“How language shapes thought and thought shapes language” is an age old question in linguistics and philosophy. I’m not in any position to give a definitive answer, nor, I suspect, is anyone else. Having taught math and physics at the university level, I am willing to offer some thoughts about how the language of mathematics and the symbols and glyphs used to turn mathematical concepts into written words shape how people think and solve problems.
In this blog I will be focusing in the concept of infinity and the philosophic implications that come from using it. But before I get to infinity directly, I would like to discuss, by way of a warm-up exercise, how the use of the symbol ‘x’ throws off a lot of beginning students.
When describing a function or mapping between two sets of real numbers, without a doubt, the most common notation that teachers use is to allow the symbol ‘x’, called the independent variable, to be any member of the initial set, and the symbols ‘y’ and ‘f(x)’ to be the corresponding member of the target set and the function that generates it. The symbolic equation ‘y = f(x)’ becomes so rigidly fixed in some students minds, that the idea that the symbols ‘x’ or ‘y’ could be replaced with any other symbol, say ‘y’ and ‘z’, never occurs to them. I myself have experiences of students coming and asking if their book has a typo when it asks them to solve ‘x = f(y)’ or integrate ‘f(y) dy’ or the like (once this happened while I was out to dinner at Olive Garden with my family – but that is a story for another day).
There is no easy way to fix this problem as there is a kind of catch 22 in the teaching of mathematics. One on hand, the mapping between sets exists as a pictorial relation between ‘clouds’ and ‘the points within them’
without the need for written glyphs. One the other hand, an initial set of well-defined symbols keeps initial confusion to a minimum and allows the student to focus on the concepts without all the possible freedom of choice in notation getting in the way. (Note: a reader comfortable with classic philosophy may point out that a mapping between sets can be abstracted even further, perhaps to the notion of a Platonic form, but this is a side issue.)
Okay, with the appetizer firmly digesting in our minds, let’s turn to perhaps the most confusing symbol in all of mathematics, the symbol for infinity, ‘$\infty$’. This symbol, which looks like the number ‘8’ passed out after a night of heavy drinking, seduces students and instructors alike into all sorts of bad thoughts.
How does it have this power, you may ask? Well, its very shape, small and compact and slightly numberish, encourages our minds to treat it like all other numbers. There are literally countless examples of infinity masquerading as a plain number, much like a wolf in sheep’s clothing. One of the most egregious examples is the innocent-looking expression
\[ \int_0^\infty e^{-x^2} dx = \frac{\sqrt{\pi}}{2} \]
where ‘∞’ is compared side-by-side with the number ‘0’. There is perhaps a more palatable way of writing the integral as
\[ \lim_{a \rightarrow \infty} \int_0^a e^{-x^2} dx = \frac{\sqrt{\pi}}{2} \]
but it still looks like the number ‘a’ can be thought of as becoming or approaching ‘∞’. A seasoned practitioner actually knows that both expressions are really shorthand for something much more involved. I will summarize what this more involved thing is in one short and sweet sentence. Infinity is a process that you have the freedom to perform as many times as you like. Or even shorter: Infinity is an inexhaustible process.
Take a moment to think that through. Are you back with me? If you don’t see the wisdom in that maxim consider either form of the integral expression listed above. In both cases, what is really being said is the following. Pick an upper bound on the integral (call it ‘a’ to be consistent with the second form). Evaluate the integral for that value of ‘a’. Record the result. Now increase ‘a’ a bit more, maybe by doubling it or multiplying it by 10 or however you like, as long as it is bigger. Now evaluate the integral again and record the result. Keep at it until one of several things has happened: 1) the difference in the recorded values has gotten smaller than some threshold, 2) you run out of time, or 3) you run out of patience and decide to go do something else. The term infinity is simply meant to say that you have the freedom to decide when you stop and you also have the freedom to resume whenever you like and continue onwards.
If you are new to calculus, you will no doubt find this short sentence definition somewhat at odds with what your instructors have told you. Where are all the formal limits and precise nomenclature? Where is all the fancy machinery? If you are an old hand at calculus you may even be offended by the use of the words ‘process’, ‘freedom’, or ‘inexhaustible’. But this sentiment is exactly at the heart of the Cauchy delta-epsilon formalism, and the casual nomenclature has the advantage of ruthlessly demolishing the ‘high-brow’ language of mathematics to bring what is really a simple idea back to its rightful place as an everyday tool in the thinking person’s toolbox.
On the other hand you may be thinking that everyone knows this and that I am making a mountain out of a mole hill. If you fall into that camp, consider this video about the properties of zero by the Numberphiles.
I must admit I like many of the Numberphile’s videos, but this one made me shake my head. They allowed language to affect their thinking, and they were seduced by the evil camouflage powers of infinity. They go to great trouble to explain why you can’t divide by zero, and they note that people say “isn’t dividing by zero just infinity?” and they point out it isn’t that simple.
The problem is, it is that simple! Dividing by zero is infinity as understood by the maxim above. The Numberphiles prove this fact themselves. At about a minute into the video, one of their lot begins to explain how multiplication is ‘glorified addition’ and division is ‘glorified subtraction’. The argument for ‘glorified subtraction’ goes something like this.
If one wishes to divide 20 by 4, then one keeps subtracting 4 until one is lefts with a number smaller than 4 (in this case zero). The number of times one engages in this subtraction process is the answer, with whatever piece left over being the remainder. So dividing the number 17 by 5 is a shorthand for subtracting 5 from 17 three times and finding that one has 2 leftover. So one then says 17/5 = 3 with a remainder of 2.
The same bloke (I use bloke because of their cool English or Australian or whatever accents) then says that 20 divided by 0 goes on forever because each time you subtract 0 you are left with 20. Here then is the inexhaustible process that lives at the very heart of infinity. Sadly, while he looks like he is about to hit the bull’s-eye (at 2:20 he even says infinity isn’t a number), his aim goes horribly awry at the last moment when he objects to saying that the expression ‘1/0 = ∞’ can’t be true because one could then go on to say ‘1/0 = ∞ = 2/0’ from which one can say ‘1=2’.
This is, of course, a nonsensical objection since the expression ‘1/0 = ∞’ is a shorthand for saying ‘the glorified subtraction of 0 from 1 (in the sense used above) is an inexhaustible process.’ It is no more meaningful to say that this process is the same as the ‘glorified subtraction of 0 from 2’ as it is to say that ‘1/0’ is the same as any other inexhaustible process, like halving a non-zero number until you reach zero.
The fact that the words ‘0’, ‘1’, and ‘∞’ and the sentence ‘1/0 = ∞’ result in an illogical conclusion is an important warning about the power of language to shape thought. The Numberphile guys had all the right ideas but they came up with a wrong result.