The patient readers may learn something about the discrete structure of spacetime at very short distances.

...

Yesterday Cumrun Vafa gave a cute colloquium at Harvard. The relevant papers are

Cumrun worked with a PowerPoint presentation that included a lot of animated pictures. As a true artist, he drew mathematics (for example, the integral signs) by hand. Well, we will have to fix this imperfection. ;-) Cumrun described Isaac Newton, apples, Albert Einstein, general relativity, and so forth. I assume that most readers don't need to hear this part. As an anonymous pal below has pointed out, Cumrun Vafa has also shown photographs of string-theoretical experiments taken at Harvard, including the pants diagram encoding the stringy interactions. :-)

**John Wheeler and foam**

The first observation related to his main topic was the concept of quantum foam - a weird, bubbling, fluctuating, topologically non-trivial picture how Wheeler imagined that the spacetime looks at the Planck scale, about \(10^{-35}\) meters. Relatively speaking, the quantum fluctuations become very large at very short distances. In general relativity, the geometry is dynamical, and therefore the distances - but even the topology - fluctuate frenetically.

String theory is a consistent theory of quantum gravity and it allows us to calculate finite scattering amplitudes (from infinity to infinity), but it is not straightforward to "see" how the quantum foam at the very short distances looks like according to string theory. The reason is that the strings' typical length is close to the Planck scale, but it's not quite the same thing. The typical size of the oscillating strings is\[

l_{\rm string} = \frac{l_{\rm Planck}}{ g^{a}}

\] where \(a\) is a positive exponent. Because the coupling constant \(g\) is much smaller than one in the regime where we can make easy and accurate enough calculations based on strings as fundamental objects (because their interactions, measured by \(g\), are weak), you can see that \(l_{\rm string}\) is much longer than \(l_{\rm Planck}\). Strings regulate the space at the string scale. Geometry is heavily modified by new string physics already at \(l_{\rm string}\), and you don't really see what's happening at much shorter distances such as \(l_{\rm Planck}\).

You may also argue that \(g\) may be of order one and then the string scale and the Planck scale coincide. That's correct, but if \(g\) is roughly equal to one (or higher), then strings are strongly coupled (they split and join frequently) and they're not the best degrees of freedom to describe physics - and it becomes more difficult for us to say what the physics actually looks like.

Nevertheless, one can ask: can you apply more sophisticated tricks to see what's going on at the Planck scale? Cumrun et al. gave the answer "Yes" in a simplified version of string theory called topological string theory. (Note that this blog recently discussed other works by Vafa et al., for example the so-called topological M-theory as well as the paper on the "entropic principle".) This theory has no propagating degrees of freedom in the bulk; it is nevertheless analogous to the "full" physical string theory. In fact, topological string theory may also be viewed as a tool to calculate a subset of physically interesting results in the full string theory - those that are "holomorphic" and whose values are "protected" by supersymmetry (which means that an easy calculation is guaranteed to be exact because supersymmetry ensures the cancellation of all corrections).

**Geometry of the problem**

OK, so take topological string theory in \(\CC^3\), i.e. a flat six-dimensional space with a complex structure, and compute its partition sum - which is about the best thing you can do with this theory. The partition sum may be interpreted in the full theory as a prepotential, roughly speaking, and it is a function of the string coupling \(g\) as well as the parameters that describe the shape of the six-dimensional spacetime (Calabi-Yau); the latter variables are missing in the case of \(\CC^3\), and therefore the partition sum is a function of \(g\) only. According to the approximation of Einstein's low-energy gravity, the space looks like the naive space we know. However, according to perturbative topological string theory, the space is slightly modified near the origin (yes, in this particular context the translational symmetry is broken).

Instead of the complex coordinates\[

z_1, z_2, z_3

\] of the space \(\CC^3\), use their squared absolute values\[

p_j = |z_j|^2, \quad j=1,2,3

\] and the corresponding angles\[

\theta_j; \quad z_j = \sqrt{p_j} \exp(i\cdot\theta_j)

\] The complex structure on \(\CC^3\) also includes the information about a symplectic structure (the kind of structure that determines the Poisson bracket on phase spaces), and \(p_j\) and \(\theta_j\) are canonically conjugate, much like the position and momentum in quantum mechanics.\[

{\rm Symplectic} = \sum_{j=1,2,3} \dd \theta_j \wedge \dd p_j

\] That was boring. Now realize that \(p_j\) are non-negative numbers, and therefore they "live" in an octant of a three-dimensional real space. At each point of this space, you have a three-dimensional torus spanned by the angles \(\theta_j\). This is an awkward way to draw a six-dimensional space - the method is called toric geometry. At the boundaries where at least one of the quantities \(p_j\) vanishes, the corresponding circle \(\theta_j\) shrinks to zero size, which is necessary to assure that the six-dimensional space has no boundaries.

Once we described the six-dimensional space in this weird "toric" way, we may calculate the partition sum using perturbative topological string theory. We will find out that it is dominated by a real two-dimensional surface\[

\Sigma

\] embedded into the three-dimensional space parameterized by \(p_j\). It almost copies the boundaries of the octant. However, near the corner, the surface deviates a little bit and becomes smooth. (You need to look at the papers to see a picture.) It deviates roughly by a distance related to \(l_{\rm string}\).

Imagine that you calculate the partition sum. In perturbative string theory, it is a power expansions that involves various Riemann surfaces of genus \(h\) where \(h\) plays the role of the exponent in the perturbative expansion (the number of loops). You re-express the complete sum in terms of a different variable \(q\) related to the string coupling \(g\) by \(q = \exp(-g)\), and what you get is\[

Z(q) = 1 + q + 3q^2 + 6q^3 + \dots

\] I don't want to write all the terms because I don't have an infinite amount of time right now, but the funny thing is that you can show that the full result is McMahon's function which can also be written as an infinite product:\[

Z(q) = \prod_{n=1}^{\infty} (1-q^{n})^{-n}

\] That's very funny because the same function appears in statistical physics of a melting crystal; you can call this other context "condensed matter physics" which shows that a piece of sugar is made of quarks that are made of strings that are oscillations on spacetime that is made of sugar again. ;-)

Why does the function appear in the context of crystals? Imagine an infinite crystal made of atoms (cubed lattice) and try to melt its corner. The rule of the game is that you can only remove the atom whose three faces are visible if you look from the direction of the corner of the crystal - I mean those three faces that contain the equally oriented corner of the cube (atom).

This condition can also be described by saying that the missing atoms in the corner of the octant must look like a "three-dimensional Young diagram" - which is a three-dimensional counterpart of the usual Young diagram, or equivalently, a set of boxes such that if you slice them, you obtain a decreasing sequence of two-dimensional Young diagrams.

In other words. Start with the full octant; it's the term \(1\) in McMahon's function. The only first atom you may remove is the atom in the corner; that's the term \(q\) in McMahon's function. Then you have three choices how to remove two atoms; that's the term \(3q^2\). Then you have six choices (\(6q^3\)), and so forth. The numbers of choices exactly give you the coefficients in the Taylor expansion of McMahon's function. In the context of statistical physics, \(q\) is interpreted as \(\exp(-\mu/kT)\) where \(\mu\) is some kind of chemical potential paid for removing one atom.

The physical picture is clear. The preferred shape of the surface Sigma discussed above is just a macroscopic, averaged approximation of a microscopic, "atomic" shape that is allowed to oscillate. The size of the "stringy shapes" is of order \(l_{\rm string}\) while the atoms of the crystal are much smaller: \(l_{\rm Planck}\) which in this case is of order \(g\cdot l_{\rm string}\) where \(g\) is a small value of the string coupling constant.

In the thermodynamic limit, the typical shape of the atoms that contribute to the partition sum of the melting crystal approach the shape Sigma. Moreover, every configuration of the atoms may be associated with a unique, non-trivial topology that asymptotically looks like \(\CC^3\), but one that contains non-trivial handles and bubbles in the middle; the general geometry may be obtained by a sequence of "blowups" - which is a process that removes a point from the manifold and replaces it by a complex projective space (a kind of sphere).

Let me re-iterate the interpretation of the partition sum again. There are two ways how to calculate the partition sum in topological string theory. One of them is a Taylor expansion in the string coupling constant \(g\); the powers of \(g\) are called \(h\) and the coefficients come from the Riemann surface (worldsheet) of genus \(h\). String theory expansion sees no discreteness whatsoever.

The other way is to sum over all possible allowed "bubbling" geometries. Each of them is uniquely associated with a configuration of atoms - a three-dimensional Young diagram or a melting crystal. Although the bubbling geometries are smooth, they can be described by discrete data. The discreteness is one of the features that distinguishes Cumrun's crystal from the bubbling AdS spaceof Lin, Lunin, and Maldacena which is otherwise remotely similar; in the latter (AdS) case, the corresponding topologically non-trivial geometries are described by a continuously infinite number of continuous parameters. On the other hand, the values of \(p_j\) in the crystal geometry are kind of quantized because they're the canonical momenta associated with the periodic coordinates \(\theta_j\) and everyone who knows quantum mechanics on a circle can say what it means for the momenta. Note that the crystal does not directly live in the six-dimensional space but rather in its three-dimensional simplified "toric" caricature.

**Generalization to physical quantum gravity**

The main question is, of course, whether these intriguing dualities between different systems only hold for topological string theory, or whether they can be extended to the realistic, physical string theory - the theory that contains the Standard Model, gravitons, as well as all other qualitative ingredients to match the real world. That's of course a big question. Cumrun says that he sympathizes with the "words" that various discrete people - like loop quantum gravity or perhaps even Ed Fradkin ;-) - are using; they just use the wrong math.

However, in the full physical string theory, we must first of all deal with a richer spectrum of different scales; \(l_{\rm string}\) and \(l_{\rm Planck}\) are not the only possibility. Various probes, objects, D-branes, other branes are sensitive to distances of different orders which indicates that if the crystal picture may be generalized at all, there will probably be a large number of subtleties. Also, in the full string theory one wants to restore the full rotational (and Lorentz) invariance, and therefore the "arbitrary" separation of the coordinates to complex coordinates (and subsequently to their absolute values and angles) is not doable. It is also reasonable to expect that some quantization rules from topological string theory won't hold in the full string theory, and the discrete character of the description will disappear anyway.

The frequent readers of my blog know that I am convinced that there exist very general arguments indicating that no discrete structure embedded in spacetime can be compatible with a Lorentz-invariant theory that includes local excitations. But I am open-minded about some completely new loopholes that may avoid these problems. However, all the examples of discrete spacetimes I have seen are incompatible with the desired low-energy limit.

## snail feedback (30) :

Hey Lubos,

you forgot to mention that Cumrun used the occasion to present the first real picture of closed strings in a "pants diagram transition", taken at a Harvard lab.

Before I forget, my headhunters have finally found someone whose musical talent rivals yours:

http://www3.ns.sympatico.ca/lyle_24/myhero.swf

Sorry pal, I was going to offer the job to you...

Incidentally, what did you do to Sean Carroll? Reportedly he's been crying his eyes out ever since you argued with him on his blog.

Thanks! Fortunately I am now on Linux that does not communicate with the sound card properly. :-)

Our latest battle with Sean was about his article that has humiliated a survey done by some libertarian scholars who showed a ridiculously small percentage of the right-wing scholars at Stanford (and Berkeley). Well, I am pretty sure that it's even worse at Harvard.

Lawrence Summers, a powerful mind that has worked for Dukakis, Clinton, and other liberals, is the ultimate symbol of the courageous rightwingers at Harvard. He has had a hard time in the last month or so - try to guess how does the life of the *real* rightwingers at Harvard looks like. ;-)

Sean has explained the low percentage of Republicans by the lack of innate aptitude of the Republicans, and their inability to work the whole week. Which is a fine conjecture except that it's obviously wrong.

I offered a different explanation. Sean wanted to assure everyone that the Republicans are definitely not discriminated against in the academia at all. So he banned half of Harvard University from posting to his blog. Every time you try to open his blog from the same computer that I can also use, you get a message "Banned by webmaster. Your posts won't appear."

This is the democracy, free exchange of ideas, and diversity as designed by far left-wing policymakers.

OK, so I picked a different computer than the banned ones - of course, there are hundreds of such computers and he is completely naive if he thinks that he can stop the diffusion of information just like Stalin did, but without the military on his side ;-) - and I wrote him that his behavior was Stalinist because it was Stalinist.

I think that the lack of intellectual diversity in the Ivy League is a serious problem. It is particularly pronounced in the social sciences and humanities - the same places where one can also think that it is a result of discrimination of politically inconvenient people.

On the other hand, there are of course other reasons why conservatives tend to avoid the academia. Many of them prefer the commercial sector; private think tanks, and they just don't like the environment that is virtually controlled by the left wing (academia).

This comparison of the female minority and the right-wing minority is very interesting and striking.

There is roughly 10% of Republicans in the Academia. I know very well that most of them are completely quiet and they just hide themselves if a political discussion appears.

I know that many people think that it's OK, but it's like forcing black women to paint their faces white and attach a fake penis every day. It's just not right.

You can also compare how a Republican vs. female candidate would be dealt with. The former would be painted as a dirty ally of oil industry and imperialism ;-), while the latter's imperfections would immediately be forgotten.

One can watch the real world in which many professors - even physics professors - find it completely natural to propose statements by the faculty (e.g. a letter to the newly accepted grad students) that would say "the faculty of the physics department does not agree with the Bush administration".

The rightwingers in the academia have become such an anomaly that many people don't even want to consider the possibility that they exist. ;-)

Imagine that someone would propose "the faculty of the physics department does not agree with the African Americans, and it finds the statements made by women wrong" or something like that.

And of course, the intellectual diversity is more important than the races and genders - the latter are mostly pretty irrelevant superficial details, at least for an individual, but the essence of the people is about something else than the color of their skin. Especially at a University whose goal is not to compare the skin all the time, but to deal with the ideas.

Intellectual diversity is important and it's lacking.

Thanks for sharing all these thoughts with us. It's quite interesting and it is important that whiny liberals like Sean Carroll get some headwind. His censorship attempt is so pathetic -- but I couldn't care less, because I find his blog quite boring anyways.

Liberals can afford to be so aggressive in academia because they have a majority there. Trust me, silently there is a substantial number of people who abhores these wanna-be good-doers.

By the way, you started your blog only after they hired you as assistant professor, right? Coincidence or strategy?

Thanks for your comments.

There is no correlation between either of these things! I actually started the Czech blog lumo.blogspot.com in 2003, but because the number of visitors was very small, I did not update that blog too often.

I completely forgot what was the immediate reason to start with the English one. Let me look it up...

Well, it looks like an uninteresting beginning. Apparently in October someone complained that I was not editing my Czech blog, and there was probably some inspiration to start with mine - I guess the goal was to compensate for the opinions of Peter Woit? ;-) I had to see some blog in the blogspot.com domain (Sean?) to decide that this was a usable format.

I am not interested in any kind of cersorship or any invain attempt to block different opinions. For this I feel sorry for Sean Carroll, regardless of the opinion he holds. More effective means of censorship than blocking internet IPs had been tried for thousands of years and they all failed.

It's even sillier what www.physicsforum.com is doing to me. When I visit anonymously I can at least read everything there. When I login with my registered username, which I know they have banned, I can not see any page there at all. Once I clear out all the cookies I can see it again. Isn't it stupid?

I am a bit puzzled by what Lubos is doing in his daily Harvard life. Obviously you spend a lot of time on stuff like figuring out "abracadabra" or what the next Mersenne number is, arguing with Sean Carroll and make him cry, teasing with libertarian girl and things like that. And publish maybe two or three papers a year jointly with others, and doesn't seem to be teaching much either. That does not look like a new hired assistant professor who is supposed to be struggling for tenureship. I would thought junior people like Lubos would be working 25 hour days and sweat like slaves before getting their tenureship. How come life is so easy for Lubos?

Quantoken

> How come life is so easy for

> Lubos?

Because he is smarter than you!

Anonymous said:

"> How come life is so easy for

> Lubos?

Because he is smarter than you!"

No, don't compare with me :-) I think my life is much easier than his. At least living in a house is more comfortable than living in an apartment. I am just trying to compare Lubos with the

generally miserableacademical lifes.Dear quantoken,

you find it silly that physicsforum is banning you? Well, how about your claims that you have solved a million dollar millenium problem in 2 trivial lines for a reason? Or your bizarre number games that you claim exlain fundamental particle masses? You are the prototype of a crackpot and the only one who doesn't know it is you. Maybe you should follow the advice you undoubtely receive every day and look for professional mental help.

You are so disconnected from reality it is just about as scary as it is annoying.

Best wishes,

Dan

Come on Mr. anonymous, so you think it's NOT a disconnect from reality when 20 years of reasearch by some of the most intelligent mind failed to produce a single experimentally verifiable calculation. But it is a disconnect from reality when I correctly predicted the CMB temperature, solar radiation constant, radius of the universe, and even neutron mass to the precision of 10 decimal places?

http://quantoken.blogspot.com

You think it is a number game to have calculated the correct neutron and muon mass to that kind of precision? I know full well what is numerology and what is not! You are so far fetched from the reality.

As for the millinium math problem. I did not claim to have solved the math problem. I merely pointed out that it is a fake problem and has no relationship with the real physical mass gap.

I challenge you, or any one, to come up with a calculation that results in a mass, using only hbar and C, and no other dimentionful physics constants. Feel free to us any arbitrary dimentionless parameter you want to insert, and you can write in any mathematical form or shape.

You are welcome to continue the discussion on MY BLOG, not here. I assume you no message will be erased just because you have discussed a different opint of view. Certainly vulgarity will always be prohibited, though.

Quantoken

Dear quantoken,

the only thing "dimentionless" I see is your "reasearch".

I find your invitation to discussions on your blog about as attractive as an invitation to lunch in a public restroom.

What in the world gives you the self-confidence to judge the work of the most brilliant contemporary physicists and mathematicians and compare it to your vacuous delusions?

Please, lower your self-confidence to a level roughly compatible with the feedback you get from your peers.

Thanks,

Dan

Lubos:

Please delete Mr. "Anonymous" Dan's comment above. It's just personal attack and completely off topic. I am not going to lower myself to that level. But if you allow this to continue off track it's not going to look pretty on your BLOG.

BTW this guy created a username called "Dr. psycho" or sort of thing, and repeatedly posted the same vulgar message on my BLOG who knows how many times, maybe a couple hundred? Let me keep counting if he is going to repeat it a thousand times :-)

Quantoken

"What in the world gives you the self-confidence to judge the work of the most brilliant contemporary physicists and mathematicians and compare it to your vacuous delusions?"

Because I am able to point to the works of the most brilliant scientists, and plainly point out what is right and what is wrong.

An example would be the famous Bohr-Einstein light-box debate. Both of them made fatal logical mistakes as I discovered in my first year in College.

That simple fact was NEVER realized by some of the brightest physicists today, and never even documented in the text books. That says the state of intelligence of the science community in general.

Lubos, or any one, do you know why both Bohr and Einstein was wrong? Care to explain your opinion? I shall write a BLOG entry on my BLOG probably tonight.

Quantoken

Dear Quantoken,

sorry but Dan's message is pretty decent, and he may have been one of the first people who offered you a friendly mirror to look at yourself.

I hope you'll survive! ;-)

All the best

Lubos

Lubos:

Sorry for you for what you considered decent and keep on your Blog. But no, I will not be hurt by dirty words. If anonymous dan is going to repeat the same cheap boring vulgarity on my BLOG a 1000 times, I will just hit a delete key a thousand times. If he does it enough times and I am tired of hitting the delete key, I can write an autobot to do it automatically for me.

But

don't you feelsomething a bit odd in the mental state of someone repeated posting meaningless vulgarities like that?Seriously what do you think about my comment that both Bohr and Einstein made serious mistakes in their famous debate? You must think that's total nonsense, right?

Being someone working to find a way of unifying GR and QM, you must know how hard it is to associate the two together. Isn't it

too triviala feat for Bohr to associate GR with the QM uncertainty principle, by using the GR argument to defend the QM uncertainty principle against Einstein?Don't you see something wrong? At least Bohr himself was so uncomfortable with his own arguments that he had the drawing of the lightbox hanging next to his deathbed in his final moments, wondering what was wrong.

Well I guess you really don't understand it.

Quantoken

Dear quantoken,

I won't be shy to tell you the truth about the crackpot you are. Nevertheless, I have never posted a word on your blog and never will. This fact is correlated with my preference of having lunch in a dining room.

If someone posted demeaning things about you on your blog, then it might be a sign of yet another sane and friendly person out there. Or, more likely, it's your alter ego that's trying to clue you in? This would keep the number of different individual visitors of your blog at the expected value of one.

Dan

Dear CIP,

your postings have been erased because they were multiply copied and off-topic.

I am not sure whether the question is related to melting crystals ;-), but I certainly don't think that all terrorists must be treated the same way, and it has never been my job to plan their execution or torture.

But yes, those who are threatening the very basics of our civilization must be dealt with efficiently.

All the best

Lubos

To the previous anonymous:

Well, Mr(s). Political-Correctness, this blog is privately owned and not free of opinion. If you can't tell the difference between the hecklers and crackpots on the one hand and the people willing and able to make contributions to the discussion on the other, that's your problem. Gladly, Lubos proves to have a fine sense for this.

Best wishes,

Michael

Quantoken is perfectly able and willing to make physics contributions to the discussions, as you can check for yourself. On the other hand, the attitude of other physicists towards him is nasty indeed. He is an unwelcome member of most blogs and forums and people feel free to insult and put him down.

It may be worth expounding on how QCD, a theory with no dimensional parameter, acquires a scale perturbatively and how a mass-gap might arise non-perturbatively.

Classically, chiral QCD has no dimensionful parameters, but however, it acquires a conformal anomaly after quantization. This anomaly can be traced back to the regularization process. Most regulators have a cutoff scale, which obviously breaks conformal invariance. During renormalization, we have to take the scaling limit which simply means we fix the infrared properties of QCD. This shows up as the running of the coupling constant. Even though dimensional regularization does not have a cutoff scale, in the limit as the dimension goes to 4 (don't ask me what the nonintegral dimensions mean), the power dependence of the coupling strength as a function of the scale turns into a logarithm. Classical QCD is only conformally invariant in four dimensions. Unfortunately, a logarithm needs a distance scale to be well-defined.

Try looking into N=4 QCD.

Anonymous said:

"Unfortunately, a logarithm

needs a distance scaleto be well-defined."Well said. A distance scale. This is what I have been saying when I discussed the existence of mass gap. You need THREE rulers to measure nature. You already have C, the natural convertion ratio between distance and time. You have hbar, the quantum of action. So you need a third ruler. It can be either a mass scale, or a length scale, or a time scale. But you need such a scale to complement hbar and C to be able to make definite measurements.

The G would not qualify for this third ruler although it does provide the right unit to complement hbar and C. That's because G is the scale of the curvature of the universe so it is NOT a microscopic physics constant. The established theories meet all sorts of problem of wrong scales because they mis-used G for the third ruler, those includes the cosmological constant problem, fine tuning problem. vacuum energy problem, etc. etc.

Mass gap could not have raised from Yang Mills theory because the Yang Mills theory does NOT contain this third ruler which is needed to obtain a definite value of mass. You can not obtain mass value from any combination of just hbar and C. See:

http://quantoken.blogspot.com

Quantoken

To Dan and Lubos,

I am a complete outsider here (too fat to be stringy) but, or therefore, I see fit to make the following general comment:

I remember when a person who proposed the use of amorphous silicon crystals to directly convert the sun's rays to electricity was poo-pooed by those in the expert (academic) community who *knew* that such a thing could not work. :]

There are always new things to be learned. And NOT necessarily

from lessons we expect to have.

How about someone learning something from his own nastiness, sometime! :\

To Dan and Lubos,

I am a complete outsider here (too fat to be stringy) but, or therefore, I see fit to make the following general comment:

I remember when a person who proposed the use of amorphous silicon crystals to directly convert the sun's rays to electricity was poo-pooed by those in the expert (academic) community who *knew* that such a thing could not work. :]

There are always new things to be learned. And NOT necessarily

from lessons we expect to have.

How about someone learning something from his own nastiness, sometime! :\

I'm probably taking my life in my hands in saying this. But I don't think the characterization (or should I say caricature) of Derrida and deconstruction is correct. Unfortunately many in the humanities have seriously misread and distorted Derrida. However physicists should have some sympathy. Think what humanity majors have done to the 2cd Law of Thermodynamics, to the uncertainty principle or to Goedel's Theorem. Talk about abuse.

If you are interested, here's a few quotes from Derrida correcting misunderstandings. He's a difficult philosopher to understand, and one must read him carefully. Unfortunately most who read him start with his works from the mid 70's onward. But those presuppose a familiarity with his earlier work. If you don't have that they'll frankly typically seem like nonsense. His earlier stuff is sadly ignored far too often by humanity majors (although the deconstruction fad is now long over). Further it is written as more traditional philosophy engaging in figures like Husserl, Peirce, and others.

Not that this has much to do with your discussion of deconstruction which I found fascinating.

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