Sunday, May 1, 2011
A New Philosophy of Mathematics
A New Philosophy of Mathematics
L. Edgar Otto (The PeSla) May 1, 2011 Eau Claire, Wisconsin
New mathematics, new philosophy of mathematics, perhaps a new philosophy itself. What sort of idea is an original idea- that which comes first? That which cannot cite some earlier work? That which stabilized the chaos and variety in the multiverse? Did Einstein not publish his discovery with other papers or abstracts of abstracts to show he spoke the language and played the game of physics, with no one to reach with him into the contemplations- he there first?
Since him, as it dawned on some others, space and time never to be seen again as separate things, independent and absolute, but one entity. These over the span of spacetime to dominate the wider scale- as so our wider span of thought the human scale. And yet, discovery or invention, how do we measure in individual contribution as to the when? With what reference frame he who comes first can be said in a world of time without simultaneity?
As far as the string and other physics of our time we are in a position much like that before Einstein. We regard the idea of independence of the background as clarification of the scientific stance toward other confused notions- or say that on some basis the mathematics of vectors apply, the universe of universal law, objectively and dispassionately, double blind, of sufficient sampling and luminosity of that light we discover in our arcs for motion pictures, our drinks of medications and meditations with mind. What is true and cannot be made untrue yet surpassed in the philosophy of science no sooner is grasped as a powerful idea than it becomes, especially in the west where we are rewarded with great machines of planet forming, it brings up the age old questions of what brainwashes our thoughts and divides our minds to which we are only too happy to claim them for our credentials, to use them even if know partially without their higher implications as clubs in debates with the clamoring crowd coming to our castle in fear of the monster of Frankenstein and in no mood to joke for things solid science keeps us strong against the madding wave of superstition, fission in bubbles of beer, the Young Frankeinstein.
So, we are on the verge of a new physics, and new age perhaps for mathematics. It is as influential on future mathematics as that result of our understanding of space time to which the mathematics had to catch up to the theory as did the experiments.
In particular, we sometimes separate what is useful to do so but it is not the end all of theory. Many want to go back to put things together again, to unwind the marriage of space and time and find a new love, a new beginning and look at absolutes. Other just want to undermine the theory established no matter which side of some political fence they are on. Science while it should heed moral law or at least consider the consequences of its discoveries, should not be the weapon of politics alone.
A first such principle, and a powerful one, is how current theoreticians consider the idea of groups and apply them to physics. We have separated the idea of groups from the idea of sets so to establish and apply by various theories some pattern of symmetries. Symmetries said to be the proof of such groups and sets. There is even a longing for some sort of super-symmetries, and these as well behaved as their theories of dimensions to which they may claim surface things project into the volume things and as a hologram explain other things emerging- so in the identities of all of this we derive the idea of conservation, some spinning objects of spacetime. Henceforth or for this century at least, let us not keep the Group ideas and the Set and symmetry ideas separate, for at least in the depth they are one entity.
The second thing is our crude way of counting the dimensions, to count the vertices as from one corner then develop the network outward or count the symmetries from all the vertices- the so called stabilizers and orbits, again separate functions, and the endless play of the search for the true meaning of the continuity of sub sets.
I must ad a third thing, our understanding of primes as a most powerful concept in abstract algebra. To understand what it cannot do or what it does not describe in the real world of things is to come closer to a more unified idea of physics and the philosophy of our being in this world. Such stabilized and orbiting partitions and separating of some notions to discover or make them absolute is not as simple as our idea of counting the dimensions this way, we stuck in a corner or floating in a disembodied cloud- numbers are much more subtle, especially for smaller scales.
It seems reasonable, to say in a cube we have three edges and eight points for the rotations and all that of twenty four. It may seem fanciful but reasonable and intelligible to say in a hypercube we have for edges x three planes x 16 for 192 rigid rotations... thus the general formula n! x 2^n for the full group with inversion- all very reduced and intelligible. But who is to say that between for such objects there are four and not three from some perspective and not even considering inversion thru a center, a higher dimension really in the count? At what point do our assumptions exclude into the compacting down of space the extra elements or those implied as shadows in the distance we imagine in the count and balancing the books of particle energies for our physics?
What is the Knights tour but he the piece of resistance, the four knights who left the full deck long ago to joint the old gods and dreams of higher space? What left then as they move outward to the next major arcana of a hidden structure and nucleus of three space but holes to be filled, to move between from one realm to another, to chase and replicate the joker as well in the deck, resistance, inertia, that which from a wide view or sense of feeling animals see the world as familiar and intelligible and cannot just be reduced to dust, or become a fog of dreams, but a clear path thru the depth and span of dimensions.
We cannot just define ourselves, nor the particles and fields, as just material items nor as mindful stuff alone- nor suggest these the same substance. Such a world would no more be dynamic as we observe it, and diverse, than to insist that zero plays an intelligible role in the group and set ideas not beyond its vanishing.
How we see these simple math games and profound and powerful ideas is to a great extent, while we are free to play on our own and develop the pecking order on the playground, will influence how we see or refuse to see ourselves.
In this new philosophy of mathematics we could find a new view of ourselves with an intelligible way to negotiate the magical and the real of numbers as existent that we are intelligibly one with the idea of being, its space and time and spacetime, its realization that we are one with number and can be numbered (in a way of course this bit of wisdom even from the gods does not insult the intelligence nor quality of their creation if not they but just our gardeners for our souls), unique and to some extent that uniqueness in theory is preserved- the truths of geometry too- that if the numbers exist in an ideal way independent of us- we are ideal too and they are not as independent, necessarily, as we imagine in some earlier philosophy of mathematics to which the concepts of what anything is cannot be so contained.
Just to bring this idea home a little, the consideration of how things move abstractly, like the knight moves, the changing between signs, the parity, the bare analogy to those concepts of what is existent or shadows, real or ideal, intelligibly finite or that implication of a vague wider sea to which we now depend mostly on the relation to methods of chance, these justify our thoughts and methods like the Fourier analysis, a view really that thinks of waves, of heat, of oscillations in the motions and so on, that divides space in depth and span into the various trees rooted or not into some general structure, like the quasicontinuum, to that we justify our concepts of extended motion, energy, force like things and mass beyond a small discrete step- we in fact justify our quantum formulations- but we need not be trapped into the paradigm where it is on one hand a stepping stone and on the other a great suspected solver of all paths of the intelligible as we ourselves find a more certain view, ideal and pointless, between existence and non-existence that not necessarily something independent of us and within itself, we the observers and observed on the scale of our own background- and perhaps final choice to uplift some aspect of our beliefs or faiths and skepticism so as to make the leap into what we wish and intuit in this- I to tolerate the diverse and precious minds and lives of others maybe did not need to make such a leap in order that after a long quest for what we hoped to find we found something better and freer and the mandate to do things as if for so high a purpose- a distant God in the background, the work of gods as a creative force, not the oppression of such narrow views of chance and necessity, in which the shamans of logic know better among themselves the limit of the game and the self-healing motions on the believers if they who lost their faith continue the ritual, logic to catch up again to the mathematics.
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Interestingly, in my stray thoughts of last night- of course the papers thinking of the 7 hexagons to cover the torus surface, and in a way the same sort of intelligibly connected topologies that suggest such a role for carbon in the expression from some zero vertex in topology as a useful study, or that the number of divisions of it in three space is that 13 is the number of sections- all of this should be taken in the light of the new mathematics, geometry and numbers from a slightly higher vision to which even the six points of this genus where the fox and corn and chicken, and the boatman conscious of this to cross them if he can go against the stream of what is accessible in the infinity of the Tower of Hanoi so to find beyond mere gears and clockworks the other solutions to which it is hard for mere machines to program to do in steps and to see.
But I would be remiss if I did not mention, and in detail and recalled the gist of this post and the understanding, that this did not come from a deeper dream after reading a few of our theorems in an old moldy book I found before my move soon to come- thus I felt compelled for a further post, and that it is the hope of the sun in May.
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This comes close to my sentiments today, with synchronicity from a fellow poet blogger:
In search of Spacious Poetry by David King
Tate Modern's Turbine Hall:
a space to test conceptual powers -
almost to destruction.
How can an individual fill a space like this?
Though few are called and fewer chosen,
art eschews the easy victory,
prefers to fail at the impossible -
the task a metaphor for that
which faced the Great Creator when
the world was yet an empty shell.
But here in Turbine Hall is space
an ocean might not fill,
space that awaits a single concept, not a world
of possibilities. The thing about
this eeriness is what has gone:
the turbines that once powered a neighourhood.
It's these the installations must replace.
So much has vanished from our world,
our culture, our environment. From us.
We need to give the space that's left
a shape, dimensions, to define its emptiness.
can make the poet in us all
Yet poetry deserves its Turbine Hall,
a fairy godmother to call
and wave her magic wand,
something gargantuan in us,
in what we stand for,
that only poetry could fill. But
what a vastness and what poetry!
Mankind perhaps would end up looking small,
creation grow beyond today's imagination -
but individual man walk tall.
Do not suppose in saying "vastness" I mean "long".
Words do not fill such spaces by their numbers,
but by a coefficient of expansion. This,
the soul, and only it, can give.
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And Sultan Ratrout sent me a translation of his inspirational poem (on facebook):
The Sunset Journey !
- Its the sunset time, The time for self-confrontation , a time not to escape from Allah ! - Confront yourself in front of Allah, today you are alive , tomorrow you are dead ! - There is Allah waiting you in heaven , raise your hands up and supplicate whatever you want ! - Go deep down in yourself , contemplate what you did, take advantage of time, not unlike that time that wasted. -Don't argue about your fate that has been written ages ago, but learn from your past sins with regret - Address Allah in eyes full of tears, and pray with an honest intention and piety ! - What a beauty stemming from Tasbeeh and istighfar (Seeking forgiveness) phrases, There is no god except Allah, he is the one who saves you from depression and falling down! - Beg Allah in this period of self-frankness, Seek forgiveness from him and from those whom you oppressed ! - Clean yourself of grudges and bad intentions, People change even they have the most stubborn opponents ! - Don't get desperate ! Allah's mercy is unlimited, Always be optimistic and always raise your hands to him !
Vintage Thoughts and Truths - alas, we but frail humans not quite able to reach into such heights and believe. leap into the faith, transcend mere fate and chance, save for grace and mercy- and the promise actually when the heavens are unified, that in some ways tomorrow we are alive as to some degree today we are dead.
Reply by The Pe Sla
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Ulla sent more links and a general statement:
Adel Bilal, Introduction to supersymmetry (2001) ArXiv hep-th/0101055,
They are invariant under local supersymmetry. Because of the underlying supersymmetry algebra, the invariance under local supersymmetry implies the invariance under spacetime diffeomorphisms
I reply here to this-
of course under locality any such concept of averaging or invariance or the condensation of entities even if chiral depends on the math of the compass of primes and their distribution over our familiar real spaces of fact and perception. This still but the germ of a new theory where clearly our ideas of what is the stablizers at infinity (for we are trying to view that sphere of an infinite radius with infinite dimensions, possibly discrete in some view that is nowhere while our center singularities are everywhere- to go a little beyond Pascal who was beyond St. Agustine's two space... and of course we will have to go beyond n-space concepts one day if we extend this sort of analogy.
Not sure what you refer to here as invariant. Those papers look to me long winded and from my view very much outdated but perhaps seems revolutionary for their time- in any case beneath all the symbols and equations are simple numbers they do not seem to be able to count consistently. How could such intricate minds find dead ends and slip from theory into a morass? I wonder if they will ever see it as a waste of time- or for that matter if we take the time to learn this agenda and totally understand the papers wonder if it was worth the effort.
But I do not know to which post or comment you are referring to here- I just posted what is likely my last for awhile in which I took the last step to firm some of the themes of the blog up. But thank you for the links- interesting to me from a historical point of view really.
No damn cat, no damn cradle (strings...) well, all such ideas have their place- but until some of them deal with a better view of at least dimensions they remain controversial. And if we are to justify our use of such things in biology (DNA etc) then we have to go to a better philosophy of mathematics and the tools of it.
From a science fiction view- sooner or later we become ascendant beings as we go thru the wormholes of the ancients- but with such speculation I have to see the work, not just say that certain octonions were explored and shown only relevant to certain spinors and so on (as Matti said) if that is what you refer to..
Have a nice May Day... it is going to be 25 tonight and seems a flashback to late November, may snow tomorrow.
I already miss the clarity of Kea's post, to which when she makes a clear statement as to the absurdity of some methods- it rings true.
You all can email me any time, thanks for the links.
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Footnote: Omega Minus- (I refer to Lubos post here which always seems to praise Jewish physicists only and not the work dispassionately) this discouraged my independent research for a long time and I see no good reason to be grateful for those who developed this sort of physics, they also set back research and for what- some sort of tribal world view. It is high time, if indeed we want to see science rise again among all cultures- and I doubt this will happen at the rate things are going now- we took the best from all such cultures and faiths for our physics and not worship the false saints of our forefathers and mentors to the detriment of our own ability to think and discover for ourselves and not just use the knowledge we stumble upon that is so inadequate because it was a stumble- and then to praise it as the sum total of human knowledge.
But like politics, such philosophy will not let us make rational decisions soon nor even those to which with a tendency we gravitate in our genes. Not as long as we persist in but one of the many faith based physics of limited views and inability to see our common grounding.
That the untruths are glossed over, talked down, forgotten in time to be followed again by the fresh new masses- does not make things true nor democratic the new paradigm.
Even the great Goethe, as brilliant as Newton, seems to have had his imaginary playmates- the imagining of friend before him and their expressions as he told them his problems. But since "He who has a lot within needs very little without" to quote the old genius one can only wonder to what extent this imaginary friend we call the internet if we need outside help will respond to us without anger and dismissal but with soul. Let those who can see thru the screen to the true heart unto the degree he can of another so see. And to those who cannot quite see may the wisdom of the mind not stifle your creativity, and we bow down on our knees in respect for our prayers for human courtesy and civility.
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