Comments on Bharath’s letter

1) Observation 1 that Bharath summarized by the formula

n2 = (n – 1)2 + (2n – 1) (*)

is correct because if we simplify the right side we have

(n – 1)2 + (2n – 1) = n2 – 2n + 1 + 2n – 1 = n2

2) In general it is hard to prove theorems or solve problems about the integers because they have neither adequate nor valid axiomatization. This is also true of the real number system because two of its axioms – the trichotomy and completeness axioms – are false. Counterexamples to them were constructed by Brouwer and Banach-Tarski, respectively. This accounts for the failure to resolve Fermat’s last theorem (FLT) and Goldbach’s conjecture in this system. I improved the situation by constructing the new real number system (consisting of decimals) under very simple set of only three axioms without these false axioms. Then the integers are embedded in it as the space of integral parts of the new real numbers. The real numbers are also embedded in it and, therefore, they form a subspace of the new real number system. The new real number system is finite but unbounded, has no contradiction, has natural ordering (the real numbers have none) and enriched by the unique new nonstandard numbers d* and u* called dark and unbounded number, respectively. It resolves FLT with countable counterexamples, establishing FLT to be false (Nonlinear Studies, Vol. 5, No. 2, 1998). It also yields a proof of Goldbach’s conjecture (Applied Mathematics and Computation, Vol. 138, 2003). Although various aspects of the new real number system are partially discussed in the above papers and Applied Mathematics and Computation, Vol. 30, 2001, its full and integrated development appears, for the first time, in the paper, The new nonstandard analysis, Nonlinear Analysis and Phenomena, Vol. 2, No. 1, for release this month (if technical printing and formatting problems are resolved). These subjects are also introduced in my websites:
http://www.users.bigpond.com/pidro/home.htm

http://home.iprimus.com.au/pidro/

3) Regarding randomness,
Every physical system and events including the brain are subject to the laws of nature. Therefore, the occurrence of an event is not random. However, when huge number of small events occurs such as collision of atoms in atmospheric phenomena, the situation approximates what I call chaos, that is, mixture of order none of which is distinguishable. Such physical events only approximate randomness because each event is subject to the laws of nature.

4) How about non-physical things like numbers. First of all, numbers must be well-defined by a consistent set of axioms. For a new real number to be well-defined, every digit must be known or computable. Based on this constructivist requirement the irrational numbers, except for a few, are ill-defined. There must be some precise algorithm for computing every digit because a new real number is determined by its digits. Can we command the computer to pick out a number at random? I don’t think so for two reasons:

(a) the computer is subject to the laws of nature and

(b) the command or instruction already induces bias on the choice.

However, random choice can be approximated by running the basic digits at great speed cyclically in any order and instructing computer to dip into and pick out a digit almost periodically, meaning, increase the frequency by little amount (otherwise, there is the chance the same number may be picked out each time). A number of any digits can be formed this way and it would approximate a random number.

5) The rules of inference that determines one’s logic or way of thinking are learned like values by formal training and experience. Concepts are encoded in the sensation regions of the cortex as network of neural clusters with specific vibration characteristics. They are all connected to the Creative-Integrative region of the cortex just below the top of the cranium. Logic is encoded as network of neural connections to the sensation regions’ stored information (in the appropriate network of encoded neural clusters) all connected to the Creative-Integrative region. When a network of neural clusters in the Creative-Integrative region vibrates, representing thought, it activates (vibrates) the appropriate the appropriate neural clusters in the sensation regions via resonance to recall the relevant concepts in accordance with the logic concerned. Each knowledge system has its own logic well-defined by its axioms.

-E. E. Escultura
Institute for Advanced Studies and Department of Physics
GVP College of Engineering,
Visakhapatnam, 530041
A. P. India