Math can prove NOTHING about the real world without first scientifically proving the assumptions and axioms that underlie the mathematical model. This is the great weakness of math. The weakness of mathematical logic is best illustrated with the 2300 year saga of Euclid's fifth postulate. (and yes, it does mean something for trading)
<b>Euclid 5th Postulate and the Errors of Intuition:</b>
The history of Euclid's 5th postulate illustrates the problem of trying to apply the abstractions of math to the real world. For those that do not know, back around 300 BC, Euclid created 5 postulates (core truths) that underpin all other developments in planar geometry. The first 4 postulates are short, simple sentences that defined points, lines, circles, angles, etc. The fifth postulate was uglier -- a long-winded, inelegant definition of parallel lines.
For centuries, mathematicians have sought to prove the fifth postulate from the first 4. Their arguments for their efforts had been that: 1) the properties of parallel lines are obviously physically true; and 2) the fifth postulate is too kludgy to be a postulate (even Euclid himself did not like this postulate). But, try as they might, these learned souls could not find a way to PROVE the fifth postulate from simpler axioms.
One tack taken by the mathematicians was to show the outrageously absurd implications of assuming that Euclid's 5th postulate was false. The result was a counterintuitive world with "curved" straight lines, parallel lines that diverged or converged, and angles that did not add up.
Thus, mathematicians invented a bizarre world of so-called non-Euclidean geometry. Although non-Euclidean geometry grossly violated the physical sensibilities of everyone, it did not violate the logic of the other 4 postulates. By the mid 1800's non-Euclidean geometry was a oddity of only academic interest.
Then along came Einstein, whose work on relativity and the nature of space-time used Riemann's work in non-Euclidean geometry. Einstein's theory suggested that the physical universe might be non-Euclidean! This was first proved in 1919 -- showing that "straight" lines of starlight curved as they passed close to the gravitational influence of the Sun. Thus, the patently absurd became the scientifically accepted. The evidence that the universe is non-Euclidean is evidence of how wrong mathematical and physical intuition can be.
<b>Conclusion:</b>
My point, for those that have stuck with me on this long essay, is that math and the real world have nothing to do with each other until science connects the two (and science is a fickle matchmaker because it can easily disconnect the two at any later date). Math is a virtual world populated by assumptions, idealizations, and their implications. Mathematical constructs are abstractions -- convenient mental artifacts that may (or may not) bear a passing resemblance to reality. (At some level, I would argue that ALL applications of math to the real world are mere examples of the curve-fitting that traders know to avoid.)
In the case of geometry, all of the mathematical PROOFs that everyone learned in high-school geometry are true in the mathematical world but NOT true in the physical world. In the physical world, they are only approximations (admittedly, they are extremely good approximations). The math of the markets is the same. And in the gap between the assumed mathematical ideal of the market and the actual physical reality of the market lies the potential for profits for traders.
Postulating high profits for some traders,
Traden4Alpha
<b>Euclid 5th Postulate and the Errors of Intuition:</b>
The history of Euclid's 5th postulate illustrates the problem of trying to apply the abstractions of math to the real world. For those that do not know, back around 300 BC, Euclid created 5 postulates (core truths) that underpin all other developments in planar geometry. The first 4 postulates are short, simple sentences that defined points, lines, circles, angles, etc. The fifth postulate was uglier -- a long-winded, inelegant definition of parallel lines.
For centuries, mathematicians have sought to prove the fifth postulate from the first 4. Their arguments for their efforts had been that: 1) the properties of parallel lines are obviously physically true; and 2) the fifth postulate is too kludgy to be a postulate (even Euclid himself did not like this postulate). But, try as they might, these learned souls could not find a way to PROVE the fifth postulate from simpler axioms.
One tack taken by the mathematicians was to show the outrageously absurd implications of assuming that Euclid's 5th postulate was false. The result was a counterintuitive world with "curved" straight lines, parallel lines that diverged or converged, and angles that did not add up.
Thus, mathematicians invented a bizarre world of so-called non-Euclidean geometry. Although non-Euclidean geometry grossly violated the physical sensibilities of everyone, it did not violate the logic of the other 4 postulates. By the mid 1800's non-Euclidean geometry was a oddity of only academic interest.
Then along came Einstein, whose work on relativity and the nature of space-time used Riemann's work in non-Euclidean geometry. Einstein's theory suggested that the physical universe might be non-Euclidean! This was first proved in 1919 -- showing that "straight" lines of starlight curved as they passed close to the gravitational influence of the Sun. Thus, the patently absurd became the scientifically accepted. The evidence that the universe is non-Euclidean is evidence of how wrong mathematical and physical intuition can be.
<b>Conclusion:</b>
My point, for those that have stuck with me on this long essay, is that math and the real world have nothing to do with each other until science connects the two (and science is a fickle matchmaker because it can easily disconnect the two at any later date). Math is a virtual world populated by assumptions, idealizations, and their implications. Mathematical constructs are abstractions -- convenient mental artifacts that may (or may not) bear a passing resemblance to reality. (At some level, I would argue that ALL applications of math to the real world are mere examples of the curve-fitting that traders know to avoid.)
In the case of geometry, all of the mathematical PROOFs that everyone learned in high-school geometry are true in the mathematical world but NOT true in the physical world. In the physical world, they are only approximations (admittedly, they are extremely good approximations). The math of the markets is the same. And in the gap between the assumed mathematical ideal of the market and the actual physical reality of the market lies the potential for profits for traders.
Postulating high profits for some traders,
Traden4Alpha
