Friday, January 28, 2011
Questioning Our Philosophy of Mathematics
Questioning Our Philosophy of Mathematics
Today, a couple of poems- one that certainly was a mix of the snow series and arithmetic- possibly a result of the "running" of two creative moods at the same time.
I did some reading in Elementary linear algebra in a complimentary copy of a discarded book and some things dawned on me- after all it has been half a century or so since I had some brief contact with them going thru the motions of solving systems of equations with determinants and what not- but not in the way done in this book- nor its separation and emphasis on the more abstract use of matrices.
But such thoughts casually in the background of my thinking have raised some rather interesting questions on our philosophy of mathematics.
Note: Lubos has an especially interesting post today (maybe he is coming around to this wider view or role of the geometries) and Kea has discerned and counted some things of which by a different path and philosophy I found does indeed apply to asymmetric values and mirror directions of counting over a totality of systems.
Lampion: 01-27-11 In the presence of a central inversion (a singularity) of a structure, or as that singularity is approached and passed thru, we find some sort of mirror image which changes the directions and thus the chirality of the of these structures. This of course to be seen in the higher dimensions, their relationships, and so forth... [What is this sort of information and where does it go?]
What made me consider this [and what is conserved or invariant- in ideas of fluid turbulence] was the mapping of the idea of incidence matrices for the Soma cube. Such a model suggests to me a certain distinction between consciousness and awareness in that these abstract structures do not necessarily define consciousness but in our consciousness we do seem to make models that match the physical word and the understanding of them is something of which we may use for deeper comprehension.
I am not emphasizing the Soma cube especially beyond other such models- even though it is intriguing that Piet Hein invented it while attending a quantum lecture by Heisenberg- I would like to see the content of that lecture. I first thought to study this cube thinking that the Egyptians had toys with wheels and axles long before they used anything but rolling logs to move adult sized things. What significant toy might be in our civilization that could have more advanced use?
Apparently, this skeleton of our matrix notation covers a very wide scope of mathematics. It seems to underlay and contain most everything. I am amazed that determinants can be used to solve some sets of differential equations. I am amazed that they can be used to integrate and give volumes of vector parallelepipeds. I am amazed that these terms (some of which I assumed the meaning taken from the context) cover such things in a simple form to convey say, relativity. (skew symmetric is an example of how different the terminology can be perceived.) Where Lubos states in the article geometry gets hard in the remote distance- that is the beginning of deeper insights- that is a reconsideration of the varieties of the proofs of the Pythagorean theorem and its variations applied to dimensions of reference frames.
Now the nature of reading for me is not an instant continuity of comprehension, especially if that goal of comprehension is to direct in dialectics to some view or if the assumption is go thru the motions and the understanding will come. Some ideas are more simple than we have imagined and some much more complicated. So it seems there are levels of reading of which our initial comprehension radically transforms and exceeds what we thought was a complete understanding. Alas, I do not know what in the outer reaches of dimensions awaits in this complexity. What understanding I have is most likely equivalent to a hand held calculator. Of course one needs a BS in mathematics not using the calculator as a teaching tool as much as to know what buttons to push already! This paragraph applies to philosophy itself way beyond our considerations of mathematics.
I have said that I took a cue from Fermat with his recondite observations of what happens in number theory- some regard him as an amateur and a second rate mathematician. Well, I feel I am in good company then. Looking back from new connections of the background of trivialities I wonder, given his initial insights for space itself so limited in holding the perception but so much wider in the application to the idea of dimension- that after all a simple proof of his last theorem in not far away. The discrete can give us hard geometry also on the familiar scale of things and high vague languages we set up as models and for communication. After all, to some approximation of the abilities of a calculator in the distant higher calculations it will spew out that a cube can be the sum of two cubes- infinite descent is a sort of mirror of which such proofs seem to apply but lose comprehension in the process of applying it. This, like infinite regress where the exact analogies of geometry may apply, is in the end a matter of taste.
Lampion 01-27-11b The issue is our ideas of the discrete and continuous- I question, given a wider view of the role of matrices (that some can be unsolvable yet not describe some physical reality) if in general the square can be a sum of two squares even if the sides of the triangles are say triangles. I wonder if we can really say this of fractals on the sides of a triangle if they are not on the same level of being a fractal. Virtual photons can have a structure that suggests they can have a discrete collective measure of contributions by looping.
A fractal after all, complex or in some other form of pure recursive shapes, is that quantum idea of fractional dimensions. The question of fractional charges or not will be found an inherent property of the general arithmetic and geometry as dimensions.
[the mirror ideas, as made manifest, I thought I would designate as smell- thus there are caramel, melot, and horse flesh (to which like said of Fuller someone would call his touch geometry- Singh defines geometry as touch and sight- his work was horse hockey- but alas, very high grade horse hockey.) but these poetic ideas are not things I am sure I need to incorporate into the body of my mathematical philosophy.]
I am amazed, but not surprised, that these elementary matrices can work with probabilities such as Markov chains. I am not surprised because after all Pascal devised and applied all this because of theories of probability.
Now in the quasic grid, I leave it again up from yesterday if anyone wants to print one off but it would take 64 sheets of them to describe the 16 dimensional case. I know in the 9 dimensional case what I was looking for, the validity's of sorts to which the 24 is so important in three and four space, was not quite the answer I expected. It is always possible my own level of comprehension will be viewed as below some higher level of what can be obtained by self or others. I realized that some of the numbers I have called quasic or rather important for quasic structures are really the simple 1's and 0's of the diagonal matrix notation. 16 or 17 depending on how we count would be included in 8 or 2^8th dimensional resolution of the grid. Thus what we do on the line of numbers, one side of the quasic grid, is what we do with matrices- now what of the quasic plane itself and its numbers?
In another important realization is the idea of complementary or inverse matrices that is intrinsic to the quasic encoding- as if these, the absolute change in coordinates of 1 and 0, rather than these taken as absolute and then operations require multiplicative and additive distinction, are in a sense real and absolute space. However, in reducing things to one quadrant, or to one octant, we should take the others into consideration as we explore the spaces of interrelated higher dimensionality.
I found it rather interesting, the incidence matrices used for programming and to which I realized could be considered quasic coordinate notations, and yes there are more complex forms of this, the K circuits and so on of Coxeter, which considers all the subcells of things connecting or not- that we have to define a node as not connected to itself. Perhaps there are intermediate ideas of such connectivity in higher spaces, perhaps this again is a problem of looping virtual photons or even how in some way things can connect intelligibly at a generator of the null. We can proceed with caution in handling such cases but one day have to understand them from a more general view- not just the ability to build up, from flat brane like planes computation from a seemingly connected string at some point, or to compute in so many of some preferred step the end of calculations from the corner of a structure.
It is ok in such a view of matrices to perform a reductionism of the local and global information of general matrices if the result is one of a logic and its philosophy- for in doing so we at the same time expand to the theories of everything.
I suppose that covers about everything in my casual stray thoughts- I am most amazed that I can come here almost every day with so little to say then somehow type out so much. Walking, or taking a break outside, moments of easy thinking, something that I developed writing poetry in the street and but rarely having to stop and sit on the sidewalk to write a line down lest I forget- I mean, it just takes practice to have a global survey of what you recall and want to say and it gets easier.
* * *
Feathers, Seeds and the First of Stars L. Edgar Otto 01-27-11
Overcast the sky's pink glow
unnaturally returns the city's light
That ate the Milky Way long ago
The air so thick I drink it in refreshing me
I did not notice when the first icy star fell
Noticed at some point the slippery road
The profusion of them in all directions of the wind
the loon peeped, owl screeched, unconcerned as if
Snow itself was but a sandy beach
The Masked bandit peeked out from the rusting gutter drain
warily eyes me holding still, sniffed, went back again
His underground highways, snowmobile trails and rain
Tomorrow piles up mud and slush again until great floods
Nature bursting at the banks, brings back the rainbows.
* * *
Pebbles before Sand, Raindrops before Snow L. Edgar Otto 01-27-11
My calculator seems alive tonight
less the wires and glass bead game, an abacus
How I wanted one, then crude and costly prototype
freedom from the slide rule and when
They ruled the world, the first word of my first born
Oh, it needed sunlight or carbon and zinc to feed its
eerie dimming day glow eyes to help me think, discern the skies
Do long divisions and square roots not by hand so wise a babe
no more than I understanding the spewing out at creation
Reaching the pesky inverse of zero
Although it could not dance beyond two to the three three oneth power
The fractal shadow ghost of Ramanujan welcomed its poetry lost while its program error.
* * *
Perhaps it was not the dead end of trying to apply across my quasic grid the idea of the direction in one way a factorial thing and in the other way the powers of things, for to solve this we needed to work in a much higher concept of the complexity of space. Of course this apparently is what some now explain as one of Ramanujan's insights on numbers where they do seem rather fractal.
In the notation of the incidence matrices for the C++ program to generate the Soma cubes from the x direction there cannot be more than three 1's in the rows over the whole of the seven pieces. But this property in the quasic grid, that and the chirality of it in the incidence and direction of the pieces, tends to establish the four quadrants in the formalism that applies in even and odd spaces but especially to the squares of powers of this fundamental 3+1 or 2+2 Fermat-Ramanujan structure and how we understand, distinguish, or get around the relevant bilateral symmetries across the quasic grid- that important for the quasi-reductionism at least in lower orders of quasic space for the gene reading.
Of course a clue or insight such as Kea's in her post today may be a geometrical version of this number thing where vast generalizations await us and perhaps a still deeper understanding of particles and dimensions. How else could any of the powers of things act as strings without the strings? How else can such ideas as the nature of point like nodes in general space imagine or see things in the quantum world like in a sense all electrons are one electron?
In questions of the surface of that which in volume, scaleless, we note the structured central spacious or not singularity is it not obvious that somewhere the nodes can be of more than a plane? And yet if the ideas of quasics apply to such planes in what sense can we intertwine and relate to the general natural space wherein we may now describe the totality itself as naturally quasic? This perhaps a whole new area for inquirey to make what is still intuitive concrete.
Somewhere, as in my poem and apparently science news concerning the first stars, this sort of geometry in the background slowly develops them until their profusion. As vast as this vision of higher space is- in my remote awareness of it I must note also that the panorama of it can be as much humbling.
* * *