How can Infinity exist?

[D_Winds]

For something to exist, does it not that mean it has to be obtainable to some degree? In other words, if something exists, then it means we can get it on some way or form, but if infinity cannot technically be reached, then how does it exist? This question has been bugging me for some time now, and now i feel that i can disprove the existence of infinity, SO PLEASE PROVE ME WRONG.

This discussion may end up leading to a non-mathematical concept.........

 

[stapel]

Please note that FreeMathHelp is a free tutoring service, not a discussion site. The volunteers help students work through specific homework exercises; we do not do research or write papers, nor does the site offer peer review. I apologize for any confusion.

For technical discussions of the concepts of infinity, etc, you might want to try a "discussions", "philosophy of math", "foundations", or other related site.

....PLEASE PROVE ME WRONG.

Once you find an appropriate forum for your debate, please note that you will need to provide your claimed proof, as nobody can "prove you wrong" when you haven't yet done anything but make a statement of your feelings. A good place to start will be with your definitions of "infinity", "existance", and "reaching" a thing.

My best wishes to you.

Eliz.

 

[tkhunny]

I like this question and can provide a strongly mathematical reply. You can bring it up in your Philospohy class and maybe scare someone.

What is your definition of "infinity"?

Often, when "infinity" is used, it simply is code for a more general idea that something increases without bound or cannot be reached.

If one "blasts something to infinity", we mean it is really, most sincerely gone, we do not mean it actually went anywhere.

If someone lives to the age of "infinity", we do not mean they have been around for a very, very, very long time. We mean only that the present age nearly is incomprehensible.
If we go to summer camp and make "an infinity of those" boondogles, again, it is not such an incomprehensible thing, it is only a number so large that we are troubled by it.

The concept of "infinite" is also subject to cretain finite limitations. The most useful definition I use is simply "big enough" or more simply, just "enough".

For example, when speaking of human life span, what is "infinity" in light of this vague definition? 150? 200? Maybe 1000 if you use Biblical accounts. Why does infinity have to be any greater than 1000 when refering to human life span? Use 10,000 if you like. It's just as unlikely. I am nudging into the mathematical concept of the Supremum. Perhaps you would like to read up on that a bit.

 

[daon]

Let me answer your question with more questions: Does a line exist? Does a circle exist? What about the number "0"... and dare I ask it... a negative number?! *gasp*. If "infinity" does not exist, neither does "6" or "-3". All convey an idea. Its all made up! You could imagine the confusion of a student first hearing of these "imaginary numbers" which are really all in the same as "regular numbers", just having a different application.

Just as the concept of debt uses these things called "negative numbers," theres also a use for infinity: Clearly there is "no single biggest number" that can represent the number of points on a line segment. Thats one way to interpret "infinity," and there are many others.

Calling \(\displaystyle \infty\) a number (in the extended real numbers) makes math and in turn, science, easier, so we use it. Thats how all math is. Someone invents things and says "take these to be true and to exist without question." From there we say, "what else is true if this is true." Sometimes what this "playing around" spits out tends to change the world.

edit: May want to move this to "Odds and Ends."

 

[pka]

The question of ‘infinity’ for philosophy it is simple: “It is a place where mathematicians hide their ignorance”. Just look at Zeno’s paradox.
But Cantor returned us to the mathematician’s Garden of Eden.
The idea of infinite countability and even uncountability is part of the same question about non-finite sets.
Yes, this is an unsettled question. But its answer is far more than most can understand.

 

[cadamcross]

Helping a student get his head around infinity is math help just as much as factoring polynomials.

Zeno's paradox hasn't been a paradox since Newton's time, and I disagree with the idea that infinity is a contrivance used to hide the ignorance of mathematicians.

The others have pointed out that a notion of infinity arises in many contexts. I often have to warn my calculus students against naïvely plugging infinity into functions to calculate limits. They want to treat it like a number, but infinity is not a number. Infinity describes a process. It is not a positive description. It is a negative description. It means nothing more or less than "not finite". It exists in the sense that it describes certain other ideas that have been useful to us. For example, there are certain computer programs that will not halt in finite time. This is easily seen. Write a program to find the largest even number or largest prime number. The program will never halt and say, "I found it." Infinity captures this idea -- that the program will not halt in finite time.

Infinity is used in the same sense when people say that the limit of a function f(x) as x approaches c is infinity. This only means that the function increases without bound near c. Again, the function never attains infinity.

People do sometimes work in number systems containing an object called infinity (the extended reals, say). However, infinity always behaves in a fashion that is dissimilar to the numbers. For example, infinity - infinity is indeterminate. It is not a number like three or like negative three were like the imaginary square root of negative one. All of those are contained in fields.

To any who read this, if you do want to learn more about the concept of infinity I recommend studying set theory, the branch of mathematics created by Cantor. If you want to learn more about numbers, find a book on abstract algebra -- the study of groups, rings, and fields.

 

[stapel]

Helping a student get his head around infinity is math help just as much as factoring polynomials.

Yes; helping a student who wants to learn is indeed "math help". On the other hand, being ordered to do a peer-review (of a proof in a paper which has not been provided) is not. :wink:

It's one thing to help a student who expresses a genuine request for help with learning a new and complex concept; that "math help" process can be quite a lot of fun; and the student's growth, zeal, and ambition can be a joy to witness.

It's quite another thing to be summarily directed to engage, yet again, with someone who appears only to wish to be combative, someone who is demanding that we re-prove modern mathematics (proof theory, foundations, definitions, etc) to his satisfaction. :shock:

It would have been so much nicer had the original poster exhibited interest in the former.

Eliz.

 

[galactus]

This is a fun topic to explore. You can google Cantor, transfinite numbers, continuum hypothesis, and many other terms and lots of stuff will pop up.

Try looking up Hilbert's "Paradox of the Grand Hotel". That was always one of my favorites.

 

[cadamcross]

The will and strength to question authority combined with the intelligence to do so in a way that makes sense -- this is part of what makes a good mathematician. When I was a youngling I did not accept the idea that one cannot divide by zero. I was certain division by zero must yield infinity. I had not yet met the ideas of limits and multiplicative inverses. My point is only that anyone with the brains to question authority should be encouraged. Being able to do so politely and eloquently comes with age and experience.

I also think infinity is a fascinating topic. The idea that there could be different sizes of infinity as described by Cantor is on the main reasons why I decided to study math in college. It's poetry.

 

[tkhunny]

There is a grave and important difference between an honest questioner and a combative crackpot.

An honest questioner may ask anything desired and expect a reasonable reply, even from an "authority". These honest questioners can be, and often are, very hopeful students of mathematics.

It is a very rare combative crackpot that can be brought to a useful understanding of anything. Every once in a very long while, one of these combative crackpots can be salvaged.

One must decide where to spend one's time, effort, bandwidth, etc...

 

[cadamcross]

There is a grave and important difference between an honest questioner and a combative crackpot.

An honest questioner may ask anything desired and expect a reasonable reply, even from an "authority". These honest questioners can be, and often are, very hopeful students of mathematics.

It is a very rare combative crackpot that can be brought to a useful understanding of anything. Every once in a very long while, one of these combative crackpots can be salvaged.

One must decide where to spend one's time, effort, bandwidth, etc...

Is a person a crackpot merely because he is combative? Is he a crackpot because he questions received wisdom? If that's the case, it seems to me that most of human progress occurs by the efforts of combative crackpots. If on the other hand you define a crackpot to be a person whose ideas are useless because they are so strange, then I don't think you can tell who the crackpots are until a hundred years after they are dead. Mainly, I think mathematicians use the word crackpot to describe people who think they can trisect the general angle with a straight edge and compass (and the like). As for me, if a crackpot comes to me claiming he can trisect a general angle, I will point him in the direction of field extensions and possibly to someone who knows more about them than I do. If he argues with me even thought he doesn't know what he's talking about, then I will likely walk away.

 

[stapel]

There is a grave and important difference between an honest questioner and a combative crackpot.

Is a person a crackpot merely because he is combative?

If a complete stranger stalks up to you, blocks your path, and starts screaming at you, demanding that you justify your existance, and requiring that you disprove his (unstated) claims and beliefs, whilst never leaving the imposition of his faith, he is not "merely questioning received wisdom". His hostility is not meant to be productive, and is entirely outside of his rights: he's demanding (unearned) respect (respect that he will not reciprocate), and he won't regard you to have "open-mindedly" "discussed" the "issues" until you relent and accept his definitions, his logic, and, basically, his "religion". You are not allowed to "bring in" dictionaries, history books, or logic that he does not accept; you must submit to his attacks and eventually to his claims. Because, at heart, the "issue" isn't the point. He's only interested in his authority. :shock:

When the original poster claimed to have proved that infinity doesn't exist (perhaps because it isn't a counting number? or because it isn't a location? or because it isn't a physical object?), and demanded that we submit to his (unstated) non-mathematical proof or else, within his (unstated) non-mathematical rules, prove it wrong, his demands and context seemed fairly familiar. :roll:

On the other hand, someone with an honest question is an honest questioner, and his approach and tenor are quite different. Somebody who has carefully considered an issue and is respectfully requesting to discuss his thoughtful questions tends to approach one in a manner quite unlike the above, since the point is the discussion and growth, not him being "better" or "smarter" than you. Why anybody would expect such a person to express this intellectual interest in attacks is unclear to me...? But years of experience (decades, for some of the folks around here) has led many to understand that your wished-for situation is, well, fairly mythical.

On a mathematical note, "infinity" in a set of concepts, methods, and definitions. It isn't a cultural encrustation or a mere folkway, whose baleful impact on society "should" be "questioned", whose "age-old" "authority" is putting a brake on human progress. So I'm not sure how a political or sociological debate would relate...?

In the case of mathematical terms, either you accept logic and the current definitions, or else you accept logic but reconstruct things (proving their validity and usefulness as you proceed) from your own new definitions (which you provide in all completeness) -- or else you don't accept logic. I'm sorry, but it isn't "repressive" or "a block to human progress" or an attack on one's civil rights not to wish to waste one's limited time and resources in an attempt at a logical discussion with somebody who doesn't accept logic.

My apologies for any offense my "political correctness" (and age) may cause.

Eliz.

 

[tkhunny]

Is a person a crackpot merely because he is combative?

No.

Is he a crackpot because he questions received wisdom?

No.

If on the other hand you define a crackpot to be a person whose ideas are useless because they are so strange, then I don't think you can tell who the crackpots are until a hundred years after they are dead.

"Strange" is not a good definition.

I think mathematicians use the word crackpot to describe people who think they can trisect the general angle...then I will likely walk away.

Good. That is one way to identify such. There are others. For example, any presentation that includes power, glory, and maybe a God Complex, doesn't really need much discussion to discern, no matter what the claims. In the case of this original poster, there was a request, pointed out by Elizabeth, that appears to be entirely disingenuous. Several thoughts have been presented and the responses from the original poster are none. This is very common behavior of those who really do not want to be proven wrong. Generally, such will move on and try to get a rise out of their next potential victim. Seeking for glory seems to be the goal. Not finding it in one location, one must go elsewhere. The vague attempt at mathematics is merely the disguise of the glory seeker.

There are also predators. Really. There are those who hate online help and will do what they can to prevent it. They will post whatever they can in an effort to weary those who would attempt to be helpful in this medium. Learning to recognize such constitutes yet another important skill for surviving the World in which we live.

My views. I welcome others'.