A hole is defined by its boundary and volume. A hole does not exist on its own, and must be surrounded or constrained by something to fulfill its definition of a hole. A hole is not a non-value, but a negative value.

Consider a flat ground of dirt, and you decide to take a shovel and dig a hole in the dirt. What you have done is taken the equilibrium of the flat ground and disrupted it by displacing a certain amount of dirt from a certain location onto another location. The dirt you excavated has a positive value, and the hole you left behind has an opposite and equal negative value. If you take the excavated dirt and place it back in the hole, you now have restored the equilibrium of the ground back to zero.

ground = 0
hole = -1
excavated dirt = +1

Happy to know that there are members who listen to Andalusian nanas. Hespèrion is a fantastic group, and Jordi Savall is a master. They also have interesting songs from the Balkans. :up: — javi2541997

I agree, and i'm glad to know we share a common point of resonance. :smile: :up:

Hespèrion XXI - Nana Andaluza Duerme Mi Niña


Hespèrion XXI - La Guirnalda de Rosas una Matica de Ruda

All creation or creativity involves assembling elements in novel configurations. Now, how to do it well? Some suggestions:

Consider looking to nature, and observe her creative process, where existing elements are amalgamated, fused, or 'accreted' to form new entities such as atoms, stars, planets, molecules, cells, etc. Similarly, human creativity follows this pattern, albeit with thoughts as the building blocks that are 'accreted' together to generate innovative ideas and creations.

Long ago, i came across a story about an alchemist who employed a creativity technique consisting of two phases. The initial stage, was the absorption phase, and it involved the alchemist immersing himself entirely in his chosen subject. This entailed extensive reading and engaging with individuals possessing knowledge or insights related to his area of interest. This immersive phase typically lasted several months. The subsequent phase involved deep contemplation, reflection, and experimentation, during which the alchemist meticulously pondered all the information gathered during the absorption phase. It was within this contemplative stage that he purportedly experienced moments of creative inspiration and innovation.

Furthermore, adopt a multidisciplinary approach. Explore how seemingly unrelated concepts can intertwine with your creative pursuits. Embracing diverse perspectives and fields of study can often lead to unique and innovative solutions.

Most importantly have fun with it. Play with ideas as a child plays with toys, and new things will begin to occur to you. Also, always listen to your intuition, and don't let anyone shutdown your intuition; learn to listen to it closely and carefully.

Faith and reason are in complete opposition. One believes what it wants, and the other believes what it must. It is like having two masters with opposite minds, so one must always betray one to serve the other at any given time. The basis for faith is, at a minimum, desire or emotion and, at most, dogma. The basis for reason is, at a minimum, evidence and, at most, proof.

Jeremiah 17:9
"The heart is deceitful above all things"

Matthew 6:24
"No one can serve two masters; for either he will hate the one and love the other, or he will be devoted to one and despise the other."

"Man prefers to believe what man prefers to be true." - Sir Francis Bacon

Thank you, "Bohren & Der Club Of Gore - Maximum Black" has now been added to my collection.

Matthew Halsall - Fletcher Moss Park

Matthew Halsall and The Gondwana Orchestra - Journey in Satchidananda


Matthew Halsall - Daan Park


Matthew Halsall - Samatha (Poetree Remix)


Matthew Halsall and The Gondwana Orchestra - Kiyomizu-Dera

I'm sorry to say that the concept of "natural numbers" (counting numbers) should be abolished. It is logically inconsistent and causes confusion as to the true nature of number.

"The answer to the ultimate question is..........................42." — Fire Ologist

Perhaps the question to the ultimate answer is ..................... 2 × 3 × 7 = ?

Let me provide another, but different illustration that shows why counting and even numbers themselves begin at 0. Consider the two following number sequences and their logical progression:

Sequence A:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
10, 11, 12, 13, 14, 15, 16, 17, 18, 19

Sequence B:
1, 2, 3, 4, 5, 6, 7, 8, 9
11, 12, 13, 14, 15, 16, 17, 18, 19

Questions:
Which one of the above number sequences is correct, and which one is incorrect, and why?
What numbers are missing and why?
What is the relationship of the number 10 to the number 0?

programmatically in Python:

rocks = 0 # beginning at 0
while rocks < 2:
rocks += 1
print(rocks)

output = [1, 2] # correct output


rocks = 1 # beginning at 1
while rocks < 2:
rocks += 1
print(rocks)

output = [2] # incorrect output — punos


Well, what I see here is that you are saying 1+1 is not 2. So I don't know where you're coming from. — Zolenskify

In the first example, since it is starting from a value of 0, it enters the counting loop one time, adding 1 to 0 (0+1=1). Then it enters the loop again and adds 1 a second time to the last value result [1], updating the count result to [2]. This is why the first example has an output of two numbers [1, 2], because it counted twice.

In the second counting loop example, since the starting value is already 1 before the first count, the loop simply counts 1 time, resulting in one count of [2]. This loop added 1 to 1 (1+1=2), resulting in a value of [2]. That is why you only see one number as the output result [2], because it only counted one time.

In both cases, 1 + 1 = 2, but what you are neglecting to see is that the true operation was 0 + 1 = 1 (first count), and 1 + 1 = 2 (second count).

I'm coming from 0. That's where i'm coming from.

But I want to know what you think. — Zolenskify

I believe that is what i've been doing.

I do not follow how this differs from my example, only that you now add the condition that the beginning of the day is 0. — Zolenskify

This is your example:
"I pick up a rock, I have one rock. Pick up another, have two. Drop them, have none. Only started counting rocks when I picked up the first one. So, numbers start at one."

I would restate your example as:
"I have zero rocks, I pick up a rock, I have one rock. Pick up another, have two. Drop one and I have one, drop that one and I have zero, right back where I started."

But an interval of time is always passing, and can't really be counted in terms of starting and stopping. — Zolenskify

It is not time itself that stops or starts; it is you who starts counting and then stops. You are not trying to count all of time, which is infinite and thus impossible, but just the duration (temporal space or distance) of some finite phenomena. A count result can be defined as how many 'times' (time) a single count was made. Counting is an activity, which means it has a temporal dimension.

A stopwatch can do this for practical reasons, but we are then changing what time means because we are now only looking at it in terms of evaluating some other dependent variable. — Zolenskify

I don't see how the meaning of time changes when we count cycles of time. What do you mean by dependent variable in this context?

Say that we are now counting these "spaces" instead of the rocks. That "space" just becomes the object we are counting. — Zolenskify

That is precisely what i am suggesting as a representational placeholder for any object being counted. It's not a rule, but i find that conceptualizing it this way affords me a more accurate way of understanding what is happening when counting happens. There is, in any way you think about it, a kind of separation between numbers, or if not, then we would not have numbers. I think it is a more rigorous way of thinking about numbers, anchoring the concept of numbers closer to our physical experience of the world in a spatial sense.

So swapping these two objects still allows for my argument to hold. Thank you for these thoughts. — Zolenskify

For your argument to hold i believe it needs to start from 0 in order for the first count to be 1.

You're welcome, and thank you as well for your thoughts.

If roller skating is your only option, then you should probably pursue it, but you should also consider other proximally interesting options. You might want to have a non-athletic career as a backup in case you get injured in a way that prevents you from continuing a skating career.

Additionally, you should think about the possibility of AI robots taking over the sport in the near future. With rapid recent advancements in deep-skate network architecture, human skaters may soon become altogether obsolete in the sport of roller skating.

> end transmission
> exit code 1

counting models that do not start at zero are very counter-intuitive for me — Dawnstorm

As they should be.

It's not the numbers that start; it's the counting (an action that can start and end). Counting starts at either 0 or the last count result, but 0 is not a number that represents a count, which is why you don't need it to represent a counted item. Your first count is 1, and thus your first represented count starts with 1.

Perhaps it would be helpful to think about what you are counting as the space between the numbers. For example, 1 = (0 to 1), 2 = (1 to 2), 3 = (2 to 3), where each number represents the full space between one number and the next. Instead of counting points on a number line, you are counting spaces (or distances) between the numbers. Both ways work, but the latter method shows you what you are counting as represented by the spaces in between. Consider what your count would be if you had 2.5 items? Is it 2 or 3 or 2.5? Just something to think about.

Knowing any of this is unnecessary for the average person, but as a philosopher or someone who wants to know the truth of things, this model i believe provides the most insight into what is actually happening when we count.

My model for counting can be represented simply as:

x = 0 (or the sum of the prior count)
x + 1 = 1 (x = 1) # count 1
x + 1 = 2 (x = 2) # count 2
...

This, in my view, is the most basic and universal model (algorithm) for counting, and any other model is either derivative or a simplification of this model. My (preferred) counting model yields an accurate count in any case you might apply it to, with the only modification, if necessary, being to the initial condition or quantity. Other models may work fine in some cases but not in all cases.

Another way to understand why one must start counting from 0 to count correctly:

Let us assume that we already have a count of 2 rocks. Instead of counting up to 2, we will count down 2 times (the number of rocks we have). Beginning at 2, we discount 1, leaving 1. Then, we count down 1 more time, and we reach 0. A down count of 2 brings us to 0. If we want to get back to 2, we count up 2 times (an equal and opposite operation) from 0 to 1 to 2.

Where you begin counting, and the first count are two different things. They are not the same. Everyone counts this way, but not everyone realizes they are starting from 0.

if I count 2 rocks do I count from 0 to 2 or from 1 to 2 — Dawnstorm

I think of it like this: before i count, i place my finger on 0 in the number line, and when i make my first count, i move my finger to 1 on the number line, and so on. That 0 tells me what i have before i start counting. If i place my finger at 1 on the number line before counting, then for my first count, my finger moves to number 2 on the number line. That 1 tells me what i had before i started counting. So the process of counting is adding to the prior count. Sometimes that count is 0 (no count), and sometimes it's more than 0.

(< 3) and (<= 2) are essentially the same in the context of the counting loop. Let's try (<= 2):

With print statement before the count:


output = [0, 1, 2] # incorrect output


output = [1, 2] # correct output


In this second counting loop, the number of rocks you start with is already 1, and you are printing your first "count" without having counted yet. So, the loop actually just counts 1 time to get to 2. Although the output is apparently correct, the logic behind the count is not. The loop would function as an accurate rock counter nonetheless.

With print statement after the count:


output = [1, 2, 3] # incorrect output


output = [2, 3] # incorrect output


The reason i place the print statement inside these loops is so that we can see the process of counting as it happens. If i place the print statement outside the loop after it is done counting, then the result will be the same for both counting loops.


output = [2] # correct final output


output = [2] # correct final output

If i place the print statement before the count (rocks += 1) then:


output = [0, 1] # incorrect output



output = [1] # incorrect output


Both counting loops result in incorrect counts. The logic is that one should state the count after it is made, not before. Yes, it is context-dependent. When the context is counting rocks, then obviously the first loop i wrote is correct because it correlates with the results we get when we naturally count for ourselves. This is not the case for the other counting loops.

The relevance lies in the logic, not the programming language. There is a right and a wrong way to count. When counting rocks, it is essential to establish whether there are already rocks present. If i have 2 rocks and then pick up and count another rock, i will have 3 rocks (the count begins at 2). Conversely, if i don't have any rocks and then pick up and count 1 rock, i will have just 1 rock (the count begins at 0).

I have never used MATLAB, so i can't comment on what it's doing. If you have a code snippet that i can look at, that would be helpful. MATLAB isn't a general-purpose programming language, but if MATLAB uses 1 as its base index, it must mean that it doesn't represent a count until the first count is made. In Python, the count begins at 0 before the first count is made. I'm sure that MATLAB uses a 0-based index under the hood and creates a 1-based index after the first count.

And the fact that one piece of code counts "correctly" and the other "incorrectly" (like ChatGPT says) begs the question as to why one is right and the other wrong, which is the thread's topic. — Lionino

The reason one is right and the other is wrong is that one starts from 0 and the other starts from 1. If i wanted to count rocks with an app that uses the example code that starts with one, then my app would count rocks incorrectly. It's just that simple to verify, like counting horse teeth.

Also, notice that the output count did not include 0, although it's where it began counting. We forget 0 because it's usually invisible as our reference point before a count can begin. When you count, what you are doing is adding to 0, and you must start at 0, or your count will be off by the number that you started counting from.

I simply told the program to count 2 rocks, and yes, i told one to start from 0 and the other to start from 1. If you look closely, you will notice the two code examples are identical except for the starting or initial condition.

You are correct in that both code samples are logically correct in their execution, but as in logic, one began with a different 'premise' than the other. And we know that if you have an incorrect premise, then your conclusion would be incorrect no matter how flawless the execution of the logic.

The two code samples illustrate what happens when you start your count from 0 versus 1. Both cannot be correct for use in counting rocks.


Here is ChatGPT:
which one of these code examples is the correct one for counting rocks?

The first code example is the correct one for counting rocks. It initializes the variable "rocks" to 0 and then uses a while loop to increment the value of "rocks" until it reaches 2, printing the current value of "rocks" in each iteration. The output of this code is "1, 2." Here's the breakdown of the code:



This code will output:
1
2

The second code example initializes the variable "rocks" to 1 and then uses a while loop to increment the value of "rocks" until it reaches 2, printing the current value of "rocks" in each iteration. The output of this code is "2," which is not the correct one for counting rocks.

There is nothing correct or incorrect about the output, it is just the way you programmed it and how Python works. — Lionino

What is your criteria for correct or incorrect? Are you saying it doesn't matter how i program it, that i can choose either one, and it'll be as good as anything else? If that is the case then it would appears that no one is correct or incorrect about anything, because it is just the way they said something. Really?

programmatically in Python:


output = [1, 2] # correct output



output = [2] # incorrect output

Every computer programmer knows that counting begins at 0.

In this context counting physical objects like rocks can be a bit confusing to some because it's not always clear what exactly is being counted. On the other hand, counting time is more straightforward.

For example, when counting days, day 1 is considered at the end of the day, with the beginning of that day being counted as 0 and ending with count 1. The next day begins at 1 and ends at count 2. Therefore, 1 is the first complete count, but for this to be true, the count must begin at 0.

When counting rocks, what is actually being counted is the space the rock occupies. This can be seen as the space between 0 and 1 being counted as 1 (the counted entity is contained between 0 and 1). If a rock didn't occupy any space, there would be nothing to count, as there can't be a rock that takes up no space.

Just finished watching the series last night, and i do recommend:

Another:

Suppose that the universe has infinite space, and let's also say that there is an infinite number of particles in this space. For there to be space between the particles, would that not make space a bigger infinity than the infinite number of particles in the infinite space?

The statement that some infinities are bigger than others comes from set theory. The OP talks about infinities in the context of counting procedures. These are two different concepts of infinity. — DanCoimbra

Most of what i think about infinities comes from my own intuitions, so forgive me if i sound a bit ignorant of the well-established terms and procedures involved.

The "counting procedure" aspect is what i relate to the temporal sense of speaking or thinking about it. The other side seems to be more spatial in character, which instantiates an infinity all at once, outside time, so to say. It's just something i noticed recently and thought it might be useful to know when thinking about infinities. There are probably proper terms for these distinctions, and if there aren't then there should be.

Welcome to TPF! :smile:

Yea, like i said im not up on all the terminology. I'm a little bit motivated now to look a little deeper into it, because i do find it interesting. I'm going to look up some of these concepts you mentioned like transfinite cardinals, and ordinal arithmetic. But i'd like to ask.

What was i describing in my last example about the infinite hotel. What is the correct terminology for what i described?

Is it?

That's not Hilbert's paradox. — Michael

Yes, i am aware of that, but i didn't see the point in describing something one could just read anywhere. I was trying to show a different way of conceptualizing different sizes of infinities. That's all, but i'm more interested in if my example is a reasonable one or not.

Another example that just came to mind related to Hilbert's infinite hotel thought experiment.

Consider a hotel with an infinite number of rooms, all of which are occupied. Due to the infinite guests, there are no vacant rooms. However, if each room in the hotel were to magically double into 2 rooms, the hotel would then have an additional infinity of rooms to accommodate an extra infinity of guests. Although the number of rooms seems the same in both cases, the capacity differs in some sense. In the first case, no more guests can be accommodated, while in the second case, an additional infinity of guests can be accommodated. This doubling (spacial sense) can continue (temporal sense) indefinitely in both time and space.

If there is no end to something, how can another thing with no end be twice as large as it? Don't they both have no ends? — Philosopher19

I believe the the concept of infinity is often misunderstood because it can be applied to different contexts, such as time and space, which are not necessarily equivalent. To explore the differences in sizes between different infinities, let's consider a few thought experiments that illustrate how infinity can vary in magnitude.

First, imagine you have achieved immortality and are presented with two options: to receive $1 every day forever or $1 every year. Intuitively, you would choose $1 every day because, over the same infinite duration, you would accumulate more money. This illustrates that while both options extend to infinity in time, the rate at which you receive money differs, leading to a larger "size" of wealth in one scenario over the other.

Now, let's consider a spatial analogy. Imagine two pipes, both of infinite length, but one has a diameter of 1 inch and the other has a diameter of 10 inches. Despite their lengths being equally infinite, the pipe with the larger diameter has a greater volume. This demonstrates that even with one dimension being infinite, other finite dimensions can contribute to a difference in "size" or capacity.

Interestingly, if we were to expand the diameter of the pipe to infinity as well, we would lose the essence of what makes a pipe a pipe. To maintain its identity, certain characteristics, like diameter, must remain finite. This constraint allows us to differentiate between pipes of different diameters, even if their lengths are infinite.

Lastly, consider an infinite number of pencils, each 6 inches long, laid end to end to form a line of infinite length. If we compare this to another line composed of an infinite number of 3-inch pencils, both lines would stretch to infinity. However, if you were to take one pencil from each line, there would be a clear difference in their lengths. This paradox highlights that while the total lengths of both lines are infinite, the "size" of their components is different, and this difference is observable when comparing individual elements.

So, the concept of infinity can indeed vary in magnitude depending on the context. Temporal infinity can differ based on the rate or frequency of an event, while spatial infinity can vary when other dimensions are considered. These examples show that not all infinities are created equal, and it is the nuances in their properties that allow us to distinguish between them.

When considering a sequence of numbers, such as those between 0 and 4, where there are an infinite number of elements (including fractions and irrationals) between 0 and 1 and every other natural number in the sequence, the set of numbers between 0 and 2 would contain more elements than between 0 and 1. If one were to continuously move through this sequence of numbers, they would never reach the number 1 since one would have to move through an infinity of fractions first. In contrast, if the infinite sequence consists simply of natural numbers, reaching 1 would happen almost instantly.

This comparison illustrates that the "distance" covered between 0 and 1 in the first example is different from the "distance" between 0 and 1 in the last example. It can be concluded that the first example which includes the rationals and irrationals represents a larger infinity than simply an infinite series of natural numbers.

Does this explanation make sense? I admit that I haven't spent much time studying the intricacies of infinities and am not completely familiar with the technical terms and notations that mathematicians typically use to discuss these concepts.

Yes, if you were to measure both distances at a specific point in time, but outside the context of a finite time measurement, the distance is probably equally infinite for both.

I think that to make sense of infinities, one has to have a system for extracting their finite properties, as I mentioned in my prior post, or by looking, for example, at the difference between one element in a sequence and the next, which has a specific finite value. This specific value for example can be considered a fundamental component of a periodically regular sequence, by which any periodically regular sequence can be constructed, including infinite ones.