Will's Blog: Permission or Forgiveness?

[Proxima]

I'm not really following this argument (that, of course, doesn't necessarily mean it's wrong). To give an idea of what my current understanding is: If I had to compare the area of the levels, my instinct would be to say that the infinity*infinity level is larger than the infinity*160 level (I'm intentionally avoiding terms like "cardinality" here, since I'm not quite sure I understand how it works with infinite sets). My reasoning would be that, suppose we compare an infinity*160 level with an infinity*320 level. In the x-direction, each level would be infinite. Pixels are countable; if we are at pixel n, there exists a pixel n+1 and there aren't any pixels in between (real numbers, by comparison, are uncountable; while you can compare which of two real numbers is larger, there is no "next" real number, since if you tried to define one, you'd always be able to insert another one in between). So there should be a one-to-one correspondence between pixels in each level in the x-direction. But the second level has twice as many pixels in the y direction for each pixel in the x-direction, thus the second has a greater area. An infinity*infinity level would have an infinitely greater area than either.

Yes, this is very tempting but wrong.

Forget about area for now and concentrate on numbers, because that's simpler.

Suppose you have a list of items and you want to know how many items are on the list. You can write "1" against the first item, "2" against the second, and so forth. If you eventually stop -- for example at "8" -- and that is the last item, then you know there were eight items on the list.

A variant of the same process works for infinite sets. With infinite sets, of course, there is no "last item" and the process will never stop. But if I can list the items of some set in a patterned way, so that I can see that I can label the first item 1, the second item 2, and so on, and match every single item with a unique positive integer, then the set must be the same size as the set of positive integers.

Because we can pair the positive integers with the even positive integers (1 -> 2, 2 -> 4, 3 -> 6 and so on), the set of positive integers and the set of even positive integers are the same size. You might think the set of positive integers is larger, since it includes all the even positive integers and also the odd positive integers, but they are the same size.

Another way to put this is that an infinite set can be the same size as a subset of itself. In fact, that is how mathematicians prefer to define "infinite".

Now, how about the set of all integers (positive and negative)? Is this larger than the set of positive integers, or the same size?

We can try to answer this by listing the integers: 0, 1, 2, 3, 4, 5... But we'll never get to the end of the positive integers and get started on the negative ones, so this list doesn't include all integers. Instead, we should list them in an order like 0, 1, -1, 2, -2, 3, -3... Now all integers are on the list, so we can see the set of all integers is indeed the same size as the set of positive integers. In other words, when enumerating the items in an infinite set, the order can matter.

Similarly, your 160 x infinity and 320 x infinity levels are the same size. You can't prove this by looking along the x-direction first, because you'll never get to the end of the first row and get started on the second. Instead, go down the first column, then down the second column, then the third, and so on. Each column is finite, so in both cases, this order of counting the pixels will take in all of them, proving that in both cases, the set of pixels in the level is the same size as the set of positive integers. And two sets that are both the same size as N must be the same size as each other.

What about the infinity x infinity level? In this case, we can't go down the columns either, but we can start from a corner and enumerate the pixels in a zigzag pattern (see my post on the previous page). Therefore this level, too, is the same size. (This supposes that the level has a corner and continues to infinity only rightward and downward. What if it's an infinite plane with no edges at all? Well then, we can start from an arbitrary pixel and enumerate the pixels in an outward spiral.)

* * *

You mention the fact that the real numbers are dense (between any two distinct reals, there is another real). This is true, but oddly enough it's a red herring. The rational numbers (numbers that can be expressed as fractions) are also dense: between any two distinct rationals, a and b, there is the number (a + b)/2, which is also rational. You would think that there must be more rationals than integers...

Amazingly enough, this is wrong, the rationals are countable. This is quite easy to understand if you think again about the infinity x infinity level. Suppose we label one axis "numerator" and the other "denominator"; then any rational can be paired with a unique pixel on the level, and we already showed how to count those.

Or we could just enumerate the rationals directly: 1/1, 1/2, 2/1, 1/3, 2/2, 3/1, 1/4, 2/3, 3/2, 4/1.... In other words, I start by listing the only rational with numerator + denominator equal to 2, then those with numerator + denominator equal to 3, and so on. This list will hit every positive rational sooner or later, and it's easy to tweak this method to cover the negative rationals as well if you like.

* * *

As for real numbers between 0 and 1, compared with real numbers between 0 and 2, both sets are uncountable, so we can't compare their sizes by enumerating them. However, we can still put them in correspondence with each other, showing that both sets have the same size as each other (and both are larger than the set of positive integers). To do this, pair every real number x between 0 and 1 with the number 2x between 0 and 2. Again, this illustrates that an infinite set can be put into correspondence with a subset of itself.

[ccexplore]

Cantor's cardinality measure is a specific way to extend the intuitive concept of "size of a set" from finite sets to infinite sets, where human intuition is far less reliable and certainly lacks rigor.  It specifically avoids the concept of "counting" (which is obviously problematic for infinite sets) in favor of a simpler notion of bijection, aka 1-to-1 mapping.  Two sets, be it finite or infinite, have the same cardinality if there exists a 1-to-1 mapping between elements of one set to those of the other.  For finite sets, Cantor's cardinality is equivalent to the intuitive notion of the size of the set as measured by number of elements.

Cantor's cardinality measure does have the property that for infinite sets, it is indeed possible for a proper subset of the set to have same cardinality as the entire set itself, something that's not possible with finite sets.  At the same time, Cantor also proved that not all infinite sets will have the same cardinality--the set of real numbers is for example provably not possible to create a 1-to-1 mapping with the set of natural numbers; any attempts to do so will always result in a real number that is not included in the mapping (eg. via the diagonal argument).

Cantor's cardinality measure works well because it assumes very little about the set.  Your example with level areas for example is in effect imposing a two-dimensional arrangement of the elements of your set of pixels, and similarly puts restrictions on how you perform the mapping of pixels between the two arrangements.  I'm not saying you can't necessarily come up with a mathematically consistent measure in your case, but even if you can, it will likely be more restricted in applicability compared to Cantor's cardinality.

[ccexplore]

You've said this quite a lot in your post. I realise it was something of a rant, and clearly something you're passionate about, and ultimately it's good to have your engagement. However, since you feel quite happy to reiterate this statement again and again, I'll take issue with it: if you have failed to recognise the meaning in something that someone has taken the time to formulate and share, that's on you.

Maybe "meaningless" is too strong a word, but I think the main point is that you are trying to refer to various concepts and terminologies that you have little understanding of.  It is a little hard to take your initial post seriously when it demonstrates a great deal of confusion and misunderstanding about the mathematical things you try to refer to, and at the same time phrased in such ways as to seem to assert decades of work by world-class mathematicians are complete garbage.

And I'm sorry to say, just because someone happened to spend time to formulate or share something doesn't mean the thing that was formulated or shared necessarily had to make much or indeed any sense.  Example.  Of course I'm not saying what you're doing here is anything like that, but it's just as much on you as it is on the readers to help come to a mutual understanding.  And poor uses of mathematical concepts and terminologies is unsurprisingly likely to become a roadblock to such mutual understanding.

The question here would be: if mathematics is a system of understanding and interacting with reality in the abstract, as useful and accurate as it tends to be, is it not still dependent on its creators (i.e. the human race, specifically mathematicians) for its development, definition and maintenance?; and, that being the case, is it not therefore limited by current existing limits of our understanding?

Doesn't that go for any human discipline?  "Understanding" for humans is fundamentally a mental process.  It will always be limited by our brains.

If so: then, surely, further investigation and development is needed - and this may involve consideration of radical or seemingly nonsensical ideas (although not necessarily) in order to expand.

If not: then how can we prove it?

As I recall, the historical development of set theory and its treatment of concepts like infinity was specifically in service of mathematics.  At the time, mathematics like calculus already had concepts that touched on infinity, but the treatment of those things tended to be very hand-wavy.  The development of set theory came about partly to help addressed those parts of mathematics that lack rigor.

Mathematics is based on proving other statements to be true starting from a small set of statements accepted as "ground truth", aka axioms.  One unfortunate property of logic is that if you ever end up with a statement that can be demonstrated as both true and false, then the entire system falls apart because you can take that true-and-false statement, and applying the standard operations of logic (deduction etc.) will actually allow you to then prove any other statement to be both true and false.  And thus the system becomes useless.  So mathematicians need to be much more conservative--the number of axioms should be kept low to help avoid tripping yourself over a self-contradiction, and then try to prove everything else as either true or false as followed from the small set of axioms plus the application of standard logic like deduction etc.

That's not to say you can't end up with results that seem "radical" or "nonsensical".  Mathematics explore a lot of things that are so far beyond our everyday experiences that it is expected to be counterintuitive from time to time as a result.  Here's an example.

It's good to recognize that because understanding is limited by our brains, it is quite easy to arrive at counterintuitive results especially when exploring areas of study that are so far removed from everyday experiences, but just because our intuition may be ill-adapted at processing the results doesn't mean it's false or incomplete.  After all, a dog tends to chase its own tail because perhaps it doesn't understand that tail is his, but that obviously don't mean the tail is in fact not the dog's; it just means his brain is not equipped to properly process the concept of self.

I'm saying that I am investigating the concept of infinity from a mathematical point of view and finding only questions, rather than answers. I've updated the OP so it's worded more carefully and in a more open, question-asking tone so as not to cause further irritation or misunderstanding.

Again, I'm not suggesting that maths isn't the best tool, I'm simply asking the question as to whether or not it is. Perhaps I worded the question as a somewhat ignorant-sounding statement. My bad, I apologise if that's how it came across.

Since you ask, I would say: I don't know what the best tool is, that's why I'm investigating the subject.

<snip>

Perhaps this is true. But my point is more that I do not believe that mathematics is the best way to interact with it, whatever it is, whether it's a magical, mystical force of reality or a boring, easy-to-understand force of reality.

This feels like a very personal question.  Understanding by whom and for what purposes?  Everyone's brains work differently and I don't really know what works for you and to what degree.  At the same time, set theory and its treatment of certain concepts like infinity is simply what mathematicians have found useful over the years as applied to their every day work on mathematics.

I'm not even sure I understand what it means to say to "interact" with "infinity".  Isn't it just an abstract concept?  What does it mean for you to "interact" with "fairness" for example?  In our everyday physical world we do not actually directly experience anything infinite--the examples you come up with invariably relies on infinite time or possibly infinite space, neither of which any human can directly experience.  Though perhaps they are easier or less confusing to mentally process, but probably only if you avoid thinking too deeply about it.

I'm also not sure why you want to be picking on mathematics on this.  There is a much richer, longer history and literature from philosophy on the exploration and discussion of infinity, that you seem completely oblivious to.  Maybe explore that a little bit too and see if it is more useful for your personal understanding?

By the same sword, I cannot abide when people hold onto "established results" so rigidly and stubbornly that they refuse to open their mind up to any possible alternative perspective, however reasonable/absurd it may be. So, at least we feel the same way about that, if only from different perspectives!

For the record, though: I am not a flat Earther! There is far too much evidence to prove that Earth is a sphere, if not a perfect one. However, I do like the idea of Antarctica being a huge ice wall surrounding the Earth, and explorers seeking to find what's on the other side of it. That kind of idea yields great stories!

Kinda funny you mentioned "flat earther", because in some ways you kind of sound like one, and I think you recognized it or you wouldn't have brought up the flat earther thing even though that had never come up in the discussion.

The problem is this is only a matter of perspective in the sense that your perspective of the mathematical treatment seems so incomplete and confused.  In some cases I think you just read way more into something than what it actually means.  Cantor's cardinality measure is just some measure you can apply to both finite and infinite sets.  It is defined based on the notion of 1:1 mapping (aka bijection) as opposed to any intuitive notion of "counting".  For finite sets it happen to be equivalent (ie. produces the same results so to speak) to counting number of elements.  For infinite sets, it is not surprising that you will end up with results unique to infinite sets, it's nothing more than infinite sets being fundamentally different from finite sets.  Maybe it's less confusing for you if you just forget Cantor's cardinality measure as being somehow equivalent to "size", and just note that it happens to behave like "size" when it comes to your everyday finite sets, but otherwise just think of it as a specifically defined measure in mathematics and don't insist on trying to mesh it against your everyday intuition of size when it comes to infinite sets.  After all, seems rather a little rigid maybe to insist on a perfect mesh with intuition for a concept like infinity that is already so far removed from everyday experience?

it's trying to use tangible, rigid concepts and theories to express the intangible.

I don't understand why you seem so rigidly insist on infinity necessarily being "intangible", or why the mathematical treatment is "rigid" (or even if it is based on whatever definition you are applying, why is that a problem).  There seems to be a mental bias in your current thinking that I'm not completely grasping.

[namida]

Kinda funny you mentioned "flat earther", because in some ways you kind of sound like one, and I think you recognized it or you wouldn't have brought up the flat earther thing even though that had never come up in the discussion.

WillLem wasn't the first to bring this up; ∫tan x dx was:

I apologise if my rant comes off as being confrontational or overly hostile. But I cannot abide when people who clearly have no idea what they are talking about decry well established results in any given area. It's this kind of self assured superiority that causes flat earthers, and other nonsensical beliefs.

[kaywhyn]

Wow, all the fellas above can definitely explain all this way better than I can. Did you gentlemen study math in college/university? Or is math just something that you always had a passion for and love studying in your free time? I have always loved math as a child, and I was a math major in both my undergrad and in grad school. I was trying to respond earlier with an example for WillLem on how infinite sets can have different cardinalities. The one I used was the set of natural numbers and integers, only to find out in my research, to my surprise, that they have the same cardinality. So, it looks like I don't know as much about set theory as I thought I do. Indeed, Proxima just explained why they do by means of a bijection between the two existing, so by definition they have the same cardinality. The example I used in my only post to this topic so far was the set of naturals and the even positive integers having the same cardinality, as you can simply take an element from N and pair it with the even integer from Z that's twice it. So, looks like I explained that correctly as confirmed by Proxima. To be fair, I did mention that set theory was something I didn't take or really study as a math major. When it comes to set theory, I only know the basics, i.e, definition of a set, element of a set, what it means for two sets to be equivalent, and cardinality, among other things. I'm only vaguely familiar with infinite sets and proving their cardinalities are the same or not depending on if a bijection exists through reading about it in my free time and not through an instructor in a math course. Proxima, tan, and cc did a great job with their explanations, so I'm still able to follow pretty much all of them.

[Flopsy]

I also did Mathematics at University, I graduated back in 2008 kaywhyn.

It's nice to see who knows about the subject.

I don't really want to get involved in this discussion however because it's not really something I understand myself to be honest.

[Proxima]

Just a hobby for me. I also loved mathematics when I was a child, and grew up reading popularisations such as the books of Martin Gardner and Raymond Smullyan, and later Hofstadter's Gödel, Escher, Bach. In particular, since we've been talking about infinity, Smullyan's book Satan, Cantor and Infinity contains a lot of these arguments, presented through conversations between a Sorcerer and his pupils, who start off not knowing much about the subject and go through the common misunderstandings.

When I reached 16, everyone around me expected me to go on to study maths at university, but I was clear that it wasn't for me. I didn't want to become too specialised; I wanted to learn more about other aspects of the world. I chose to study Oxford's famous PPE degree, and went on to an MA in Philosophy.

[kaywhyn]

I also did Mathematics at University, I graduated back in 2008 kaywhyn.

It's nice to see who knows about the subject.

I don't really want to get involved in this discussion however because it's not really something I understand myself to be honest.

That's awesome, Flopsy. I was in my 2nd year of college when you graduated from university. That was also the year that I switched my major from biochemistry to math. I thought I wanted to do pharmacy, but I gave up on that once I took my second chemistry college course. I enjoyed chem a lot in high school, but it was just too much for me in college. Math was probably more difficult, but at least I was studying something I've been enjoying since childhood.

I know what you mean, though, as I'm not that well-versed in set theory, in particular proving things about infinite sets. I'm just familiar with them enough to be able to follow the explanations given.

Just a hobby for me. I also loved mathematics when I was a child, and grew up reading popularisations such as the books of Martin Gardner and Raymond Smullyan, and later Hofstadter's Gödel, Escher, Bach. In particular, since we've been talking about infinity, Smullyan's book Satan, Cantor and Infinity contains a lot of these arguments

When I reached 16, everyone around me expected me to go on to study maths at university, but I was clear that it wasn't for me. I wanted to learn more about other aspects of the world. I chose to study Oxford's famous PPE degree, and went on to an MA in Philosophy.

How awesome, Proxima! Thanks for clarifying. I would had guessed you studied math in college/university due to your depth of knowledge in the subject when in fact that you didn't. It's great that you were able to take something you read as a kid and apply it here. I definitely commend you for being able to take up reading. I never was, and still aren't, much of an avid reader myself. I just can't bring myself to read books.

I would like to point out that besides infinity, there are plenty of other nuances in math, but we all probably already know this. I have taken so many math courses and seen numerous examples in various mathematical topics in the last 12-13 years (I studied math for both my undergrad and grad school) to know that mathematicians definitely mean business when it comes to being precise in mathematical definitions, especially relating to mathematical pedagogy, the art of teaching math. In my experiences, students will take what the teacher says very carefully, and this can lead to confusion. For example, think back to how you were taught multiplication in elementary. It's likely that you were taught multiplication to mean repeated addition. 3 x 5, for instance, simply means 3 + 3 + 3 + 3 + 3. So, multiplication is simply a faster way of doing repeated addition. Similarly, -3 x 5 means to add -3 five times, i.e, -3 + (-3) + (-3) + (-3) + (-3). Ok, all well and good, but can you apply this same definition to, say, -3 x -5? Try as you might, you won't succeed, because what exactly does it mean to add -3 negative 5 times? So, we can't use the elementary definition of repeated addition that we learned for multiplication for this problem. This simply means that while the common definition of multiplication that we learned in elementary applies to many problems, it doesn't work for all, especially when negatives are involved. Since we only learn about positive integers at first, this isn't a problem, but once you start learning about negatives, this is where previously learned mathematics possibly start breaking down, like it does here for -3 x -5.

One way is to use a pattern. What's 3 x -5? That's -15. How about 2 x -5? That's -10. Now 1 x -5. That's -5. If you look at the answers, every time we decrease the first number by 1, the answer increases by 5. Continuing this pattern, you can conclude that -3 x -5 = 15. However, it is important to note that this is not a mathematical proof that two negatives multiplied together yield a positive. Rather, it is a way to convince oneself that - x - = a positive. If I remember correctly, the proof relies on the fact that 0 times any number equals 0 and the distributive property. Similarly, there is a proof that 0 times any number equals 0 (I saw this proof in an upper division math course in college), so this isn't something special that mathematicians agreed on just for the sake of it and that we should just take for granted. There's an actual mathematical proof of this fact!

Another good example is when we were taught how to order numbers. We all agree that 2 < 5, since when we learned to count, we know that 5 comes after 2 on the number line. However, when it comes to fractions, 1/2 > 1/5 even though we know that 2 < 5. This right here seemingly violates what we were taught previously on counting and ordering and will definitely confuse a child (I know I was when I was first taught fractions). Fractions is where previously taught mathematics breaks down. Once you consider fractions are different, eg, a special kind of number with two parts (right now I'm blanking out on how my college instructors taught us how fractions are defined, I think they might had used the word "number" in there), a numerator and denominator, then you'll see that once again that you can't apply similar concepts that you learned about ordering the counting numbers (the natural numbers) to them. What exactly does it mean to order things with two parts in them that mean different things (numerator is only a part, while denominator is the whole)? So, yet another instance where previously taught math concepts break down!

Seeing how this topic started with infinity, it can be seen the same way. However, as some have correctly pointed out, infinity is not any particular number, but rather a concept of things going on and on without end. You can treat infinity as though it is a number so that you'll be able to apply the same mathematical properties as you do with actual numbers, but infinity is a special case where known mathematical properties need to be slightly tweaked so that we can more easily work with infinity but at the same time they still hold true for numbers.

[WillLem]

We define aleph-0 to mean "the cardinality of the set of positive integers". Any set that can be put into one-to-one correspondence with the set of positive integers has this cardinality. There you go, I defined it

A basket can be made that is large enough for 3 oranges, so I can understand 3 oranges belonging to a set. However, there is no basket that can be made that is big enough for (every-positive-integer) oranges, so how can every positive integer belong to a "set", as we understand it?

Suppose it has a top-left corner and continues right and down to infinity.

I do understand your explanation, and can reasonably visualise how each pixel in one example would have a corresponding pixel in the other, the same as how positive integers can be mapped to positive even numbers 1-1. I totally understand that part of it.

What I don't understand is... how can something that is infinite have "a top-left corner"?

So I guess this is the "WillLem is insulting all the mathematicians out there" topic.

It's kind of annoying that it's been taken that way, I didn't set out to insult anyone; such is the wonder of internet communication. For what it's worth, I apologise for any offense I may have unintentionally caused.

You seem to argue the set of all natural numbers is abstract and does not "exist in the real world", but does either of your examples actually exist in the real world either?  The picture of the corridor is almost certainly of a real corridor that is in fact not infinite, you are only imagining in your mind that it goes on forever.

Mathematics only implies that the list of positive integers goes on forever, usually using an ellipsis(...) It, in fact, doesn't, as such a list only exists in the imagination, which makes it exactly as valid as an imaginary infinite corridor.

And don't be so sure about the world being analog either.  Some physicists postulates that space and time themselves might be subjected to the laws of quantum mechanics and therefore may actually be quantized, such that there may actually be a minimum length/duration enforced by the laws of physics.

I'm guessing you're referring to Planck length here. This is something I don't even pretend to fully understand, and I'm sure that greater minds than mine have come up with plenty of reasons why this should be accepted as the smallest possible length in the observable physical world. No dispute there at all; I can accept that there is a possibility that the world is indeed digital and not analog, and that this would render certain aspects of my initial premise somewhat redundant.

You mentioned music and art, but I'm having trouble following what examples you have mentioned that are actually coming from those disciplines and relating to infinity.  I'm rather surprised you didn't mention philosophy.  The nature of infinity had actually been historically discussed as philosophy far, far longer than the modern mathematical set-theoretic treatment had existed.

Absolutely - reference to philosophical concepts as well as mathematical ones is equally valid and interesting for this discussion. As for artistic examples, I'd say Escher has produced some stunning works that are both mathematically and artistically concerned with the concept of infinity. Specifically, Relativity and Waterfall would be great examples of this, but there are others.

It may not be visually measurable, but if there is truly no way at all (not even non-visually) to measure the spinning, then how do you prove that it is even spinning at all?

This is a brilliant question, and one that I can't answer. It's fascinating though, and I love to think about questions like this from time to time.

Suppose we have an infinite set that represents real numbers between 0 and 1

What causes my confusion about real numbers belonging to a set at all is this:

Start at 0. Now count to 1, but don't miss out any numbers in between.

Where do you start?

This is what I meant by "arbitrary counting distance" in my OP: we can manually create a starting point for such a process, because we can define numbers such as 0.5, 0.237987, 0.09719682649862389749283496 and so on.

But... where exactly is the first of these numbers?

We also use 0 as a convenient starting point for a lot of things, for example:

Instead, we should list them in an order like 0, 1, -1, 2, -2, 3, -3... Now all integers are on the list

But this brings me back to my question as to how something that's infinite can have a "top left-hand corner"...

Cantor's cardinality measure is a specific way to extend the intuitive concept of "size of a set" from finite sets to infinite sets, where human intuition is far less reliable and certainly lacks rigor.

I can't find a smiley for this. Cantor was a human, so any ideas he may have established are, by definition, based on human intuition!

It is a little hard to take your initial post seriously when it demonstrates a great deal of confusion and misunderstanding about the mathematical things you try to refer to, and at the same time phrased in such ways as to seem to assert decades of work by world-class mathematicians are complete garbage.

I have taken care to retract any such perceived assertions, as they were not intended to have been taken this way at all. I am not so arrogant as to suggest that mathematics is "complete garbage" - your words, not mine - in fact, I am certain that far greater minds than mine have spent far more time than me thinking about these things.

Is it wrong for me to question their findings, though, limited though my own understanding may be?

set theory and its treatment of certain concepts like infinity is simply what mathematicians have found useful over the years as applied to their every day work on mathematics.

The longer something has been established as "truth", the more necessary it is to question and re-evaluate it.

I'm not even sure I understand what it means to say to "interact" with "infinity".  Isn't it just an abstract concept?

I guess I mean how can we make sense of it, explore it, when it's seemingly unexplorable. As you've touched upon: nothing in the physical world, particularly humans, can experience the infinite except in our imaginations. We seem to agree about this.

I'm also not sure why you want to be picking on mathematics on this.  There is a much richer, longer history and literature from philosophy on the exploration and discussion of infinity, that you seem completely oblivious to.  Maybe explore that a little bit too and see if it is more useful for your personal understanding?

I'm not "picking on mathematics", and have already apologised for and edited the tone of my original post. Please can we move on from that aspect of this discussion now.

Philosophy is certainly an area I'm also interested in. I began this discussion from a mathematical point of view because that's the side of it I understand the least, but I'm more than happy for this discussion to develop into something more philosophically-orientated.

Kinda funny you mentioned "flat earther", because in some ways you kind of sound like one, and I think you recognized it or you wouldn't have brought up the flat earther thing even though that had never come up in the discussion.

Actually (as Namida also pointed out - thanks!), ∫tan x dx was the one who mentioned flat earthers, in relation to the regrettable way I initially worded my questions about mathematical understanding of infinity. To re-interate: I am not a flat-earther, nor do I subscribe to any other such belief system.

I have to say, I had no idea that my post would come across this way or cause people to take offence. I am simply asking questions, that's all. I do not claim to have greater knowledge, or assuredness of a particular belief system - far from it: I'm essentially open-minded in my thinking on most subjects, and merely wish to seek further understanding.

The problem is this is only a matter of perspective in the sense that your perspective of the mathematical treatment seems so incomplete and confused.

Correct: that's why I'm discussing it - to further complete and clarify my perspective. So far, the discussion is having that effect. I now understand far more about concepts such as bijection, cardinality and set theory than I did before.

I'd just like to close this post by clarifying a few things for the purposes of the discussion moving forward:


1. I have nothing against mathematics, mathematicians, or indeed any school of abstract thought (actually, I happen to be subscribed to Numberphile on YouTube and I regularly watch and enjoy their videos, and find them very fun and educational). I am simply interested to understand it better, and have come up against questions to which the answers have caused further confusion, in some areas. In other areas, I now feel more enlightened thanks to people's explanations and the progress of this discussion.

2. I know that infinity is not a number. I am asking about the relation of the concept of infinity to the mathematical system of numbers.

3. I think you're all awesome.

[Proxima]

A basket can be made that is large enough for 3 oranges, so I can understand 3 oranges belonging to a set. However, there is no basket that can be made that is big enough for (every-positive-integer) oranges, so how can every positive integer belong to a "set", as we understand it?

A set, being a mathematical object, doesn't have to exist in the real world. As I said earlier, when mathematicians talk about a cube, for example, they mean an idealised cube with perfectly straight and exactly equal edges, even though such an object can't exist in reality.

What I don't understand is... how can something that is infinite have "a top-left corner"?

Imagine an infinite grid of squares. Imagine that one particular square is black. Then imagine that every square, continuing rightwards and downwards from that square, is black. Now the black area continues to infinity in two directions (right and down) but has a definite top-left corner; an ant walking over the black area leftward or upward would eventually reach the border with the white area.

This is the 2-dimensional equivalent of the distinction between the positive integers (starting at 1 and continuing to infinity in one direction) and the set of all integers (continuing to infinity in both directions).

Mathematics only implies that the list of positive integers goes on forever, usually using an ellipsis(...) It, in fact, doesn't, as such a list only exists in the imagination, which makes it exactly as valid as an imaginary infinite corridor.

Everything mathematicians talk about is imaginary. If you don't accept imaginary things as valid, you can't begin to do mathematics.

You also can't begin to make bank transfers, because that is asking someone to manipulate imaginary concepts on your behalf. Your bank account isn't a Gringotts vault containing actual physical money....

What causes my confusion about real numbers belonging to a set at all is this:

Start at 0. Now count to 1, but don't miss out any numbers in between.

Where do you start?

You are confusing the concepts of set and series or sequence.

A set is a collection of objects, without regard for order. The set {5, 6} containing only the numbers 5 and 6 is the same as the set {6, 5}.

A sequence is a progression that starts with a first term, then a second term and so on. For example, the sequence 1, 4, 9, 16... of square numbers, in which the nth term is n^2.

In many cases, these concepts overlap. We can talk about the sequence of square numbers when we think of them (or perform mathematical operations on them) as a sequence, such as finding a formula for the nth term. We can also talk about the set of square numbers (the collection of all square numbers, without regard for order) when we want to think about them as a set, for instance to ask "what is the cardinality of this set?" (how many square numbers are there?)

In other cases, it's important to draw the distinction. The real numbers are a set, but not a sequence. There is no "first term" or "second term". You cannot count them.

The longer something has been established as "truth", the more necessary it is to question and re-evaluate it.

Mathematicians are not unfamiliar with this principle.

[Simon]

Happy to see so many here with a grounding in set theory.

I've enjoyed formal mathematical education. Still, our uni never offered even a basic course on axiomatic set theory. This baffled me to no end, for several reasons:

1. Set theory (really the study of infinite sets) is enormously interesting by itself. Fascinating questions arise immediately and naturally, already during the first few weeks of first-semester mathematical training.

2. The tooling is surprisingly useful in related research. Topology, group theory, abstract algebra, ... everywhere, there will be questions of cardinality. Occasionally, transfinite induction or Zorn's lemma appear in proofs, and it becomes much more natural to use them yourself.

In universities, it's common to spoon-feed every first-semester mathematics student Zorn's lemma without proof. But it's not satisfying.

Much more rarely but not impossibly, independence results appear in the related fields during research, but attacking those would be beyond a basic course in set theory anyway, so their proofs are perhaps rightfully skipped.

3. Such internet discussion 20 years ago was one of the many reasons I studied mathematics and not physics. The problems in mathematics sounded much more interesting.

Given the lack of courses at university, I've read set theory books in the evenings during my undergrad studies. In 2008, Deiser's Einführung in die Mengenlehre was an ideal entry-level text, it read like a criminal novel despite staying 100 % formal in its proofs. Around 2012, this dovetailed into Kunen's Set Theory with my goal of at least a rough understanding of Gödel's and Cohan's CH results.

In 2014, I sacrificed time off my PhD to hold my own one-semester course in axiomatic set theory. It was a seminar with the students giving 90-minute talks that I would design and select literature for. The seminar became a smash hit, I had to assign two students to every talk to accomodate everyone. By the middle of the seminar, we had introduced von-Neumann ordinals and cardinals, and my wish is that at least these ideas have stuck with all participants.

To this day, I get questions for when that seminar repeats, and it's always sad to tell them it was a one-shot. >_>



The answers in this topic give solid explanations. I have practically nothing to add yet that wouldn't go over the top or be confusing. Still, I'll happily follow this.

I had been reluctant to reply because the unedited OP reeked like a troll. It wished to discuss the real numbers, noble goal. Then the obvious attack fails, normal and interesting. Then it suggests that mathematics is not the best tool for a problem that can't even be stated outside mathematics.

There is no slur at all in pursuing art. That's merely unlikely to yield understanding of the reals, which seemed like the point.

It's possible to reject infinite sets and still do mathematics, but that didn't become popular in 20th-century-mathematics. The theorems of ZFC would then still be true, but considered uninteresting.

-- Simon

[WillLem]

Thanks for the explanation Proxima, that helps to clear a few things up. It's obvious to me now that I was perhaps visualising it differently, but I understand the crucial differences between a "set" and a "series" better now.

Then it suggests that mathematics is not the best tool for a problem that can't even be stated outside mathematics.

Philosophy?

The concept of infinity was philosophical before it was mathematical. Granted, mathematics has so far been the tool of choice for discussing, comprehending and otherwise dealing with the concept, but I am asking whether or not it's the best choice.

Note: asking, not suggesting either way.

Anyway, if people are just going to continue to take what I've said as an attack, belittle me for my lack of formal mathematics education, and ignore the fact that I have edited my OP having learned from my mistake, then I no longer have any interest in discussing it here.

[Simon]

Philosophy? The concept of infinity was philosophical

I'm happy to learn more here. I merely didn't see any examples and don't have much experience here myself. I hope that explains my math-slanted reply.

take what I've said as an attack

With "attack", I meant: method to solve the given problem (which was to show the uncountability of the reals).

Reworded: The first attempt to solve the question about the real numbers fails, which I call "normal" because I expect this at that time halfway reading through OP given the methods of the attempt, and which I call "interesting" because I expect good discussion following. I did not take edited OP as a potshot against established mathematics, you had made that clear already in the days since.

80 % of the edited-as-is OP is still math. Thus, please make it very clear where that mathematical context is over.

The problem is that otherwise, reading OP allows one to assume you wish to solve the uncountability of the reals with music/art, when instead you wish to learn how music/art/philosophy have dealt with not-necessarily mathematical inifinity, which indeed happens to fascinate also non-mathematicians.

and ignore the fact that I have edited my OP

I have seen the edit. I wrote that the unedited OP discouraged me from replying on day 0.

It still sounds to me like you want to understand the uncountability of the reals. Your replies to other people also suggest interest in distinguishing countable and uncountable sets.

-- Simon

[ccexplore]

Mathematics only implies that the list of positive integers goes on forever, usually using an ellipsis(...) It, in fact, doesn't, as such a list only exists in the imagination, which makes it exactly as valid as an imaginary infinite corridor.

It goes on forever in that for any positive integer you can come up with, I can add 1 to it, to get a different positive integer that is bigger than the one you give me.  This is what mathematics really mean by the list "go on forever".

The ellipsis is just a shorthand notation.  It is actually not how one would rigorously formulate all the positive integers; after all, the next number after "1, 2, 3, ..." might not be 4, it might be 5 for example if you are actually talking about the Fibonacci numbers.  The notation using ellipsis may be convenient, but clearly imprecise.  Instead of example members plus ellipses, to be rigorous you would just provide a formula that says you start with 1, and then you can keep creating the next integer by adding 1 to the previous.  You don't need any ellipses and there is no ambiguity.

Yes, you can call it imaginary in the same way as the imaginary infinite corridor.  My question is, it seems like you object to the imaginary status of the list of all positive integers more than you object to the imaginary infinite corridor, is that so and if yes, why is that?

Suppose we have an infinite set that represents real numbers between 0 and 1

What causes my confusion about real numbers belonging to a set at all is this:

Start at 0. Now count to 1, but don't miss out any numbers in between.

Where do you start?

Ok, I think you have a confusion between the concept of a sequence vs a set.  A sequence is ordered and countable.  A set is not ordered and doesn't have to be countable.  You can define a set purely based on a formula that can describe, given something, whether that something belongs to the set or not.

As already explained by others, it is actually impossible to list out all the real numbers, or even just the ones between 0 and 1, in any kind of lists.  Any attempt to do so will provably result in some real number that is not part of the list you try to make.  The set of real numbers can be defined simply by the number being expressible as, for example, a regular integer before the decimal point followed by an infinite sequence of digits 0-9 after the decimal point.  Any number expressible in that form is a member of the set.

This is a completely understandable confusion given that the informal ellipsis notation for numeric sequences, is often used in place of a formal formulation, as a way to describe the membership criteria for the set.  That is, the set of positive integers is often written down as {1, 2, 3, ...}, but as explained above, it is actually rather informal and imprecise to use this ellipsis notation.  You don't have to use ellipses to define an infinite set, and for something uncountable like the real numbers, such a notation is completely useless anyway.

What I don't understand is... how can something that is infinite have "a top-left corner"?

The sequence of natural numbers (1, 2, 3, ...) has a start but no end.  It has a minimum member but no maximum; any number you can think of, I can add 1 to it to make a bigger number.  So it is unbounded (ie. infinite) but in one direction only.

Other examples abound, such as Dullstar talking about levels being infinite in one direction but finite in another.

Cantor's cardinality measure is a specific way to extend the intuitive concept of "size of a set" from finite sets to infinite sets, where human intuition is far less reliable and certainly lacks rigor.

I can't find a smiley for this. Cantor was a human, so any ideas he may have established are, by definition, based on human intuition!

Right.  At the same time, he also took care to ensure his formulations are precise and logically consistent, so that it is usable for further mathematical inquiry.  As I mentioned before, a logical inconsistency/paradox (ie. a statement both true and false) is bane to mathematical proofs, because the operations of logic can take even just one inconsistency and allow it to "prove" any other statement in the system to be both true and false as well.  A system with unresolved paradoxes cannot be used as the basis for further mathematical inquiry, especially for something like set theory that was considered for possible use as foundational basis of all of mathematics.

It dismays me that somehow the only thing you seem to read into the sentence is that I'm somehow denying the role of human intuition in any discipline.  Of course human intuition plays a part in the exploration and understanding of any topic, no one is trying to deny that.  But at the same time, surely you can agree that human intuition isn't infallible either?  That's really all I'm saying, and somehow you seem to be reading a completely different meaning into it apparently.

Is it wrong for me to question their findings, though, limited though my own understanding may be?

No, but your initial attempts, at least in their wordings, seem to show you haven't even bothered questioning your own understanding of those findings.  Shouldn't one first "question and evaluate" one's own understanding of the things one is trying to challenge, lest the attempt only wound up challenging a completely distorted version not congruent to what the findings were actually saying?

It also doesn't help much that between your questions, you talked about things like this

By the same sword, I cannot abide when people hold onto "established results" so rigidly and stubbornly that they refuse to open their mind up to any possible alternative perspective

Somehow seeming to insinuate that anyone who don't share the same confusion as you had on the details of set theory's treatment of infinity, must only be because they are "rigid" and "stubborn", as opposed to maybe other possibilities like, I don't know, maybe they understood it a little better?  Interesting that Dullstar also asked some questions here that basically seek for the same clarifications on some of the same things you were a little confused about, and yet somehow he managed to not have to bring up anything about anyone being rigid and stubborn, which is neither here nor there and certainly not helping to actually addressed the questions being raised.

The longer something has been established as "truth", the more necessary it is to question and re-evaluate it.

Sure, but at the same time, it's not like this topic is something that's decreed as unquestioned truth one day by some king or emperor, and then blindly accepted for here on out.  That's never how things work in any academic disciplines.  Right from the beginning as the initial version of the theory or framework is still being formulated and explored, things are already questioned, discussed and critiqued, and things get revised multiple times in the journey towards the formulation you see today.  People often lose sight of this, because from a textbook, of course they skip over right to what ends up being the current "final" formulation of the topic; they are not going to waste ink elaborating on all the various initial earlier formulations, the things that were questioned or critiqued and thus revised, possibly even some things that proved to be false starts and thus abandoned.  Even if you read up on additional reference materials that go more into the history and evolution of the topic, they too are liable to summarizing and not necessarily provide the full scope of everything that had happened throughout the evolution of the topic or area of study.  So far from it being unquestioned, what you see today generally had already endured years of questions and evaluations; you wound up with what had stood the tests of time.

For a topic that had been around for so long and exposed to so many people, it would be patently silly to assume that the questions one can raise haven't likely already been asked in a similar way before by someone else and then addressed, especially questions from someone who is still learning about the topic.  At the same time, don't assume that there aren't questions raised today on the topic either.  It's just that given the things that had already stood the test of time, one would likely need to question at a deeper level, at a much more particular part of the theory, in order to truly be asking something that hadn't already been asked and addressed before.  It might be hard for the average person to find examples of such questions as they likely resides in relatively obscure academic journals devoted to rather deep specializations into the topic or area of interest, but that doesn't mean no such questions are raised.

I'm not even sure I understand what it means to say to "interact" with "infinity".  Isn't it just an abstract concept?

I guess I mean how can we make sense of it, explore it, when it's seemingly unexplorable. As you've touched upon: nothing in the physical world, particularly humans, can experience the infinite except in our imaginations. We seem to agree about this.

I guess I don't really see how it is "unexplorable".  Really, if we can talk about it, isn't that enough to make it explorable?  Academic studies are full of things that don't have direct physical manifestations in the real world anyway.  Geometry may be based on our experiences in the real world, but there is no actual physical manifestation of an ideal geometric line of 0 thickness and infinite length.

At the same time, just because something is grounded in the physical world, don't assume that our intuition and initial understanding of it will necessarily be more reliable and accurate.  After all, Aristotle the ancient Greek philosopher believed that it is the natural order for heavier things to fall faster than lighter things.  And honestly, that matches our everyday intuition pretty well, so pretty much everyone quickly believed it to be true.  It took a surprisingly long time before people finally discovered that, no, in fact that's not true.  In the absence of air, all objects would actually fall at the same speed regardless of their weight.  It is actually the air providing air resistance that results in the differences one observes in the speed of falling, and in turn the air resistance depends on other properties of the object besides its weight/mass.

[ccexplore]

It's possible to reject infinite sets and still do mathematics

Thanks Simon for bringing this up, I guess there's something for me too to read up more on.

So as one can see, academics are far from being rigid unquestioning sycophants, when it comes to complex topics like infinite sets.  The above topic actually also touches nicely on the fuller history of the development of mathematics in the 19th century.  Read the "history" section of the article, and you see that the development was far from a few person coming up with some ideas, and then everyone else just blindly accepts from here on out.  (Then again, no one can possibly seriously believe that's how academic pursuits ever work in the real world, right?)

The topic of philosophy of mathematics might also be of interest, given some of the things that have been brought up.  I'm certainly no expert in this area though, I can only offer the Wikipedia article as a starting survey for people to further explore as they see fit.