B Help answer student question about infinity?

[Mark44]

Well I guess I'm trying to make a philosophical point whether valid or not it seems like it is to me.

Whatever point you're trying to make isn't based on logic. As you say, you're not a math person, so IMO, you're in over your head here.

The point has already been made but isn't it just nonsense to say "the probability of choosing one item from an infinite set is zero?"

@PeroK gave a good explanation of why this is not nonsense in post 17. Did you read it?

If you draw a line of longitude and latitude according to the coordinates that intersect at that point, the lines must have some thickness to create a point thus the point must have some dimensions to exist. Maybe you can conceive of a line with no thickness or a point with no dimensions but it can't exist--be on the map--unless it has dimensions. edit: or can you even conceive of one really?

You've totally missed the point of my analogy. It was not whether lines have thickness or points are dimensionless -- it was a thought experiment to provide an example of an infinite length that had a starting point.

Ok but from what little I've checked it seems a set has no real definition in mathematics or a very broad one so maybe that's the argument we're having. I'm defining it as a group of things. If a set is a group of things and there are no things then there is no set.

"From what little I've checked" is key here. I agree that sets are loosely defined, for the reason that the most basic terms can't be defined by using more basic terms. Empty sets are allowed to provide for completeness. An empty set can be combined (the operation is called a union) with any other set so that the result in a new set that is identically the same as the one you started with. This is the set analog of 0 being the additive identity in arithmetic. I.e., for any number x, 0 + x is identically equal to x.

That is a question that is discussed here and elsewhere occasionally. It is not a simple question. I personally have no problem with saying that a number between 0 and 1 has been selected using a uniform distribution, even though there are uncountably infinite numbers there, each with a zero probability.
I know that many people disagree with me, many who may be much smarter than me.

Again, PeroK gave a good explanation of why you if the numbers are assumed to be uniformly distributed, then the probability of picking a single number can't be larger than zero.

If I'm understanding...isn't "countably infinite" a contradiction?

No, not at all, and this goes back to what you said about not being a math person.

How can you say an infinite quantity of things is dense or not?

You can say it if you know how the term "dense" is defined. Here's a link to a wikipedia page on this subject - https://en.wikipedia.org/wiki/Dense_set. The set of positive integers, {1, 2, 3, ..., n, ...}, is an infinite set, but this set is not dense in the real numbers. On the other hand, rational numbers are dense in the real numbers.

Once you get beyond the concept of finite sets, you learn that there are different "sizes" of infinite sets. Here are several infinite sets: the positive integers, the set of even positive integers, the rational numbers in the interval [0, 1], the real numbers. Although it seems counterintuitive, the first three sets are the same "size" in some sense, while the fourth set, the reals, constitute a much "larger" set.

But I'm not a math person anyway agree to disagree I guess.

Then your position is at odds with everyone in the world who has studied mathematics beyond the first year level in college.

What's the process to match them up? Ok let's start 1 goes to 1/a billion, 2 goes to...wait wait we missed one go back. Ok start 1 goes to 1/a billion billion 2 goes to....wait wait stop again, we missed one start again!

Let's looks at a simpler example, with one set being the positive integers and the other the set of even positive integers. If I can demonstrate a one-to-one pairing between the two sets, then the sets have the same cardinality, the actual term that is used to compare sizes of infinite sets.
In the following, the first number is one of the positive integers and the second is one of the even positive integers.
1 gets paired with 2
2 gets paired with 4
3 gets paired with 6
...
n gets paired with 2n
...
If you tell me an element of the positive integers, I'll tell you what even integer it gets paired with. OTOH, if you tell me one of the even integers, I'll tell you the integer that it got paired with.

Is there a ratio between them that for every integer there's so many natural numbers? No because for every 1 integer there's infinite natural numbers.

???
I have no idea what you're saying here.

Just my opinion just seems like an error of reasoning maybe.

Yes, you are making many errors of reasoning.

 

[Blargus]

Let's see if this professor also gets deleted here as an "illegitimate source:"

"The Hilbert hotel, in my view and in the view of a mainstream of mathematics is not a true paradox in the sense of one that is reveals a deep contradiction in our conceptions. It is merely an appearance of such a problem. It requires us to reconfigure our default understanding of infinity. Much of what follows implements that reconfiguration. However for a minority view in mathematics, matters are otherwise. This minority view is not to be dismissed lightly, since it is held by some distinguished mathematicians. The dissenting view falls under the rather loose banner of "finitism." There are many versions of it. What unites them is a distaste for the actual infinities (as opposed to potential infinities), that are so readily embraced in the Hilbert hotel and other related analyses. Even Hilbert himself seems to have had some sympathies for a version of finitism. To say more goes well beyond what can be responsively covered here. If the view has a rallying cry, it might be Leopold Kronecker's 1886:


"Dear God made the whole numbers; everything else is the work of man.""

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh
https://sites.pitt.edu/~jdnorton/teaching/paradox/chapters/infinity_paradox/infinity_paradox.html

 

[renormalize]

This minority view is not to be dismissed lightly, since it is held by some distinguished mathematicians.

John D. Norton
Department of History and Philosophy of Science
University of Pittsburgh

John D. Norton is indeed a distinguished professor in the Dept. of History and Philosophy of Science, but he's no mathematician. It would be helpful if you could provide some citations to recent publications from "distinguished mathematicians" that espouse this "minority view".

 

[Blargus]

John D. Norton is indeed a distinguished professor in the Dept. of History and Philosophy of Science, but he's no mathematician. It would be helpful if you could provide some citations to recent publications from "distinguished mathematicians" that espouse this "minority view".

I guess I could try though isn't that a job for the "math people?" Or people who can make any sense at all of math publications? I have a hard enough time with pubmed.

 

[Drakkith]

I guess I could try though isn't that a job for the "math people?" Or people who can make any sense at all of math publications? I have a had enough time with pubmed.

Generally it is the responsibility of the person proposing a position to provide references supporting that position when asked to do so. If, by your own admission, you cannot make sense of math publications then I recommend not supporting a minority or fringe position on a topic. You can, of course, always ask for help finding and understanding such references, but it's a little odd to try to argue for a minority position that you yourself don't understand well enough.

 

[Blargus]

Generally it is the responsibility of the person proposing a position to provide references supporting that position when asked to do so. If, by your own admission, you cannot make sense of math publications then I recommend not supporting a minority or fringe position on a topic. You can, of course, always ask for help finding and understanding such references, but it's a little odd to try to argue for a minority position that you yourself don't understand well enough.

You guys. It was told to me that my understanding on infinity or not believing you can make an infinite set was ridiculous, irrational, illogical and not held by anyone who has studied math beyond the first year. But hey I appreciate you trying I guess and the conversation helped me understand what my "problem" is.

Rather than continue to argue as I am not able to do with math I attempted to show by "legitimate sources" that this is actually a controversial topic in the history of mathematics at least at the turn of the last century and into the 20th.

So if you guys wanna not take that point because I can't find a 2023 journal article of Gobbledegook Theory Volume 80 then me and Leopold Kroeneker apparently will bid you adieu.

ATM I'm glad I can't live in "Cantor's Paradise" and I wonder if anyone should.

 

[Drakkith]

Rather than continue to argue as I am not able to do with math I attempted to show by "legitimate sources" that this is actually a controversial topic in the history of mathematics at least at the turn of the last century and into the 20th.

That's fine if you want to discuss the history of this topic, but those views are a century or more out of date now, so why bring them up? There have been many, many mathematicians in the last hundred years just as smart, productive, and knowledgeable as Cantor, Hilbert, and other mathematicians from the late 1800's and early 1900's, and the vast majority of them accept Cantor. So you can either believe that all of these people are absolutely and utterly wrong, or you can accept that maybe the situation is a bit more complicated than you understand and accept that they probably know a few things you don't.

 

[Mark44]

Rather than continue to argue as I am not able to do with math I attempted to show by "legitimate sources" that this is actually a controversial topic in the history of mathematics at least at the turn of the last century and into the 20th.

Correction to the above: "this was actually a controversial topic" up until about 100 years ago. Only a few years before that many people also believed that travel faster than 60 mph would be fatal to humans.

I read the link to the Claes Johnson blog post that you linked to before it was deleted. It's interesting to me that this 2009 post of Johnson's had all of 2 votes. It's not surprising to me that computational mathematicians have a distaste for infinity, inasmuch as computers are limited to dealing exclusively with a finite set of numbers. I did find it surprising that Johnson included a quote from the author Jorge Luis Borges.

One of your "legitimate" sources was Kronecker, whose main quibble seems to have been that Cantor did not show the fractional numbers in complete form in his list. Every rational number can be represented in decimal form in which
a) some pattern terminates at some point, with all subsequent digits being 0 (e.g., 1/8 = 0.12500000...) or
b) some finite-length pattern repeats endlessly (e.g., 1/9 = 0.11111...). I doubt that even Kronecker would have a problem with the "dots" here.
However, it is well-known that irrational numbers such as $\sqrt 2, \pi$, and others have decimal representations that never repeat and never terminate, so Kronecker's complaint that the numbers that Cantor listed weren't complete seems baseless to me. There is a reason that Cantor is so well-known more than a century after his death, while the only thing Kronecker is remembered for (at least by myself) is the Kronecker delta function.

Minor point: the last century was the 20th. We're now in the 21st Century.

So if you guys wanna not take that point because I can't find a 2023 journal article of Gobbledegook Theory Volume 80 then me and Leopold Kroeneker apparently will bid you adieu.

"Don't let the door hit you in the butt on your way out."

 

[Blargus]

the vast majority of them accept Cantor

Oh so now it's the "vast majority" not everyone who studies math and all mathematicians?

However, it is well-known that irrational numbers such as...pi , and others have decimal representations that never repeat and never terminate, so Kronecker's complaint that the numbers that Cantor listed weren't complete seems baseless to me.

Doesn't seem like you're seeing the issue clearly. My understanding is that you seem to think like the math prof in the video about the diagonal proof that "you will eventually get to every number." IMO this is a flagrantly wrong view of the infinite and if it is accepted by Cantor as it seems he does in statements it is wrong and he is misleading everyone "the vast majority."

Decimal representations that never terminate or repeat have a "complete" form as decimals? Are you saying pi in decimal form can be totally expressed by any method? Not potentially expressed to any desired length, totally expressed to the last decimal? Just NO for the last time by definition it cannot. It seems horrifying that this is apparently being argued against. It's a philosophical point that is there in calculus limits which protects it and if math has divorced itself from this point it is going the wrong way.

Minor point: the last century was the 20th. We're now in the 21st Century.

Oh 20th didn't go so well and 21st isn't either. Opinion of a fool!

"Don't let the door hit you in the butt on your way out."

Ouch but that makes it easier. Gonna have to discipline myself now to not come here anymore goodbye.

 

[Klystron]

Lest quibbling be the last word on teaching infinity, George Cantor remains a rather tragic historical figure, attacked on many sides during his lifetime not only for his mathematics.

Depending on the biographer, Cantor attended Lutheran church but based on his name (and intellect IMO) was ethnically Ashkenazi. Grist for the mill based on his time and place.

Some mathematicians and math students including myself and several of my professors hold him in high regard for his contributions to human knowledge, often in the face of adversity and intense resistance to transfinite concepts.

Coincidentally, my early Summer reading includes new novels by my previous university math professor turned SF writer, Rudy Rucker. "White Light" features Georg Cantor as a supporting and explanatory character having afternoon tea with Albert Einstein and David Hilbert, naturally on the veranda of Hilbert Hotel. Einstein relates humorous anecdotes while Cantor pours.

 

[mishima]

It's kind of ironic how this thread has detoured. Blargus is very similar to the student in my high school class, attempting to understand rigorous mathematical concepts with everyday language like the one which prompted this thread.

The student is of course forgivable, who has not even completed studies in trigonometry but nonetheless finds topics like infinity worthy of thought. And wouldn't it be educational heaven if that student had a teacher who could somehow 'skip' all the interim courses and jump right to the heart of the matter, somehow delivering a knockout explanation without all the background theorems and corollaries?

I am not such a teacher, but I think its important to remind such students that applied mathematics is a thing. These concepts, however counterintuitive and bizarre sounding, lead to the real solution of human problems in fields like engineering. What a coincidence it would be if these rigorously defined concepts turned out to be fatally flawed (as the student might argue), and yet the technology they have spawned continues to function well enough.

I would further challenge such students if they wish to counter the gobbledygook 'established' definitions they have not studied, then to prove their independent, new definition's value by producing a superior technology capable of solving novel, human problems.

 

[Blargus]

Blargus is very similar to the student in my high school class, attempting to understand rigorous mathematical concepts with everyday language like the one which prompted this thread.

The student is of course forgivable, who has not even completed studies in trigonometry

I completed calculus maybe you should play the student and think about what a limit is saying as the variable "approaches infinity." APPROACHES get it? "Potential infinity" never "actual infinity" maybe?

I am not such a teacher, but I think its important to remind such students that applied mathematics is a thing.

Apparently applied mathematics took the fork in the road and avoided Cantor's paradise and became the mathematics that made computers. But someone doesn't want you to read that.

I would further challenge such students if they wish to counter the gobbledygook 'established' definitions they have not studied, then to prove their independent, new definition's value by producing a superior technology capable of solving novel, human problems.

I'm gonna make everyone's day. Did you ever build a "Bedini SG?" Or did you watch the Myth Busters and hem and haw about how it's "proven wrong" and the fools that believe it and not analyze their attempt or see what the "free energy crazies" said about it? You're welcome I said the F.E word I await my fate.

 

[mishima]

Well, let me try putting it another way.

Math is a tool. Do I need to know the intricacies of electrodynamics to drill a hole in a piece of wood with an electric drill? If you have an alternative theory that counters electrodynamics, does it matter if you can't drill a better hole? If you can't drill a better hole, what is the value of your new, alternative theory?

The analogy being, if you think you have a better concept of infinity, ok, what are you doing with it to make your different idea matter? Can you do everything engineering currently does, plus more? That would be a great advancement, if so.

 

[Blargus]

Well, let me try putting it another way.

Math is a tool. Do I need to know the intricacies of electrodynamics to drill a hole in a piece of wood with an electric drill? If you have an alternative theory that counters electrodynamics, does it matter if you can't drill a better hole? If you can't drill a better hole, what is the value of your new, alternative theory?

The analogy being, if you think you have a better concept of infinity, ok, what are you doing with it to make your different idea matter? Can you do everything engineering currently does, plus more? That would be a great advancement, if so.

Sounds reasonable to me. Sorry I'm skeptical what did Cantor's concept of infinity saying apparently we can count an uncountable set do for math's practicality? I mean really. I can't talk about his math fine probably never will but just this concept? Wouldn't engineers would say pie in the sky, if I need a concept of infinity give me calculus's concept of infinity like limit "as it goes to infinity" because my set I'm working with is things nuts and bolts ones and zeroes and sets of things always have a finite number in it no matter how large?

I get the impression that this is the stuff Claes Johnson is saying about math education.

 

[mishima]

So, if I'm getting you right, you are questioning if engineers actually use calculus. Further, or more specifically, you are asking for a few examples of how the concept of infinity is used to solve real human problems? I'm sorry if I am mistaken, that is how I am interpreting your last text. I just want to be sure this is what I'm reading, and that I am getting you correctly.

You are skeptical that these strange concepts find application in the real world and actually lead to the creation of technology, as I have boldly claimed without evidence?

 

[Blargus]

So, if I'm getting you right, you are questioning if engineers actually use calculus. Further, or more specifically, you are asking for a few examples of how the concept of infinity is used to solve real human problems? I'm sorry if I am mistaken, that is how I am interpreting your last text. I just want to be sure this is what I'm reading, and that I am getting you correctly.

You are skeptical that these strange concepts find application in the real world and actually lead to the creation of technology, as I have boldly claimed without evidence?

No, I'm saying it seems to me philosophically Cantor wrongly redefined the concept of infinity and I believe it differs from what I have had some distant experience in with calculus which seems ok with the examples of limits. And his redefinition seems flatly wrong, denegrates the use value of it and looks like it leads math on a mad journey unmoroored from reason where "the probability of choosing a member of an infinite set is zero" or "some infinite quantities can be measured" or as the math prof in the video said "eventually we'll get to every one of the numbers in this infinite set." FWIW.

 

[mishima]

I see. So you are questioning set theory, but not calculus. You concede that the concept of infinity, at least as it is employed in calculus, is a useful tool for solving human problems.

I admit I am far from an expert in set theory, that is why I made my original post requesting clarification. The rigorous definitions are very precise and counterintuitive! Set theory was the hardest class I ever took in college (I majored in astrophysics), and I still struggle with it in self-study. However, to the extent I studied calculus, I can see how most of the major ideas of calculus can be constructed with proofs using the ideas and notation of set theory. It seems if I abandon set theory as you are requesting, I must also abandon calculus.

Can you connect your new idea of infinity to calculus in a way that is better than what set theory can do?

 

[Blargus]

I see. So you are questioning set theory, but not calculus. You concede that the concept of infinity, at least as it is employed in calculus, is a useful tool for solving human problems.

In thinking about this argument yes it seems to me that limits using infinity make sense and are probably useful. But seems to me for use value the simpler the better maybe algebra is more useful than calculus?

Can you connect your new idea of infinity to calculus in a way that is better than what set theory can do?

No. Is it a new idea of infinity or is it the old idea before so-called superior modern understandings? I'm really just standing by a philosophical point can't talk about the math will never be able to most likely. Except maybe limits. Limits don't measure infinity they measure something as it "gets to infinity" right? By their definition infinity is untouchable. But it seems set theory is saying we will measure infinity. Let's take down infinity and make it serve us. No.

The rigorous definitions are very precise and counterintuitive!

From what I read briefly online don't quote me it was a debate around set theory whether or not to abandon or lessen the value of intuition in math. I think it said Kroenicker or Poincare said no. Lessen the value of intuition in math? Hmm.

If you studied math in college I can't really discuss or understand these things in a way you or people here do. But FWIW my opinion on this stuff I would suggest that maybe you could be a "finitist sympathizer" in math if only in secret?

 

[Mark44]

Oh so now it's the "vast majority" not everyone who studies math and all mathematicians?

I'm excluding cranks such as the one you brought to the table, namely Claes Johnson. I don't include another you cited, John D. Norton, who is not a mathematician.

Doesn't seem like you're seeing the issue clearly. My understanding is that you seem to think like the math prof in the video about the diagonal proof that "you will eventually get to every number."

I think that you are the one who is not seeing the issue at all. You question the method of enumerating rational numbers but admit that you don't understand how it was explained.

"The snake" is an interesting way of ordering rational numbers I guess not that I totally get it seems like you can't order them at all as a total group.

The above confirms to me that you don't understand what you saw in the video. The scheme you describe as a "snake" does not order the rational numbers; it merely lists them.

I challenge you: please tell me one rational number that won't be enumerated in this enumeration method.

IMO this is a flagrantly wrong view of the infinite and if it is accepted by Cantor as it seems he does in statements it is wrong and he is misleading everyone "the vast majority."

It takes a very high level of self esteem to argue that a particular point is wrong when you have no understanding of the explanation you're disputing.

Seems to me that just because you can seemingly order them on a number line in between integers doesn't mean there's more, because both go on forever.

Here's another thing you didn't understand. No one is ordering anything on the number line between integers.

Decimal representations that never terminate or repeat have a "complete" form as decimals? Are you saying pi in decimal form can be totally expressed by any method? Not potentially expressed to any desired length, totally expressed to the last decimal? Just NO for the last time by definition it cannot. It seems horrifying that this is apparently being argued against.

You apparently misunderstood what I said, which was NOT that pi, $\sqrt 2$, and other irrational numbers have terminating decimal expansions.

It's a philosophical point that is there in calculus limits which protects it and if math has divorced itself from this point it is going the wrong way.

I'm not sure you understand calculus limits, either, despite your claim of having taken a calculus class way back when.

Apparently applied mathematics took the fork in the road and avoided Cantor's paradise and became the mathematics that made computers. But someone doesn't want you to read that.

Apples and oranges. As I said before, computers deal with numbers that can be represented using a relatively small number of digits. The IEEE 754 standard for floating point arithmetic that is universally used in computing touches on the concept of infinity, but only as a number that is either larger than any positive number that can be represented, or smaller than any negative number that can be represented.

I'm gonna make everyone's day. Did you ever build a "Bedini SG?" Or did you watch the Myth Busters and hem and haw about how it's "proven wrong" and the fools that believe it and not analyze their attempt or see what the "free energy crazies" said about it? You're welcome I said the F.E word I await my fate.

"Free energy" is one of several topics explicitly no allowed here at Physics Forums. See what happens if you bring it up again.

Sorry I'm skeptical what did Cantor's concept of infinity saying apparently we can count an uncountable set do for math's practicality? I mean really. I can't talk about his math fine probably never will but just this concept?

This has nothing to do with practicality, but so what? There are a lot of topics in advanced mathematics that have no connection with the practical. On the other hand, the algebra developed by George Boole was considered to have no practical application at the time. But with the advent of digital computers, suddenly Boolean Algebra had a great many applications. Another area, vector spaces, must have seemed to be impractical at the time, but have since found applications in cryptography and error-correcting algorithms for CDs and DVDs.

No, I'm saying it seems to me philosophically Cantor wrongly redefined the concept of infinity and I believe it differs from what I have had some distant experience in with calculus ...

Well, that's a problem, trying to use philosophy to understand or dispute mathematics. Logic would be a better tool to use, together with some basic understanding of what was being discussed.And unless I miss my guess, the "distant experience" was a long time ago.

And his redefinition seems flatly wrong,

Cantor did not redefine infinity; his work extended the idea of infinity to include different levels of infinitude.

denegrates the use value of it

??? - no idea of what you're trying to say.

and looks like it leads math on a mad journey unmoroored from reason where "the probability of choosing a member of an infinite set is zero"

As I said before, what you're calling a "mad journey" was explained very carefully much earlier in this thread.

"some infinite quantities can be measured"

There was no mention of measuring infinite quantities. The discussion was about enumerating and comparing different infinite sets.

as the math prof in the video said "eventually we'll get to every one of the numbers in this infinite set."

Again, I challenge you to tell me a single number that he won't get to. In my reply I'll tell you where that number is and how many steps it takes to get to it.

Since this thread has degenerated into much nonsense by a new member, I'm closing it.

@Blargus, if you want to reply to my challenge, you can do so my sending me a PM.