Book review: Love and Math

“There is a secret world out there.” This is the beginning of Edward Frenkels book about his mathematics and his story of how he fell in love with it. Popular books about mathematics are rare compared to areas of science such as particle physics, cosmology or even biology. It is hard to write a mathematics book that will appeal to the masses. You cant really play the trick of skipping all the equations or the details because these are really the essence of what makes mathematics so beautiful to those who master it.

Even rarer are such books written by the people who are at the bleeding edge of current mathematical research. There are some great maths books by Marcus de Sautoy, Ian Stewart, Simon Singh etc. , While some of these authors are maths professors their popular books cover stories of mathematical problems solved by others. I have always found that the most engaging books in popular science are the ones written by those who were closest to the discoveries themselves and this book is an excellent example.

In “Love and Math” Frenkel recounts his voyage of discovery with details of the maths and the equally fascinating story of his passage through the education system of Russia in the 1980s where he faced ridiculous obstacles placed in his way simply because his family name is Jewish. Despite glowing exam results from high school he was not permitted to attend Moscow University and has to settle for another college more geared to industrial engineering . Luckily such difficulties were compensated for by a system of informal mentoring by some of Russia’s greatest mathematicians that supported the most promising young students like Frenkel.

The tale of his progress from school to Harvard professor is interwoven with potted lessons in group theory as he had to learn it to solve the problems posed by his mentors. These are aimed at non-experts. For someone like myself who is already familiar with the standard methods but not with all the recent progress this is light and enjoyable reading right up to the final chapters where he described his work with Ed Witten on geometric Lamglands. I cant say how a complete novice would find it but young math students would surely find inspiration and useful knowledge here and others can skip the details and enjoy the human side of the story.

The book ends with a chapter about his controversial short film “Rites of Love and Math.” This is said to have made Frenkel something of a sex symbol among mathematcians, certainly a new idea. Unfortunately the film is not available through the inline rental services I use so I cant tell you any more about it. Here is the trailer from his youtube site.

36 Responses to Book review: Love and Math

  1. Sexual analogies ring a bell close to our hearts that despite the vagueness most people seem to understand. Why? Imagine an individual is an n-dimensional bilaterally symmetric complex geometry limited by the properties of math at the octonions. What happens if two lucky souls are superimposed to mixed groups? Especially if in a weightless attractive environs. But what a new geometry outside oneself to explore, rationally as Greek science and mystically with a sense of beauty.

    The light from inside the jack-o-lantern carved with our happy faces may not keep Nature there very well, no matter how permanent our markers or tattoos we try to solve on the surface- Plato’s cave and all that or some mythic lovers long to be one again. Or how we may say hidden in our psyches some Freudian slip that makes the interpretation of our lonely dreams described by Lamb-glands, Vive l’difference as we share wisdom and beauty in the Maths and hold to a dream that She is our universe and everything.

  2. John Rennie says:

    As someone who is interested in maths, but doesn’t want to become a mathematician, I found the book fascinating and I strongly recommend it for all us maths amateurs. Apart from the human interest, Frenkel explains various aspects of the Langlands programme in accessible terms, which is not an easy task!

    Having said this, I imagine most of us will get a bit lost in the last few chapters as Frenkel lets his enthusiasm cary him away. Oh well :-)

    I’d say the book is pitched at about the same level as Yau’s book The Shape of Inner Space (if you haven’t read this I heartily recommend it). Both books would be trivial for the professional mathematician, bewildering for the beginner bust just right for us interested amateurs.

  3. Tienzen (Jeh-Tween) Gong says:

    Thanks for this great book review. I especially enjoyed the video. Although life is the carrier of love, the death can never destroy love. My love on math is only about one question.

    Is math the human construct and only as the *tool* for describing the laws of Nature? Or, the laws of Nature are embedded in math?

    For me, the laws of Nature must be embedded in math. Then, the task is to *derive* all known (including the unknown) laws of Nature from *math*. As the math universe is *infinite* while the nature universe is finite, the task becomes concretizing infinities (the countable and the uncountable). In the book “Linguistics Manifesto (115 pages, ISBN 978-3-8383-9722-1)”, it shows two concretizing processes.

    One, embedding the countable infinite in finite — this is done by trisecting a (any) angle. The following is quoted from that book.

    If a thing “TA” is the product of an infinity to finite transformation, then TA is the concrete representation of that infinity. Can we find such a TA?

    If such a TA does exist, it must be produced with “infinite” steps. Then TA and infinity (TA) are two sides of the same coin. That is,
    From TA, we get infinity (TA)
    From infinity (TA), we get TA.
    In mathematics, there are two kinds of infinity, countable and uncountable. There is a commonly accept belief that there is no way to trisect an angle in Euclidean geometry. Yet, a trisected angle is always a physical reality. In fact, if we can evenly divide an angle, we can always trisect that same angle with the following process.
    Divide angle A evenly.
    Divide the two angles evenly again.
    The top and the bottom angles are now 1/4 of A. As they are symmetrical, we will look the top angle AT only.
    The middle angles are divided evenly again. They become 1/8 of A.
    Let AT plus 1/8 A, and it is larger than 1/3 A.
    Divide the 1/8 A evenly again, 1/16 A.
    Let AT (1/4 + 1/8) minus 1/16 A. “AT” is now smaller than 1/3 A.
    the above process goes countable steps.
    Then, AT = 1/2 – 1/4 + 1/8 – 1/16 + 1/32 – 1/64 + 1/128 – 1/256 + 1/512 – 1/1024 + 1/2048 -… +…
    = .33349 – … + … = .3333333333333….. = 1/3

    With countable steps, the above process trisects the angle A “exactly”. That is,
    with countable infinity, we get AT. With AT, we get countable infinity.

    With the above trisecting angle procedure, we know that 1/3 = 0.333333…, that is, 1/3 has countable number of digits. In mathematics, we do know that the number pi (=3.14159…) is a normal number, that is, it has uncountable number of digits in it. Then, how can we reach pi (precisely)? Well, we can reach it with the following equation.

    Equation A:
    (The circumference of a circle with a radius of 1/8) = 1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11 + 1/13 – … + … (with “countable” infinity steps)
    = pi / 4

    That is, the number pi (which contains uncountable number of digits) is reached with a “countable” steps. So, the uncountable infinity can be renormalized with countable infinity, and countable infinity can be renormalized with the trisecting angle procedure.

    Note 1: There is something interesting in equation A. The circumference of a circle with a radius of 1/8 (with an even number denominator) can be reached only with a sequence of odd number denominators.

    Note 2: The area of this circle = pi / 64; then, the number “64″ must be fundamental for this uncountable infinity renormalization process.

    In the two equations above, the creation process (from infinities to finites) depends on two very important numbers [3 (or 1/3), 64 (or 4^3)]. Indeed, these are the numbers for the Alpha-equation below.

    Beta = 1/alpha = 64 ( 1 + first order mixing + sum of the higher order mixing)
    = 64 (1 + 1/Cos A(2) + .00065737 + …)
    = 137.0359 …

    A(2) is the Weinberg angle, A(2) = 28.743 degrees

    The sum of the higher order mixing = 2(1/48)[(1/64) + (1/2)(1/64)^2 + ...+(1/n)(1/64)^n +...]
    = .00065737 + …

    The physics underneath this Alpha-equation must be the fundamental for all laws of physics. This truth sits here silently, blocking all detour attempts. This is my love on math.

  4. Tienzen Gong… that last post has a lot which is relevant and may begin to answer if math is related or is independent from physics (and in what sense of certainty to logic)… the fine structure constant is indeed regarded as a value to understand notably by Dirac and Sir Arthur Eddington and his 64 four coordinate quantum relativity of 1929 in the Fundamental Theory. The question is also if there is something between the countable and uncountable, structurally anyway… is not the gene code 64 as a system? The Egyptians used 256/81 as a better approximation of pi. So as they say we need to take a little longer in understand the finite. What is left dividing the whole by the Horus eye except finer and finer descent into divisibility or something left over that belongs to the gods? Can the number of protons in the universe be finite, something like 2 to the 137 times as a factor in the equation or is that somehow a first approximation although and exact number? You as well as others will sooner or later understand what you suspect is there in a better and unified physics.
    As to the theme of this topic thread… it inspired a poem on my blogspot http://dreamikins.blogspot.com/2013/10/love-in-icy-cold-of-space.html and this is one of the stanzas:

    What in your ankle sent Frenkel wandering
    in wonder far from the arms of Milky Way
    His magic marker and equations on your
    belly down to the end your belly button

    Let us not dismiss too readily the wisdom of the East for their I Ching system is also based on 64 or 384 lines if you count the rotations of a hypercube… but why not 256 of 8 lines we we can read the DNA also in a 256 base system in our lab experiments.

    • Tienzen (Jeh-Tween) Gong says:

      @L. Edgar Otto, “… by Dirac and Sir Arthur Eddington and his 64 four coordinate quantum relativity of 1929 in the Fundamental Theory. … Let us not dismiss too readily the wisdom of the East for their I Ching system is also based on 64 or 384 lines, …”

      I have deep respect for those old wisdoms. But, no, please don’t mix up my discussion here with them. I am talking only about physics, about the following issues.

      a. The theoretical base for the free parameters (the Cabibbo and Weinberg angles, Alpha, mass-charges, etc.)
      b. Quantum / gravity unification
      c. String unification (the theoretical base for the Standard Model particles)
      d. Planck data (dark mass and dark energy)
      e. Mass-rising mechanism [the article “Top Ten Higgs Boson Facts (by Sean Carroll, at http://www.preposterousuniverse.com/blog/2012/11/13/top-ten-amazing-higgs-boson-facts/ )”]. One its fact is as below.
      (8.) Your own mass doesn’t come from the Higgs.
      f. Etc.

      Of course, my approach to the above issues are significantly different from the current paradigm. The major difference is about the view on *MATH*. In the past 500 years, the Math was viewed as a human construct. Every great mathematician in the past created a set of lego pieces and built some great artistic wonderful structures. Yet, incidentally, many of these artificially structures are great *tools* for describing the laws of nature. This view was the history and cannot be arguing against. But, my view on math goes beyond that. That is, the physical universe is only a *subset* of the math universe, and all laws of physics (nature) can only be the emergent of laws of math. So, all known laws of physics (nature) must be directly *derived* from the laws of math, and all unknown laws of physics (such as the issues listed above) must also be derived from the *pure math*. Of course, if I cannot actually show the *derivations* of those laws from pure math, it will just be a personal opinion (simply nonsense). But, I have showed the calculations of the Cabibbo/Weinberg angles and Alpha. I also showed the calculations for dark mass and dark energy and the string unification. I think that these are enough as appetizers. Investigating the *physics* underneath those appetizers is much easier than building a new collider while I personal do not against for building a new collider.

      My project has only a very few steps.
      i. Concretizing the infinities (both countable and uncountable) with *physical* processes (the trisecting an angle and *dividing a pie*). This concretizing process is the *creation* process, and the Alpha-equation is this creation equation, as the anchor or the lock for the process.
      ii. Second tier concretizing process (the rising dimensions) — this is completely different from the ultra-dimension of M-string theory. This dimension is defined with Georg Cantor’ theorem ( In 1870s, Georg Cantor proved that every n-dimensional space can always be brought into a one-to-one correspondence with the one dimensional line, that is, one dimensional line can give rise to n-dimensional space. See http://prebabel.blogspot.com/2012/04/origin-of-spatial-dimensions-and.html ). The number *64* becomes (4^3). That is, 3 spatial dimensions while each carries 4-sub-dimensions; total of 64 dimensions. Then, these 64 dimensions are divided into two groups, 48 matter (anti-matter) dimensions and 16 vacuum dimensions (the dark energy). Yet, with the inter-dependent relations, these 64 dimensions are reduced to 11 dimensions [7 colors (packed and hidden) and 4 expressed].

      This 4-sub-dimensions form a higher symmetry than the Standard Model symmetry, that is, the SUSY (with s-particle) is ruled out. Then, the Quantum/gravity unification and the mass-rising mechanism are truly the same issue. I have showed that the ħ (Planck constant) is, in fact, the gravitation force for the ground state of a hydrogen atom (see http://www.prequark.org/Gravity.htm which also discusses the mass-rising mechanism).

      “Can the number of protons in the universe be finite, …”
      The entire physical universe is finite.

      • Gong, I am not trying to dispute your interesting methods and researches… again I am just a poet – I thought this sort of intuitionist post of yours allowed a little wider metaphor of things not yet known. It is said that the order of the hexagrams is forgotten and known only by the gods- yes, there is old wisdom and surely even mentioning the I Ching adds to the claim of scientist dismissed and not taken seriously. What of Plato’s sacred number 192… or ancient babylon’s 240 (as in the recent discovery that many 8 dimensional spheres surround another). To assert that 16 of the 48 explains dark energy or matter is a good guess and hint but way too premature as I do disagree.

        if the universe is finite, entirely, thus its averaged parts finite I might reply that the universe itself is a giant collider acting in the same way where time itself is that to where we aim our smaller energies and measure things as if our LHC a telescope. In any case the world is also quasi-finite and infinite.

        But we can ask simpler if controversial questions like why is there motion short of trivial chance in the now in the I ching, or if one line is yin or yang, or just yang right, left, or non-right. Or that DNA may have evolved from a precursor of 16 codons.

        It is thrilling to imagine a line segment, a plane, an infinite expanse of infinite space can map as one counting… but what do you say about aleph 2 of which it counts a larger number curves possible drawn on a plane…or in wider dimensional space? and what about aleph 3 and so on…

        So just think of my comments like a very old Chinese poem of which we have long lost the meanings but it is saved for the calligraphy’s sake.

        Thank you for the dialog, and good inquiring!

  5. orwin says:

    I came across the traces of the Pythagorean art of Physiognomy in W.D. Ross’ catholic translation of the Aristotelian library. They make no pretense of attributing it to Theophrastus or Stabo after Aristotle, which means no post-Aristotelian language. We know that Aristotle was in Italy with Plato on route to Sicily, collecting evidence for his cyncial view of the Pythagorean legacy, and likely collected some texts to substantiate his position to skeptical students. But I would attribute On Physiognomy to Simmias and Cebes, companions of Socrates when he was starting to worry about his mental stability, and told off by him for passing old Theban lore as Pythagorean. How that must have hurt! – and then Socrates was more or less crucified for showing it.

    There’s a horrible scar on the later philosophy: Ibn Sina (Avicenna) died of colic, having failed on “Physician, Heal Thyself!” and when C.S. Peirce got back ot the second-order logic he pioneered, he had an attack of the same (he called it Grippe, German for gripe) writing a key paper and sprayed it with errors.

    Now i can tell you, you have to mind:

    1. pH balance [Deformation algebra!]
    2. Sodium/Potassium balance [Atomic Numbers!]
    3. Static balance [& how you throw your weight around...]

    That gives those blessed Numbers as the Aristotlian ‘middle term’ in the logic of it all, which Speusippus and Xenocrates after Plato, looking back to Pythagoras, interpreted as Forms in Plato’s sense. Aristotle had zero algebraic intuition (like Kant), made no sense of this and tried to cut the math out of the Pythagorean legacy.

    That leaves us with no leads on the relevance of old Arabic bonesetting = al-gbra “to restore the missing part” => algebra. But its cure-critical to actually unfreze odd bones in writs and ankles because frozen they subvert your internal pressure-sense and hence thermodyamic regulation. Leaving you to bounce around between Osteopaths, Chiropractic, Homeopathy and smelly weed (which does enhance peripheral circulation, but packs an insult to the heart => paranoia/bad trip).

    What more could the Ancients have known? Just the effects of how you allign the nozzle of your alchemical Pelican with the Earth’s magnetic field, on whatever you are distilling. Since the Compass comes to us from ancient Feng Shui (lo Pan), the marker is five elements/five humours which you do find in Posidonius and Avicenna on faculties, evidently from Arabic alchemical tradition. But to mention that in academic circles is suicide.

    A bad trip for all of us, given healthcare overehads.

    • Orwin… creative, insightful, and brilliant Sometimes I wonder how future thinker will look back on our contemporary version of all this… hopefully we will still have the capacity for romance and continue to pass the baton of wisdom and love. This as a direction of research or trying to make sense of steps in history of thinking tells me things I did not realize how little in it I know, we all have our roles I suppose. But if we include issues of the algebra again how can I not see a link to the five fold wisdom of the East and the ten fold wings as metaphors of say string theory?

    • Tienzen (Jeh-Tween) Gong says:

      @orwin,

      Very good article. But, I am not sure of getting your point. If you are hinting that my posts above are similar to those numbers of ancient, I would like to make my point more clear. By the way, I am a true expert on the East culture, especially on Yijing and the Five Walks (fivefold wisdom of the East). If you and Otto are interested in those, I will definitely share with you some knowledge of them which you cannot get anywhere else. Now, back to physics and math.

      For the past 100 years, physics is very successful while *ignoring* some other obvious *facts*, the life and the math (using it only as tools). The exclusion of these two facts from the scope of physics was necessary tactically, as the laws of physics are thus far seemingly unable to encompass them two. But, by excluding them in *principle* (such as using Boltzmann Brain as a possible cause for the rising of life), physicists are then fooling themselves, as this Nature consists of, at least, three parts.
      a. The physical universe (not including life),
      b. Lives,
      c. Numbers.

      Then, there are two possibilities. There are three different sets of laws for these three parts. Or, those three are governed with a set of unified laws. For the current physics paradigm, it has chosen the former. On the other hand, I have selected the later (they are unified). With this choice, the other two facts (lives and numbers) become the *check points* for forming the physics theory.

      Life has two very important attributes, the individuality and the need of processing information (needing a computing device). Thus, a physics theory which give rise to life must encompass,
      i. A four color system (red, yellow, blue, white) or (A, G, T, C) which gives rise to individuality.
      ii. A computing device — abacus, counting straws or a Turing machine.

      Numbers encompass infinities (countable and uncountable). Thus, a physics theory must also encompass those infinities (in a concretized form, at least), not just cut-them-off by using the renormalization.

      With the above overview, now we have two types of physics.
      A. Human physics — physics laws *discovered* by human.
      B. Nature physics — physics laws created by Nature.
      These two types of physics are significantly different in the aspect of the emerging processes. The human physics comes about in piece meal. The Nature physics must come as an axiomatic system expression, that is, the axiomatic physics (see http://prebabel.blogspot.com/2012/04/axiomatic-physics-final-physics.html ). This axiomatic physics goes way beyond some numbers. I have used some numbers (the Cabibbo and Weinberg angles, Alpha and Planck data) as the showcases for it, because there is no arguing about whether we have matched those numbers or not.

      On the same token, there are two types of math.
      C. Math of Nature — giving rise to a physical universe (concretizing the infinities).
      D. Math of human — inventing some lego pieces and building some wonderful lego structures.

      That is, the current *love* on math can only be the half way to the true ecstasy, at the best.

      • Gong,

        Not sure we are close to the topic or not…but if no one minds. Surely science must have some relation to heart, I mean if love is mutual, abstract, or an individual thing. I would not mind first hand information on the non physics issues (but not here maybe) too big a project.

        Now I agree with you on the re-normalization method as not deep enough but what will replace it? Your statement on the Boltzman brain- what do you offer to replace that sort of idea or can it ever be resolved?

        I find this statement interesting: ” (A, G, T, C) which gives rise to individuality” it gives a certain sense of stability of choices if 4 and there is room in arbitrary coloring for 5 (on a plane) but to assert individuality is to need 5 (no I ching link intended) or if mirrored as complex numbers 10…(no lovers finding the other half of each other or sting theory intended). For in mammal cells we can have less than 2^5 cells become clones but removal of one greater kills it as an individual. Around 256 it is not clear if the egg seeks the womb wall or it is a matter of chance gravity as that experiment vanished with the loss of the shuttle (as far as I know it was not tried again).

        Do you mean, vaguely I perhaps am reading it… dark matter stuff is in and lets physical and mental things emerge from the world of pure mathematics? If we insist on the core physicality of three space as physics being primary then we can say that this much is within the realm of standard science,. But if DM is part of a unified conception then how is Boltzman which is between the other fermion or boson staistics not a unity of dark energy flatland or maybe half the ecstasy of understanding the Higgs? If it does relate then why not a Boltzman brain where things can change to seem to deviate from unity to emerge or devolve? What pure math maps this out? I meant to review you comments on transcendental numbers btw..

        One thing for sure is that China and the United States have a healthy streak of pragmatism of all nations in the world today.

        BTW Wineberg does not have all the significant angles nor the idea of a Turing code of sufficient complexity in varied forms can be shown to be a multiversal rather than universal code. Our thoughts and dreams emerge all at once also while some axioms emerge in one piece also. I do like your philosophy of mathematics, and in this age of theory we do need things brought down to earth again as if our technology is not growing rapidly…and the certainty of applying it hopefully as well.

        And the 6 compactification manifolds seem to me missing a couple of such symmetry dimensions as well the Bell ideas on the nature of non-locality. Also of the hierarchy of dimensionless constants we need combinations theory between them as it seems that the shift of neutrino values is not simply a quantum phenomenon, as we solve these issues of what is given whole and continuous or what may be constructed as discrete.

      • Tienzen (Jeh-Tween) Gong says:

        @L. Edgar Otto, “And the 6 compactification manifolds seem to me missing a couple of such symmetry dimensions as well the Bell ideas on the nature of non-locality.”

        I have no idea about what the “6 compactification manifolds” of yours is. You are making many preconceiving conclusions in your own discussion.

        Today, the math is very much different from the physics. The validity of math needs not be verified by any *external* physical reality, that is, it is in fact de-linked from the physical universe in principle although it is the best *tool* for describing the laws of physical universe. So, my questions are:
        a. Is the math we know of today (human math) mainly the human construct (as the lego games)?
        b. Is the human math the re-discovery of the Nature math, similar to the human physics which is the re-discovery of the Nature physics?
        c. If there is indeed a Nature math, what is its *mission*?
        d. Is the human math converging to the Nature math? Or just going all over the map, as the scope of the Nature math has infinite degree of freedom?
        e. Is the Nature math governed by a set of unified laws which also govern the physical universe and life?
        f. Many more.

        The above questions are simple and reasonable, and they should change our view about math completely just by those questions, regardless of their answers. By choosing a *unified* view, the math becomes the *check-point* for any physics theory, and the structure of the laws of physics also must be the structure of the Nature math. The symmetry structure of the laws of physics are indeed the properties of many math structures. Steven Weinberg said, “String theory is attractive because it incorporates gravitation, it contains no infinities, …”. Yet, with this unified view, the physical universe (although totally finite) must encompass all infinities which are the essential parts of the math universe. And, this becomes a very important check-point for any physics theory. Thus, the concretization of infinities in a physics theory becomes the central point. Yet, we don’t see any such a connection between the human physics and the human math. The major problem comes from the definition of *continuity* in math, which sweep the essence of the Nature math under the carpet. The key essence of the Nature math can be expressed with one simple equation.
        A – b = 0 but A is not b.
        What does this mean?

        The *numbers* are in *principle* (with some exceptions) not *reachable* by all finite *means* (arithmetic / algebra operators, etc.), such as (1/3) = .333… = .3C (C as countable digits) ; Pi = 3.14159… = 3.14159+ (+ as uncountable digits) ; etc. . This *unreachable* fact is permeated all over the places. The simplest example is the prime numbers which are unreachable by the *multiplication* operation. Thus, when b being a *touching* number of A [that is, their distance (A – b) = 0], but A is not b. That is, b is unreachable by all means, and the best way to identify b is A+ or A-. Thus, the majority of numbers must be *color* coded [let, A as yellow, A- as red and A+ as blue]. Then, there are *reachable numbers* (the white).

        As numbers must be a colored-system, it could be the base for the life-color system (A, G, T, C) which gives rise to individuality. In fact, life has a third attribute, the immortality which needs 3 additional codes (colors) — G1, G2 and G3 [G1 as M (male), G2 as F (female) and G3 as K (kids)]. If we can show that the numbers are in fact a 7-color system, then the Nature math is the base for life. The Chapter 7 — Colored Number of “Super Unified Theory (US copyright TX 1-323-231)” gives a detailed description about numbers which are a 7-color system.

  6. Gong, great questions of which I indeed have had some assumptions to consider or even recognize where I accepted them. Roughly, as in the idea of Dirac of a filled vacuum… I tend to make a subtle distinction between the number 0 and the empty set Null in such matters of distinguishing the indistinguishable. But I am not saying it all can relate to a sort of mirror dark like space to which we can imagine (as some do, Pitkanen lately, and some popular speculations by M. Kaiku who bids the listeners to ask questions (If anyone can explain dark matter they will get the Nobel Prize… a carrot to inspire new scientists… or that what we see is temporary in the cold endless expansion of space and time… but some may say that some “human form” or consciousness as dark physics could persist beyond the brief era of visible and hot space.) Hey, spelling is not my strong point… I mean the Yau 6D manifolds that assume stretching out to infinity and thus a complete picture to which what we see in the 7? manifolds ignores an eighth at least in such compactification of which geometry can be colorizes to show some things as Conway suggests the case with numbers. Does it not take 7 colors for the torus or its volume into 13 parts…is this not a starting point of intelligible counting and coloration? But I would think such ideas although modern put scientist off much more than say references to ancient wisdom. What would we do with a theory of such mirror or dark particle like relations that also has the limits and defects of our long attempts to support the standard theory and not see thru a layer at least of quantum foam and vibrations to find even beneath dimensionless strings these questions of the null? Our use of axioms for a theory is much wider in possibilities than how they are uniquely applied to physical models possible or yes the divisions (be they logic issues of addition or multiplication as somehow different in effects and phenomena) may indeed point toward a convergence of the way you have divided things into human and nature math… still, can we do otherwise? Thank you for your well thought out and sharply focused comments.

    • Tienzen (Jeh-Tween) Gong says:

      @L. Edgar Otto,

      Thanks for your comment. This is Philip Gibbs’ blog, and I must not talk too much here. Thus, this will be my last comment here on this math issue. Instead of answering your questions, I can make my point clearer by commenting on Steven Weinberg’s comment on the issue of multiverse. He wrote, “Inflation is naturally chaotic. Bubbles form in the expanding universe, each developing into a big or small bang, perhaps each with different values for what we usually call the constants of nature. The inhabitants (if any) of one bubble cannot observe other bubbles, so to them their bubble appears as the whole universe. The whole assembly of all these universes has come to be called the “multiverse. … Such crude anthropic explanations are not what we have hoped for in physics, but they may have to content us.” (http://www.nybooks.com/articles/archives/2013/nov/07/physics-what-we-do-and-dont-know/ ).

      Let’s discuss two bubbles only.
      a. The bubble (us) which has three well defined nature constants [e (electric charge), c (light speed) and ħ (Planck constant)]. Then, these three constants are locked with a pure number, the Alpha.
      b. The bubble (A) which has some different nature constants [d, f, g and more], and we do not know anything about these constants. It might also be locked in some way, or might be not. If it does, we can also call it the Alpha [bubble (A)].

      Thus, the Alpha should be a function of F [something (some nature constants perhaps), bubble-specific factors].

      For the Alpha [bubble (us)], it is a lock which locks three bubble (us)-nature-constants. Yet, if we can show that the Alpha [bubble (us)]-equation is completely independent from any bubble (us)-factor, then it should be a bubble-independent parameter.

      The definition for bubble is that two bubbles must not have any connection. If there is a connection between two bubbles, they are in fact a single bubble. Thus, as soon as we can show that Alpha [bubble (us)]-equation is bubble-independent, there is no need for us to content with the bubble(s) issue anymore. The concretization in the Nature math is indeed fulfilling a *mission* of bubble-busting.

  7. http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes linked from there sorry I forgot I changed my copy paster to slow things down.

  8. http://michaelnielsen.org/polymath1/index.php?title=Bounded_gaps_between_primes sorry I meant this one I recently changed my copy clipboard paste to slow things down. linked from fg vixra org

  9. orwin says:

    Yes, we do need to distinguish between the Art of Mathematics and physical quantity, which we approach through science. And physical quantity is certainly a problem in dimensions. To medite these through concretization is the Classical way, but Plato disapproved and it went underground, with Pythagoras, alchemy and actually medicine, with all the tales from the Old Silk Road.

    I must emphasise this: ONLY concretization grips the logic of instantiation, and logic is critical for the way ahead, with representation and all the difficulties of our finite discourse in infinite ranges of number.

    When I say the Classical way, I mean by the association of rational with reason and irrational with passion, and Scorates insisting that the soul encompasses both and we need to take care of our souls, in a heartful way. People may be cynical about Fraenkel’s movie, but it actually runs inthis spirit, in a light-hearted and moder way, and I think that’s gret, given all the heavy dogma and academic turf-minding, cutting knowledge into properties which must be policed against deviance.

    But I’m surprised to hear you say, Tienzen, that science has opted for pluralism. There is the old traditional/Cartesian dualism, but much discussed here is firmly in the shadow of Monism, the doctrine of Ernst Haeckel, William James and Paul Carus’ journal. Both seem to me very problematic.

    This is all of long-standing interest to me, with a reasearch focus on ancient and alternative medicines, and new science rendering them more intelligible and accessible.

    From the reaction to Plato’s dogma, we have the classic debate on mixtures: is a perfect mixture possible? Statistically, no, for reasons close to Crowley’s interest now. But a crystal matrix grows near-perfect, and the interesting point is that it has to GROW that way.

    Isn’t this top-down causality intervening in physical chaos in a manner rather like life? Can we settle on a pluralism of causalities within one unvierse of discourse?

    All the Lie algebra, from crystallography, in current physical theory speaks ot me of a potential for more than the physical in the old sense, and of philosophical challenges that physicists are failing to manage. Peter Woit now says there’s really no creative, path-breaking teaching going on in physics: so I’m in no mood to back off now….

    • orwin says:

      Afterthoughts: the top-down/bottom-up is not enough – its just systems egineering, which hasn’t broken the logjam. There’s also interactive causality.

      Descartes was not actually a traditional dualist, and the Queen of Sweden told him off for that! Traditional dualism merges growth and spirit, muchas Galen had Nature as a “principle of growth and increase” – hence pantheism, Spinoza, etc. But for science today its more relevant to distinguish first- and second-order processes, allowng for emergences at the second order (syncategoremata/impredicate definitions/self-organization).

      Then the whole picture is just more complex,and you get something like:

      [2 ^ 3]co-ordinate = [3 ^ 2]projective,

      intoducing a central projective “point of view”. You can parse that as the observer, in a Grassmannian manifold, in which torsion is not defined and there is no distinction yet between vectors and pseudovectors, which involves physical dimensions.

      Yes, the Grassmannian is ObserverSpace, a space of logical extensions, and he took it into philology and linguistics! And its not enough for physics – you can go badly wrong thinking it is – into actual “pseudo-science” confusing vectors and pseudovectors in tacky paradoxes, etc.

      Which explains the current interest in conformal invariance. But don’t go takign that for granted…!!

    • Tienzen (Jeh-Tween) Gong says:

      @orwin, “But I’m surprised to hear you say, Tienzen, that science has opted for pluralism. There is the old traditional/Cartesian dualism, but much discussed here is firmly in the shadow of Monism, the doctrine of Ernst Haeckel, William James and Paul Carus’ journal. Both seem to me very problematic.”

      I myself is a philosopher and have deep respect for all those great philosophers in the history. My book “The Divine Constitution (US copyright TX 3 292 052)” is all about philosophy. But, I do not want to mix the philosophy here for *now*, as it is all about the math and physics. And, I do not see this as a pluralism.

      As I have said that the *mission* for the math of Nature is to derive all laws of Nature (physics). Thus, I must show at least one solid example, in addition to the deriving of the Cabibbo / Weinberg angles and the Alpha.

      The essence of the math of Nature is all about infinities and the pathways of their concretization. That is, the key equation is,
      A – b = 0, but A is not b.
      This means that most of numbers are unreachable by finite means (arithmetic and algebra operations), as every *finite* does carry a tail with infinite digits (either countable or uncountable). Those unreachable numbers must be color-coded, such as, b = A (red) or = A (blue). Yet, there is always a number C, and
      A – C > 0
      The smallest C cannot truly be determined with finite means. But, in principle, there is always *a* smallest C in the physical universe *with* finite means. That is,
      A – C = g
      Although we do not know the exact value for g, g is larger than 0 (g > 0). Yet, for the *physical* universe, the g can actually be determined. Let,
      X-axis as space, thus, the (delta S > g).
      Y-axis as momentum, the (delta P > g).
      So, (delta P) x (delta S) > g^2

      In physics, the photon is the medium for causality (see Constants of Nature, http://www.prequark.org/Constant.htm ). Thus, the smallest *deterministic* unit in the physical universe is (photon / c), c is the light speed. That is, in the physical universe, g^2 = (photon / c).

      Yet, photon is the result of the interaction of e (electron).
      So, g^2 = (photon /c) = (e^2/c), e is electric charge.
      In the article “The Rise of Gravity and Electric Charge, ( http://www.prequark.org/Gravity.htm )”, the e-charge is
      e (charge) = (L * C)^(1/2) = [(1/2) ħ * C]^(1/2); L the angular momentum, C light speed, ħ (Planck constant).

      So, g^2 = ħ * C / C = ħ,
      Thus, (delta P) x (delta S) > g^2 >= ħ

      Now, the uncertainty principle of physics is the direct consequence of the *Nature math*, the essence of infinities and of unreachable of numbers. Yet, this major essence of the *unreachable numbers* is swept away in the human math by the concept of *continuity*. That is, the human math is completely unaware of this *Nature math*. Thus, the current *love* on math can only be the half way to the true ecstasy.

      • orwin says:

        I don’t get numbers with uncountable digits, not even pi: that won’t square with Cantor’s diagonal argument, and the second-order arithmetic that followed. The point is rather the combinatorial power of the digits, and whether they show any redundancy. So I think you are expecting too much of them, in a way that blurs the distictions between digits and letters, events and contingencies, data and context.

        Context-sensitive languages cover everything we can talk about, but I don’t know of any sense in which numbers as such are context-sensitive. Physics then follows numbers, with concretization, with the important implcation that reality can’t be just physical!

      • orwin says:

        One could say you have a transfinite cylyndrical algebra, which is a logical construct and can carry the load, but the transfinite case is off the map. This is about physics not absorbing where math can go now, hooking into visible trends like category theory or Langlands… physical science hasn’t got going with interdisciplinary work, so life science streeks ahead now.

        arxiv: 1302.1368v1 for a very readable historical introduction…

        One can say that the Turing machine is a continuity theorem for computation, but the happening stuff now takes off from his reaction-diffusion equation, into enzyme kinetics: where the topology of the enzyme matters, you are looking beyond Chern-Simmos, Calabi-Yau, etc.

        Acetyl kinetics involves the Lambert W-function, which takes you right up to transcendentals. But the biological opposite, methylation kinetics, involves the whole cell, inheritance, and even mercury, yes good old alchemical mercury, colloids, Mie…way off the map there.

        We are so far now from having a physical language for all this, looks ike we’d better hitch a ride on cyindrical algebras and the like.

  10. L. Edgar Otto says:

    Orwin. Good post. I have new insights on pesla blogspot. And fb. We are in the trancendental beyond that in our candy corn universe. I ask you (only) to maybe evaluate, Philopher-Scientist.

    • orwin says:

      This ia a wrap on what I’ve been seekign all along – the biological imprint of that elusive factor of viscosity we call stringiness. So take it eaasy, Edgar, – this is a big one – I’ll be back after a rejig: I don’t like being gravatard, and heath seekers are being relentlessly targeted (its a civil rights issue by now) and all those intrusive traces are now, of coure, snagging healtcare.gov, and its a national crisis with international resonances.

      So let Google teacheeads do overtime and Oracle’s Ellison chew on his paycut – he didn’t get the real sac, so he can count himself lucky, but that also means he still hasn’t got the score….

      For Physicists: just get this: supersymmetry works in 1-D and can only work that way – that should be clear by now. Now face the fact that Nature was there long before you and made the most creative use of this opportunity, which we know as the double helix of DNA. But there’s also the elementary respone in a lipid bilayer, key to the motility of a single cell. Involved here is the Casimir force, as originally noticed by Newton, and now known to set the height limit on plant transpiration: just why trees don’t grow to the sky!!

      What I demonstrated with this insight, with Senna leaves: much more than a peristaltic laxative. If you are constipated, it will act as well known, but after drinking, chase it with a uriary alkalizer. Or on an empty stomach, you can get a mouth purge: gargle with alkaline mineral water.

      Does that help you understand Geeks better?

      That instrusive hypnotic can be fluxed off with spirit alcohol and black heat (a waffle iron or similar does very well). Now think deeper, and realise that your lungs are a potent ion exchgange and very useful for detox. That way you can get a rhythm reboot for your muscle system: chase with strong coffee, which carries alkaloids with the caffeine to the same muscles.

      • L. Edgar Otto says:

        Orwin.
        Santayana said man’s achievement in architecture was taking the Euclidian plane and standing it on end. You are so right. The evidence is in the nano experiments. Virus dogma is flawed & yes political. We are not weeds & insects to modify blindly. We will find wonderful new technology when we make a plane of such supersymmetric 1D strings & lay it down again. Excellent & important work let us know. Physics is fundamental & sexy. Not evolution control by retarded social agendas.

      • orwin says:

        With the fullness of meaning comes the sadness of how people shy away from it: “you can’t think without words…”; “all ideas are images…”: oh yes, just how to auto-lobotomize yourself, switch of half your brain, dopier than the seal who at least switches over…

        Which is why art now gets meta-representational, like when they do calligraphy over Banksys.

    • orwin says:

      Dedekind’s approach to ideals was contested by Kronecker, who took a relational view, like C.S. Peirce, and that tension blowed up into algebraic vs computational and again Bourbaki vs Wittgenstein, all over theory and methodology and metaphysics today…

      Jeremy Avigad, Methodology and metaphysics in the development of Dedekind’s theory of ideals, April 13 2005.

      What I can add just now is the historic fact that Greek philosophy already did the splits over this problem, and the only philosopher who saw the pattern was George Grote, and Radical and a banker, and in fact the granddaddy of the Masters of the Universe today. Grote wrote the modern classic History of Greece, and was promptly forgotten in academia, as the Whig/Radical party of his day dissolved into Labour and Social Democrats…. No he’s not on Google Books ex some university library, just on Gutenberg!!

      George Grote, Plato and the Other Companions of Sokrates, vol 1. Project Gutenberg EBook.

      The radical dissent from the Academy of Plato ran through forgotten trails of Megarian philosophy to the scarcely remembered Eritreans, and then into the gutter. I mean on into the heyday of the Cynics, these beggar philosophers would bust into Anthenian homes and rip the larnies to pieces for being poncy snots, complete hypocrites, logical idiots, etc. It was non-violent, but brutally confrontational. And a century later, with Mithradites I at war with Rome, and an agent of his in Athens, he brewed up a revolution, which took the city for Mithradites as Sulla’s Roman army entered Greece.

      Now the Academy had slipped away into exile in Rome, and you can read this sob story on Wikipedia (Platonic Academy) about Sulla’s seige and how he cut down the trees in the Academy and the Lyceum to make seige engines, and, so they say, laid waste to the place, so that was the end of the Academy (sob!).

      BULLSH*T!!! Sulla carefully packed up the libraries, and shipped them back to Rome, where they were painstakingly catalogued: Diogenes Laertius then quotes precise numbers of lines in the extant manuscripts!!

      And next thing, an Imperial edition of Plato, by Thrasyllus, working for the Emperor Tiberius. So you seriously believe they let the succession of the Academy break in the meanwhile???

      So why don’t we hear about it? Because the Library concerned was dedicated to Apollo, and stood behind the cult of the deified Caesar, and was hated by every independent-minded intellectual since Marc Anthony murdered Cicero… Tacitus shredded them in his histories…etc.

      Uh, please understand that the constitution of the Academy survived, specifying a corporate body dedicated to the worship of the Muses, which COULD NOT JUST FAIL UNDER ROMAN LAW. INDEED, WE ALSO KNOW THAT THE EMPEROR JUSTINIAN FINALLY TOOK IT UNDER JUDICIAL MANAGEMENT.

      Why? Having just written up the Institutes of the Roman Law his credibility was very directly at stake!!!

      >>Because the Imperial Cult of Apollo was not a worship of the Muses!!!!
      >>>And some serious scriptorium then got behind a scrappy book on the Liberal Arts called The Marriage of Mercury and Philology, WHICH THEN BECAME THE MOST SUCESSFULL TEXTBOOK OF ALL TIME!!!

      DUH!

      And the neo-imperial Academy/university near you is LIVING A FAT LIE, IN INSTITUTIONALIZED DENIAL OF ITS OWN ROOTS.

      I’m not saying this for any trivial reason: the problem now is that we can have very little confidance in Academic capability in the urgent task of taming the so-called Masters of the Universe. This is now a happening problem: the Academic Economics Nobels are not coping….

  11. Thanks, Orwin. A most insightful post reply. Of gods and physics in Time’s Midsummer’s Night dreams… Academia.

    • orwin says:

      A neat mathematical realization of your 16-D vision:

      http://arxiv.org/abs/gr-qc/0611050

      A theory in which points, lines, areas and volumes are on the same footing is investigated. All those geometric objects form a 16-dimensional manifold, called C-space, which generalizes spacetime. In such higher dimensional space fundamental interactions can be unified a la Kaluza-Klein. The ordinary, 4-dimensional, gravity and gauge fields are incorporated in the metric and spin connection, whilst the conserved gauge charges are related to the isometries of curved C-space. It is shown that a conserved generator of an isometry in C-space contains a part with derivatives, which generalizes orbital angular momentum, and a part with the generators of Clifford algebra, which generalizes spin.

      Its amazing that technology is headed there already, for the next-gen Internet pipes, carrying data in orbital am. So this is the holographic/mind-actuality potential, the ever-persistent sense in which reality is intelligible.

      • crow says:

        Quite interesting paper here http://arxiv.org/abs/1301.7725 From the Virasoro Algebra to Krichever–Novikov Type Algebras and Beyond

      • Orwin, Thanks for the in depth reply and link. You certainly understand the issues. To see things this 100 yr old way is Ok but needs generalization and perhaps my almost 50 yr old method. It says this direction needs research so it is not a full picture (nor is mine that hints of even a deeper pictute “beyond”.

        Compactification and Ciifford dimension ideas are close to mine but with more mirroring…
        I find the term c for Clifford conflicts with C. For creativeness field of Hoyle.

      • orwin says:

        Thanks for that link, crow: starting from poles, it looks to bridge the gap between Dedekind and Kronecker/potential theory/relational logic. We need pioneering ventures like that, to open new approaches to transcendentals, where Edgar keeps showing us up for being unimaginative. Now I’m full of the squeezed, flattened, pressured feeling of all the restrictive visions one was given in place of education.

        And have a deep lead I must think about: the proper background to Zipf’s law is in Grassmann, but his forgotten detour into linguistics, Sanskrit and the Vedas, where he at last found academic recognition.

        Now Zipf’s work provoked a wrangle between Mandelbrot and Herbert Simon, which ended in a stand-off, because neither had the math to get above the tension between first-order and second-order solutions. Nor any inkling of Grassmann in back.

        So Samuelson of economics ill-fame did a fabulous fudge and became the guru of the Monetarists and the Republican fundamentalists, and that’s just where we’re stuck: a new stand-off between Herb Simon’s leftie Scientific Management and nostalgia for the Gold Standard/Golden Age….

        The interesting point is the distribution for Zipf’s law is a finite approximation to: the zeta! there’s a new ladder to transcendence! and how any physicist can venture into life science without considering Zipf’s law escapes me.

      • crow says:

        orwin, do martingales appear in physics? u should install retroshare, its better than the blog format. as should all ya’ll

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