Why is blue blue?
Because it is light with a wavelength of 450 - 475 nanometers?
No, because its definition is light with a wavelength of 450 - 475 nanometers.
Why is a prime a prime, even even, and odd odd? Because of definitions.
After reading over every post -again- it seems a lot clearer. Perhaps I simply was lost in a deep fog of thought that would not let me see the simplex. So, that said, am I correct in simply saying "We only use integers, because they work. To include fractions get's us no where and the bylaws of the game shall for ever more exclude non-integers from being involved in primes."
suppose you attended a school that had a school uniform. suppose further that one part of the school uniform was a plain navy blue blazer. for the years you attend the school, you dutifully wear the navy blue blazer, every day.
then one day, you find yourself in a clothing shop, and since you are need of a new blazer, you ask the clerk to show you the navy blazers. the clerk informs you they are out of navy, and then asks you: "how about a grey one instead?". and you think to yourself: "why would i ever want a grey blazer?".
it is much the same with numbers. most people use a single number system all their lives. whether it is the accountant or salesclerk with their two-place decimals, or perhaps the computer programmer who only has room in the code for 14 decimal places, or just the simple shepherd, who needs only numbers as large so as to count his flock. or perhaps the electrical engineer, who begrudgingly accepts the imaginaries (which he denotes as j, rather than i). each of these number systems has its own set of rules, and as long as one stays entirely within one system, one comes to view these rules as sacrosanct.
but in truth, there are *many* number systems, in fact there are families of number systems, each of which has many members. some are finite, some are uncountably large, some of them contain other number systems as "subsystems", some defy all ordinary experience.
but the concept of "prime" makes sense for a rather large class of number systems (the term for such systems is called "domains"), and under certain circumstances, we can factor things into "primes" in these number systems. so it's not *just* that "the concept of "prime" works in the integer number-system, it works in many others as well. it's an idea that lends itself well to generalization, in areas as diverse as cryptography, to the proof of Fermat's Last Theorem, and other areas of higher mathematics that you needn't worry about.
in general, one has something like C (i am using capital letters to dissuade you from thinking these stand for any kind of "familiar" thing), and one wants to know if C splits into two smaller parts, like A and B: C = AB. one would like A and B to be "the same sort of thing" that C is...we don't want to have to switch to "a different set of rules". of course, if we CAN do such a thing, it is natural to ask if A and B might themselves be so split. at what point do we stop, which things are "unsplittable"? because those things, if we understand them well, we can "reassemble" into big complicated things like C.
this is, for example, what we do when we factor polynomials: we take a polynomial of large degree (which might be very hard to understand), and chop it up into very simple polynomials, like x - 2, which is *much* easier to deal with. it's much like splitting up a complicated job into preparation, initial attack, establishing a rhythm, winding down, clean-up. it's easier for our minds to deal with small bite-sized pieces.
"fractions" allow us to solve more kinds of problems that integers do, but in a sense, the rational numbers are like a soup: none of them stand out from the other ones. you can always chop a fraction into a fraction of a fraction, and it's still just a (smaller) fraction. with integers, primes are like numbers that say: the buck stops here. they're *special*.
in all honesty, i don't know how or why it is that we hit on that particular idea to single out. it was before my time (primes have been objects of curiosity since antiquity). but it turns out to have been useful, in many cases, the size of something has to do with "how much overlap" two numbers have (imagine this: you have two wheels, turning at different speeds, and you want to perform some action when two points on each wheel are "lined up". if wheel A is turning once every k minutes, and wheel B is turning once every m minutes, and they start out "in-sych", they will line up again after the least common multiple of k and m minutes. this number depends on the greatest common divisor of k and m. in particular, if k and m are different primes (like 2 and 3), the least common multiple is always k*m, which makes things "easier to compute"). comparing this numerical "overlap" can be done by looking at the primes that factor into them, and seeing which ones are the same. for example, going back to the "two wheels", if k = 4, and m = 6, we don't have to wait for 24 minutes, it turns out that 12 will be enough.
or: let's say you're someone who ships small items in boxes, like watches, or jewelry, and you want to decide how items to pack per layer in a box (so that you can order the right-sized shipping box to pack them in). 17 would be a bad choice, because no matter how you split it into rows, you're going to have an empty space (unless you use long, skinny boxes with just room enough for 1 row). 25 is a better choice, you can layer them 5 by 5, but perhaps 24 offers you more options (4 by 6, 3 by 8, or 2 by 12). this kind of problem doesn't lend itself well to fractions: sawing a (small) box in half to make it fit in the big box, would be counter-productive (and no doubt lead to an unhappy customer, at some point).
I feel that the even more I consider it, the even more shed I am. I feel that I simply don't understand the concept of amounts. How to influence those "things" called numbers, I can understand, however exactly what it is I am controling and just what I am doing to them, I do not. I am just a monkey that learn to relocate symbols around in a particular online game.
Why's blue blue?
Because it is light using a wavelength of 450 - 475 nanometers?
No, because its definition is light having a wavelength of 450 - 475 nanometers.
Why is a prime a prime, even even, and odd odd? Because of definitions.
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