Writing this blog, I get lots of email. One of the things that I get over and over again is a particular kind of cluelessness about the idea of infinity. I get the same basic kind of stupid flames in a lot of different forms: arguments about Cantor’s diagonalization; arguments about
calculus (which I’ve never even written about!); arguments about
surreal numbers; and worst of all, arguments about nullity.
There’s a fundamental bit of foolishness that underlies all of the flames. Infinity is not a number. It’s a mathematical concept
related to numbers, but it is not, not a number.
The most recent version of this is an email from last week, titled “Nullity and concerning your ignorance of it”. It’s pretty typical of the basic confusion that follows from treating infinity as a number.
I was searching the internet for stuff on Nullity, this new number that I noticed you think is crank. Since internet arrogance and ignorance are at an all time high, I thought I might share some explanation of it as I’m not sure if my response was presented on your site.
First off, 1/0 is infinity because if you were to divide 1 into zero pieces, you would need to divide an infinite number of times. Same goes for -1/0 .
Now, what is interesting about infinity is that when you add or subtract a number from it, infinity remains. When you subtract infinity from infinity, something else happens entirely: the two concepts stack up with each other. Thus, [1/0 - 1/0] gives you 0/0 as the denominators need not change, obviously. For simplicity’s sake, just think of the positive and negative signs as cardinal directions when dealing with infinity.
Now, what this means is that both the positive and negative directional infinities are being represented, and that is the totalization of the catesian plane, or nullity. You can think of it as the limit to the cartesian plane, or all of the cartesian plane, or none of the cartesian plane… whatever suits your ship. Nullity is the summation of all sets of infinity.
Hope this helped.
You can see the confusion right away: “1/0 is infinity because …”. Bzzt. No. 1/0 is not infinity. 1/0 is nothing. 1/0 isn’t defined in our number systems: it’s not a number. In fact, it’s not just not a number, it’s nothing. It’s a meaningless expression. Asking what 1/0 is is like asking “What’s the square root of a nice juicy plum?”. Or what predicate makes the logical statement “∀x: P(x)∧¬P(x)” true?
If you treat infinity as a number, you fundamentally break everything that makes arithmetic work. For example, the most basic definition of numbers that I know of is Peano arithmetic. Peano arithmetic is a set of axioms that defines how the natural numbers work. It’s the set of axioms that are typically used as the fundamental basis of a formal definition of numbers. One of the Peano axioms says that for every natural number, there is exactly one natural number that is its successor; and every natural number except zero is the successor of exactly one natural number.
What’s the successor of infinity? Or to phrase it a slightly different way (by using the Peano definition of addition), what’s ∞+1? As my clueless correspondent says, “what is interesting about infinity is that when you add or subtract a number from it, infinity remains.”
So, ∞+1 = ∞. And ∞+1+1 = ∞. And ∞+1+1+1 = ∞.
And there went the Peano axioms, right out the window. The failure of
the Peano axioms isn’t some trivial, obscure theoretical issue. If the field axiom fails, then every proof about the natural numbers, every statement about how the natural numbers work, loses its validity. Nothing is safe. 1+1=2? Nope: field axioms say that if x=y, the
x+z=y+z. Let z=∞. Then ∞+1+1 = ∞ + 2
. ∞+1 = ∞, so then ∞+1 = ∞+2. Remove the ∞ from both sides,
and 1=2. But wait, you say, you can’t remove the ∞ from both sides! In infinity is a number, if 1/0=∞, then yes you can.
The natural comeback to that is something like “Well, so ∞ isn’t a natural number, but it’s still a number.”
Still no good. First, we normally define numbers using the Peano naturals as a starting point. But even if we don’t, if we start with some other construction, most of the math we do with numbers ultimately relies on the fact that numbers form a field. Whether you’re looking at
rational numbers, real numbers, complex numbers, or whatever, they form
a mathematical structure called a field. Fields are defined by
a set of fundamental axioms. If infinity is a number, then the field axioms
fail – and if the field axioms fail, then pretty much everything that we do with math – every proof about numbers, every numerical fact,
it’s all rubbish.
The typical comeback to this is something like “So ∞ isn’t a number, but it’s still something, and 1/0 = ∞.” Nope, still no good. The field axioms define division in terms of the multiplicative inverse,
and both multiplication and the multiplicative inverse are closed – meaning that you can’t get a value outside of the real numbers from anything defined using multiplication or the multiplicative inverse in the real number field.
But wait, you might say, I distinctly remember talking about infinity in calculus class: limx→∞1/x=∞!
Limits aren’t really talking about numbers, they’re talking about curves. When we talk about infinity in limits, we’re talking about trend lines, not necessarily numbers. Some limits
trend towards a number; some limits don’t follow a trend at all;
and some follow a trend towards an unbounded increase. That last one
is what we mean when we say a limit trends towards infinity.
There’s an easy illustration of what I mean when I say that a limit
talks about the trend of a curve. Think of one of the simplest curves: y=1/x.
What’s the limit of 1/x as x approaches 0? That’s not a meaningful question. You need to state from which direction you’re following the curve. If x≥0, then limx→01/x=+∞;
if x≤0, then limx→01/x=-∞. We’re not talking
about numbers there, but about the direction of an unbounded trend. When
we say “+∞” there, we’re talking about the fact that the curve increases without bound in the positive direction. When we say that the
limit is -∞, we’re not saying that the curve converges on a specific number called -∞; what we’re saying is that the curve increases without bound in the negative direction.
∞ isn’t a number. If it were, it would break the fundamental axioms that define numbers. Nullity is even worse; it breaks even more of the fundamental axioms of math. The guy who came up with
nullity is a true idiot: in the presentation shown by the BBC in which
he demonstrates the supposed properties of nullity, he uses several steps that only make sense under the field axioms, which are violated by nullity. So the results are fundamentally wrong in a very strong way: if you define nullity as he defines it, then you can use the existence
of nullity to prove statements like 1≠1.
It just doesn’t work. There’s no way of taking values representing true infinity, and turning them into numbers. You can do things like
Cantor’s transfinite numbers, but they get very strange very quickly – and
they don’t work the way that we expect numbers to work – for example, you need to distinguish between cardinal numbers and ordinal numbers, and you don’t get fractions. And they don’t even really behave like
infinity! For example, if we treat infinity as a number, there’s no number
greater than infinity. But for Cantor’s transfinite numbers, if N is a transfinite number, there’s always another transfinite number larger than N.
And even with transfinites, you can’t cause an expression like 1/0 to
have any meaning. 1/0 is fundamentally undefined. It’s not ∞. It’s not ω. It’s not nullity. It’s not anything. The moment you see something use the statement that 1/0=∞, you know that they’re an idiot who doesn’t really have the slightest clue of what they’re talking about.