Here's two quotes for consideration.

Intuition and Logic in Mathematics by Henri PoincarĂ©

On the other hand, look at Professor Klein: he is studying one of the most abstract questions of the theory of functions to determine whether on a given Riemann surface there always exists a function admitting of given singularities. What does the celebrated German geometer do? He replaces his Riemann surface by a metallic surface whose electric conductivity varies according to certain laws. He connects two of its points with the two poles of a battery. The current, says he, must pass, and the distribution of this current on the surface will define a function whose singularities will be precisely those called for by the enunciation.

Felix Klein on intuition

It is my opinion that in teaching it is not only admissible, but absolutely necessary, to be less abstract at the start, to have constant regard to the applications, and to refer to the refinements only gradually as the student becomes able to understand them. This is, of course, nothing but a universal pedagogical principle to be observed in all mathematical instruction ....

I am led to these remarks by the consciousness of growing danger in Germany of a separation between abstract mathematical science and its scientific and technical applications. Such separation can only be deplored, for it would necessarily be followed by shallowness on the side of the applied sciences, and by isolation on the part of pure mathematics ....

In context of examination while not mathematically trained I was always curious about the process unfolding.... the foundations and their beginnings. The Sound Of Billiard Balls

Dirac became proof for me of the issue of being abstract while needing the image to go with it? Symbolically recognized while analytically described. So his axiomatic stance lead me to question why how and why Feynman designed his "word art(feynman diagrams)?"

Part of finding this truth is a deep examination(deep play) of what has been perpetuated so far and a meta look synopsis at how to gather and explain it so as to move on.When one is doing mathematical work, there are essentially two different ways of thinking about the subject: the algebraic way, and the geometric way. With the algebraic way, one is all the time writing down equations and following rules of deduction, and interpreting these equations to get more equations. With the geometric way, one is thinking in terms of pictures; pictures which one imagines in space in some way, and one just tries to get a feeling for the relationships between the quantities occurring in those pictures. Now, a good mathematician has to be a master of both ways of those ways of thinking, but even so, he will have a preference for one or the other; I don't think he can avoid it. In my own case, my own preference is especially for the geometrical way.Paul Dirac

The deeper truth is an image that has to be explained? Part of our innateness has left an impression on the soul and recognizing the "time capsule" that is mandala in origins, is the method by which the soul engages what explodes back into their consciousness? This arises from a subconscious level and so too having traveled there you recognize what happens when you touch the very core of your being?

While I may refer to the geometric as inherent in such a truth in expression as some light behind us shining our shadow on the cave walls, these geometries can be covered by ancient designs and can lead the soul back to this beginning?

While you were looking out there, you were looking inside.

*******

**See**: Understanding our Angels and Daemons

http://empslocal.ex.ac.uk/people/staff/mrwatkin/isoc/jungianNT.htm

ReplyDeletea Jungian perspective on the use of 'emotional' language in descriptions of number theoretical phenomena

Thanks Ulla,

ReplyDeletehttp://empslocal.ex.ac.uk/people/staff/mrwatkin//isoc/jungianNT.htm

I list some information here about Riemann Hypothesis.

I try and explain this feeling as a Toposense

Stanislas DehaeneLike Lakoff, I am convinced that cognitive studies of mathematics will ultimately provide beautiful examples of the limits that our brains impose on our thoughts. As I tried to show in The Number Sense, we have very strong intuitions about small numbers and magnitudes, which are provided to us by a specific cerebral network with a long evolutionary history. But one could probably write another book describing the limits on our mathematical intuitions. Take topology, for instance. At home, I have a small collection of extremely simple topological brainteasers. Some of them (essentially made from a metal ring and a piece of string) are strikingly counter-intuitive ‹ our first reaction is that it is simply impossible to remove the ring, but of course it can be done in a few moves. Thus, our sense of topology is extremely poor. Yet it's easy enough to imagine a different species that would have evolved a cerebral area for "topo-sense", and for which all of my brain-teasers would be trivialSee:From: Stanislas DehaeneDate: 3-28-99

Best,

Maybe our extremely strong focus on left brain has something to do with poor topology sense?

ReplyDeleteMath is like vision? It is not mainly cortical? And our brains are extremely plastic and UNLEARNS, select away.

So just to say we have bad sense for topology don't explain anything.