This question probably falls more in the philosophical arena than the practical, but I didn't see any available "math philosophy" forums, so here goes. I've always found it curious why, out of all the possible ways to partition a number, is standard division a/b considered "natural." That is, why is it intrinsically more significant to partion a value into a set of equal sized parts, rather than any of the other possible partitions? Without this "equal size" constraint, prime numbers loose their magical significance, fractions become a small subset of the total set of possibilities etc... Just wondering what are the arguments for basing pretty much all mathematics on this one seemingly arbitrary choice.
I wasn't aware that it was! "Partitioning a" and "dividing a by b" are completely different things. It is probably true that dividing into equal parts is easier and so used more often but I don't know that it would be considered "more natural". Of course, I also wasn't aware that prime numbers had any "magical" significance!
FYI, we have a Philosophy Section, and one of the subforums is "Philosophy of Science, Mathematics, and Logic"
Well math isn't "based on division" as I implied, but division is definitely one of the core tools. As we currently use math, it would be mighty hard to get by without it. Not so different really. As currently used. a/b is just a shorthand for "the partition of 'a' that contains 'b' identical parts. The more expansive interpretation a/b = the partitions of 'a' that contain 'b' parts is never used. <snip> How about "profoundly mysterious?" Even after thousands of years of study by history's greatest mathematicians, we still haven't found a generating function for primes. We have many methods to test for primality, but none to generate the nth prime directly.
Generally division is taught as a / b = c by elementary examples of such sort. It's not well defined as to what b is. Is it volume, partition, or what it stands for. It's taught as two numbers dividing to produce c. But in fact I found division to have a dual nature. One form can like you are looking at partition, or group into even portions. And second forms specific sets of volume. Let me explain with this example of pie division: Division into Partitions: 12 oz of pie / 2 parts = 6 oz in each part Division into Volume: 12 oz of pie / 2 oz = 6 partitions of pie or 6 groups of pie One creates the other. And without proper definition, do formulas like a / b = c , produce a volume or a partition of volume? And it is important because you cant divide a partition into a volume. You cant take 6 groups of pie and divide them by 2 oz. You can only further divide partitions into smaller partitions like 6 groups / into 2 groups which will leave you with 3 groups ( in two partitions ). In general our lack of definitions within our functions and examples lead one to believe all numbers can calculate endlessly to produce one final answer. But in reality, certain numbers can not calculate with other number by definition. Simply put, you can't multiply 6 apples by 2 apples. But If you remove the definition of apples from the numbers, you get 12 something.
Surely division a/b=c is defined so that a=bc; i.e. that division is the inverse of multiplication. No one really chose a specific way to partition a into b parts, but merely so that b lots of c will give us a again.
It's not always though. In very concrete terms, sort of like man's first forays into exploring numbers, multiplication of positive integers always yields other positive integers, while division of integers sometimes yields these oddball relations like 9/2, 16/3 etc which have no integer solutions. In order to work with these odd relations, you have to invent a completely new kind of number (fractions) If you limit yourself just to integers, these oddballs represent an intrinsic limitation of division. So, confronted by this limitation, there are two possibilities: Preserve the process of division and invent a new class of numbers (fractions). or, Preserve the integers and invent a new kind of process (partitioning). This sort of primitive-man math was what inspired the original post. Basically, why did we choose to go the "invent a new number" route rather than the "invent a new process" route? I realize that once math graduated from purely concrete tallying methods, all this territory was eventually explored in depth, but math is just a hobby for me and I enjoy rambling walks down these little side paths : )
Talking about integers, this is decidedly not the case, see http://mathworld.wolfram.com/PartitionFunctionP.html However, this isn't the sort of thing that ends up being used often in the 'real world', not nearly as much as the usual division. If you wanted to write a given positive real number a as a sum of b positive real numbers (b a natural number of course), this isn't a very exciting problem as there'd always be an infinite number of ways of doing this and unless you have some preference for which one to pick doesn't give a unique result. Division gives you a prefence by partitioning into uniform blocks. With any old partition, the blocks aren't necessarily going to satisfy anything nice. I'd also hesitate to label partitions as a more expansive interpretationfor a/b as this will only be defined for natural numbers b. It's more general in the sense it would spit out many answers including the one you'd get with division (when it's defined), less general as it's not defined for as many b's.
From my perspective as a math hobbyist, I'm taking the caveman approach to all these fundamental mathematical building blocks and asking really basic stuff like: Q: "Why exactly did we decide to classify fractions as numbers?" A: "Some fractions yield natural numbers (6/3, 22/11, 35/7) so it seems sort of logical that 11/5, 18/7, 24/9 must also be numbers, just a different kind." This choice has proven undeniably useful, but in some 'real world' contexts (like the interaction of atomic particles) you have to wonder, "Why is it that the fluid dynamics in a vigouously shaken bottle of water is orders of magnitude beyond the modeling capability of even our most powerful computers?" "Do atoms actually behave acording to the rules we've defined for fractional, real and imaginary numbers, as they go about their business?" Or are they behaving according to integral rules we don't yet understand? The same questions also go for other invented number types like negatives, irrationals, reals, imaginaries and trancendentals. All of these were invented to overcome limitations in our ability to calculate "unique results." The premise that the "real world" has this same preference for unique results strikes me as a bit suspect. Did you actually mean to define "a" as a "positive real number" or were you also defining it as a natural like "b"? I think naturals is what you meant but just wanted to make sure. If so then there are a finite number of solutions to a + b = c. I never stated it, but I was operating from this first principles/caveman perspective to the different types of numbers mentioned above. Assuming that fractions, negatives, reals etc are useful workarounds for calculation problems, but not necessarily evidence of "mathematical truths."
I meant it exactly as I worded it. If a is any positive real number, and you are trying to write it as a sum of b positive real numbers ("b" counting the number of terms in this partition, therefore it's a natural number otherwise it doesn't make sense) then there are infinitely many ways to do this, eg. a=2, b=3 then the following are partitions of a into a sum of three real numbers: 2=1+.5+.5 2=1.5+.3+.2 2=.988+.6+.612 2=sqrt(2)+.5+(1-.5-sqrt(2)) etc. The distinction between this and the what's covered in the link I provided is losing the restriction on the partition, that is not forcing the 'parts' to be naturals.