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RECENT POSTS IN THIS TOPIC

John R. Cox: on 8/7/17 at 21:32pm UTC, wrote Nicely written introductory piece. Even makes me curious. Topology has...

Anonymous: on 8/7/17 at 3:45am UTC, wrote Comment on this Article Please read the important Introduction that...

STEVE A JEFFREY: on 9/23/11 at 7:36am UTC, wrote The Grand design states that in 10 dimensions only circular orbit can exist...

Sridattadev: on 9/12/11 at 19:51pm UTC, wrote Dear Cephin, Pick any number and when we deal with that number alone...

Sridattadev: on 9/12/11 at 19:35pm UTC, wrote Dear Cephin, "Absolute" truth is absurd to our senses which deal with...

Cephin: on 7/25/11 at 20:26pm UTC, wrote Sridattadev: 0/0 is undefined; your equation 0/0 = 0 is false....

Sridattadev: on 7/21/11 at 14:29pm UTC, wrote Dear All, A simple mathematical expression of 0 = infinity can be proved...

Michael Jeub: on 6/16/11 at 5:23am UTC, wrote I once thought that Ricci Curvature was like Cat III and scalar curvature...



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September 22, 2017

ARTICLE: The Art of Math [back to article]
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Roy Johnstone wrote on Oct. 22, 2010 @ 06:10 GMT
I like the idea of this sort of non-mathematical representation as I think it may be the only way to advance our understanding of "time" in our theories, insofar as the way we treat "time". Graphic representations such as category theory may help to clarify and provide a working method for, the fact that what we call "time" should only be treated as internal to the system of interest and defined *only* by components/observables of the system. Not as an external, absolute quantity based on "clock time".

It also seems like a good way of doing "built-in" dimensional analysis!

There is actually a novel version of category theory proposed by one of last year's FQXi essay contestants, Lev Goldfarb, which he calls "Evolving Transformations Theory".

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Anonymous wrote on Oct. 22, 2010 @ 06:29 GMT
Anyone who wants to see some more technical details can go here.

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John Baez replied on Oct. 22, 2010 @ 06:30 GMT
Anyone who wants to see some more technical details can go here:

http://golem.ph.utexas.edu/category/2010/10/the_art_of_
math.html

this post was moved here from a different topic

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Eckard Blumschein wrote on Oct. 22, 2010 @ 07:27 GMT
Mathematics is not just an art but to some extent also a useful tool. Did technology benefit from what belongs to the category theory on higher level as for instance theory of distributions, aleph_2 or hyperreal numbers? Perhaps it did not yet and will never do. I which John Baez good luck. In general, shouldn't those who are highly gifted and well educated in mathematics so try and live up to the obligation checking foundational questions of appropriateness for physics?

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Lawrence B. Crowell wrote on Oct. 23, 2010 @ 02:57 GMT
I am somewhat familiar with Grothendieck topos theory. I have also read Baez’ papers that connect the Cayley number system of reals to octonions and Clifford algebraic extensions with superymmetry. I have a number of technical questions with those, but will defer them until later. To me the central question is whether category theory indicates whether Heisenberg groups have some categorical equivalence with light cone structure. In other word I suspect that quantum mechanics and general relativity are equivalent in a categorical system.

Lawrence B. Crowell

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T H Ray replied on Oct. 23, 2010 @ 23:13 GMT
Lawrence,

Do you mean equivalent or dual? It's impossible for quantum mechanics and general relativity to be equivalent in any theory. I can only see them dual under isomorphism in a scale invariant background space. I don't see how category theory can solve the problem of background independence.

Tom

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Lawrence B. Crowell replied on Oct. 24, 2010 @ 18:06 GMT
It does sound crazy to speculate that general relativity and quantum mechanics are equivalent in some ways. To say they are dual is another way of maybe saying equivalent. In an S-dual setting with a magnetic monopole the electric and magnetic fields are completely equivalent in form.

Cheers LC

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Clinton Kyle Miller replied on Oct. 24, 2010 @ 21:05 GMT
I think, as you suspect, that if you design these two theories into a categorical system they could be made equivalent. That system may be able to "unify" these two apparently different descriptions of matter, however, it obviously wouldn't be able to unify matter itself. Thus, that equivalence would be akin to simply saying that matter is matter. The real trick, in my opinion, is to find some kind of relationship between quantum mechanics and general relativity that can be rendered as something that is measurable--rather than trying to find a way to equate these two models.

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amrit wrote on Oct. 25, 2010 @ 18:14 GMT
Time exists, time is real,

Time is numerical order of change.

Time is exclusively a mathematical quantity.

Mathematic is real……

attachments: 1_Time_measured_with_Clocks_12.10..pdf

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JOE AVERAGE replied on Mar. 13, 2011 @ 06:23 GMT
EInsteins time is not defined as earth orbit.

But by defining time as a 360 day orbit we can convert it to 365 days.

And get time that is six minutes shy of sidereal time per year.

This adds up over millions of light years.............

Time is an absolute if you use this clock on mars and on earth this clock will work the same on both mars and earth.

attachments: 3_simple_clk7.zip

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Lawrence B. Crowell wrote on Oct. 29, 2010 @ 14:34 GMT
Tom,

The underlying question is whether gravity can at all be quantized in a complete form. To be honest I don’t think so. The AdS spacetime, with its Einstein spacetime boundary and equivalent QCD or CFT physics, I think emerges as a phase transition. The AdS spacetime in two dimensions is an SL(2, R) group, the conformal group of quantum mechanics, and which defines classical flows that obey the sine-Gordon equation. Zamolodchikov proved how this system is S-dual to Fermi-Dirac Lagrangian with a quartic potential. The quartic potential defines Bogoliubov coefficients, which define the thermodynamics of Hawking radiation. If this is the case gravitation is formally only a semi-classical field, computable to a small finite loop order. String theory tends to reflect this to a degree, where the string defines a quantum correction to a background. The loop variable approach runs into some difficulty as well, where the Imirzi-Barbero parameter ambiguity with gamma = 3/8 instead of 1/4, and the break up of Lorentz symmetry at small distances (ruled out observationally) suggests that something funny is going on as well.

Underneath the string are D0-branes or partons. For an open strung the endpoints are analogous to quarks in a meson and the string is composed of D0-branes analogous to gluons. The closed string is analogous to a glue-ball observed in di-jet hadron scattering. String theory emerged as a way of working hadron physics, where the Hagedorn temperature was the phase transition to the quark-gluon plasma phase. Similarly the Hagedorn temperature T = 1/L_{string} is a phase transition temperature from the underlying structure of partons, partons as Fermi-Dirac particles plus bosons --- a sort of SU(N) QCD theory, to the extended structures as strings and Dp-branes for p > 0.

As for Lisis’ theory, the biggest problem is that he frames the graviton with other fields in a way the Coleman-Mandula theorem rules out. Lisi’s theory is clever in a formal sense with representation theory, but it suffers from this and other difficulties. The only way to frame the graviton with other fields is with supersymmetry, which is a compelling reason for thinking SUSY is real physics.

Cheers LC

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T H Ray replied on Oct. 29, 2010 @ 16:44 GMT
I never said supersymmetry isn't real physics. In fact, I think supersymmetry is demanded by real physics -- not as a primary phenomenon, but a result.

Whatever quantum field theory has to recommend it as a background independent theory, I think at this point we have to recognize the field source as hyperbolic and extradimensional in principle while remaining true to relativity -- i.e., by not allowing the field an independent physical reality.

Tom

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Lawrence B. Crowell replied on Oct. 31, 2010 @ 16:38 GMT
At the risk of making some mistake I am going to extraploate things a bit. The background turns out to be somewhat fixed, or determined by other physics. The AdS/CFT correspondence holds that the AdS_{n+1} has a boundary defined by a horizon-like condition on one of the time variables. This construction is an Einstein spacetime E_n with a conformal quantum theory that has the same content as the AdS. The AdS_4 and AdS_5 are particularly interesting, but let me look at AdS_4. This has a boundary that is a 2 space plus time boundary. The BTZ black hole in 2+1 spacetime has zero entropy with maximal BPS charge. In the interior of the AdS_4 spacetime the BTZ black hole has a correlated extremal black hole with AdS_2xS^2 structure. This is the extremal condition on an ordinary black hole in 3+1 spacetime. The AdS_2 ~ SL(2,R) which is the elementary s = ½ conformal group of quantum mechanics. Further, the AdS_2 spacetime, the Poincare disk defines geodesics which preserve volume in a manner that obeys the sine-Gordon equation. The dual to this is a Thirring model of a Fermi-Dirac field. This is very interesting, for it means there is naturally embedded in the AdS spacetime a supersymmetric structure that is inherited at the boundary. In fact Maldecena’s theory is naturally N = 4 supersymmetry.

The boundary of the AdS_{n+1} is a spacetime of zero or positive constant curvature. If the boundary has zero curvature the AdS spacetime is a cylinder, where the boundary is the spacetime and the interior the holographic content. The soliton and Fermi-Dirac field contents are projected onto the boundary as supersymmetric partners. These fields have enormous quantum fluctuations near the Planck scale, and this leads to the 120 order of magnitude problem with the cosmological constant. However, these enormous quantum fluctuations may only exist in this interior region. A particle may access this enormous quantum vacuum by transforming to its supersymmetric partner, but it transitions through supersymmetry space or equivalently through this interior. This only happens at 10^{-17}cm or the TeV scale in energy. At lower energy physics on the boundary does not interact with these quantum fluctuations because supersymmetry is broken and decoupled from this vacuum gemish. Supersymmetry generators Q are such that Q^2 = 0, for they are fermionic. Q^2 = 0 defines a cohomology, so there is some topology here. The breaking of supersymmetry is equivalent to a surgical change in topology, and the violation of an Euler index. The Euler index is a quantum number, and if we are to presume quantum numbers (information) is conserved it must be countered elsewhere. It is assumed on the boundary in the transition of the spherical space to a flat space. So the can is peeled open and the outer boundary stretched out into an infinite spacetime. The AdS Poincare disk is rolled out onto the Poincare half-plane.

Cheers LC

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T H Ray replied on Nov. 1, 2010 @ 14:29 GMT
Lawrence,

I think we are very close, though our mathematical methods differ.

If you mean by Euler index, the Euler characteristic (Xi) or Euler number, then sidebar 2 (pp 32-35) in "time barrier" explains the internal quantum fluctuation and topology with surgery that you describe:

With the 6 points of a 3-ball embedded on the 10 dimensional manifold, just a single quantum fluctuation at the sub Planck scale on the interior point of the 3-axis intersection (which we may interpret as a quantum field potential at the string scale) is destabilizing, resulting in an interior split of the Xi = -3 object (hyperbolic)into a normalized Xi = 2 + 2 (parabolic) by the Banach-Tarski construction. We find then that the interior plane, by fixed-point hyperbolic projection is open (flat, infinite) and the external manifold (Xi = 2) is closed (finite, bounded at the Planck energy), implying particle properties. So we have a fully relativistic model of spacetime-particle dynamic interaction, with quantum field string energy in the 10-dimension limit.

The deeper implication is that spacetime structure itself originates at the sub Planck scale. IOW, we don't need the assumption of an infinitely dense matter state at the origin (big bang). All we need is a self organized state of space and time; i.e., spacetime is a structure of measure zero in an n-dimensional scale invariant quantum field. This gives the big bang cosmology a probability 1.0, independent of any specific spacetime point. Which means that time alone seeds particle action at the string scale -- the 2 + 2 topology of disjoint 3-balls is self-sustaining in a scale invariant evolving complex system, finite and unbounded, consistent with GR. I show a very slight information loss by n-dimensional dissipation, because time (and gravity) is identical to physical information and the time metric is n-dimension continuous by analytic continuation.

I am pretty confident that my model can also deal with the mass hierarchy problem. Randall and Sundrum have already shown that the problem can be met by an extradimensional theory in a fully relativistic theory. And that's where I think we're all in agreement -- that any comprehensive unity of QM and GR has to be fully relativistic. My own aesthetic sense is closer to Smolin, though -- I want full background independence as well.

Tom

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T H Ray wrote on Nov. 23, 2010 @ 13:32 GMT
To continue the discussion about the distinctions among duality, equivalence and independence:

Having just finished _The Grand Design_ by Stephen Hawking and Len Mlodinow (it's a good book, and the following is an observation, not a criticism) -- I had marked a passage early on where the authors, explaining their concept of model-dependent realism, implied the "duality" of classical and quantum theories (p. 58), "Dualities like this -- situations in which two different theories accurately describe the same phenomenon ..."

Classical relativity and quantum mechanics do not, however, describe the same phenomenon. Relativity describes continuous spacetime; quantum mechanics describes discrete events in which time plays no role. The theories are independent, not equivalent or dual.

I think that what makes this other than nitpicking, is that dualities that actually do "describe the same phenomenon" -- such as point-line duality in projective geometry, the duality of differentiation and integration in the calculus, and string theory dualities among types -- are invertible; i.e., one gets smoothly from one mathematical description to the other.

Though I appreciate Hawking's concept of model-dependent realism as a useful tool for crafting mathematically consistent theories within given domains -- I think it gives too little credit to mathematical ingenuity, in which we have seen as in the above examples, a progressive unity of methods.

Tom

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nirvikar wrote on Nov. 25, 2010 @ 21:50 GMT
Time is Numerical Order of Material Change

attachments: Time_is_a_Numerical_Order_of_Material_Change.pdf

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JOE BLOGS wrote on Jan. 30, 2011 @ 09:41 GMT
skip to main | skip to sidebar

BLOG THE BOOK 2

Sunday, February 21, 2010

BLOG THE BOOK 2

If as I believe my thoery is true and it will be accepted by one compentant judge it will be a considerable step in science.Charles Darwin............No that I say Darwinism is right just that I think this quote applies equally to a thoery of ex-nihilo creation.

My theory...

view entire post


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Peter David Mastro wrote on Mar. 1, 2011 @ 11:48 GMT
This is absolutely great. From the perspective of an artist, if there is one thing that science needs it is a better understanding and better tools to visualize.

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JOE AVERAGE wrote on Mar. 13, 2011 @ 06:18 GMT
Simplify the problem it is just a very accurate clock..............

I like this.

But complicate the problem and it becomes a theory of everything.

Convert a 360 day circular earth orbit in 11 dimensions to a 365 day eliptical orbit in three dimensions.

And you have your three dimensional string theory.

Reverse the equation to convert Einsteins time/space to 11 dimensions.

See if it matches string theory see if one cna be converted into the other...........

Steve.

This works out really well.

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Sushant goel wrote on Jun. 10, 2011 @ 06:19 GMT
It is an enlightening article. Since I have completed my UG in physics recently.It would be immensely helpful for me to understand and most important to visualize physics in a broad way. Since the idea of picture can include each and every description of any physical process we can easily formulate equations. I believe this way should pave way towards new philosophy in physics and mathematics as common

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Michael Jeub wrote on Jun. 16, 2011 @ 05:23 GMT
I once thought that Ricci Curvature was like Cat III and scalar curvature was like Cat II and Metric like Cat I....and I like the idea of getting spaces such as metric, Banach and Hilbert into cat I, cat II and cat III....It seems to have an intuitive feel to it and allows for representation freedom. I have not read up on category theory but dream about pictorial programmers being influenced and directed to program in a virtual graphic video space ruled by category theory. I will have to read up more on category theory, it is most interesting.

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Sridattadev wrote on Jul. 21, 2011 @ 14:29 GMT
Dear All,

A simple mathematical expression of 0 = infinity can be proved as follows.

0 + 0 = 0

0 - 0 = 0

0 * 0 = 0

0 / 0 = 0

and so on....

Zero remains constant in relation to itself, no other integer can satisfy all the conditions.

I will use the character "~" to represent infinity and express the following

~ + ~ = ~

~ - ~ = ~

~ * ~ = ~

~ / ~ = ~

and so on....

also infinity is similar to zero and remains constant in relation to itself.

This simple truth of 0 = ~ will solve all compplex mathematical equations pertaining to quantum mechanics where infinities occur, all one has to do is to replace infinity with zero.

who am I?

zero=I=infinity

Love,

Sridattadev.

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Cephin wrote on Jul. 25, 2011 @ 20:26 GMT
Sridattadev: 0/0 is undefined; your equation 0/0 = 0 is false. Algebraically, you are asking to find the number x such that 0 = 0*x (or equivalently, 0/0 = x). Since any real number x can solve 0 = 0*x, it is clear that x is not a specific number, such as zero. For example, you could write 0 = 0*4 and 0 = 0*0, which translates to 0/0 = 0 = 4, which is absurd.

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Sridattadev replied on Sep. 12, 2011 @ 19:35 GMT
Dear Cephin,

"Absolute" truth is absurd to our senses which deal with relative reality. The same absolute truth can be deduced mathematically as follows as well.

If 0 x 0 = 0 is true, then 0 / 0 = 0 is also true

If 0 x 1 = 0 is true, then 0 / 0 = 1 is also true

If 0 x 2 = 0 is true, then 0 / 0 = 2 is also true

If 0 x i = 0 is true, then 0 / 0 = i is also true

If 0 x ~ = 0 is true, then 0 / 0 = ~ is also true

It seems that mathematics, the universal language, is also pointing to the absolute truth that 0 = 1 = 2 = i = ~, where "i" can be any number from zero to infinity. We have been looking at only first half of the if true statements in the relative world. As we can see it is not complete with out the then true statements whic are equally true. As all numbers are equal mathematically, so is all creation equal "absolutely".

This proves that 0 = i = ~ or in words "absolutely" nothing = "relatively" everything or everything is absolutely equal. Singularity is not only relative infinity but also absolute equality. Everything emerges from nothing. Everything might mean something different relatively, but it is the same thing absolutely.

Love,

Sridattadev.

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Sridattadev replied on Sep. 12, 2011 @ 19:51 GMT
Dear Cephin,

Pick any number and when we deal with that number alone it has no significance it is just absolute and just a number. And all numbers when dealt this way are the same and that is the truth of mathematics. Only when you think of a number in a relation (relativity) to something else it acquires a special meaning. Realization of the self is like relating the number to itself, its equality. Similarly every point in space-time or universe is the same absolutely. Singularity or Universal I or conscience is that point.

Love,

Sridattadev.

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STEVE A JEFFREY wrote on Sep. 23, 2011 @ 07:36 GMT
The Grand design states that in 10 dimensions only circular orbit can exist and these are unstable..

And this applies to atoms too atoms would be unstable in higher dimensions and the electrons would either spiral into the nucleus or fly off into space.

If atoms cuold not exist in 10^500 other universes each with it's own set of rules.

Then life cannot exist in another universe.

Does that mean life was intelligently designed. What do you think.?

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Anonymous wrote on Aug. 7, 2017 @ 03:45 GMT
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John R. Cox wrote on Aug. 7, 2017 @ 21:32 GMT
Nicely written introductory piece. Even makes me curious. Topology has become something of a unification of mathematics with its role in computer sciences, and I have just read that the techies are now looking at 'the 2nd Billion' smart-phone users in emerging countries with high rates of poverty and illiteracy and the developing of 4Gig cheap phones using non-alphabetic symbology and/or voice commands. The n-category café sounds like the route this interface might take, and evokes a sci-fi-like future of global communication that obliterates the language barrier with intuitive glyphs which might "replace(s) equations with isomorphisms" at the interface of the programming code and the machine code in the chip architecture which is the real physical (and topological) algebraic operating system in any electronic device. After all its called a computer because all it really does is math.

Hi Kids! I've been looking in occasionally, just got personal aging things prioritized and don't want too many irons in the fire (or my pitifully small bank account) and had to change my e-address after marketing intrusions from one of the political campaigns. Hello to all, reads like you are all well enough to tangle. Best Wishes, jrc

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