Many posts refer to paradoxes and I'd like to make some general comments here.
The Achilles paradox is reduced to an infinite series whose sum "is finite". This remains true in math paradigm alone as energy per infinitesimal step of space is never considered or coming into the picture. So a neat integration on dx integrates to a finite x is a
mere notion bereft of reality. Here the infinitesimal dx is a valid axiomatic notion within
the math paradigm.
1) If the problem is taken to the physical paradigm it differs (thus does not remain the same problem) because per infinitesimal step an infinitesimal energy (due to motion) is associated with the step. Here the infinitesimal energy may be an INVALID axiom; first nature must be investigated to demonstrate such infinitesimal energies. Until then no assumption of it's continuum can be taken as a starter (or axiom).
2) Physical steps can sum to finite energies. This is got by assuming discrete space with discrete jumps from one step to another accompanied by finitely small quantum of energy. The sum is then physically finite.
Which of 1) or 2) is correct is a matter to be resolved in reality investigations. The correct physical axiom can then be used to build a math base.
Such a stance bepeaks of a potential demise of differential calculus in representing "truth" and that it generates differential equations mimicking reality must be a fluke; stated otherwise mathematic results mimic reality only approximately. See:
http://www.dc.uba.ar/people/profesores/becher/ns....
by G. Chaitin (the Omega man).
The axiomatic basis of mathematics when used to represent reality must be based on real cognizable axiomatic entities of the Universe. Using, say, discrete manifold topology would be considered a fair starting point to build a math genuinely representing reality after experiments reveal (if at all) that space is discretized (as suggestive of string theory). For Plank length and time see and http://superstringtheory.com/basics/basic4.html
Lastly, the A and B series of Julian Barbour's "no time hypothesis" is seen here:
http://www.evcforum.net/cgi-bin/dm.cgi?action=msg...
indicative that time could be a (mere) human construct.
It is then possible that motion is energy manifested onto the illusion of space and that the Universe exists as different states corresponding to entities having energy, consciousness and free will all together as attributes of entities existing as events.
The Achilles paradox is reduced to an infinite series whose sum "is finite". This remains true in math paradigm alone as energy per infinitesimal step of space is never considered or coming into the picture. So a neat integration on dx integrates to a finite x is a
mere notion bereft of reality. Here the infinitesimal dx is a valid axiomatic notion within
the math paradigm.
1) If the problem is taken to the physical paradigm it differs (thus does not remain the same problem) because per infinitesimal step an infinitesimal energy (due to motion) is associated with the step. Here the infinitesimal energy may be an INVALID axiom; first nature must be investigated to demonstrate such infinitesimal energies. Until then no assumption of it's continuum can be taken as a starter (or axiom).
2) Physical steps can sum to finite energies. This is got by assuming discrete space with discrete jumps from one step to another accompanied by finitely small quantum of energy. The sum is then physically finite.
Which of 1) or 2) is correct is a matter to be resolved in reality investigations. The correct physical axiom can then be used to build a math base.
Such a stance bepeaks of a potential demise of differential calculus in representing "truth" and that it generates differential equations mimicking reality must be a fluke; stated otherwise mathematic results mimic reality only approximately. See:
http://www.dc.uba.ar/people/profesores/becher/ns....
by G. Chaitin (the Omega man).
The axiomatic basis of mathematics when used to represent reality must be based on real cognizable axiomatic entities of the Universe. Using, say, discrete manifold topology would be considered a fair starting point to build a math genuinely representing reality after experiments reveal (if at all) that space is discretized (as suggestive of string theory). For Plank length and time see and http://superstringtheory.com/basics/basic4.html
Lastly, the A and B series of Julian Barbour's "no time hypothesis" is seen here:
http://www.evcforum.net/cgi-bin/dm.cgi?action=msg...
indicative that time could be a (mere) human construct.
It is then possible that motion is energy manifested onto the illusion of space and that the Universe exists as different states corresponding to entities having energy, consciousness and free will all together as attributes of entities existing as events.
