On the definition of number

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Divine Insight
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On the definition of number

Post #1

Post by Divine Insight »

What is the definition of Number?

I'm talking about the concept of number in general.

Is there only one definition?

Are there many different definitions?

If there are more than one definition then shouldn't you need to tell me which definition you are using every time you give me a number? Otherwise how would I know which definition you are using?

In a recent video math course I watched the definition of number was said to be a "Moving Target".

Does this make any sense?

Do we know what numbers are or not?

What is your definition of number?

Please feel free to offer either a formal definition of number or an intuitive definition, or both.
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Post #2

Post by keithprosser3 »

If there are more than one definition then shouldn't you need to tell me which definition you are using every time you give me a number? Otherwise how would I know which definition you are using?
As children we learn the meaning of the word 'number' the same way we learn the meaning of other words, such as dog or chair. It seems we all have an innate ability to identify 'common features' and to classify things accordingly. Thus we can classify an object as (for example) a chair or a dog, even if we have never seen that particular chair or dog before. Thus we develop an intuitive sense of number as the common feature of a collection of three chairs and three cats.

That intuitive meaning of 'number' has nothing to do with reading a dictionary definition nor even being told explicitly what 'number' is supposed to mean. But that intuitive meaning is adequate for most - perhaps nearly all - our communication needs.

I would guess most people go through school without ever having been introduced to a formal definition of a number. Rather their intuitive understanding of the word 'number' is gradually broadened by being introduced to 'negative numbers', fractions, decimals and maybe even binary and complex numbers. We end up knowing what a number is the same way we know what a chair is... we can't say what it is we recognise about an object that we use to identify it as a chair. We recognise something as a chair because it looks like a chair! Similarly we know what a number is because it looks like a number, without necessarily knowing what it is about 'the square root of -1' that makes us identify it as a number.

Of course a certain type of philosopher is not satisfied with intuitive meanings but want everything defined dictionary style. I am all for being precise, but I don't think a dictionary definition of what 'number' means is all that important. I managed quite a few decades without a formal definition and I can't say it worries me unduly now.

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Post #3

Post by Divine Insight »

keithprosser3 wrote: As children we learn the meaning of the word 'number' the same way we learn the meaning of other words, such as dog or chair. It seems we all have an innate ability to identify 'common features' and to classify things accordingly. Thus we can classify an object as (for example) a chair or a dog, even if we have never seen that particular chair or dog before. Thus we develop an intuitive sense of number as the common feature of a collection of three chairs and three cats.
That's exactly right. We were typically taught the idea of number using "Flash cards" with pictures of individual things on them. This is in fact, the correct idea of number. It's a comprehensible and meaningful idea. In fact, it's really the only idea we need.

Moreover, an important thing to realize here is that we relied heavily upon a prerequisite idea of what it means to recognize an individual thing. In other words, throughout this whole process, and ability to recognize a quantity of ONE was paramount.

After all, if you can't recognize the completeness of "One Thing" then trying to comprehend many things would be extremely difficult.

Another thing to realize here too is that the concept of number is a concept of "many or few" individual things. It's a concept of many or few objects, each of which has been recognized to have a property of Oneness.

Later, we moved on to flash cards with "numerals" on them. We then learned the names of the number quantities in terms of these numeral names. But the numerals themselves were not representing the actual ideas of quantity. The idea of quantity was the original idea of collections of individual things. That is the actual concept of number.

It's intuitive, comprehensible, and easy to understand. But unfortunately this idea is quickly overshadowed after the numeral names for numbers are introduced and we start to think of numerals as "numbers' which of course they aren't. They are just the symbolic names of quantities. Number is the idea of quantity. The idea of many or few individual objects.
keithprosser3 wrote: That intuitive meaning of 'number' has nothing to do with reading a dictionary definition nor even being told explicitly what 'number' is supposed to mean. But that intuitive meaning is adequate for most - perhaps nearly all - our communication needs.
I would argue that the intuitive idea is the correct idea of number. Other definitions often obscure this original concept. Especially things like the introduction of negative "numbers". That's already wrong. There is no such thing as a negative quantity. A quantity is just a collection of things. Whether it's viewed as positive or negative quantity has to do with relative properties associated with other things, such as other collections of things, or ownership of a collection of things, or a vector direction associated with a collection of things.

Teaching us that "Negative Numbers" exist in their own right is already a bad move, because it truly misses the point of what negativity actually means.

Negativity is a relative property between a collection of things and some other external circumstance. It's not an absolute property of quantity. It's a relative property that requires more information to make sense.

So we're already "off track" the moment we start teaching people that there is such a thing as a "negative number".

We simply don't need to do that. Instead we could teach them the TRUTH and explain what negativity truly is. It's a relative property that a collection of objects (a quantity of objects) can take on with respect to some other contextual situation.

That's what it TRULY is, why not teach it as such? :-k

And the same is true of imaginary numbers as well. It's just another vector like negative only in another direction altogether. But the fundamental idea of "number" that undies the whole thing hasn't change one iota.
keithprosser3 wrote: I would guess most people go through school without ever having been introduced to a formal definition of a number. Rather their intuitive understanding of the word 'number' is gradually broadened by being introduced to 'negative numbers', fractions, decimals and maybe even binary and complex numbers. We end up knowing what a number is the same way we know what a chair is... we can't say what it is we recognise about an object that we use to identify it as a chair. We recognise something as a chair because it looks like a chair! Similarly we know what a number is because it looks like a number, without necessarily knowing what it is about 'the square root of -1' that makes us identify it as a number.
Yes, but what we end up doing is having a bunch of "funny looking numerals", that we don't really understand outside of a whole bunch of rules and regulations. It's no wonder that so many people dislike mathematics. They soon get bogged down in a bunch of rules that don't intuitively make sense to them.

They begin to wonder what those original flash cards that had groups of objects painted on them have to do with something like the "square root of -1".

How do you take the square root of a minus cat for example? And what does it even mean to have a minus cat? Is it any wonder why people drop out of the maths in droves?

There simply isn't any need for this. We can still talk about the square root of -1 one in a meaningful way. But instead of pretending that it's "just another number", we could actually explain what it means is a way that is indeed intuitively comprehensible. It's actually quite easy to understand this concept entirely intuitively if you just take the time to realize that the imaginary symbol i that pops out of this is NOT a number, but rather it's a direction vector.

Sure we can associate this vector with a unit quantity of ONE. But the i itself is really just a vector, no unlike a negative sign. It's easy to comprehend and understand if taught for what it actually is instead of just pretending that it's just another number. That's just not necessary at all, and it can actually be quite confusing for someone who was trying to imagine the concept of number to be an idea of many or few. Now it's the idea of the square root of -1? What? :shock:

I mean, sure I understood the concept too when I studied the maths, but that's only because I quickly realized that it's just a vector. It's NOT a new idea of quantity at all. In other words, it's not a new idea of number. It's a new direction vector. That's all it is.

Why not teach it for the TRUTH of what it is? That's all I ask.
keithprosser3 wrote: Of course a certain type of philosopher is not satisfied with intuitive meanings but want everything defined dictionary style. I am all for being precise, but I don't think a dictionary definition of what 'number' means is all that important. I managed quite a few decades without a formal definition and I can't say it worries me unduly now.
It's my view that if a concept can be made intuitive and understood intuitively, then why bother making it ambiguous? :-k

Moreover, I would argue that if a concept cannot be intuitively understood then it is not a comprehensible concept. After all was does comprehension mean if not to intuitively understand something?

If we don't intuitively understand something, then we are lying to say that we comprehend it. At best all we can say is that we accept a bunch of rules that lead to conclusions that we don't truly understand.

Because after all, comprehension of something requires an intuitive understanding of it. If we can't say that we intuitively understand it, then we're not being honest if we claim to comprehend it.

What I am saying is that almost all of mathematics could indeed be intuitively understood if it was both taught, and restricted to addressing quantitative concepts as we intuitively recognize them in the real world.

Why we have chosen to make things more ambiguous than necessary is beyond me. We don't need negative "numbers" all we need are sets that have relative negative vector associated with them. And this is a truly easy idea that any grade school child can easily intuitively grasp. Even kindergarten children should be able to grasp this idea if taught correctly.

So much of mathematics could be easily defined, taught, and understood as very simple intuitive ideas with no "Abstract Magic" required.

It would actually be a better mathematics if constructed in this way as well.
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Post #4

Post by keithprosser3 »

Are you aware of this guy? LEJ Brouwer

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Post #5

Post by Divine Insight »

keithprosser3 wrote: Are you aware of this guy? LEJ Brouwer
Yes, but I don't agree with his concept of intuitionism.

I also don't agree with the concept of axiomatic systems based on an idea of "Pure" mathematics or a concept of "Pure" number.

The very concept of "pureness" is a concept of removing the idea of quantity from the idea of the thing that is being quantified.

This is why the mathematical community fell in love with Cantor's "Empty Set" idea. Cantor begins with the idea of an empty set which is totally removed from any physical concept and he builds a purely abstract notion of numbers from there.

However, this violates the very meaning of the concept of number that we intuitively learned.

What did we learn as children? We learned that the concept of quantity (or number) is an idea of a collection of individual things. This concept requires that we can recognize that individuality of the things in a collection.

The idea of "Pure Mathematics" or "Pure Number" is to remove the concept of quantity from the individual things being quantified. In other words, we're going to remove the "physical dependency of the quantitative individuality of things" from the equation altogether and replace that with an abstract notion of a "pure individual thing". That abstract notion became the notion of an "Empty Set". A collection of things which, by definition, does not contain a single solitary individual thing.

And this abstract notion of a collection of things which does not contain an individual thing becomes the model of a "pure mathematical individual thing".

Which is absurd.

Then we have the empty set being equated to Zero = 0.

One becomes the set containing the empty set, 1 = {0}.

We might intuitively think that two would then be the set containing two empty sets, 2 = {0, 0}

But no, that won't work in Cantor's magic land. We need to have two different objects before we can move forward so officially 2 = {0, {0}} The set containing the empty set, and the set containing the empty set. Or for more convenient notation, 2 = {0, 1}

And it grows from there.

3 = {0, 1, 2} and so on.

This actually starts to look like something as we build up. But it really doesn't work. It actually violates the laws of addition of sets. there is no way that you can add the sets 1 and 2 together and come up with set three using the rules of set addition.

Moreover, all this has done is remove the idea of physical quantity (which is based on the actual property of individuality of the things being quantified) and has created a totally abstract idea of individuality that has no real meaning at all.

It moves from being a quantitative idea to a qualitative idea. And the original idea of quantity is lost.

If we want to speak of number as being a quantitative property of a collection of things, then we can't toss out the all-important property of individuality of the things being quantified. But this is precisely what is done in Empty Set theory.

~~~~~

Here's another thing that's kind of interesting too. Godel's incompleteness theorem would not apply to a mathematics based on the correct idea of quantity. Because it would no longer be a self-referenced system which is paramount to Godel's proof.

Georg Cantor's Empty Set Theory is self-referenced and therefore is vulnerable to Godel's incompleteness theorem.

So there's yet another reason why keeping the idea of quantity firmly connected to the objects that are actually being quantified is paramount.

We must always keep track of that all-important property of individuality and how it relates to the things being quantified. We lost that with Cantor's Empty Set theory.

This is why a "Pure Mathematics" could never predict the complementarity of Quantum Mechanics. But a mathematics that never loses sight of the importance of the property of individuality of the things being quantified could predict that complementarity would be inevitable at some point.

A rock solid mathematics could have predicted Quantum Theory before physicists actually discovered it. Although they may not have been able to predict the exact value of Plank's Constant. But they could have predicted that it necessarily must exist at some level.

A "Pure Mathematics" could never do that. Because "Pure Mathematics" has removed the concept of number from the individual things that are actually being quantified. Cantor created a phantom "Individual Thing" in his idea of an "Empty Set".

It's a "Phantom" because it's an entirely qualitative idea rather than an actual quantitative idea.

Cantor basically divorced mathematics from describing the actual quantitative nature of reality and twisted it into becoming a totally abstract notion about a qualitative nature of reality that doesn't even have any actual meaning with respect to the quantitative nature of our physical world.

He totally removed mathematics from what it had originally set out to study.
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Post #6

Post by Excubis »

Since I cannot sleep. A number is an value integer placement, basically it is a symbol used to characterize a value/amount, primarily used in mathematics and statistics.

Now to find a value/amount can definitely be thought of as a moving target, I would suggest that is why ratio's and differential equations are so prevalent in real world physical mathematics. This is also another aspect of the measurement problem, uncertainty, and chaos, all 3 present a fundamental problem in extrapolating consistent value amounts.

First in measurement, the issue is comparison, you always need more than 1 to find proper continua, or axiom, (rule of formula, how it works). Now if that which is being measured is moving too fast, has extreme mass, or really small (quantum) it is often that once you make one measurement and move to the next the first one is now invalid. Weird stuff. This is were you get Schrodinger's cat from. Although some think it only applies to the collapse in a wave function that is no longer the case. Blackhole oscillations, mass of space, dark energy, others...

Uncertainty, applies in similar fashion but only to the quantum as the measurement problem did at one time and gave rise to observed uncertainty. This is about independent measurements of velocity at a very minute scale. Basically x point of a wave pattern and y point cannot be exactly known with precision only probability. So if say a wave (think water) in a very very small container, now the container is bumped by unknown source and you wanted to know how much energy was now it the water you would measure wave height vs distance between each wave peak over time traveled. Yet if the container was soo small and wave lasted less than say a nano second( it is faster in reality) you cannot take multiple measurements in order to extrapolate precise values, but can surmise probability while only having a rough estimate of energy in time, as long as peak is measured (wave apex).

Chaos theory, has 5 postulates 1)butterfly effect: small changes in a system can have grand sweeping affects. 2) Unpredictability: All complex systems have unknowns therefore we cannot logically predict all outcomes in complex system. 3) Turbulence: two variable in a complex system that share same position in origin of space and time will undoubtedly end up in vastly different position over time in a complex systems. 4) Fractal: a geometric system that repeats in a simple process/rules that will promote by variation od a overall imprecision. Best example weather.

Okay tired now.
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Re: On the definition of number

Post #7

Post by jcgrey87 »

There are so many different meanings when it comes to numbers. I personally love reading about the hidden mysticism of numbers and what they actually mean and not just the standard version of they mean nothing that kids and people learn when they are in school. I feel like there are so many hidden secrets when it comes to numbers. I usually use some articles that I really like as reference (You guys can check it here: https://www.sunsigns.org/angel-number-55-meaning/ ) I like them because there is a lot of useful info that we can use and I feel like it lifts the vale a little bit between here and there.

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