Category Archives: Mathematics

Further maths

We said said we want to generalise the set of complex numbers, but we derived only the minimum  set of numbers necessary for our generating function, namely multidimensional numbers with integer coordinates. So our M4 set are only the Gaussian numbers.

In order to represent complex numbers we need fractions and real numbers. We can easily extend our scheme to define these and other useful sets. Even though not necessary for our purposes, I think it is interesting.

I’ll outline how to do that in a way that is hopefully easy to follow, but not very rigorous. For this post to make sense you must have read and understood the maths section.

Chains and fractions

Let us consider any two consecutive numbers on any chain.

f1

Let us say, for simplicity that the blue arrow corresponds to the open function +1, although of course the same applies to any other chain.

First we introduce an intermediate number.

f2

The green arrow would then correspond to the chain +1/2. It is defined by d22(x)=inc(x).

We can also go via two intermediate numbers:

f3

In this case the brown arrow corresponds to +1/3, which is the chain defined by d33(x)=inc(x).

With three intermediates:

f4

The red arrow corresponds to +1/4; it is defined by the chaind44(x)=inc(x).

Since d44(x)= d42(d42(x)), it holds that d42(x)=d2(x).

Continuing this way we can make fractions with arbitrary denominators, but this is unsatisfactory because as we increase the number of intermediates we lose the ability to express some numbers we could do previously. For example, in the diagram above with two intermediates we can express the number 1/3. Increasing the resolution to three intermediates loses this ability again.

Instead we now add intermediate chains and keep the previous ones.

We start as before:

f5

Now we add two intermediates, but keep the previous intermediate order:

f6

For the next step with three intermediates you might expect three new numbers, but you only get two, because we have one of them already; two applications of the +1/4 chain is equivalent to one of the +1/2.

f7

This diagram shows the fractions ¼, 1/3, ½, 2/3, and ¾, but we can combine the chains to express other fractions. With a finite depth of nested intermediates we have generated the set Qn of fractions, where Qnis the set generated by repeated application of chain d to 0 and where dnn(x)=inc(x).

Hierarchies of Infinities

If we allow infinite applications of functions then we can generate new sets: hierarchies of infinities and infinitesimals as well as real numbers.

Infinities

Any number that can be reached by a finite number of applications of an inc function to zero is finite. Conversely, numbers for which this is not true, we call “infinite”.

“Finite” and “Infinite” are properties of the numbers, like “even” or “odd”.

We have denoted repeated applications of an open function to an object with a superscript. For open function f and all objects p:

f0(p) = p
fn+1(p) = f(fn(p))

This defines the positive integers n as a finite applications of f. Remember that we use the shorthand notation  f3(p) for f(f(f(p))).

We pick any infinite n and call it ∞.

Hilbert’s Hotel

All the usual chain and loop axioms apply to infinite numbers.

loopinf

“Infinite” is a property of a number. The number ∞ – 1 is less than ∞, but still infinite.

The number ∞ + 5 is a number 5 applications of the inc function away from infinity the same way that 1000 + 5 = 1005.

This is contrary to the usual treatment of infinity that treats a set of infinite numbers as a number and takes ∞ + 5 = ∞.

In Hilbert’s Hotel an infinite number of rooms are all occupied. A new guest can be accommodated by moving all guests up one room. An infinite number of new guests can also be accommodated by moving all the guests from room n to room 2n, freeing all the odd numbered rooms.

This only works when you confuse the property “infinite” with the number “infinity”.

Hilbert’s hotel is as much a paradox as saying that my two-seater sports car has an even number of seats ergo it can accommodate 4 people.

If the hotel has ∞ number rooms and a new guest arrives then we have ∞ + 1 guests. If an infinite number of guests arrive then we end up with 2∞ guests. We can’t just conjure up new rooms and all the usual rules of arithmetic still apply.

Real numbers

We can express any fraction with a finite depth of intermediate numbers.

Transcendental numbers, such as π or√2, cannot be expressed as a finite fraction. We can approximate them to an arbitrary accuracy using fractions with bigger and bigger denominators, meaning more and more intermediate numbers, but to express a transcendental number accurately we need an infinite number of intermediates.

ff1

The blue arrow depicts an inc function again.

The grey arrow is the function d(x) , where Δ = d(0)  is an infinitesimally small number, such that it takes infinity applications of the grey functions to advance the distance of a blue function.

This way we can use our chain functions to also generate the real numbers.

Automorphisms

Consider the structure of the functions +1 and +1/2.

ff2

The +1 chain function is shown in blue and the +1/2 chain function is shown in green.

Now consider the functions +1 and +2:

ff3

The +2 function is shown in orange.

You will notice that these two structures are isomorphic, i.e. there is a one-to-one mapping between them.

In other words the blue function is to the green function like the orange function is to the blue function.

Hyperreals and Infinities

Here is the diagram from above again:

ff1

Looking at the grey d function (adding an infinitesimally small value to a number), we see an automorphism similar to the one shown in the previous section. The grey function is isomorphic to the blue function.

If we consider the grey function to be +1 then the numbers on the blue function chain will be a series of infinities. These have been studied by Cantor. If we look upon the chain function d(x) as inc(x) then the numbers 1,2,3,… become ∞, 2∞, 3∞, … After an infinite number of infinities at ∞∞ we arrive at a number that Cantor says has the cardinality of aleph1.

We can keep going scaling up infinities. There is never a final biggest number that cannot be exceeded.

We can of course apply this mapping the other way and consider the blue function to be in which case the numbers generated by the grey function might be related to surreal and hyperreal numbers. Maybe a mathematician reading this could confirm or refute?

More sets

We can apply infinitesimals and infinities to loop functions, too. I don’t know whether these sets have already been explored.

A loop function with an infinite order is not the same as a chain function. A loop function returns to the first value after infinite applications, while chain functions continue to make new elements even after infinite applications or more.

For x ≠ 0
rot(x) = x
inc(x) ≠ x

 

Rotational numbers

In this post I’d like to talk about the mathematics section. In particular I would like to address some common objections.

Rotational numbers are not an advanced concept at the cutting edge of mathematical research, but a basic number set that, in my view, was passed over in the relentless drive towards abstraction. In fact it is telling that the professional mathematicians I discussed this with often seemed to prefer to derive this as a special case of a higher level concept, like a module over a ring, rather than accept the elegant derivation from first principles.

Other common objections are (1) concern about the absence of multiplication, and (2) discomfort about using the output of an open function to define new sets because it is contrary to the requirement that functions must have a well defined domain and range.

Multiplication is essential to many areas of mathematics and we learn it at primary school so it maybe helpful to take a step back and review what it is in essence.

We use unary inc open functions to generate the natural numbers. We can create a new binary function that repeats the application of inc. Clearly the second argument, the loop counter that determines the number of repetitions, must be a positive integer. We arrive at the binary addition function. We find that the function is commutative, meaning that we can swap the loop counter and the first argument and get the same result. The inverse function to addition is subtraction. This is not closed in the natural numbers. We extend the range set for subtraction and arrive at the set of integers. We can retroactively extend the domain for addition to include negative numbers. We don’t want to break commutativity, so we invent a rule to allow negative numbers as the second argument (the loop counter).

Now we repeat the process and repeat adding the same number to itself and arrive at multiplication. We find that this is also commutative. The inverse function is division which again is not closed under the integers, so we need to extend the range set to the fractions. Again we extend the domains for addition and multiplication to include fractions. Because we want to keep commutativity we allow fractions as the loop counter argument.

We repeat the process again and arrive at potentiation. This is no longer commutative, so we get two inverse functions: roots and logarithms. The inverse functions again require a range extension to real numbers. Retrofitting fractions and real numbers to addition, multiplication and potentiation results in some rules that don’t appear intuitive at first sight, but that work to maintain the expected laws.

The next level is tetration or the power tower. Again, there are two inverses. Finding sensible domain extensions becomes quite difficult from this point and is an active area of research.

The point of all this is to demonstrate that defining new sets as the range of functions is perfectly natural and that multiplication is merely the hyper2 function that has many applications and regularities.

The conventional insistence of all unity roots being complex numbers rather than defining them to have independent dimensions (where the complex numbers are the order 4 special case) stems from the desire to maintain general multiplication.

In the mathematics section we don’t strictly generalise complex numbers, but Gaussian numbers, because we only require integer coordinates. The same methods can be used to derive real numbers and some other interesting sets. I’ll do that in the next post.