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started universal exceptionalism
related discussion is taking place on g+ here
I have to admit: much like invocations of anthropic principles, I’ve never understood this type of argument or how it has (or even could have) any explanatory force.
It is a philosophical sentiment, not an explanation. As such, it seems to me at least as good as other philosophical sentiments that are widely being referred to, and deserves to be discussed as such. The entry is flagged as being in philosophy.
Yes, it’s a sentiment. As such it seems to me very close to a religious sentiment, and in fact it’s not clear to me why religious sentiments aren’t at least as good. :-)
Todd, just so I understand, are you objecting against the existence of the entry?
(I used the word ’explanatory’ because that’s how authors like Stephen (whoops, make that Richard) Dawkins, I think in his God Delusion book, seem to use anthropic principles: there are why questions followed by phrases like ’because of the anthropic principle’.)
I’m not sure I’m objecting. Probably no more and no less than I would object to entries that discuss religion, until I understand better what the function is besides general discussion.
In this case the funcion is to record references, Baez and Witten. I see the entry as being on par with an entry on, say, positivism, which I’d figure should be an uncontroversial entry to have on a wiki that claims to be, to one third, about philosophy.
Fair enough. So let me take any hints of objection off the table, and simply revert to what I said in #2, that I just don’t get the physico-philosophical sentiment. But as a mathematician, I find that exceptional structures are exceptionally worthy of study, if only because their perceived ’exceptionalism’ is a sign that we to understand them better, i.e., as entirely natural.
One should also perhaps cite anti-examples, like Lisi’s $E_8$ theory.
I just don’t get the physico-philosophical sentiment
Sure, that’s why we run into problems whenever it comes to philosophy of this speculative kind, that it is based on supersensous intuition and robs us of the paradisic situation that modern rigorous mathematics brought about: the ability to find undoubted agreement.
Still, since it’s part of human thought, and since at the creation of this wiki we decided that it is to span the whole reach from philosophy via mathematics to phyiscs, it seems at points be desireable to me to record what sensible people have voiced in such regard, wherever it touches on relevant mathematics. That seems to be the case here.
And I certainly share the feeling that it is most tiresome to indulge in this speculation if it doesn’t proceed to something more tangible. The philosophy that I like to record on the $n$Lab is always that which I presently see relevant to the mathematics that I am looking into. In this case it’s the maths discussed in the thread from higher to exceptional geometry?.
David: I was thinking a little while ago of the same (anti-)example.
wherever it touches on relevant mathematics
That’s a key phrase. (And by the way – I wouldn’t be averse to entries that touch on religion, if they meet that kind of criterion. Some of the stuff on Hegel probably comes close, and that’s fine.)
The thread you mentioned at the end is of course very interesting!
Todd, okay, thanks for saying this!
Regarding the “anti-example”: this seems to be an anti-example only to a rather naive interpretation of that “universal exceptionalism” sentiment, since clearly one won’t want to read it as saying that “whatever we can cook up from some exceptional mathematical structures is relevant in physics”.
Allow me just to recall how good the much older standard reasoning is that leads to the speculation that nature may be based on an exceptional gauge group:
the potential grand unified theories that had eventually been proposed used the gauge groups $SU(5)$, then $SO(10)$ (or rather $Spin(10)$), then $E_6$. All these were motivated from detailed analysis of the experimentally observed physics, and the observation that one generation of fundamental particles fits rather snugly into certain representations of these groups.
Now the mathematically inclined observer of these developments observes that these three groups are the first three stages in a sequence that naturally continues with $E_7$ and $E_8$, since in each step there is a node added to the corresponding Dynkin diagram.
This is how the idea first came about that exceptional Lie groups might play an exceptional role in the description of nature, and while it remains a speculation, it is a rather well motivated one.
A counter position (insofar as it is one) is the (IMHO) terrible MUH: https://en.wikipedia.org/wiki/Mathematical_universe_hypothesis (and that page just looks like an advert for Tegmark’s book), and certainly one to point out is antipodal to ’universal exceptionalism’ in several ways.
Bleh, what is this??
Jürgen Schmidhuber[8] argues that “Although Tegmark suggests that ‘… all mathematical structures are a priori given equal statistical weight,’ there is no way of assigning equal nonvanishing probability to all (infinitely many) mathematical structures.” …
In response, Tegmark notes[3] (sec. V.E) that the measure over all universes has not yet been constructed for the String theory landscape either, so this should not be regarded as a “show-stopper”.
EDIT: see also this, or not.
Urs #13
hmm, I didn’t realise that about $E_4=SU(5)$ and $E_5=Spin(10)$. Given that $E_3$ is probably $SU(2)\times SU(3)$ (!), this makes sense.
yes, we have U-duality – table on this matter (I guess I just fixed one row mixup there), with more references at U-duality
this seems to be an anti-example only to a rather naive interpretation
Which makes me think that if universal exceptionalism is to succeed as a kind of heuristic (= method of discovery, hopefully using ’heuristic’ in a sense of which Andrew Stacey would approve), it needs to be augmented by other selection principles that prune out naive anti-examples. Although I couldn’t say what it was that Lisi cooked up with $E_8$ that made it fail (in the eyes of many), since my naive reading of #13 suggests that his reaching for $E_8$ was reasonably motivated. Could you shine some light on that, Urs (and sorry to ask an ignorant question)?
Something is exceptional (e.g. the exceptional Lie-groups) only ever with respect to a classification scheme. I imagine there are cases where many schemes are possible. Makes me wonder if some schemes (like the one for Lie-groups) are in some way natural, and what natural means here. That it’s formulated using relatively shortly formulated predicates?
re # 18:
Todd, the proposal you are asking about differed from the traditional theory of GUTs in that it meant to unify not only the gauge groups of the three gauge forces, but in addition the 4d Lorentz group (meant to stand for the remaining “gauge” force of gravity) into $E_8$. At least one way of making this precise was proven in arXiv:0905.2658 to have no realistic solutions. In reaction to that the opinion was voiced that it should be possible to tweak some assumptions such as to get something viable after all, but if so then more convincing of the expert community is still necessary than the exorbitant public media attention might suggest. Generally, beware of the extreme distortion that public attention is causing in high energy physics these days. The incidents that David is pointing to in #14 and #15 are another crass example.
Maybe I should add that proposals for concrete variants of the general idea of exceptional GUTs are a dime a dozen, they are being proposed, tested against available consistency checks, discarded, refined, reproposed all along. Public media is generally unaware of this process, since usually it happens in articles with sober titles. Rather recently for instance arXiv:1412.4776 appeared, which I thought stood out as being in pretty good shape in view of available experimental data, but no doubt this will not be the last word and will be further tweaked, especially as new experimental data comes in. HEP phenomenology is a business of endless trial and error, even when one focuses on one paradigm, such as exceptional GUTs.
Re #20: don’t worry, I wasn’t taking Lisi seriously or paying attention to public buzz; I just wanted to hear what you had to say about the proposal in general terms. Thanks!
I finally recovered the quote by Witten that I was after which more explicitly has the idea of “reality is exceptional”: it’s on p. 13 of arXiv:hep-ph/0201018 and goes like so:
…arise in compactifying from eleven to four dimensions on a compact seven-manifold X of G2-holonomy. This seems like an interesting starting point for making a model of the real world, which is certainly exceptional…
David Corfield kindly pointed out two further relevant quotes, now from Pierre Ramond. I have added them to the entry.
You added that Vafa observation:
A related comment in the context of F-theory GUT phenomenology is in (Vafa 15, slide 11).
Would it be reasonable to distinguish this kind of observation from the Baez-Witten-Ramond variety since it appeals to an empirical measurement of a coupling constant?
The more speculative philosophical route sounds to me somewhat in line with that kind of argument that took the Principle of Sufficient Reason to justify the variational approach to mechanics: There must be a reason for how a system behaves as it does. It results from selection of an extremal value of some quantity.
Would it be reasonable to distinguish this kind of observation from the Baez-Witten-Ramond variety since it appeals to an empirical measurement of a coupling constant?
Yes, that’s why I didn’t add the quote, but just a pointer.
The more speculative philosophical route sounds to me somewhat in line with that kind of argument that took the Principle of Sufficient Reason to justify the variational approach to mechanics: There must be a reason for how a system behaves as it does. It results from selection of an extremal value of some quantity.
I think in view of this and in view of the comments that Todd made, it is important to notice that what Witten and Ramond express in the quotes collected in the entry is not a vague “the world is governed by exceptional structures!” which could mean anything and nothing. Instead, what they express happens within a conceptual framework that is already known to qualitatively correctly model reality (GUTs, string theory). Within that framework, they find a remaining choice of mathematical structure, the compact gauge group. They say: if all these choices are mathematically possible, but some are mathematically exceptional, isn’t it somehow plausible that if one of them is singled out by not just being mathematically possible but also physically realized, that it is the exceptional ones?
That’s still not a proof of anything (and it is worth noticing that Witten never goes around proclaiming this as a principle, he just has some passing comments to this extent) but it is much more constrained than the idea that “anything anyone can construct using $E_8$ will be relevant to physics” or the idea that one might be able to decude the laws of physics from such reasoning.
That all said, maybe I may be excused to remark that I started this here because elsewhere I was running into a route that does have the feel of a more accurate mathematical incarnation of these sentiments.
What I wrote wasn’t an accusation about philosophical speculation, if it came across that way. If the Principle of Least Action owes anything to the Principle of Sufficient Reason, then that has to be one of the most fruitful pieces of philosophical thinking ever.
If the Principle of Least Action owes anything to the Principle of Sufficient Reason,
But does it? I feel this is a dubious claim.
A derivation of the principle of extremal action from more primitive principles that I believe in is that discussed at prequantized Lagrangian correspondences in this section. This may be said to be neatly motivated from modal homotopy type theory. That is the context of Classical field theory via Cohesive homotopy types (schreiber).
But does it? I feel this is a dubious claim.
You mean historically? Just search for Maupertius and “sufficient reason”.
Maupertius is generally credited with being first to a principle of least action, though some claimed Leibniz (most associated with Sufficient Reason) was there first.
But does it?
Historically, maybe? Compare the development of Euler characteristic in terms of genus via all the attempted proofs that $\chi=2$ for “all” polyhedra, as told by Lakatos.
Let’s see, the “principle of sufficient reason” states that “nothing is without reason”, according to Wikipedia.
This principle seems to imply deterministic laws of nature, but it would not seem to imply that these say that physical processes extremize an action functional.
So Leibniz tends to use Sufficient Reason it to explain how God must have arranged the Universe. He combines it with the Principle of the Best: the kind of explanation required by Sufficient Reason for how God arranged this world is that it is the best of all possible worlds. The sufficient reason for any contingent truth is that it is for the best in some sense. It’s not hard then to see how one adapts this, say, to think that of all possible paths for light to follow, God must arrange things to work most effectively.
Maupertius does this Wikipedia
The final stage of his argument came when Maupertuis set out to interpret his principle in cosmological terms. ‘Least action’ sounds like an economy principle, roughly equivalent to the idea of economy of effort in daily life. A universal principle of economy of effort would seem to display the working of wisdom in the very construction of the universe. This seems, in Maupertuis’s view, a more powerful argument for the existence of an infinitely wise creator than any other that can be advanced.
He published his thinking on these matters in his Essai de cosmologie (Essay on cosmology) of 1750. He shows that the major arguments advanced to prove God, from the wonders of nature or the apparent regularity of the universe, are all open to objection (what wonder is there in the existence of certain particularly repulsive insects, what regularity is there in the observation that all the planets turn in nearly the same plane – exactly the same plane might have been striking but ’nearly the same plane’ is far less convincing). But a universal principle of wisdom provides an undeniable proof of the shaping of the universe by a wise creator.
Hence the principle of least action is not just the culmination of Maupertuis’s work in several areas of physics, he sees it as his most important achievement in philosophy too, giving an incontrovertible proof of God.
Okay, so my answer to your question in #9 “You mean historically?” is “No.” :-) I mean it intellectually.
Maupertius’ reasoning reminds me of a phase that I went through when I was a child; where one day I tried to convince myself that the reason that stuff drops to the ground is because… it just has to be that way, it is just the most natural way for things to be. For one day I was convinced that I had solved this problem for myself.
(Now that I say this, I remember another day I had the theory that everything on earth that is fluid is so because it contains water. I think I got to that point by asking about the nature of milk. When I mentioned this theory to my father, we happened to be on the sidewalk and just this moment we passed a small oil spill next to a car, that was shimmering in rainbow colors. No, my father said, and pointed to it, oil contains no water, instead it repels it. I was very impressed.)
An empirical way to see that Maupertius’ reasoning has a gap is to observe that a frequently asked question on online physics forums is why on earth it is the action $E_{kin} - E_{pot}$ that is being extremized, instead of, say, the much more evident seeming total energy.
I have seen that a lot and if pressed will google for good examples. A quick search gives this example.
Seeing what Urs is doing with growing branes from a superpoint at geometry of physics – fundamental super p-branes, how about a speculative explanation for the appearance of exceptional structures in physics as resulting from this growth? We have a template in the Cayley-Dickson construction applied to $\mathbb{R}$, where after three steps we’re left with the precarious octonions, which can just about manage to form a projective plane by no more. The octonions are linked to many kinds of exception.
But until now and M-Theory from the Superpoint, we never seemed to hear of a direction of flow, and so above it’s just Nature choosing an exceptional structure. Now we seem to have Nature unfolding until it can go no further, resulting in the exceptional. Or would this be to take the talk of growth and unfolding too literally?
By the way, do we expect a network of relations between exceptions, or could there be something more like a prime mover? The octonions occur all over the place. But maybe something deeper drives their appearance.
It seems that, mathematically, the process that we see here is a kind of “consecutively invariant Whitehead tower”. It would be useful if we could pin this down better.
An ordinary Whitehead tower of some connected homotopy type is obtained by the the iterated presciption
find the universal 2-cocycles of $X$, pass to their homotopy fiber $\hat X$;
find the universal 3-cocycles of $\hat X$ and pass to their homotopy fiber $\hat {\hat X}$;
find the universal 4-cocycles of $\hat {\hat X}$ and pass to their homotopy fiber $\hat{\hat {\hat X}}$
and so on.
Clearly in the brane bouquet something very similar happens, but there are two modifications:
in each stage we look for invariant cocycles, namely those cocycles wich are invariant with respect to the maximal subgroup of the automorphism group such that there are any non-trivial invartiant cocycles at all;
we also allow steps where we pass from a super-homotopy type $X$ to its “type II version” $X \underset{\overset{\rightsquigarrow}{X}}{\coprod} X$ (here $\overset{\rightsquigarrow}{(-)}$ denotes the bosonic body).
Apart from these two modifications, the process is the same as before: find your cocycles, extend by them, find the cocycles on the extension, extend by those, and so on.
Thus it feels like it ought to be true that the brane bouquet is a special case of a general process that ought to go by some canonical homotopy theoretic name, it feels like there ought to be a crisp statement of the form “the brane bouqet is precisely the equivariant super-Whitehead tree of the superpoint”. But I am not sure yet.
Is there a dual, Postnikov-like description of what you said?
Hmm, do I even understand the relation in the ordinary version? E.g., at Whitehead tower
the Whitehead tower may be constructed by taking each stage $X^{(n+1)} \to X^{(n)}$ to be the homotopy fiber of the corresponding map into the $(n+1)$st stage of the Postnikov tower.
So $X^{(1)} \to X^{(0)}$ is the homotopy fiber of the map to the first stage of the map to $X_1$. Then that is to be related to finding
the universal 2-cocycles of $X$, pass to their homotopy fiber $\hat X$;
By the way, Google thinks there is one appearance of “equivariant Whitehead tower”, in TMF0(3) Characteristic classes for String bundles.
Hmm, do I even understand the relation in the ordinary version? E.g., at Whitehead tower
the Whitehead tower may be constructed by taking each stage $X^{(n+1)} \to X^{(n)}$ to be the homotopy fiber of the corresponding map into the $(n+1)$st stage of the Postnikov tower.
The second “$n$” here was a typo, I just fixed it. (This is stated correctly at two places further down in the bulk of the entry.) The map whose homotopy fiber is $X^{(n+1)} \to X^{(n)}$ is to some Eilenberg-MacLane space, hence really is a cocycle in ordinary cohomology. If however we allow “non-abelian cohomology”, then the whole map $X^{(n+1)} \to X$ is the homotopy fiber of a map from $X$ into a Postnikov stage.
In the brane bouquet this phenomenon corresponds to how the consecutive extensions by ordinary cocycles such as
$\oplus_{p \; even} \mathfrak{D}p\mathfrak{brane} \to \mathfrak{string}_{IIA} \to \mathbb{R}^{9,1\vert \mathbf{16}+ \mathbf{16}*}$are equivalently given by a single “non-ordinary” cocycle
$\mathbb{R}^{9,1 \vert \mathbf{16}+ \mathbf{16}*} \longrightarrow \mathfrak{l}(KU/BU(1))$The following diagram illustrates these relations between Whitehead, Postnikov, and the consecutive ordinary extensions (every square and every composite rectangle here is homotopy Cartesian, the pasting law rules supreme):
$\array{ & \mathbf{\text{Whitehead tower}} \\ &\vdots \\ & B Fivebrane &\to& \cdots &\to& * \\ & \downarrow && && \downarrow \\ \mathbf{\text{second frac Pontr. class}} & B String &\to& \cdots &\stackrel{\tfrac{1}{6}p_2}{\to}& B^8 \mathbb{Z} &\to& * \\ & \downarrow && && \downarrow && \downarrow \\ \mathbf{\text{first frac Pontr. class}} & B Spin && && &\stackrel{\tfrac{1}{2}p_1}{\to}& B^4 \mathbb{Z} &\to & * \\ & \downarrow && && \downarrow && \downarrow && \downarrow \\ \mathbf{\text{second SW class}} & B S O &\to& \cdots &\to& &\to& & \stackrel{w_2}{\to} & \mathbf{B}^2 \mathbb{Z}_2 &\to& * \\ & \downarrow && && \downarrow && \downarrow && \downarrow && \downarrow \\ \mathbf{\text{first SW class}} & B O &\to& \cdots &\to& \tau_{\leq 8 } B O &\to& \tau_{\leq 4 } B O &\to& \tau_{\leq 2 } B O &\stackrel{w_1}{\to}& \tau_{\leq 1 } B O \simeq B \mathbb{Z}_2 & \mathbf{\text{Postnikov tower}} }$Re #36, that doubling of a super-homotopy type is surely suggestive of some connection with the Cayley-Dickson construction, at least through the early stages. And the change of pattern at 9,1 relates to the octonions losing nice properties? (There’s also that variant of the construction which leads to split forms of the algebras.)
How general is Cayley-Dickson? I see someone applying a form to a non-associative unital algebra, $A$, with involution defined over an arbitrary unital, commutative and associative ring of scalars, $K$, with a choice of cancellable scalar. There’s a $\mathbb{Z}_2$ grading on the resulting double. Is there a natural fit between the super-world and such doubling? Does Cayley doubling crop up in the world of ring spectra?
Hmm, what do people get up to: Superalgebras of (split-)division algebras and the split octonionic M-theory in (6, 5)-signature
we introduce the whole set of (split)-division algebras through the unified framework provided by the (generalized) Cayley-Dickson doubling construction.
I added another quotation:
If the current M-theory is a unique theory, one should expect it to make use of singular, non-generic mathematical structures. Now it is known that many of the special objects in mathematics are related to octonions, and therefore it is not surprising that this putative theory-of-everything should display geometric and algebraic structures derived from this unique non-associative division algebra.
Re #36, that doubling of a super-homotopy type is surely suggestive of some connection with the Cayley-Dickson construction, at least through the early stages.
Yes, certainly. As one unwinds (as John Huerta unwinds, that is) how the trunk of the brane bouquet proceeds, then these doublings correspond to the Cayley-Dickson constructions.
I view it this way: It was clear all along that the Cayley-Dickson construction knows something about supersymmetry and the stringy spacetimes, but this left open two problems: why consider star-algebras and their CD-doubles in the first place, and why stop the CD-process at some point?
Now with the bouquet, these two questions are answered. We see (that’s how I view it anyway) that those algebras are not the truly fundamental agent here. While they happen to neatly encode the crucial relations, the true fundamental concept is the progression of universal invariant (higher) central extensions of super Lie algebras. That this happens to be accompanied by division algbras for parts of the journey is a useful fact, but division algebras are not conceptually what drives this process.
And the change of pattern at 9,1 relates to the octonions losing nice properties? (There’s also that variant of the construction which leads to split forms of the algebras.)
Something like this.
Hmm, what do people get up to: Superalgebras of (split-)division algebras and the split octonionic M-theory in (6, 5)-signature
Thanks for the pointer, I’ll have a look.
If
the true fundamental concept is the progression of universal invariant (higher) central extensions of super Lie algebras,
other data points would help. A minimal variation might start out from $\mathbb{R}^{0|3}$. What other starting points?
Somebody should look into it. I wish I had more time.
If the Cayley-Dickson process is a consequence of “the true fundamental concept” (#42), is it the case that the construction of the Albert algebra, as made up of 3-by-3 hermitian matrices over the octonions, is also a shadow of the more fundamental progression?
I have talked with John Huerta about this in the context of the rooted brane bouquet, and there are other people who have speculations about the Albert algebra and spacetime, but for the moment it seems to be inconclusive.
Where $2 \times 2$ hermitian matrices over real normed division algebras are identified with spacetime in dimensions 3,4, 6 and 10, the Albert algebra is identified with the direct sum of a) spacetime of dimension 10 and b) one real irreducible spinor representation in that dimension and, finally, c) one scalar.
Viewed with the string theory folklore in mind, then this kind of data would naturally appear in “heterotic M-theory” (the scalar being the scale of the orbifolded 11th dimension).
However, it is unclear how the Albert algebra is, apparently, trying to say that the 10+1 bosonic dimensions together with the 16 fermionic dimensions of the heterotic supermanifold should be unified to one single bosonic thing of dimension 27.
On the other hand, there is a latent speculation that bosonic string theory with its 26 spacetime dimensions should have an analogue of what is 11d M-theory for the superstring in 10d, hence should have a 27 dimensional non-perturbative completion, “bosonic M-theory”.
So maybe the Albert algebra is trying to give us a hint how bosonic M-theory and ordinary M-theory are supposed to unify.
At this point this is pure numeralogy. Which doesn’t imply that it’s wrong.
Here your previous insistence to explore the bouquet emerging from $\mathbb{R}^{0\vert 3}$ comes into play, for if the Albert algebra is to appear in in a way analogous to how the $2 \times 2$ Hermitian matrix algebras appear in the brane bouquet over $\mathbb{R}^{0 \vert 2}$, then it ought to be over $\mathbb{R}^{0 \vert 3}$.
In principle this is a straightforward computation. Regard $\mathbb{R}^{0 \vert 3}$ as a super-Lie algebra and then consecutively do the following: compute the semi-simple part of its external bosonic automorphism group (i.e. the semi-simple part of the quotient of all autmorphisms by the stabilizer of the bosonic part), then compute the maximal central extension invariant under this group, then repeat.
Maybe after four steps one gets an $\mathbb{R}^{27|something}$ whose underlying bosonic thing is the Albert algebra. Somebody should check.
Ok, how about this for the first step?
Like with $\mathbb{R}^{0|2}$ in Prop 7 of your paper, we just need symmetric bilinear forms on $\mathbb{R}^3$. There are 6 of these, so the underlying super-vector space of the maximal central extension is $\mathbb{R}^{6|3}$. Then we’d need the induced super Lie bracket.
Why do I have a sneaking feeling that we should be ’doubling’ up to the octonions and only then ’tripling’. Can one think about extending $\mathbb{R}^{9,1|16+16+16}$, perhaps with some bars?
Hmm, 3-component Cayley spinors in
Could that $\mathbb{R}^{6|3}$ be $\mathbb{R}^{4,2|3}$? It would be nice to see $SO(4,2)$ appear. [Maybe 3,3 is more likely.]
Could that $\mathbb{R}^{6|3}$ be $\mathbb{R}^{4,2|3}$? It would be nice to see $SO(4,2)$ appear.
The bilinear pairing from fermions to bosons in $\mathbb{R}^{6 \vert 3}$ – when regarded as the maximal central super Lie algebra extension of $\mathbb{R}^{0 \vert 3}$ – is equivariant under $SL(3,\mathbb{R})$. One fermionic dimension down we have the exceptional isomorphism $SL(2,\mathbb{R}) \simeq Spin(2,1)$, but is $SL(3,\mathbb{R})$ related to $Spin(4,2)$?
Also, if the pattern continues to the octonions, then, by the first article by Dray and Manogue that you linked to, one will find $SL(3,\mathbb{O}) = E_6$.
Also, the determinant of matrices is what equips the space of $2 \times 2$ matrices with a bilinear structure, which the happens to coincide with the Minkowski metric for hermitian matrices with coefficients in real normed division algebras. But for $3 \times 3$ matrices it gives a trilinear form.
For the hermitian $3 \times 3$ matrices with coefficients in the octonions, that trilinear form is something like $(c,v,\psi) \mapsto c \eta(v,v) \pm \eta([\psi, \psi],v)$, hence something like the dilaton-corrected metric plus the 10d SYM 3-cocycle. These are all very suggestive ingredients, but I am still unsure what to make of it here in this form and context.
But since the automorphisms of this trilinear structure is $E_6$, and given the resemblance of all the ingredients to heterotic M-theory data, maybe the answer is to be found when looking at embeddings of $E_6$-GUT into heterotic string theory. There is a fair bit of discussion of these, but I don’t see it all fall into place yet.
$SL(4,\mathbb{R})$ is a double cover of $SO(3,3)$ and $SU(2,2)$ is a double cover of $SO(4,2)$ (here), if that helps.
Really ought to be doing something else, but from Rigidity of an Isometric SL(3,R)-Action:
$\mathfrak{so}(3, 3)$ is isomorphic, as a $\mathfrak{sl}(3, \mathbb{R})$-module, to $\mathfrak{sl}(3, \mathbb{R}) \bigoplus \mathbb{R}^3 \bigoplus \mathbb{R}^{3 \ast} \bigoplus \mathbb{R}$
Somehow having a “critical opinion” on a philosophical sentiment does not quite parse.
But is “sentiment” an accurate term to refer to plausibility assessments? Moore even makes a prediction.
Isn’t the issue more the value of this particular critical opinion, which aims to lessen the degree of plausibility towards this expectionalist idea?
Maybe it’s instructive to think of Lisi’s $E_8$ theory, which a priori is following the universal exceptionalism drive, but doesn’t (following Jacques D’s assessment) at all work. One might say it’s too naive, but it’s only after analysis that shows it doesn’t properly reproduce the Standard Model particle spectrum.
Kelvin’s knotted vortices also come to mind. It’s probably true that knots were the only stable, topologically nontrivial structures known at the time, and so it was not silly to ask that atoms, whose stability was an open problem, could be explained thereby.
The particular opinion that Daniel dropped, is, as far as one can discern, critical of strategic and political decisions in contemporary high energy particle physics, and it seems a typing error to oppose this to the content of an entry on the philosophical stance expressed by universal exceptionalism. By the same logic this would be “critical” of absolute idealism, which may make the absurdity more manifest.
I suggest: Let’s add pointers to coherent authors taking stances opposite to universal exceptionalism, possibly but not necessarily from this century, authors who don’t get their concepts mixed up already in the title of their books, and maybe give them their own entry even, which we could then cross-link with “universal exceptionalism”. To some extent such an opposite stance is already discussed at multiverse, but we could bring this out more clearly by having more references and possibly a dedicated entry (possible title: universal mess?! :-)
Just checked with Igor Khavkine, and he promptly points me to a text rejecting theoretical aestheticism, that is worth citing (in this entry or in a new one to be pointed to from here):
Lots of good quotes in there.
At the beginning of this entry I have added the line
This (i.e. universal exceptionalism) may be regarded as a facet or version of theoretical aestheticism, in contrast to empiricism.
Then I created some minimum at empiricism and moved those “critical opinions” to there.
I was trying to quote various bits from Born’s Experiment in theory and physics, but the electronic copies which I found don’t seem to have the text recognition required to copy-and-paste, and I don’t have the time or energy now to re-type it all. But if anyone knows how to get the actual text (as in: characters, ASCII code) it would be very worthwhile.
For the moment theoretical aestheticism redirects to universal exceptionalism, but eventually if would deserve its own entry.
Is the relation between aestheticism and exceptionalism so strong? I mean why should one necessarily view exceptional structures as any more beautiful than infinite families? I could imagine someone taking them to be ugly glitches.
Surely there are two separate ideas here. One, epitomised by Dirac,
God used beautiful mathematics in creating the world.
The other takes our universe to be exceptional since it exists, and so to be represented by an exceptional structure.
Sure, Please feel invited to edit further!
So your point is that the universal exceptionalist paraphrases Dirac as saying
God used monstrous mathematics in creating the world.
;-)
Sounds like a Cathar viewpoint. But I guess it would be more likely for an exceptionalist just to want to keep God and beauty out of it.
I was just joking in #64, of course
it would be more likely for an exceptionalist just to want to keep God and beauty out of it.
and I suggest we keep God and Beauty strictly out of everything here. This whole business about beauty-or-not is just a gate for the trolls to fall in and drown any substantial discussion. Much better to stick to concepts like exceptional structures, which have a definition to hold on to and find agreement about.
so I should stick to my own suggestion: have removed mentioning of “theoretical aestheticism”.
Odd in a way, isn’t it, for people to make the connection between our universe being exceptional since it exists and exceptional structures. Putting things the other way, to what extent does the multiverse explain the appearance of exceptional structures? So ’everything’ is tried out in some universe or other, and we, of course, live in one with the complexity to give rise to intelligent life. But what do exceptional structures have to do with making intelligent life more likely? Why can’t some member of an infinite family be equally as complex?
In a way, your ’out of the superpoint’ story has something going for it here in that there’s a process driving a series of extensions which terminates somewhere, and termination happens at the exceptional where the process breaks down. Even the sporadic groups have a description as the end of a line of extensions, as discussed back here.
Re your first paragraph: This is the kind of stuff to speculate about at a camp fire over a beer, but as long as we don’t have more insight, there is nothing much of substance to be said here. One needs to put these questions aside and make progress where progress is possible at this point.
(At times I was wondering, watching a whole chunk of the professional physics community go down that road, if maybe it’s indeed a consequence of a lack of those campfire nights in the modern urbanized world, hence a lack of appropriate channel for thinking these thoughts, which made them go astray this way as professional physicists.)
I do like to understand better what you are reminding me of regarding how to find the sporadic finite simple groups here. We should have a note on that in soem nLab entry!
Added
and quotation
Physicists who wonder about the ultimate structure of our universe can be divided into two classes. The first class believes that our universe is chosen at random from a huge ensemble of possible universes. The second class believes that our universe and the fundamental laws that govern it are based on some beautiful and exceptional mathematical structure. String theory provides evidence for both points of view… The authors of the present paper are philosophically inclined towards the point of view of the second class of physicists and are thus interested in studying special points in the landscape of string compactifications that involve beautiful and exceptional mathematical structures. Certainly among these special points the ones associated to moonshine are amongst the most beautiful.
Thanks.
By the way, I came to think that the page title “universal exceptionalism” is no good:
The point of the philosophical sentiment discussed in the entry is not that the exceptional is universal but quite on the contrary, that the exceptional singles out that which exists from that which does not.
Can we find a better term? How about “existential exceptionalism”? Or “exceptional existentialism”?
I agree that the name is no good. The “universal” is presumably trying to refer to the form of the universe rather than that something holds everywhere, but that’s not typical usage.
To my ear “existentialism” has too much Sartrean baggage to be used.
How about “natural exceptionalism” or maybe just plain “exceptionalism” ?
By the way, I never saw why the contrast to be made is with empiricism, as at the end of the Idea section. I can see that there’s a kind of rationalism to it, that we have reason to believe something on a priori grounds, but it’s not as though the people listed there don’t also think that observations and experiments will decide.
How about “natural exceptionalism”
Okay, but then how about “exceptional naturalism”?! That would seem to make sense, in view of the established meaning of “naturalism”.
Regarding the remark on empiricism:
Yes, this needs rewording. The issue to be expressed here concerns research outlook on unknowns: The exceptional naturalist expects currently unknown fundamental physical nature to eventually turn out to be “exceptional”, whereas the, let’s see: positivist(?), expects nothing in particular.
What brought me back to this just now was a wonder about the exceptionality inherent in hypothesis H. Of course, there’s that deriving from the brane bouquet framework, but presumably there should be a way to see J-twisted 4-Cohomotopy as exceptional directly.
Looks like we overlapped, let me know what you think of the suggestion in #73.
Regarding the 4-sphere: Lots of special aspects of it in low dim topology, like being the base of the quaternionic Hopf fibration, the twistor fibration, etc. Next we’ll bring out its role via the Arnold-Kuiper-Massey theorem
Re #73
… the established meaning of “naturalism”.
Naturalism comes with a complex history too. When I read this word, the first impression I have is close to this variant from SEP: Naturalism:
They urged that reality is exhausted by nature, containing nothing “supernatural”, and that the scientific method should be used to investigate all areas of reality, including the “human spirit”.
So it’s a belief that the natural sciences provide the only methods to come to know reality.
Yes, and seems to fits well with what the entry is about: Scientists in search of the a scientific “theory of everything”. Certainly naturalism is understood by default by all the authors cited there.
Hmm. Do we know their views on the relationship between physics and consciousness/history/literature/art/… ? I don’t see why someone’s beliefs about the form of an ultimate physical theory need dictate their views on mind, etc.
And here I though that “theory of everything” is too blunt a term. Maybe it should be called “theory of everything – and we do mean everything” :-)
I look forward to ’J-twisted cohomotopy and the role of Verdi’s operas in the formation of Italian national identity’.
But seriously I don’t see why the ’exceptional’ aspect has to come along with any kind of physicalism (SEP: Physicalism).
I am surprised to learn of your assumption that modern physicists don’t believe that there is anything but physics running the universe.
That’s certainly the idea of any “theory of everything”, such as putative M-theory, that it would underly all of physics, thus all of chemistry, thus all of biology, thus all of psychology, creativity, …. In a word: everything. You’d appear like a time-traveller out of long gone magical eras if you suggested to physicists that their theories apply to everything except Verdi’s operas.
Regarding the entry, I think we do need a qualification to “exceptionalism” since we don’t mean mathematical exceptionalism, nor american exceptionalism, nor, … but a specific idea that relates exceptional mathematical structurs to physical reality.
If “exceptional physicalism” seems more approriate than “exceptional naturalism”? The latter sounds better to me, but what do I know.
According to this list, Juan Maldacena is a Christian, as is Christopher Isham.
Maldacena is a member of the Society of Catholic Scientists.
Our messages are crossing.
I am surprised to learn of your assumption that modern physicists don’t believe that there is anything but physics running the universe.
Let’s get the quantifiers right here. I’m only claiming that I don’t rule out some modern physicists believing that there is something other than physics running the universe. Is the Kantian idea wholly dead amongst all physicists that there’s a way to understand the world as it appears to us, and a different way to comprehend what Kant puts as God, free will and immortality? Is Maldacena going to take his Catholic faith as something run by physics?
Re #75
Regarding the 4-sphere: Lots of special aspects of it in low dim topology, like being the base of the quaternionic Hopf fibration, the twistor fibration, etc. Next we’ll bring out its role via the Arnold-Kuiper-Massey theorem
Is it envisaged that you will make contact with the intricacies of the differential structure of 4-manifolds, such as the diffeomorphism group of the 4-sphere?
Hi David, only just came online late today. I just sent you an email, please have a look. Thanks!
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