This is an allegory; let the reader understand.
It’s important to recognize the distinction between accuracy and precision in mathematics. Take the value of pi for example. Is it 3? 3.14? 3.14159? 3.1415926535? Well, in each case, the answer is yes and no. Pi is an irrational number with never-ending decimal places which cannot be fully expressed numerically. Because of that, pi isn’t really 3, 3.14, or 3.14159. However, we can say that 3, 3.14, 3.14159, and so on are all accurate representations of pi because in each number, the given digits speak the truth assigned to them. Where they differ is in how precise they are. The more decimal places there are, the more precise the value is. Nevertheless, they’re all imprecise to some extent because the decimal places go on to infinity–far more than can be written down by mere humans.
So how precise does a person’s value for pi need to be? It really depends on a variety of factors. It depends, for example, on how precise your measurements are. When making calculations, you should always keep the concept of significant figures in mind because you shouldn’t claim greater precision than you’ve been given. It also depends on what you’re doing with pi, for some applications require more precision than others. It even depends on how educated you are. It’s proper for college students to know pi more precisely than kids in elementary school. Ultimately, answering the question of how much precision is required necessitates good judgement in each individual case, for to whom much is given, much is expected.
Because of this, I think It’s very good for mathematicians to argue about pi so they can learn it’s value more accurately and more precisely. Iron sharpens iron, after all, and truth is a noble pursuit. What’s more, if you’re going to build anything circular, you need to make sure you’re working with an appropriate value for pi–something that’s both accurate and precise enough for your project. Otherwise, you’ll end up with a deformed circle or a different shape entirely. Accordingly, argument is also important when false mathematicians promote inaccurate values–they need to be corrected.
But not all kinds of arguments are helpful. And people who fail to remember the distinction between accuracy and precision are doomed to suffer an eternity of making stupid arguments about the value of pi. For example:
“Pi is not 3! It’s 3.14159! Anyone who says pi is 3 has denied mathematics!”
This is not so. 3 and 3.14159 are both accurate values for pi. The latter is merely more precise. It’s entirely inappropriate for Dr. 3.14159 to be telling Mr. 3 that he’s wrong. It’s good for a man to have greater mathematical precision, but he should not be lording it over anyone. Rather, he should patiently and humbly lead people from 3 to 3.1 or from 3.1 to 3.14 as needed.
He should also keep in mind that not everybody needs the same level of precision. The man who is fabricating sophisticated scientific devices with circular components needs more decimal places than the man who is building a hula hoop. We all have different callings and different gifts. Accordingly, we should respect those who express accurate values with different levels of precision.
“Pi is not 3.14159! It’s 3.0! The 3 is the only important thing, and anyone who looks beyond the decimal point is just an arrogant egghead trying to confuse the issue!”
This is not so. On the contrary, 3.0 is actually an inaccurate value for pi and needs to be corrected. True, 3 and 3.0 may be numerically equal. But in its virtuous humility, 3 does not reach for greater precision than it possesses. In its pride and arrogance, 3.0 insists on specifying digits of which it’s ignorant, and consequently falls into error. Mr 3 should not be contemptuous of Dr. 3.14159 or scorn his precision. Not only is that precision useful in certain applications, anyone who is interested in geometry will naturally want to learn more about pi. Those with greater intellectual gifts will usually learn its value more precisely.
If you aspire to greater precision, then work hard to learn the decimal places accurately. Otherwise, remain silent about them. Once again, all would-be mathematicians from greatest to least need the humility to recognize different gifts and callings.
“How dare you reject my favorite mathematician who says that pi is 9.14159! So what if he has one little digit wrong? He’s GREAT about the decimal places!”
This is truly foolish. The decimal places may look very very similar, but 9.14159 is a wildly inaccurate value for pi! Unfortunately, it’s quite easy for precision-loving mathematicians who spend their lives calculating decimal places to forget how important the whole number is. After all, they learned the whole number long ago, and now concern themselves with “deeper” matters. But deeper in this sense is a matter of precision while the basics are a matter of accuracy. By their very nature, some digits are far more important than others. If you get the whole number wrong, it doesn’t matter how many decimal places of pi you have correct. Building anything circular based on that value would be an absolute disaster.
You might think yourself wise and sophisticated for getting decimal places right, but never take the whole number for granted. The things we take for granted are precisely the things we lose over time. And nobody should ever trust a mathematician who let themselves get that far off track–not until they’ve recognized and publicly acknowledged the reality and magnitude of their error.
“Look: mathematicians throughout history have never really agreed on the value of pi. Some say 3, others 3.14, still others 3.14159, and so forth. Some have even said it’s 3.142! Clearly, pi has no definitive value. So why should anyone have a problem with me saying pi is 89.4?”
This too is foolish. Our limited ability to express the value of pi precisely does not mean we cannot express it accurately or that all inaccuracies are equal. Of course those kinds of disagreements will happen among those mathematicians who strive for greater precision. That’s all part and parcel of working out calculations. But we shouldn’t confuse disagreements in how many decimal places to use or when to round with disagreements over the value of pi.
What’s more, we shouldn’t assume that differences between mathematicians about pi means that any and every value is fair game. Just because answers with different levels of precision are all accurate doesn’t mean there are no inaccurate values. Anyone giving a value like 89.4 is not talking about pi, but about another number entirely. Trying to pretend their value is actually pi is inherently deceptive. And that deception will be a catastrophe for anyone who believes that lying mathematician and then tries to make something circular.
It may be the case that nobody can fully understand the value of pi, but that does not excuse deliberately misunderstanding it.