Infinity

So far I’ve blogged only about the finite variable and a topological structure (FCA) on it:

• a variable is a set of exclusive values, created by a selection process
• variables live in a lattice of set containment (topology, FCA)
• an extensive value is a variable by itself

The last kind I explore in a separate measure blog.

Here I want to ponder over infinity, how it can be integrated into the current understanding and what good it is for.

What is infinity?¶

Finite variables have finite information. Do infinite variables have infinite information? But that would mean a computer taking up all the universe would not suffice to store the values of an infinite variable or select a value thereof. Infinity in this sense does not exist and therefore one can also not make any statements or conclusions that involve infinity.

The universe can be regarded as a parallel computer consisting of a myriad of, but still finitely many, selection processes happening in parallel. Each of these selections are from finite variables.

We, as an organism that survived by evolving adaptability through a brain able to map the world and simulate it, need to be able to generate internal variables that can be mapped to the real ones.

Paradigm change to algorithms: generated variables

Instead of regarding actual variables with a finite number of values we now start to look at algorithms to generate variables. E.g. \(ℕ\) is generated by looping +1 (=(+1)*). With this the size of a variable (space complexity) can be moved to the time domain (time complexity). This is not only a feature of the information processing brain but also of the world itself (\(ΔEΔt≥h\)) (action, h).

A generated variable is versatile. It can be generated to the appropriate size, whatever the physical variable asks for.

Infinity is a construct of the mind and not of the physical reality. Nothing in reality is infinite, neither time nor space nor anything else. Infinity is not applicable to real variables, but only to the generation of variables, i.e. to its generation algorithm The infinity that is meant in mathematics is a loop in an algorithm that is ended, when the wanted magnitude or precision is reached.

Our numbers are such a generated variable. It is an algorithm to create a multitude mappable to all kind of physical variables. It is infinite, but this only means that our only limitation is the available time or space:

• numbers of \(ℕ\), generated by \((+1)*\), can be selected (written down) only if not too large
• additionally a number in the continuum, the real numbers, \(ℝ\) can be selected only with a limited precision.

Infinity

Infinity is a practical notation (for the deferred decision about the stop time) of an algorithm with at least one loop.

The continuum \(ℝ\) is not only generated by an algorithm but also consist of algorithms. Operations on extensive physical values (quantities) are mapped to operations on the numbers and then made part of the numbers to form reusable algebraic structures like groups (addition) and fields (addition and multiplication).

• Addition to the size of an extensive value
• Multiplication: Independent extensive values are variables independently selectable from. One can form the cartesian product. Multiplication gives the size of \(AxB\).

So numbers are algorithms and it turns out that some of them do not have finite time complexity. For example, \(√2\), the diagonal of a unit square, is an algorithm which involves an infinite loop (open loop). That the algorithm never ends is synonymous to: \(√2\) does not exist in \(ℚ\), but by including such algorithms we make the completion of \(ℚ\), which is then called the real numbers \(ℝ\): \(ℝ=ℚ∪𝕁\). The irrational numbers \(𝕁\) and rational numbers \(ℚ\) are dense in \(ℝ\).

Why include infinite loops?

They in principle never end and thus they need to be aborted, and then we are still in \(ℚ\), i.e. the limit points practially don’t exist. We can get out of this dead end by not talking any more about the limit points but rather about the algorithm: \(√2\) as an algorithm is different than \(1.4142135623730951\).

By including the algorithms the statements given by the algebraic structures become more general. We have less limitations when making new calculations (closure).

Because the limit points can actually never be reached, i.e. do not exist, one must be careful about calculating with “algorithms”, i.e. in \(ℝ\), when dealing with \(0\) and \(∞\). \(0/0\) can be any number, but it is still the foundation of calculus. It matters how one approaches 0. L’Hôpital’s rule and in general asymptotics helps then to find the limit. The same is true for \(∞\) (infinitely large). \(∞/∞\) can be any number as well. By a linear definition \(0 = \lim_{n→0} n\) and \(∞ = \lim_{n→∞} n\) one could write \(∞/∞=1\), \(0/0=1\) and \((1+1/∞)^∞ = e\). Because \(1/∞\) is approaching \(0\) slower than linearly, we have, \(0∞=0\). \(∞+1=∞, `3∞=(3+0)∞=3∞\). But to acknowledge L’Hôpital’s rule we would have \(3∞ ≠ ∞\) and \(3\cdot 0≠0\), which is not consistent with \(ℚ\), thus \(0/0\) and \(∞/∞\) is not defined in \(ℝ\).

\(ℚ\) vs \(ℝ\)

In \(ℚ=\{a/b|a,b∈ℤ,b≠0\}\), if one allows \(a_i∈ℤ\) and \(b_i∈ℤ\) to go to \(±∞\) in any possible way, also nonlinearly, i.e. if one admits sequences in this way, then one defines only a subset of \(ℝ\). The reason is not so much proofs like

\(√2=a/b ⇒ 2=a^2/b^2 ⇒ a^2\) even \(⇒ a\) even \(⇒ b\) even \(⇒ a/b\not\inℚ\)

which hinges on

if \(a^2\) is even then \(a\) is even

and it is questionable, whether that can be done if we can’t reach \(∞\). Remember: \(∞+1=∞\).

It is rather the comprehensive definition of \(ℝ\) to comprise all thinkable sequences that approach a number as one equivalence class.

Infinity and Information¶

The information of a variable is basically the number of values. If the values of the variable are generated by an algorithm, then one moves the complexity (the size, the information) to the number of time steps needed to generate the values.

algorithmic complexity

Kolmogorov complexity looks only at the length of the algorithm and neglects the time. In the general descriptive complexity theory the time is considered, though, via complexity classes.

The information of an infinite variable in bits is always infinite, but one can further classify them via their algorithmic complexity, i.e. via their number of nested endless loops.

• \(ℕ\) has one loop and so do \(ℤ\) and \(ℚ\), because they have a bijection to \(ℕ\).
• \(ℝ\) has two nested loops, infinitely dense-in-itself and unbounded. One can think of writing a real number with infinite but countable binary digits and thus can conclude that the cardinality of the real numbers is \(2^ℕ\). Every ever so small interval in \(ℝ\) is also of the same size. See also continuum hypothesis

axiom of choice

It is not possible to choose all elements from an infinite variable, because it would need infinite information and/or infinite time. One therefore resorts to make choice an axiom to still be able to reason about such sets.

Practically the information of a infinite variable depends on when one chooses to stop the infinite loop, i.e. on the precision. When modelling reality in computer software the integer type or the floating point type is chosen according to the needed precision. If they are not enough, one can use arbitrary precission libraries.

Infinity and Topology¶

In \(ℝ\) even a normal \(1\) means \(1.\bar{0}\). The latter is an algorithm that never ends. The short \(1\) chooses an element of \(ℕ\) with variable length coding. The presence of infinitely close other numbers in \(ℝ\) asks for a method to distinguish the \(1\) from them. This is done by an algorithm that produces numbers ever closer to \(1\). At a certain step, e.g. \(1.000n\), the \(n\) is chosen to be \(0\), instead of \(1\) to \(9\). Every step generates a variable to allow a further choice. This infinite loop is called open.

open

open can be interpreted as open ended infinite loop. Open ended in the sense that we will decide later, when to get out of it.

The objective of an open loop is to define an element \(x\) by approaching it. The intermediate steps of the loop form a neighborhood of \(x\). \(x\) is defined by the algorithm and is included in the set: topological closure.

The definition of elements is given by the separation axioms:

• T0 defines a point \(x\) (element, value) via neighborhoods as described
• T1 is when, of any two points, one has a neighborhood not containing the other
• T2 is when any two points can be distinguished by disjoint neighborhoods. Every filter and every net has a unique limit.

If the closure of a set has missed a point and that point is separated from the closed set by a neighborhood, then this is a regular space. A regular space is metrizable.

Cauchy convergence and completeness is generalized with the uniform space that is built on top of axioms equivalent to a pseudometric, where two elements not necessarily need to have a distance.

If in the topology we have a metric a neighborhood conveniently is defined as an open ball.

All these concepts to define closeness can be visualized with a finite FCA lattice and then be generated to finer ones ad infinitum.

space vs time

Neighborhood is normally mapped to our sense of physical space, but this provokes the misleading idea that more is selected at the same time. It is better to map a neighborhood to our sense of time, because that better depicts the fact that the points’ only reason to occur together, is the selection process itself. We look at one selection process at a time. open makes this selection process an open ended loop. It stands for the older notation \(x = \lim_{n→∞}x_n\). “\(f: A→B\) is continous if an open \(X⊂B\) has an open \(f^{-1}(X)⊂A\)” is the same as \(\lim_{n→∞}f(x_n)=f(x)\). With a metric one can also say “For every \(δ>|f(x_n)-f(x)|\) there is an \(ε>|x_n-x|\)”. Note also that neighborhood does not imply nearness in the metric sense, but rather in the set containment sense.

compact

Every open cover has a finite subcover.

With this property connects the infinity with the finite and thus allows to make global statements about the set that at the infinitely close (local) wouldn’t have much meaning, because infinity can never be reached.

That a bounded set is closed or vice versa can be proved with compactness.

Usefulness of Infinity¶

• One such use mentioned already is to have a versatile multitude to map all kind of real variables to.

• Variables do not exist alone. They exist because of other variables. The functional dependence is a general characterization of the system, not so much the size of the variables (information), which changes from system to system. With the infinity as defined here one makes a simulation with generated variables to the wanted precision. The algorithm needs less memory and simulations can be done with minimal generated variables. So this analytic description altogether saves a lot of memory.

• One does not need to use by chance numbers

  When a physicist liberally uses infinity in his description of the world, this is an idealization justified by the wanted precision. For example an infinite distance could be a few centimeters when describing an atomic scale phenomenon. It is this idea that makes him use \(∞\) instead of a by chance distance like 2cm or 3cm.

• One can make more general statements. Such statements are shorter, i.e. need less space (information, complexity).

  In a general statement the precision is unknown and so the decision about it needs to be deferred.

  For example in a NaCl crystal the diagonal will need one precision for KBr another one.

• There is often no finite algorithm to describe certain things, like the length of the diagonal of a square (\(√2\)).

• Trial and error is a basic principle because it follows from selection. This is an infinity iterative algorithm that is stopped when content with the result. Obviously this has many applications.

From Variable to Structure¶

Variable is the name I have given to the basic building block of information in that previous blog. Variable and value can often be identified with set and element, concepts ubiquitous in mathematics. This makes it obvious that mathematics is all about information. How could it not be so!

The most basic step in information processing is:

• Exclusive selection of a value from a variable.

Exclusiveness

For a information processor the exclusiveness of values of the variable mapped internally results from the actual exclusiveness of the real variable by approximating the information content of the latter. Instead of having the \(\{\text{yes},\text{no}\}\) choice (variable) for every element, i.e. \(I=n\), we can do with \(I=\log n\) such bits. That saves memory.

Sets in mathematics can be a together of anything. One can combine them and intersect them. There are subsets and power sets. The elements must be distinguishable, though. So there is the basic idea of selection in the set. But it is often not made fundamental enough. The variable/value idea gives a motivation to the language of sets.

In FCA one starts with sets of intents shared by a set of objects (extent). But the set is formed by selecting an intent in or out, i.e. by a binary selection. Then subsets are formed by intersection (which means union of extents).

Variables do not exist alone or outside any context. No, they are the consequence of selection, and that asks for a criterion, which comes from another variable (unless we are looking at the uncertainty principle in quantum mechanics). There is an identity that links certain variables, like a very specific triangle links the lengths of its sides. This is the object in the extent in FCA. The sides of different triangles are not linked. Of most importance is functional dependence. The function represents the identity. This is an idea upon which the mathematics of \(quantum mechanics\) is based, and it allows to reconcile infinity with quantum.

Subsets will arise via such dependencies as described in FCA. They are also a means to do selections in steps, and these steps do not need to complete (infinity). One can also simply combine separate selection processes by some criteria and thus introduce subsets that are distinguishable, but possibly overlapping. In FCA the overlapping parts of an intent become a separate intent.

Context

Exclusiveness depends on the context. In FCA there is a context concept. It combines those elements that occur together into objects. Objects (extent) that share elements (intent) form an (extent,intent)-node. Such a node can be viewed as a different and more specific context concept of a variable: Combined with the values of a variable, this node give rise to new nodes: {context}x{values of variable}. More generally every Cartesian product gives rise to variables. Subsets of the Cartesian product are also called relations.

The lattice has the nodes as it’s elements. Each node is a dual view on a set of intents or extents. In mathematics one embraces more often the topology view, but it is basically the same thing, just that one looks at all atomic elements and adds on top of it a set hierarchy. This additional set hierarchy is a topological structure on the set.

A set with additional structure, like a topology, is a space. Space/point corresponds to variable/value.

With the topology, variable/value can also be topology/set in topology. In the lattice view this corresponds to lattice/node. The topology \(T\) consists of many variables, determined by the context, as described above.

Topological vs. programming structure

Topology and FCA form a set structure, which is a set hierarchy, which is a hierarchy of variables.

Although more general, this notion also applies to programming. In the programming language C, struct names a set of variables. An important difference is the “variable” in C and other programming languages is only the bits that allow to choose a value of the actual variable. The actual variable is specified outside the program, in the documentation or by implicit knowledge of the programmer.

Extensive Variable¶

There is an important difference between

1. a variable, whose values are atomic by nature
2. a variable, whose values are sets, i.e. variables themselves.

The first is a point space. A topology gives a point space a way to distinguish points (separation axioms, infinity) and a sense of distance between points (metric).

The second kind is no variable any more, if one value is contained in the other (chain in a partial order), because the exclusiveness has gone. But if the sets are exclusive and form a Cartesian product with some other variable(s), then they form a “variable of variables” kind of variable.

If the value is a variable it is an extensive value of an extensive variable. Only such values motivate certain operations:

• addition
• multiplication: If two such extensive values (= if the values of two such variables) form a Cartesian product. A Cartesian product requires orthogonality. The more general concept of the wedge product has non-orthogonality included.
• inner product as degree of non-orthogonality

The fact that often only the size of extensive values is mentioned (by some measure: count, length, area, volume, weight,...) should not let forget that by nature it is nevertheless a set.

To summarize: The variable and the selection of values is physical. It occurs in the real world. It occurs in the computer and in our brain, because they are part of the real world. Still, the latter two’s purpose is to predict the outside world and thus that world needs to be simulated. For that purpose internal variables are mapped to the outer real world variables. More variables form a structure, in the mathematical and in the programming sense. Structures can be reused by mapping some of their variables, e.g. input and output variables (indirection). There are also variables and structures that arise from saving memory or processing time. Extensive values are themselves variables even if only represented by their size via a number.

Natural Information

I want to compare the number of combinations of n bits:

2ⁿ

with growing processes, like with accruing of capital with annual compounding

(1 + (i)/(100))ⁿ

or the especially interesting natural growing

e^x = lim_{n −  > ∞}(1 + (1)/(n))^nx = lim_{m −  > ∞}(1 + (x)/(m))^m

Mathematics describes reality. So the similar formulas should have similar, though abstracted, realities: This similarity can be found in the context of information.

Note

I often use the variable - value pair. Value, though, has the connotation of additiveness. This is not implied in my usage, but also not excluded. Variable - state, or variable - element would be alternatives.

The key to compare them is to understand information in the shape of bits as a growing process.

The bit values are identifiable, have a real existence in the computer. Every bit variable in a set of bits is also identifiable. Let's not name the bit variables and bit values, though, but instead only look at the size of distinguishable bit value patterns. Every bit increases this size by 1 times what is there already. Let's denote this aspect of the bit by (1 + 1) to emphasize that an additional 1 is added to the one there already. The parentheses make this an operator, an element of the number set Q, the set where the number also includes the multiplication operation, thus making it an operator. The operator's effect is to duplicate the distinguishable multitude. n repeated applications of (1 + 1) produces a multitude of size

(1 + 1)ⁿ = 2ⁿ

Every new bit is compounded to the existing combinations.

The entry in this multitude is used to represent a real value in a real variable of size C < 2ⁿ. The information measure for a real variable, when dealing with bits, is the number of bits n = log₂C needed to create the multitude of size C. This measure arises from the wish to create a multitude, using bit variables, that can be mapped to the values of the real variable. The real information is the size of the actual variable we map to.

Note

If we start from a number of variables, the exponential function gives the number of value combinations. If we start from a number of values, the logarithm gives the number of variables needed to represent it.

For interest calculation we look at an amount of money (the 1), which is deposited in the bank with interest i. After n years the 1 has grown to

(1 + i ⁄ 100)ⁿ = qⁿ

q is not 2, normally just a little above 1. The corresponding "information" measure in a financial context of interest i would be the number of years, or whatever unit of compounding period one chooses to use.

The essential difference with respect to bit information is that what is added is a fraction of what is there. But then, fraction is actually just a matter of units.

The units of living organisms are cells and the ultimate units in the real world are the quantum particles. Both of them are small compared to the things around us. And with such small units one can also compound arbitrarily (infinitely) often:

lim_{m −  > ∞}(1 + (x)/(m))^m = lim_{n −  > ∞}(1 + (1)/(n))^nx = e^x

In the first equality we see that, given a certain growth, varying the compounding steps amounts to varying the growth rate.

Note

Actually in the financial world the real compounding takes place in very small steps, just that the bank forwards them to the customer in larger units of time for several reasons.

x is the information in the natural information unit nat. Basically we split up the size of the variable to infinitely many infinitely small fractional variables, whose size are just a very little bit larger than 1.

What is special about e^x is that its change is also e^x. It is a consequence of the infinite amount of compounding steps and that ∞ − 1 = ∞:

(de^x)/(dx) = ∞(1 + (x)/(∞))^{∞ − 1}(1)/(∞) = (1 + (x)/(∞))^∞ = e^x

So to summarize:

• The actual information is the size of the real variable.
• The logarithm x = log_bC arises from the view of this size by combinations of constituent or representative variables like bits (b = 2), years (b = q), trits (b = 3), nats (b = e), digits (b = 10); cells, atoms, quantum particles, ...

This change of perspective is done very often in physics. We can think of it whenever we see an exponential function or a logarithm in a formula.

In programming the size of a program can also be measured

• by the number of variables (assumed independent by a good design, and only those, which actually contain information, i.e. assume more than one value)

  This is a relevant measure for design and reviewing of source code.

• the number of value combinations throughout the lifetime of the program

  This is a relevant measure when testing a program.

Note

Following these thoughts helps to feel at ease with the two ways to view information:

• the number of representational variables (information measure)
• the number of values (states) of the actual variable

Relate health issues first to your diet

The human species started out of Africa and spread over the whole planet. At those times the human world population was small compared to today. The roaming and scattering was done in small groups. As with all species, once separated through distance, independent breeding and environmental differences in selection our development forked. But we all did quite well. The populations grew and some groups became peoples, the first civilizations burgeoned. This was facilitated by domesticating animals and farming the land. From there on we talk about history. But the time before, the prehistory, was a lot longer. It is there that genetic differences developed.

In the long prehistory time humans adapted genetically to their environment, especially in what we ate. The tribes following the herds to the north fed on meat mostly, especially during the long winters. Those in the warm regions ate more plants and fruits because they were available either throughout the year or at least during summer.

When the populations grew, the peoples spread and met again: They waged war, fought for resources, mixed genetically again. These are time periods we learned to survey in our history education, periods of 2, 3 thousand years. That is no time compared to the hundreds of thousand of years before that. That is no time for genetic adaption, which counts in generations. The history time accounts for a hundred generations, the prehistory time for 10000 generations.

The history time brought us in today's small world for the 7 billions of us. We now can reach every spot of the world within a day. We mixed up and we mix up more than ever today, genetically, culturally and, in regard to the topic, especially in food culture.

We may be born into a "homogeneous" culture, but actually might not be so much in a genetic sense, because the mixing went on for some time already. So we start to live in a food culture that is good for some of us, but is bad for others. Some of us can drink milk and eat sweets throughout their life, others get intolerant to the lactose in the milk and can't eat fruits when they become adults. Others report issues when eating meat.

Health issues were not very much linked to food throughout history, because there was not the awareness brought by through the scientific method. Surely some individuals found out by observing themselves and couldn't explain why everybody else seemed to have no problems at all. Still today diets are advertised as the good diet for all. But it is not the case. We are different in what we can eat and in what we can't, even if we live side by side.

Were I was born people have a saying: "You eat what is on the table". I was born on a little mountain farm and I know of grandfathers who had stomach issues and died early. These people lived before the modern world had reached the village. They lived of the farming, and a lot of it was grain. The land was quite split already and rather small. So one had to be parsimonious to get along though the winter. One could not slaughter too often and so one ate grain products mostly. Their health issues were not associated to what they ate. They themselves were not aware of it and the people around them would also not have accepted special food for one of them.

All people have a long history of natural genetic selection behind them and are or were well adapted to their environment. This means that they also felt well, felt lively and full of energy. The fast change, compared to evolutionary times, in the food culture though cultural mixing and systematic creation of new food products, made some of us feel tired most of the times.

If you have health issues, mentally scan your diet first and do trial and error with food until you feel better. Observe your digestion.

A doctor will not see how you feel day in, day out. In general one cannot give away responsibility to others, also not to a doctor. Doctors often don't search carefully, because they don't have time, because they are in a routine to walk people through their practice. Don't become a congestion of the health system and don't waste money for doctors or health gurus.

But what is a health issue in the first place? If you feel tired and can't concentrate although you have slept enough, you might think, that's the way you are. But think again, think of what you ate, Tiredness is the first symptom and can last for decades, possibly accompanied by red eyes or light skin issues. The next symptoms show a lot later, maybe when you are 40. You get swollen eyes, more visible skin issues. But then your inner organs and all the rest of your cells have a long time of suffering behind them and have spent their very resourceful ability to cope with the poisonous milieu you have kept them for too a long time.

That is a big problem about eating related issues. It takes a long time until the symptoms become serious. So people tend to neglect them.

Take the first signs serious. I've heard that from an old professor who was shaking like hell while lecturing. Ultimately the nerves are the most precious part that will cease functioning: shaking, dementia an the like. Of course everything will cease functioning finally at one time. It is just that eating related issues tend to be a slow decline.