|Theo van Doesburg, 'Counter-Composition VI' 1925|
The Problem with Mathematics I
The "Geometrification" of Phenomena
Let us be both precise and revealing about this particular and remarkable union of Materialism and Idealism - made possible by Pragmatism - "If it works, it is right!"
The clearest, useful cases occur with the quite evident geometrical advantages of Graphical representations of whole data sets, taken from real World experiments, in up to three dimensions (using the actual three dimensions of Real World Space) to handle up to three variables across a given range of real instances.
Such a clever capture of a whole-set of related instances within a single situation, which maintains its relations in spite of quantitative changes in its evident contributions, clearly has advantages, and, pictorially, delivers the whole measured range simultaneously, "effectively" revealing the same thing as does a fitted-up formula can do covering the same real data, but is definitely a more informative representation.
|Parabolas: These two graphs are not to scale and are here just to illustrate the two types of graph explored below|
CASE ONE: is when raw data is directly plotted in a graphical form.
CASE TWO: is when a pure abstract Form, defined by an Equation, fitted up to that data, is similarly plotted.
Now, mathematicians deal exclusively in Generalised Forms - a mixture of Variables (representing measureable as varying quantities), and Constants (which can actually be a vast range of values, giving a host of versions of the very same Form, without changing the relational nature of that represented Form).
These two come from two different Worlds - the data-based plot is from Reality (the real World - though usually extensively farmed), while the formula-based plot is from Ideality ( the World of Pure Forms alone)! And, of course, they have to be related in some way, to make the latter Purely formal Relation helpful for the real world situation - captured via its measured data.
Initially, of course, they are merely approximately the same, but are clearly resonating-analogues of one another, in the mind of the observer of them both, but the astute investigator recognises this, and is determined to tailor, or fit-up, the easier-handled, general abstract form to fit the collected data.
And, such tailoring is a purely mathematical operation! The original General Form has, as yet unknown, constants within it represented usually by placeholder alphabetic letters. The tailoring process involves substituting sets of data, representing a measured real world instant, into the general equation giving a formal relation in only the constants. Other sets are then substituted-in giving different versions, in which he unknowns are, once again, the constants.
Using the technique of Simultaneous Equations, these constants can be calculated from the combined use of the various versions, and substituted back into the general form to produce a particular instance or Equation representing (reasonably well) the original real world data. But, it is STILL an idealised form!
It is by no means identical with the dealt-with Real World situation, OR EVEN the full content "resident" within that measured data. And, perhaps, most important of all, it behaves very differently indeed "in the limits".
For, while the real world situation flips into another totally unpredictable situation in Reality, the Equation merely blows-up, via asymptotic approaches to Infinity or Zero: and these Singularities mean absolutely nothing in the real world - merely indicating "the usefulness over!" So, any assumption that the Equation (so-obtained) is the true encapsulation of the real world relation is, most certainly, incorrect.
Indeed, it is usually much worse even than that! For, data taken directly from totally unfettered Reality can only very rarely be treated in this way. More often, the only way to deal with the situation is to quite radically farm it to produce more conducive and informative data. The real world situation is changed - by removing any evidently confusing factors, and then also controlling others, until a possibly targeted factor becomes clearly displayed, and only then is the required data-set extracted. But, even that is usually insufficient for reliable data: so several sets of measurement are taken over the exact-same sequence of the independent parameters, and averages taken of all the results.
This technique removes other still-there variable effects, and it is only this Well-Formed-Data that is finally used in the above-described techniques.
As this theorist has explained at length, elsewhere, what is then used is both from a highly simplified & stably-maintained context, and fitted up to an idealised mathematical form (suitably adjusted to be as close as possible to the farmed data). It is pragmatically useful, if used in the same context to that from which it was extracted. BUT, does not behave the same over all extended situations. AND, even more crucially, gives major errors when used theoretically in more complex reasoning.
IMPORTANT NOTE Though not the main purpose of this paper, it must be mentioned here that the main philosophical error involved, in both sides of this inter-discipline co-operation, is the adoption of the Principle of Plurality, which is never true of Reality-as-is, where Holism certainly dominates. Plurality, in fact, only-ever-holds in strictly-stable natural and artificial situations, and, of course, in wholly idealised discipline areas such as Mathematics.
And finally we must emphatically differentiate between Graphical Dimensions and Real World Dimensions.
For, the former are in Ideality and can be as many as there are varying parameters, whereas the latter are limited by Reality itself to ONLY THREE! As soon as the wedding to Formal Equations ousted Causal Explanations in Physics, the legitimisation by reference to Reality was lost, and replaced by the Rules of Mathematics alone. Dimensions, beyond 3, were objectified by various ruses totally illegitimately.
Of course, the practitioners involved would never agree to this point, primarily because it would undermine their whole "world". They worship Form so finally, that they study purely formal equations in preference to studying Reality itself. "A blackboard-and-chalk will suffice!"
David Malone and Chaos Theory
Computerised Solutions, The Nature of Mathematics & The Necessary Revolution in Philosophy (Special Issue 46)