Experimental Mathematics and the Power of Computing

"But at the same time, the field of mathematics grew and deepened so much that today some questions appear to require additional capabilities beyond the human brain."

.. capabilities beyond the human brain .. required ..

"There is a growing consensus that human minds are fundamentally not very good at mathematics, and must be trained," says Bailey."

... must be trained ..

"Given this fact, the computer can be seen as a perfect complement to humans---we can intuit but not reliably calculate or manipulate; computers are not yet very good at intuition, but are great at calculations and manipulations."

.. we can intuit .. but not reliably calculate or manipulate .. computers are not yet very good at intuition .. but are great at calculations and manipulations..

.. artificial intelligence .. a computer .. computations .. algorithms .. that can intuit ..

.. the article puts .. a new perspective both .. on human brain .. and computer .. which I find .. deep and insightful .. down to earth .. mother-earthly talk ..

imperative .. the outcome .. a perspective shared and spread .. upon which .. social structures .. in its ever wider sense .. are worth erecting ..

"Although mathematics is said to be a "deductive science", mathematicians have always used exploration, whether through calculations or pictures, to test ideas and gain intuition, in much the same way that researchers in inductive sciences carry out experiments. Today, this inductive aspect of mathematics has grown through the use of computers, which have vastly increased the amount and type of exploration that can be done."

.. exploration .. intuition-driven .. unyielding .. and shamefacedless .. deeply entrenched

.. to test ideas and gain intuition .. intuition .. upon a ..better and ..better ground .. what has been ploughed .. by testing .. and keep testing ..ideas ..

..intuition .. its incisive surgical tools .. the raw power of our brain's neurons .. one that can be likened .. if not being .. quantum computation proper ..

"Computers are of course used to ease the burden of lengthy calculations, but they are also used for visualizing mathematical objects, discovering new relationships between such objects, and testing (and especially falsifying) conjectures."

... falsifying .. the dreaded tenet .. of reductive science .. faltering .. a Damocles sword .. a deterrent .. to keep at bay .. intuition ..

.. the necessity .. the forced .. ideas .. theories out of them .. to give the mechanisms .. the processes .. to falsify self

"A mathematician might also use a computer to explore a result to see whether it is worthwhile to attempt a proof. If it is, then sometimes the computer can give hints about how the proof might proceed. Bailey and Borwein use the term "experimental mathematics" to describe these kinds of uses of the computer in mathematics."

.. proof .. based upon .. on what it is known .. what is known .. out of concepts .. 'proven' to hold .. a poor grasp on reality ..

concepts that muddle ..attempts to elucidate .. the true inklings of reality ..

Chaotic Logic, Ben Goertzel, page 23,

"In mathematics, "chaos" is typically defined in terms of certain technical properties of dynamical systems. For instance, Devaney (1988) defines a time-discrete dynamical system to be chaotic if it possesses three properties: 1) sensitivity to initial conditions, 2) topological transitivity, and 3) density of periodic points.

On the other hand, the intuitive concept of chaos -- apparent randomness emergent from underlying determinism -- seems to have a meaning that goes beyond formal conditions of this sort. The mathematical definitions approximate the idea of chaos, but do not capture it.

In physical and mathematical applications of chaos theory, this is only a minor problem. One identifies chaos intuitively, then uses the formal definitions for detailed analysis. But when one seeks to apply chaos theory to psychological or social systems, the situation becomes more acute. Chaos appears intuitively to be present, but it is difficult to see the relevance of conditions such as topological transitivity and density of periodic points. Perhaps these conditions are met by

certain low-dimensional subsystems of the system in question, but if so, this fact would seem to have nothing to do with the method by which we make the educated guess that chaos is present.

"Chaos" has a pragmatic meaning that has transcends the details of point-set topology."

.. formal definitions .. in all their gamut .. mathematical definitions at the forefront .. from mere tools .. absolute and unquestionable ..

.. mathematics .. its tarnished past .. overriding axioms .. clockwork orange universes .. mechanistic outlooks ..

.. its death-gripping influence on society-building .. thrown of its pedestal .. now a kitten meekly prowling .. probing nature ..

.. no reason .. to bust intuition's balls ..

.. intuition .. chaos enabled ..

.. patterns revealed .. glimpses of .. universal and ubiquitous chaos .. but there remains .. to an extent .. proof-induced ignorance ..

.. to capture the essence of chaos ..

.. quantum computation .. intrinsic within .. hidden the ultimate chaos ..

superposition .. states .. propagated .. chaos-guided .. wholly under the auspices of chaos ..

to bear in mind .. the rules extracted by observing chaos .. in near and familiar environments ..

.. applied .. inspired .. hidden variables .. to acquire a body .. a suitable ground .. amenable for probing ..

unbridled ..intuition

to unlock .. hidden rooms ..

.. to startrek self .. seeking out final frontiers .. ding ..dang ..dong ..

proofs .. often being .. the locks .. that keep away intuition .. no-trespassing rules ..ahoy .. forbidden rooms

without suggesting that either .. can not live together .. fit and complete ..one ..another ..

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