taeto

On 10/6/2018 at 9:43 AM, sevensixtwo said:

Nah, you are wrong.

Three questions:

1. Who is wrong?

2. What are they wrong about?

3. What would be right?

2 hours ago, 113 said:

Who told you about dydx but hided the information about f(x+dx)−f(x)dx ?

Was it me or was it all the math books? And why should they, the mathematicians,  have done so?

If your math book suggests to compute the derivative of \(y\) as the limit of the difference quotient of \(f\), without saying first that \(y=f(x)\), then it makes sense for you to throw it in the recycling bin.

2 hours ago, 113 said:

Why can't "usual mathematics" deal with something that is infinitely small?

Maybe it can. But this is analysis, a particular branch of math that deals with functions of real and complex numbers. If you ask whether it makes sense to have a real number \(d\) with the property \(0 < d \leq x\) for all real numbers \(x\), then the answer is that such a \(d\) cannot exist, because, e.g., \(d/2\) has \(0 < d/2 < d\) in \(\mathbb{R}\) contradicts that \(d\) has the required property. 

If this is not the property that you would want an "infinitely small" number to have, then what is the property that you are thinking of? Will you evade to answer that question? 

2 hours ago, 113 said:

What is the question? Is the question "is 1/∞ either 0 or non-zero?" ?  The question does not have an unambiguous answer either 0 or non-zero, because 1/∞  is both.

Just writing up some string of symbols like \(1/\infty\) doesn't always point to something that makes any sense. I ask again, what does it mean to you? Apparently you can make sense of it, since you keep going on about it. 

2 hours ago, 113 said:

If that is a stupid thing to do, then why don't you answer the question: "is 1/∞ either 0 or non-zero?" ?

Why don't you answer the question: "what does it mean?"? 

2 hours ago, 113 said:

Yes. But I am not talking about sets,  because maybe it is not necessary right now. I am talking about finite difference f(x+h) - f(x) where h is finite, as opposed to f(x + dx) - f(x) where dx is not finite. Adjective finite refers to distance in this case, finite distance.

Now it looks like you use "finite" to mean "non-zero". In the context of analysis, distance is given by Euclidean metric, and then the only distance that is not "finite" is identically 0. You end up with \(f(x+dx)-f(x) = 0\) always, independently of \(f\) and \(x\). 

2 hours ago, 113 said:

So it may look as though dx is a set for which you don't tell what is n.

The most positive I can say would be something like just forget about the actual meaning of \(dx\). In ordinary usage, the functions \(x\) and \(dx\) belong to different species, they do not allow to be composed together by the binary operation of addition. It is something like trying to add a scalar to a 2-dimensional vector. 

The most you could do is to use your expression as a notational shorthand, which substitutes \(dx\) for \(h\) and removes the needs to write the \(\lim\) symbol. Maybe you can think of any advantages in doing so. Something like how the Leibniz notation allows to write the chain rule in the form \(\frac{dz}{dx} = \frac{dz}{dy}\frac{dy}{dx}\).

1 hour ago, 113 said:

It makes no difference because y = f(x) so that dy/dx = df(x)/dx

You were hiding the information about \(y(x)\) being the same as \(f(x)\)?

1 hour ago, 113 said:

1/∞ is infinitely small. Is it 0 or non-zero?

I do not know the answer. What do you mean by \(\infty\)? And by \(1/\infty\)? Why do you think that \(1/\infty\) is infinitely small? What does it mean to you that something is infinitely small? You are saying things that have no place in usual mathematics. And do you know how to answer a question without introducing several more unknown quantities?

1 hour ago, 113 said:

The introduction of infinity brings a duality into the definition of an infinitesimal, meaning that we have to deal with objects that both zero and non-zero at the same time.

What do you mean when you speak of "infinity"? What is that exactly to you? I have absolutely no idea what you are going on about. Why would we have to deal with objects that have contradictory properties, it would seem an extraordinary stupid thing to do, no?

3 hours ago, 113 said:

I mean that dx is not finite as opposed to h which is finite. If dx is not finite, it is infinite, infinitely small.

Again now, look: the adjective "finite" in mathematics applies to sets. A set X is finite if there is a natural number n such that the "size" of X is n, which means that there exists a bijection from {1,2....,n} to X, so that you can count the elements of X from 1 to n. So correspondingly, the adjective "infinite" applies to a set for which there is no such n. Which kind of set is your dx for which the adjective "infinite" applies to it, and how can you prove this? And if so, how does this fact apply to the rest of the things that you are saying?

1 hour ago, 113 said:

Yes.  Derivative is defined as the limit of a finite difference

 

dydx=limh→ 0f(x+h)−f(x)h

 

where h is finite.

     There is a "y" on the left hand side, but an "f" on the right hand side. The equation relates the derivative of a function y of x to the derivative of a function f of x. That is not what the definition of a derivative is supposed to look like. 

1 hour ago, 113 said:

I mean that dx is not finite as opposed to h which is finite. If dx is not finite, it is infinite, infinitely small.

What kind of thing is dx supposed to be? An integer, a rational number, a quaternion, what? What do you mean when you say "infinitely small"? Can you give any examples of mathematical objects that are "infinitely small"?

58 minutes ago, 113 said:

Is it possible to define the derivative by

 

dydx=f(x+dx)−f(x)dx

 

where dx is not finite but infinitely small, infinitesimal ?

You mean \( \frac{df}{dx}(x)\). 

Maybe it is possible. But only after you explain the meaning of the expression on the right hand side. E.g. what exactly do you mean by saying that dx is "infinitely small, infinitesimal"? If there are several different dx that are "infinitely small, infinitesimal", then it would only make sense if you can prove that the value of the expression on the right hand side does not depend on which one of the possible values of dx that you apply. Obviously you have to explain what it means to do addition and division with objects that are "infinitesimal". You will have a lot of work to do before you can convince anyone that it makes sense. If you are willing to put in the work, then good luck. 

1 hour ago, taeto said:

You mean the page on "Real Number"? I would like to know where on that page you can find this? I see mention of "Dedekind cuts", but I would be concerned to define real numbers in that way, it is only one of many possible constructions of reals, and the other ones are just as good. It is not a definition anyway, only a construction. 

I decide to take this seriously, because I just see it over and over, with lots of misunderstanding mixed in.

Take another example than the concept of "real numbers".

Let us say we talk instead about "cars". We look up the wikipedia definition "A car (or automobile) is a wheeled motor vehicle used for transportation. It runs primarily on roads, seats one to eight people, has four tires, and mainly transports people rather than goods." You may agree that by this description you now have a firm basis for recognizing when some object in your vicinity is a car or not. Now someone says that Porsche has provided a construction of a car, so thereby and henceforth, the definition of a car has to be something that is made by the Porsche factories. The big news, apparently, on this forum is that: no, that is not how definitions work. 

In the case of real numbers, or in indeed all mathematical objects, the first thing that comes about is a description of what are the supposed properties of such a thing. In the case of real numbers it happens to be not the number of wheels etc., but things like properties of addition and multiplication, ordering properties, and more. When you have those properties sorted out, you are ready to distinguish between objects that are real numbers and those that are not. If somebody says, wait, Cauchy has a construction of real numbers, so we should take the definition of real numbers to be equivalence classes of Cauchy sequences, then that is BS, it is still not how it works. Not with cars, not here either.

2 hours ago, sevensixtwo said:

Since you seem to have a brain and are not trolling me like user name "stupid idiot," what is your opinion on an implication that infinity is a real number?

Do not worry. Nobody in this forum will troll you, the moderators are too strict. 

You seem confident when you use the word "infinity" that it means something specific to you. To me it means nothing in particular. For someone who works with elliptic curves, infinity is just an extra point that gets added to get the full group structure. It could be that infinity is something like the set with one element \(\{0\}\). Of course it is the name of the symbol \(\infty\) that gets used all the time in indefinite summations and integrals, and in limit calculations. It is just a letter in that context. So if you say "let infinity be a real number", then I am cool with that, except I would find "let x be a real number" easier to live with.

But in contrast you mean something precise and absolute when you say "infinity"?  

2 hours ago, sevensixtwo said:

I agree that the rules you have proposed lead to the implication that infinity is a real number but I am not willing to concede that these rules are part of the definition of a real number.  For that, you would need to present some third party source which say the rules you posed are part of the definition of real numbers rather than merely some properties which can be derived by only considering real numbers in the neighborhood of the origin.

The wikipedia page says that \((\mathbb{R},+,\cdot)\) is a field, in particular \((\mathbb{R},+)\) is a group, which implies the three properties that I mentioned, if you add a simple notational convention. I do not know if you will consider the defining properties of a group as only "derived". 

2 hours ago, sevensixtwo said:

Until you do, I will stick with the definition in my real analysis book.  If you have such a book it is probably in yours too: real numbers are cuts in the real number line. 

No, I have to admit that I never came across such a definition. What is the name of your book? Maybe I will recollect.

2 hours ago, sevensixtwo said:

I just tried to google it and Wolfram doesn't define real numbers but I found the same definition on Wikipedia.  I can post the link to Wikipedia if you want.

You mean the page on "Real Number"? I would like to know where on that page you can find this? I see mention of "Dedekind cuts", but I would be concerned to define real numbers in that way, it is only one of many possible constructions of reals, and the other ones are just as good. It is not a definition anyway, only a construction. 

1 hour ago, sevensixtwo said:

but I do not see how that implies that infinity is a real number.  My definition of a real number is the one I learned in the first five minutes of the first lecture of my first semester of undergraduate real analysis: a real number is a cut in the real number line, only that and nothing more.  All other definitions have to do with fields of numbers and other more complicated things that exceed the definition of a real number.

     I do not recognize your definition of a real number. But it is not serious, because after all the precise definition does not matter, so long as we agree about the properties of real numbers.

So at which step do I go wrong in the following:

1. if x is a real number, then -x is a real number,

2. the difference y-x is the same as y+(-x),

3. if x and y are real numbers, then x+y is a real number?

Because if these are all correct, then the difference y-x between real numbers x and y will always be a real number.

1 hour ago, studiot said:

As I understand neighborhoods as applied to the reals, the infinity and its neighborhood arises not as an endpoint to the real line but as a result of a limiting process

In one  of the first posts I already asked him to define a neighbourhood of infinity, which he did by the example of the complex plane, where it is given as the unbounded region outside a circle of radius R.

The real line is special, since there are two regions defined by a circle \( \{-R,R\}\), a negative and a positive neighbourhood of infinity. In higher dimension the neighbourhood is always a connected set.

But I suspect that he is thinking of a neighbourhood of infinity as something different yet.

9 hours ago, sevensixtwo said:

That doesn't follow.  If you think it does, please give two real numbers whose difference is infinity.  Before you do that, please consider that if you define a real number as numeral rather than as a cut in the real number line I will have to remind you that the definition of a real number is that it is a cut in the real number line.

I just pointed out that \(\widehat{\infty}\) is the difference between \(b\) and the real part of \(z_0\), and deduce that \(\widehat{\infty}\) is also real.

Are you sure about your definition of a real number, because it sounds circular?

9 hours ago, sevensixtwo said:

If you would have read the paper, you would have seen that "b" is a real number.  There is no real number greater than infinity.  

Isn't \( b - \widehat{\infty}\) the real part of your \(z_0\)? If \(b\) is also real then \(\widehat{\infty}\) is the difference of two real numbers, hence real. It is what you call "infinity"? But for any real number there are larger real numbers. 

And anyway, the question is why the real part \( b - \widehat{\infty}\) of your \(z_0\) is not \(1/2\). Is it because it is not real?

14 hours ago, sevensixtwo said:

If you would have read the paper, you would have seen in the first few lines that infinity is "+/- infinity" and if you were familiar with analysis at even the undergraduate level you would know that these are the endpoints of the extended real line.  You might feel more at home in the less advanced forum because these symbols I use are already quite basic.

I believe the point is that it is already known that every nontrivial zero has real part strictly between 0 and 1 (the 'critical strip'). So where you say that there are nontrivial zeros with real part equal to \(-\widehat{\infty}+b\), we all see that as a consequence, \(b\) must lie strictly between \(\widehat{\infty}\) and \(\widehat{\infty}+1\). How do we make sure that \(b\) is not exactly equal to \(\widehat{\infty}+1/2\), in which case you would just have a point on the critical line, and not a counterexample?

9 hours 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 would argue that you cannot refer  to "the neighbourhood" of infinity, when your definition of such a neighbourhood clearly depends on a positive real number R, unless you fixate the value of R. Grammatically correct would be "a neighbourhood" otherwise. But since every nonzero complex number belongs to some such neighbourhood, it would seem to add no further information.

Thank you Strange for the references!

So we know only a little bit about \(\widehat{\infty}\) from that paper, in which it is a central quantity.

Then for the OP: How can we learn enough about it to be able to read past the third

line of the paper, where it first appears?

Could you present an explicit example in which \(\infty\) and \(\widehat{\infty}\) both appear in the roles

that they are supposed to occupy in this paper? 

1 hour ago, sevensixtwo said:

I think the numerical value of the radius of the neighborhood of the origin might not exist due to equation (1).  What is your opinion regarding the question you asked?

The value of R goes into the definition of "neighborhood" in the beginning. Are you saying that this value does not exist, hence the definition is meaningless? If you think that the paper begins with meaningless definitions, then why would you present it here in the forum?  

      So in the article, the real numbers are divided into those x with |x| at most R

and those with |x| > R. Does the numerical value of the radius R matter at all? It

is not specified in the OP article? 

       What is the meaning of \( \widehat{\infty} \)?

How is "neighborhood of infinity" defined?

25 minutes ago, Carrock said:

The problem is that he claims that each of the uncountably infinite points in that (finite or infinite) universe 'expanded' into a finite volume. That results in a universe with an uncountably infinite volume. Not possible.

Can you make this precise? Would you say that if we 'expand' each point of the uncountably infinite set R into a line of uncountably infinite many points, thereby naturally expanding R into R^2, then we have  done anything impossible? Because R has measure zero in R^2? The cardinalities are still preserved. 

8 hours ago, beecee said:

an infinitesimal fraction of the infinite Universe

If you want to throw in a notion which serves to invalidate your entire explanation, then this is a pretty good choice by the author of that piece.

1 hour ago, Carrock said:

 you can't expand an uncountably infinite number of points into a universe of uncountably infinite volume.

Surely you cannot expand anything other than an uncountably infinite collection of points into a universe of infinite volume (though the same would be true for any positive volume). Other than the trivial copout to expand by adding a sufficient amount of enough new points to do the trick, of course.

2 minutes ago, DannyTR said:

I only have a problem with the ‘axiom of infinity’; the rest of ZFC I’m not questioning. Calculus worked just fine for 300 years before the invention of infinite sets. I think the limit concept is fine as that can be defined with just potential infinity. 

Excuse me. Please explain how the Axiom of Infinity has any bearing on the definition of a limit in Calculus??

And since you will not be able to do that, please explain why you thought that was the case.

1 minute ago, Strange said:

I had never heard of “finitism” before. If anyone else is in that fortunate position, you can confirm your worst fears about philosophy as an academic subject here: https://en.m.wikipedia.org/wiki/Finitism

I do have a lot of respect for proper philosophy. But not this sort of junk. 

I have not checked that particular wikipedia page, which I admit could possibly be infested with crackpot contributions. But finitism is basically a respectable theory, just to check how much one can get away with without the possibility of something like "infinite sets". Obviously it is not a lot, as it bans you from doing arithmetic, calculus, complexity etc. But in some contexts of fundamental maths it is a worthwhile study.

Of course the cranks, like the OP, hate this theory. They want to argue that all of ZFC is "wrong", without admitting to any finitistic presumptions.  

38 minutes ago, DannyTR said:

Actual Infinity is difficult to define because it does not exist exist mathematically or in nature; just exists in our heads as a (flawed) concept.

Then why do you want to discuss about it all the time? Noone else has an idea what it is about, and neither do you.

Do you happen to know about the theory of "True Arithmetic"? It has only natural numbers 1,2,3... in it, and all kinds of true theorems on addition and multiplication of those numbers. It is also known as the standard model of the natural numbers. There are no infinite objects in this theory. It is a part of mathematics, but would you approve of it nonetheless? 

19 minutes ago, Carrock said:

Everything I was going to post has been posted while I was composing my response...

except which is 'Actual Infinity' supposed to be?

The number of integers, the number of points on a line... etc

It is Saturday evening here; everyone has a lot of time on their hands, sorry

DannyTR already argued why there cannot be any  number larger than all other numbers, which, assuming the total ordering of numbers, is the same as saying that there has to be infinitely many numbers. By inference he is not disputing that fact.

A, ehm, mathematically inclined person might speak of the "cardinality" of the set of integers, the set of points on a line. Rarely of the "number". But you hint at the possibility that the OP is thinking about this "Actual Infinity" as a number?  

9 minutes ago, DannyTR said:

That falsehood has made its way from the abstract study of sets into the non-abstract study of the universe and IMO is causing much confusion and time wasting.

Q.E.D. as witnessed by this thread: OP's confusion and everyone else's time wasted.

4 minutes ago, DannyTR said:

The definition of a set in set theory is also wrong (polymorphism).

That is pretty hilarious. I wasn't sure before that you are just trolling. Just for fun though this question: if the objects of set theory should not be called "sets", then what is your alternative suggestion?

8 minutes ago, Carrock said:

From   https://en.wikipedia.org/wiki/Axiom_of_infinity linked from the previous reference:

The inductive set is familiar. How do we get from there and to a definition of "Actual Infinity"?