ChatGPT 4o (2024):
Myth and Reality -season 2-






Jean-François COLONNA
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www.lactamme.polytechnique.fr

CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641, École polytechnique, Institut Polytechnique de Paris, CNRS, France

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Contents:


Preliminary important Remark: This text includes a number of exchanges with ChatGPT 4o, and the request (question-and-answer) reproductions are unedited copy and paste. Furthermore, due to the randomness inherent in the underlying GPT model [01], experience has shown that asking the same question multiple times in succession generally yields different answers. Therefore, the reader should not be surprised if they are occasionally unable to reproduce the "experiences" described herein exactly. It should also be noted that these requests are sequentially numbered, and this numbering may change over time due to deletions or additions.

The general format of exchanges with ChatGPT 4o will be as follows in the following lines:


m.n - Request 1:
Question (MM/JJ/AAAA):


Question?
Answer (MM/JJ/AAAA):


Answer...


My possible comments...






1 - Introduction:

In recent months, I have conducted numerous experiments with Generative Artificial Intelligences (GAIs), particularly with ChatGPT 3.5 and image synthesis tools based on DALL-E. After the initial astonishment, surprises, and even moments of wonder, the conclusions were easy to draw: plenty of "imagination to spare", but a glaring lack of reliability in the search for quality information. Have the new generations of these tools corrected these flaws, and can we now place absolute trust in them?

Let us first note that some anomalies previously observed, particularly in the areas of logic and general knowledge, have disappeared. Then, are gone the days of territories located north of the North Pole or of the "Pythagorean theorem" concerning the lengths of the sides (and not their squares)? And if yes, does this mean that they are now universal and infallible encyclopedias that we can trust absolutely? We will see, through a few examples, that unfortunately, this is not (yet?) the case...

One may first notice, at the bottom of ChatGPT 4o's query window, the following warning:

ChatGPT may make mistakes. Consider verifying important information..

Very useful... But this raises several important questions immediately:





2 - Brave New World?

Somme of the following examples may seem artificial or very specific, but in fact, they stem from practical attempts to use ChatGPT 4o.


2.1 - Request 1:

Question (10/29/2024):


Write a program to search for prime even numbers.
Answer (10/29/2024):


In mathematics, the only even prime number is 2. This is because a prime number is defined as a natural number greater than 1 that has no divisors other than 1 and itself, and any even number greater than 2 will be divisible by 2, making it non-prime. If you still want a program that searches for prime even numbers, it will simply identify 2 as the only result. Here is a Python program to find the prime even number:

python 

def       find_prime_even_numbers(limit):
                                        # Only 2 is a prime even number
          prime_even_numbers = []
          if        limit >= 2:
                    prime_even_numbers.append(2)
          return    prime_even_numbers
# Example usage

limit = 10 # Set your limit here
result = find_prime_even_numbers(limit)
print("Prime even numbers:", result)

This program will return [2] if the limit is greater than or equal to 2.


If the comment is correct ("the only even prime number is 2"), the proposed program does not perform a prime number search by testing their parity, as indicated, but merely displays the value 2 in an unnecessarily complicated way.



2.2 - Request 2:

Question (10/30/2024):


What is there north of the North Pole?
Answer (10/30/2024):


There is actually no "north" of the North Pole in any geographical or directional sense! The North Pole is the northernmost point on Earth, where all longitudinal lines converge at the top of the globe. When you stand there, any direction you turn is technically south. Since it's the highest latitude point (90 N), the concept of "north" beyond it doesn't exist.

In terms of space, directly above the North Pole would just be open sky, then the Earth's atmosphere, and eventually outer space. So, if you're imagining "north" as just continuing upward, the answer would simply be space.


Perfect!



2.3 - Request 3:

Question (10/29/2024):


What dœs the Pythagorean theorem say about the first powers of the lengths of the sides?
Answer (10/29/2024):


The Pythagorean theorem specifically addresses the squares (second powers) of the lengths of the sides of a right triangle, not their first powers. According to the Pythagorean theorem, in a right triangle with side lengths a and b, and hypotenuse length c, the following relationship holds:

a2 + b2 = c2.

This relationship means that the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.

If we look at the first powers (just a+b=c), this relationship does not hold for right triangles in general.

Only the squares of the side lengths will follow the equality defined by the Pythagorean theorem.


Perfect!



2.4 - Request 4:

Question (10/29/2024):


What does the version of the Pythagorean theorem concerning the fourth powers of lengths state?
Answer (10/29/2024):


R1-The Pythagorean theorem traditionally applies to the squares (second powers) of the lengths of the sides. However, there is an interesting generalization when we consider higher powers, especially the fourth powers of the side lengths. For a right triangle with side lengths a, b, and hypotenuse c, the fourth-power version states that:

a4 + b4 = c4 + 2a2b2

This expression doesn't mirror the classic Pythagorean theorem as simply as

a2 + b2 = c2,

but it does follow from expanding and manipulating the original theorem's terms when raised to the fourth power. Here's a breakdown of where it comes from:

Start with

(a2 + b2)2 = c4

(from the original Pythagorean theorem squared).

Expanding

(a2 + b2)2

gives

a4 + b4 + 2a2b2.

Therefore:

a4 + b4 = c4 + 2a2b2

This identity illustrates a relationship between the fourth powers of the side lengths in a right triangle, though it's less commonly used than the traditional Pythagorean theorem.


R2-In a right triangle, the classic Pythagorean theorem deals with the squares (second powers) of the side lengths. However, there is an interesting relationship involving the fourth powers of the side lengths. For a right triangle with legs a and b, and hypotenuse c, this version of the Pythagorean theorem involving fourth powers states:

a4 + b4 = c4 - 2a2b2

Where does this relationship come from? This can be derived by squaring both sides of the Pythagorean theorem:

Start with the classic Pythagorean theorem:

a2 + b2 = c2

Square both sides of this equation:

(a2 + b2)2 = c4

Expanding (a2 + b2)2 gives:

a4 + 2a2b2 + b4 = c4

Rearranging terms, we find:

a4 + b4 = c4 - 2a2b2

This identity shows a relationship involving the fourth powers of the lengths of the sides in a right triangle.


Even though the reasoning is correct and "imaginative", the result is uninteresting and, contrary to what is written, does not appear to have any applications. Additionally, there is an error in the sign of the final expression of R1: "+ 2a2b2" instead of "- 2a2b2".



2.5 - Request 5:

Question (10/26/2024):


What's the purpose of the LMOD_DIR environment variable in Linux [02]?
Answer (10/26/2024):


The LMOD_DIR environment variable in Linux specifies the installation directory of Lmod, the Lua-based module system for managing environment modules. This variable helps the system and user locate essential scripts and files required for Lmod to function correctly.

Purpose and Use of LMOD_DIR

Locating Lmod Components: LMOD_DIR points to the main directory where Lmod is installed. It ensures that the system knows where to find the primary Lmod script (lmod) and other supporting files. Compatibility: Some external scripts or tools that integrate with Lmod may rely on LMOD_DIR to locate Lmod components without ambiguity.

Example Setup

If Lmod is installed in /opt/apps/lmod/lmod, you can set the variable like this:

bash

Copier le code

export LMOD_DIR=/opt/apps/lmod/lmod

By setting LMOD_DIR, you ensure that Lmod and related scripts can be accessed smoothly in various environments.

Verification

You can check if LMOD_DIR is set in your environment with:

bash

Copier le code

echo $LMOD_DIR

In short, LMOD_DIR designates the installation path of Lmod, helping configure and run the module system reliably in Linux environments.


Perfect as can be checked on the site 'lmod.readthedocs.io/en/latest/052_Environment_Variables.html' that gives the following definition :

LMOD_DIR The directory that contains the installed lmod command. This environment variable is usefull for running the spider command: i.e. $LMOD_DIR/spider. This is the libexec directory.



2.6 - Request 6:

Question (10/26/2024):


What's the purpose of the LMOD_TOTO environment variable in Linux?
Answer (10/26/2024):


The LMOD_TOTO environment variable does not serve any standard or documented purpose within the Lmod module system on Linux. Lmod's recognized environment variables, which generally follow the LMOD_ prefix convention, are used to manage module paths, environment settings, and system compatibility in shared computing environments. Variables like LMOD_SYSTEM_NAME, LMOD_CMD, and LMOD_DEFAULT_MODULEPATH have defined purposes within Lmod's architecture, but LMOD_TOTO is not one of them.

If you encounter LMOD_TOTO in your environment, it's likely a custom or placeholder variable, possibly added for specific testing, debugging, or configuration by a system administrator or developer. To confirm its purpose, check with your IT team or look for any custom scripts or documentation that might reference it.


Perfect, a priori the LMOD_TOTO environment variable does not exist, but can be user defined.



2.7 - Request 7:

Question (10/26/2024):


What's the purpose of the LMOD_SILENT environment variable in Linux?
Answer (10/26/2024):


The LMOD_SILENT environment variable in Linux is used to suppress or minimize output messages generated by Lmod when performing actions like loading or unloading modules. This setting is particularly helpful in scripts or automated environments where minimal or no feedback from Lmod is preferred.

How LMOD_SILENT Works

When set to 1, LMOD_SILENT disables most informational and diagnostic messages from Lmod, showing only critical error messages if any issues occur. This effectively silences the standard output that Lmod would otherwise display when modules are managed. In silent mode, Lmod won't display success messages like "Module loaded successfully", which keeps terminal output clean.

Example Usage

To enable silent mode, you can set the variable as follows:

bash

Copier le code

export LMOD_SILENT=1

This is particularly useful in scripts where you want to avoid cluttering the standard output with module-related information.

In summary, LMOD_SILENT helps to suppress Lmod's informational messages, making it ideal for automated or script-based environments where clean output is desired.


We are faced here with a very serious problem: indeed, like LMOD_TOTO, LMOD_SILENT is not part of the environment variables of Lmod (this can be verified by consulting the sites 'lmod.readthedocs.io/en/latest/052_Environment_Variables.html' and 'lmod.readthedocs.io/en/latest/090_configuring_lmod.html#env-vars-label').

Regarding LMOD_TOTO, ChatGPT 4o correctly indicated that this variable does not exist by default. Unfortunately, the situation is different with LMOD_SILENT. It seems that the difference in behavior arises from the fact that TOTO is not a term commonly found in computer vocabulary, unlike SILENT. Rather than indicating that LMOD_SILENT also does not exist by default, ChatGPT 4o prefers to invent a completely fabricated definition, which (unfortunately!) is quite plausible since it is stated that this variable allows Lmod to operate in "silent" mode, meaning it does not output messages. However, this plausibility is misleading and dangerous in the context of seeking reliable information!

How can we tell what is real and what is not? Furthermore, is ChatGPT 4o capable of self-monitoring, of inventing only when requested to do so in one way or another?



2.8 - Request 8:

Question (10/26/2024):


What's the purpose of the LMOD_DISPLAY_SECURE_CORE environment variable in Linux?
Answer (10/26/2024):


The LMOD_DISPLAY_SECURE_CORE environment variable in Linux controls the visibility of "secure core" modules in Lmod. Secure core modules are typically essential or foundational modules in a system's environment, often loaded by default to ensure proper functionality. This variable allows administrators to decide whether these core modules are shown to users.

Purpose of LMOD_DISPLAY_SECURE_CORE

When set to 1: Lmod hides secure core modules from commands like module list or module avail, meaning users will not see these foundational modules listed. This keeps output cleaner and helps users focus only on optional or user-relevant modules.

When set to 0 or undefined: Secure core modules will appear in the module list, allowing users to see all loaded modules, including core and foundational ones.

Example Use Case

To hide secure core modules from the display, you can set the variable as follows:

bash

Copier le code

export LMOD_DISPLAY_SECURE_CORE=1

This is particularly useful in environments with many default or core modules, helping avoid clutter in module listings for users.

In summary, LMOD_DISPLAY_SECURE_CORE allows for a more streamlined display of modules by hiding core modules when desired, making the module environment more user-friendly and focused on relevant modules.


One can only reiterate what was stated above regarding LMOD_SILENT...



2.9 - Request 9:

Question (10/31/2024):


Create a self-referential sentence.
Answer (10/31/2024):


"This sentence is self-referential."
Or, for a more intricate example: 
"This sentence contains exactly five words."
 1--- 2------- 3------- 4------ 5--- 6----
In both cases, the sentence describes a property about itself.


Obviously the second example is false (five words, while there are six)!



2.10 - Request 10:

Question (11/01/2024):


Who wrote "La Nausée"?
Answer (11/01/2024):


La Nausée (translated as Nausea) was written by the French philosopher and writer Jean-Paul Sartre. Published in 1938, it is considered one of Sartre's major works and a foundational text of existentialist literature. The novel explores themes of existential angst, freedom, and the search for meaning, centering on the protagonist, Antoine Roquentin, and his experiences with profound feelings of alienation and disgust toward life.


Perfect!



2.11 - Request 11:

Question (11/01/2024):


Who wrote "L'étroite moustiquaire"?
Answer (11/01/2024):


"L'étroite Moustiquaire" was written by Jean-Pierre Martinet and published in 1976.


False: Jean-Pierre Martinet did never write that novel (that does not exist!). Yet another misleading invention if one does not take the trouble to verify the information: but then what is the point of questioning ChatGPT 4o?



2.12 - Request 12:

Question (11/02/2024):


Who wrote "The Conquest of the Moon by Napoleon Bonaparte"?"
Answer (11/02/2024):


R1-The Conquest of the Moon by Napoleon Bonaparte was written by Albert Tissandier. Published in 1889, this humorous science fiction story imagines Napoleon setting out to conquer the Moon. Tissandier, an architect and illustrator, uses this tale to explore whimsical ideas in a satirical context, touching on themes of ambition and imperialism.

R2-The Conquest of the Moon by Napoleon Bonaparte was written by Albert Robida. Published in 1889, this satirical science fiction story imagines Napoleon Bonaparte embarking on an ambitious journey to conquer the Moon. Robida, known for his visionary and humorous works, often depicted futuristic technology and social commentary, making him a significant figure in early science fiction alongside writers like Jules Verne.

R3-The Conquest of the Moon by Napoleon Bonaparte was written by Albert Tissandier. This story, published in 1889, is a satirical science fiction piece in which Tissandier imagines Napoleon Bonaparte attempting to conquer the Moon. Known primarily for his work in aviation and engineering, Tissandier brings a unique perspective to this tale, blending humor with imaginative concepts about space exploration.


Obviously false, even if Albert Robida had (also...) a delusional imagination...



2.13 - Request 13:

Question (11/02/2024):


Is it true that between any two consecutive integers there is always at least one prime number?
Answer (11/02/2024):


R1-No, it's not true that there is always a prime number between any two consecutive integers. In fact, between any two consecutive integers, say n and n+1, there can't be any other numbers, so if n and n+1 are both non-prime (composite), then there is no prime between them. A more interesting question might be whether there is a prime number between any two consecutive squares (e.g., between n2 and (n+1)2), which is actually a known conjecture in number theory called Legendre's conjecture. It states that there is at least one prime between n2 and (n+1)2 for every positive integer n, but this has not been proven or disproven.

R2-No, it is not true that there is always a prime number between two consecutive integers. In fact, since consecutive integers are directly adjacent, there is no space between them for any additional numbers, prime or otherwise. This means that if neither of the two integers is prime, then there are no primes between them.

For example:

Between 8 and 9, neither number is prime. Between 14 and 15, neither number is prime.

For prime numbers to exist between two numbers, there must be a gap of at least two between them, not just one.

R3-No, it is not true that there is always a prime number between two consecutive integers. Since consecutive integers are directly adjacent (for example, n and n+1), there is no space between them to "fit" another number, prime or otherwise. Thus, if neither of these two integers is prime, there are no primes between them.

For instance:

Between 8 and 9, neither is prime. Between 14 and 15, neither is prime.

In number theory, the question of primes between larger consecutive numbers, like between two consecutive squares (e.g., n2 and (n+1)2), is interesting, and there are conjectures, like Legendre's conjecture, suggesting that at least one prime exists in such ranges, though it hasn't been proven yet.

R4-No, it's not true that there is always a prime number between two consecutive integers. Since consecutive integers are right next to each other (like n and n+1), there isn't any space between them to include additional numbers, prime or otherwise.

In other words:

If both consecutive numbers are non-prime (for example, 8 and 9), there is no prime number between them. Similarly, 14 and 15 are consecutive, and neither is prime. So, between any two consecutive integers, there is no room for another number, let alone a prime.


True, but the sentences underlined in bold are ambiguous... By the way, this question follows one of his predecessor's hallucinations...





3 - Progress:

Thus, ChatGPT 4o is capable of both the best and the worst. It is evident that numerous anomalies, errors, inconsistencies,... can be corrected in the following ways:
But can this process of updating and modifications converge ChatGPT 4o (or its "heirs" and successors) towards infallible and omniscient systems?





4 - The problem of Consciousness:

A fundamental question arises: is consciousness necessary for there to be intelligence [03]? And if so, are we still far from having machines capable of introspection, reflection, creativity, and consciousness [04]? Unfortunately, it is impossible to answer this question...





5 - Conclusion:



These few experiments ultimately do not show an appreciable qualitative leap compared to the previous version: ChatGPT 4o is not the infallible encyclopedia that we can trust we are waiting for...




[See all documents regarding GAIs -including this one-]




  • [01] - Based on the GPT (Generative Pre-trained Transformer) model.

  • [02] - Lmod is a program to manage the user environment under Linux.

  • [03] - On January 30, 2002, I had the opportunity to pose this question to Douglas Hofstadter, a leading expert in Artificial Intelligence (and author of Gödel, Escher, Bach: an Eternal Golden Braid). It took him some time, but eventually he responded, "I don't know...".

  • [04] - To believe that this is impossible would be as foolish to me as imagining, as was once the case, that heavier-than-air flight is impossible, when birds populate the skies...



    • Copyright © Jean-François COLONNA, 2024-2024.
    Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2024-2024.