**Abstraction**: giving rise to**Science****Idealization**: giving rise to**Mathematics**

But why should attention be restricted to the creations of Nature? Why not take a look at our own creations: human made Technology? Some cameras are capable to "see" in the infrared domain. Our radio telescopes are even capable to "see" the radiofrequencies of far away galaxies. Far more common and well-known everyday abstractions of reality are performed, however, with measuring devices like rods for the abstraction of lengths, clocks for the abstraction of time intervals. But these measuring devices have become more and more self supporting these days. When coupled with digital computers, human interaction is hardly needed anymore. An example is the well known device for the abstraction of weights: the balance. It's further development has resulted in the fully automated Mettler Balance, which is only one of the many examples though. All the apparatus make an abstraction of reality, which is thus a

From the Mettler balance example, we see that abstraction results in numbers. Let us not be bothered by the question whether these numbers are numbers in a mathematical sense. Likewise it is assumed that the outcome of an abstraction can be another rudumentary mathematical object, like for example a naive "set". Or even a complex number - as far as the latter is concerned, suppose that our measuring device returns an ordered pair or real numbers. And let us not forget geometry. Many laboratory devices come with a screen for displaying graphs. Among the more advanced possibilities offered by modern instruments is three dimensional modelling of the data received. The possibilities only seem to be limited by lack of imagination. It doen't seem very wise to exclude anything a priori and it's better to decide not to do so either. In short, we consider any "primitive mathematical notion" as a possible product of abstraction, which is thus a product of

Before proceeding any further, something has to be explained about a nasty
habit of *some*, (if) not all, theoretical mathematicians.
I want to talk about their **systematical disdain for** that continuing
activity of physicists, called **measuring**. Let it be stated firmly that
those theorists, in general, don't understand quite much about
performing experiments in the real world. Especially, they are not very much
aware of the fact that there exists a highly fundamental phenomenon out
there, which is inevitably and intimately related to any physical observation
of the continuum, inasmuch as to any realistic abstraction from it: the quite
**limited accuracy** of any measuring equipment. Always resulting, not in
*one*, but in several, distinct numerical values. For one and the same
physical quantity, we find a *whole range of rational numbers*, instead of
just one real. It is common practice to replace that range by a mean value and
kind of a deviation from the mean, called the measurement **error**. But the
word "error" alone seems to sound already most frightening to some.
Yet the whole secret of the continuum is in the **error** accompanying the
rational numbers, when abstracted from reality.

But, errors with rational numbers are by no means the exclusive intellectual
property of physicists. Not anymore. Because, with the advent of *digital
computers*, mathematicians have become experimentalists too ! Almost every
mathematician has a personal laboratory at his disposal these days. And - not
surprisingly - with these digital working places, the same kind of erroneous
behaviour is experienced as within the laboratories of physicists.

Yet, strangely enough, among the most challenging idealizations are not found in pure mathematics, but in theories of Physics. Most famous are the idealizations called Gedanken Experimente (Thought experiments which were carried out by Albert Einstein in order to establish the basic foundations for his (Special) Theory of Relativity.

In "The Theory of Heat Radiation" by Max Planck, Wien's Displacement Law (chapter III) can only be derived under the following conditions:

But let's go back to the lowest level for a moment. We have seen that mathematicians have their own laboratories these days, installed at modern personal computers. These Math Labs, essentially, exhibit the same sort of peculiarities as the laboratories in physics. And, in very much the same way, the need for idealization is felt here too. The reason is that there exist two basic limitations with any digital mathematical experiment:

- in the Discrete domain: the limited size of (the set of) whole numbers
- in the Continuous domain: the limited size / accuracy of real numbers

- the limited number of words in computer memory
- the limited number of bits in a computer word

- an unlimited number of words in computer memory
- an unlimited number of bits in a computer word

After some thought, it becomes evident that the three idealizations - unlimited # words, bits and processing power - cannot live without each other. It may be concluded that the idealizations which are needed with pure mathematics are involved a great deal with the

With the above in mind, it's easy to see how the reverse process of idealization,

But, materialization being difficult is not the same as: finding that the way back to earth comes to a dead end altogether. If the practitioners of Pure Mathematics are allowed to pick up their ideas out of the blue sky, then it cannot and it will not be guaranteed that materialization is possible indeed. This makes pure mathematics vulnerable to the criticism that some of its ideas may not be useful, meaning that there is no way, at all, to turn them into a useful application. These ideas just end where they are. They are doomed to remain ideas. There is no way back to the abstractions and, via the abstractions, back to the real world.

There is no recipe for idealization. Thus what it means can only be shown by giving good and bad examples of it. Among the good examples are: Euclidian geometry, which can be materialized to Computer Graphics; Classical Calculus, which can be materialized to Numerical Analysis. Bad examples will be singled out in the future, but only after their badness has been causing much harm to the whole of mathematics. Among these bad examples of (irreversible) idealizations are the Transfinite Cardinals and Ordinals. Also some constructs related to the classical concept of continuity have to be considered as inappropriate idealizations, because they are void of any counterpart in real world matters. Point Set Topology comes into mind. Anyhow, it is concluded from the above that the whole realm of Mathematical Idealizations can be splitted up in two distinct parts:

- Idealizations which can be materialized. They constitute the useful parts, Snippets of Pure Applicable Mathematics.
- Idealizations which can
**not**be materialized. Deemed to be useless, it seems, for now and for all centuries to come.

There are NO Infinities in Abstract Science. (See The Physics of Infinity)

Conclusion: Infinities do only exist as Idealizations in the human mind.

So we have a three-level system of the World that concerns us, Mathematics as an activity in the real world (Math Lab), and as an activity of the Mind, and everything in between:

- Laboratory. Or material reality. Here it is where our computers live in. And program development. Number Crunching activity is all over the place. The numbers here appear in two flavours: 32-bit integer and 64-bit floating point. There are a few exceptions to this rule, but they are rare. Experimentation also takes place here, like finding large prime numbers for the purpose of encryption. But all kind of physical measurements as well.
- Abstractions. Here is the place where all Scientific Theories start.
Sometimes they stay here forever. There is also quite some mathematics here,
especially finitistic mathematics. Like Turing machines, computable functions,
a lot of Number Theory, Galois Fields. The whole machinery of Constructive
Mathematics (and maybe Intuitionism as well) is living here. Logic may be quite
different from the standard at this place: the law of excluded middle is
questionable. Numbers have a finite precision, but this precison can be much
larger than can be realized in any conceivable computer. And, oh yeah, all of
the
*infinitesimals*, as employed for example by physicists. are to be conceived as abstractions. They will only become idealizations - and only eventually - in a later stage (e.g. when they become part of an integral). - Idealizations. Cantorian mathematics is here in the first place. But also Euclidian Geometry, Point Set Topology, the irrational numbers. Set Theory is the governing principle here. In short: it's the heaven of mainstream Mathematics. But the ideal pendulum is part of the heaven's clock as well :-)