Understanding Infinity

[Sisyphus]

I once met Neil deGrasse Tyson (I wanted to get a picture with him), and I overheard him having a conversation with another guy about multiple universes, infinity, and so on. I only heard the end of the conversation, but I remember Neil mentioning how the human brain isn't really efficient at comprehending infinities. I don't know why, but I kept thinking about that afterward. I didn't disagree with him, but I guess it just made me think about infinity more than I have in the past.

Then I remembered reading about our universe, and how it's possible that it's been around forever in one form or another. It seemed weird at first, because it's difficult to imagine no beginning and no end to something. It seems as though there should be some point in the past where everything originated.

I thought of time as a number line, with zero in the center. Zero being the origin, with infinity stretching out in front of and behind it. So, in this way, it made me think that any point in time can act as zero on the number line, making any point in time the origin depending on the perspective of the time frame. I don't know how accurate this is in viewing our universe, but it helped me look at infinity in a different way than I had before. It also made me wonder that if the universe has no center, and if time is just a dimension of space (space-time), then time wouldn't exactly have an origin either. That's probably way off though, right?

[Natural ChemE]

I've just spent half of the day crunching modeling equations. I've been through countless* infinities so far.

When you do Calculus, you take infinitely small lengths, areas, volumes, etc. to add up. And you add 'em up infinite times. Afterwards, when you've solved the differential equations, you have all of these arbitrary constants that you have to solve for - often by using boundary conditions as something goes to infinity.

About five times so far today, I've split up a finite expression into an infinite series. Just easier to work with those sometimes, particularly when you can make truncating assumptions.

And, hell man, thanks to the Dirac delta functions, I've even stuffed an infinitely high function into an infinitely narrow location (such that the sum under the boundary was unity, of course).

Point being, I can't empathize with infinity being a difficult concept. If anything, it's the simplifying tool that makes our lives so much easier.

*Because it's just plain easier to say "countless". Gotta love infinity approximations!

[Sisyphus]

Well, if more people were experienced with advanced mathematics, then concepts like infinity probably wouldn't be so difficult for most people to grasp, at least in certain contexts.

[realcomfy]

While I too use mathematical expressions, constructs and the like involving infinity in my day to day work, I find the conceptual idea of infinity a little more abstract than just some convenient tool that I use to do calculations. Sometimes we get so caught up with the formalism of the mathematics that we forget to pause and think about the underlying meaning of the expressions we so cavalierly throw around. What is a delta function? It is a physically ridiculous idea. However, it does a really nice job of explaining things on a scale where we can make some limiting approximations (for example, delta functions can be used to model the charge of an ion on a macroscopic scale because when your thinking about a bulk material, the spatial extent of the charged particle is so small it makes no practical difference). That doesn't mean that there is an implicit understanding of the concept of infinity. It doesn't mean that the full scope of what infinity represents has been condensed into a nice, packaged representation that succinctly demonstrates all that there is to know about it. All it means is that there is a technique that a physicist came up with to nicely describe some physical phenomena, and that technique happens to use the idea of infinity.

I think the idea behind infinity is quite a bit more complex than simply throwing out some mathematical terms and explaining how they can be used. I'm not saying that I can illuminate your minds with some especially clear explanation of what infinity means, what it "actually" represents, or its true manifestation in the physical world, but I am encouraging you to think a little more about the concept. Maybe I'm way off base here, maybe infinity isn't described beyond its mathematical usage, but I don't think so. So what do you think about when you really consider what "infinity is".

I especially liked the thought of a number line with an infinite number of digits flowing out in either direction. Imagine that the zero represented the now and that everything negative represented the past and everything positive represented the future. There is no beginning and no end. There are an infinite number of now's on the number line, each one of them just as meaningful as the last, and without which there could not be a next. Conceivably, some action at zero could have an effect that propagates out to positive infinity (just as some action at negative infinity could have had an effect that propagated out till now, something like the butterfly effect). I'm not saying determinism, I'm saying that without the big bang, we wouldn't exist.

When you consider that some action performed in THIS moment could somehow affect some future moment, an infinite distance away (using the number line analogy), and that your conscious mind will likely never have any knowledge of the outcome, then it does seem like there is some conceptual idea that the human mind cannot easily grasp. We can't imagine what the universe will be like in 14.5 billion years from now, and even less so in the infinite future (if it could still be called a universe). I think there is a lot to the idea of infinity, and one shouldn't presume to have it all wrapped up because they know math.

[xcthulhu]

I've just spent half of the day crunching modeling equations. I've been through countless* infinities so far.

When you do Calculus, you take infinitely small lengths, areas, volumes, etc. to add up. And you add 'em up infinite times. Afterwards, when you've solved the differential equations, you have all of these arbitrary constants that you have to solve for - often by using boundary conditions as something goes to infinity.

Well... while a lot of calculus courses aimed at the physical sciences are taught like this. Is it really that intuitive?

Let be an infinitesimal number such that but for all integers we have that . Now, is apparently an infinitary number, since it's greater than every integer. Is ? Does adding one to an infinite number make a difference?

You probably know that the integers are part of the real numbers. If we allow for infinitesimals and infinitary numbers, is there an infinite number such that every positive integer evenly divides that number? If so, is there a least such infinitary number? Is every infinite number divisible by some integer less it, or are there things like infinitary primes?

The above touches on a rich and complex subject - for those interested in a more in depth response I have posted one here: posting.php?mode=post&f=19]

However, I will just point out that in thinking about this, I agree with the original post. If you ask the right questions, infinity reveals itself to be one of the most complicated concepts in mathematics...

[Mirthe]

ty first of all for this thread :-)

i am no scientist, no mathematician, i can but guess what novelties you learn each day :-)

i am perhaps more a wonderer, i wonder often, as in reading what you wrote, and you trying to calculate the infinity, or should i say infinitively*gg

and i would like to know how you see the coherence with our past, i mean if you look in literature, art, architecture, all that man has done up until now.. there is an ever reoccurring theme in those, may it be love, pride, an everlasting yearn for gain, from the oldest novels we can read, until now, for us, as people that is a long time.. longer than anyone who lived then, could foresee. Sadly or perhaps astoundingly.. those things havent changed much, it seems to go on indefinitely, as in we are humans..therefore.. *g

Is it then truely possible that you come up with a brand new idea about infinity , or is it a new understanding about how it works.. both equally interesting:-)

I dont know, but would like to :-)

kind regards,
Mirthe

[realcomfy]

Hi Mirthe,

Welcome to the forums.

We we talk about infinity in the contexts that ChemE mentioned (calculus, infinite series and delta functions) we are using the infinity as an approximation. Some functions such as exponents or sines and cosines can sometimes be difficult and clusmy to work with in the equations we are trying to solve. However, clever mathematicians of the past have found ways to approximate such functions using infinities, and these approximations are often much easier to work with.

I don't think anyone here was trying to redefine infinity or provide some new interpretation of what exactly it means. But keep asking questions and perhaps someone can enlighten you about what it is we DO know about infinity.

[seeker]

Sisyphus,

I'm with you: infinity makes no sense to me. We can imagine it mathematically but we can't really imagine it emotionally.

For example: Imagine a universe that has no beginning and no end. We're talking infinite time.

Now imagine a universe that began at some point and may end at another point. What was there before reality began? Nothing? And afterward?

In fact, the very idea of existence is mind boggling. Why should anything exist? And how could existence begin? Or end?

To me, the idea that time had a beginning is, well, mind-boggling. And the idea that time is infinite is equally mind-boggling. Frankly, either is the stuff of nightmares. It suggest that humans are simply not equipped to deal with reality.

[linford86]

Seeker -- I haven't the faintest idea what you mean when talk about imagining something emotionally, especially mathematical concepts like infinity. For me such things are emotionally neutral, unless you're talking about the emotional reaction that I have to doing mathematics. I like doing mathematics. Others dislike doing mathematics (or claim that they do; most people I've met who say they dislike mathematics don't really know much about it and probably haven't ever seen something really interesting in mathematics.) But for some reason I don't think this is what you mean. I think you mean something else, but I don't know what that would be.

[realcomfy]

Seeker -- I haven't the faintest idea what you mean when talk about imagining something emotionally, especially mathematical concepts like infinity. For me such things are emotionally neutral, unless you're talking about the emotional reaction that I have to doing mathematics. I like doing mathematics. Others dislike doing mathematics (or claim that they do; most people I've met who say they dislike mathematics don't really know much about it and probably haven't ever seen something really interesting in mathematics.) But for some reason I don't think this is what you mean. I think you mean something else, but I don't know what that would be.

But then again people can surmise from a very short interaction with a concept, be it mathematics or swimming or talking to rednecks, that they don't enjoy it and it is likely that prolonged contact will only lead to further consternation. So its likely that a number of people that don't like mathematics have a reason for doing so. I don't think you can say people that don't like math do so just because they don't know how to do it. But that just seems like philosophy.

[linford86]

I can say it because there was a time when I disliked mathematics. I hated arithmetic, long division, etc. It was only when I saw a little bit deeper, and a little bit past my secondary school teacher's knowledge that I really saw what mathematics was all about. I can also say it because I've taught introductory physics (which is supposedly math heavy) and I know when my students start to hate the subject and the visceral reactions they have.

You are right that people can often surmise whether or not they enjoy a subject from a short interaction with it. It has been my experience, however, that most people never see or understand real mathematics even for a short time; they instead only see the brutal hack job that somehow passes for mathematics in elementary and high school education. Since they are never actually exposed to mathematics, even for a short time, they simply don't have the sort of short interaction that you're talking about.

[realcomfy]

That's a good point, the environment in which a person is exposed to mathematics has a lot to do how they feel about it. For instance I had a tutor during highschool that was both competent and interesting, so I was drawn to the subject and was able to do well in it, while a friend of mine had a gradeschool teacher that taught math with a laser pointer from the back of the room and he has been completely turned off to the subject since then. I have to agree with you that a deep understanding of the subject often provokes a deep respect and almost a kind of reverence in extreme cases, but that is true of many different occupations.

Say you don't like woodworking. You took shop class in highschool and just couldn't see the point of drilling all these silly holes and making sure the measurements were just right. The shop teacher just didn't explain anything well and failed to inspire and motivation to do the work. It makes perfect sense that I wouldn't pursue a career or even a hobby attitude toward woodworking, but maybe that's just because I never had the real satisfaction of seeing a beautifully crafted bookend. Then there are people who had good shopteachers and satisfy themselves with a life of crafting bookcases and coffee tables and have no idea why everyone doesn't own a lathe and a drill press.

I wouldn't expect a wood worker to be able to talk about the intricacies of infinity in an especially refined and "deep" sort of way, although the woods do breed philosophers, but when you are beginning to think about it and getting used to the different definitions and ideas out there, it makes sense to talk about the different situations it is applied to (cosmology, time, math, etc.) and confusion is an emotional reaction that makes sense.

[Sisyphus]

I also had a bad experience with math growing up, and for a long time I believed I couldn't do it. Recently, I've found that isn't the case. I've been taking college math courses and have been getting A's. Although, even though I'm doing well, I feel like I don't fully understand the math I'm learning. I can remember how to do math problems through repetition, but there's a lot of stuff going on behind the scenes that I'm not aware of.

[Lincoln]

That's not so unusual. I've always been awesome at math, but I didn't really understand integrals until a junior level physics class. Sometimes you just have to turn the crank a lot and then the underlying stuff sneaks up on you.

[edy420]

I used to think I had infinity within my imaginations grasp, mostly because of that figure eight sign that lies sideways.

But then I asked myself what is infinity subtracted by 1?
My head just exploded and now my tiny imagination fears the very thought of infinity.

If any shape or symbol is considered a representative for infinity it shouldn't be two dimensional because even 3 a dimensional shape is misleading when thinking about infinity.(an infinite dimension symbol maybe?)

[Louis_B]

Infinity is great stuff isn't it? Its not so bad if you dont think about it in terms of math. Imagine our universe, say for the sake of arguement it is expanding in all directions at light speed. Imagine you are in another universe watching it grow..Like throwing a rock in an ocean on a planet made entirely of water..It doesn't matter where the centre is, there's probably nothing there anyway. Once the singularity event occured, the rest is history!! And for all you math guys, math isnt going to describe infinity, its not a math problem, its an existential one!!

[Giacomo]

I once met Neil deGrasse Tyson ...

Then, read this article on NOVA: A Mathematical Perspective

I remember you telling this community you started to learn Calculus. Very cool!

You have noticed that in Calculus there are 2 parts: differential calculus and integral calculus.

And that limits give us a firm basis on which to examine the infinite and the the infinitesmial.

Have you started to study these gems: the fundamental theorem of calculus, the chain rule, L'Hopital's rule, Taylor series?

Calculus is very useful tool to get a sense of infinity. But that's not enough!

Read an introduction to Set Theory. For example, this

I also recommend this book : A Brief History of Infinity: The Quest to Think the Unthinkable


The book explains another paradox: a shape that has a finite volume, but infinite surface area.

[Sisyphus]

Thanks for the links. Actually, Calculus is this semester. I just finished Trigonometry last week. I did read a book on the origin of Calculus, though. It was pretty interesting. Hopefully, I can do as well in Calculus as I did in my other math classes.

[Lincoln]

Very important bit of information. The first 2-3 weeks of calculus is conceptual, important, hard to understand, and you will likely never use it again (unless you're a math major.) If the whole limits thing makes your eyes spin, that's OK. Once you get to differentials, it's all about execution and not theory. I wasn't ready for the first few weeks when I first encountered calculus in high school. But after that, it was pretty trivial.

Just FYI.

[Natural ChemE]

I'd like to stress Lincoln's point that Calculus is very important.

Some of Calculus is just tedious bull. For example, when you get to your second semester of Calculus, you'll learn things like how to take the integral of sin⁷ * cos³. You'd probably be just fine just learning enough to pass the test on those parts.

But the basic concepts - "what's a derivative?", "what's an integral?", "how do they work?", "how do we use them?" - are vital to understanding modern analytical thought. These are the concepts that you want to know like the back of your hand. And if you really want to get an answer to the more tedious questions that they'll put you through in the second semester of Calculus, you can just use a computer program like Mathematica or even just Wolfram Integrator (which you should probably bookmark now).

And while Calculus II is a bit of a waste of time, Calculus III (or "Multivariate Calculus") gets back to the useful conceptual stuff again. And after Calc 3, it’s off to Differential Equations or/and Linear Algebra land with you! (Unless you do graduate work in science, or you're a Math or Physics major, you'll probably never go passed basic Diffy Q or/and Linear.)

[Sisyphus]

So, is there a lot of Trigonometry in Calculus? The Trig class I just took was a compressed summer course and online. I regret not taking it as a normal semester class now. I got a B, but I don't feel like I have such a firm grasp on it as I do with Algebra.

[Natural ChemE]

Basic Calculus is very simple. You can mix Trig with Calc (and frequently do in application), but Calc doesn't require Trig.

You don't need to know Trig to do Calc anymore than you need the quadratic equation to solve for x in sin(pi)=x.

Most of the base elements of Calculus are concerned with either one of two questions:
-What is the derivative of a function?
-What is a function the derivative of?

[Lincoln]

The first semester of Calculus has many facets. In my experience, the first couple of weeks is the limit concept. This appears on one test, with one problem on the final. If you get it, excellent. If you don't, you can still do great in calculus. After that, there is the fundamental concept of a derivative. This is actually quite important, but my experience tells me that most students don't actually "get it" for a long time. The final thing is the mechanics. This is most of the class. Given a function, find the derivative. The rules for taking the derivative of a polynomial are x. For an exponential, the rules are y. Trig functions, the rules are z. For composite functions, you do this thing. For functions of functions, you do the chain rule. Etc. A lot of students are good at this, but don't have any clear idea as to the physical (or even mathematical) significance of what they are doing.

Getting back to the trig thing...it doesn't appear much in calculus. This is especially true for the trigonometric identities. The concept of a tangent is implicit in calculus. This is equivalent to the "rise over run" of slope from high school algebra.

Unless you're planning to be an engineer...mechanical or electrical, or a physicist, I wouldn't worry too much.