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Alex, Why Don't You Answer When The Questions Get Tough?

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Posted by Paul R. Martin on June 21, 2001 17:59:03 UTC

Hi Alex,

Here are a couple of questions you avoided which were about to disappear in the quicksand of this forum. I would like to hear your answers to them.

1. Of the circles, or at least the approximations of circles, we find in nature, they are all produced by some physical system such as compasses and paper, plotters and paper, bubbles and water, etc. Some of them, for example the ones produced by a plotter, may have been generated with the aid of an equation of a circle, or an algorithm which uses the equation of a circle.

But it is the evolution of the physical system that produced the circle. The equation by itself cannot produce a circle without some kind of physical system acting on the medium on which the circle eventually appears.

Even if the equation of the circle could somehow spontaneously generate a circle, the equation itself needs to be somewhere, doesn't it? How can there even be an equation for a circle without some physical system embodying the symbols of the equation?

The same questions apply to your 1=object, 0=no object formula. Even if it could exist in the absence of anything physical or real, how can it spontaneously begin to act and create anything physical or real?

2. You claimed that the axioms, 1=yes object, 0=no object, are sufficient to develop all mathematics and all physics. I don't think those axioms are enough to develop much mathematics at all. I think mathematicians agree that the Peano axioms are the bare minimum that can categorically generate arithmetic. Set theoretic axioms can also be used but they also go beyond your basic definitions of 0 and 1. To get any structure at least as complex as arithmetic, you need to choose some additional axioms, which are arbitrary. You get a different structure depending on what additional axioms you choose.

It turns out that the universe appears to have a great many correspondences with the system of arithmetic. It also has a great many correspondences with the system of analysis, which extends the arithmetic of natural numbers to the calculus of irrational and even complex numbers. However the correspondences with analysis break down in physical systems which are not classical. For example, the smooth continuity of analysis is broken in the quantum world.

The same is true of geometry. The axioms of Euclid work very well as long as the physical system is nearly flat. If things get too big, or involve relativistic effects, Euclid's axioms no longer work and we have to choose Minkowski's or someone elses non-Euclidean geometry.

I don't think you can find any set of axioms that are natural and inviolate from which all of the mathematics that can describe the world can be derived. That's what Euclid, and even Pythagoras before him thought, and they were proved to be in error by Lobachevski and others in fairly recent times.

So my question to you, Alex, is how can you develop the mathematical system that physics rests upon from your two simple axioms alone?

If there are other axioms required, which I think all mathematicians would say there are, how did the particular axioms that happen to correspond to the physical world get chosen?

Warm regards,


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