 1/2^n on (1,infinity)=1.

Any geometric series of that form equals 1. Which makes perfect sense if you think about it for even half a second. The 1/2+1/4+1/8... geometric series is just .111... in base-2. And that's also 1, the same way .222... in base-3 is 1. And .333... in base-4 is 1. And so on and so forth. It's all fucking 1.

Anonymous at Sat, 8 Jun 2024 00:29:29 UTC No. 16220973

>>16220812
actually 0.999...=1.111... in base 9 in the same way that 0.AAA...=1.111... in base 10, for reasons that i'd hope should be obvious with a bit of thought

Anonymous at Sat, 8 Jun 2024 00:32:01 UTC No. 16220976

>>16220358
>What really bothers them is that it would make mathematics too easy, and put them out of business.
what the fuck is with schizos and their tax money possibly going to wards mathematics?

Anonymous at Sat, 8 Jun 2024 00:36:23 UTC No. 16220985

>>16220913
>So how would you write .999... in base 9?
Depends which .999... you are talking about. There are many ways to reach .999... and not all of them are equal.
>muh series
They approach 1. Their limit is 1. A function (or series) never reaches its limit; it is bounded by its limit.

Anonymous at Sat, 8 Jun 2024 00:37:32 UTC No. 16220987

>>16220913
>So how would you write .999... in base 9?
.888...
obviously

Anonymous at Sat, 8 Jun 2024 00:39:42 UTC No. 16220991

>>16220973
So you just don't know how positional notation works, huh?

1 in base 9=1 in base 10.

You are arguing that .999... in base 10 is >1 in base 10, you fucking idiot.

Anonymous at Sat, 8 Jun 2024 00:54:52 UTC No. 16221008

>>16220985
>A function (or series) never reaches its limit; it is bounded by its limit.
Incorrect (setting aside the fact that you forgot limits don't have anything to do with infinity, eg the limit of 5 as x approaches 7 is, obviously, 5). Convergent series DO reach their limit definitionally. It just takes infinitely many steps to do so.

Surely you've heard of the tortoise paradox?

Anonymous at Sat, 8 Jun 2024 01:01:35 UTC No. 16221016

>>16220987
That's the sum of 8/9^n on (1, infinity), not the sum of 9/10^n on (1, infinity).

Anonymous at Sat, 8 Jun 2024 01:13:00 UTC No. 16221027

>>16220991
>So you just don't know how positional notation works
of course i do
a*base^2+b*base^1+c*base^0+d*base^-1+e*base^-2+...
and 0.9...=1 due to the fact that positional notation involves sums so, im amazed that you took from my writings that i attempted somehow to disprove the obvious, lots go over why 0.999... in base 9 is equal to 1.111... in base 9 though the analogue in base 10(here we take A=9+1 due to backporting notation from hexadecimal):
0.A=1.0, 0.AA=1.10, 0.AAA=1.110, 0.AAA=1.1110, and as such 0.AAA...=1.111...
the added zeroes at the end of the finite ones are to make the pattern that i said should be obvious to see with a bit of thought, well, visible

Anonymous at Sat, 8 Jun 2024 01:16:06 UTC No. 16221034

>>16221016
nigga, you know that in base 9 we use {0,1,2,3,4,5,6,7,8} to write numbers, right?, do you have trouble comprehending that in binary we use {0,1}, or in ternary {0,1,2}?, how the hell do you fail lo see the pattern?

Anonymous at Sat, 8 Jun 2024 01:23:52 UTC No. 16221038

>>16220141
what resource do you recommend students learn calculus from?

Anonymous at Sat, 8 Jun 2024 01:29:32 UTC No. 16221045

>>16221027
0.X in any base will never before greater than 1 when translated to another base.

🗑️ Anonymous at Sat, 8 Jun 2024 01:59:45 UTC No. 16221063

>>16221027
>.A=1
.A=.91666...

As I said, you just don't know how positional notation works.

>>16221034
I see the pattern. I discussed the pattern earlier. Proving why the pattern holds is legitimately more difficult than proving that .999...=1 which is how you should be writing it in any base. Writing .999... in base 10 as .888... in base 9 to avoid conceding the point is asinine.

Anonymous at Sat, 8 Jun 2024 02:01:38 UTC No. 16221066

>>16221027
>.A=1
.A=.909090...

As I said, you just don't know how positional notation works.

>>16221034
I see the pattern. I discussed the pattern earlier. Proving why the pattern holds is legitimately more difficult than proving that .999...=1 which is how you should be writing it in any base. Writing .999... in base 10 as .888... in base 9 to avoid conceding the point is asinine.

Anonymous at Sat, 8 Jun 2024 17:36:08 UTC No. 16222350

Why are they like this? Why can you cancel dt? This is from a different book (Shankar's Fundamental physics I). Does he view dt as an infinitesimal? Is that why he cancels them out?

Anonymous at Sat, 8 Jun 2024 18:15:05 UTC No. 16222439

>>16218636
i'd recommend "non-standard analysis" by abraham robinson if you're actually wanting to understand infinitesimals as a mathematical object. iirc (its been over a decade since i've visited the book), the infinitesimals are part of the set of hyperreal numbers as an extension of the real numbers, and there's a function for extracting only the real number component from the hyperreal expression when the only hyperreal component are infinitesimals.

Anonymous at Sat, 8 Jun 2024 18:34:34 UTC No. 16222474

>>16222350
>Why can you cancel dt
You can't because dx/dt isn't a fraction. It's not canceling in the traditional sense. It's rewriting the equation as something necessarily true and useful based on prior information. It just looks like canceling. Hence why it's "viewed with suspicion", he's basically just saying "trust the math". It's similar to how you can, but shouldn't, make logical leaps in formal logic based on prior determinations. It wouldn't work in a formal proof, but for reaching right conclusions, it's perfectly valid.

tl;dr somebody else did the work, don't worry about it

Anonymous at Sat, 8 Jun 2024 18:59:14 UTC No. 16222533

>>16218636
The definition of the infinitesimal derivative involves only taking the non-infinitesimal part of what you get.

the Standard Part function is defined to be, st(a + b\epsilon) = a
and this makes the derivative

d/dx f(x) = st( f(x + \epsilon) - f(x) )

Anonymous at Sat, 8 Jun 2024 19:35:07 UTC No. 16222623

>>16222439
Which one between Infinitesimal Calculus (Henle & Kleinberg) and Keisler best exposes infinitesimals?

Anonymous at Sat, 8 Jun 2024 21:41:41 UTC No. 16222911

>>16222623
Not familiar with the former but I like the latter

Anonymous at Sat, 8 Jun 2024 21:47:38 UTC No. 16222922

>>16221016
9*10^-n on (1,infinity) is not the only way to get .999...
As already stated, not all .999... are equal.
Some .888...s in base nine are equal to some .999...s in base ten.
9*10^-n on (1,infinity) in base ten and 8*10^-n on (1,infinity) in base nine can both be written as .999... in base ten or as .888... in base nine even though they are not equal (in the realm of hyperreals). And obviously "10" in the former means ten whereas in the latter "10" means nine.
>>16221066
>>16221066
>avoid conceding the point is asinine.
There is no need for me or >>16221034 to concede to you when we are right and you are wrong.
Will you AT LEAST admit that .999... is not equal to 1 in the realm of hyperreals, and that not all .999...s are equal to each other in the realm of hyperreals?

Anonymous at Sat, 8 Jun 2024 22:51:53 UTC No. 16223040

>>16222922
.999.... is equal to 1 in the hyperreals.

the sequence of .9, .99, .999, ... is not the same as the constant sequence .999..., .999..., .999..., ...

.999... is just 1 without even introducing the hyperreals.

the non-constant hyperreal sequence .9, .99, .999, ...
is "infinitesimally" smaller than 1
But .999... is just 1 since the real number .999... is equal to 1.

Anonymous at Sat, 8 Jun 2024 23:24:58 UTC No. 16223088

>>16220116
the epsilon-delta definition uses the existential quantifier
the existential quantifier is not well defined

Take Chaitin's constant (in base 2), then C is a sequence of 0's and 1's.
But because of how it is defined, there is some point, beyond which we can never prove a bit is 0 or 1. pic rel
So there is some maximal S = .01001100000... which has only finitely many 1s, so that |C - S| is as small as provable, but non zero.

Then if I take an epsilon smaller than that error, you claim "there exists" some extension S' of S for which |C - S'| is less than |C - S|. Since Chaitins constant is a real number, we can apply the existential operator in the definition to "realize" such an S'.

This isnt a contradiction, since the existential operator cannot actually be used to show the existence of anything. But this does mean, that the usual conception of the existential quantifier is not well-defined, and is completely meaningless.

Anonymous at Sun, 9 Jun 2024 13:51:06 UTC No. 16224403

This problem had already been solved by Tony Hawk in 2020.

I believe the basic theory describes:

1/0 does not. (Not exist)

But

The [distribution] of 1 / 0 is described as infinity

Because

0 does equal + - 1

Given all your previous recursive bullshit

1 [distributed] over 0.0000000000001 = X

Therefore tendency is infinity not DNE. (False)

The actual proof goes further but I cannot be bothered to repeat

Anonymous at Sun, 9 Jun 2024 13:53:59 UTC No. 16224408

:range of points between 1 and 0

Is infinity

I believe there is further literature for Tony Hawks original proof but you will have to find it on your own

Etc etc etc

Anonymous at Mon, 10 Jun 2024 06:05:24 UTC No. 16226271

I learnt this from Stewart Calculus

Anonymous at Mon, 10 Jun 2024 10:46:10 UTC No. 16226570

>>16218636
https://calculusmadeeasy.org/

Anonymous at Mon, 10 Jun 2024 14:58:42 UTC No. 16226854

>>16220058
It's a metaphor; I thought the context would make the intent obvious. I'm just saying the relationship between 0.999... and 1 can be understood as similar to the relationship between 2x+dx and 2x.
0.999... = 1 = lim dx->0+ of 1-dx.
2x = lim dx->0+ of 2x+dx
One-sided limits used here to make it sensible. The point is to assuage the OP's concern that it's approximately equal to by hopefully relating it to something that the OP might be familiar with already. I'm not claiming that 0.999... approx 1.

Anonymous at Mon, 10 Jun 2024 17:27:21 UTC No. 16227075

>>16219264
>take the limit to 0 and kill the term because it's... infinitesimal. It's approximately equal to in the same sense that 0.999... is approximately equal to 1.0.
You were almost correct until you added this retarded bit

Anonymous at Mon, 10 Jun 2024 17:43:29 UTC No. 16227108

>>16220020
>Telling them that you can split dy/dx into a dy and dx like it's a fraction isn't technically accurate
What do you mean by 'not technically accurate'? If you treat dy and dx as infinitesimals, nothing should stop you from doing what you describe. AFAIK, modern infinitesimal approaches are equivalent to ε-δ, even they are resented by some.

Mechanics folks certainly seem to love working with infinitesimals. Perhaps your use case is a good demonstration why

Anonymous at Mon, 10 Jun 2024 18:29:15 UTC No. 16227180

>>16218636
In some number systems (i.e. the surreal numbers), you can't just automatically discard the dx part.

Anonymous at Mon, 10 Jun 2024 20:15:38 UTC No. 16227344

>>16218636
An infentesimal = x
x != 0
x*x = 0

This should clear things up

Anonymous at Mon, 10 Jun 2024 22:05:26 UTC No. 16227513

Suppose some function [math] f(x) = y [/math].
Vary the ind. variable by some increment dx, such that [math] f(x+dx) = y+g(y) [/math] where I posit g(y) as a undeter. function of y.
The function g(y) can be deter. by substracting f(x) from f(x+dx).
This results in: [math] f(x+dx)-f(x) = g(y) [/math].
Taking f(x) to be a variable here, [math] f(x+dx)-f(x) [/math] must correspond to some variation df.
Ergo, we have [math] f(x+dx)-f(x) = df(x) [/math] where the differential df(x) depends on the function f(x) and the step width dx.
This means that [math] df(x) = g(y) = dy [/math]. Thus the function g() is nothing other than a differential operator.
Df(x) = dy is a function of dx, as all the independent variables x have been cancelled out by subtraction.
This must be the case because otherwise [math] f(x+dx)-f(x) = h(x,dx) [/math].
If we take dx = 0, [math] f(x)-f(x) = h(x) != 0 [/math] which violates the definition of a function.
Concordantly, dy only contains dx, dx multiplied finitely often by itself or dx multiplied finitely often by x.
Therefore dy = h(dx,x*dx...). The function already found by subtraction.
For small values dx -> [math] dx^{0.2} > dx^{0.5} > dx > dx^2 > dx^3... [/math].
Thus, the increment with the smallest exponent becomes the dominating term in the function.
For any irrational function, dx can only have natural numbers as exponents. This is true for roots as well.
If it didn't, [math] df(x) = (x+dx)^n - x^n [/math] where n < 1, could be expressed in the form [math] df(x) = g(x)*dx^m [/math], for m < 1.
Multiplying by [math] 1/m [/math] yields [math] df(x)^{1/m} = ((x+dx)^n - x^n)^{1/m} = g(x)^{1/m}*dx [/math], which would mean that [math] y^{1/m} = g(x)^{1/m}*x [/math] or [math] (y/g(x))^{1/m} = x [/math].
This only makes sense for n = m. In this case, [math] ((x+dx)^n - x^n)^{1/n} = dx = (x+dx)^{n/n} - x^{n/n} + k(x,dx) [/math].
[math] k(x,dx) [/math] would have to be zero for all values. Only posisble if [math] 1/n = 1 = n [/math]

Anonymous at Mon, 10 Jun 2024 22:07:30 UTC No. 16227519

>>16227513
[math] y^{1/m}=g(x)^{1/m}∗x [/math] or [math] (y/g(x))^{1/m}=x [/math]
should read
[math] y^{1/m}=g(x)^{1/m}∗dx [/math] or [math] (y/g(x))^{1/m}=dx [/math]

Anonymous at Mon, 10 Jun 2024 22:11:56 UTC No. 16227527

>>16227513
It is advisable to look to Euler and Lagrange for those who are interested in the arithmetic foundation of infinitesimal calculus. The derivative of a function is nothing other than the function df(x) for which increments other than dx, in this case dx^2,dx^3 etc, are ignored since these end up being infinitesimally small compared to dx.

Anonymous at Tue, 11 Jun 2024 00:10:39 UTC No. 16227691

>>16227108
>If you treat dy and dx as infinitesimals, nothing should stop you from doing what you describe.
Okay, but the thing is they *aren't* infinitesimals in that context. They're differential forms. dy/dx isn't a fraction. It behaves like a fraction in certain limited contexts, but that's all.

Anonymous at Tue, 11 Jun 2024 01:08:26 UTC No. 16227757

>>16218636
the language i like the most is 2x + dx, where dx is arbitrarily small without being 0, and for brevity dx is omitted

Anonymous at Tue, 11 Jun 2024 08:41:35 UTC No. 16228130

>>16227691
In what contexts would differential forms be distinct from corresponding infinitesimals? Is there a context where dy/dx would be something distinct from a fraction of infinitesimal quantities dy and dx? I have been led to believe there is no such context, at least not in the sense that they would produce different results to evaluate.

I might well be wrong, my practical understanding of the matter comes from applied mechanics, rather than formal mathematical education (of which I have only vague memories)

Anonymous at Tue, 11 Jun 2024 08:59:09 UTC No. 16228134

>>16227075
Turns out my post is a litmus test for poor reading comprehension and crippling autism. I can express a continued fraction as a limit as well. I'm just saying it's a similar scenario, not that 0.999... neq 1.

Anonymous at Tue, 11 Jun 2024 19:02:18 UTC No. 16228857

>>16228130
>In what contexts would differential forms be distinct from corresponding infinitesimals
(dy/dx)^2 isn't dy^2/dx^2 as an example.

Anonymous at Tue, 11 Jun 2024 22:37:16 UTC No. 16229306

>>16228857
While
[eqn] [y'(x)]^2 = (\frac{dy}{dx})^2 = \frac{dy^2}{dx^2} [/eqn]
holds true with infinitesimals, it is not the same as
[eqn] y''(x) = \frac{ d[ y'(x) ] }{dx} = \frac{ d[ \frac{dy}{dx} ] }{dx} = \frac{d^2 y}{dx^2} [/eqn]
(last part of which takes some manipulation to make sense of)

Anonymous at Wed, 12 Jun 2024 03:42:37 UTC No. 16229721

>>16220352
>Errett Bishop.
>https://en.wikipedia.org/wiki/Criticism_of_nonstandard_analysis
>Katz & Katz (2010) note that a number of criticisms were voiced by the participating mathematicians and historians following Bishop's "Crisis" talk, at the American Academy of Arts and Sciences workshop in 1974. However, not a word was said by the participants about Bishop's debasement of Robinson's theory. Katz & Katz point out that it recently came to light that Bishop in fact said not a word about Robinson's theory at the workshop, and only added his debasement remark at the galley proof stage of publication. This helps explain the absence of critical reactions at the workshop. Katz & Katz conclude that this raises issues of integrity on the part of Bishop whose published text does not report the fact that the "debasement" comment was added at galley stage and therefore was not heard by the workshop participants, creating a spurious impression that they did not disagree with the comments.
well, it seems that someone is a shifty cunt

Anonymous at Wed, 12 Jun 2024 03:48:51 UTC No. 16229733

>>16227344
dual numbers != hyperreals

Anonymous at Wed, 12 Jun 2024 06:32:02 UTC No. 16229876

I don't think the infinitesimal numbers of non-Archimedean ordered fields are a good model of how physicists and other applied math people think about dx. When a physicist says something like [math]dx = d(\frac12 a t^2) = at dt[/math], the change in time dt is just a very small real number, and the equation is understood to only hold true in an approximate sense. For people working in such a field, equations only holding true in an approximate sense is the rule rather than the exception.

Anonymous at Wed, 12 Jun 2024 15:05:44 UTC No. 16230406

Correct me if I'm wrong since I'm a bit rusty but can't we say dx is the denominator of the limit definition of differentiation (lim x->a (f(x)-f(a))/(x-a)). From this we can directly prove that the lone dx=lim x->a (x-a)=0 by epsilon delta (shown below)

Fix epsilon>0 and let delta=epsilon. Notice that if 0<|x-a|<delta, that |(x-a)-0|=|x-a| (since our limit is 0) is less than delta. Thus we see that |x-a|<epsilon for every epsilon>0, proving that dx=lim x->a (x-a)=0 and that 2x+dx=2x+0