Water is not continuous, because it is made of molecules. Even if we ignore the fact, "Water is continuous." is mathematically nonsense.
Topics
About: elementary school mathematics
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Water Is Not Continuous
- 2: Continuousness Cannot Be Defined as a Property of Set
- 3: Continuousness is About Measure
- 4: What Is Discrete Measure?
- 5: Continuousness Is Not About Covering a Line
- 6: Continuousness May Be About Map
- 7: What Should Be Taught to Low-Level Schools Students
Starting Context
- The reader knows the background of this site.
Target Context
- The reader will know what 'discrete' or 'continuous' means in mathematics.
Orientation
There is an article on becoming a benefactor of humanity by being a conduit of truths
Main Body
1: Water Is Not Continuous
Stage Direction
Special-Student-7-Hypothesizer has a paperback in his hands, turning over some pages back and forth.
Special-Student-7-Hypothesizer
This book, an enlightening book on mathematics (a good one in that it arouses interests in mathematics, whatever we say about it hereafter), says that there are discrete amounts and continuous amounts, and cites water as an example of continuous amounts.
Special-Student-7-Rebutter
I do not understand a bit: while any chunk of water is a set of molecules, how is water a "continuous amount"?
Special-Student-7-Hypothesizer
Obviously, the author ignored the fact that water was made of molecules.
Special-Student-7-Rebutter
Then, what water was supposed to be?
Special-Student-7-Hypothesizer
Water seems to be have been supposed to be what the ancients imagined to be.
Special-Student-7-Rebutter
What did the ancients imagine?
Special-Student-7-Hypothesizer
Well, ... I do not know how to state it.
Special-Student-7-Rebutter
But modern mathematics says that the chunk of water is made of some points anyway, even if the points are not suppose to be the molecules, does it not?
Special-Student-7-Hypothesizer
I think so: the only way of describing the chunk of water by modern mathematics is to regard the chunk of water as a set of some points, whether the set is a set of finite number of molecules or a set of infinite number of points, as far as I know.
Special-Student-7-Rebutter
And how is the set of infinite number of points "continuous amount"?
In the 1st place, what does "amount" mean? Is the set itself "amount", is the number of points "amount", is the volume of the chunk "amount", is the mass of the chunk "amount", or what?
Special-Student-7-Hypothesizer
The book seems to be saying that the chunk of water itself is a "continuous amount".
Special-Student-7-Rebutter
I do not understand a bit.
Special-Student-7-Hypothesizer
The book compares the chunk of water with a set of apples and says that water is continuous because while each element of the apples set is separated and independent from the other elements, that is not the case for the chunk of water.
Special-Student-7-Rebutter
I disagree. Each point of water is separated from any other point: for example, denoting any point as \((x, y, z)\), any other point will be \((x', y', z')\) such that \((x, y, z) \neq (x', y', z')\), and while the distance of the 2 points, \(\sqrt{(x' - x)^2 + (y' - y)^2 + (z' - z)^2}\), is positive, how are the 2 points not separated? Besides, what does "independent" mean? Any distinct 2 points of any set should be independent, because they are different points: otherwise, they would be the same point.
Special-Student-7-Hypothesizer
What the author wanted to say seems that between any distinct points of water, there is another point of water.
Special-Student-7-Rebutter
That is the case also for the rational numbers set (I am not talking about the real numbers set) with the canonical order, but mathematics does not call the rational numbers set "continuous", as far as I know.
Special-Student-7-Hypothesizer
The book also says like "Water can be divided infinite times to still be water and when any 2 chunks of water are united, the union is seamless.".
Special-Student-7-Rebutter
I sincerely disagree: any single point of water cannot be divided, while any infinite set of apples can be divided infinite times: for example, supposing that the apples are indexed with the natural numbers, divide the set into the even-indexed set and the rest; divide the even-indexed set into the 4-multiple-indexed set and the rest; and so on; seamless? but the distinction of whether a point is from the 1st chunk or from the 2nd chunk still exists, and I call the distinction "seam": it is just a matter of that we think of the imaginary partition board in the united water.
Special-Student-7-Hypothesizer
Certainly, whether a set can be divided infinite times is just about whether the set is infinite; whether a set is seamless is just about whether the union is of sets with some common points: if there is no common point, the union has the seam defined by the origins of the points.
Special-Student-7-Rebutter
Then, how is the chunk of water "continuous amount"?
Special-Student-7-Hypothesizer
The book also says that the elements of the apples set can be counted, while the elements of the water chunk set cannot be counted.
Special-Student-7-Rebutter
Huh? Are we still talking about continuousness? That was about the distinction between countable sets and uncountable sets.
Special-Student-7-Hypothesizer
Mathematics distinguish countable sets and uncountable sets but does not call uncountable sets "continuous".
2: Continuousness Cannot Be Defined as a Property of Set
Special-Student-7-Hypothesizer
Continuousness cannot be defined as a property of set, because any pure set is just a collection of elements without any relation between elements assumed.
Special-Student-7-Rebutter
The word, "continuous", insinuates relations between elements, but the concept of 'set' has eliminated relations between elements.
Special-Student-7-Hypothesizer
In mathematics, abstraction is crucial: once any concept has been defined, it is important that any undue property that has been eliminated from the concept is never surreptitiously assumed.
Special-Student-7-Rebutter
The statement, "between any distinct points of water, there is another point of water", is not about any pure set, but about a set with an order, while the order is an extra structure put onto the set.
Special-Student-7-Hypothesizer
To distinguish concepts clearly is the 1st step for clear thinking, and mathematics is arguably the best way for teaching making clear distinctions of concepts, but lower-level schools are missing the opportunity.
In fact, whether one is good with mathematics is not about being good with numbers (mathematics is not always about numbers), but about being good with making clear distinctions of concepts.
In fact, arguing with a mathematics-hater is very futile, because he or she incessantly confuses things and his or her arguments are just muddles.
Special-Student-7-Rebutter
And there are so many mathematics-haters ...
Special-Student-7-Hypothesizer
Continuousness is not about set or about set with order: certainly, we can think of whether another element exists between any 2 distinct elements in a set with an order, but mathematics does not call that property "continuousness".
Special-Student-7-Rebutter
I understand that all cannot be taught in lower-level schools, but I claim that no lie should be taught in whatever level.
Special-Student-7-Hypothesizer
Not teaching something is OK, but teaching a lie is not OK, while "Water is continuous." is a lie.
Special-Student-7-Rebutter
Such lies may be "easy" for many unenthusiastic students, but sincere students will be confused by such lies, because lies do not make sense if sincere students contemplate them.
Special-Student-7-Hypothesizer
Education on the Bias Planet is giving priority to pampering unenthusiastic students over not-confusing sincere students, because unenthusiastic students are many while sincere students are very few: it is called "democracy".
3: Continuousness is About Measure
Special-Student-7-Hypothesizer
In fact, continuousness is about measure.
Special-Student-7-Rebutter
What is 'measure'?
Special-Student-7-Hypothesizer
Any measure is an extra structure put onto a set that (the extra structure) is defined like this: we think of a \(\sigma\)-algebra, which is a subset of the power set (the set of all the subsets of the set) of the set, and map to each element of the \(\sigma\)-algebra an element of \([0, \infty]\), while the \(\sigma\)-algebra and the map have to satisfy certain conditions.
That may sound enigmatic to many students, but for example, the method of taking volumes of water chunks is a measure.
The reason for choosing a \(\sigma\)-algebra instead of using the power set itself is that we do not need to measure all the subsets (so, we call each element of the \(\sigma\)-algebra "measurable subset").
Special-Student-7-Rebutter
That is the reason why I asked what "amount" meant: "Water is continuous." is nonsense because any measure is not specified.
Special-Student-7-Hypothesizer
Note that we can think of many measures for the same set: we can take a measure that takes volumes, a measure that takes masses, the measure that counts numbers of points (called "counting measure"), and so on; if we choose different units of volumes or masses, the measures are different.
Special-Student-7-Rebutter
Can we take the counting measure for the chunk of water while the set is uncountable?
Special-Student-7-Hypothesizer
Yes we can. The counting measure maps to each infinite measurable subset \(\infty\) and maps to each finite measurable subset the number of the points, so, the set's being uncountable does not prevent the counting measure from being valid.
Special-Student-7-Rebutter
So, although the book is as though the chunk of water cannot use the counting measure, that is not true.
Special-Student-7-Hypothesizer
That is certainly not true, although most people do not use the counting measure for daily purposes.
Special-Student-7-Rebutter
And how is a measure 'continuous'?
Special-Student-7-Hypothesizer
Any measure is continuous if and only if each single point subset measures 0.
Special-Student-7-Rebutter
That is rather unexpectedly simple.
Special-Student-7-Hypothesizer
The volume measure for water is indeed continuous, because the volume of each single point subset is 0.
On the other hand, the counting measure for the chunk of water or the apples set is not continuous, because each single point subset measures 1.
Special-Student-7-Rebutter
So, "Water is continuous." is indeed a nonsense, because the chunk of water can have a non-continuous measure.
Special-Student-7-Hypothesizer
As continuousness is about measure, we need to specify what measure we have chosen.
What the book should have said is "For water, humans usually choose continuous measures, practically speaking.", while the reasons why such a measure is continuous the book cites (like "points are separated"; "infinitely divisible"; "elements are uncountable") are all wrong.
4: What Is Discrete Measure?
Special-Student-7-Rebutter
What is discrete measure? Is every non-continuous measure discrete?
Special-Student-7-Hypothesizer
No.
'Discrete measure' is defined to be any measure such that there is a countable measurable subset whose complement (which is guaranteed to be a measurable subset by the definition of \(\sigma \)-algebra) measures 0.
Special-Student-7-Rebutter
In the case of the counting measure for the apples set, ...
Special-Student-7-Hypothesizer
As the whole set is countable, the "countable measurable subset" can be taken to be the whole set, and the complement is the empty set, which measures 0. So, the counting measure for the apples set is discrete.
Special-Student-7-Rebutter
So, the counting measure for each countable set is discrete.
How about the counting measure for the chunk of water?
Special-Student-7-Hypothesizer
There is no such a countable subset, so, is not discrete.
Special-Student-7-Rebutter
So, the counting measure for the chunk of water is not continuous or discrete.
Special-Student-7-Hypothesizer
Also the volume measure for the chunk of water is not discrete.
5: Continuousness Is Not About Covering a Line
Special-Student-7-Rebutter
What the word, "continuous", reminds most people of may be 'covering a line'.
Special-Student-7-Hypothesizer
Ah, but mathematics does not call it "continuous", as far as I know.
And note that "divisible infinite times" or "having a point between any 2 points" does not imply 'covering a line'.
In fact, the rational numbers set on a line is 'divisible infinite times' and 'has a point between any 2 points', but it does not cover the line.
And note that the real numbers set is just a set and the points do not have inherent locations: if you imagine that the real numbers set is on a line, that is just because you are imagining so: you can also imagine the real numbers scattered around the universe.
Special-Student-7-Rebutter
When you talk about the real numbers set on a line, you are talking about the Euclidean metric space, not about the pure real numbers set.
Special-Student-7-Hypothesizer
Mathematics talks about 'connected metric (or topological) space' but not about "continuous metric (or topological) space", as far as I know.
Any connected topological space (any metric space is canonically a topological space) is defined to be a topological space that is not the union of any disjoint nonempty open subsets, which may not be understood by people who have not learned topology, but you should at least know that criteria like "divisible infinite times" do not define 'connectedness'.
Of course, 'covering a line' is an important issue that historically prompted real numbers, but that is different from being continuous.
6: Continuousness May Be About Map
Special-Student-7-Rebutter
There is also the concept of 'continuous map', right?
Special-Student-7-Hypothesizer
Yes, we have talked about continuous measure so far because the book should have meant it, but "continuous" is used in some other cases.
"continuous map" is arguably the most famous one of them.
Special-Student-7-Rebutter
What does that mean?
Special-Student-7-Hypothesizer
Continuousness is in fact talked about each point of the domain of the map, and means that for each neighborhood of the value point on the codomain, a neighborhood of the domain point is mapped into the codomain neighborhood (the knowledge of 'topology' is required in order to understand that). Continuous map means that the map is continuous at each domain point.
Special-Student-7-Rebutter
How about "discrete map"?
Special-Student-7-Hypothesizer
I have not heard that concept.
7: What Should Be Taught to Low-Level Schools Students
Special-Student-7-Rebutter
But what should be taught to low-level schools students? I mean, should they be taught the exact conditions for \(\sigma\)-algebra or measure?
Special-Student-7-Hypothesizer
I do not particularly say so.
In fact, the concept, 'continuous', does not need to be introduced to them.
Special-Student-7-Rebutter
Does it not?
Special-Student-7-Hypothesizer
The real purpose is just to convince them that natural numbers are not enough; they need fractions and decimals. So, just convince them so, without introducing 'continuousness'.
Special-Student-7-Rebutter
Certainly, 'continuousness' is not indispensable for introducing fractions and decimals.
Special-Student-7-Hypothesizer
As an example, line segment length is better than water volume, because water really consists of molecules, which complicates the situation.
Special-Student-7-Rebutter
But students need to know about water volume anyway.
Special-Student-7-Hypothesizer
Students definitely need to know about approximations. While water really consists of molecules, we make an approximation when we measure the volume of any chunk of water.
Special-Student-7-Rebutter
Yes, 'approximation' and 'error' may be the most important concepts that even low-level schools students need to learn.
Special-Student-7-Hypothesizer
While any chunk of water is made of some molecules and the molecules are moving around, what is indeed the volume of the chunk of water? ... However we exactly define 'volume', the volume is really wobbling, and your measured volume has an error, partly because the volume is wobbling and partly because your act of measuring is inaccurate.
Special-Student-7-Rebutter
It is crucial to understand that that does not mean that the reality itself is ambiguous: you cannot say that the reality is ambiguous just because you cannot measure the reality accurately.
Special-Student-7-Hypothesizer
Unfortunately, that is what is not understood even by many renowned scientists.
Special-Student-7-Rebutter
What should be taught about 'set'?
Special-Student-7-Hypothesizer
Any set is just a collection of some elements with no extra property assumed, and when you measure a set, you need to define an extra structure, which even low-level schools students need to learn.
Special-Student-7-Rebutter
At least, the students need to understand that the chunk of water does not automatically dictate any measure: there can be a volume measure, a mass measure, the counting measure, and a fancy measure someone can concoct, and they need to specify the measure they are going to talk about.
Special-Student-7-Hypothesizer
That is really a matter of of course: without our choosing how to measure the chunk of water, the value of the measuring does not appear, but that knowledge seems to be supposed to be difficult to be understood.
Special-Student-7-Rebutter
As there can be multiple measures for a set, the concept, 'set', has to be established independent of any measure. Without understanding that way of thinking, mathematics cannot be really understood.
Special-Student-7-Hypothesizer
In fact, that way of thinking is the thing that students most need to learn, but they are being pampered on the excuse, "They won't understand anyway.".
