I was thinking the other day about the difference between rational and irrational numbers, and wondering whether the distinction between them is created by leaving out discussion of precision.

So for instance people are calculating trillions of digits of pi (http://www.numberworld.org/misc_runs/pi-12t/).

However I found discussion of the limitation of pi based on planck's constant on quora:


Philosophically, it seems like PI the mathematical value is different from PI the engineering value.

Another area where I've seen this is in banking/lending. When you look at your bank account, it states the balance to two decimal points, whereas internally, from my experience banks keep account values to ten or more decimal points for the purposes of compounding interest.

Beyond the first hundred digits of PI, are the next trillion digits "real"? Similarly, are the digits of your bank account past the 2nd decimal place "real"? Saying "I have exactly two hands" seems to me to be "real". But saying "I walked exactly two miles" or "I poured exactly 1 teaspoon" seems like needs to include a precision metric with it.

So I think we need a philosophy of precision. Does this exist? I can't find any evidence of it on the internet

  • Keeping bank account values beyond the precision of the currency would be meaningless, but they use a lot more intermediate precision when doing some calculations.
    – user207421
    Aug 24, 2019 at 6:58
  • What is a hand? What has the quantity to do with utility? Does size of hand strength shape dexterity matter? In what way? When? Why? ... :-) Aug 24, 2019 at 12:59

4 Answers 4


You can address this issue through significant figures, which indicate the level of precision with which a value has been measured. If I say that I walked 2 miles, that's a rough estimate - I'd be justified in saying that if I had walked anywhere from 1.5 to 2.4 miles. If I say that I walked 2.000 miles, that is a much more precise number - I'd only be justified in saying that if I had walked between 1.9995 and 2.0004 miles. Precision will be dictated by your measurement tools and what's needed for your domain. Pi, for example, has an infinite number of digits, which are "real" and truly do exist, but most of them are completely unnecessary for any practical application. Using only 40 digits of pi will allow you to calculate the circumference of a circle the size of the visible universe with an error the width of a hydrogen atom.

Proper application of significant figures will allow you to determine the level of precision of any measurement, indicating that the significant figures are indeed "real". Improper application of significant figures will get you long decimals where you don't have any business claiming such precision - those decimal values are not "real", they're numerical artifacts that result from not understanding how to propagate the limitations of your tools. We can calculate pi to an infinite number of decimal places, but at some point, the limit of precision is your measuring stick and not the value of pi.

  • You hit it head on with "... but at some point, the limit of precision is your measuring stick..." Mathematics can define an irrational number, but this number is always going to reduce to the finite when you apply the measuring stick. This is ringing "Heisenberg Uncertainty" bells. Very satisfying! Aug 23, 2019 at 17:06

It think there's something confused in this question. Mathematical constants like pi are exact. Pi is the ratio between the circumference and diameter of a perfect circle on a flat plane, and the fact that we cannot calculate an exact value for it does not mean that the constant itself is not exact. Planck's constant might suggest that we cannot have a mathematically perfect circle in the real world, but that doesn't affect the value of pi in the least.

Your question seems to amount to: "Is an exact value 'real' if we only ever use approximations to it?", and I don't know quite how to approach that.

  • Nice! Physics constants are not numbers. They are a fuzz around a math number Aug 23, 2019 at 17:27
  • Actually, my question is the opposite: Are digits of PI that go past all possible applicability in the physical universe we live in "real?" Aug 23, 2019 at 18:46
  • 3
    Well, they are 'real' in the sense that they measure exactly how far from the 'true' value of pi your approximation is. I mean if you were to say that pi was '3.14' for your practical purpose, then I would know that your practical purpose was insensitive to true pi by approximately .0015926 (roughly .05%). Besides, the idea of 'past all possible applicability' is meaningless; who knows what l'applicabilities' will be required or available in the future? Aug 23, 2019 at 19:11

I'm not sure about your approach but I'd agree that the issue of precision is an important philosophical topic that deserves more attention.

I don't know of anyone who explores this as a philosophical issue. It is not just to do with Pi. No location on the number line can be identified with precision if all locations are infinitely divisible.

EDIT: Doh! I forgot the mathematician and physicist Hermann Weyl, who I believe deals correctly with these issues. He makes clear the necessary links between mathematics, metaphysics, experience and Reality. The mathematics of his book The Continuum was mostly beyond me but it includes a valuable discussion of its philosophical implications.

I think I had a senior moment. There is also Tobias Dantzig, a mathematician admired by Einstein. Here he is introducing the issues.

"Herein I see the genesis of the conflict between geometrical intuition, from which our physical concepts derive, and the logic of arithmetic. The harmony of the universe knows only one musical form -the legato; while the symphony of numbers knows only its opposite, -the staccato. All attempts to reconcile this discrepancy are based on the hope that an accelerated staccato may appear to our senses as legato. Yet our intellect will always brand such attempts as deceptions and reject such theories as an insult, as a metaphysics that purports to explain away a concept by resolving it into its opposite."

  • This seems true intuitively, but it is not. 1/9 is on the real line, but is a non-terminating decimal. In fact, Cantor's definition of the reals, opposed to Dedekind's, makes use of infinite sequences. Precision is a topic of the philosophy of math.
    – J D
    Aug 23, 2019 at 17:23
  • @JD I draw no line between metaphysics and mathematics, which is why I believe this topic is not just important.but that study of it can reveal the true nature of Reality. I've edited my answer because I remembered who I should have mentioned. .
    – user20253
    Aug 23, 2019 at 23:31

The matter of precision is handled philosophically in the discussion of the infinite. In mathematics, for instance, precision is handled by the concept of the limit. Limits are particularly important in the philosophy of mathematics in topics such as the philosophy of set theory or analysis.

If you are genuinely interest in this topic, I would recommend one of my favorite books, The Philosophy of Set Theory by Mary Tiles which starts off in Chapter 1 discussing the infinite universe and outlines such philosophical positions as strict finitism which rejects the infinite, classical finitism which rejects absolute infinity, but accepts potential infinity, and relates the concepts of continuity and infinity, both important not only to the development of set theoretic thought, but also functions and differentiability. Later in the chapter Xeno's paradoxes are handled, and the end of the chapter explores the universe, absolute infinity, unbounded surfaces, and touches on Newtonian models of the universe.

It should be noted that these ideas, like the concept of space-time and Minkowski space are central concepts of study also in the study of the philosophy of science which asks questions, what is the fundamental nature of space and time? For instance, is the gravitational field and disturbances called gravitational waves the fundamental "stuff" of the universe?

If you're interested in understanding precision philosophically, you have to study these topics.

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