This was more of a mathematical question, but because it’s closed and people say it’s philosophical I post it here:

Someone tells you “my pen is 9 cm long”.

How can that be? Can you really have a pen of length 9 cm? I mean, in the manufacturing process you would somehow have to measure EXACTLY 9 cm, which seems impossible considering you also have 9.00000000028288282828 and 9.00000000001 and 9.000000000000000000029838 and so on, infinitely many real numbers infinitely close to 9 cm.

For any other physical measurement too. All the measures we attempt to make (9 cm, 23.5 kg, 2.29 m) just seem like approximations of these idealized measurements: 9, 23.5, 2.29, etc.

  • Are all measurements in the real world, like the ones above, just approximations of an idealized theoretical value?
  • Is it possible to get an exact idealized value in the manufacturing process?
  • All measurements have finite precision, yes, and there are theoretical limits to what that precision can be due to quantum effects. Whether there is something "ideal" to be "approximated" depends on whether one believes in existence of platonic forms.
    – Conifold
    Mar 2 at 8:45
  • No, measurement is a physical process Mar 2 at 8:57
  • 2
    It's probably the other way around -- our ideal models are simple approximations of the real world. The aptly named ideal gas is an example of such a model. Same true with measurements -- in reality, there is no such thing as an ideal circle, or 9 cm pen. Still, those ideal models are useful (because simple to understand) approximations. Mar 2 at 10:08

8 Answers 8


This is a fundamental question of philosophy1

From Plato on there is a strong division in philosophy between the sensible (= sense-able) and intelligible worlds. Made more demarcated in the last ½ millennium in European philosophical schools — the Empiricists and the Rationalists.

Its most reasonable to assume that there are these two worlds

  • Things in the physical world are the empirical facts
  • Logic/math are the archetypal rational truths

They map to each other... kinda, sorta... but not perfectly

Yeah, it seems as though 9 is a »clean« number and 9.0000000something-or-other is not. And so the 9.0000xx is the approximation and the 9 is the exact one.

This is the rational prejudice

As Yuri points out in a comment one could as well put it the other way

The fundamental underlying truth (fact?!) is that the mappings either way are not exact.

Towards seeing more "exactly"(!) this inexact mapping first some...


The notation for the closed interval [a,b] is used to denote the set
{x | x ∈ ℝ, a-ε ≤ x ≤ a+ε} where ε is some small positive real number "to be understood from context"

It will be convenient for us to make a related notation, say «a,ε» for the interval [a-ε,a+ε]

You can think of «a,ε» informally as An ε-width fuzz around a

Let's agree to call «a,ε» a fuzzy a, and ε...

  • maximum accuracy of the instrument — if you're a scientist
  • the fuzz-factor — if you're a mathematician


Your question

  • In the rational world numbers exist directly, ie. by definition. So they are and can only be exact or non-fuzzy
    9 is exact but so is 3.141592 up to however far you wish to go
  • In the empirical world numbers exist indirectly via observation and inference. All numbers here are fuzzy or inexact
  • For the mathematician, the fuzziness has an exact definition («a,ε»)
  • For the scientist, mathematics is just a tool like a telescope, and caring about exactness way outside the fuzz is a fuss!
  • The fuzziness ε is the least count of the measuring device... at best
  • "At best" because instruments can break and give wildly off readings
  • Or we use computers where ℝ is approximated by a strange thing called float. Here's some python showing that float is not ℝ
    >>> 0.1 + 0.1 == 0.2
    >>> 0.1 + 0.1 + 0.1 == 0.3
  • Quite simply there are "daemons" messing around the empirical world

In short experimental scientists have many adventures!

A modern further refinement of the 2 worlds, rational and empirial which are often preferred today are the three worlds of Popper and Penrose

1 I'm surprised how poorly it's answered so far


When using a ruler to measure your pen, the minimum uncertainty is the smallest tick on the rule. So the proper way to report a measurement is to include the uncertainty: 15 cm +- 0.5 mm.

Manufactured parts are built within tolerances that allow creation of "identical" parts. So despite the fact that the parts are not identical, they are interchangeable provided they are within tolerance.

A blueprint may show the "ideal" pen but the tolerances in the blueprint keep it real.


A measurement is a physical process that produces a record of the value of some physical quantity. A record is instantiated in a physical quantity whose value can be copied indefinitely often. So the theory of measurement is constrained by physics.

A pen is made out of atoms. Atoms are composed of protons and electrons whose position is described by quantum observables that have a spread of values. Quantum theory will give you an expectation value for those observables but not a single value. The spread of those values for position and momentum is typically very small on the scales of everyday life because of quantum decoherence:


In both theory and practice the idea that a measurement gives a single well-defined real numbered value is false. For some manufacturing processes, such as manufacture of a pen, the scale over which the measurement value is spread is negligible compared to any length scale we care about. For other manufacturing like electronics the spread may be relevant:



All measurements taken on analog aspects of the material universe are necessarily imprecise, as the measurement devices contain the same classes of uncertainty that the objects themselves do.


"In this sort of predicament, always ask yourself: How did we learn the meaning of this word ("good", for instance)? From what sort of examples? In what language-games? Then it will be easier for you to see that the word must have a family of meanings."

-Wittgenstein, in Philosophical Investigations

The issue is just about context, and especially implicit context. Your claim about the pen is different in a lab, vs a school room. That's because it's meaning is not just in your utterances, but in the whole of the language games involved.

The key to understanding this is to go beyond objectivity vs subjectivity as a binary option, to recognising intersubjectivity. Money is fake, a collective fiction, but the strength we hold it with makes it real to us - until a bank-run, or hyperinflation. Our cognitive furniture is not founded on direct contact with the world (ie noumena), which are unreachable. But instead on how much of the world we share. Discussion, experiment, agreements, suspending disbelief, are all mechanisms to share experiences. Certainty can only relate to how precisely we define context.

Errors are a deep problem in science, and the Uncertainty Principle arises from the fact there is a limit to the smallest possible transfer of information (ie, the photon or other massless particles) and below a certain scale such probes disturb dystems by more than they reveal. Such fundamental limits to information basically give rise to all the complexities of quantum physics. So no, there is no exact value that can be known seperate to context.

The SI unit of length has been defined in such a way that light crosses a given distance in a given time, making a yardstick other lengths are compared to. But then you have to define time. And:

"Time exists in order that everything doesn't happen all at once…and space exists so that it doesn't all happen to you."

-Susan Sontag

Which is to say, it's deeply linked to how our minds and perceptions work.

In summary:

"We see things not as they are, but as we are."

-Anais Nin, in Seduction of the Minotaur


NIST TECHNICAL NOTE 1900 (105 pages):

Simple Guide for Evaluating and Expressing the Uncertainty of NIST Measurement Results


Measurement is understood in a much wider sense than is contemplated in the current version of the International vocabulary of metrology (VIM), and is in general agreement with the definitions suggested by Nicholas and White (2001), White (2011), and Mari and Carbone (2012), to address the evolving needs of measurement science:

Measurement is an experimental or computational process that, by comparison with a standard, produces an estimate of the true value of a property of a material or virtual object or collection of objects, or of a process, event, or series of events, together with an evaluation of the uncertainty associated with that estimate, and intended for use in support of decision-making.

In general taking two or more actual measured values generates discrepancy, and therefore uncertainty, concerning the true value sought to be measured. I think we use ideal values in the math symbols to describe the measurement uncertainty using statistical models which are also ideal. The so-called true value is never known and is only produced by applying a statistical model to the actual measured values.


No, you can't manufacture things with exact dimensions in the sense that you mean. If you take a pencil, for example, its dimensions will vary with environmental factors, such as humidity and temperature, as will the dimensions of the machinery making the pencil. Even if you could control for those, both the pencil and the machinery are made of quantum particles, and it is impossible to nail down their positions to arbitrarily small degrees of uncertainty. An example of the state of the art in precision measurement of distance is LIGO, which I recall can detect changes in distance to twenty or more decimal places.


Someone tells you “my pen is 9 cm long”.

How can that be?

Reasonable people all say that the pen is 9cm long whenever the measure of the pen says the pen is 9cm long.

I don't believe for a moment that anyone not somehow insane would insist that when the measure says 9cm, then it has to be that the pen is really exactly 9cm long. Reasonable people, and most people are reasonable,¹ don't waste their time nit-picking over actual measurements.

If we measure things, it is when size matters and sometimes precision matters as well, and in our age of science and technology, people sometimes have to go to extremes to improve the precision, but that still doesn't mean that any of those involved would claim that their measure gives the actual length of whatever they have measured. People have long understood that any measurement is subject to the precision of the measurement apparatus and of the technology.

They accordingly will use whatever apparatus and technology available to get the precision they want to have, but no reasonable person would come to think that their measure is absolutely exact.

It is also interesting to note that while we can imagine a pen, we cannot imagine a pen of exactly 9cm. We can say that this or that real pen is maybe or probably or possibly 9cm long, but our guess is just that and only as good as anyone else's, and any pen that we conceive in our mind as a matter of fact is never actually 9cm long because things in our mind cannot be measured, certainly not in the same sense that things in the real world can be measured.

So our measures are what they are and nobody reasonable cares that they may not correspond to some "ideal". All that we need is that the precision be good enough, and this is because measures only really exist in the real world. We can of course always pretend that the pen we have in mind measures exactly 9cm long, but this would just be complete nonsense.

It is also interesting to note that you cannot prove that a pen measured at 9cm long is not exactly 9cm long. You can always take another measure, but this isn't going to prove that the first measure is somehow false. It will just prove that the second measure perhaps is different from the first. It will also not prove that the second measure is exact. More precise perhaps, exact of course not.

So we may have something like an ideal pen in mind, but no ideal pen will ever replace real pens, and whether real pens can be exactly 9cm long doesn't matter for anyone, at least for anyone reasonable. I don't remember anyone wasting their time measuring the pen that they meant to buy to make sure it was 9cm long.

That being said, any object in the real world has exactly the length it has.

Too bad we cannot even figure it out!

(1) Citations:

“Most people are reasonable. That's why they only do reasonably well.” -- Paul Arden (1940–2008) director of Saatchi and Saatchi.

"Most people are reasonable, thereby inhibiting progress." -- Paul Rulkens, a business consultant of KPMG.

"Most people are reasonable but sometimes, for whatever reason, somebody will be hang up on a position and you just cannot negotiate your way out of it." -- Richard Flinton, Chief Executive, North Yorkshire County Council.

"Regardless of this I always tend to work from the position that most people are reasonable and can be reasoned with. That is to say they are open to discussion, can evolve their thinking and that as colleagues you can generally find a way to work with each other that produces reasonable outcomes." -- Natasha Norton, senior executive at KorumLegal with 20 years experience in technology & management consulting.

"The vast majority of people are reasonable, supportive and mutually co-operative, because those are the qualities which helped our species to develop." -- Mark Errington, studied at the university of Birmingham.

"The point is most people are reasonable. They do not destroy the environment." -- Loraine Margaret Braham is an Australian politician, a member of the Northern Territory Legislative Assembly from 1994 to 2008, representing the electorate of Braitling.

And Google Scholar found for me 145 occurrences of the phrase "most people are reasonable"!

  • 2
    "...most people are reasonable..." citation needed :)
    – Conrado
    Mar 2 at 14:57
  • @Conrado "citation needed" This is a reasonable request. Mar 2 at 16:34

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