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As is now understood, line A and line G, can be measured as 90 degrees from the equator line and thus are parallel lines, and yet they can eventually interface with each other if the surface is curved [see image].
The problem with assumptions such Euclid's is that they are ultimately only true in an artificial situation and are not reflective of how such things will interact in reality. Also, there is no way to predict from Euclid's fifth law, how two parallel lines would interact in a three dimensional space, let alone in n dimensional space as described by the physicist Bernard Riemann.
Ironically, in his book, von Neumann likens his "pioneering" work in the field of game theory to that of what physicists have been doing for centuries, that is, mathematical formulations that represent, albeit simplified, the "laws of nature," concerning matter and energy. However, von Neumann again showcases that he has no comprehension as to what constitutes the foundation for such "laws of nature."
In his Hypotheses that Lie at the Foundation of Geometry and other works, Riemann rigorously developed the notion of an anti-Euclidean physical space time shaped not by linear dimensions of an "x,y,z grid", but rather dimensions defined by an ever growing array of discovered physical principles such as magnetism, light, heat, gravity, sound, etc.- each organizing principle being ironically characterized by both a finiteness and unboundedness with quantized least-action pathways discoverable therein.
According to Euclid's logic, you never "see" two parallel lines intersecting and thus it is unfathomable that they ever could intersect. His "rule" was based off of shared assumptions of what we "think" we are observing in such phenomena, however, this is not necessarily reality and it certainly does not translate to a "rule" that governs all.
By von Neumann acknowledging that he himself is heavily relying on his so-called "self-evident" truths in simplifying human behaviour, he is asserting an outcome, he is not proving the outcome's natural occurrence.
The Robinson Crusoe example in Monetarist Economic TheoryAccording to von Neumann, the Robinson Crusoe example was used by the Austrian economic school to model an individual's behaviour towards maximizing pay-off in an environment (in this case an island) where the resources available to you are set and limited.
There are many problems with this, but the most unforgivable one is the assumption of a set, limited and unchanging reserve of resources available to the individual. In other words, the Austrian school of economy and von Neumann with them, consider Crusoe's deserted island as the perfect case study for a limited resource, zero-sum game scenario.
Ironically, this assertion is entirely missing the point of what actually occurs in Daniel Defoe's story of "Robinson Crusoe," and causes one to wonder whether these theorists ever read the book or rather read a two line cliff's notes synopsis.
Henry C. Carey, Lincoln's economic advisor, would say in his book "Unity of Law" (1872):
"Crusoe having made a bow, had thus acquired wealth; that wealth exhibiting itself in the power obtained over certain natural properties of wood and muscular fibre, thereby enabling him to secure increased supplies of food with greatly diminished expenditure of labor. Having made a canoe, he found his wealth much increased, his new machine enabling him to obtain still further increase of food, and of the raw materials of clothing, at still decreased cost of personal effort. Erecting a pole on his canoe, he now commands the services of wind, and with each and every step in this direction finds himself advancing, with constantly accelerated rapidity, toward becoming master of nature, and a being of real wealth and power."
Does that sound like the description of a "limited resource," "zero-sum game" scenario? In other words, where is the "set" limit? The limit is constantly being readjusted to what the individual creates which changes his/her relationship to the "utility" of the resource.
For example, the resource wood, depending on the individual's innovation can be used to keep one warm and dry, cook food, create weapons, create shelter, create a ship for travel etc. etc.
The existence of yet-to-be-created potential thus offsets the entire system of von Neumann because his system has no way of predicting potential, i.e. qualitative transformations, nor how it will affect behaviour.
If you cannot predict future qualitative change, which is ongoing, such as the discovery of electricity or the creation of man-made Plutonium and other transuranic elements, or the potential waiting to be unlocked by the very feasible fusion plasma torch that can turn landfills into resource mines, how can you assume a defined set limit or even a defined zero-sum game as a "self-evident" truth when you cannot even predict what is the limit?
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