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Why do we measure quantities like length, time, mass in terms of a discrete quantifiable unit (for eg: length in metres or planck lengths, if one scales down to that level) ? Is it possible to quantify physical dimensions as a continuous real entity? I'd really appreciate it if someone would kindly break this down for me and explain to me in the simplest way possible. I apologize if I cannot get my question through to the members of the community due to my lack of knowledge of technical terms: I am a mathematics undergraduate student, so kindly bear with me.

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Consider the unit of length. Bring two points A and B arbitrarily close, to say 1 planck unit. Now halve that distance (theoretically of course), and keep halving them such that they become closer and closer still. Do this arbitrarily many times. Note the distance is still not zero. But there is no quanta of measurement as it is much smaller that the quantifiable physical quantity. So how do we overcome this hurdle of defining length arbitrarily close without using quanta of measurement?

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closed as unclear what you're asking by AccidentalFourierTransform, ZeroTheHero, knzhou, Jon Custer, Yashas Mar 20 '17 at 2:54

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Can you give an example of what you mean by a "continuous real entity?" The general answer to your question is "no, we don't do that and/or don't want to do that," but if we understood what you thought the alternative was, we would be able to better tune the answer to your way of thinking. For example, it's quite clear that we can measure real values with our units (such as $\sqrt2 \text m$), and we have units like radians where irrational number creep in all the time. What sort of measurement are you thinking the alternative might look like? $\endgroup$ – Cort Ammon Mar 19 '17 at 18:13
  • $\begingroup$ @Cort Ammon: As I said, I find difficulty in explaining using physical/mathematical technicalities. But consider this: Is there a way such that we can measure physical quantities using a limit-point-like approach: Like, can we find an arbitrarily small ε for which |a-b|<ε (ε>0) ? $\endgroup$ – evil_potato Mar 19 '17 at 18:20
  • $\begingroup$ Are you looking at measurement, which is the act of using physical devices to actually do measurement, or are you looking at units of measure. I, as a non mathematics student, read what you just wrote as similar to how we determine the mass of something using a balance scale. With some small epsilon of error, you can find a collection of weights whose mass is equal to that of your mass-under-test. $\endgroup$ – Cort Ammon Mar 19 '17 at 18:23
  • $\begingroup$ Okay, another attempt at explaining: Consider the unit of length. Now we measure length from A say to point B in metres. Fine. Now bring the two points A and B arbitrarily close, say 1 planck unit. Now halve that distance (theoretically of course), and keep halving them such that they become closer and closer still. Do this arbitrarily many times. Note the distance is still not zero. But there is no quanta of measurement as it is much smaller that the quantifiable physical quantity. So how do we overcome this hurdle of defining length arbitrarily close without using quanta of measurement? $\endgroup$ – evil_potato Mar 19 '17 at 18:33
  • $\begingroup$ So we usually handle quantities (the official term for this) as a real number and a unit, such as "3.4" and "meters." We handle the sorts of issues you describe by packing them into the real number portion and leaving the unit alone. By doing this, our system of measuring quantities is not quantized at all, even though our choice of units may be. I believe these sorts of fun quantities appear in black hole mechanics, where you have a singularity that you can approach in a limit-like fashion. $\endgroup$ – Cort Ammon Mar 19 '17 at 18:56
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You've either totally garbled your understanding of definitions or you've answered your own question.

By definition, a unit is something which is fundamentally non-continuous. A unit is either:

a) Something that is considered single and complete in itself. It is absolutely individual but can make up parts of a greater whole. In other words, it is an element, a building block upon which to consider other greater things.

b) A set quantity which is deliberately chosen as a standard in terms of which other quantities may be expressed. It then becomes convention and scaling occurs from there.

To quantise something is to move it from the realm of the continuous to the realm of the quantised, to declare the bare minimum. In physics, this is called the "Hypothesis of Quantisation" and is based on the minimum physical entity that can interact. How could it be possible to quantify anything as a continuous, real function ? The very concepts are antithetical to each other. In fact, one could argue that your understanding of real numbers in mathematics is based off of a discrete concept of a unit (in this case, the number one) when your parents first show you a singular thing as one. You understand two as double that of one and by extension, a half as one split into two.

Why physicists choose units for measurement is because it is both much, much easier and more intuitive to measure something as a multiple of a unit as opposed to trying to define it in a continuous sense. I am not aware of any measurable unit that is not discrete. I'm not sure how one would even begin to measure length as a continuous function without some mind-bending paradox occurring. That is not to say that continuous things or a continuous concept is not useful in physics (electric flux, Maxwell's Laws for instance).

Your thought experiment whilst interesting and stimulating, doesn't hold much appreciable application (I think) because there's a limit to what we can measure or what is useful. Sure, you can cut something up arbitrarily many times in your head but you can only do it so many times in real life. Consider Zeno's Paradox of Achilles and the tortoise; it is both stimulating and interesting to think of but in real life Achilles wins the race. So the model of quantising length into something (be it the yard, metre or Planck length) which we can then measure, see, scale and think with is demonstrably much more useful as a predictive tool. Now why we choose what we choose as a unit is a slightly different question.

On another note, I think why you're getting downvoted is a combination of you not looking up certain fundamental parts of physics and not understanding technicalities before you post a question. General and broad philosophy-type questions aren't quite what this place is about, I think.

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I feel that conceptually, your question can be broken down to "Why do we think that we need Reals (or Complex numbers) to describe nature?" and "Is this really so, given Quantum Mechanics?" (I hope describes what you're aiming at.)

But to answer your first questions, why do we use a discrete units like 1 meter as a unit to measure distance?

Out of convenience. It's just an arbitrary length that can be put between your hands which happens to have a ~1:40.000.000 relation to the earth circumference [citation-needed]. But this is entirely arbitrary, for example others use feet, A.U. or light-years. You could use the Planck Distance to measure the length of you car, but this is inconvienent due to the large 10-to-the-power-of-something factor involved.

When it comes to describing nature, physics uses the formalisms of mathematics to describe the laws it discovers. Counting things involves natural numbers, dividing things up involves Rational numbers. By the way, the division of 2 (halving) you describe, does not leave Rationals.

But we discovered, that Rationals are not enough. Throwing a ball involves parabolas, celestial mechanics involves other conic sections. This means you're into solving things like x^2=2, which has no solution in Rationals and you're into Reals (\sqrt{2} being irrational).

Then we discovered infinitesimals and differential equations and saw that they describe nature accurately enough for all practical purposes (and you're into analysis over Reals and Complex numbers):

Nowadays gravity, mechanics, eletromagnetism, all other forces and all quantum phenomena are described using differential equations and the frameworks of analysis over continuous fields (Reals & Complex numbers).

It is this success, that lets us assume that the physical dimensions are equivalent with a continuous line of Reals. (It's a tremendously useful assumption, by the way.)

Only recently (~100 years ago) did we discover, that some things in nature happen in discrete steps and we have created theories describing this - again using the continous reals (of dimension 2, i.e. complex numbers). These theories, to the best of my knowledge, do not require to abandon the assumption that space and time are discrete. Neither do they have consequences, that may render this assumption false. Neither do they give us experiments, which we could carry out today, which could falsify this assumption. Nor do we have an alternative theory, which describes nature more accurately using a discrete formalism.

So "Is this really so?" - time will tell.

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In the world of dimensional analysis, there are three key concepts:

  • Dimensionality - These are concepts like "length" and "time duration."
  • Units - These are concepts like "meters" or "inches" or "seconds," each of which has an associated dimensionality
  • Quantities - These are measurable values which consist of a number paired with a unit such as "3 meters" or "2.45 microfortnights."

All of these concepts have been invented because they are convenient. Its convenient to be able to think of "units which measure lengths," where length is a dimensionality. It's convenient to be able to do "dimensional analysis" to see that you made an error. In theory you could do all of physics without any of these concepts, but it is inconvenient to do so. The concept of dimensionality is too helpful in guiding us along the right path.

Quantities are a very old concept. They are fundamental to adapting the abstract concept of numbers into the real world. They divide up a value such as "the length on this wall" into two pieces: a number (such as 2.4) and a unit, which is an easily reproducible length. Nowadays these units are all things like meters and seconds which have very exacting specifications, but in the past we'd also use units like "armspans," which are technically different from person to person but are often similar enough to be useful.

One of the very powerful features of this is that it permits you to leverage all of the power of numbers. When we discovered irrational numbers, we also discovered the capability to talk about irrational quantities, even if we may not be able to perfectly construct any arbitrary irrational quantity.

So, to your question, the typical approach would be to do all of the complexity of managing numbers within the number portion of the quantity. If you need real numbers, complex numbers, or even odd numbers like infinitesimals, you can do so without requiring any fancy behavior with the units.

The only case I can think of where you might need to have a concept of a "continuous unit" might be analytic continuation to extend real equations into the complex plane. In such a process you might find the need to have the concept of a continuous unit as you move along a sheaf. In practice, however, it is more common to do all of that in the pure abstract mathematical world and then determine how to relate it back to physics afterward.

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