A same length is measured 1.00 m, 3.28 ft, e.t.c., so what?
Topics
About: elementary school mathematics
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: 'Measurement' Is a Basis for Applying Mathematics to the Reality
- 2: Measurement Is to Represent an Aspect of the Reality by a Mathematical Entity
- 3: Let Us Measure a Length
- 4: But How Does Relativity Change the Situation?
- 5: The Reality Is No Uncertain
- 6: Measurements Are Harassments
- 7: Isn't There a Misunderstanding About 'Scientific Method'?
- 8: A Bad Moral Implication
- 9: Conclusion
Starting Context
- The reader knows the background of this site.
Target Context
- The reader will know the relation between the reality and measurements.
Orientation
There is an article on becoming a benefactor of humanity by being a conduit of truths
Main Body
1: 'Measurement' Is a Basis for Applying Mathematics to the Reality
Special-Student-7-Hypothesizer
As I look at a Japanese elementary school mathematics curriculum, children 1st learn natural numbers with their additions, subtractions, and multiplications, and then, their divisions together with fractions and decimal fractions, but intermingled with the process, children learn measurements.
Special-Student-7-Rebutter
Theoretically speaking, learning 1st collections (and maybe sets), the natural numbers set, the integral numbers set, the rational numbers set, and the real numbers set seems logical, but the curriculum does not go that way.
Special-Student-7-Hypothesizer
In fact, children do not learn negative numbers until junior high school, and as far as I judge from the curriculums for elementary school, junior high school, and high school that I am looking at right now, children do not seem to learn collections until high school.
Special-Student-7-Rebutter
Really? How could children learn functions or probabilities without the concept of collection?
Special-Student-7-Hypothesizer
I do not know; I am looking only at the curriculums not at any textbook, so, the concept of collection may be being cursorily introduced somewhere mingled in a topic.
Anyway, I wanted to say that 'measurement' is being introduced at the 1st stage of mathematics educations.
Special-Student-7-Rebutter
Well, 'measurement' does not seem any indispensable part of mathematics, or is 'measurement' really a part of mathematics?
Special-Student-7-Hypothesizer
We are not talking about the measure theory in mathematics, which is about measuring some subsets of a set, while 'measurement' here is about measuring the reality, which means appointing a number to the reality.
Special-Student-7-Rebutter
'measurement' here does not seem any part of pure mathematics, but is a basis of applying mathematics to the reality.
Special-Student-7-Hypothesizer
The reason why 'measurement' is learned at the 1st stage is that the school mathematics is not meant to teach pure mathematics but to teach applied mathematics.
Children learn natural numbers in order to count some apples (counting is a kind of measurement) and learn decimal fractions in order to measure the length of a side of a desk.
So, school mathematics does not follow the logical order of the construction of pure mathematics but jump to some urgent applications and conveniently pick up some mathematical entities necessary for the applications.
2: Measurement Is to Represent an Aspect of the Reality by a Mathematical Entity
Special-Student-7-Hypothesizer
In general, 'measurement' is to represent an aspect of the reality by a mathematical entity (typically, a number).
For example, for the reality in which a box contains some apples, a measurement gives 5.
I said "an aspect", because there are many boxes and many non-boxes in the reality and the measurement looked at only the specific box and cared only about the quantity of what are inside the box: we cannot exactly represent the whole reality by anything.
So, not only measuring some lengths by some rulers and measuring some weights by some scales but also countings are measurements.
3: Let Us Measure a Length
Special-Student-7-Hypothesizer
Let us measure the length of a side of a desk.
Special-Student-7-Rebutter
Do we need to?
Special-Student-7-Hypothesizer
Ah, that may sound like just a guff, but is really an important question; certainly, we do not need to measure the length, at least, as far as the desk is concerned: the desk exists without being measured, or more generally, the reality exists without being measured. We will measure the length just because that suits our convenience.
Special-Student-7-Rebutter
I am making sure of that point, because there are some people who are claiming "There is no reality except measurements.".
Special-Student-7-Hypothesizer
According to that claim, the desk did not exist before the human measured the desk, and at the instant the human measured the desk, the desk popped up in the reality, and furthermore, at the instant the human abandoned measuring the desk, the desk puffed out from the reality, ..., is that any serious talk?
Special-Student-7-Rebutter
Those people are adamant.
Special-Student-7-Hypothesizer
... Let us talk about it later.
Anyway, we use a ruler to measure the length.
Special-Student-7-Rebutter
Or a tape measure.
Special-Student-7-Hypothesizer
Yes, but that is not any issue here, so, let us call it a ruler.
An issue is that the ruler may be a meter-ruler, an yard-ruler, or any some-other-unit-ruler.
Accordingly, the measurement may be 0.91 (m), 1.00 (yard), or something.
Special-Student-7-Rebutter
With an error.
Special-Student-7-Hypothesizer
Yes, with an error, because the ruler is not so accurate (the production is not so perfect) and is deformed somewhat by temperature, moisture, gravity, e.t.c., your eyesight is not so accurate, e.t.c..
Students learn that it is crucial to specify the possible error in any measurement.
Anyway, the measurement depends on the unit you choose.
So, measurements are relative with respect to units.
But that does not mean that the reality is relative.
In fact, if you have changed from meter to yard, the measurement has changed, but the length is not changed just because you have chosen a fancy unit.
Special-Student-7-Rebutter
The seed of misleading is the definition of 'length'.
Special-Student-7-Hypothesizer
To simplify the argument, let us forget Relativity throughout this section.
Then, I mean by 'length' how much the 2 points are separated in the space in the reality.
Just because you have used a yard-ruler, the 2 points have not become less separated.
In fact, if you thought that the desk had shrunk just because you used a yard-ruler, you would be crazy; or when a man lives 1,000.000 km separated from his love, if he thought that adopting a unit, perhaps called "yearn", by which he was 0.000,001 yearn separated from his love, would shrink the distance to his love from him, he would be a romanticist.
Likewise, measurements' having some errors does not mean that the reality has some errors.
If you thought that the reality had an error just because the production of the ruler was not perfect and your eyesight was not accurate, you would be crazy, and egocentric, in addition.
Special-Student-7-Rebutter
I talked about "The seed of misleading", because if someone meant by "length" 'measurement of the length', "lengths" would be relative with respect to units.
Special-Student-7-Hypothesizer
We mean by "length" 'how much the 2 points are separated in the space in the reality', and when we mean 'measurement of the length', we use "measurement of the length".
In fact, calling 'measurement of the length' "length" is a sloppy abbreviation, and some people refuse to refrain from using such sloppy abbreviations, which causes confusions.
When a measurement of the length of a side of a desk is 1.00 m and a measurement of the length of a side of another desk is 1.00 yard, we do not say "The 2 desks have the same length.".
4: But How Does Relativity Change the Situation?
Special-Student-7-Rebutter
It is good so far for non-Relativistic cases, but how does Relativity change the situation?
Special-Student-7-Hypothesizer
Relativity does not say "the reality is relative" at all, although it is rather prevalently misunderstood to be saying so.
Special-Student-7-Rebutter
What does Relativity say about "the length of the side of the desk"?
Special-Student-7-Hypothesizer
Relativity says that the universe is a 4-dimensional spacetime.
Note that the spacetime itself is not relative in any way.
But when you begin to talk about a "space", what is the "space"?
Special-Student-7-Rebutter
The "space" seems to mean a 3-dimensional space.
Special-Student-7-Hypothesizer
When you talk about a 3-dimensional "space", you mean a cross section of the 4-dimensional spacetime.
But you can choose infinitely many cross sections of the 4-dimensional spacetime.
So, the "space" depends on your choice, which means that "space"s are relative with respect to your choices.
If someone finds difficult to imagine the 4-dimensional spacetime, he or she could imagine the spacetime as a 2-dimensional plane, as a metaphor.
Then, a "space" would be a line of your choice on the plane.
The 2 ends of the side of the desk draw 2 so-called "world line"s in the spacetime, and when you choose a "space", each of the world lines intersects the "space" at a point, and when you say "how much the 2 points are separated in the space in the reality", "the 2 points" are the 2 intersections.
As the 2 intersections depend on your choice of the "space", the length depends on your choice of the "space", which is what Relativity says.
Special-Student-7-Rebutter
Whatever "space" you choose, that does not change the reality at all, which is parallel to that choosing a unit of length does not change the reality.
Special-Student-7-Hypothesizer
'choosing a "space"' is typically related with choosing a 3-dimensional coordinates system: the 3-dimensional coordinates system at a "time" is really a cross-section of the 4-dimensional spacetime, which is called "space", and the 4-dimensional spacetime is sliced into such parallel "space"s with different "time"s.
And some 2 3-dimensional coordinates systems moving against each other have sliced the 4-dimensional spacetime in some different directions.
Special-Student-7-Rebutter
Is it not bad to adopt a 3-dimensional coordinate system for the 4-dimensional reality?
Special-Student-7-Hypothesizer
In fact, it is misleading if not absolutely bad: people tend to imagine as though "the space" is developing through "time", but your "space" is just your choice, not any objective existence.
Now, you should understand why the 2 coordinates systems that are moving against each other measure the side of the desk differently.
In the 2-dimensional spacetime metaphor, the 2 ends of the side of the desk draw some parallel world lines on the 2-dimensional plane (let the plane be the x-y plane and let the 2 world lines be x = 1 and x = 2), and the 1st coordinates system is the bunch of y = t lines and the 2nd coordinates system is the bunch of y = x + t' lines.
Then, the 1st coordinates system sees at t = 0, the 2 ends of the side of the desk at (1, 0) and (2, 0), while the 2nd coordinates system sees at t' = 0, the 2 ends of the side of the desk at (1, 1) and (2, 2).
So, the 1st coordinates system talks about the length between (1, 0) and (2, 0) and the 2nd coordinates system talks about the length between (1, 1) and (2, 2), and it is not mysterious that they report some different measurements, because the 2 coordinates systems are talking about the different pairs of points.
Special-Student-7-Rebutter
As a caveat, the 4-dimensional spacetime has the Lorentzian metric, not the Euclidean metric.
Special-Student-7-Hypothesizer
Yes, "the Euclidean metric" means that the length between \((x_1, x_2, x_3, x_4)\) and \((x'_1, x'_2, x'_3, x'_4)\) is \(\sqrt{(x'_1 - x_1)^2 + ... + (x'_4 - x_4)^2}\), but "the Lorentzian metric" does not say so (we will not discuss here what exactly it says).
So, in the 2-dimensional metaphor, the length between (1, 0) and (2, 0) is not necessarily 1 and the length between (1, 1) and (2, 2) is not necessarily \(\sqrt{2}\).
So, we are not saying "the length between (1, 1) and (2, 2) is larger than the length between (1, 0) and (2, 0)" but are saying just that it is natural that the 2 coordinates systems report different measures.
Special-Student-7-Rebutter
But the reality as the 2-dimensional plane is not concerned with what coordinates systems a crazy human has chosen or what measurements are with respect to such coordinates systems.
Special-Student-7-Hypothesizer
As some 2 rulers with different units measure the desk differently, some 2 rulers moving against each other measure the desk differently, which is no mystery: it is about that just like the 2 rulers with different units are different rulers, the 2 rulers moving against each other are different rulers, and it is natural that some different rulers measure differently.
In fact, an observer (a coordinates system) flies by the earth with a high speed relative to the earth and measures the length between the 2 lovers as 1 cm, but how is that a consolation for the poor romanticist?
5: The Reality Is No Uncertain
Special-Student-7-Hypothesizer
Another prevalent misunderstanding is "the reality is uncertain, by the Heisenberg uncertainty principle".
No.
The Heisenberg uncertainty principle is claiming that any simultaneous measurement of each of some pairs of observables is uncertain.
Special-Student-7-Rebutter
What are uncertain are some measurements not the reality.
Special-Student-7-Hypothesizer
And there is also a prevalent misunderstanding that the break of Bell's inequality has proved "the uncertainty of the reality".
No.
The break of Bell's inequality is also suggesting the uncertainty of some measurements.
Special-Student-7-Rebutter
We need to distinguish the reality and its measurements.
Special-Student-7-Hypothesizer
The reality exists without being measured and does not need to be measured.
For example, did the Big Bang need to be measured for it to exist?
Was the Big Bang uncertain until a human measured it?
Special-Student-7-Rebutter
Has the Big Bang been measured even now?
Special-Student-7-Hypothesizer
It seems impossible to fully measure the Big Bang, while some humans have measured a tiny part of the aftermath of the Big Bang.
So, does the Big Bang become somewhat a little more certain at the instant a human measures a tiny part of the aftermath, retroactively?
Special-Student-7-Rebutter
In fact, any measurement is really an aftermath of what was intended to be measured in the reality.
Special-Student-7-Hypothesizer
When a human tries to measure an aspect of the reality, the human puts an apparatus near the aspect, and the apparatus and the aspect interact somehow, and the result of the interaction is the measurement.
If you claim that the aspect is determined retroactively because of the interaction, that is a violation of causality.
Special-Student-7-Rebutter
I thought that even quantum mechanics did not refute causality.
Special-Student-7-Hypothesizer
If the laws that govern the reality are probabilistic, that interaction is probabilistic, and the measurement is not the exact representation of the aspect.
Bell's inequality is based on the assumption that measurements are the exact representations of the aspects of the reality, and its break is suggesting that the assumption is wrong.
Note that Bell's inequality was introduced for the purpose of supporting Mr. Einstein's claim, "God does not play at dice", and naturally supposed that the laws of the reality were deterministic, which would imply that measurements as interactions were deterministic, which the results seem to have refuted.
Although there are some prevalent misstatements like "The reality is not determined until it is measured.", absolutely no, 'measurements are not determined until the measurements are performed' seems a more correct statement, which is natural because any measurement is a physical phenomenon and as far as the laws of the reality are probabilistic, the outcome of the measurement is not determined until the outcome happens.
6: Measurements Are Harassments
Special-Student-7-Hypothesizer
Let us think of an allegory.
You, an Earthian, are captured by an alien species.
The alien species feel only 'fear' and 'anger', and so, they understand only 'fear' and 'anger', in fact, they cannot ever imagine whatever else except 'fear' and 'anger' exist in the reality.
You feel 'love', 'pity', 'joy', 'sadness', e.t.c., I hope.
The alien species measure you, which is to determine whether you are in 'fear' or in 'anger', for the alien species, because that is all they can ever imagine to exist.
So, the alien species harass you to force you to show 'fear' or 'anger', and they are satisfied when you show 'fear' or 'anger' saying "Oh, this maggot was in fear!".
They invent Bell's inequality and conclude that your state was not determined until they measured you, and they even begin to claim "We determine the reality!".
Special-Student-7-Rebutter
Of course, you were not in fear until they measured (harassed) you.
Special-Student-7-Hypothesizer
The alien species's conceptions are only "in fear or in anger", and when you are in pity, they declare "This maggot's state is uncertain." and claim "The reality is not determined until we measure it.".
Special-Student-7-Rebutter
"state" by them is the state brought about by their harassment, and certainly, the "state" does not appear in the reality until they perform the harassment.
Special-Student-7-Hypothesizer
And if your reaction to the harassment is probabilistic, the "state" was certainly not determined until they performed the measurement.
But that does not mean you had not been in an absolutely definite state of pity before the harassment was performed.
Special-Student-7-Rebutter
We should be aware that human concepts like "position", "spin", e.t.c. are derived from pathetically limited human daily experiences, and Earthians who are trying to measure the reality in such concepts are like the alien species who are trying to measure you in 'fear' or 'anger'.
7: Isn't There a Misunderstanding About 'Scientific Method'?
Special-Student-7-Hypothesizer
Isn't there a misunderstanding about 'scientific method'?
Certainly, science values and requires measurements, because just willfully claiming something without any measurement does not give any credibility.
But science is also based on the assumption of the existence of the objective reality.
In fact, if there was no objective reality, what would we measure?
I thought that as we supposed the existence of the objective reality, we were trying to measure the reality. If measurements are relative and uncertain, that is not convenient for us, but we must accept it, and we try to approach the objective reality as near as possible, if not we reach the objective reality, through trial-and-errors of making hypotheses and checking them with measurements, which is the scientific method, I thought.
The scientific method is never "There is no reality except measurements.", I thought.
Special-Student-7-Rebutter
If "There is no reality except measurements", why don't we just do willful measurements to get desired results? We would not need to pay any respect to non-existent reality, so, any willful measurement would be as good as any measurement.
Special-Student-7-Hypothesizer
'accuracy' is the difference between the objective reality and the measurement, and if there was no objective reality, we would not need to care about any accuracy, because there would not be such thing as 'accuracy', because there would be not anything with which the measurement is compared for accuracy.
So, we would concoct an apparatus that would give '1' anyway, and we would find a golden theory, "Measurements are always '1'.". ... Why not? As only we were creating the reality, why wouldn't we create the willful reality?
Or you should just stop any measurement, and there would be no reality, and "There is no reality." would be the absolutely complete theory.
Special-Student-7-Rebutter
Sun will burn out in short time, Earthians will perish with it, and just because pathetic Earthians will perish on a pathetic remote tiny planet, will the reality disappear because measurements will not be performed anymore?
8: A Bad Moral Implication
Special-Student-7-Hypothesizer
In fact, "There is no reality except measurements." is the root of every evil.
"More people came to my inauguration than to the predecessor's!" would be an "alternative truth", because that perception was a measurement and there would be no objective reality to refute the measurement.
There would be no lie, because there would be no objective reality to compare with for the accuracy of the statement.
So, a villain would claim anything willfully with indemnity.
Sounds familiar?
Blackguards cite some misleading YouTube videos or TV programs that claim that Relativity and Bell's inequality have proved "the relativity and uncertainty of the reality" to justify themselves, saying "So, there is no objective truth, so, whatever I say are as truthful as whatever anyone says.".
...
Instead, we are supposed to accept the existence of the objective reality and be humble that our present claim is somehow relative and uncertain, so, we need to be ready to listen to claims by others to remedy our claim to make it more impartial.
Special-Student-7-Rebutter
Relativity and Bell's inequality are in fact some opportunities for us to be humble if they are understood properly, but with their understood improperly (via improper promotions), they are enticing absolute arrogance.
9: Conclusion
Special-Student-7-Hypothesizer
After all, what good children need to engrave in their minds at this point is the relation between the reality and measurements.
The reality exists independent of any measurement and measurements are relative and uncertain, without that presumption, any science, including not only natural sciences but also social sciences, would become non-science.
Special-Student-7-Rebutter
While there are many people who claim that they do not need to study mathematics ("because mathematics is not any useful for their daily lives", they say), there are some things understood best (or probably only) via mathematics.
