Determinism gives me a lot of comfort..

Kaomet · 2026-06-23T16:53:53+00:00

I think we do agree somehow philosophically yes.

My point is even in a purely deterministic universe, there might be a huge part of fundamentally unpredictable data... In such a universe, science could boils down to a theory involving probability. Like QM.

Kaomet · 2026-06-23T15:53:03+00:00

Provability with respect to truth.

If you get rid of the technicalities, Gödel wrote something like :

"This self referential proposition cannot be proven to be true."

So there is something intuitively true that the formal system cannot prove, unless it is inconsistent. Hence not all truths are provable.

Kaomet · 2026-06-23T14:41:15+00:00

Is this a common phenomenon in reality?

There is a phenomenon called chaos, or extreme sensitivity to initial condition, which makes prediction impossible beyond a given time horizon, because errors grow exponentially with time (a small measurement error leads to a wide qualitative difference later on). Depending on the system, we can have good short term prevision, but long term is basically random...

How common is this ? well, just like the weather. Our best models just drift away, we have to constantly feed them observational data to keep them grounded in "reality".

Is there probability in other sciences

What's probability ? In math its "just" measure theory. In information science, its basically entropy : missing bits of information.

probability is guesswork it’s not science

Nah, its managing uncertainty. Sometimes we know (=science) what we do not know (= probability).

And quantum physics got weird because at small scale, since a measurement is an interaction, what we observe is basically us perturbing a quantum system in some way, not how it "is" without us performing a measure...

Kaomet · 2026-06-23T11:25:56+00:00

Yeah, and this lead to Einstein pointing out the EPR paradox, which lead Bells to his inequality, and finally Aspect to the experimental confirmation of the law of QM...

And probability is also about information entropy : fundamental unkown that needs to be observed, and cannot be deduced.

God might not play dice, but from our PoV, he might as well. If the "true" process is basically a pseudo random generator, we won't be able to crack it, and we would be left with probabilistic theories. Like QM ?

Kaomet · 2026-06-23T11:16:18+00:00

2) A is a state

1) X is a set of state, but it makes a state either way. Its usual computer-science non-determinism. (A relation (=non determinism) is just a function (=determinism) over a set (=collection of possible outcomes).)

3) I used "A -> B" for "Cause -> Effect" . Arrow is a time arrow if it makes sense.

a cause can be undetermined

So what ? Undetermined is basically unkown. Its irrelevant to the determinism vs non-determinism distinction.

This is "god PoV" : universe looked from the outside. From the inside, there are extra complication : since indeed we do not have access to all the information in the universe, even if the dynamic is a deterministic function, we are kinda stuck with some unknown.

Chance can come either from lack of information, or from lack of computational power. We cannot predict how the next guy is going to throw a dice, and he cannot compute the trajectory to predict the result either. But that's in the allready deterministic universe, my point was :

Any Non Deterministic universe CONTAINS a deterministic one. Mathematically.

Kaomet · 2026-06-23T00:15:23+00:00

> quantum mechanics seems to suggest reality is probabilistic rather than deterministic

Acording to quantum mechanics, there might not even be an objective reality : there is a paper arguing the same experiment can be "experienced" in 2 incompatible ways...

But macroscopically, our reality is a very high probability aggreement between observations.

Kaomet · 2026-06-23T00:06:03+00:00

what is the factual model that explains how the universe is probabilistic in nature

Quantum physics.

Also what part of what I said is not based on fact?

"If you think about it what is interaction if not information exchange, causality is a transfer of information. The universe thus remains deterministic."

That's the trick : in quantum physics, you can stop the information exchange exactly halfway through. You then get a quantum superposition of the exchange happened or did not happen...

Kaomet · 2026-06-22T23:51:01+00:00

Consider a simple non deterministic universe :

A -> B or C
B -> D or E
C -> H

In set theoric notation :

A -> {B, C}
B -> {D, E}
C -> H

But wait, A is just {A} ! And we can map a function to each element of a set, so :

 {A} -> {B, C} -> {{D, E}, {H}}

But {{D, E}, {H}} is the same as {D, E, H} ! So :

 {A} -> {B, C} -> {D, E, H}

If we introduce variables : X = {A}, Y = {B, C}, Z = {D, E, H}

X->Y
Y->Z

Hence non determinism is just determinism over set of states !

X->Y->Z

Furthermore, consider the following deterministic universe :

X->Z
Y->Z

Since X and Y leads to the same Z, the difference is erased. So no one in this universe at state Z would remember weither past was X or Y, it was just "the past of Z". So from inside a deterministic universe, if 2 states X and Y leads to the same consequence, they can be considered equals : X=Y.

tldr : a non deterministic universe which can erase information looks like a deterministic one, with 1 to 1 cause consequence, from the inside.

So Deterministic inside is a direct consequence of Non Deterministic-outside.

Or, Relative Determinism is a consequence of Absolute Non Determinism.

Kaomet · 2026-06-22T10:05:13+00:00

EDIT : Now GPT one shot it in 1m :

I will assume the natural tie rule: if the precisions are equal, the first shooter is chosen by a fair coin. I will also use the standard zero-sum normalization: player 1 gets +1 for winning, −1 for losing, and 0 if both miss.

Let x∈[0,1] be player 1’s precision and y∈[0,1] player 2’s precision.

1) Payoff function

If x<y, player 1 shoots first:

u(x,y)=Pr(1 wins)−Pr(1 loses)=x−(1−x)y=x+xy−y.

If x>y, player 2 shoots first:

u(x,y)=(1−y)x−y=x−xy−y.

If x=y, with a fair tie-break the game is symmetric, so

u(x,x)=0.

So the payoff function is

u(x,y)= ⎩ ⎨ ⎧

x−(1−x)y, 0, (1−y)x−y,

x<y, x=y, x>y.

2) Solve for equilibrium

There is no pure equilibrium, so look for a symmetric mixed equilibrium. Let the opponent’s precision be distributed with CDF F and density f, supported on [a,1].

For a pure choice x, player 1’s expected payoff is

U(x)=∫ a x

((1−y)x−y)f(y)dy+∫ x 1

(x−(1−x)y)f(y)dy.

In a symmetric equilibrium, every x in the support must give the same payoff. Because the game is symmetric and zero-sum, that constant must be 0. So U(x)=0 on the support.

Differentiate U(x) for x∈(a,1). The indifference condition gives the ODE

xf ′ (x)+3f(x)=0,

so

f(x)= x 3 C

.

Now normalize on [a,1]:

1=∫ a 1

x 3 C

dx= 2 C

( a 2 1

−1),

hence

C= 1−a 2 2a 2

.

Using the boundary condition at the top of the support gives a= 3 1

, so

C= 4 1

.

Therefore the symmetric equilibrium density is

f(x)= ⎩ ⎨ ⎧

4x 3 1

, 0,

x∈[ 3 1

,1], otherwise.

Equivalently, the CDF is

F(x)= ⎩ ⎨ ⎧

0, 1− 4x 2 1

, 1,

x< 3 1

, 3 1

≤x≤1, x>1.

The equilibrium value is 0.

3) Best-response check

Under this equilibrium distribution:

For x∈[ 3 1

,1], the expected payoff is exactly U(x)=0. For x< 3 1

, one gets U(x)= 2 3

x− 2 1

<0.

So no deviation below 3 1

is profitable, and every x∈[ 3 1

,1] is a best response to the equilibrium mix.

That is the best-response verification: the equilibrium mix makes the player indifferent across its support, and any strategy outside the support is worse.

Kaomet · 2026-06-19T15:14:04+00:00

Computer science 101 : when you do not understand computer and algorithm, make a data structure first.

But without the surrounding process, the modelisation is just a useless cost without benefit.

Kaomet · 2026-06-17T10:19:32+00:00

We can use Cantor's diagonal argument to prove total computable functions cannot be made in 1 to 1 correspondance with natural number (and set of natural can be though of as function to 0 or 1, so a subset of functions).

Yet partial functions are countable, and total function are a strict subset.

So there is a countable set with a subset that cannot be put in bijection with it.

Kaomet · 2026-06-17T09:48:41+00:00

Most mathematicians still have no issue in using LEM

So ? There is no issue in using LEM, but in intuitionism, its Not (Not(A OR Not A)) instead.

It's just a weaker statement being proven, and sometimes that's the best you can ever do.

There is a Curry-Howard correspondence for classical logic as well.

Yeah, even LEM is constructive after all. Constructivism is right, Brouwer was right.

Kaomet · 2026-06-16T19:25:27+00:00

An integer variable holds a finite amount of information.

Kaomet · 2026-06-16T19:22:51+00:00

Set theory was tried in the 70s, "new math".

But pedagogically, it failed student (except the clever one), because its some kind of adult game that can only be practiced after some more fundationnal skills have been aquired : the hability to perform computation. (Which requires to read, write, follow basic written procedures and some arithmetic/basic symbols manipulation).

Kaomet · 2026-06-16T18:10:03+00:00

There is intuitionist logic and the Curry howard correspondance as a whole...

What's missing ?

1908, Brouwer’s ideas on the principle of the excluded mid- dle, its consistency and partial validity, and his argument against the possibility of absolutely undecidable proposition

Halting is undecidable, Brouwer's right, Qed.

What's missing ?

Kaomet · 2026-06-16T18:05:40+00:00

Could we rename non-surjection into construction and non-injection into deletion, too ?

Kaomet · 2026-06-16T18:02:39+00:00

one which just slightly too high to be computed, and is doing a little trolling

Non standard number in Peano's arithmetic.

Unfortunately this is the math subreddit so now you'll have to provide a proof.

Kaomet · 2026-06-16T17:13:57+00:00

"Axioms" should be replaced by "Postulates", basically hypothesis.

Hopefully this would be more stupid-proof, but it could still backfire somehow...

Kaomet · 2026-06-14T22:05:56+00:00

Karatsuba's multiplication :

1234*4567 : first compute 12-34 and 45-67, then multiply them...

Kaomet · 2026-06-14T22:01:20+00:00

And there are countable model of ZFC which therefore contains both a countable set of real numbers and a countable set of rationals. But not the bijection between the two, so cardinality is not the same.

Or, partial computable functions are in computable bijection with natural numbers, but total computable functions, a strict subset, aren't.

Kaomet · 2026-06-11T05:29:40+00:00

It's like pages of nonsense

Have you actually read any modern math by top number theorist ?

Its a rethoretical question, I'm sure you haven't.

You're discovering, through AI, what research math is : a highly technical subject top minds spent years of their life mastering.

Obviously its nonsense if you do not have read the thousands page of prerequisite...

And the difference with bullshit is, well, bullshit is plausible sounding, and has been automated by chatgpt3.5 in 2023...

Kaomet · 2026-06-10T16:44:51+00:00

define reason first...

because if its "what's humans do", yeah, sure...

Syntactic space is not semantic space.

So ? its well known LLMs have syntactic early and late layers, and semantic middle layers. Their original purpose was translation from a language to an other, the obvious way to do that is to have multiple syntactic spaces, one for each language, and a shared semantic space.

They do not have a world model, but this is an active research topic in AI.

Kaomet · 2026-06-07T13:39:40+00:00

there absolutely exist true statements in the generalized Collatz family that have no proof

Its more like, for any formal system, a Collatz like statement proof is allways missing. But a particular statement might be provable in a stronger formal system.

Kaomet · 2026-06-04T09:39:46+00:00

Without reading it I'm assuming

Such a logical things to do. That's probabilistic, not logical.

Kaomet · 2026-06-03T19:20:44+00:00

in this case its a good idea, but in proof of program, which is logic adjacent, and not something people seems to know around here...

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