Another challenge for the conventionalists: the monster group

[Bruno Marchal]

Hello,

Some have defended conventionalism in mathematics. I shown that hard to sustain in recursion/computability theory, and thus arithmetic. Here something which shows that it is hard to maintain conventionalism in the study of finite symmetries.

Groups (mainly set of symmetries) can be decomposed into some composition of “prime groups” (called simple group).

Who is the guy who decided that a all finite simple groups belong to either 18 infinite families of groups, except for 26 exceptional one, the sporadic groups, which does not, and who decided conventionally that the biggest one is Monstruously big, the Monster, which has

808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

elements.

All groups can be represented by a group of matrices, with the coefficients belonging to some field (usually the complex numbers), with the usual product of matrice. A field is itself a special sort of double group. (Not to confuse with quantum field, of filed of forces).

The minimal dimension needed for that representation is the dimension of the group. It is dimension of the space in which the element of the group represent the symmetries. 

The Monster group has dimension 196,883 with the matrix coefficient taken in the field of complex numbers, but it has dimension 196.882 on the field z_2 with two elements {0, 1}.

Who decided that the dimension of the monster group is 196.882. Divine convention? Could a God makes this in another way?

It might play some role in physics, notably conformal fields, strings,… (cf Munshine).

A rather nice video on the Monster group is:

https://www.youtube.com/watch?v=mH0oCDa74tE

As I have explained, the non algorithmic distribution of the codes of the total computable function is enough kicking back for me to be realist on arithmetic, but the Monster group presents, I think, some difficulties for the conventionalist too.

Bruno

[Brent Meeker]

Are you claiming someone had to decide on these numbers?  The whole point of mathematics is that these are logical implications of axioms, and the axioms are quite simple and easily thought of.  So if the axioms are conventions and the rules of inference are conventions, then conventionalism is true.  Of course one could argue that axioms and rules of inference are not arbitrary conventions...they are grounded in biological evolution.

Brent

[Bruno Marchal]

On 14 Sep 2020, at 00:19, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 9/13/2020 9:45 AM, Bruno Marchal wrote:

Hello,

Some have defended conventionalism in mathematics. I shown that hard to sustain in recursion/computability theory, and thus arithmetic. Here something which shows that it is hard to maintain conventionalism in the study of finite symmetries.

Groups (mainly set of symmetries) can be decomposed into some composition of “prime groups” (called simple group).

Who is the guy who decided that a all finite simple groups belong to either 18 infinite families of groups, except for 26 exceptional one, the sporadic groups, which does not, and who decided conventionally that the biggest one is Monstruously big, the Monster, which has

808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

elements.

All groups can be represented by a group of matrices, with the coefficients belonging to some field (usually the complex numbers), with the usual product of matrice. A field is itself a special sort of double group. (Not to confuse with quantum field, of filed of forces).

The minimal dimension needed for that representation is the dimension of the group. It is dimension of the space in which the element of the group represent the symmetries. 

The Monster group has dimension 196,883 with the matrix coefficient taken in the field of complex numbers, but it has dimension 196.882 on the field z_2 with two elements {0, 1}.

Who decided that the dimension of the monster group is 196.882. Divine convention? Could a God makes this in another way?

It might play some role in physics, notably conformal fields, strings,… (cf Munshine).

A rather nice video on the Monster group is:

https://www.youtube.com/watch?v=mH0oCDa74tE

As I have explained, the non algorithmic distribution of the codes of the total computable function is enough kicking back for me to be realist on arithmetic, but the Monster group presents, I think, some difficulties for the conventionalist too.

Are you claiming someone had to decide on these numbers?  The whole point of mathematics is that these are logical implications of axioms, and the axioms are quite simple and easily thought of. 

Yes indeed. They are “definitional” axioms.

So if the axioms are conventions and the rules of inference are conventions, then conventionalism is true. 

Not at all. You can say that the choice of axioms/definition is conventional (the rules of inference here are informal), but the non conventional part is in what follows from them, which was not asked for. If we could change the definition so that the weirdness disappear, we would have done this, but the notion of symmetry which motivated them would be lost, so we have to live with the Monster (and can expect some role of it in physics).

Of course one could argue that axioms and rules of inference are not arbitrary conventions...they are grounded in biological evolution.

Exactly.

And then with mechanism, the biological evolution will be grounded in quantum mechanics, and quantum mechanics will be grounded in the unavoidable hallucination of the numbers). Mechanism generalises Darwin by explaining the original of the physical laws from the psychology of number or combinators. The non conventional aspect of the combinator rules, or of elementary arithmetic is in the fact that they are Turing-universal, and we get the same physics independly of the choice of the universal system. With or without invoking biology, conventionalism cannot make sense, except for the choice of axioms, assuming some free will :)

(That conventionalism is the same as the decision to study Mars instead of Uranus, or choosing in a menu…)

Bruno

Brent

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[Lawrence Crowell]

A compendium on this topic is Sphere Packing, Groups and Lattices by Conway and Sloane. Before we rush off into doing physics with the monster group or monstrous moonshine it is best to consider the E8 group, the Jordan 3×3 matrix of E8 groups J^3(O), and the Leech lattice Λ24 . The E8 group by itself is not an effective group for physics. The reason can be seen from Bott periodicity, where the cyclicity by 8 with doubling from reals (a unit), to complex numbers (2) to quaternions and octonions that octonions or E8 are real valued. To get an effective theory of physics one must employ E8×E8. where we think of these as a pair similar to a pair of reals in a complex number. This defines a standard Lagrangian type of physics and is the heterotic string. To get additional topological cyclicity terms, analogous to edge states in topological insulators and fractional Hall states, in a Chern-Simons Lagrangian this can be extended to the Jordan 3×3 matrix of E8 groups  There a system of roots and weights with this that is 196,556 dimensions called the Leech lattice. Before charging off into the actual Fischer-Greiss group or the monster, there is a lot here to consider.

The Leech lattice embeds into the Matheiu group M24 with 196,883 elements and which also defines the automorphism on the FG monster. It might be said the monster group is a symmetry system that keeps the elements of Λ24, J^3(O) and M24 constant. So if all of physics we measure is determined by this large group, extended all the way to the sporadic group M24, there is behind it all this monster group. There is further a connection to number theory called monstrous moonshine of Borcherds. We might further consider the Langlands program which is a deep connection between algebraic geometry and number theory. I have been working on the geometry and information theory of entanglement, and this has a duality with the symmetry groups of gauge fields, which have a Bott periodicity structure. Behind all of this there is the role of the monster group, which acts in some way to fix these structures, in a way to maintain a vacuum symmetry. 

LC

[Lawrence Crowell]

On Monday, September 14, 2020 at 4:27:48 AM UTC-5 Bruno Marchal wrote:

On 14 Sep 2020, at 00:19, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:

On 9/13/2020 9:45 AM, Bruno Marchal wrote:

Hello,

Some have defended conventionalism in mathematics. I shown that hard to sustain in recursion/computability theory, and thus arithmetic. Here something which shows that it is hard to maintain conventionalism in the study of finite symmetries.

Groups (mainly set of symmetries) can be decomposed into some composition of “prime groups” (called simple group).

Who is the guy who decided that a all finite simple groups belong to either 18 infinite families of groups, except for 26 exceptional one, the sporadic groups, which does not, and who decided conventionally that the biggest one is Monstruously big, the Monster, which has

808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000

elements.

All groups can be represented by a group of matrices, with the coefficients belonging to some field (usually the complex numbers), with the usual product of matrice. A field is itself a special sort of double group. (Not to confuse with quantum field, of filed of forces).

The minimal dimension needed for that representation is the dimension of the group. It is dimension of the space in which the element of the group represent the symmetries. 

The Monster group has dimension 196,883 with the matrix coefficient taken in the field of complex numbers, but it has dimension 196.882 on the field z_2 with two elements {0, 1}.

Who decided that the dimension of the monster group is 196.882. Divine convention? Could a God makes this in another way?

It might play some role in physics, notably conformal fields, strings,… (cf Munshine).

A rather nice video on the Monster group is:

https://www.youtube.com/watch?v=mH0oCDa74tE

As I have explained, the non algorithmic distribution of the codes of the total computable function is enough kicking back for me to be realist on arithmetic, but the Monster group presents, I think, some difficulties for the conventionalist too.

Are you claiming someone had to decide on these numbers?  The whole point of mathematics is that these are logical implications of axioms, and the axioms are quite simple and easily thought of. 

Yes indeed. They are “definitional” axioms.

The sporadic groups, more bizarre than the exceptional G2, F4, E6, E7, E8 groups, have a hierarchy relation shown in the diagram below. This is not entirely understood and there are the orphaned groups, shown without color and their connections unknown. The number of papers written to understand these could fill a small library. For physics the M24 group is seen as the largest red Matheiu group where these connect to the Conway group Co1 and the Fischer group Fi24 and then to the monster at the top. The red groups are Matheiu groups, green and Conway groups and blue the Fisher groups. 

Where these number come from is difficult. The M24 embeds the the Leech lattice and the J^3(O). This matrix contains 3 octonions or E8s and is

|x        O_1   O_2*|

|O_1*   y      O_3  |

|O_2   O_3*    z    |

where this 3×8 + 3 = 27 element system with the Freudenthal determinant conditions is 26 dimensional and defines a 26 dimensional Lorentzian manifold for the bosonic string. The triality system is supergravity with a 3-diagram for O_1 connected to edgelinks for O_2 and O_3* which are supersymmetric. This then embeds into M24 which requires an additional 327 elements, and this is figured out by working out how to minimally embed the Leech or J^3(O) .

There is much not known, and if you read Conway and Sloane you find that the book has few theorem-proofs. Some have complained the book is like the bible, lots of statements and few proofs. However, if proofs were included in what is known the book would be a library, and there are many conjectures on this that are as yet not proven. The role of these orphaned groups is a mystery, though some progress has been reported.

LC

[Brent Meeker]

On 9/14/2020 2:27 AM, Bruno Marchal wrote:
>> So if the axioms are conventions and the rules of inference are
>> conventions, then conventionalism is true.
>
>
> Not at all. You can say that the choice of axioms/definition is
> conventional (the rules of inference here are informal), but the non
> conventional part is in what follows from them,

??  That's like saying that painting stop signs red is a convention, but
the fact that stop signs are red is not.

Brent

[Lawrence Crowell]

The number 196883 and 196884 comes from the Klein j-invariant function that for q = e^{2πiτ} is such that 

j(τ) = q^{-1} + 744 + 196884q + 21493760q^2 + 864299970q^3 + ...,

where the numbers 744 define the number of elements in three E8s in the Jordan J^3(O) and 196844 is the number of elements in M24 or dim(Λ24) + 328. The various powers q^n are the n grades in the general Fisher-Griess algebra, where the first is the 196884 or 196883 of the monster. This first term 196884q is the cornerstone of moonshine theory. The monster with 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements is just the first term in q. Mathematics in its purest form has even larger groups corresponding to higher powers in q.

LC

[Bruno Marchal]

Thanks Lawrence. That’s interesting stuff...

Fiction is 42.

Truth is 24.

But is it 6x4 or 8x3?

:)

An interesting book, which gives the wrong, but interesting, feeling that group theory is basically equal to physics, like with Noether theorem, is the book by Cvitanovic (Group Theory), which shows amazing relation between the additive theory of numbers (partition theory) and the exceptional groups, notably E8, but unfortunately, it presupposes too much of quantum field theory, and I go very slowly through it. The author use interesting sort of “Feynman diagram”.

… but where is my exemplar now? …

Bruno

<SporadicGroups.png>

 

So if the axioms are conventions and the rules of inference are conventions, then conventionalism is true. 

Not at all. You can say that the choice of axioms/definition is conventional (the rules of inference here are informal), but the non conventional part is in what follows from them, which was not asked for. If we could change the definition so that the weirdness disappear, we would have done this, but the notion of symmetry which motivated them would be lost, so we have to live with the Monster (and can expect some role of it in physics).

Of course one could argue that axioms and rules of inference are not arbitrary conventions...they are grounded in biological evolution.

Exactly.

And then with mechanism, the biological evolution will be grounded in quantum mechanics, and quantum mechanics will be grounded in the unavoidable hallucination of the numbers). Mechanism generalises Darwin by explaining the original of the physical laws from the psychology of number or combinators. The non conventional aspect of the combinator rules, or of elementary arithmetic is in the fact that they are Turing-universal, and we get the same physics independly of the choice of the universal system. With or without invoking biology, conventionalism cannot make sense, except for the choice of axioms, assuming some free will :)

(That conventionalism is the same as the decision to study Mars instead of Uranus, or choosing in a menu…)

Bruno

Brent

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<SporadicGroups.png>

[Bruno Marchal]

?

It is more like saying that the axioms of arithmetic, and the definition of prime numbers, are conventional but that the truth of Euclid’s statement “there is no biggest prime” is not; or that the definition of group is conventional, but that the classification of finite simple groups is not.

Bruno




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[Brent Meeker]

On 9/15/2020 5:04 AM, Bruno Marchal wrote:
>> On 14 Sep 2020, at 19:32, 'Brent Meeker' via Everything List <everyth...@googlegroups.com> wrote:
>>
>>
>>
>> On 9/14/2020 2:27 AM, Bruno Marchal wrote:
>>>> So if the axioms are conventions and the rules of inference are conventions, then conventionalism is true.
>>>
>>> Not at all. You can say that the choice of axioms/definition is conventional (the rules of inference here are informal), but the non conventional part is in what follows from them,
>> ?? That's like saying that painting stop signs red is a convention, but the fact that stop signs are red is not.
> ?
>
> It is more like saying that the axioms of arithmetic, and the definition of prime numbers, are conventional but that the truth of Euclid’s statement “there is no biggest prime” is not; or that the definition of group is conventional, but that the classification of finite simple groups is not.

What is entailed by a convention is conventional.

Brent

[Lawrence Crowell]

Where the conventions really end is beyond the linear term in the Klein j-invariant

j(τ) = q^{-1} + 744 + 196884q + 21493760q^2 + 864299970q^3 + ...,

where the lattice systems for the linear term involves Jacobi θ-functions span 24 dimensions, the root space is 196884 and the whole space of elements that keep this space invariant is this crazy number ~8.1x10^{51}. This part is known. Anyone with the temerity to climb to the quadratic term? The lattice system will be 48 dimensions, the P48 lattice, and the super-monster group with 21493760 roots and some ungodly number of elements. Then one can go to the cubic term to a lattice P72 or P64(+8) or some such construction. At p48 there are open questions on just the lattice space, where the automorphism systems is a mystery.

In some sense we have to go there. For quantum complexity this extends beyond classical complexity C ~ exp(k log(2)), for k elements.. This is just e^S for the entropy is k log(2), Quantum complexity is a sum over the power set of this, think of the superposition and entanglements of all possible states, and complexity is e^{e^{k log(2)}}. The size of Hilbert space is then enormously large, say for a black hole with entropy 10^{72}bits/string. The quantum complexity is far more vast. If one is working a theory of quantum gravitation then for a sufficiently small black hole the linear term in the Klein j-invariant is sufficient, maybe far larger than needed. However, if one wants to consider the quantum complexity of a black hole we need to have some trail blazed far beyond the domain of the monster group or monstrous moonshine. We can of course qualitatively say the j-invariant is a Fourier series over an infinite Hilbert space and so there is no bound on quantum complexity. However, we are groping largely in the dark. 

However, to pull back a bit from this, a theory on how to describe quantum entanglements with Jacobi θ-functions is maybe a good start for  this. There is a vast frontier here.

LC

[Bruno Marchal]

On 15 Sep 2020, at 12:57, Lawrence Crowell <goldenfield...@gmail.com> wrote:

The number 196883 and 196884 comes from the Klein j-invariant function that for q = e^{2πiτ} is such that 

j(τ) = q^{-1} + 744 + 196884q + 21493760q^2 + 864299970q^3 + …,

The most famous modular form, probably. 

where the numbers 744 define the number of elements in three E8s in the Jordan J^3(O) and 196844 is the number of elements in M24 or dim(Λ24) + 328. The various powers q^n are the n grades in the general Fisher-Griess algebra, where the first is the 196884 or 196883 of the monster. This first term 196884q is the cornerstone of moonshine theory.

OK.

The monster with 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 elements is just the first term in q.

What do you mean by this?

Mathematics in its purest form has even larger groups corresponding to higher powers in q.

But not larger finite simple group! Some would argue that Nature is full of infinite group, from R to sophisticated Lie groups.

Mechanism is still very far away, but the presence of what I have called here (many years ago) LASE (Little Abstract Schroedinger Equation; p -> []<>p, for p sigma_1, et “[]” being a material mode, with “& p” or “& <>t") do provide some evidence for a physical core which is highly symmetric … at the (logical) bottom. We get the symmetries, the quantum structure, but not yet Hamiltonian, or space, energy or particles. But to get the qualia, we have no choice; we have to derive the quanta from (Gödel-Löbian) universal machine self-reference. Then, the presence of subtle mathematics in physics gives hope that a lot of theoretical physics will help to finish the work, in some futures.

Bruno

LC

On Monday, September 14, 2020 at 12:33:02 PM UTC-5 Brent wrote:

On 9/14/2020 2:27 AM, Bruno Marchal wrote:
>> So if the axioms are conventions and the rules of inference are
>> conventions, then conventionalism is true.
>
>
> Not at all. You can say that the choice of axioms/definition is
> conventional (the rules of inference here are informal), but the non
> conventional part is in what follows from them,

??  That's like saying that painting stop signs red is a convention, but
the fact that stop signs are red is not.

Brent

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[Bruno Marchal]

My point is that the definition, the choice of axioms, etc. can arguably considered conventional, or even contingent, relying on our goal, which can be local and related to some human interest. Why study symmetries? Why study smooth varieties, or why study Saturn, Mars, … Where the convention stops is when we derive the consequences of the axioms, or the truth about the objects that we defined. It is the same in math and in physics.

A witnessing of this is the use of computer in sorts of mathematical experimental exploration, like just observing where the Riemann critical zeros are. We have looked at billions of them, and they are all well situated on the critical line, confirming the Rieman Hypothesis “experimentally”. Now mathematician want proof, but, even if we find it, the result can hardly be described as conventional. If I was, let us convene that all zeroes behave well, and let us convene to send me the one million of dollars prize :)

Bruno

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[smitra]

But it was not possible to explain this all that well in the time limit
of ten minutes. What is missing from the talk is that the aggregate of
percolation clusters of organics ends up implementing a very
sophisticated machine learning system that ends up building the
machinery of life step by step. But that's way too technical to explain
in a conference talk, I'll have to do that in a paper.

Video: https://vimeo.com/458077141

Abstract:

https://meetingorganizer.copernicus.org/EPSC2020/EPSC2020-785.html

The mathematician John von Neumann, through his work on universal
constructors, discovered
a generalized version of the central dogma of molecular biology biology
in the 1940s, long
before the biological version had been discovered. While his discovery
played no role in the
development of molecular biology, we may benefit from a similar
mathematical approach to find
clues on the origin of life. This then involves addressing those
problems in the field that
do not depend on the details of organic chemistry. We can then consider
a general set of
models that describe machines capable of self-maintenance and
self-replication formulated in
terms of a set of building blocks and their interactions.

The analogue of the origin of life problem is then to explain how one
can get to such
machines starting from a set of only building blocks. A fundamental
obstacle one then faces
is the limit on the complexity of low fidelity replicating systems,
preventing building
blocks from getting assembled randomly into low fidelity machines which
can then improve due
to natural selection [1]. A generic way out of this problem is for the
entire ecosystem of
machines to have been encapsulated in a micro-structure with fixed inner
surface features
that would have boosted the fidelity [2]. Such micro-structures could
have formed as a result
of the random assembly of building blocks, leading to so-called
percolation clusters [2].

This then leads us to consider how in the real world a percolation
process involving the
random assembly of organic molecules can be realized. A well studied
process in the
literature is the assembly of organic compounds in ice grains due to UV
radiation and heating
events [3,4,5]. This same process will also lead to the percolation
process if it proceeds
for a sufficiently long period [2].

In this talk I will discuss the percolation process in more detail than
has been done in [2],
explaining how it leads to the necessary symmetry breakings such as the
origin of chiral
molecules needed to explain the origin of life.



[1] Eigen, M., 1971. Self-organization of matter and the evolution of
biological
macromolecules. Naturwissenschaften 58, 465-523.

[2] Mitra, S., 2019. Percolation clusters of organics in interstellar
ice grains as the
incubators of life, Progress in Biophysics and Molecular Biology 149,
33-38.

[3] Ciesla, F., and Sandford.,S., 2012. Organic Synthesis via
Irradiation and Warming of Ice
Grains in the Solar Nebula. Science 336, 452-454.

[4] Muñoz Caro, G., et al., 2002. Amino acids from ultraviolet
irradiation of interstellar ice
analogues. Nature 416, 403-406.

[5] Meinert, C,., et al., 2016. Ribose and related sugars from
ultraviolet irradiation of
interstellar ice analogs. Science 352, 208-212.


Saibal