Tuesday, November 19, 2013

About numbers: A series from notes about infinity, I

Today was a fun day. Philosophy professor and best-selling author Peter Boghossian, Manual For Creating Atheists, invited me to speak with his Atheism class at Portland State via Skype. The conversation was a lot of fun, and I hope I'm right in feeling that it went really well--infinity is a fun concept to introduce to students at all levels of mathematical background, including little to none. Not knowing fully what to expect, I had prepared some notes ahead of time that ended up seeing little or no action, so I've decided to turn them into a short series of blog posts about infinity.

The original notes were geared toward addressing five questions, four being of general inquiry to get Boghossian's students thinking about numbers and infinity. The remaining question focused more narrowly on an essay from Evangelical Christian apologist William Lane Craig's Q&A portion of his Reasonable Faith website, which was also assigned as background reading for the discussion. Here is a link to Craig's Q&A, #325, "God and Infinity." Please note that my responses to the questions assumed the background reading, and thus some address Craig in addition to being straight answers on the given questions.

In this series, I'll break up my notes into a few posts, first some that deal with my responses to these questions and then some that address Craig's Q&A response directly. I anticipate four posts on this topic, this being the first of those.

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1. Where do numbers come from, and what are they?

Craig calls numbers “useful fictions” and “abstractions,” and I would argue that this is a correct assessment.

Let’s talk about where numbers come from, though, so we can understand why. Numbers come from abstracting the common enumerative properties encountered when counting the contents of collections that contain the same number of things. That’s a lot of words that sound intense but can be captured pretty easily by a pretty simple idea (that rapidly starts getting complex the more general we get).

Simply put, numbers are for counting; that’s where they come from. If I have three of something, say three pens, three balls, three houses, three cars, three stars I’m looking at--three of anything--then we can identify the “common enumerative property” of threeness that enumerates each of those collections. The things don’t even have to be the same kind of things. I could have a pen, a ball, and a house, and those three things grouped together still exhibit the property of threeness.

Notice that threeness is not a property of the things being grouped together but rather of the collection of the things. Now notice further that the collection of things that isn’t itself a "thing," as nothing requires us to physically group the objects together.

The notion of the group of things is kind of an abstract construct that ties the three things together. Maybe it’s three balls in a pile, three letters in a word, three words in a phrase, three sentences in a paragraph, or a group of three objects--like the pen, ball, and house--that have no obvious connection other than that they’ve been grouped together. Maybe there are no objects at all, like the grouping of three numbers, three thoughts, three ideas, or three groups of other things. What we’re experiencing here is the process of abstraction, and the group, which is an abstract notion applied to things, is what has the property of threeness.

Hopefully, it feels very much like “threeness” is an abstraction on its own merits, without even having to remind ourselves that it applies to every collection of three objects in exactly the same way. This is what is meant by the common (as in, shared in common) enumerative (meaning, "that counts the things") property that defines what a number is. Numbers are abstractions for counting. Thus, Craig calling numbers "useful fictions" and "abstractions," is pretty near the mark, even if those two things aren't philosophically identical.

Understanding numbers

To address the number directly, "three" is the idea that expresses the property of threeness. This is true for all other counting numbers as well: four expresses fourness, one thousand expresses one-thousandness, and so on.

This understanding of numbers can be taken right down to the barest bones of how we understand numbers. If we count three things, what we have is one thing, plus another thing, and then those together plus yet another, symbolically maybe expressed with three capital letters I: III, as in Roman numerals. I+I+I=III; that is, one plus another plus yet another is what exhibits "threeness," the property that is held in common between all collections of three things (or abstract non-things).

We can see this property is held in common by all collections of three objects by using a one-to-one correspondence in which we match each of the prototypical capital letters I in III with each of the things in our collection of three things. They match one thing to one thing, leaving none out on both sides of the matching, so we say the match is "one-to-one." If two collections of things can be matched one-to-one, we agree that they answer the "how many?" question identically, using the technical term cardinality and the informal phrase "number (of)" to express that idea. For example, the cardinality of the set {a,b,c} is three, or we might say that the number of items in that set is three. Cardinality means "number of" in all finite cases but is more general and can thus account for infinitely large sizes as well.

What about fractions and decimals?

So, at bottom, numbers are for counting, but they also do more than that. What about measurements, for instance, like 15.2 inches? All of the measurements we’ll ever make, because all of our measuring devices are only so accurate, will return fractions of whole numbers of units that could be expressed in terms of whole numbers of units. 15.2 inches is 152 tenths of an inch, for example. This feels like cheating, but we have to put the decimal place somewhere.

Another way to look at this is that if we are given a unit of some kind, we can either count the unit itself or break the unit into equal-sized pieces to create a new unit and then count those. This will provide us with all of the fractions (rational numbers) of that unit if we so desire, thus defining the rational numbers. For instance, 47/23 is the proportion that represents taking an initial unit of measurement, breaking it into 23 equal sized pieces, and then counting out the length of 47 of those smaller pieces. We have 47 things, where each thing is the original (arbitrary) unit broken into 23 equal pieces.

What about irrational numbers?

Things get more complicated when we seek to include irrational numbers, like the famous constant pi that measures the ratio of the circumference of a circle to its diameter, but these numbers can be accounted for using the calculus. In very brief, we can understand these numbers to be predictions of what we would measure if we were to continually make better and better measurements to more and more decimal places, though they themselves cannot be expressed in numbers of units.

For whatever it is worth, for these numbers to make sense, since each has an infinitely long decimal expansion and cannot be written as a fraction of whole numbers (by definition), we need to have the notion of infinity at our disposal. Without it, the best we can do is approximations of these irrational constants (which implies no perfect circles exist and that a perfect square cannot have its diagonal measured out exactly--uncomfortable consequences).

To talk about numbers leads us to wonder about infinity.

Already, then, we hit two ways in which we need to invoke infinity to understand what we're working with. Since there is no largest natural number--any claimed largest is smaller than itself plus one--we need infinity to account for all of the natural numbers. Additionally, to understand real numbers at all, we're committed to them having infinitely many decimal places. We will leave off here, though, with our discussion of what numbers are.



2. What does infinity mean, and is it a number?

Infinity can be taken to mean “without limit” or "an inexhaustible potential for more." The formal definition of infinity now is more precise but maintains this notion as a corollary. The current definition is that a set is infinite if when an object is removed from it, the resulting subset is in one-to-one correspondence with the original set. Thus, to jump ahead, infinity is not a number because numbers do not keep the same value after having one subtracted from them.

The modern definition of infinity is not as simple to understand as it may seem at first. The definition says that we are able to take an infinite set, say the natural numbers for instance, and take away some element(s) from it to get a new set that can be put in one-to-one correspondence with the original. Remember from above that a one-to-one correspondence means that every element in the one set can be matched uniquely with an element of the other set so that neither set leaves any out--the elements are matched one-to-one.

One-to-one correspondence

This is very difficult to conceptualize, but here is the general idea. Imagine that we use the natural numbers, {1,2,3,4,...} and then the set of natural numbers modified so that we have removed the element 1 from it, {2,3,4,...}. We can create a one-to-one matching between these two sets by matching up 1 with 2, 2 with 3, 3 with 4, and so on, like this:
1---2
2---3
3---4
...
Notice that in the left-hand column, we will eventually see every natural number, {1,2,3,...}. Likewise, on the right-hand column, we will eventually see every element in the set of modified natural numbers {2,3,4,...}. What this means, really, is that however far out we look on either list, we will find a number in each column corresponding to the stated sets along with a one-to-one matching between the two sets.

Generally speaking, we're matching up the natural number n with the number n+1 in the set of modified natural numbers. Because this pairing is unique for each and every natural number with a unique match from the modified natural numbers, we have a one-to-one correspondence. This shows that the cardinality, which is the generalized notion of "how many," for the two sets is the same. In plain language, we have taken an infinite set (the natural numbers), removed an element (1), and ended up with a new set that is the same cardinality (same answer to the "how many are in here?" question). Thus, we can say that these sets are infinite in cardinality and that they have the same cardinality.

Infinity is not a number

Thus, infinity does not work like a number and, more plainly, is not a number. Given any number, subtracting one makes a smaller value. Given infinity, subtracting one gives the same value. In reverse, we see, then, that adding one also won't change the value of infinity. Doing so over and over won't change the value either, so we could add or take away as many as we wanted from infinity, so long as that value is finite, and would not change the quantity present by doing it. As it turns out, multiplying and dividing do not change the value of infinity either, though it starts getting a bit far afield to go into the details of this fact (the interested can imagine removing all of the even numbers from the set of natural numbers and looking for a one-to-one correspondence between the natural numbers and the leftovers, the odd numbers).

Infinity, then, can be seen as being "an inexhaustible potential for more," or it can be seen as "that which is so 'numerous' that removing one doesn't change the quantity." I'll close the discussion on this question here.

Isn't this supposed to be about God?

Well, yes. Besides the fact that Craig's Q&A behind all of this is titled "Infinity and God," I'd like to point out that it is Catholic dogma that God is intrinsically infinite. Other theologians throughout time, including presently, identify infinity with God, either by simply calling God "infinite" without clarification or by giving him properties that are infinite in scope like omnipotence and omniscience. Thus, theologians seem to insist that to talk about God with any authority would require some understanding of the infinite. Hopefully already that makes my readers think that maybe, given what we just covered, the theologians' hands are tied by this insistence. At any rate, that's a big part of what I wanted to show in my new book, Dot, Dot, Dot, that created the invite that led to the notes that led to this post.


There will be more! In the next post in this series, I hope to answer questions 3, 4, and 5 from my original list:
3. How can infinity be thought of as a quality and as a quantity?
4. Can the quantitative aspect of infinity be removed from the concept of infinity?
5. How is Craig's thinking muddled on the idea of "potential" infinities on a future timeline if God is supposed to be separate from the universe and eternal? Particularly, how is Craig treating God in a local frame when his God hypothesis requires a global one? (See Craig's Q&A #325 for context.)

In the meantime, do me a favor and consider picking up Dot, Dot, Dot: Infinity Plus God Equals Folly, and do yourself a favor and get Peter Boghossian's Manual For Creating Atheists.


Edit: The next post in this series is available here.

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