Mathematics and the human brain
I observed Prof. Neil Turok excellent lecture in
https://www.youtube.com/watch?v=JV7K8CvA26I where he mentioned at the end of it the notion that the smallest (for example, a point) and the infinitely large (let's represent it by at least non-composed endless straight line), may be two simple extreme expressions of a single substance.
I wish to share with you my view of this notion.
Mathematics of the past 200 years is mostly learned and done by using verbal-symbolic brain skills.
In order to be clear about my arguments about, so called, Traditional Mathematics, I claim that it is mostly done by persons that believe that mathematical objects are platonic, which means that they are independent of our brain-skills.
In this case we can ask, how can we deduce about these platonic things, in the first place?
My answer is that our brain is essentially inseparable of reality, yet easily can become unaware of this inseparability by partially using its skills.
In order to be aware of this failure The Axiom Of Mathematics is:
Given X (reality) no sub-X (partial expression of X), is X (reality).
According to latest scientific researches ([1],[2],[3],[4],[5],6]) visual_spatial AND verbal_symbolic brain skills are activated during mathematical work.
I wish to share with you some model (some analogy based on visual_spatial AND verbal_symbolic brain skills) that demonstrates the linkage between Logic and Physics.
The attached schema (
https://c1.staticflickr.com/4/3929/15309487328_d8023ed9f0_b.jpg) illustrates the possible associations between Bose-Einstein statistics (even function) and Fermi-Dirac statistics (odd function) and the mathematical concepts of contradiction (expressed as sine-like wave) and tautology (expressed as cosine-like wave).
By this model contradiction is the separator among universes, where each universe has its own tautology (its own natural constants and, so called, the laws of nature).
The simplest form in this model is illustrated by non-composed endless straight line, which is essentially not the sum of its vibrations simply because being non-composed endless straight line is independent of its sine-like/cosine-like vibrations, yet they are depend on it.
This asymmetry of dependency defines a non-entropic realm since no amount of vibrations is the non-composed line in itself.
Being aware of the non-composed line during its vibrations, is actually Unity consciousness that is, in my opinion, the ultimate goal of life phenomena, that has no limits to its developed expressions, exactly because no amount of vibrations is the non-composed line in itself.
By not being limited by contradiction and tautology, life phenomena is beyond AND among the expressed universes, such that they become harmonious organs of a one organism, which is ever developed exactly because no organ harms the developments of the other organs, since the development is naturally derived from cross-universes' source, which is their cause.
From this notion, Mathematics is a fulfilled tool only if it supports the development of Unity consciousness among the expressed universes, such that they become ever developed organs of one harmonious self aware realm.
As I see it, Contemporary Mathematics that defines collections of infinitely many members by using fixed sizes like |
N|, |
R|, or defines the sum of infinitely many numbers, or multiplication of infinitely many (0,1] members by a single finite result, blocks the ever developed nature of multiplicity and the direct awareness of Unity as the cause of multiplicity, which can't express the full potential of Unity, since, unlike Unity, multiplicity is limited by contradiction and tautology.
I wish to clarify something.
Given a non-composed endless straight line and a point not on that line, there are endless larger non-composed circles that are smaller than that line, and there are endless smaller non-composed circles that are larger than that point.
Yet
pi is a proportion among the endless larger and the endless smaller non-composed circles.
So a fixed value can be related to infinitely many things as long as it is not used to define their amount, their sum are any other fixed value that contradicts their property of being endless larger or endless smaller things.
For more details, please search for Nicholas of Cusa.
Here is an example of a mathematical work that is done by using visual_spatial AND verbal_symbolic brain skills:
Please observe the diagram in
http://farm5.staticflickr.com/4015/4430320710_686e9e991b.jpg.
X>0 AND X>2(a+b+c+d+...) are inseparable in the following diagram (where X is the constant positive value of the infinitely many orange figures in that diagram, whether they are straight (the first orange figure) or bent (the rest infinitely many orange figures, which their endpoints are projected upon the straight orange figure, and define 2(a+b+c+d+...) as an endless collection of added positive values that are < X, exactly because no infinitely many orange figures with constant value X>0 can have value 0)).
By using mathematics that is done only by verbal_symbolic brain skills, X=2(a+b+c+d+...), which means that X>0 AND X>2(a+b+c+d+...) are separable in the considered diagram, which (in my opinion) it is the result of separating visual_spatial and verbal_symbolic brain skills during mathematical work.
As I see it, Mathematics is actually fulfilled in case that (by using visual_spatial AND verbal_symbolic brain skills) it also goes beyond contradiction and tautology, and enables one to directly be aware of Unity as their cause (or in terms of Physics, it enables one to directly be aware of Unity as the cause of Bose-Einstein statistics (even function) and Fermi-Dirac statistics (odd function)).
Direct awareness of Unity is not achieved at the realm of thoughts process, which is closed under multiplicity.
As I get it, one enables to transcend multiplicity by being aware of finer vibrations' levels of thoughts' process without being restricted to their semantics (they are taken only in terms of different levels of vibrations)) which enables quantum leap to the non-composed source of multiplicity and quantum leap back to multiplicity.
More such bidirectional quantum leaps are exercised in one's brain, more one's brain enables the awareness of Unity as the source of multiplicity during daily life.
In my opinion, Mathematical science has to deeply be involved in such exercises in order to really be fulfilled, where reasoning at the level of thoughts (which is restricted by contradiction and tautology, like any multiple phenomena) as currently done by Contemporary Mathematics, can't fulfill the actual abilities of this science among human beings.
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[1]
https://software.rc.fas.harvard.edu/lds/wp-content/uploads/2010/07/dehaene1999.pdf
[2]
https://www.youcubed.org/think-it-up/visual-math-improves-math-performance/
[3]
http://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0148981&type=printable
[4]
https://www.omicsgroup.org/journals...-and-learning-2168-9679-1000325.php?aid=80807
[5]
http://bernhard-hommel.eu/Effectiveness of visual and verbal prompts.pdf
[6]
http://www.mitpressjournals.org/doi/abs/10.1162/089892903321107819#.WMfarIVOIeF
