Maths

The real questions are what underlies math (patterns of actual reality, of course) and why it is what it is.

For example, it is important to realize that the most fundamental operation is plain addition \(+\), and everything else follows from it.

A number is a genralized abstraction about universal property of a pile. Any pile, or pile of anything. It is how many more than one (or how many ones) is in it. A pile comes prior to associations and the abstract number line.

Notice that a pile is a generalized abstraction itself. The most general one. With little more imagination a single piece is a pile of just one, and, stretching imagination even more, when that one is taken away an “empty pile” is what “remains”.

This is how the mind of an external observer produces its “abstract concepts” and this is the universal basis behind the Set Theory, and thus behind all of mathematics - the mind of an external observer creates abstract concepts and superimposes them back upon What Is.

When abstract concepts are proper generalization - it is knowledge, when they are abstract bullshit - it is Maya (a literal weil that hides and distorts reality from itself). Unnecessary, redundant abstractions is the root of all evil.

Addition, as an operation, is a generalization of a universal notion of putting together. \(H\) and \(O\)’s come (or being added) together to form an \(H_2{O}\) molecule. And this “addition” is associative (but is not a closure).

Multiplication is a repeated addition when we multiply by a Natual Number. It is even deep in a language - multiple times. Multiple times what? Add a number to itself.

When we multiply by a Real Number (especialy less than \(1\)) this operation is scaling and when the numner is less than one is best understood in terms of division which is, splitting into equal parts (with a possible remainder).

Notice that here equal parts is crucial - it corresponds to adding to itself multiple times.

When we scale by, say, \(0.5\) we scale by a half, which means division by \(2\). And this is the basis of the abstract notion of percentage. By convention we split into \(100\) equal parts and then take some.

In this case one splits a given number in \(n\) equal pieces, selects some \(m\) pieces (out of \(n\)) and substracts from the number. When \(m = n\) it is just \(1\) - no scaling. When it is \(0.5\) we subtract a half.

This, by the way, is where the general notion of a weighted sum comes from, which is, arguably, is the most common generalized abstraction - one adds up scaled terms (each term scaled by its own miltiplier (or a scalar), which could be \(1\) - no scaling is done - an ideintiy of multiplication, or \(0\) - turns term to “nothing”, and zero is identity of addition).

And a polynomial is a specialization of a weighted sum, by adding more rules of what kind of terms should be added together, and that some terms may be negative. And then the whole hell of algebraic numbers broken lose.

Notice also that we could say than numbers had a strucure - they could be viewd as a sum of other numbers, a product or even a polynomial.

Primeness of a number means that it is NOT some other number added to itself any number of times (there is always some reminder left). And that is just it. Everything else is bullshit. Just not a multiple of any other number - a pile which cannot be divided into any number of equal parts (smaller piles of equal size).

In everything in Nature a more than one is an aggregate and it has its own particular structure - how exactly one and another one held together.

This is the universal basis of typing - not everything could be added to everything else and resulting aggregates is NOT the same.

Molecules (aggregates of atoms) are (literally) structureally typed. So are aggregates of molecules (linear or other kinds of structures). Typing is an universal notion and Lisp guys got it right back then.

BTW, especially for Haskell guys, this is a literal crossing of a type-boundary - individual “free” atoms became “bound” within a structure which has its own properties, bigger that merely a sum of its parts. The old-school guys of lambda Calculus, Combinatory Logic and early Lisps were onto something fundamental, and yes, this is what a Monad is metaphysically - once atoms became parts of a molecule they “crossed an abstraction boundary (to an external observer) and actually became parts of something else (became bound)” while still remaining what they are (unchanged and immutable).

Mother Nature does untyped programming. Actually, it does structural typing and does structural pattern-matching which is what old Lisp guys did with type-tagging and type-casing and math guys did with plain old structural equality which is, for us, a simultaneous traversal - the core of universal pattern matching of the ML family of languages, but comes “naturally” in the Universe.

• \(+\) is associative on any Set of Numbers, which implies a closure property on a Set under this operation.
• \(*\) (multiplication) is just repeated addition to itself. Multiplication distributes over addition precisely due to associativity of addition - every part of a sum is scaled in the same way and implicitly added (back) together.
• \(n^m\) (exponentiation) in turn is repeated multiplication by itself Its properties are due to its “nature”
• \(-\) (substation) is an inverse (or a reverse) of addition - just taking out what has been added (put together)
• \(/\) (division) is an inverse of multiplication, and from it some additional properties of whole Numbers could be derived.

It is said that Natural Numbers have structure because they could be imagined as sums of other numbers (there are infinitely many of sums for every number if we consider negative integers).

With products it is different - some Natural Numbers are NOT products of other numbers (lesser than itself). This is used to be a mysterious fact, but it only means that a pile cannot be taken apart into any number of smaller piles of an equal size (and this - of an equal size - is an important notion) which, in turn, means that they are NOT sums of any number of equally sized parts greater than 1 - which is just a special kind of a sum.

There is nothing too deep about primes. First of all, they all are odd (not multiplies of 2, which implies not of any even number, and yes, 2 is very special), and by the same logic, they are not multiplies of any other primes (less but not equal than itself).

What does it mean that a number is not a multiple of anything? It is not rectangles if we consider an abstract shape of a product - some leftovers (from a smaller rectangle) always “feels off” (or a bigger rectangle lacks a few “pieces” to be completed).

A sum, by the way, could be viewed as a “linear structure”.

And that is it - a pile cannot be taken into any number of equal parts (some remainder will always be left) and there is no way to spread the pile in a rectangular shape of any height (or width) - height 1 is a line.

So, when we reduce products back to repeated sums it will tell us that a prime number is not a sum of any number of equally sized parts greater than 1.

The sieve algorithm (which could be visualized as crossing off multiplies of a number lesser than it on line) sums it up and operationally defines everything that is there about primes. Geometric visualizations with piles (angles) and rectangles are just fancies.