wtf

21 minutes ago, studiot said:

I am waiting for your definition of an infinitesimal, that I asked for a while back.

I offered my best one.

Going to a grammar school I did Latin. (That was the English definition of a grammar school)

evanescent increments was not translated by 'some historian'.

It was part of a very famous attack on Newton by the Church of his day.

 

Yes I believe I said something similar in my first post.

I don't know if you mean  Δ ?

But this is connected to Newton thinking like a physicist (Which he was in all but name)

William Playfair, the accredited inventor of line graphs, pie charts etc was just being born when Newton was in his grave twenty odd years.
Newton and his contemporaries worked from tabulations.
Newton developed an advanced calculus of finite differences, characterised by the use of upper case delta to denote a finite difference.
These were fixed values and most decidedly not infinitesimal; they were (and still are) sometimes quite large in value.

Newton used these to fill in or interpolate gaps in his tables, but I thought you knew all this.

So it is not a great step from big(ish) differences to small differences characterised by lower case delta, and thence to differences as small as desired.

Later mathematicians extended this idea to the 'epsilon - delta' construction you will find in many modern higher level texts on analysis; again I'm sure you already know this.

 

 

> I don't know if you mean  Δ ?

Studiot what is the curly delta?? I've asked four times. I don't understand your notation.

If the evanescent increments was written as criticism of Newton, you can hardly use it as evidence of what Newton himself thought. 

In fact Newton's description of "ultimate ratio" sounds suspiciously like the modern definition of limit. As [math]\Delta[/math]x and  [math]\Delta[/math]y get "closer and closer" to zero, their ratio  [math]\frac{\Delta y}{\Delta x}[/math] reaches its ultimate ratio. That's the informal way of thinking of a modern limit.

This is very different than regarding As [math]\Delta[/math]x and As [math]\Delta[/math]y as ever being infinitesimal. On the contrary; at any time, they are NOT ZERO. They're strictly positive.

By the way an infinitesimal is a positive quantity that's strictly less than 1/n for every positive integer n. Sometimes it's less than or equal so that zero is regarded as the only real infinitesimal by some authors. 

28 minutes ago, studiot said:

 

I have never tried to read Leibnitz.

What do you understand by evanescent increments?

https://evanescentincrements.wordpress.com/about/

 

Maybe I'm wrong about some detail but

I don't know where the attribution of the ratio being the fluxion has come from, my sources seem to clearly indicate that Newton considered  δx   and 

 δy as fluxions.

 

 

 

Studiot you still haven't told me what the notation δy means. I don't know what you mean by that notation. 

Secondly I don't think we could have a sensible conversation about what Newton meant when he wrote something down in Latin that some historian translated as evanescent increments. We don't know what Newton was thinking. He was most likely thinking like a physicist. "It doesn't make mathematical sense but it lets me explain the apple falling on earth and the planets moving in the heavens by the same simple principles. So I'll just use it, and let the mathematicians try to sort it out for the next couple of centuries."

I do know that over his career, he explained his fluxions in several different ways. That shows he was well aware of the logical problem of a lack of rigorous foundation. However, my understanding is simply that whereas Leibniz was "Infinitesimals, dude!", Newton was more like "top and bottom close to zero, ultimate ratio is what I call the fluxion." 

But we're not historians of science. A lot of people have written a lot of books about every detail of Newton's thought. In the end all I want to know is what you mean by curly delta so I can have some idea what you're talking about. 

Um ... just realized this. Do you mean delta-x and delta-y? What's usually marked up as [math]\Delta[/math]x and [math]\Delta[/math]y?

10 hours ago, studiot said:

 

 

 

I most definitely said that dx and dy are not infinitesimals.

What is your definition of an infinitesimal by the way?

As to Newtons and his fluxions he published a book about them.

 

The important point about making the distinction is that the dy and dx in

dydx

are considered inseparable (In Mathematics) and maths teachers take great pains to emphasise this.

 

 

Studiot, I was very confused by your post.

First, yes dx and dy are not infinitesimals. I misread that part of your post.

But you said that "The quantities   δx   and  δyare infinitesimals. (Newton called them fluxions, not infinitesimals)"

I have two problems here. One, what are δx and δy? I looked back through the thread and could not find that notation defined. Clarify please?

Second, Newton called the derivative a fluxion. dx and dx aren't fluxions. The limit of delta-y over delta-x is the fluxion. Of course Newton didn't have the formal concept of limit but his intuition was pretty close.

Then you tried to argue that Newton wrote a book on fluxions. Um, yeah, he did. What does that have to do with what we're talking about? What we call the derivative, Newton called a fluxion. Neither derivatives nor fluxions are infinitesimal.

Finally, Newton tried several different approaches to clarifying what he meant by (what we now call) the limit of the difference quotient. He did NOT really espouse infinitesimals in the same way Leibniz did. That's the part that is historically arguable -- what Newton thought about infinitesimals. 

To be clear:

* Fluxions are derivatives, not infinitesimals. (And fluents are integrals).

* Newton didn't really use infinitesimals as such in the strong way Leibniz did. 

* Newton wrote books. But fluxions aren't infinitesimals. Nor did Newton think about dy and dx as infinitesimals. Not (as I understand it) in as explicit a way as Leibniz did.

22 hours ago, discountbrains said:

Oh my goodness. No, no, no. I'm not trying to prove anything about countability. I mean use the same method Cantor used and construct a numer that's not a member of ypur list. If the number is not in the list how can we find its place in the list? How can we compare it to any of the numbers?

Typo above 'A + {_{a1, a2, a3,...}}' should be A = {_{a1, a2, a3,...}}. I have vision problems. And, yes, 1,3,5,... is also clearly a well ordering and so are many other sequences due to the definition of WO..

You are missing the point. Just because a set is well-ordered does NOT mean it looks like a1, a2, a3, ... First, the a1, a2, a3, ... order is the usual order type of the natural numbers. There are a lot of other well-ordered set. Secondly, enumerating a1, a2, ... implies that your ordered set is countable. But a well-order of the reals is an uncountable well order. It starts as a1, a2, a3 ... but after the dots there's a lot more stuff.

You said a while back that you don't want to learn about ordinals, but that's exactly the study that would clarify many of your misconceptions. A well order of an uncountable set can't be notated a1, a2 ... It BEGINS that way, but after the dots there are many more reals.

If you diagonalize any countable list of reals, you'll find a real not on the list. 

5 hours ago, studiot said:

 

The quantities   δx   and  δy

are infinitesimals. (Newton called them fluxions, not infinitesimals)

 

 

 

Studiot you are most definitely wrong about that. Since there are no infinitesimals in the real numbers, how could that possibly make sense? dx and dy are differential forms. They are not infinitesimals in the modern view. Nor were they ever. Nor did Newton think they were, although the historical evidence for that proposition can be argued. But mathematically, dy and dx are not infinitesimals.

ps -- Sorry what? What is δ? I must be misunderstanding you. Off my game. (THAT WAS A JOKE!! FROM NOW ON I WILL CLEALY NOTE MY JOKES AS SUCH)

1 hour ago, discountbrains said:

OK, here we go. I'm just answering wtf at this point. Let A = (0,1) with the ususal ordering. Yes, it seems obvious like he says if a set A is WO by some other ordering  then it can be able to be written as A + {_{a1, a2, a3,...}}. This means it can be arranged in one of Cantor's array like things. 

No. When you well-order the reals, the reals are still uncountable. 

Jeez guys I thought it was inadvertently funny, not a deliberate insult.

5 hours ago, taeto said:

Fair enough. I forgot about the exact title. The thread is divided among contributors who either do not know what is in the thread or do not remember what it is about. Plus Strange and studiot, they are always on the game.

Studiot is British. "On the game" is quite the insult. I believe you called him a prostitute or a pimp. Studiot please confirm or clarify my understanding. 

1 hour ago, discountbrains said:

Ok, the question just popped in my head. I didn't give it any thought. There are, I think, many ways to construct a bijection between N and Q.

"Nice to know that those with greater depth of knowledge of these matters than mine have my back." Yes, I know its universally true its always safest to be on the side of conventional wisdom. I ought to try it sometime, but if everyone did this there would be no advances in anything.

I actually have your back on that last point. Einstein was told that hundreds of physicists disagreed with his theory. He said: "If my theory is wrong, it would only take one."

I agree with you that it doesn't matter who agrees with whom. If the whole of established mathematics believes one thing, and you believe another, and you have a proof, then you are right and they are all wrong. 

So ... have you got a proof?

1 hour ago, discountbrains said:

I quite well understand this. Of course they are mutually exclusive. Also, its hard to imagine how any set that is dense with respect to < or about any ordering could be ordered in such a way  that it can be enumerated by 1,2,3,... There are just too many numbers that still pop up in between. I was going to add in my post above that intuitively it seems you can pick one number out of any set in a collection of sets. But, actually doing this is something else. For one thing, which number are you going to choose? Maybe I should ask another question: Can one show that any set dense with respect to one ordering is therefore dense for all possible orderings?

 And, yes studiot, I also think I've found the best way to understand a new concept is to see a few examples of it. Its maddening sometimes to try to figure out what a theorem is saying by just looking at it alone.

> I quite well understand this. Of course they are mutually exclusive. Also, its hard to imagine how any set that is dense with respect to < or about any ordering could be ordered in such a way  that it can be enumerated by 1,2,3,... There are just too many numbers that still pop up in between. 

Ah! But I just explained that. We know there's a bijection between the naturals and the rationals. So to put a well-order on the rationals, we just enumerate them q1, s2, q3, ... where qn is f(n) where f is the bijection.

Going the other way, we may put a dense order on the natural numbers by simply using the inverse of that bijection, call ig g. So g(q) is some natural number. So order the naturals n, m according to whether g-inverse(n) and g-inverse(m) are < or > to each other in the rationals.

That's why this example is so important. It shows how we may reorder a given set to have strikingly different properties.

>Can one show that any set dense with respect to one ordering is therefore dense for all possible orderings?

On the contrary we can prove that false. Any set may be well-ordered. That's the well-ordering theorem. Given any dense set you just well-order it. That's the point of the entire thread, right?

And of course a MUCH SIMPLER case is that the rationals, which are densely ordered, may nevertheless be well-ordered using any bijection with the naturals. And we do not need to rely on the well-ordering theorem for this fact. That's important. The well-ordering theorem requires the axiom of choice. But establishing a bijection between the naturals and the rationals does not. So your claim is falsified by THE MOST OBVIOUS EXAMPLE.

That's another reason this example is so helpful. It's like the well-ordering theorem but without having to invoke the awesome power of the well-ordering theorem! That's why it's of such great interest.

9 minutes ago, studiot said:

 

 

Be aware that wtf was refering to the rationals (symbol Q) not the integers or natural numbers, (symbols Z and N respectively).

Although the rationals are countable they are also dense, which the integers and naturals are not.

In this discussion it's important to note that the rationals are dense in their usual order; and that the naturals are well-ordered in their usual order.

Consider that we may use any bijection between the naturals and the rationals to induce the order of one onto the other. In that way we may impose a dense order on the naturals; or a well-order on the rationals.

Another way to look at this is that the naturals and the rationals are actuall the same exact set. Their order properties depend only on the order we impose on them. If we impose the rational order, our set looks like the rationals. If we impose the naturals, our set looks like the naturals. But it's the same underlying set.

I'm not claiming this as a "fact," as if there could even be any facts about abstract entities such as sets. Rather, I'm presenting this as a point of view that gives insight into the nature of ordered sets. 

It's also worth noting (again) that the properties of density and well-order are mutually exclusive. An ordered set may be one or the other, or neither. But it may not be both.

59 minutes ago, taeto said:

Strange: there is no proof. It is just an illustration of how the Collatz sequences work, using the standard tree representation.

I suggest that it is allowed to discuss a problem in the forum without making a claim to have solved it, no?

Thread title is "a possible proof." I haven't followed the thread in any detail.

48 minutes ago, phyti said:

This is the file.

 

Collatz 2018.pdf

I read the conclusion. You never even claim to have solved the problem. 

3 hours ago, Edgard Neuman said:

I have a nice video from numberphile that says 1+2+3+4+5+ ....  = -1/12 .. but because ... = ... +1 ? Isn't it 
1+2+3+4+5+ ....  = -1/12  +1   ? I'm confused 

 

I'm sure Numberphile explained that this is an INFORMAL way of looking at something much more subtle. You need to use the analytic continuation of the Riemann zeta fuction via a process called zeta function regularization, to show that zeta(-1) = -1/12. You can CASUALLY think of zeta(-1) as sort of looking like the formal expression 1 + 2 + 3 + ... but it actually isn't. That series diverges to infinity. 

It's sad that so many people have gotten confused by this video. Numberphile is usually reputable but they unleashed a lot of confusion online about this issue.

https://blogs.scientificamerican.com/roots-of-unity/does-123-really-equal-112/

https://plus.maths.org/content/infinity-or-just-112

4 hours ago, discountbrains said:

Can't find an 'edit' button. Ran out of time yesterday. wtf kept correcting me saying look at an example like the natural numbers and observe these clearly can be WO. I was very surprised at his answer since it sounds like he is a professor or something. This is clearly incorrect: the thing is like what I said above which is 'not finding one example of a set WO'ed by some.<*'. The example he gives is not dense anyway.

 

Astonishing. 

* I told you at least three times that I invited you to look at the example of the naturals since they would help you gain insight into the problem you're attacking. That you continually fail to comprehend this point is mystifying to me. I say again: The way you attack a hard problem is to start by analyzing simpler instances of similar problems. If you don't want to get this, so be it.

* You say I am "incorrect" that the real numbers can be well ordered; but that they can be bijected to the rationals, which are NOT well-ordered. You say that is incorrect. Please provide a proof or explanation. 

* Being well-ordered and being dense are mutually exclusive. I thought I pointed this out several times. I thought you UNDERSTOOD this several times. As an exercise -- again, to help you build intuition -- I suggested that you write down a formal proof. I reiterate that suggestion.

Since you reactivated this thread you haven't provided a proof or even any additional argument.

6 hours ago, Edgard Neuman said:

but you can't have n -> n*2  or n-> n+1  because if you assume that  you have ALL the elements from ℕ in the left set,
some elements from the right set are always bigger, so they are not in the ℕ you started with (unless you didn't have ALL, which is a contradiction with the hypothesis, and in that case, the bijection wouldn't be decided)
 

I don't understand this point at all. 

Let's look at a simpler case. We have two sets A and B.

A = {1, 2, 3, 4, 5} and B = (2, 4, 6, 8, 10}.

It seems perfectly clear that we can biject the elements of A to those of B, despite the fact that the numbers in B are larger than some of the ones in A. And it's true that some of the elements of B are not in A. Why does this trouble you?

7 minutes ago, pengkuan said:

This means that the set of natural numbers {1,2,3…} contains only numbers with finite number of digits. Suppose the number of members of N is n.

Then, the set of even numbers corresponds to 10…101010, with n digits.

When you evaluate that string using standard positional notation, how do you do it?

4 hours ago, studiot said:

I suggest you read the standard Oxford University text at this level for half a century

That'll keep him busy.

I'm only following sporadically but this is what I'm confused about from the beginning.

* If [math]\widehat{0} = 0[/math] as has been recently acknowledged by OP; and

* If correspondingly, [math]\widehat{\infty} = \infty[/math]; and

* If the symbol [math]\infty[/math] is identical to [math]+ \infty[/math] in the extended real numbers; which I believe the OP acknowledged a while back;

* Then the expression [math]\widehat{\infty} - b[/math] is undefined. It can't be sensibly defined. The exposition so far has not defined it.

Has the OP addressed this issue yet?

Also, I'm still confused about the neighborhood of infinity. In the real or extended real numbers, we can certainly call the set of reals larger than some fixed real, a neighborhood of infinity. And if we consider the set of reals whose absolute value is greater than some fixed real,then that would be a neighborhood of infinity in the space where you identify plus and minus infinity of the extended reals. In other words it's a circle, but with the point diametrically opposed 0 called infinity. It's a regular circle but with a funny metric.

The two-dimensional analog of that construction shows how to regard the complex numbers as a sphere. If you take the complex plane you can add a single point called [math]\infty[/math], which can be visualized as the north pole of a sphere. This is called the Riemann sphere, https://en.wikipedia.org/wiki/Riemann_sphere

It's like going "out to infinity" in every direction and sewing together the boundary at infinity to make a sphere. Since there's no order on the complex numbers, there's only one point at infinity. Another name for this construction is the one-point compactification of the plane. If you know what a compact set is, you know the plane's not compact. But using this construction you can add a single point and make it a compact set.

If you draw a circle on the sphere centered at the north pole, you could legitimately call that a neighborhood of infinity of the complex numbers. It's the same as if you consider the set of all complex numbers whose modulus, or absolute value, is greater than some fixed nonnegative real number. 

Now as I understand it, the OP does not mean any of those things. I still don't understand what "the" neighborhood of infinity is; and I also don't understand why OP calls it "the" instead of "a", since in the contexts I mentioned, there are many neighborhoods of infinity.

1 hour ago, Country Boy said:

Originally Newton tried to define the derivative as "dy divided by dx" where dy and dx are "infinitesmals" but was not able to give a rigorous definition of "infinitesimal".  

I do not believe this is historically accurate. Leibniz used infinitesimals directly. Newton referred to the "ultimate ratio" of delta-y over delta-x as both become closer and closer to zero. Newton's idea has more in common with the modern viewpoint than Leibniz's. That's my understanding. There is historical debate around this topic.

3 hours ago, sevensixtwo said:

real numbers are cuts in the real number line. 

But this couldn't be. You are defining the real numbers as cuts in ... what? In the real number line. Where do those real numbers come form before you have defined them? Your definition is circular.

Of course the answer is that the real numbers are cuts in the rationals. You should demand your money back from the university that sold you a course in real analysis, since you clearly didn't learn anything.

Just now, sevensixtwo said:

You are probably not in the intended audience for the paper then.

It's true. I'm not a member of the empty set.

Why don't you just explain yourself?

1 hour ago, Strange said:

I am not a mathematician and am not qualified to comment on the content of the paper, but I dod have a (perhaps naive) question. The paper says: 

As the reals are uncountable, can one make a statement comparing the numbers in one subset with those in another like this? Just curious...

It's certainly true in the usual interpretation of "a" neighborhood of infinity; if by that we mean the set of reals whose absolute value is greater than some fixed value. In other words there's a bijection between the set of reals x with |x| <= 1000 and the set of reals x with |x| > 1000.

But the OP seems to have some other definition in mind for "the" neighborhood of infinity, the definition of which he has not given with sufficient clarity.

1 hour ago, sevensixtwo said:

No.  The meaning is that ascribed to it in the sentence which describes it in the body of the article and the author, who is definitely me, has also provided a citation for anyone who cannot parse "all canonically non-standard properties except additive absorption."  However, anyone who cannot parse that statement is probably not able to parse the similar statements in the citation.  On the other hand, if you wanted to make the argument that something which has not been previously invented cannot be invented then I would be open to reading your argument.

I couldn't parse it. Can you please explain it? What do you mean by "canonically non-standard properties?" Are you referring perhaps to the nonstandard real numbers aka the hyperreals? If so, you should definitely say so.  And even with that interpretation I have no idea which canonically non-standard properties you're referring to.

2 minutes ago, Strange said:

 

What's an "oid"? 

An old is what the kids call some of us these days.

https://www.urbandictionary.com/define.php?term=An Old