Monday, October 09, 2017

What does cognitive representability really mean?

I had a debate with Santeri Satama about the notion of number leading to the question about what cognitive representability of number could mean. This inspired writing of an articling discussing the notion of cognitive representability. Numbers in the extensions of rationals are assumed to be cognitively representable in terms of points common to real and various p-adic space-time sheets (correlates for sensory and cognitive). One allows extensions of p-adics induced by extension of rationals in question and the hierarchy of adeles defined by them.

One can however argue that algebraic numbers do not allow finite representation as do rational numbers. A weaker condition is that the coding of information about algorithm producing the cognitively representable number contains a finite amount of information although it might take an infinite time to run the algorithm (say containing infinite loops). Furthermore, cognitive representations in TGD sense are also sensory representations allowing to represent algebraic numbers geometrically (21/2) as the diameter of unit square). Stern-Brocot tree associated with partial fractions indeed allows to identify rationals as finite paths connecting the root of S-B tree to the rational in question. Algebraic numbers can be identified as infinite periodic paths so that finite amount of information specifies the path. Transcendental numbers would correspond to infinite non-periodic paths. A very close analogy with chaos theory suggests itself.

See the article What does cognitive representability really mean?

For a summary of earlier postings see Latest progress in TGD.

Articles and other material related to TGD.

1 Comments:

At 1:50 PM, Blogger Santeri Satama said...


Empirical science can postulate and study cognitive mathematics naturally as time-dependent observation events - e.g. physical notion of 'number' or most fundamental mathematical relation as relational spatiotemporal bivectors < and > (observation event of 'more' and 'less'; number and antinumber). Quadratic area <> ("planar dot product") thus suggests itself as the most basic element of cognitive mathematics which can be also interpreted as a planar projection of geometric observation event that self-contains the three point minimum of area as well as "observer point".

Notion of time-independent mathematics - especially in the sense of metalanguage of timespace - is by it's nature non-empirical, metaphysical and theological. By definition there is no empirical way to show that e.g. non-demonstrable real numbers can perform basic arithmetic operations and form a field, such claim must be axiomatically postulated as pure matter of faith, as belief that there is divine mind beyond capabilities of human cognition where real number arithmetics can and do actualize. We should, however, make here very careful distinction between pure mathematics and applied mathematics of approximate measurements, and a) theories containing to applied mathematics that don't presuppose platonistic theology and use e.g. notion of infinite set only heuristically and instrumentally, and b) theories that give ontological interpretations to divine aspects of time-independent theories of mathematics.

This fundamental distinction does not aim to exclude theological notions of mathematics from more general philosophical or more narrow discussion of empirical science, but to clarify basic concepts and categories and make e.g. discussions about nature of causality and causal power of mathematics more fruitful and communicative.




 

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