Understanding advanced mathematics of economics and physics for ordinary people.
The Mengerian revolution, which we call the Marginalist revolution, occurred when the people of the period applied calculus ( the mathematics of “relative motion”) to what had been largely a combination of accounting and algebra.
20th century economics can be seen largely as an attempt to apply the mathematics of relative motion (constant change) from mathematics of constant categories that we use in perfectly constant axiomatic systems, and the relatively constant mathematics of physical systems, to the mathematics of inconstant categories that we find in economics – because things on the market have a multitude of subsequent yet interdependent uses that are determined by ever changing preferences, demands, availability, and shocks.
Physics is a much harder problem than axiomatic mathematics. Economics is a much harder problem than mathematical physics, and before we head down this road (which I have been thinking about a long time) Sentience (the next dimension of complexity) is a much harder problem than economics.
And there have been questions in the 20th century whether mathematics as we understand it can solve the hard problem of economics. But this is, as usual, a problem of misunderstanding the very simple nature of mathematics as the study of constant relations. Most human use of mathematics consists of the study of trivial constant relations such as quantities of objects, physical measurements. Or changes in state over time. Or relative motion in time. And this constitutes the four dimensions we can conceive of when discussing real world physical phenomenon. So in our simplistic view of mathematics, we think in terms of small numbers of causal relations. But, it does not reflect the number of POSSIBLE causal relations. In other words, we change from the position of observing change in state by things humans can observe and act upon, to a causal density higher than humans can observe and act upon, to a causal density such that every act of measurement distorts what humans can observe and act upon, by distorting the causality.
One of our discoveries in mathematical physics, is that as things move along a trajectory, they are affected by high causal density, and change through many different states during that time period. Such that causal density is so high that it is very hard to reduce change in state of many dimensions of constant relations to a trivial value: meaning a measurement or state that we can predict. Instead we fine a range of output constant relations, which we call probabilistic. So that instead of a say, a point as a measurement, we fined a line, or a triangle, or a multi dimensional geometry that the resulting state will fit within.
However, we can, with some work identify what we might call sums or aggregates (which are simple sets of relationships) but what higher mathematicians refer to as patterns, ‘symmetries’ or ‘geometries’. And these patterns refer to a set of constant relations in ‘space’ (on a coordinate system of sorts) that seem to emerge regardless of differences in the causes that produce them.
These patterns, symmetries, or geometries reflect a set of constant relationships that are the product of inconstant causal operations. And when you refer to a ‘number’, a pattern, a symmetry, or a geometry, or what is called a non-euclidian geometry, we are merely talking about the number of dimensions of constant relations we are talking about, and using ‘space’ as the analogy that the human mind is able to grasp.
Unfortunately, mathematics has not ‘reformed’ itself into operational language as have the physical sciences – and remains like the social sciences and philosophy a bastion of archaic language. But we can reduce this archaic language into meaningful operational terms as nothing more than sets of constant relations between measurements, consisting of a dimension per measurement, which we represent as a field (flat), euclidian geometry (possible geometry), or post Euclidian geometry (physically impossible but logically useful) geometry of constant relations.
And more importantly, once we can identify these patterns, symmetries, or geometries that arise from complex causal density consisting of seemingly unrelated causal operations, we have found a constant by which to measure that which is causally dense but consequentially constant.
So think of the current need for reform in economics to refer to and require a transition from the measurement of numeric (trivial) values, to the analysis of (non-trivial) consequent geometries.
These constant states (geometries) constitute the aggregate operations in economies. The unintended but constant consequences of causally dense actions.
Think of it like using fingers to make a shadow puppet. If you put a lot of people together between the light and the shadow, you can form the same pattern in the shadow despite very different combinations of fingers, hands, and arms. But because of the limits of the human anatomy, there are certain patterns more likely to emerge than others.
Now imagine we do that in three dimensions. Now (if you can) four, and so on. At some point we can’t imagine these things. Because we have moved beyond what is possible to that which is only analogous to the possible: a set of constant relations in multiple dimensions.
So economics then can evolve from the study of inputs and outputs without intermediary state which allows prediction, to the study of the consequence of inputs and the range of possible outputs that will likely produce predictability.
in other words, it is possible to define constant relations in economics.
And of course it is possible to define constant relations in sentience.
The same is true for the operations possible by mankind. There are many possible, but there are only so many that produce a condition of natural law: reciprocity.
Like I’ve said. Math isn’t complicated if you undrestand that it’s nothing more than saying “this stone represents one of our sheep”. And in doing so produce a constant relation. all we do is increase the quantity of constant relations we must measure. And from them deduce what we do not know, but is necessary because of those constant relations.
Math is simple. That’s why it works for just about everything: we can define a correspondence with anything.
Curt Doolittle
The Propertarian Institute
Kiev Ukraine