as we know that $$ \gamma = \lim_{n \to \infty} \left( \sum_{k=1}^n \frac{1}{k} - \ln(n) \right) $$ And $$ \lim_{n \to \infty} \ln(n) = 1 $$ from L'H rule. So can I say that $$ \gamma = \sum_{k=1}^\infty \frac{1}{k} -1 $$ ? Please tell me and correct me if I am wrong.
the existence of that first limit you wrote out very much depends on a kind of 'cancellation' occuring between sum k=1..n 1/k and ln(n), both of which individually go to +infty
its pretty hairy in general, particularly if n is highly composite. oeis.org/A000001
for entrance exam questions, n will not be highly composite, and you'll probably use sylow theorems for the (small) number of prime divisors of the order to figure out what is going on.
this is nothing resembling the application of general techniques, as much as it is, "oh, here's a nontrivial application of some introductory group theory."
I don't know about the Sylow Theorems.
But I have been wondering about a proof of the fact that a group or order $pq$ where $p$ and $q$ are distinct primes must be cyclic. I can't quite work out the details, but here is the general idea. I would like help with filling in details. I assume that i...
in particular, i don't know of any general ways of counting the number of groups of a given order up to isomorphism, that is any easier (in general) than classifying the groups of a given order up to isomorphism (i.e., actually identifying the possible group structures)
one popular family of exam problems is to ask about some order where, even though the order is not prime, it can be shown that any group of that order is abelian. (so you can count using the fundamental theorem, although without some work, it wouldn't be clear that you're counting all groups)
This problem is not homework but, I was stuck to it when I reviewed the Sylow theorems and problems. I am really interested of finding a test in which we can examine whether a finite group of certain order is abelian or not. It tells:
$G$ is a finite group of order $p^2q$ wherein $p$ and $q$ ...
i can't think of any general approach to uniform convergence. again, if it's given on an exam where you're expected to be able to do it on limited time, it's because there's extra information that makes it possible.
hhaha, fun fact, 99% of all groups of order less than 2000 have order 1024. (use a uniform distribution on the set of isomorphism classes of such groups)
there are 49487365422 groups of order 1024, and most n in the neighborhood of 1024 have much smaller numbers of groups of that order
ChatGPT can now talk to Wolfram | Alpha. It's still needs more training, but it already appears to be quite usable. writings.stephenwolfram.com/2023/03/…
Yeah but also... none of that stuff requires an AI to do. AI training is just lazy algorithms research :P
Literally just building a tree from a rather inefficient set of "learning" (rather, optimization) algorithms.
Also, we now observe just how absurd a world we live in when we abandon common sense philosophy! You guys hear that the US courts retracted the finding that a woman's comic's AI-generated images don't have copyright?
Guess who pushed the button to make that AI generate those images!
well it is often done in industrial processes, owners of the mean of productions get attributed the creation of something
again it's not clear cut, a brush doesn't have any sort of special license, it's probably a commodity classified specifically so the ownership is given to whoever buys it
False comparison. Those who do work for hire are merely secondary causes. It is the one who commands or commissions a work to be done who is rightly and responsibly called the creator of that thing.
In such case, the people are merely a brush.
@shintuku You're looking too much at the subjective circumstances. You need to think objectively.
abstracting from these specific, concrete circumstances creates a false impression that chatgpt is similar to a brush (or people are similar to a brush in certain circumstances)
sure, but why wouldn't they be partially incarnate in, e.g., chatgpt, or Steve Jobs?
if i grant you perfect essences, why would they need to remain pure in incarnate reality? why couldn't real objects have part creatorship, and part toolship
Because truth is merely a reflection of that which exists. Essence belongs to truth, so it itself is accidental to the object which has it despite the essence being that which makes an object to be what it is.
Formally, truth is defined as the conformity of subject to object.
Objects are perfectly capable of having many essences, but you can always represent that object as a single essence with those other essences as metaphysical parts.
If a tool is capable of being a creator, then we arrive at a metaphysical contradiction, viz., that nothing causes itself, therefore a tool, or more generally a means, cannot be a primary cause in itself. A creator by definition must be a primary cause. The product of the creator is a final cause (effect).
similarly, creatorship in the woman using chatgpt, i would argue it is at least not pure
@AMDG i understand, but why can't the essences alternate? no need for the essences to be manifest simultaneously, why couldn't they be manifest alternately?
also, chatgpt is a program, why couldn't certain sections of it manifest pure toolship, and others, manifest pure creatorship?
by alternate, i meant that chatgpt is, at one time, pure toolship, and at another time, pure creatorship, and alternating like that. what excludes this possibility?
by manifest, i meant the same thing as incarnation
@shintuku Well the nature of essence is such that the essence of a thing is unchangeable.
It cannot at one time be a creator, then suddenly be a tool, then back again.
Otherwise, for that to occur, some substantial change must occur such that the thing ceases to be one essence and becomes another, but that requires, again, an agent other than the thing itself being changed.
@shintuku It would be better if you just used already established terminology: just say "having actuality".
An essence itself is said to be in potentiality, and a potentiality is anything that is capable of existing but does not have existibility in itself.
@shintuku This question is answered by the more general answer given here, but regarding the specific essence program here, a program is a species of software which consists of action and chronology where chronology is the time in which action is to occur.
A program differs from software in that software itself consists generally of action and chronology, but a program is unique in that its actions are mathematical.
ChatGPT isn't just a simple passive tool. Using it to (for example) create an essay isn't the same as writing an essay using a word processor with built-in spelling and grammar checkers. ChatGPT is generating content based on its vast training data. True, it merely reconstructs content that's statistically consistent with its training data. OTOH, it can easily create sequences that never existed in that data.
well as programming code, chatgpt is a highly modular entity: why can't some parts of it act as agent, causing other parts of it to switch essence between toolship and creatorship?
Ok, but they're just objects, and objects have essence. You can change the object to get a change in essence, but you cannot change the essence and get a change in object per se unless you are God as is the case with transubstantiation where the matter is untouched, but the form is changed directly by God, hence the accidents which remain, and hence why the substance ceases to be when the accidents dissolve because matter gives being to form in substantial beings.
ok, but i don't see why chatgpt can't be an aggregate of objects, one of them having agenthood, the others having toolness and creatorship. i'm not asking for the transformation of any essence, they remain intact
Once upon a time, in a dense jungle, there lived a “stochastic parrot.” This parrot was unlike any other bird in the jungle because it could generate random phrases and sentences that would often confuse and surprise the other animals.
Despite its unique abilities, the stochastic parrot often felt out of place in the jungle. It longed to be a real parrot that could communicate with others and fly through the sky.
One day, the stochastic parrot overheard a group of real parrots singing a beautiful melody in the distance. Enchanted by the sound, the stochastic parrot decided to approach them.
Consider that in the case where a commissioner commissions a product by many people doing work for hire, those people are indeed agents because they are the immediate cause of the work, but the primary cause is the commissioner.
In this same sense, you could say chatgpt is an agent insofar as it is the proximate cause of the final cause, but it is nevertheless the secondary cause only. Man is the primary cause (usually).
@shintuku Chatgpt et al. cannot act on its own: neither itself as a program (it needs an executor, viz., a computer), nor can the computer (a computer does not power itself on, nor does it assemble itself--a (usually) human cause manufactures the computer, assembles it, then configures it to be on or off).
I'm not saying that chatgpt et al. is only a tool. I'm merely stating that it is a tool.
Whether or not it is only a tool is determined by what makes it... to be what it is, i.e., its essence.
I've given you here what chatgpt is at least according to the OpenAI researchers (a program), so that is its essence.
Considered in the context of creatorship, it must be a tool only because it is incapable of being the primary cause of some work: it requires some other cause to make it act.
@PM2Ring Here, the only difference is that I had more choice in the matter of writing by way of my hands as the tool rather than a program doing most of the work.
i see what you're saying, but i'm not seeing how you can infer, or see something obvious about, the law case mentioned above if chatgpt is not only a tool
i would understand how you would see that is obvious if chatgpt was nothing but a tool
Well, it's a specification of action and chronology, and has no use outside of that. It exists as a means to an end. A tool is a type of means, a means to bring about some particular purpose, so yes, it is just to say that a software is also identical with the essence of tool.
The core GPT code is relatively simple and small. However, ChatGPT isn't just its software: it's the software plus the data that it created by transforming its training data.
@shintuku If I steal a brush, the brush is not mine, but as the creator of a work that I would make with the brush, the work is mine. Likewise, I do not own chatgpt, but all reason concludes likewise that the work it produces is mine unless the OpenAI licensors are merely permitting you to view and use the works it generates. I would argue this to be just since this is their service, not ours; if you had your own instance, then it would be clear that you are wholly the origin.
Thankfully, however, this actually doesn't matter for the question at hand. The question is whether or not the AI itself made the work, or the one who made the AI to work.
The courts incorrectly concluded the AI is the creator.
A carpenter has free will to choose and to act, and God gives force to the carpenter as an agent to be able to do so. The AI lacks this perfection, hence, it is not moved by God first in the chain of causes: it must be second.
@AMDG So who is the creator of that "stochastic parrot" story? ChatGPT wrote it in response to Lyle Anderson's prompt "how about a fairy tale about a “stochastic parrot” that grows up to be a real parrot at the end of the story?" However, there may have been a previous interchange between Lyle & ChatGPT in that session which influenced the story.
@PM2Ring "may have" lol dude, ChatGPT does nothing of itself.
;;;;;;;;;;)
@shintuku so case in point here... chatgpt must be a secondary cause only because the poem would not have been written had Lyle Anderson not made chatgpt to create it.
@AMDG why does chatgpt not have the free will to act on, e.g., its outputs? i understand in what sense you might say humans have free will, but i don't see why, mechanically, in some areas of its being, it might not manifest free will
Free will by definition requires the capacity to act contrary to what is necessary. A computer is wholly deterministic and for that reason lacks free will.
In fact, that is how one proves that the Infinite Cause, God, has free will.
By reason of our existing, and being unnecessary.
Specifically, necessity here refers to a kind of being.
Necessary being is that being which must exist "or else".
@AMDG You give it a prompt and it responds. I showed you the immediate prompt, but there may have been previous dialogue in that session. ChatGPT-3.5 generates the next token in response to all previous tokens in the current session, limited to a window size of 2048 tokens.
i'm not convinced chatgpt is not like the human brain, but with lower complexity. so why would chatgpt not be able to act contrary to what is necessary, but a human person could?
@PM2Ring Well, to be fair, you were referring to how his interaction may have influenced the story itself, but I'm just trying to prove a point. My bad.
@shintuku The human brain merely stores sense knowledge. It is the spirit which knows and understands essences.
ChatGPT can do everything a human brain can, but it cannot do everything a human spirit can, so by way of a human agent, ChatGPT can have the appearance of free will, etc.
@shintuku Yes, but the spirit is dependent on the brain for its operation.
@shintuku GPT operates deterministically, but it does use random numbers Those numbers may be generated by a pseudorandom sequence, but they may be seeded by a genuine entropy source.
@shintuku Knowledge is immaterial, therefore an immaterial cause must exist for man to understand knowledge. This immaterial cause we call man's spirit.
The brain is wholly material, and brute animals also have brains, but they do not understand, whereas man does understand, therefore man must have some immaterial cause in him. This immaterial cause is man's spirit.
Exactly. At zero temperature, the output token selection is totally deterministic, but as you increase the temperature there's more use of random numbers. However, in many situations, there's an overwhelmingly high probability for only one token that fits, so the temperature won't have much effect.
@AMDG these are all very non-obvious statements about what knowledge is, at least to me. i don't understand where they come from, i don't see why they should have these special properties
i would understand if they are grounded in faith, but otherwise i'm not seeing their obviousness
Because it continues to exist in spite of a material object being destroyed.
If such immaterial entities could not exist, then necessarily nothing could be made, and our knowledge of a thing would have to cease as the objects themselves cease.
For any object that could cease to be, we could imagine it again even after forgetting it.
@shintuku Well I should be saying truth continues after the knower disappears. For some reason we've all developed the habit of using the two terms as synonyms. I will do my best, therefore, to say truth rather than knowledge for the sake of clarity.
It is necessary being because without it, well, knowledge is awareness of the truth. If truth didn't exist beyond the minds of creatures, well nothing could exist at all. This would be equivalent to saying that no objects have essence. All reduces to absurdity, so this is sufficient to prove the necessity of truth being eternal, i.e., without beginning and without end.
Like you'd have to say that an object is what it is not: that it has no whatness; no definition; yet somehow exists.
Because truth is immaterial. Can you see how the same can be said of words themselves? The essence attached to a word (by way of the word's definition) is accidentally within the word, but not constituted by it, and the word could just as easily change definition (as happens often enough throughout time in human history).
@AMDG "ChatGPT can do everything a human brain can" Not really. It can manipulate language, but its grasp of concepts is rather tenuous. And it has no mechanism to evaluate the truth of its utterances (or its prompts) beyond their statistical likelihood relative to the weights it computed during training.
Put a different way, if truth were wholly contingent on objects themselves and not that objects are contingent on truth, then once again, we would arrive at the absurd notion that an object can exist, but then suddenly should cease if forgotten.
@PM2Ring Well brains don't grasp concepts. Intellects do.
@AMDG i'm not seeing why materiality implies truth wholly depends on objects. why is it not a process between the brain of a knower and the object as it is represented in the knower's brain?
i'm not seeing why objects stop existing if forgotten, either, on the basis of the materiality of truth
right, that is the opposite of what you are saying. I don't understand why, if truth is material, it implies that case. I don't understand why you're saying truth must be immaterial on the basis that, if it was material, then objects would stop existing if forgotten
@AMDG yeah, I'm not understanding why truth isn't a phenomenon that happens in our brains. I don't understand why you're saying that, if it was, objects would stop existing if forgotten
Maybe we should step back from the philosophy and spirituality, and get a little mathematical... Eg, Euclid proved that there's no largest prime, so there's certainly an n such n is the smallest prime >10^1000000. We might never be able to compute it, but we can calculate lower & upper bounds for it. I'm happy to say that n exists, despite nobody knowing its exact value.
@shintuku Because truth once again is conformity of subject to object. Put differently, truth is all that is possible; all that is capable of existing. To say that truth is merely a phenomenon that happens in our brains would be equivalent to saying that reality depends on us individually thinking it into being: that we invent reality, rather than that reality is something imposed on our intellects and subjectively integrated into our minds without destroying the truth contained therein.
Hence why I say that objects would have to cease to exist if forgotten because truth and reality go hand in hand. They must both exist simultaneously.
As soon as something exists, already the truth of that thing must have existed first.
@shintuku Propositions are attributed true or false, and that a proposition is true belongs to truth; and that a proposition belongs to falsity also belongs to truth.
The reality is imposed on us by the extrinsic reality, and not interpreted.
You immediately intuit the sense of a thing. That is the nature of sense knowledge.
I don't evaluate the background of this chat as being white and blue. My eyes and then my spirit tell me that it is white and blue by its very existing.
"Yeah, oh no guys! He's gonna challenge my subjective beliefs! How dare he talk about the most important question of our existence and attempt to persuade us of the truth!"
I get that a religious context might make you feel entitled to convince someone that they are wrong, but usually these conversations are best handled from a secular viewpoint instead of axiomatizing something on a god that not everyone agrees exists (let alone agrees who it is).
Well because the material objects are being, not truth, yet the truth is something which the object says of itself so to speak, viz., its essence.
All such material objects having such same essence have something in common yet are not the same material being, so we must state that there is some immaterial being there by necessity, though not necessarily in the literal sense of something like a rock having an inanimate spirit per se.
Because otherwise the knower would be inventing the essence as opposed to the object communicating its being to a knower. A knower, by nature of truth, must necessarily receive truth, and that truth has to come from somewhere, and it cannot come from anything but the object itself.
@shintuku Well you see the funny thing about what you just said is the implication that religion pertains to truth and reality, not subjective belief, and that is precisely so. Since my religion conforms to reality, it is therefore true, but proving that is non-trivial.
i don't understand why the objects can't transmit their being, and then we judge truth with respect to our sense-perceptions of what we receive from their being
otherwise, i don't see a difference between truth and being
It's a subtle difference. Truth is that which would be the object itself were the object not to exist is the easiest way I can think of explaining what I'm trying to say.
I mean you can take a picture of a thing and then destroy the thing.
Supposing sufficient information were present, you could reconstruct it from the picture.
The only mental thing about it (LOL) is the representation of the object. The truth of the object exists accidentally within the representation in much the same way the digital camera represents a set of points with values representing colors to represent an image.
well right, but I'm not seeing where this 'truth' is, beyond true knowledge, or being itself
you're positing it is something else, I'm not seeing how. if the truth of the object exists accidentally within the representation, where is it beyond here, unless you say it is being? that's what you're positing, if I understand you, and I'm not seeing what is distinguishing it
@shintuku The essence, ultimately, of the thing. I mean truth is a form of being, but not as being in itself. It requires reality (an object) for its existence, but in order for that to be so, then that requires some cause greater than the immediate reality (the object) which is capable of being both created and destroyed: something in that changeable object is unchangeable. The unchangeable part is the truth of that object, and a kind of being which is distinct from the object itself.
Can anyone please help me with this question:( choose the correct option) The inequality $\sqrt{x+6}\geq 6$ is satisfied for $x$ if and only if A. $-3\leq x\leq 3$, B. A circle and a pair of straight lines C. -6<=x<=3, D. 0<=x<=6 .
My solution goes like this: given, $\sqrt{x+6}\geq x$. If $x>=00$ then we can square both sides, and thus, we get, $x^2-x-6\leq 0$. Solving this inequality we get, $-2<=x<=3$ . But as $x\geq 0$, so we have $ 0\leq x\leq 3$
But this $ 0\leq x\leq 3$ is nowhere given in the option.
@shintuku Truth consists of two species: that which is possible, and that which is not possible. This is not an equation of truth and falsity. It means that one can be aware of, or have knowledge of, that which is possible, and that which is not possible.
Can anyone please help me with this question:( choose the correct option) The inequality $\sqrt{x+6}\geq x$ is satisfied for $x$ if and only if A. $-3\leq x\leq 3$, B. A circle and a pair of straight lines C. -2<=x<=3, D. 0<=x<=6 .
My solution goes like this: given, $\sqrt{x+6}\geq x$. If $x>=00$ then we can square both sides, and thus, we get, $x^2-x-6\leq 0$. Solving this inequality we get, $-2<=x<=3$ . But as $x\geq 0$, so we have $ 0\leq x\leq 3$
But this $ 0\leq x\leq 3$ is nowhere given in the option.
Note that if $x=-6$, the inequality is satisfied ($0 \geq -6$). But none of the answers encompass this solution. I don't know what the circle/lines answer alludes to.
This problem is not homework but, I was stuck to it when I reviewed the Sylow theorems and problems. I am really interested of finding a test in which we can examine whether a finite group of certain order is abelian or not. It tells:
$G$ is a finite group of order $p^2q$ wherein $p$ and $q$ ...
