1.5.5 - Summary & Conclusion

1.5.5

SUMMARY&CONCLUSION

SECTION I

In "Section I : Fundamentals of the counting numbers", I first introduced the counting numbers as an intuitive interpretation of "number sense". From our vague concepts of distinctness, and collections, early humans came up with the idea of numbers, and used various tally methods to record them. Underlying the concept of tallying is the idea of one-to-one correspondence.

With this basic understanding of how to extract the abstract concept of "number" from a number of objects, humans further developed notations and language in order to record such counting numbers.

Finally we went over decimal notation, and the way that languages all over the world express these numbers.

After this I presented my own designations for the first 1099 counting numbers, and lastly we went over numbers which often arrise, small primes, and elementary arithematic.

What is there to be learned from all of this ? Firstly, there are many ways to express the same fundamental idea. All of these forms of expression are equally valid. Furthermore there are potentially many many more ways that the counting numbers could be expressed that humans have as yet to devise. Each form of expression highlights different aspects of the number concept, so I take a wholistic view, that some knowledge of each is important to get to the fundamental idea underneath. Remember that expressions like 1,2,3, etc. are just designators, labels. The true abstraction is something even more subtle. Do numbers really exist ?

Originally ancient people didn't think of numbers as abstractions independent of reality. You could have 3 sheep, or cats, or dogs , or apples, but you couldn't simply have "3" . Such an idea has no tangible meaning. Yet humans were able to extract something common to all sets of 3 objects, ie. the number 3.

While it's true that the objects we count themselves exist, the way we precieve them is of our own devising.

For example, take the 3 apples on the kitchen counter. It is our mind which treats them as a group. In what way are they seperate from all the apples on the earth ? In fact, in what way is any collection of objects truly a collection , could it be that each apple exists independently of all others ? Is it fair to lump distinct objects together and treat them as "whole" ?

On the other hand, what is it really that seperates things ? Sure everything is seperated by space and time, yet they are all part of space-time. Eastern philosophy sometimes holds that the whole universe exists as one. That all things are just one continuous existence, and the seperation we preceive between things is only how we choose to see it. If everything is one collective whole, on what basis do we isolate "some" objects and consider them together ?

Am I saying that concepts like "collection" have no meaning ? No. But I do think it's important to realize that to a certain extent it IS a concept, and a question as to the reality of "collection" is unanswerable. A collection is simply a useful mental tool that humans use to conceptualize their universe. Because of this, we don't need to strigently define what makes a collection. The person who precieves the collection is the one determining what is and is not part of that collection. In the case of the 3 apples on the kitchen counter, one is choosing to only concern oneself with "apples on the counter", and ignore apples in the refrigator, and the nearby supermarket, across the globe, etc. But it IS a choice. It depends what kind of collection we wish to consider.

Likewise, numbers are a tool, and we are not concerned whether they represent a concrete reality or not.

In a way this is an important point to drive home for the rest of this website. Because numbers are "ideas", very large numbers are no less "real" than the countable ones ! They don't need to represent an actual number of objects, and the fact that they seem to utterly transcend known reality should not phase us.

The great thing about ideas, is that anyone can come up with them at anytime. I created the "unique names" because the possiblilty to do so existed. But there is another point that I've been trying to make throughout the writings. Ultimately, our notations and language are limited in their ability to capture the concept of number in their entirety. It's true that some notations ( decimal notation for example ) are superior in expressive power that other notations, but this doesn't mean that the other notations don't provide us with insight into the human mind and how it tries to encapsulate "number".

While a list of seemingly endless unique names is entirely impractical, it does reveal the difficulties inheritent in attempting to name larger and larger numbers individually. This is kind of like a metaphor for what is to follow. While counting the normal way by forming groups of ten may seem superior, it to would eventually reach a point where essentially one would be forced to "invent names", "patterns", and so on.

In the end, it is NOT possible to construct a counting system in which every number has a short name. English numbers, for all there cleverness, essentially are built on a comprimise. It provides short names for certain special large numbers such as "quintillion",

while others will neccessarily be assigned sprawling and confusing names ...

" nine hundred thirty seven quadrillion five hundred and two trillion three hundred and eighty six billion eight hundred and eleven million twenty three thousand seven hundred and nineteen"

This proves a practical solution, because when numbers get this big, we rarely need to be that accurate. Emphasis shifts from a "direct count" to only a concern of "magnitude".

So maybe we can not have a short name for every number, but could we have a short name for larger and larger numbers, all the way to infinity ? Clearly that too would be impossible. As we will learn in the next section, even the sequence of -illion numbers ...

million , billion , trillion , quadrillion , quintillion , etc.

will eventually crash.

The first Chapter of "Section II : Astronomical numbers and the -illion series" has been completed. The chapter , entitled "Large numbers in Physics" goes over the kinds of numbers that occur in science and theoretical physics. You can jump right to the next section by clicking the link below...

Home>1.5>

END