“To understand this for sense it is not required that a man should be a geometrician or a logician, but that he should be mad.” - Thomas Hobbes

Mentally Divergent

Greetings. That is, to anyone who might be reading this. I’m known as ObsessiveMathsFreak to some, and this is, as you’ve probably guessed, my “weblog”. I intend this weblog to be a sort of dumping ground for any interesting mathematical ideas, problems and solutions I come across. My own interest is towards applied mathematics and mathematical physics, so this paticular mathematics site will be of that “flavour”.

I had intended to write my first entry on a completely unrelated topic, but after some consideration, I thought it might be best if I began by explaining the context of the quote on the site logo, by the 17th century English philosopher, Thomas Hobbes.

The quote is Hobbes’ now famous response to an apparent mathematical paradox proposed by the Italian physicist Evangelista Torricelli, of ‘Torricelli vacuum’ fame. In modern mathematical notation, the apparent paradox ran as follows.

Evangelista Torricelli
Consider the graph of the function \frac{1}{x}. Torricelli first asked; what is the area under this graph, and above the x-axis, from x=1 to \infty? This area is denoted by the shaded region in the following graph.

Graph of Area Under 1/x

To obtain this area, we use the fact that the area under a graph, down as far as the x-axis, A is given by the formula \textstyle A=\int{}_1^{\infty} \frac{1}{x}\:dx

Using the fact that the anti-derivative of \frac{1}{x} is \ln(x), \textstyle A=\ln(x)\mid^{\infty}_1\Rightarrow A =\ln(\infty)-\ln(1)

\ln(x) keeps on getting bigger and bigger forever, i.e. grows without bound, and so informally we can say that \ln(\infty) \equiv \infty. Also \ln(1)=0 and thus we find that A\equiv \infty. Shockingly informal mathematics of course, but the result is valid. namely that the area under the curve is “infinite”.

Now for the paradox. Torricelli next asked, what is the volume of the “cone” generated by rotating this area about the x-axis? The cone is, poorly, shown from a slightly tilted perspective in the following diagram. We might be quick to say that “obviously” the volume is infinite, but lets calculate it anyway.

1_over_x_rotated.png

This cone, sometimes called Gabriel’s Horn, is a “Solid of Revolution”, as it is obtained by revolving the area under the curve around the x-axis. We can obtain the volume of this solid by the following method. Consider the area of one of the circles in the cone shown above. Remember the cone is viewed here from a slightly tilted angle, so we are looking “down” this transparent cone.

The radius of the circle at a distance x along the x-axis is simply the height of the rotated area at that point, i.e. \frac{1}{x}. So the area of one of these circles is simply A_c(x) = \pi r^2 = \pi (\frac{1}{x})^2 = \frac{\pi}{x^2}.

So to obtain the volume of the entire cone, we can simply integrate the areas of these circles A_c(x) from x=1 \mbox{ to } \infty. So the volume of the cone \textstyle V = \int_1^{\infty} A_c(x) \:dx = \int^_1^{\infty} \frac{\pi}{x^2} \:dx = \pi \int_1^{\infty} \frac{1}{x^2} \:dx .

The anti-derivative of \frac{1}{x^2} is \frac{-1}{x}. And so
V = \pi (-\frac{1}{x}|^{\infty}_1) = \pi (-\frac{1}{\infty}-(-\frac{1}{1}))

\frac{1}{\infty} \equiv 0 as \lim_{x \to +\infty}\frac{1}{x} = 0, and so

V = \pi ( 0 +1) = \pi

So, apparently, by revolving an infinite area about the x-axis we obtain a finite volume V=\pi. Faced with this proof, Hobbes could respond only with wit, and so we have;

Thomas Hobbes

“To understand this for sense it is not required that a man should be a geometrician or a logician, but that he should be mad.”

Why the paradox? Well, paradoxes are quite common in mathematics as soon as the concept of “infinity” is introduced. This is partly because of a poor understanding of infinity, and of course the fact that infinity is not a number, and so to talk of it in a mathematical context is to invite peril. Can we rectify this? Partially.

Rigorously speaking, it is incorrect to write \textstyle A=\int_1^{\infty} \frac{1}{x}\:dx. It is more correct to write this formula as \textstyle A=\lim_{R \to \infty}\int_1^{R} \frac{1}{x}\:dx. Working down as before, we would obtain;
A=\lim_{R \to \infty}\mbox{ }\ln(R).

As R increases, we can see, beacuse of the behavior of \ln(R), that A will keep on getting bigger and bigger, i.e. grows without bound, as we increase R. So we can say that with “infinte” R, the area A would be “infinite”.

Applying the same logic to the previous volume equation, we find that \textstyle V = \lim_{R \to \infty} \int_1^{R} A_c(x) \:dx = \pi(1-\frac{1}{R}). As we increase R, \frac{1}{R} becomes closer and closer to 0. Therefore, we can say with “infinite” R, the volume of the cone would be \pi.

This in some small way resolves the paradox. Instead of considering the infinite, we simply consider the case where x “approaches” infinity. In this case, the area grows without any upper bound, but the volume grows without ever being able to exceed a finite value. There is no value of R for which the volume will be greater than \pi

Why has this happened? Essentially it has to do with the “dimension” of the quantities we are considering. Length has a dimension of 1, i.e metres. Area has a dimension of 2, i.e. metres squared. Volume has a dimension of 3, i.e. metres cubed. In this case, as we increase the dimension, we decrease the significance of the far flung portions of the integral to the total sum.

To obtain the area A, we add up or integrate, “strips of length” which are of height \frac{1}{x}. To obtain the volume, we add up, “sheets of area”, the circles, which are of area \pi\frac{1}{x^2}. The key point here is that, as x becomes very large, \frac{1}{x^2} \ll \frac{1}{x}. The numerical size of the “sheets of area” we add, is far far smaller that the numerical size of the “strips of length” we add. So much smaller in fact, that where the “strips” become an ever increasing area, the “sheets” simply decrease too quickly to ever bring the volume up beyond a value of \pi.

This paradox seemed a good way to start a weblog, and explain the quote. The moral of the story, is essentially Hobbes’ age old warning. Mathematics, in its rawest form, inevitably requires a certain amount of mental unsoundness on the part of those who dabble in it. And so, the tone of the site, is firmly set.

Posted in Mathematics on Thursday, April 13th, 2006 | 13 Comments »