The Science of Pi

At the recent Loftus / Rauser debate in Calgary I got to ask a question during the Q&A. Loftus is a quintessential Atheist in his unwavering faith in Science (the religion popularly referred to as “Scientism”) and he repeatedly framed his comments and arguments through the lens of Science. So great is his worship of Science that he seemed to be working with the idea that if something cannot be proven through the Scientific method then it must be fairy tales or falsehoods. These are not his exact words, but seems to convey his general message.

If one could demonstrate that humanity possesses knowledge that is acquired through means other than Science then much of Loftus’ bluster is deflated. My question was posed in order to draw attention to the fact that we know a lot of things in life through non-scientific means, such as the fact that Pi is an irrational number (I also mentioned my wife’s name as another piece of non-scientific knowledge, but he didn’t run with that).

Gaging his response the point was obviously lost on him (in his defense it had been a long evening and my question was second to last, and a little abstract) but I wanted to elaborate the concept a little bit through a brief blog article.

Pi – the basics

This is really going back to the basics, but for those who have forgotten, the number “Pi” (it’s a Greek letter) is the ratio between a circle’s circumference and it’s diameter.

Pi = Circumference / Diameter

It holds true for any circle and the value of the number Pi is important for a wide range of scientific and engineering applications. Given its importance people want to know what the value is. So, what is Pi? Well, it’s 3.14.

No, actually, it’s 3.141. Well, to be more precise it’s 3.1415. Or, 3.14159. Well, actually, it’s 3.141592.

Ok, I know there are a lot of nerds out there who have memorized Pi to, perhaps, 100 decimal places but I only have 3.1415927 memorized. That’s plenty close enough for the engineering I do. But notice something about Pi. The number itself is what Mathematicians call an “irrational” number. No, that does not mean it’s a female number (ok, I’m in hot water now) what it means is that the number is not an integer (like 3.00000…) nor do the decimal places repeat themselves (as in 3.141414141414…). More broadly, an irrational number is a number that cannot be expressed as a fraction of integers (as in 22 / 7 which is a somewhat close approximation to Pi).

But perhaps the most striking thing about Pi is not that it is irrational, but that we know it is irrational. We are not merely guessing or assuming, we actually know this for a fact! It would be one thing, for instance, to notice that Pi has a lot of non-repeating numbers after the decimal place and think to oneself “well, it must be irrational” which is a reasonable inference, but it is quite another thing to claim to know that it is irrational beyond any shadow of a doubt. To have proven that it is irrational!

But it gets even weirder! Not only do we know that it is irrational, we do not even know what all the numbers after the decimal are! How can we know that it is irrational if we only know, say, the first million decimal places (or so)? Isn’t it possible that starting at the million and first decimal place it repeats all the previous million decimal places? No matter how many decimal places somebody discovers, is it not possible that they start repeating from that point on? Perhaps it is vanishingly improbable, but can we say, with absolute certainty, that we know that it is impossible that they start repeating? How can we know that fact without knowing what all the numbers are?

How do we know? Mathematics! Here’s a Wikipedia page describing this for those who care to know the details. I’ll admit that the proof is beyond my knowledge of Mathematics. The key point in all of this is that we have irrefutable proof – absolutely certain knowledge – that Pi is an irrational number. This is the highest possible form of knowledge; well beyond theory, speculation or even highly probable inference. This is knowledge.

Ok, so we arrived at that knowledge through Math, but I’m sure the Scientists have solved that issue on their own, too. Loftus would have us believe this, at any rate. How would Science deal with Pi?

Pi, by Science

Loftus argued that we can solve for Pi through the Scientific Method. What, exactly, is the Scientific Method? Surprisingly it is a matter of some dispute among Philosophers of Science (and Scientists themselves). Rauser touched on this during the debate but his very legitimate point seemed lost on Loftus. However, roughly speaking, Science looks something like this:

  1. I have some idea / hypothesis.
  2. I do some test to prove / disprove my hypothesis.
  3. The test must be repeatable by others (and preferably “double blind”).

So how would a Scientist go about solving for Pi? As Loftus started describing before he went off topic, we started discovering Pi by measuring round objects. We measured their circumference and we measured their diameter. Do the math. How hard is that?

Unfortunately he derailed his answer within a few moments of starting to answer my question and he went off on very weird tangents after that, but let’s follow his initial train of thought for a moment and give him (and his sacred Scientists) the benefit of the doubt. What tools and supplies would we need to figure out what Pi is, using nothing but Science?

First, we need a circle. No problem, let’s get an orange and cut it in half. There you go; you wanted a circle I just gave you two! Well, that’s not really a very good circle because you might be squishing it a bit as you hold it and take measurements. A lot of circles in nature are either not quite perfect circles (even planet earth is more like a slightly squished ball, actually) or they are not easily measured (like a floating bubble of soap suds). Humans are capable of making some pretty impressive circles using technology like a metal working lathe or something. I have a steel shaft on my desk that I cut from a metal lathe. Other tools create even better circles than my handiwork.

But this raises an important first point; is there a “perfect circle” (and I mean perfect down to the micro-atomic scale) anywhere in the physical universe? If there are no perfect circles in our physical universe then the experiment is doomed to failure from the start. That’s our first problem but, surprisingly, that is the least of our difficulties!

Here’s the bigger problem; measurement. Oh, come on; it cannot be that hard to measure it. I just pull out my handy-dandy pocket tape measure and do some measuring. I used my pocket tape measure to get the dimensions of the steel shaft on my desk. The circumference is 10 cm and the diameter is 3.1 cm. Given these dimensions, Pi is:

Pi = Circumference / Diameter

Pi = 10 cm / 3.1 cm

Pi = 3.225

Oh, hold on, I must not have measured correctly. Just a minute, let me measure again. Ok, circumference is actually a little under 10 – let’s call it 9.99 – and the diameter is more like 3.2 (I must have measured wrong the first time). This gives:

Pi = 9.99 / 3.2 = 3.12

Um, actually, the 3.2 is a little high. It was probably more like 3.18 or so. Let’s try again.

Pi = 9.99 / 3.18 = 3.141509

Hey, that’s much better. But it is still wrong. I got the first four decimal places right, but it’s wrong after the “5.” But if I had a better measuring tool then I could get a better result. True, but how good does the tool have to be? The question comes down to, “how many decimal places of precision does your instrument provide?” If our measuring tool is only accurate enough to give us the integer and no decimal points then we would end up with this result for the steel shaft on my desk:

Pi = 10 / 3 = 3.3333…

That’s pretty lousy. What if we can do the first decimal place? Then we get:

Pi = 9.9 / 3.1 = 3.1935…

Just for fun let’s suppose I could get the measurements down to 10 decimal places. Assuming the circumference was, in fact, exactly 10 cm, the diameter, to 10 decimal places, would be 3.1830988618. With these measurements we get a value of Pi that is:

Pi = 10 / 3.1830988618 = 3.141592653627205666

But Pi is actually 3.1415926535897932384… This level of precision gives us 10 decimal places correct and then it gets messed up after that.

This points us to our substantially bigger problem; more accurate instruments will give us more accurate results for the value of Pi, but instrumentation is the limiting factor for the accuracy we can possibly get through Science. A typical tape measure could hit one or two decimal places. A high precision measuring device could probably hit 3 or 4 decimal places. Suppose NASA could do 10 decimal places. Even 100 decimal places. Imagine if they were amazing enough to give us 1000 decimal places! That would be impressive (and certainly beyond the capability of any present-day instrument) and probably represents an incredibly generous upper limit to how accurate Science can go.

But – prepare to be amazed – these guys have managed to calculate Pi to over 51.5 billion decimal places (did anybody check their results for accuracy??). I assure you they did not use Science to come up with 51.5 billion decimal places for Pi because there is no instrument, nor any instrument we could possibly conceive of, accurate enough to produce such results. If humanity knows the number of the one billionth decimal of Pi it must thank something other than Science for that knowledge. We must thank Math for such knowledge.

No instrument could possibly give us Pi through measurement of a physical circle to more than a few dozen decimal places, and I am being generous. Furthermore, as I mentioned, this entire discussion assumes we were able to find a perfect circle in the first place. Good luck. No, it would seem that Science is not the right tool to use if you want to solve Pi to billions of decimal places.

Pi, the irrational number

But perhaps you noticed that I did a bait and switch. Most of this discussion has been about solving Pi to this many or that many decimal places. That was not my original question. What I asked of John Loftus was how could Science prove that Pi is an irrational number. I did not ask how Science might solve Pi to 2 decimal places, or 10 decimal places, or even 51.5 billion decimal places. Remember, even though we do not know all of Pi’s decimal numbers, we do know – we are not assuming or inferring, we actually know through verifiable proof – that Pi is irrational.

The only way that Science could possibly prove that Pi is irrational would be to first solve Pi to all of its infinite number of decimal places, then demonstrate that they never repeat themselves, and then conclude that it is irrational. And, of course, the experiment that you conduct to achieve all of those infinite number of decimal places must be repeatable for somebody else with a lot of time on their hands and a twisted sense of their life’s purpose. Math, on the other hand, proved that Pi is irrational without solving all of the decimal places, which would be impossible anyway. The proof (according to the Wikipedia site) appears to fit on a single piece of paper.

Not only is Science woefully ill-equipped to provide knowledge about Pi to 51.5 billion decimal places, Science could not possibly prove that Pi is an irrational number in the first place. Put bluntly, Humanity possesses knowledge which the Scientific Method could not possibly provide. This is not to say that all Mathematical knowledge is beyond Science to speak to. Science works to verify simple math functions like 1 + 1 = 2. Find an apple. Find another apple. Two applies. Viola! But what about imaginary numbers? Can Science give us any meaningful information about the square root of -1? Or what about infinites? Scientists would have to find an infinite number of something in our universe (they are stored right next to the perfect circles, just behind the unicorns) in order to conduct repeatable tests on that infinite set of… whatever. But if something does not exist in the material universe then Science cannot study it. Yet we have substantial knowledge about these numbers through Math. In this case our discussion has focussed on Math, but many other fields of human knowledge exist that Science cannot reasonably inform. Philosophy, Ethics (apologies to Sam Harris), Theology, Art, Economics, Literature; the list goes on.

Science is a wonderful tool and absolutely the best means to discover truth in its field of study, but outside its field of study (irrational numbers are but one such example) it is simply the wrong tool to use. Trying to use Science to discover Mathematical truths is like using the odometer on your car to acquire knowledge about the reproductive processes of trees. However, if Science is not the only means of acquiring knowledge then the Scientism that Loftus embraces is unwarranted. Humanity does, in fact, possess knowledge that is beyond the domain of Science to even begin to address. Dismissing this or that truth claim just because it cannot be proven “scientifically” is about as meaningful as dismissing the claim that Lenin was the leader of Russia because it cannot be proven through oncology.

Scientism, it turns out, is just as irrational as Pi and we can prove it!

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About Paul Buller

Just some guy with a variety of eccentric interests.
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4 Responses to The Science of Pi

  1. Dr. Michael W. Ecker says:

    You missed the point, and you do not have the correct definition of irrational number. It is a number that cannot be expressed as a ratio of two integers. Not a ratio. Not rational. Get it?
    We know pi is irrational from mathematical reasoning. The question of the digits not eventually repeating in blocks misses the point. It’s also NOT the definition.
    If you wish, I can explain more.
    Dr. Michael W. Ecker
    Associate Professor of Mathematics
    Pennsylvania State University
    Wilkes-Barre Campus
    Lehman, PA 18627
    (affiliation for ID purposes only)

    • Paul Buller says:

      So when I said, “More broadly, an irrational number is a number that cannot be expressed as a fraction of integers (as in 22 / 7 which is a somewhat close approximation to Pi)” you are right that I should have said a “ratio” of integers, not a “fraction” of integers. My bad. Thank you for graciously correcting me.

      Given this correction, could you help me understand how this corrected definition of “irrational” undermines my primary point that science could never demonstrate that “pi is an irrational number?” I do not see how it could. That conclusion was derived through the field of mathematics, not by positing an hypothesis, formulating an experiment, obtaining repeatable results, etc. Even with your correction, I believe my main point stands.

      Your thoughts?

      • Dr. Michael W. Ecker says:

        Paul: I do not share your artificial distinction between science and mathematics. Both – unlike, say, pure religious faith – employ logic. It is reason, rationality, that is the point here. In fact, mathematics has been called the queen of the sciences. Some say we are mathematical scientists. And so on.

        The point is that you are attempting to drive a wedge between math and science — for your purpose of damning science, it seems. No good. It is logic that proves pi is irrational. To argue that it is not science is an attempt at sophistry, and frankly, a disingenuous one at that.

        I think part of your problem stems from an incomplete concept of what distinguishes the scientific method from non-scientific ones and anti-scientific ones. The primary point is that what is studied can be replicated, and ultimately, can be verifiied or falsified. Something that is offered that can never be tested or a hypothesis that can never be negated fails outside the purview of science.

        The issue about the digits of pi, in fact, is settled. It is true that only rational numbers have their decimal expansions eventually repeat in blocks, such as 1/1100 = 0.0009090909… (ad infinitum), where the block is 09 eventually. Because we know that pi is irrational, it follows that the digits can never repeat in blocks beyond some point as they do with rationals.

        Thanks, by the way, because this reminds me to stress this in my Calculus II this semester when we get to infinite series. It seems that you nonmathematicians need to hear these things.

        Best wishes, Mike
        Michael W. Ecker, Ph.D.

        (The writer is Associate Professor of Mathematics at Penn State, Wilkes-Barre.)

        • Paul Buller says:

          It seems you cannot be bothered to respond to what I have actually written. I am beginning to wonder if you actually read it at all. You claim, “you are attempting to drive a wedge between math and science — for your purpose of damning science, it seems” despite the fact that I have very clearly stated, “Science is a wonderful tool and absolutely the best means to discover truth in its field of study.”

          I am an engineer working in the field of machinery vibration. Have been for 15 years. I have to understand both the physical systems (piping, compressors, etc) as well as somewhat “complicated” mathematical concepts such as fourier transforms of wave functions. Surely my knowledge of math pales to oblivion in comparison to yours – this isn’t a pissing contest – but to suggest that I would denigrate the two fields that make my livelihood possible at all, and profoundly interesting to boot, is a straw man of apocalyptic proportions.

          You invent criticism and “damning” where there is none; explicit or implicit. Furthermore, you continue to churn out concepts and claims that do not, actually, address anything in my article. You point out that something needs to be testable, repeatable, etc, in order to be “Scientific.” I agree. In fact – and I’m starting to sound like a broken record here – that is exactly what I said in my original article, if you would bother reading it. And your comments about irrational numbers not repeating is, again, something I said in my original article. Your comments do not in any way undermine my primary thesis, nor do they give any hint that you have actually read what I wrote.

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