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?

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?

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?

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.