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:
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.
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
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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:
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.
<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>
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 + …,
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.
LCOn 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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