/sci/ - Science & Math

>>15140542
>Is modern math a joke?
Not the word I'd use.
It just has some trends and is arguably more boring for it.
Set theory etc. aside, the dominance of algebraic geometry and topology is sickening. Those are relevant topics to study, but they are 10x overrepresented as to what they should be.

>>15140561
To be fair, the logicism developments from around 1870-1910 are still seen by many as "the foundation"

https://arxiv.org/pdf/2008.11245.pdf

>the paper is actually a 250 page book
I shiggy

>>12214123
Let's turn it into another question.
Consider a finite group [math]G[/math], i.e. a set [math]S_G[/math] of size [math]g=|S_G|[/math] and a particular, fixed, group structure [math]*_G[/math].

Here's a list of the first such groups.
https://en.wikipedia.org/wiki/List_of_small_groups

The number of groups per finite cardinality [math]n[/math] is

[math](\sigma_n)_n=[/math]
1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 13, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, 1, 2, 3, 4, 1, 6, 1, 52, 15, 2, 1, 15, 1, 2, 1, 12, ...

The fractions of those values over [math]n[/math] are

[math](\tfrac{1}{n}\sigma_n)_n=[/math]
1.0, 0.5, 0.333, 0.5, 0.2, 0.333, 0.143, 0.625, 0.222, 0.2, 0.091, 0.417, 0.077, 0.143, 0.067, 0.875, 0.059, 0.278, 0.053, 0.25, 0.095, 0.091, 0.043, 0.625, 0.08, 0.077, 0.185, 0.143, 0.034, 0.133, 0.032, 1.594, 0.03, 0.059, 0.029, 0.389, 0.027, 0.053, 0.051, 0.35, 0.024, 0.143, 0.023, 0.091, 0.044, 0.043, 0.021, 1.083, 0.041, 0.1, 0.02, 0.096, 0.019, 0.278, 0.036, 0.232, 0.035, 0.034, 0.017, 0.217, 0.016, 0.032, 0.063, 4.172, 0.015, 0.061, 0.015, 0.074, 0.014, 0.057, 0.014, 0.694, 0.014, 0.027, 0.04, 0.053, 0.013, 0.077, 0.013, 0.65, 0.185, 0.024, 0.012, 0.179, 0.012, 0.023, 0.011, 0.136, 0.011, ...[/math]

But since we fixed the group operation on [math]S_G[/math] in advance, it seems a little more complicated to compute the chance of a random subset [math]S_A\subset S_G[/math] equip it with [math]*_G[/math] restricted to [math]S_A[/math] forming a group. Correct me if I'm wrong and it's simpler.

In the original subgroup-subset variant, we can ask the possibly complicated question given B, I think, since some groups will be more likely to be subgroups than others

For n<100, say, we have all the data to answer all questions (even brute forcing it), so all these have concrete answers.

>>12214123
Let's turn it into another question.
Consider a finite group [math]G[/math], i.e. a set [math]S_G[/math] of size [math]g=|S_G|[/math] and a particular, fixed, group structure [math]*_G[/math].

Here's a list of the first such groups.
https://en.wikipedia.org/wiki/List_of_small_groups

The number of groups per finite cardinality [math]n[/math] is
[math](\sigma_n)_n=[/math] 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 13, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, 1, 2, 3, 4, 1, 6, 1, 52, 15, 2, 1, 15, 1, 2, 1, 12, ...
The fractions of those values over [math]n[/math] are
[math](\tfrac{1}{n}\sigma_n)_n[/math] = 1.0, 0.5, 0.333, 0.5, 0.2, 0.333, 0.143, 0.625, 0.222, 0.2, 0.091, 0.417, 0.077, 0.143, 0.067, 0.875, 0.059, 0.278, 0.053, 0.25, 0.095, 0.091, 0.043, 0.625, 0.08, 0.077, 0.185, 0.143, 0.034, 0.133, 0.032, 1.594, 0.03, 0.059, 0.029, 0.389, 0.027, 0.053, 0.051, 0.35, 0.024, 0.143, 0.023, 0.091, 0.044, 0.043, 0.021, 1.083, 0.041, 0.1, 0.02, 0.096, 0.019, 0.278, 0.036, 0.232, 0.035, 0.034, 0.017, 0.217, 0.016, 0.032, 0.063, 4.172, 0.015, 0.061, 0.015, 0.074, 0.014, 0.057, 0.014, 0.694, 0.014, 0.027, 0.04, 0.053, 0.013, 0.077, 0.013, 0.65, 0.185, 0.024, 0.012, 0.179, 0.012, 0.023, 0.011, 0.136, 0.011, ...

But since we fixed the group operation on [math]S_G[/math] in advance, it seems a little more complicated to compute the chance of a random subset [math]S_A\subset S_G[/math] equip it with [math]*_G[/math] restricted to [math]S_A[/math] forming a group. Correct me if I'm wrong and it's simpler.

In the original subgroup-subset variant, we can ask the possibly complicated question given B, I think, since some groups will be more likely to be subgroups than others

For n<100, say, we have all the data to answer all questions (even brute forcing it), so all these have concrete answers.

>>8214237
>>8214272
How complicated is to make a proof depends on the specification of the type, and then on what the compiler can do automatically. And, thus, when two proofs differ is determined by the system.
You basically say that two proofs can be different by some silly addition, so how can there only be one proof
E.g.
>You may proof that (1+2)^2 equals 10-1 by reducing the left side to 9 and then the right to 9,
>or by reducing the left side to 9, proving Fermats little theorem and then going back to reducing the right hand side to 9!
But a proof corresponds to a term of a type, and those are specified exactly, and the equality is also just a type.

Consider the logical truth
(¬A ∧ ¬B) ⟹ ¬(A∨B)
>If 'A isn't true' as well as 'B isn't true', then 'either A or B is true' is false.

In terms of types, using the notation

A⟹B ≡ A → B (functions taking terms a:A to a term of B)
¬A ≡ A → 0 (functions taking terms a:A and not terminating)
A ∧ B ≡ A × B (pairs (a,b) )
A ∨ B ≡ A + B (tagged terms such as left(a) or right(b))

this translates to the type

((A → 0) × (B → 0)) → (A+B) → 0

Look at OP's pic again, in particular the LAM rule. It reads
[math] \dfrac {x\,:\,X\, \vdash\, t\, : \, T } { (\lambda x. t)\, :\, X\to T } [/math]
"if you can get a term t of T, possibly using an x of X, you can get a function term of X→T writen with the lambda notation"

To construct a term of X → T, corresponding to an implication, we can only follow this rule. This what it means to get a term of this type
(remark: and for general equality, the rules are much harder - that's why Martin Löf only did it in the 70's. However, as an example, the equality for numbers, as you e.g. see in >>8209921 is much simpler)

The constructor for the A+B type goes like
"if you got either a:A or b:B, then you get a term of A+B and it known if it's in the left or right slot and what it was"

Good summer, well-formed people,

I was pointed into the direction of this book (1) and thought I'll open a thread on all things related. It looks pretty comprehensive and clear, and the first edition from 2012 is actually downloadable.
Besides, I'm considering delving into Idris (3) in the upcoming months and the main developer is publishing a book this summer as well (4).

(1)
https://ncatlab.org/nlab/show/Practical%20Foundations%20for%20Programming%20Languages
(2)
http://www.cs.cmu.edu/~rwh/plbook/book.pdf
(3)
http://docs.idris-lang.org/en/latest/tutorial/index.html
(4)
https://www.manning.com/books/type-driven-development-with-idris

[math] (1+x) (1+x^2) (1+x^4) [/math]

[math] = (1+x^2) (1+x^4) + x (1+x^2) (1+x^4) [/math]

[math] = (1+x^4) + x^2 (1+x^4) + (x+x^{2+1}) (1+x^4) [/math]

[math] = (1+x^4) + (x^2 + x^{4+2}) + (x (1+x^4) +x^{2+1} (1+x^4)) [/math]

[math] = (1+x^4) + (x^2 + x^{4+2}) + (x+x^{4+1} + x^{2+1} + x^{4+2+1}) [/math]

[math] = 1 +x+x^2+x^3+x^4+x^5+x^6+x^7 [/math]

The point is that at n=3 numbers, each number in the scheme 2^n (so 1 and 2 and 4) never added to itself, give you
1
2
2+1
4
4+1
4+2
4+2+1

similarly, for n=2 and the scheme 3^n & 2·3^n (so 1 & 2 and 3 & 6 and 9 & 18)
1
2
3 (not 1+2, which is excluded as they come in one sum)
3+1
3+2
6
6+1
6+2
9 (not 3+6, which is excluded as they come in one sum)
9+1
9+2
9+3
9+3+1
9+3+2
9+6
well and so on.
You take a grid like 1,2,4,8,16,... or 1,3,9,27,... and provide stuff so that everything up the the next number can be reached.

>>7973361
Here's something I came up with, a method of translating any infinite product to an infinite sum:

[math] \prod_{k=0}^\infty b_k = \prod_{k=0}^{M-1}b_k + \sum_{n=M}^\infty(b_n-1)\,\prod_{k=0}^{n-1}b_k [/math]

E.g.
[math] (1-x)^{-1} = 1 + \sum_{n=1}^\infty x^{2^n} \, \prod_{k=0}^{k-1} (1-x^{2^k}) [/math]

[math] (x+1) x^2+(x+1) \left(x^2+1\right) x^4+(x+1) \left(x^2+1\right) \left(x^4+1\right) x^8+x+1 + ... [/math]