{∅, {∅}, {∅, {∅}}} : Rage Against the Meat Grinder

Thanks a lot.I genuinely   like these kind of problems.As van Gogh said-I am seeking, I am striving, I am in it with all my heart.

The few versions I launched late last night were running through the night, but were only up to about the millionth prime (there are 11262113374 primes less than 11^11 = 285311670611), so it would be 11262 times as long as "all god damn night").  I had to hunt down some special function for finding the square root of Big Integers so that I (the machine, actually) could "check primes" faster.  The changed code is below.  I have to venture out to the grocery store ... not much sleep, so I have to be careful not to get into any altercations with any of my beloved fellow nervous wrecks.

I just launched this program, and it is already zooming into the 10 millionth prime in about 5 minutes!   I think that, by the time I return, it will be done, and I will have a time elapsed for you.   
I am leaving the code above since it did work ---- and I will run it one week just to see how much longer it takes using the basic "check_prime()" code.
____________________________________________________________________________

#include <numeric>
#include <functional>
#include <cmath>      // for pow(11, 11) = 285311670611
#include <vector>
#include <iostream>
#include <gmpxx.h>   // g++ -lgmpxx -lgmp -g product_sqrt.cpp -std=c++17 -o product
#include <chrono>
#include <unistd.h>

using std::cout;
using std::endl;

mpz_class prod (const std::vector<mpz_class> v)
{
    mpz_class accum = 1;
    for (const auto& p : v)
        accum *= p;
    return accum;
}

int check_prime(mpz_class n)
{
   return mpz_probab_prime_p(n.get_mpz_t(), 15);  // see Notes below (Number Theory)
}
 
int main ()
{
    auto start = std::chrono::steady_clock::now();
 
    std::vector<mpz_class>  primes; 
 
    // std::cout << "Maximum size of a 'vector' is " << primes.max_size() << "\n";

    mpz_class count = 0;
    for (mpz_class i = 2; i < std::pow(11, 11) ; ++i)    // 11^11 = 285311670611
        if (check_prime(i)) {
              cout << "\nprime[" << count << "] = " << i << '\n';
              primes.push_back(i);
              ++count;
        }
       
    cout << "\n\nThere are " << count << "prime numbers multiplied together!";
    cout << "\n\nBig Product is " << prod(primes) << endl;

    auto end = std::chrono::steady_clock::now();
 
    cout << "Elapsed time in seconds : "
        << std::chrono::duration_cast<std::chrono::seconds>(end - start).count()
        << " sec";

    cout << "Elapsed time in minutes : "
        << std::chrono::duration_cast<std::chrono::minutes>(end - start).count()
        << " min";

    return 0;
}
/* Notes

sage: prime_pi(11^11) = 11262113374
11^11 = 285311670611

Number Theoretic Function: int mpz_probab_prime_p (const mpz_t n, int reps)

    Determine whether n is prime.
    Return 2 if n is definitely prime,
    return 1 if n is probably prime (without being certain),
    or return 0 if n is definitely non-prime.

    This function performs some trial divisions, a Baillie-PSW probable prime test,
    then reps-24 Miller-Rabin probabilistic primality tests.
    A higher reps value will reduce the chances of a non-prime being identified as “probably prime”.
    A composite number will be identified as a prime with an asymptotic probability of
    less than 4^(-reps). Reasonable values of reps are between 15 and 50.
 */
_______________________________________________________________

By the way, Holden, can you imagine how big this product is?   It gotta be Bernie Sanders' HUGE.
More than HUGE!  It's a Big Mother Fuckin' Integer, baby!   (sorry, no sleep!)

I can't wait to see it.   Stay tuned, weirdos.   

Yes,it's a very big number.Maybe someday I will be able to write programs like you to solve such problems.
But for the present,this is all I have:

https://i.postimg.cc/mgtfJL4Q/IMG-20200423-114046.jpg

Note that the author of "Goethe and Schopenhauer on Mathematics," this Arnold Emch, may have had strong gort-tendencies (believing strongly that that which is so, is so).

He had wife and two sons, one of which became a well-known "management consultant" - whatever the fuck that's supposed to be.     (I have never heard of him, but - then again, who the hell am I?  I'm just some old suicide who has lived long beyond what would be expected (beyond 50, that is). )

Maybe it has been a desire to understand more math(s), to rejuvenate my interest in programming by focusing primarily on mathematics-oriented and educational code, that has enticed "the will to live" to endure the redundancy of "doing what we do to stay alive" (much of which is beyond our personal control).

If the food trucks stop hauling the animal parts to the spooky stores, then I don't know what we become.  We certainly will see, won't we?

On some level I realize that much of what I think I am (a personality? a soul?) is just the energy from food consumed, and that a starving me might be so drawn inward that articulating myself might not be possible.

Essentially, as long as I am an Eater-of-Food, I am full of shit.   When the animal body dies, no more food, no more stomach, no more shit ...

 

I have taken your advise of focusing on math problems with regard to which I have "some hope of understanding them" .
This approach has helped me quite a bit.

This book you mention sounds very interesting to me.

Yes, Holden, it is  melting my brain even just in the Introduction.  I will have to devote at least an entire notebook just to this collection.  I will want to read notes from it somewhere besides attached to a terminal or reading device.

I am tired and a little emotionally drained (not too bad), and yet still parts of me have a gut intuition that this book will have a great impact on me as I have been alarmed throughout my life by certain comments I've read from the 1900's disparaging to Schopenhauer's mathematical insight.  I have never felt qualified to defend him, or where even to begin.  Fortunately, we have "elder brothers" (and a handful of sisters, i suppose) who have stepped up to throw this wonderful pie into the face of The Church of Reason, whose representatives have conveniently written Schopenhauer off as "the artists' and musicians' philosopher."

Well, I would not mind a strong dose of Schopenhauer's genius.  Indeed, logarithms may have become like poetry to me, but only as a personal technique for warding off decades of general dissatisfaction.  Sometimes I am able to capture a mood where I feel fortunate or blessed to have gained a bit of familiarity with the small but fundamental branches of mathematics I have studied.

I wish I could be to young mathematics students what Schopenhauer was to me, even though I do not see Schopenhauer as one who has had any mathematical influence upon me other than to remind me what drudgery arithmetic is, and how a man can actually fry his brain via number crunching.  Schopenhauer knew he was better off playing the flute!   

Don't get me wrong, I do sometimes slip into my number-crunching (manic? WTF) modes, but it is usually kind of disturbingly stimulating, in that I become entirely obsessed and engrossed in the computations.  It is similar to a drug-induced trance. 

No matter what level of understanding of mathematics or calculating and computing I have developed over the decades, I am fully aware that Schopenhauer's grade of intelligence and "spirit" or "character" was superior to mine.  That is, I have no problem with being under any kind of delusions that I have deeper insight just because I have been exposed to more recent methods of analysis (which were being developed in Schopenhauer's time).  No, and AGAIN NO!   I am very grateful to those who authored the contents of  Language, Logic, and Mathematics in Schopenhauer 1st ed. 2020 Edition. (editor: Jens Lemanski) !  Right out of Germany ... maybe some of the authors are researchers who had access to Schopenhauer's research "library" and "notes" ...

I know Schopenhauer had a different kind of mind than my own, and that he possessed an enormous wealth of knowledge, not to mention the confidence to question authorities that I would be too intimidated by to ever even think of doubting.

Maybe I am setting myself up for a disappointment, but this is quite a rare collection of essays.  I am intrigued.  I will read into the night, having finally put away the math texts for the night.

I know the human world is filled with misery, want, and need, and I am not ignoring the ever-present annoyance of the burden of my own existence, but kindling such interest in two areas I have been influenced to believe were not even remotely related.   Now I come to find out that there are others ... others who have delved into it.   

Maybe Hell really is about to freeze over.

We were scratching at this very surface a while back with your Schopenhauer's Philosophy of Mathematics.  I was even gathering specific parts of Schopenhauer's opus when we were zeroing in on those gems ... if only because they were so few and far between. 

It is great to know someone else has been eagerly leaning in this particular direction, just waiting for some guidance and fellow-students-of-life to forge a bit of a trail into that woodlands.

The freaking bloody Berlin Lectures are out of the bag, and they thought nobody in the world would notice. 

Again, I feel we are blessed to have the haven/fortress prepared for paying respect to those who might greatly benefit, spiritually and even emotionally, by having access to lectures Schopenhauer intended for the most genuine students.   That "Prince Vault" has yet to be translated, and to remains as mysterious to me as, say, an old Sanskrit text might be to Holden.  I am linguistically far removed from understanding the German language.  I struggle just to remember the numbers in German.  I will be delighted to take note of some of the more significant German words that I am sure to be exposed to in the readings.

Those lectures might reveal the details I crave!  I'm sure some of the Berlin Lectures will make it into this or future incarnations of this or similar scholarship.

His wished for his published (paid for out of his own inherited pocket) work to be received by lone scholars throughout several ages, and did not wish to bore the reader or tax his memory with mathematical abstractions or chains of formal logic diagrams.   Maybe someone inspired by this collection will be motivated to fund such research and translation ...

To have something out of Academia, at least, unknowingly respond to our willy-nilly inquiries, our little elephants in the room ... might qualify as uncanny.  Deeper Understanding would be appreciated, especially for one who goes from one set of [school mathematics] exercises to the next ... it is cool to step outside of the structure and method, an opportunity to reflect upon the foundational genesis of logic, the queen of mathematics --- maybe  the combination of Fourfold Root + both volumes of WWR really does reflect the world as it is, but with the abstract symbols of language, logic, and mathematics.   These are all from the first of the four parts of WWRv1.   Hmmmm ...

I had definitely paid much more attention, especially when going back over (and over) ideas pertaining to salvation, to Book Four, on Ethics, so this book will challenge me to reflect upon the very parts of Schopenhauer's system which I may have paid least attention to, due to the literary slander against him in this respect from the early 1900's.  They were bad-mouthing him, Schopenhauer, that is, making him out to be mathematically naive or immature, rudimentary, even.  Ha!   Without logic, there is no mathematics or philosophy. (opinion or fact? I am not certain.)

The World as Will is Blind Appetite, Hunger, Thirst, Lust ... It is what it is.  Chaotic, irrational, absurd.   The world of logic is an abstract world of symbols, but we take it all so much for granted that we don't appreciate that Schopenhauer's quest for an intuitive understanding of the Thing-in-itself is what he was attempting to shoot into the future for the benefit of mankind.  He possessed the confidence to create a kind of "holy book."  He would often mention the realm of dreams, where the most abstract sorts of meaning and logical relations might be represented by spatial relations.

I suppose this editor is going to be "taking us to school."  That is, it has the potential to ignite, not only more Schopenhauer scholarship, but also may breathe new life into those parts of Schopenhauer's system that place knowledge in the realm of intuition.

I am reaching an age where forced friendships reveal themselves as such, and rather than cursing Fate, I may simply acknowledge the nature of this world, the nature of other human beings, the unpleasantness of my own moody and irritable Thingy-in-Itselfy.

No One Knows The Big News :
we ourselves are the dark light



This type of scholarship is the type of material I wish to spend my energy on, not on people I feel obligated to because of past history.    When we walk in the woods with others, there comes a time in the middle of the night when we wander off our separate ways.  Maybe the mental world of ideas is similar to those woods in that, well, when it comes to thinking about ideas or the very structure of our mental faculties, we have no other choice but to think for ourselves.  No one can do your thinking for you, and, truth be told, as Ligotti points out in the song, No One Knows the Big News, not many others are concerned with the big news (in the world of ideas).  Their heads are just too heavy with so many plans and schemes, thousands of tasks that will not allow them to focus on something that is so strange, anything that is so uncertain.  They have no time to confront some ultimate revelation.  They have no desire to find out some incredibly big news.  Such a thing would take everything they know and arrange it in another way altogether, telling a story so different than the one that is already familiar to them.  No one knows the big news.  Yet the big news is always there.

In a thread called,
Arctangent Without a Calculator ...Why Bother?,

Mathematical reasoning, Schopenhauer argued, is fundamentally different from ordinary logical or syllogistic reasoning in being based on intuition or construction, not on deduction from premises to conclusion; and accordingly Schopenhauer advocated the revision of Euclid, who, he believed, mixes the genuinely geometrical with the spurious logical proof. Schopenhauer even offered specimens of the right kind of proof. While the idea is interesting, I feel daunted by complexity of the problem S. has raised.Do you think his work on the logical foundations of mathematics has any value to-day?

I think that, if I invest a little time at the end of each night to read through, reread, and take notes from the recently mentioned collection, I will be able to respond in a more intelligible manner.  Your question is a great motivation for zooming in on those particular parts of his work, as well as being on the look out for the rare scholarship going on in this area.