Infinity: Paradoxes in logic, math, physics, and blog posts

I would like to be somewhat less than infinitely boring. So I have no intentions of referring to limits at infinity from calculus, or infinite divisibility of some unit of distance or time. I would like to bring something new to the discussion (or at least some new perspectives) and would like others to add if they are willing (questions, comments, complaints, other examples, comments about my mental health, etc.). So while I will introduce the topic with some “formal” stuff, I intend to get into physical infinities and more interesting topics and hope others will find them interesting as well.

I will begin with the not-first-step.

#1 Not-the-first-step

This is an example a bit like infinitely dividing some time interval but it is different enough to be included and is a good starting point for that which is to come. Most of us here probably remember something about the real number line. That’s the one that includes counting numbers, fractions, and numbers like pi (infinite non-repeating decimals). And we use it all the time when we try to figure out how much something costs if it is on sale for 10%, or using a recipe that calls for 1 parts cyanide and 2 parts arsenic, etc. Most of us are also used to thinking about numbers on the number line in terms of an order: 2 is bigger than 1, we’d rather win a million dollars than ten, etc.

Here comes the not-first step. Say I’m at 0 on the number line. I want my “next step”, or the next number, to be the one number next in line (for the counting numbers, that would be 1, followed by 2, 3, 4, etc.). What is my “stepping” number? I can choose a really small number like 0.00000001. However, I can extend infinitely the number of 0’s I place in front of that one. There is no “next” number.

But, you might say, this is just because you’re using the real number line and that involves infinite decimals. Ok, I won’t. Let’s say I change the number line and exclude all non-rational numbers. Can I now take my next step? No. Recall that a decimal like .01 is a fraction: 1/100. I can continue to infinitely extend the number of 0’s as I did before and never use an irrational number.

And to end the number line example, let’s think about an irrational number for a second. Since pi is probably the most familiar, let’s use it. Imagine I am trying to construct my real number line, and since I know pi is a number I want to plug it in somewhere around 3.14….I have a problem. How many rational numbers are there between 3.14 & 3.141? Infinitely many. The rational numbers are “infinitely dense” in that between any two there are infinitely rational numbers. So where the hell are the irrational numbers supposed to fit?

#2 My infinity is bigger than yours

Usually, when we talk about infinity, we mean “never-ending” and we refer to one “thing”. Infinity isn’t a number. If I add 1 to infinity it is still infinity. And while philosophers and mathematicians have struggled with the notion of infinity since before Zeno and his paradoxes, they were all dealing with this one infinity. Then a guy named Cantor ruined everything. I’m going to show how.

We can all agree, I think, that the counting numbers (1, 2, 3, 4,…) are infinite. So are the integers, the rationals, and the reals. But are they are equal? Common sense tells us “duh. They are all infinite, and you can’t get bigger than infinity.”

How do we compare some collection of a number of things, like a dozen donuts or the vowels, to see if one collection is bigger (there are more donuts in a dozen than there are vowels in English)? We match them up: we can match 5 vowels to five donuts, and see that there are donuts left unmatched. Ergo, donuts win.

Imagine we do this with the integers aI the real numbers between 0 and 1. That is, we do something like this:

We assume the obvious: both collections/sets are infinite, so no matter how far we go we can always match them up: for every real number in the interval [0,1], there is a corresponding counting number. Now we create the irrational number Z=.Z1 Z2 Z3 Z4 Z5…Zn, where each “z” is not the sound of your snoring but is a digit in this non-repeating irrational number. But we do this in a special way. For the first digit Z1, we look at the first digit in the number we matched with the integer 1, and we make it 1 greater. For the second, we do the same only to the second digit of the second irrational number. Then the third, as depicted below:
We can keep doing this. We can go on forever constructing this number as we have. But the important thing is that we are ensuring with each new digit that our number Z=.Z1 Z2 Z3 Z4 Z5…Zn doesn’t appear anywhere on the list. How can that be? Both lists are infinite, yet one list has more entries! In fact, it has infinitely many more entries.

#3 What’s hotter than being hot?

Ok, enough with the pure math stuff. Most of us probably learned (even if we’ve forgotten) that there is a temperature scale, the Kelvin scale, designed such that 0 is absolute 0 (rather than the temperature at which water freezes or something). However, there is no “absolute hot”. That is, theoretically, temperatures extend infinitely in the positive direction. But so what? It’s not like there is a star out there that’s infinitely “hot”. True. But there are physical “systems” that are “hotter” than the entire infinity of all positive temperatures. I put terms like “hot” in scare-quotes because in thermodynamics heat can’t really be described by such qualities and even if it could, few physicists would want to try to feel how hot something that was a thousand degrees was or how cold something near absolute zero would be. But there are other ways of measuring temperature. For example, anybody who has put ice-cubes in a drink knows that the transfer of heat has a specific direction. It turns out that absolute zero isn’t the minimum temperature. There are negative temperatures. These negative temperatures, though, are “hotter” than the entire infinity of positive temperatures.

#4 Will too many ideas make my head explode?

Often, discussions about infinity here involve some proof of God. What they tend to share is the assumption of some concept that is then logically manipulated to be equated with god (such as the “most greatest being” or whatever). So while I’m sure not everybody here agrees, most who aren’t religiously inclined and many who are would argue that our thoughts are represented somehow by the activity of neurons in our brains. Somehow, when I think of a quote, a number, a car, or whatever the “notion” or “concept” I have in “mind” is somehow linked to the physical state of my brain. Now, I don’t know about you, but when I was a year old my vocabulary wasn’t particularly extensive. Also, my math skills weren’t that strong, I didn’t know much about history, I wasn’t all that good at physics, and in general I’d say I didn’t have many ideas or even memories. As my brain grew, so too did my knowledge. But not only was there very little relationship between this growth (in point of fact, when one is born one actually begins to rapidly lose brain cells), after it stopped growing I continued to learn new things. Where did these new ideas go?

It can certainly be said that I don’t have infinitely many ideas. But there are an infinite number of things I can think of (to see this, simply recognize that I know there are infinitely many numbers and I can think of infinitely many of them). Moreover, I am able to think of any one of these infinitely many things/ideas without acquiring new knowledge. Yet all my knowledge is fixed in a finite brain and all my ideas are somehow represented/encoded in that brain. So why doesn’t my head explode?

#5 Care for a slice of spacetime?

It’s too easy to use quantum mechanics for an example of a physical infinity. And it’s too easy to dismiss these by appealing to what we don’t know about the systems we are describing. Relativity is different, especially special relativity. It’s been confirmed by experiments since 1905 (actually, Einstein’s 1905 paper was based on previous empirical evidence; he just explained it in a way that has proved to be incredibly successful and remains integral to modern physics), it can be and has been made consistent with quantum mechanics, and it is consistent with general relativity (which is in general an extension of special relativity).

Special relativity involves a lot of concepts, but one of them is how space and time don’t exist the way we think they do, but rather are linked in something distinct: spacetime. The geometry of spacetime, and the math of relativity, enable us to show that even when two measurements disagree because they involve different “reference frames”, the laws of physics hold regardless. These reference frames, which we all have (I personally have collected 7), are really how spacetime is divided or “sliced up” from our perspective. The differences here on Earth between all of our reference frames are so slight as to not really matter. We’re all “close” enough in 4D space such that changes in any of the coordinates keeps us close. So for all of our reference frames the distances in our 4D coordinates make the way we “slice up” spacetime insignificant.

This is not true in general. For the alien invaders on their way here from a distant galaxy lightyears away, spacetime looks very different. In fact, depending upon how their fleet of battle starships moves, their “slice” of spacetime at some point could be millions of years in what, to us, is the future, while another movement could rocket them millions of years in “our past.” In fact, there are infinitely ways that their movements could slice up spacetime relative to us. Moreover, there are infinitely many ways infinitely many observers can slice up spacetime.

#6 I lied

I am going to talk about quantum mechanics. But not in the way I normally would here. I’m going to do so to come full circle (fitting, I think for the close of a post on infinity) and connect the physical back with math. In quantum mechanics, we represent “physical systems” like electrons as mathematical entities, just as in all physics (speed, for example, we represent using numbers, while velocity requires a vector). The “space” in which these quantum systems dwell is called Hilbert space and is infinite-dimensional. What does this mean? Consider the real number line again. It extends infinitely in the positive and negative direction. What about the x,y-plane we all “loved” from school mathematics classes? It extends infinitely along two directions. 3D space along three. 4D along 4. Hilbert space, then, extends infinitely in infinitely many directions.

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3 Responses to Infinity: Paradoxes in logic, math, physics, and blog posts

  1. Jeff says:

    Point #1: Not-the first-step

    This is clear abuse of the poor number line. I don’t think it was ever intended to be put under so much scrutiny and stress. Is it really infinite or are we just expanding the scope (and scale) of the number line? In other words, are we just stretching it out to hold more values than it was originally intended to hold? Do those values even exist, or are we assuming there is more to real numbers and irrational numbers are the enemy and the enemy of the number line! Kind of like entanglement, are we diving into the realm of spooky rational number action now too, with the irrational numbers doing the spooky space-time dance in infinite space?

    Good video of by Vi on anti Pi day –

    Point #2: Will too many ideas make my head explode? –

    Haha… will reading too deep into this make my head explode? =0

    As I’m sure you may already know. The modern computer was developed based on a few simple design principles. It was actually based on how people obtain, retain and process information, (think in the 1930s) which is why some parts of a computer are referred to as memory, CPU, so on and so forth. This can be related to biology, cell structures, reproduction thereof, and communication as well, but that is way beyond the point I want to make here.

    Individuality is an important trait (or quality) in humans. Just like everyone has their own unique finger prints or iris, everyone can form their own unique perspective about the world around them (and many end up coming to the same conclusions and building upon them). Everyone learns and retains information differently as well, but in my opinion it is usually information relevant to them and what they might find interesting. To draw this back to CS, there is a simple concept known as GIGO or Garbage in, Garbage out. Meaning you might keep or retain the relevant information and ignore the garbage even if it gets processed and an opinion is formulated about it. Think about reading or skimming through an online article or opinion piece. You might read it and the parts of interest might stand out, while the rest gets throw out and an opinion is formulated based on what someone might find interesting or fascinating.

    Which gets into basic topics of intelligence and making predictions, machine learning, people learning – If you know 2 of 3 values then you can determine the 3rd (x – y = b). Simplicity, and KISS – Keep it simple stupid; for designing AI and or other related design concepts in disciplines related to computer or software engineering.

    • Theoretical, particle, and similar physicists are all insane, while mathematicians are insane, sadists, or both. If you think this is an abuse of the number line, you should see what mathematicians have done to lines and curves (the poor things), or Abraham Robinson’s decision to re-introduce infinitesimals into calculus & analysis by extending the real number line to include the set of hyperreal numbers (imaginary was already taken, and I suppose “uberreal” was to German for an American mathematician).

      Do these values exist? That’s an incredibly loaded question as it includes issues from ontology (the ol’ nominalism debate, and I do mean OLD), the philosophy of logic and mathematics, set theory, and so forth. Rather than get too deep into that, there is a somewhat well-known quote from the great mathematician Leopold Kronecker: “God created the integers, all the rest is the work of man” (“Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk”, which is more literally and badly translated “The complete [set] of integers has God created, all else is man’s work”). It’s more popular thanks to Stephen Hawking, who used the first part as the title of one of his books. What even I didn’t know until a few years ago was the context: Kronecker was basically denying that there existed any irrational numbers, and in particular was dismissing a recent proof by (if memory serves) Cantor. If one traces the history of Western mathematics (acknowledging that much of its progress was made elsewhere first and either borrowed or independently discovered), one finds that this is rather the norm. Mathematicians have denied that 0 exists, that negative numbers do, that irrational numbers do, and of course the very name “imaginary numbers” suggests the skepticism with which such numbers were treated.

      The simplest answer is that unless we treat the real number line as we do (uncountably infinite and constructed most frequently in terms of limits or cuts through/on the rationals), and unless we acknowledge that the power set of a set (i.e., all possible subsets) is always strictly greater than the set itself (which means that the power set of the natural numbers has a “size” or cardinality equal to the real numbers and greater than the rational numbers) much of mathematics falls apart (not so much with the power set example, but the lack of a rigorous construction of the real number line became a serious problem for mathematicians and scientists). Without our rigorously (if horribly unintuitive) formulation of the real number line we loose most of probability theory, calculus & analysis (or at least a rigorous version), statistics, a good deal of physics, most functions, even the coordinate geometry that high school students use when working with graphs of parabolas or finding the slopes of “lines”.

      There are irrational numbers that are absolutely essential to mathematics, including applied mathematics. The numbers e and pi are of course the most well known and probably the most important, but there are others. And without e, we lose e^x (the exponential function) as well as its inverse (again, we actually loose all continuous functions, but the rational numbers are suitably dense for most purposes to get alone ok even if not on very firm ground; most of the progress in calculus occurred without a rigorous number line). In fact, if we do not allow the interval (0,1) to be uncountably infinite, we loose the normal distribution and again most of probability theory. Basically, even if one treats sets like the integers, the rationals, the reals, etc., as things that we can take or leave depending upon whether we need them (just like the way Robinson constructed the hyperreals to rigorously define infinitesimal calculus), it turns out we REALLY, REALLY need the reals as they are currently defined and all their insanely, seemingly impossible properties.

      The great thing about the modern computer is that it was “designed” mostly on paper, mathematically, before we figured out a way to implement the work by Turing and others. And despite the vast increase in computer power, the basic design principles you refer to allow us to continue to treat all computers as Turing machines, or more importantly to have computer science and computability theory (and related fields) remain unchanged by technology. In the end, the heart of any computer is the physical instantiation of logical operations: AND, NOT, NAND, etc. We can increase RAM, CPU cache sizes, hard-drive space, CPU speed, and a lot more (I’m not a hardware guy other than what’s relevant for assembler) all without requiring us to throw out the developments in computability theory, theoretical computer science, and so forth.
      KISS: the heart & soul of engineering, and kryptonite to philosophers.

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