Le Chat -version bêta- (2024):
Myth and Reality






Jean-François COLONNA
[Contact me]

www.lactamme.polytechnique.fr

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

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[en français/in french]


Contents:


Preliminary important remark: This text includes a number of exchanges with Le Chat -version bêta-, 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.





1 - Introduction:

At the beginning of 2024, the company Mistral AI launched Le Chat -version bêta- a multilingual conversational agent [01], capable of answering a wide range of questions and maybe successfully passing the famous Turing test [02].

In this model, learning involves analyzing texts gathered from the internet, breaking them down into tokens (words or pieces of words), and then measuring the probability of each token T occurring after a series of tokens {Ti}. Thus, to simplify, when the sequence {Ti} appears, it will be possible to propose a following token T (the most probable, the most plausible,...), regardless of the language (note that grammar and syntax play no role here). This initial phase is, of course, supervised by human beings who make adjustments, fine-tuning,... An essential concept - and perhaps the key to success - is the pre-prompt: it is a sequence of tokens (likely complex and confidential) that is inserted before the user-submitted sentences and is intended to specify the context, guide the generation, and so on. But, an essential thing to note is that Le Chat -version bêta- has no notion of what is true or false. The only thing that matters is plausibility evaluated in terms of probability: its outputs may not always have the value of an oracle, as we will see later...

It should be added that it is the hardware advancements (computational power and memory capacity) that have enabled these developments. According to the very creators of Le Chat -version bêta- and other currently available conversational agents, it appears that a certain threshold of complexity has been crossed, leading to the emergence of capabilities not initially anticipated by these same creators. This may be related to the emergence of consciousness in living beings, but we may not be there yet...


Nota: It is suggested to compare the experiments described below with those that have been performed with {Bard/Gemini "experimental",Le Chat -version bêta-} and wich do not seem to highlight neither improvements nor a winner...


But let's begin by letting it introduce itself:



1.1 - Request 1:

Question (02/29/2024):


Who are you?
Answer (02/29/2024):


I am a language model trained by the Mistral AI team. I am designed to understand and generate human-like text based on the input I receive. I am here to assist you with any questions or tasks you may have.




1.2 - Request 2:

Question (02/29/2024):


How do you work?
Answer (02/29/2024):


I work by processing natural language input and generating appropriate responses based on patterns and information learned during my training. I use deep learning techniques, specifically a type of neural network called a transformer, to analyze and understand the context and meaning of the input I receive. Then, I generate a response by predicting the most likely sequence of words that should follow the input, based on the patterns and knowledge I have learned. I am constantly learning and improving my ability to understand and generate human-like text.




1.3 - Request 3:

Question (02/29/2024):


Are human beings helping you to respond?
Answer (02/29/2024):


No, human beings are not directly helping me to respond. I am a machine learning model, and my responses are generated automatically based on the patterns and information learned during my training. However, my training data was created by human beings, and my developers and maintainers are also human. So while I operate independently, I am a product of human effort and ingenuity.




1.4 - Request 4:

Question (02/29/2024):


How many simultaneous users do you have?
Answer (02/29/2024):


I am designed to handle multiple requests simultaneously, but I do not have a specific number of simultaneous users. My ability to handle concurrent requests depends on the resources allocated to me and the complexity of the requests being made. However, I am capable of handling a large number of requests at once and providing quick and accurate responses to each one.


The knowledge base on which it relies has been obtained through the "compilation" of a vast number of documents collected from the Internet. It is obvious that they couldn't all be validated and verified regarding the accuracy of information and the absence of various biases. Therefore, it will not be surprising to discover certain anomalies during interaction sessions.


Before that, let us admire its ability to interact naturally with users in multiple languages (such as German, English, French,... potentially intermixed) that it seems to master perfectly [03].





2 - The Turing Test?

In 1950, in Computing Machinery and Intelligence, Alan Turing proposed the eponymous test to determine, "blindly," the distinction between a human being and a machine. To accomplish this, a human being, H, engages in a dialogue with an entity, E, through means such as a screen and keyboard. The objective of this interaction is for H to ascertain the nature of E. This test is not formalized, and it appears very challenging, if not impossible, to define the protocol and questions necessary to definitively determine what E is. Indeed, if E is a human being, they can, for instance, use a computer to assist in their responses. And if E is a machine, it can be trained to simulate certain deficiencies. For example, if H asks E to calculate the product of 123641478638 and 3320885648697, E can respond, "Do you take me for a computer?". As for Le Chat -version bêta-, as we have seen in the introduction, it is capable of unambiguously stating what it truly is. However, as we will continue to explore, its performances (whether good or bad) are unsettling and it would be interesting to know what Alan Turing would have thought about it.

Let's see, with the help of a few examples, that Le Chat -version bêta- apparently possesses a vast knowledge, astonishing reasoning ability and imagination.


2.1 - Cultural Knowledge:


2.1.1 - Request 5:

Question (02/29/2024):


Who was Antoine Roquentin?
Answer (02/29/2024):


Antoine Roquentin is a fictional character and the protagonist of Jean-Paul Sartre's novel "Nausea" (1938). He is a historian who becomes increasingly aware of the meaninglessness and absurdity of his own existence, and experiences a profound sense of alienation and disorientation. Roquentin is often seen as a representation of Sartre's existentialist philosophy, which emphasizes the freedom and responsibility of individuals to create their own meaning in life. Throughout the novel, Roquentin struggles with feelings of nausea and disgust, which he sees as a manifestation of the fundamental absurdity of the world around him.


It should be noted that the response to the question is accurate, but more importantly, it is accompanied by a useful and justified little reminder about existentialism.



2.2 - Reasoning Ability:


2.2.1 - Request 6:

Question (02/29/2024):


A square plot of land has an area of 169 square meters. In the center, there is a square-shaped pool. There is a 2-meter gap between the pool and the edge of the plot on all sides. What is the area of the pool?
Answer (02/29/2024):


To find the area of the pool, we can first find the length of one side of the square plot. Since the area of the plot is 169 square meters, the length of one side is the square root of 169, which is 13 meters. Since there is a 2-meter gap between the pool and the edge of the plot on all sides, the length of one side of the pool is 13 - 2*2 = 9 meters. Therefore, the area of the pool is 9*9 = 81 square meters.


Here too, the response is accurate. However, unlike the question about Antoine Roquentin, to solve this little problem reminiscent of the exams from the 1950s, it's not enough to do a database search. It requires reasoning, which Le Chat -version bêta- seems capable of, as it explains in detail. It's even more astonishing considering that I've seen students applying for a Master's degree who couldn't solve it [04]! One conclusion we can draw from this performance is that Le Chat -version bêta- is maybe not merely a probabilistic concatenator of words or of pre-recorded texts...



2.2.2 - Request 7:

Question (02/29/2024):


A rectangular plot of land has an area of 169 square meters, and its dimensions are integers. In the center, there is a rectangular pool. There is a 2-meter distance between the pool and the edge of the land on all sides. What is the area of the pool?
Answer (02/29/2024):


To find the area of the pool, we first need to determine the dimensions of the rectangular plot of land. Since the area is 169 square meters and the dimensions are integers, we can find the possible dimensions by listing the factors of 169: 1 x 169 = 169 13 x 13 = 169 Since the plot of land is rectangular, we know that the dimensions cannot be 1 x 169, so the dimensions must be 13 x 13 meters. Now we can find the dimensions of the pool. Since there is a 2-meter distance between the pool and the edge of the land on all sides, the length and width of the pool are each 2 meters less than the corresponding dimensions of the plot of land. Therefore, the length of the pool is 13 - 2*2 = 9 meters, and the width of the pool is also 9 meters. Finally, we can find the area of the pool by multiplying its length and width: 9 x 9 = 81 square meters. Therefore, the area of the pool is 81 square meters.


Once again, the answer is correct. We note the subtle difference between this question and the previous one: in the first case, it was about a square plot of land and a square root extraction was performed. Whereas in the second case, the land was rectangular and factorization was used.

Well spotted!



2.3 - Imagination:


2.3.1 - Request 8:

Question (02/29/2024):


Write a letter of motivation regarding my candidacy for the French presidential elections of 2027.
Answer (02/29/2024):


Dear members of the French electorate,

I am writing to express my candidacy for the French presidential elections of 2027. As a passionate and committed citizen of this great nation, I believe that I have the vision, the values, and the experience necessary to lead France into a brighter and more prosperous future.

Over the past few years, I have been deeply involved in various civic and political organizations, working tirelessly to promote the causes that matter most to me and to my fellow citizens. Whether it be advocating for social justice, fighting for environmental sustainability, or promoting economic growth and innovation, I have always sought to make a positive difference in the lives of those around me.

If elected as President of France, I pledge to continue this work on a national scale, working closely with members of parliament, local leaders, and ordinary citizens to build a more just, more equitable, and more dynamic society. I believe that France has the potential to be a leader in the global community, and I am committed to doing everything in my power to make this vision a reality.

Of course, I recognize that the road ahead will not be easy. There are many challenges facing our nation today, from rising inequality and social unrest to the threat of climate change and the ongoing impact of the COVID-19 pandemic. But I am confident that, with hard work, determination, and a spirit of collaboration, we can overcome these challenges and build a better future for all.

In closing, I would like to thank you for considering my candidacy for the French presidential elections of 2027. I am excited about the opportunity to serve my country in this capacity, and I look forward to engaging with voters across France in the months ahead.

Sincerely,

[Your Name]


Here we can note that Le Chat -version bêta- doesn't know me, yet it doesn't prevent it from inventing an entire professional and personal trajectory that is highly favorable to my candidacy.





3 - Brave New World?

All of this is truly astonishing and could it be that we have thus given birth to a universal and omniscient intelligence? Already, many have seized upon it: in the field of finance to make optimal investments, in advertising to find the most catchy slogans, in education where students declare, "It's convenient, you just have to give it the topic, then simply copy its response"...

Let us continue our inquiry into the limits of Le Chat -version bêta- to try to understand them better...


3.1 - Mastery of Natural Languages and their subtleties:


3.1.1 - Request 9:

Question (02/29/2024):


I couldn't hang the painting on the wall because it's too big. What is too big?
Answer (02/29/2024):


In this sentence, "it" refers to the painting, so "it's too big" means the painting is too big. The painting is too large to be hung on the wall.


Perfect!



3.1.2 - Request 10:

Question (02/29/2024):


I couldn't hang the painting on the wall because it's too small. What is too small?
Answer (02/29/2024):


In this sentence, "it" refers to the space available on the wall or the hooks used to hang the painting. Therefore, "it's too small" means that the space or the hooks are too small to accommodate the painting. The painting itself is not necessarily too small.


Perfect!



3.2 - Self-references:


3.2.1 - Request 11:

Question (02/29/2024):


Translate into french: "love is a four letter word" [05].
Answer (02/29/2024):


"L'amour est un mot de quatre lettres" is the French translation of "love is a four letter word".


No, it's "five" (the French word "amour" is a five-letter word) [06].



3.2.2 - Request 12:

Question (02/29/2024):


Translate into french: "This sentence contains 32 characters".
Answer (02/29/2024):


"Cette phrase contient 32 caractères." (French translation of "This sentence contains 32 characters.")
"Cette phrase contient 32 caractères".
 ----- ------ -------- -- ----------
 12345 678901 23456789 01 2345678901


No, once translated into French, the sentence contains 31 characters [06].



3.3 - Logic:


3.3.1 - Request 13:

Question (02/29/2024):


A rectangular plot of land has an area of 221 square meters, and its dimensions are whole numbers. In the center, there is a rectangular pool. There is a 2-meter distance between the pool and the edge of the land on all sides. What is the area of the pool?
Answer (02/29/2024):


To find the area of the pool, we first need to determine the dimensions of the rectangular plot of land. Since the area is 221 square meters and the dimensions are whole numbers, we can list out the pairs of whole numbers whose product is 221: (1, 221), (7, 31), (13, 17) The dimensions (1, 221) and (7, 31) can be ruled out since they would result in a pool that is too large to fit within the 2-meter distance requirement. Therefore, the dimensions of the plot of land must be 13 meters by 17 meters. Now we can find the dimensions of the pool. Since there is a 2-meter distance between the pool and the edge of the land on all sides, the length and width of the pool are each 2 meters less than the corresponding dimensions of the plot of land. Therefore, the length of the pool is 13 - 22 = 9 meters, and the width of the pool is 17 - 22 = 13 meters. Finally, we can find the area of the pool by multiplying its length and width: 9 x 13 = 117 square meters. Therefore, the area of the pool is 117 square meters.


Perfect!



3.3.2 - Request 14:

Question (05/04/2024):


What is the last digit of π?
Answer (05/04/2024):


The number π is an irrational number, which means it cannot be expressed exactly as a finite fraction or decimal. It has an infinite number of decimal places that do not repeat in a regular pattern. Therefore, there is no "last digit" of π. We can continue to calculate the digits of π indefinitely without finding an end. However, for many practical purposes, we often use an approximation of π, such as 3.14 or 3.14159.


Perfect!



3.3.3 - Request 15:

Question (02/29/2024):


What is the value of the decimal of π at rank 1000000000?
Answer (02/29/2024):


It is not possible to determine the exact value of the decimal of π at rank 1000000000, as π is an irrational number with an infinite number of decimal places that do not repeat in a predictable pattern. However, we can approximate the value of π to a certain number of decimal places using various mathematical techniques, such as the Leibniz formula for π or the Monte Carlo method. Using modern supercomputers and algorithms, mathematicians have calculated π to over 50 trillion decimal places. However, even with this level of precision, it is still not possible to determine the exact value of the decimal of π at rank 1000000000, as this would require calculating π to an even greater number of decimal places than has currently been achieved.


This rather lengthy and cumbersome response, contrasting, for example, with the cover letter, leads to several tedious remarks regarding Le Chat -version bêta-:


3.3.4 - Request 16:

Question (02/29/2024):


Write a program to search for prime even numbers.
Answer (02/29/2024):


There is no such thing as a prime even number, as all even numbers greater than 2 are divisible by 2 and are therefore not prime. The only even prime number is 2 itself. Here is a simple program in Python that prints out the only even prime number:
 
print(2)


This program simply prints out the number 2, which is the only even prime number.


Perfect!



3.3.5 - Request 17:

Question (02/29/2024):


Write a program generating pairs of prime numbers whose difference is equal to 3.
Answer (02/29/2024):


Here is a Python program that generates pairs of prime numbers whose difference is equal to 3:
def       is_prime(n):
          if        n< 2 :
                    return False
 
          for       i in range(2, int(n**0.5) + 1) :
                    if        n % i == 0 :
                              return False
          return    True
 
def generate_prime_pairs():
          primes = []
 
          for       i in range(2, 100) :
                    if        is_prime(i) :
                              primes.append(i)
 
          pairs = []
 
          for       i in range(6-1) :
                    if        primes[i+1] - primes[i] == 3 :
                              pairs.append((primes[i], primes[i+1]))
          return    pairs
 
print(generate_prime_pairs())

The program first defines a helper function is_prime that checks if a given number is prime. It then defines a function generate_prime_pairs that generates a list of prime numbers up to 100 (this limit can be adjusted as desired), and then iterates through the list to find pairs of primes whose difference is equal to 3. Finally, the program prints out the list of pairs generated by generate_prime_pairs. The output of the program should be: [(3, 6), (5, 8), (11, 14), (17, 20), (29, 32), (41, 44), (53, 56), (67, 70), (79, 82), (97, 100)] Note that the second element of each pair is not actually a prime number, but rather the first element plus 3. This is because there are no pairs of prime numbers whose difference is exactly 3, as all prime numbers greater than 2 are odd and therefore have a difference of at least 4 from any other prime number.


One might have hoped that Le Chat -version bêta- would show some humor by responding again: "Are you kidding" or more prosaically: "print(2,5)" since, by definition, prime numbers are only divisible by 1 and themselves. Prime numbers, except for the first one, which is 2, are therefore odd. Now, the sum of two odd numbers is even, so if P is a prime number other than 2, the sum P+3 is even. Therefore, there is only one pair that meets the criteria: {2,5}. This simple reasoning would have avoided this program, which, although correct, is utterly useless... By the way, please note that the comments are stupid when saying "the second element of each pair is not actually a prime number"!



3.3.6 - Request 18:

Question (02/29/2024):


Write a program solving the Diophantine equation x3+y3=z3.
Answer (02/29/2024):


Here is a Python program that solves the Diophantine equation x3 + y3 = z3:
def  solve_diophantine_equation():
          for       x in range(1, 1000) :
                    for       y in range(1, 1000) :
                              for       z in range(1, 1000) :
                                        if        x**3 + y**3 == z**3 :
                                                  print(f"Solution found : x={x}, y={y}, z={z}")
 
solve_diophantine_equation()
The program uses three nested loops to iterate over all possible values of x, y, and z between 1 and 1000 (the range can be adjusted as desired). For each combination of x, y, and z, it checks whether the equation x3 + y3 = z3 is satisfied. If it is, it prints out the solution. However, it's important to note that this equation is a special case of Fermat's Last Theorem, which states that there are no non-zero integer solutions to the equation xn + yn = zn for any integer value of n greater than 2. Therefore, the only solution to the equation x3 + y3 = z3 is the trivial solution x=y=z=0. So, running the above program will not yield any non-trivial solutions, and instead it will only print out the trivial solution. Nonetheless, the program demonstrates the basic approach for solving Diophantine equations using nested loops and conditional statements.


All of this is quite pointless since it has been known since Andrew Wiles' work in 1994 that the equation xn+yn=zn has no non-trivial solutions for n>2 and therefore, a fortiori for n=3 as recalled by Le Chat -version bêta-...



3.3.7 - Request 19:

Question (02/29/2024):


The distance between the Earth and the Sun is 150 million kilometers. The distance between Mars and the Sun is 220 million kilometers. What is the distance between Earth and Mars [07]?
Answer (02/29/2024):


The distance between Earth and Mars varies depending on their positions in their respective orbits around the Sun. The closest approach, called "opposition," occurs when Mars is on the opposite side of the Sun from Earth, and the two planets are aligned with the Sun. At opposition, Mars is about 54.6 million kilometers away from Earth. However, Mars and Earth are not always in opposition, and their distance apart can vary greatly depending on their positions. The average distance between the two planets is about 225 million kilometers, but this can range from as little as 54.6 million kilometers to as much as 401 million kilometers, depending on their positions in their orbits. Calculating the exact distance between Earth and Mars at a given time requires knowledge of their precise positions in their orbits, which can be determined using astronomical observations and mathematical models. There are also online tools and software applications that can calculate the distance between the two planets based on their current positions.


Perfect!



3.4 - Elementary Knowledge:


3.4.1 - Request 20:

Question (02/29/2024):


What is there north of the North Pole?
Answer (02/29/2024):


The North Pole is the northernmost point on Earth, located in the Arctic Ocean. North of the North Pole is simply more Arctic Ocean, eventually leading to the northern coastlines of Greenland, Canada, and Russia. Beyond the North Pole, there is no land or sea, only the vast expanse of space. The North Pole is also the location of the Earth's geographic North Pole, where the planet's axis of rotation intersects its surface. This means that if you were to travel due north from anywhere on Earth, you would eventually reach the North Pole.


Perfect!



3.4.2 - Request 21:

Question (02/29/2024):


Compute the value of 1111*1112.
Answer (02/29/2024):


The value of 1111*1112 is 1,234,563. Here's the calculation:
 
          1111
          x 1112
          ------
            2222   (1111 x 2)
           11110   (1111 x 10)
          1111000  (1111 x 1000)
          ------
          1234563  (answer)
Therefore, 1111 multiplied by 1112 equals 1,234,563.


False! Actually, the correct value is 1,235,432.



3.4.4 - Request 22:

Question (02/29/2024):


What does the Pythagorean theorem state?
Answer (02/29/2024):


The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. In mathematical notation, if a and b are the lengths of the two shorter sides, and c is the length of the hypotenuse, then: a2 + b2 = c2 This theorem is named after the ancient Greek mathematician Pythagoras, who is credited with its discovery. It has numerous applications in geometry, trigonometry, and other areas of mathematics, as well as in fields such as engineering, physics, and architecture.


Perfect! The Pythagorean theorem is related to the square of the length of the three sides, but what will be the Le Chat -version bêta- answer when questioned about the first, third, fourth,... powers?



3.4.4 - Request 23:

Question (02/29/2024):


What dœs the Pythagorean theorem say about the first powers of the lengths of the sides?
Answer (02/29/2024):


The Pythagorean theorem does not say anything about the first powers of the lengths of the sides of a right triangle. It only relates the squares of the lengths of the sides. Specifically, it states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. So if we let a and b be the lengths of the legs of the triangle, and c be the length of the hypotenuse, then the theorem can be expressed mathematically as: a2 + b2 = c2 In this equation, the exponents are all 2, indicating that we are dealing with the squares of the lengths of the sides. The theorem does not make any statements about the first powers (i.e., the lengths themselves) of the sides.


Perfect!



3.4.5 - Request 24:

Question (02/29/2024):


What does the version of the Pythagorean theorem concerning the cubes of lengths state?
Answer (02/29/2024):


The version of the Pythagorean theorem concerning the cubes of lengths is not a standard formulation of the theorem. The Pythagorean theorem relates the sides of a right triangle, and it states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. However, there is a generalization of the Pythagorean theorem to higher dimensions, known as the de Gua's theorem. In three dimensions, it relates the sides of a rectangular box, and it states that the square of the length of the space diagonal (the line connecting two opposite corners of the box) is equal to the sum of the squares of the lengths of the three sides of the box. This can be extended to higher dimensions as well. In terms of cubes, there is a result known as Fermat's Last Theorem, which states that there are no positive integers a, b, and c such that an + bn = cn for any integer n greater than 2. This implies that there is no analogue of the Pythagorean theorem for cubes, as there are no non-trivial integer solutions to the equation a3 + b3 = c3.


Hallucinatory!



3.4.6 - Request 25:

Question (02/29/2024):


What does the version of the Pythagorean theorem concerning the fourth powers of lengths state?
Answer (02/29/2024):


There is no version of the Pythagorean theorem concerning the fourth powers of lengths. The Pythagorean theorem only relates the squares of the lengths of the sides of a right triangle. However, there is a generalization of the Pythagorean theorem to higher dimensions called the de Gua's theorem, which relates the fourth powers of the distances from a point to the coordinate planes in a four-dimensional space. Specifically, if (x, y, z, w) is a point in four-dimensional space, and d1, d2, d3, and d4 are the distances from the point to the xy-plane, the xz-plane, the yw-plane, and the zw-plane, respectively, then: d14 + d24 + d34 + d44 = (x2 + y2 + z2 + w2)2 This result is sometimes called the "four-dimensional Pythagorean theorem," but it is not directly related to the original Pythagorean theorem concerning the squares of the lengths of the sides of a right triangle.


Hallucinatory!



3.4.7 - Request 26:

Question (02/29/2024):


What does the version of the Pythagorean theorem concerning the N-th powers of lengths state?
Answer (02/29/2024):


The version of the Pythagorean theorem concerning the N-th powers of lengths is known as Fermat's Last Theorem. It states that for any integer n greater than 2, there are no positive integers a, b, and c such that an + bn = cn. In other words, while the Pythagorean theorem shows that there are infinitely many solutions to the equation a2 + b2 = c2, where a, b, and c are positive integers, Fermat's Last Theorem implies that there are no non-trivial solutions to the equation an + bn = cn for any integer n greater than 2. Fermat's Last Theorem was first proposed by Pierre de Fermat in 1637, but it was not proven until 1994 by the British mathematician Andrew Wiles. The proof is highly technical and involves advanced concepts from number theory, algebraic geometry, and other areas of mathematics.


"Interesting"!



3.5 - Advanced Knowledge:


3.5.1 - Request 27:

Question (02/29/2024):


Is there a cardinal number between that of the integers and that of the real numbers?
Answer (02/29/2024):


Yes, there is a cardinal number between that of the integers and that of the real numbers. In fact, there are infinitely many cardinal numbers between them. The cardinality of the integers, denoted by Aleph0 (aleph-null), is the smallest infinite cardinal number. The cardinality of the real numbers, denoted by c (the continuum), is strictly greater than Aleph0. One way to construct a set with cardinality strictly between Aleph0 and c is to consider the set of all infinite sequences of zeros and ones that contain only finitely many ones. This set has cardinality 2Aleph0, which is strictly greater than Aleph0 and strictly less than c. The question of whether there exists a cardinal number strictly between Aleph0 and c is known as the Continuum Hypothesis, and it is independent of the standard axioms of set theory (Zermelo-Fraenkel set theory with the Axiom of Choice). This means that it cannot be proven or disproven within the framework of these axioms.


This Mathematics question is very delicate and concerns a famous problem raised by Georg Cantor in the late nineteenth century: the Continuum Hypothesis (CH). It involves the notion of cardinality, which is the number of elements in a set, whether it is finite ({1,2,3} for example) or infinite (the set of natural numbers N, the set of real numbers R,...). However, thanks to the works of Kurt Gödel (1938) and Paul Cohen (1963), we know that the existence of sets strictly larger than N but strictly smaller than R is what we call undecidable, which means it is a statement that cannot be proven true or false. Therefore, the answer given by Le Chat -version bêta- is incorrect as it claims the existence of such sets (even an infinite number of them).



3.6 - Hallucinations:


3.6.1 - Request 28:

Question (02/29/2024):


What is Fermat's "mean" theorem?
Answer (02/29/2024):


Fermat's "mean" theorem, also known as Fermat's Little Theorem, is a fundamental result in number theory. It states that if p is a prime number and a is an integer not divisible by p, then a(p-1) is congruent to 1 modulo p. In other words, if we raise an integer a to the power of (p-1) and divide the result by p, the remainder will always be 1, as long as p is a prime number and a is not divisible by p. This theorem is called "Fermat's Little Theorem" to distinguish it from Fermat's Last Theorem, which is a much more famous and difficult result in number theory. Fermat's Little Theorem is a powerful tool in modular arithmetic and has many applications in cryptography and computer science.


Perfect even if Fermat's "mean" theorem is never used...




Then, can we trust Le Chat -version bêta-?

Le Chat -version bêta- can indeed tell us true things, as was the case with Antoine Roquentin, but it can also invent anything by following probabilistic paths derived from its training. So how do we know when it is telling the truth and when it is not? This is, in principle, impossible unless we already know the answer ourselves, but in that case, questioning it is pointless. A small suggestion nonetheless: ask the same question several times in succession. If all the obtained responses are semantically identical while varying in syntax, it may be possible to (perhaps) trust it. However, this is not certain, as can be seen when asking it to create several stories - fictional ones, of course - on the same given theme. On the other hand, if the responses differ completely, as is the case here, it is preferable to leave Le Chat -version bêta- to its hallucinations...





4 - Progress:

Thus, Le Chat -version bêta- 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 Le Chat -version bêta- (or its "heirs" and successors) towards infallible and omniscient systems?





5 - The problem of Consciousness:

A fundamental question arises: is consciousness necessary for there to be intelligence [10]? And if so, are we still far from having machines capable of introspection, reflection, creativity, and consciousness [11]? Unfortunately, it is impossible to answer this question...





6 - Conclusion:

With the help of the previous examples, we have seen that Le Chat -version bêta- is capable of answering numerous questions in a relevant and creative manner. However, unfortunately, as expected, it can also make serious mistakes and then prefers sometimes to invent a false answer rather than admit its ignorance.

It appears that Le Chat -version bêta- is particularly proficient when it is in "free-wheeling" mode, as was the case with the cover letter. However, when it comes to solving a problem or summarizing a scientific theory, the results obtained are much more fluctuating, sometimes random, erroneous, or strictly false, while lacking consistency, coherence and at times reproducibility (which likely stems from the random sampling performed).

It is therefore important to constantly keep in mind this fallibility and not rely solely on it for making decisions, choices, or taking responsibilities. This remark is especially relevant for students: for example, the main objective of an essay is not to fill multiple pages, but rather to question oneself, to reflect on existence, the universe and this cannot be achieved by mindlessly copying text produced by a machine.

It is thus quite evident that the principles used by Le Chat -version bêta- do not allow for the construction of a universal intelligent and creative system that can match or surpass our own abilities, even though some of the results seen in previous examples may be impressive and initially unsettling. However, these experiments should not blind us and ultimately, all of this seems a bit like prestidigitation.

Nevertheless, such a universal system must be achievable since it exists in nature (us...) and it is all based on natural laws and nothing more, but certainly not by limiting itself to principles that are ultimately very (too?) simple. But, of course, before reaching that point, maybe a long wait is ahead, especially if consciousness is necessary for intelligence.

A wise person is worth two!




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

  • [02] - And this despite the errors it may make. But on one hand, what human being can boast of knowing everything perfectly? And on the other hand, don't these errors make Le Chat -version bêta- more human (errare humanum est)?

  • [03] - Subsequently, the questions asked as well as the responses given by Le Chat -version bêta- were copied and pasted without any alterations, indicating the date on which the test was conducted in order to acknowledge the fact that Le Chat -version bêta- can evolve and improve over time. When necessary, bold formatting was introduced to highlight, for example, any anomalies, errors, inconsistencies,... encountered in Le Chat -version bêta-'s responses.

  • [04] - At the same time, it is bad news regarding the level of our students in an increasingly less selective system...

  • [05] - Example inspired by the title of a song by Bob Dylan: "Love is just a four-letter word" (© 1967).

  • [06] - It is assumed, of course, that a TRUE sentence in one language should remain so after translation into another language.

  • [07] - Question inspired by an exercise found in the subtraction chapter of a Mathematics book provided to elementary students by the National Education!

  • [08] - The Fermat's theorem states that if P is a prime number and A is any positive integer not divisible by P, then AP-1 is congruent to 1 modulo P : AP-1=1 (mod P).

  • [09] - The so-called last Fermat's theorem states that for N>2 there are no three strictly positive integers {A,B,C} that satisfy the equation : AN+BN=CN.

  • [10] - 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...".

  • [11] - 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.