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October 05, 2011

Discontinuity

Firstly, the "Dichotomy Paradox" as highlighted by Aristotle.


The dichotomy paradox

“That which is in locomotion must arrive at the half-way stage before it arrives at the goal.”

—Aristotle, Physics VI:9, 239b10

Now let us come to DISCONTINUITY.

Let us assume that SYSTEMS flow in a smooth, continuous manner, assuming every possible numeric state while transiting from one physical state to another.

Now let's look at the following:

Notation: ≈N means "Approximately number N"

Assume that the 'continuous system' moves from ≈1 to ≈2.

To move from ≈1 to ≈2, it has to pass through 1.5

To move from ≈1 to ≈1.5, it has to pass through 1.25

To move from ≈1 to ≈1.25, it has to pass through 1.125

To move from ≈1 to ≈1.125 it has to pass through 1.0625.

AD INFINITUM.

Statement A:

"The SYSTEM passes through infinite physical states as it changes configuration."


Observation:

For passing through an infinite number of physical configurations, a system would take infinite time. You cannot have a system that can undergo infinite changes in state in finite time.

i.e., the above mentioned system cannot move at all.

The system can only move if it JUMPS and if it jumps it misses certain points. And that means it is NOT CONTINUOUS.

Moving further......


There cannot be any fixed pattern as far as the length of the jump is concerned.

The system can make only random jumps while changing configuration.

The system might jump from ≈1 to ≈1.2 to ≈1.4 to ≈1.9 to ≈2.

There is no law that it has to move in an increasing manner either.

It could jump from ≈1 to ≈1.4 to ≈1.9 to ≈1.3 and then wham ≈2.1.

It cannot cover all the numeric states between ≈1 and ≈2.

Nor can we ever predict how many jumps it will take before it reaches ≈2.

The following text is from "CHAOS: Making a New Science" by James Gleick, page 92-93.


The Noah Effect means discontinuity: when a quantity changes, it can change arbitrarily fast.

Economists traditionally imagined that prices changed smoothly - rapidly or slowly, as the case may be, but smoothly in the sense that they pass through all the intervening levels on their way from one point to another.

That image of motion was borrowed from physics, like much of the mathematics applied to economics. But it was wrong.

Prices can change in instantaneous jumps, as swiftly as a piece of news can flash across a teletype wire and a thousand brokers can change their minds. A stock market strategy was doomed to fail, Mandelbrot argued, if it assumed that a stock would have to sell for $50 at some point on its way down from $60 to $10.


Niels Bohr also talked about discontinuity, although in the context of sub atomic scales.

But there is no reason why discontinuity would exist on the quantum scale only, and not on scales we classify as the 'everyday physical world.' The stock price discontinuity that Mandelbrot talks of, is, after all a pattern in the 'everyday physical world'.

Back to Bohr & Discontinuity:

From Plato.Stanford.Edu

Bohr saw quantum mechanics as a generalization of classical physics although it violates some of the basic ontological principles on which classical physics rests.

These principles are:

• The principle of causality, i.e., every event has a cause;

• The principle of determination, i.e., every later state of a system is uniquely determined by any earlier state;

The principle of continuity, i.e., all processes exhibiting a difference between the initial and the final state have to go through every intervening state;

• The principle of the conservation of energy, i.e., the energy of a closed system can be transformed into various forms but is never gained, lost or destroyed.



Further reading:

1. "Differentiating the Discontinuous" by Mark Buchanan, Nature Physics 7, (2011)

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