B Help answer student question about infinity?

[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.