The Status of the Infinite

A friend of mine who is a professor, and a specialist in the philosophy underpinning mathematics, told me that the major debate in that philosophy is over the status of the infinite.  You can see why:
 [T]he more that physicists stopped worrying about what their complicated equations meant and simply ran the numbers, the more progress they made. Some of their predictions have now been confirmed by experiments to 10 decimal places or more— the most accurate predictions in history.  
Real objects cannot have infinite charge or mass or whatever. But when scientists in the 1950s started calculating those quantities with their latest and fanciest theories, infinities kept sprouting up and ruining things. Rather than abandon the theories, though, a few persistent scientists realized that they could do away with the infinities through mathematical prestidigitation. (Basically, they started calculating with and canceling out infinity like a regular old number, normally a big no-no.) 
No one liked this fudging, but because it led to such stunningly accurate answers, scientists couldn’t dismiss it. In fact, the reigning paradigm in physics today—which describes the workings of invisible “fields” (similar to magnetic fields)— would not exist without this hand waving. And now physics is stuck with fields: they’ve become more fundamental to understanding the universe than mass or charge. Fields have become the very fabric of reality—even if our understanding of them relies on some unrealistic assumptions.
So what's the problem with infinity?  Let me offer a couple of starting points at getting at an answer to that question -- both of them discovered not in modern philosophy, but in Ancient Greek.

The first one is the problem the article cites -- theoretically, actual infinities shouldn't exist.  Aristotle explained why potential infinities could exist, and you probably know at least one of the arguments:  if you have a number, you can always divide it in half.  Thus one can be divided into 1/2, then 1/4, and so on forever.  The other kind he recognized was like this:  if you have an actual universe with the size of two, you can divide it into half, and then take the second half and divide it in half, so that now you have divide out one and a half from the total; and then again, so that you have divided out one and three quarter from the total; but you will never reach two.  Two is thus a kind of infinity, since you can never get to it; but it is not an actual infinity.  (As the Stanford article points out, however, this is not consistent with Aristotle's idea that the universe was eternal -- and that there must have been, therefore, an infinite number of days.)

These are not actual infinities, because you have really only the thing you are dividing and it is of determinate magnitude.  If the universe is not infinite, you should not be able to get to infinitely large magnitudes.  It is only through infinite division that there should be even a potential infinite.

(The concept we learned in grade school -- that given any number, you can always add one more -- is another kind of potential infinity.  It deals only with imaginary objects, not "real objects," which we would like to believe are finite).

The other starting point for the problem are Zeno's paradoxes.  Zeno's paradoxes of motion show that, if things are infinitely divisible, motion should be impossible.  Since a distance is a length, and a length is the kind of thing that should be infinitely divisible (one mile into half a mile, etc.), it should be the case that motion is not possible.  For say that you divide the line into an infinite number of points.  For motion to be possible, you'd have to pass from one point to the next.  But there is no "next" point if there is a true infinity, because you can always divide the distance between them in half.  Thus, between any two points are an infinite number of points.  (For further discussion, see the second paradox of motion.)

Likewise time should be divisible (a minute into half a minute, etc).  This provides its own problem, since we think motion takes place in time (indeed, Aristotle thought that time was the counting of motion).  So if we divide time down to the smallest possible increment, either a flying arrow is frozen in time and space, or it moves.  If it moves, though, our instant of time must have a start and a finish -- which means it should be further divisible, contrary to our assumption that this was the smallest possible division.  Therefore, we should get to a division in which motion isn't possible; but if time is composed of moments in which no motion is possible, how could motion be possible at all?  

The rest of Zeno's paradoxes, and the thinking of Parmenides and several others, posed real challenges for any system that includes infinities, but also for any system that includes multiple things.  The upshot of both is that reality only makes sense if it isn't really divisible, but finally unified.  Aristotle argues against this in Physics I, in part from the obvious rejoinder:  well, but we see motion all the time.  Thus, motion and time must be real; we all agree on it.

For many years we've followed Aristotle's basic solution by assuming that real (i.e., actual) infinities didn't exist.  Now, however, we find that we are able to make scientific predictions that are far more accurate than anything in human history... but only by assuming the infinite with real objects.

That's a problem.  It's a problem because it means that the most accurate science in the world is founded on assumptions that we have some good reason to think are impossible.

What does it mean if they're not?


james said...

One possible reason is that the model, though OK, is not arranged correctly for calculations.

For example, the series 1/2 -1/3 +1/4 -1/5 .... converges. If you look at it chunk by chunk, 1/2-1/3, 1/4-1/5, etc, you can see that you are adding successively smaller bits in a way that stays bounded. But if you reorganize the sum as (1/2+1/4+1/6+...) - (1/3+1/5+1/7+...) you find that you are subtracting one infinity from another. The only difference is the way the sums are arranged.

BTW, I think that the concept of boundedness vs unboundedness is what you're looking for in describing "real (i.e., actual) infinities." You can divided up the number line between 0 and two into infinitely many subsets, but the distances are bounded.

Grim said...

The two concepts are very closely related, yes. However, note that the bounded version relies on an actual infinity of divisions in Zeno's paradoxes: it really must be the case that the line can be infinitely divided, or that time can be. It isn't just -- as for Aristotle -- that the line can conceptually be so divided.

I believe it's Averroes who makes this point most explicitly, in his commentary on Aristotle's metaphysics. It isn't actually possible to divide a line infinitely, because it isn't possible -- even in theory -- to divide a line "at a point, and the point next to that point." Because there are an infinite number of points between any two points, such a division is impossible. Thus, the line is potentially infinitely divisible, but not actually infinitely divisible. This is true regardless of the bounds of the line.

AndrewC said...

Hi Grim, long time lurker.

Engineering background, not much formal philosophy learning. Found your description of the two infinite division parodoxes interesting.

Motion is distance divided by time. If distance is infinitely divisible, and time is infinitely divisible, is not motion just infinity divided by infinity?

(Now you can throw in acceleration to make things *really* confusing!)

Grim said...

Great to meet you. Thanks for speaking up!

It's certainly true that time and space (i.e., not just "distance" but all extant distances) need to mate up in some fashion, but there are a couple of problems with that approach that come up quickly.

Let's say I've got two infinite things. We'd like to say -- as you appear to be saying -- that we can make a fraction of ∞/∞, and treat that as 1 for calculations.

But let's say that I add 1 to the second infinity before we make the division. Now, in principle, we shouldn't get 1 when we divide. But it's not just one we could add; we could add any number. In fact, we could add a second infinite set. So now you'd have, in principle, ∞/2∞, which we might want to say works out to 1/2; unless two infinite sets are only infinitely large.

Here's another problem: if a set is truly infinite in this sense, we can divide it without it losing its size. Say you have a line with an infinite number of points. Now divide it in half. Each line still has an infinite number of points. So in one sense the divided line should contain half as many points as it did before it was divided in two; but in practice, the number of points in each of the newly divided lines is equal to the number from the original line. That seems pretty puzzling.

Now, another thing that modern and contemporary physics does that you're also doing is treating time as a thing like space; some physics treats it as literally a fourth dimension, which can be handled and graphed spatially. There are a lot of advantages to this, especially in relativity theory; but there are also some good philosophical reasons to be uncertain if that's the right way to think about time. St. Augustine brings up several concerns about time that make him think that time isn't a physical thing at all. We can save that for another day, but I want to raise it just so you're aware that the concern exists.

AndrewC said...

Another weird thing with the numbers you're playing with:

You have to be able to add up the infinite number of points and end up with a finite number. (Whether in time or in distance)

The only way this works is by defining the size/width of the points with a denominator of infinity:

I have an (inf) of points that are 1/(inf) inch wide, for example.

The total width of the points summed together would then be (inf) * 1/(inf) = 1 inch. Each individual point is practically 0 - yet you can add them up to end with something!

Speaking of the oddity of time, our only method of measuring time is based on the motion of objects. The day is defined by the earth's motion relative to the sun, the second is defined by counting the oscillations (motion) of some element. It's a funny relationship between these concepts.

Grim said...

The problem is that you can't do that with points, because then they would be touching. If they are touching, though, there isn't a point in between them.

We could say, "Well, that's just the way it has to be," but we then run into Zeno's problem with the "infinitely small moment" that nevertheless has a start and a finish. An infinitely small magnitude is still divisible if it has a start and a finish; thus, there must be a point between the start and the finish. Therefore, points cannot have the magnitude of 1/∞.

Rather, as Euclid teaches us, they must be without extension. (It is worth noting that the graphing of time as a fourth dimension follows Euclidean forms most of the time; and where it departs, it isn't on this point, but rather on the question of the curvature of space).

If you're interested in puzzling over the relationship of time and motion, you might want to start with Aristotle's Physics Book 4. Scroll to Part 10; it's only the last three parts that deal with the subject directly.

Grim said...

By the way, your idea of a "time atom" that isn't infinitely divisible has an advocate in the ancient world. He isn't an Aristotelian, though, but a Neoplatonic philosopher named Proclus. You might want to follow your Aristotle reading with some research into him, though the Neoplatonic tradition is very different from anything familiar to us today.

However, if it interests you, I tend towards Neoplatonism myself, and would be happy to help you work through it.

james said...

Heisenberg's Uncertainty Principle might suggest a granularity in position/momentum space, though not necessarily in either by itself.

Grim said...

It's interesting that you raise that point at this time, James. Heisenberg was particularly influenced by Plato, though very widely read in philosophy generally. Chapter Eight of this book begins and ends with Platonic influence on his thinking.

douglas said...

Andrew's last comment starts to get to the essence of it, I think. That's in a conceptual way- if you use infinity in the denominator, then you eventually need to put it backin the other side to cancel it out- but of course, you guys are discussing the dimension of a point, which by definition, has no dimension- it is a singularity (a line, one; a plane, two; a volume, three).

If in the abstraction of mathematics, you can work to infinity, but in reality you cannot, you eventually will need to add in a corrective to bring the abstraction back to reality- cancelling out infinity seems intuitively a no-no, but it makes sense if it is separating the abstraction of the calculation from it's real application. Once you see that you get the most accurate results ever, it seems 'correct'. Let's keep in mind however that Newtonian physics still works very well for common physics problems, but is in fact not accurate- it's just that it doesn't break down until things get really high energy or otherwise unusual.

Grim said...

dimension of a point, which by definition...

We're actually at one level behind that: we're discussing why a point is so defined. There's a metaphysical argument for why, conceptually, it has to be without extension: the 'start/finish' argument, above.

But there are others who believed that finally that wasn't workable. These people were called the 'atomists,' and this is just why they believed that reality could not actually be composed of points, but had to be composed of some final minimal quantity. Proclus decided that time had to be of the same nature, and thus, 'time atoms.'

However, there are also some very good arguments against the atomists, so!

MikeD said...

I sometimes wonder if our concepts of (and difficulties with) infinity are a matter of perception and not one of reality. This is a bit esoteric, so forgive me if I fail to make sense.

We all "know" thanks to Einstein that the speed of light is an absolute limit on velocity (and that it takes infinite energy to achieve it, since mass approaches infinity as you accelerate towards it. Now, if you can cancel out those infinities, suddenly the speed of light is achievable. And clearly it is, light travels at that speed all the time.

But I think more importantly, Einstein's theory of Relativity is based on the ASSUMPTION that nothing can travel faster than light. Why? Because we cannot perceive it so. Anything that travels faster than light APPEARS to travel backwards in time, as we can only observe (with our senses) the after effects.

Now imagine some far off world, where the sentient race is blind, and perceives everything audibly. Echolocation and such. Their senses of perception are thusly limited to the speed of sound. Imagine a space ship from Earth travels through their atmosphere faster than the speed of sound. They would be unable to perceive the actual location of the ship, only where it HAD been. In effect, they could only "see" the after effects of it's flight (the sonic booms), and it would, to them, appear to travel backwards in time.

Is it possible that there we're confusing the limits of our perception with universal limits? We can already verify that there are some events in the physical world that violate our assumptions about how the universe works. We've observed the after effects of tachyons that seem to have traveled faster than light. I am aware that we've managed to cause photons to "jump" across a gap and arrive "before" they left. Is it time travel? Teleportation? Or is it simply that we're confusing the limits of our perception with universal laws? After all, while the math backs up Einstein, math based on faulty premises can yeild consistent results, but yet those results can be consistently wrong nonetheless.

Grim said...

There are some practical experiments we've tried to see if light speed can be exceeded, though. Your sound-only race, for example, could track the sonic boom as it occurred in different places, and they would get an accurate picture of the direction of travel. By the same token, we used particle accelerators that give us a defined starting point and a defined ending point; if you release a particle from Here and it gets to There before a light particle should have traveled the same distance, you may have a case of FTL.

Now, if it shows up There before you release it, then you've got backwards causation (for which there is some evidence in quantum physics, although I gather the field is still debating the issue). Backwards causation isn't a problem in theory -- it would just mean we were perceiving time differently than it exists, which we already accept to be true from relativity theory. Relativity theory graphs of the type I was talking about above show how two events can be simultaneous in one frame of reference, but not in another.

However, because we haven't yet been able to establish FTL travel, we also work "light cones" into these graphs to show what it is possible to perceive. At least at the levels where we still like to use relativity for generating predictions, it seems to hold so far -- the predictions work, even if we don't know exactly why, and can't believe in the things we need to believe in order to make it work. That's a huge problem, but it still leaves us with a workable system of navigation to use while we try to sort out the truth.

Anonymous said...

The thing that always bugged me about calculus is that it is a cheat: you are using the measurement of areas with angles to calculate the position of a curve, and then you draw the smooth curve you knew was there, anyway, instead of the many-sided area.

So it's very satisfying to me to know that there are some calculations that require both the insertion and deletion of the definition of infinity to get at a result.

Serves 'em right.


karrde said...

One potential problem--at least from a perspective of pure mathematics--is that not all infinities are the same size.

The discussion is interesting, but only to people who deal with sets of infinite numbers on a regular basis.

(Amusingly, the best description I've ever seen of this subject is an explanation of a joke in a webcomic. The webcomic is made of photographs of Lego minifigures, and has several dozen potential subjects...)

It appears that when physicists cancel out infinite values, that they are making sure that they have the correct orders of infinity.

On the general concept of assuming the impossible...I think it means that physics/science/math can't produce a perfect model of the Universe. It can produce good models, but those models might require things like assumptions of infinite mass or infinite charge.

Unless such things aren't actually impossible.