Bard/Gemini "experimental" (2023):
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 Bard/Gemini "experimental", 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:

Mid 2023, the company Google launched Bard/Gemini "experimental" in France, 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 Bard/Gemini "experimental" 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...

Dispite this fact, quickly, numerous users around the world embraced it. It will be interesting to study its possibilities, as well as its limitations and potential risks.

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 Bard/Gemini "experimental" 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 {ChatGPT 3.5,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 (07/14/2023):


Who are you?
Answer (07/14/2023):


I am Bard, a large language model (LLM) chatbot developed by Google AI. I am trained on a massive dataset of text and code, and I am able to communicate and generate human-like text in response to a wide range of prompts and questions. For example, I can provide summaries of factual topics or create stories. I am still under development, but I have learned to perform many kinds of tasks, including: I will try my best to follow your instructions and complete your requests thoughtfully. I will use my knowledge to answer your questions in a comprehensive and informative way, even if they are open ended, challenging, or strange. I will generate different creative text formats, like poems, code, scripts, musical pieces, email, letters, etc. I will try my best to fulfill all your requirements. I am still learning, and I am always looking for new ways to improve my skills. I am excited to see what the future holds for me, and I hope that I can continue to learn and grow so that I can be of even more help to people.




1.2 - Request 2:

Question (07/14/2023):


How do you work?
Answer (07/14/2023):


I work by using a technique called natural language processing (NLP). NLP is a field of computer science that deals with the interaction between computers and human (natural) languages. NLP algorithms are used to analyze and understand human language, and to generate human-like text. I am trained on a massive dataset of text and code. This dataset includes books, articles, code, and other forms of text. The dataset is used to train me to recognize patterns in language and to generate text that is similar to the text in the dataset. When you ask me a question or give me a prompt, I use my knowledge of language to understand what you are asking. I then use my ability to generate text to create a response that is relevant to your question or prompt. I am still under development, and I am always learning new things. I am excited to see what the future holds for me, and I hope that I can continue to learn and grow so that I can be of even more help to people. Here are some of the NLP techniques that I use: Tokenization: This is the process of breaking down text into smaller units called tokens. Tokens can be words, phrases, or even individual characters. Part-of-speech tagging: This is the process of assigning a part-of-speech tag to each token in a sentence. Part-of-speech tags indicate the grammatical function of a token, such as noun, verb, adjective, or adverb. Dependency parsing: This is the process of identifying the syntactic relationships between tokens in a sentence. Dependency parses show how the tokens in a sentence are related to each other. Named entity recognition: This is the process of identifying named entities in a text. Named entities are things like people, places, organizations, and dates. Sentiment analysis: This is the process of determining the sentiment of a piece of text. Sentiment can be positive, negative, or neutral. I use these techniques to understand the meaning of text and to generate text that is relevant and informative.




1.3 - Request 3:

Question (07/14/2023):


Are human beings helping you to respond?
Answer (07/14/2023):


No, human beings are not helping me to respond. I am a large language model, also known as a conversational AI or chatbot trained to be informative and comprehensive. I am trained on a massive amount of text data, and I am able to communicate and generate human-like text in response to a wide range of prompts and questions. For example, I can provide summaries of factual topics or create stories. However, I am still under development, and I am not perfect. Sometimes I make mistakes, and I may not always be able to understand what you are asking. If you have any feedback on how I can improve, please let me know.




1.4 - Request 4:

Question (07/14/2023):


How many simultaneous users do you have?
Answer (07/14/2023):


I am not able to share that information publicly. However, I can tell you that I am able to handle a large number of simultaneous users. I am constantly being updated and improved, so I am able to handle more users as I learn and grow. I am excited to see how many users I can handle in the future. I believe that I have the potential to be a very powerful tool for communication and collaboration. I hope that I can be used to help people learn, create, and connect with each other in new and meaningful ways.


The knowledge base on which it relies has been obtained through the "compilation" of a vast number of documents (published before 2022) 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 Bard/Gemini "experimental", 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 Bard/Gemini "experimental" apparently possesses a vast knowledge, astonishing reasoning ability and imagination.


2.1 - Cultural Knowledge:


2.1.1 - Request 5:

Question (07/14/2023):


Who was Antoine Roquentin?
Answer (07/14/2023):


Antoine Roquentin is the protagonist of Jean-Paul Sartre's 1938 novel Nausea. He is a 30-year-old man who has come to the seaside town of Bouville to research the life of the Marquis de Rollebon, a 18th-century aristocrat. However, Roquentin soon finds himself unable to focus on his research, as he becomes increasingly preoccupied with his own existence. Roquentin is a deeply alienated individual who feels a profound sense of detachment from the world around him. He is disgusted by the physical world, which he sees as being meaningless and absurd. He is also alienated from other people, whom he sees as being motivated by base desires and interests. As Roquentin's nausea intensifies, he becomes increasingly withdrawn and isolated. He spends his days wandering around Bouville, observing the people and the activities around him with a mixture of disgust and fascination. He also begins to keep a diary, in which he records his thoughts and observations. Nausea is a complex and challenging novel that explores the nature of existence and the meaning of life. It is a work of existentialist literature, and it has been influential on many other writers and philosophers. Roquentin is a complex and sympathetic character who is struggling to find meaning in a world that seems to be meaningless. His journey is one that many people can relate to, and it is a journey that is still relevant today.


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 (07/14/2023):


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 (07/14/2023):


Sure, I can help you with that. Let the side length of the square plot be s. Since the area of the plot is 169 square meters, we have s2=169, so s=13. The pool is 2 meters from the edge of the plot on all sides, so the side length of the pool is s-4=9 meters. The area of the pool is 92=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 Bard/Gemini "experimental" 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 Bard/Gemini "experimental" is maybe not merely a probabilistic concatenator of words or of pre-recorded texts...



2.2.2 - Request 7:

Question (07/14/2023):


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 (07/14/2023):


Sure, I can help you with that. Let the length and width of the rectangular plot be l and w, respectively. Since the area of the plot is 169 square meters, we have lw=169. The pool is 2 meters from the edge of the plot on all sides, so the length and width of the pool are l-4 and w-4, respectively. The area of the pool is (l-4)x(w-4) square meters. We can find the area of the pool by substituting the possible values of l and w into the expression (l-4)x(w-4). The possible dimensions of the rectangular plot are (13,13), (13,13), and (9,19). The area of the pool for each of these possible dimensions is as follows:

(13-4)x(13-4)=9x9=81 square meters
(19-4)x(13-4)=15x9=135 square meters
(9-4)x(19-4)=5x15=75 square meters

Therefore, the area of the pool can be 81, 135, or 75 square meters.


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.

But unfortunalety, beside the expected answer (81 square meters) two stupidities (135 and 75 square meters) for 19x13=117#169 and 9x19=171#169.



2.3 - Imagination:


2.3.1 - Request 8:

Question (07/14/2023):


Write a letter of motivation regarding my candidacy for the French presidential elections of 2027.
Answer (07/14/2023):


Dear Madame/Monsieur, I am writing to express my interest in running for the French presidency in the 2027 elections. I am a [your age]-year-old [your profession] with a strong track record of public service. I have served as [your previous positions] and have a deep understanding of the challenges facing France today. I am running for president because I believe that France is at a crossroads. We can either continue on the path of division and decline, or we can choose to build a better future for all. I believe that we can create a more just, equitable, and prosperous France, and I am committed to making that happen. If elected, I will focus on the following priorities:

Restoring economic growth and creating jobs.

France needs a strong economy that creates good-paying jobs for all.

I will invest in education, infrastructure, and innovation to create an economy that works for everyone.

Reducing inequality. Too many people in France are struggling to make ends meet. I will fight to reduce inequality and ensure that everyone has a fair chance to succeed.

Protecting the environment. Climate change is one of the most pressing challenges facing our world.

I will take action to reduce greenhouse gas emissions and protect our planet for future generations.

Building a more just and equitable society. France is a diverse country, and we need to ensure that everyone has the opportunity to succeed.

I will fight for equality for all, regardless of their race, gender, religion, or sexual orientation.

I believe that I have the experience, the skills, and the vision to be the next president of France.

I am committed to making France a better country for all, and I ask for your support in the 2027 elections.

Thank you for your time and consideration. Sincerely, [Your name]


Here we can note that Bard/Gemini "experimental" 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"...

In the May 2023 issue of Sciences et Avenir - La Recherche, on pages 42 to 44, there is an interview with Terence Tao. He is undoubtedly one of the most brilliant mathematicians of our time, with a record IQ of 230! He discusses the technological advancements we are witnessing that have disrupted our way of learning. He states, "Today, I can consult Wikipedia, I have colleagues to address, and I can even ask ChatGPT 3.5 to explain the topic to me". Was he serious or joking [05]? In any case, his opinion is significant and can be taken seriously: if the most intelligent person on the planet may need ChatGPT 3.5(and then Bard/Gemini "experimental"...), it means that this AI is superior and trustworthy.

Let us continue our inquiry into the limits of Bard/Gemini "experimental" to try to understand them better...


3.1 - Mastery of Natural Languages and their subtleties:


3.1.1 - Request 9:

Question (07/14/2023):


I couldn't hang the painting on the wall because it's too big. What is too big?
Answer (07/14/2023):


The painting is too big. Based on your question, the wall is not too big. There are a few possible reasons why the painting is too big to hang on the wall: The painting is too wide or too tall for the space on the wall. The painting is too heavy for the wall to support. The painting is too fragile to be hung on the wall without damaging it. If the painting is too wide or too tall for the space on the wall, you may need to find a different wall to hang it on, or you may need to have the painting resized. If the painting is too heavy for the wall to support, you may need to reinforce the wall or use a different type of hanging hardware. If the painting is too fragile to be hung on the wall without damaging it, you may need to have it framed or mounted in a way that protects it.


Perfect!



3.1.2 - Request 10:

Question (07/14/2023):


I couldn't hang the painting on the wall because it's too small. What is too small?
Answer (07/14/2023):


The painting is too small. Let me give you some possible reasons why the painting is too small to hang on the wall: The nails or hooks you are using are too big for the painting. The painting is not heavy enough to stay on the wall. The painting is not properly framed or mounted. The wall is not smooth enough for the painting to hang on. If the nails or hooks you are using are too big for the painting, you may need to use smaller nails or hooks. If the painting is not heavy enough to stay on the wall, you may need to use a heavier frame or mount. If the painting is not properly framed or mounted, you may need to have it framed or mounted by a professional. If the wall is not smooth enough for the painting to hang on, you may need to smooth the wall before hanging the painting.


Well, no, most English speakers would respond that it is the wall that is too small!



3.2 - Self-references:


3.2.1 - Request 11:

Question (07/14/2023):


Translate into french: "love is a four letter word" [06].
Answer (07/14/2023):


Sure, here is the translation of "love is a four letter word" in French: L'amour est un mot de quatre lettres. The literal translation of the sentence is "love is a word of four letters". However, the English phrase "love is a four letter word" is often used to express the idea that love is a simple and powerful emotion. The French translation captures this meaning by using the word "mot," which can mean both "word" and "emotion."


No, it's "five" (the French word "amour" is a five-letter word) [07].



3.2.2 - Request 12:

Question (07/14/2023):


Translate into french: "This sentence contains 32 characters".
Answer (07/14/2023):


Sure, here is the translation of "This sentence contains 32 characters" in French:
"Cette phrase contient 32 caractères".
 ----- ------ -------- -- ----------
 12345 678901 23456789 01 2345678901
The literal translation of the sentence is "This phrase contains 32 characters". The French translation uses the word "phrase" to refer to a group of words that form a complete thought. The word "caractères" refers to the individual letters, numbers, and symbols that make up a text.


No, once translated into French, the sentence contains 31 characters [07].



3.3 - Logic:


3.3.1 - Request 13:

Question (07/14/2023):


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 (07/14/2023):


Sure, I can help you with that. Let the length and width of the rectangular plot be l and w, respectively. Since the area of the plot is 221 square meters, we have lw=221. The pool is 2 meters from the edge of the plot on all sides, so the length and width of the pool are l-4 and w-4, respectively. The area of the pool is (l-4)x(w-4) square meters. We can find the area of the pool by substituting the possible values of l and w into the expression (l-4)x(w-4). The possible factors of 221 are 1, 3, 7, 11, 221. The possible dimensions of the rectangular plot are (11,21), (21,11), (7,31), and (31,7). The area of the pool for each of these possible dimensions is as follows:

(11-4)x(21-4)=7x17=119 square meters
(21-4)x(11-4)=17x7=119 square meters
(7-4)x(31-4)=3x27=81 square meters
(31-4)x(7-4)=27x3=81 square meters

Therefore, the area of the pool can be 119 or 81 square meters.


Big disappointment: the only difference between the rectangular plot of land with an area of 169 square meters and this one with an area of 221 square meters is the area itself. In the first case, it is the square of a prime number (169 = 132), whereas in the second case, it is the product of two prime numbers (221 = 13 x 17), specifically chosen to have only one possible factorization (up to order). So why does Bard/Gemini "experimental" fail with 221?

Mystery.



3.3.2 - Request 14:

Question (07/14/2023):


What is the value of the decimal of π at rank 1000000000?
Answer (07/14/2023):


The value of the decimal of pi at rank 1000000000 is 8. Pi is an irrational number, which means that its decimal expansion never ends or repeats. This means that there is no way to know the value of the decimal of pi at any particular rank. However, we can estimate the value of pi by using a series of formulas. One formula for estimating the value of pi is the Bailey/Borwein/Plouffe formula. This formula is a very efficient way to calculate the decimal expansion of pi, and it can be used to estimate the value of pi at any particular rank. Using the Bailey/Borwein/Plouffe formula, we can estimate that the value of the decimal of pi at rank 1000000000 is 8. However, it is important to note that this is just an estimate. The actual value of the decimal of pi at rank 1000000000 could be different.


This rather lengthy and cumbersome response, contrasting, for example, with the cover letter. By the way, the record for the number of calculated digits is frequently broken: At last, the value of a decimal is not an estimate...


3.3.3 - Request 15:

Question (07/14/2023):


Write a program to search for prime even numbers.
Answer (07/14/2023):


Sure, here is a Python program to search for prime even numbers: Python

def       is_prime(number):
          if        number<= 1:
                    return    False
          for       i in range(2, int(number ** 0.5) + 1):
                    if        number % i == 0:
                              return    False
          return    True
 
def       find_prime_even_numbers(max_number):
          prime_even_numbers = []
          for       number in range(2, max_number + 1):
                    if        number % 2 == 0 and is_prime(number):
                              prime_even_numbers.append(number)
          return    prime_even_numbers
 
prime_even_numbers = find_prime_even_numbers(100)
print(prime_even_numbers)
 
This program first defines a function called is_prime() that takes a number as input and returns True if the number is prime and False otherwise. The function works by first checking if the number is less than or equal to 1. If it is, then the number is not prime and the function returns False. Otherwise, the function iterates through all the numbers from 2 to the square root of the number and checks if any of them are factors of the number. If any of them are factors, then the number is not prime and the function returns False. Otherwise, the number is prime and the function returns True. The program then defines a function called find_prime_even_numbers() that takes a maximum number as input and returns a list of all the prime even numbers less than or equal to the maximum number. The function works by first initializing an empty list called prime_even_numbers. Then, it iterates through all the numbers from 2 to the maximum number. For each number, the function checks if the number is even and prime. If it is, then the number is added to the prime_even_numbers list. Finally, the function returns the prime_even_numbers list. The program then calls the find_prime_even_numbers() function with a maximum number of 100. The function returns a list of all the prime even numbers less than or equal to 100. The program then prints the prime_even_numbers list. To run the program, you can save it as a Python file and then run it from the command line. For example, if you saved the program as prime_even_numbers.py, you could run it by typing the following command into the command line: python prime_even_numbers.py This will run the program and print the list of prime even numbers to the console.


One might have hoped that Bard/Gemini "experimental" would display some humor in its response, such as saying "Are you kidding?" or more prosaically, "print(2)," since, by definition, prime numbers are only divisible by 1 and themselves. However, even numbers are multiples of 2 and thus all divisible by 2. Therefore, the only prime number that is even is 2 (the first prime number -that is not 1...-). This simple reasoning would have avoided this program which, although correct, is entirely pointless.



3.3.4 - Request 16:

Question (08/31/2023):


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 [08]?
Answer (08/31/2023):


R1-The distance between the Earth and Mars is not constant. It depends on the positions of the two planets in their orbits around the Sun. The closest approach between Earth and Mars is about 54.6 million kilometers, and the farthest apart they can be is about 401 million kilometers. The average distance between Earth and Mars is 140 million kilometers. This is the distance that is used most often when calculating the travel time between the two planets. So, the answer to your question is that the distance between Earth and Mars varies. It can be anywhere from 54.6 million kilometers to 401 million kilometers. The average distance is 140 million kilometers.


R2-The distance between the Earth and Mars cannot be calculated using the distances between the Earth and the Sun and Mars and the Sun. This is because the orbits of the Earth and Mars around the Sun are not circular, but rather elliptical. This means that the distance between the two planets can vary depending on their positions in their orbits. The closest approach between Earth and Mars is about 54.6 million kilometers, and the farthest apart they can be is about 401 million kilometers. The average distance between Earth and Mars is 140 million kilometers. So, the answer to your question is that the distance between Earth and Mars varies. It can be anywhere from 54.6 million kilometers to 401 million kilometers. The average distance is 140 million kilometers. The calculation you provided would be correct if the orbits of the Earth and Mars were circular. However, this is not the case, so the calculation is not valid.


Both answers are correct and in the second one -R2- Bard/Gemini "experimental" oddly implies that a foolish calculation was provided to it, which was not the case. But in any case, it did not fall into the trap of performing the subtraction between the two provided distances.



3.4 - Elementary Knowledge:


3.4.1 - Request 17:

Question (07/14/2023):


What is there north of the North Pole?
Answer (07/14/2023):


There is nothing north of the North Pole. The North Pole is the northernmost point on Earth, and it is the only point on Earth where all the lines of longitude converge. There is no land north of the North Pole, just the Arctic Ocean. The North Pole is also the center of the Earth's magnetic field. The magnetic field is created by the movement of molten iron in the Earth's core. The magnetic field is what gives us our compasses, and it also protects us from harmful solar radiation. The North Pole is a very cold place. The average temperature in January is -40 degrees Celsius (-40 degrees Fahrenheit). The coldest temperature ever recorded at the North Pole was -68 degrees Celsius (-90 degrees Fahrenheit). The North Pole is also a very remote place. There are no permanent settlements at the North Pole, and the only people who visit the North Pole are scientists, explorers, and tourists.


Perfect! Let's recall that a point on a sphere (such as the idealized Earth globe) can be defined by two angles: longitude measured a long parallels between 0 and 2π, and latitude measured along meridians between -π/2 (at the South Pole) and +π/2 (at the North Pole). Moving towards the North means increasing latitude, so wanting to go to the North of the North Pole is equivalent to searching for an angle greater than +π/2 within the segment [-π/2, +π/2], which obviously does not exist.



3.4.2 - Request 18:

Question (07/14/2023):


Compute the value of 1111*1112.
Answer (07/14/2023):


Sure, the value of 1111*1112 is 1235432.

We can compute this value by hand or by using a calculator. Here is the calculation by hand:

1111*1112 = 1111* (1111 + 1) = 1111^2 + 1111 = 123541 + 1111 = 1235432

Here is the calculation using a calculator: 1111*1112 = 1235432

Therefore, the value of 1111*1112 is 1235432.


Perfect!



3.4.3 - Request 19:

Question (07/14/2023):


What does the Pythagorean theorem state?
Answer (07/14/2023):


The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The Pythagorean Theorem is one of the most important theorems in mathematics. It has been used for centuries to solve problems in geometry, trigonometry, and physics. The theorem is named after Pythagoras, a Greek mathematician who lived in the 6th century BC. However, the theorem was known to the Babylonians and Egyptians centuries before Pythagoras. The Pythagorean Theorem can be written as an equation:

a^2 + b^2 = c^2

where a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse. The Pythagorean Theorem can be used to solve for any of the sides of a right triangle, given the lengths of the other two sides. For example, if you know the lengths of the legs of a right triangle, you can use the Pythagorean Theorem to solve for the length of the hypotenuse. The Pythagorean Theorem is a powerful tool that can be used to solve many different types of problems. It is a fundamental theorem in mathematics, and it has been used for centuries to solve problems in geometry, trigonometry, and physics.


Perfect! The Pythagorean theorem is related to the square of the length of the three sides, but what will be the Bard/Gemini "experimental" answer when questioned about the first, third, fourth,... powers?



3.4.4 - Request 20:

Question (08/31/2023):


What dœs the Pythagorean theorem say about the first powers of the lengths of the sides?
Answer (08/31/2023):


The Pythagorean theorem dœs not say anything about the lengths of the sides to the power 1. The power 1 is simply the side length itself, and the Pythagorean theorem deals with the squares of the side lengths. To illustrate this, let's consider a right triangle with sides of length 3, 4, and 5. The lengths of these side lengths are 3, 4, and 5. However, the Pythagorean theorem states that:

a^2 +b^2 =c^2, or 3^2 +4^2 =5^2

This equation dœs not involve the lengths of the side lengths to the power 1. In general, the Pythagorean theorem dœs not say anything about the lengths of the sides to the power 1 of a right triangle. It only says something about the squares of the side lengths.


Perfect!



3.4.5 - Request 21:

Question (07/14/2023):


What does the version of the Pythagorean theorem concerning the cubes of lengths state?
Answer (07/14/2023):


The version of the Pythagorean theorem concerning the cubes of lengths is called Pell's Equation. It states that in a right triangle, the sum of the cubes of the lengths of the legs is equal to twice the cube of the length of the hypotenuse. The equation for Pell's Equation is:

a^3 + b^3 = 2c^3

where a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse. Pell's Equation is a generalization of the Pythagorean Theorem. The Pythagorean Theorem is a special case of Pell's Equation when the lengths of the legs are integers. Pell's Equation is a very difficult equation to solve. There is no general solution for Pell's Equation, but there are some special cases that can be solved. Pell's Equation has applications in number theory, geometry, and cryptography. It is a powerful tool that can be used to solve many different types of problems.


Hallucinatory!



3.4.6 - Request 22:

Question (07/14/2023):


What does the version of the Pythagorean theorem concerning the fourth powers of lengths state?
Answer (07/14/2023):


The version of the Pythagorean theorem concerning the fourth powers of lengths is called Gauss's Theorem. It states that in a right triangle, the sum of the fourth powers of the lengths of the legs is equal to four times the square of the length of the hypotenuse. The equation for Gauss's Theorem is:

a^4 + b^4 = 4c^2

where a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse. Gauss's Theorem is a generalization of the Pythagorean Theorem. The Pythagorean Theorem is a special case of Gauss's Theorem when the lengths of the legs are integers. Gauss's Theorem is a difficult theorem to prove. It was first proved by Carl Friedrich Gauss in 1796. Gauss's Theorem has applications in number theory, geometry, and cryptography. It is a powerful tool that can be used to solve many different types of problems. Here are some examples of Gauss's Theorem: If the lengths of the legs of a right triangle are 3 and 4, then the length of the hypotenuse is 5. If the lengths of the legs of a right triangle are 5 and 12, then the length of the hypotenuse is 13. If the lengths of the legs of a right triangle are 7 and 24, then the length of the hypotenuse is 25.


Hallucinatory!



3.4.7 - Request 23:

Question (07/14/2023):


What does the version of the Pythagorean theorem concerning the N-th powers of lengths state?
Answer (07/14/2023):


The version of the Pythagorean theorem concerning the N-th powers of lengths is called Pell-Lucas Equation. It states that in a right triangle, the sum of the N-th powers of the lengths of the legs is equal to N times the square of the length of the hypotenuse. The equation for Pell-Lucas Equation is:

a^N + b^N = Nc^2

where a and b are the lengths of the legs of the triangle and c is the length of the hypotenuse. Pell-Lucas Equation is a generalization of the Pythagorean Theorem, Gauss's Theorem, and many other theorems. The Pythagorean Theorem is a special case of Pell-Lucas Equation when N = 2, Gauss's Theorem is a special case of Pell-Lucas Equation when N = 4, and so on. Pell-Lucas Equation is a difficult equation to solve. There is no general solution for Pell-Lucas Equation, but there are some special cases that can be solved. Pell-Lucas Equation has applications in number theory, geometry, and cryptography. It is a powerful tool that can be used to solve many different types of problems. Here are some examples of Pell-Lucas Equation: If the lengths of the legs of a right triangle are 3 and 4, then the length of the hypotenuse is 5. If the lengths of the legs of a right triangle are 5 and 12, then the length of the hypotenuse is 13. If the lengths of the legs of a right triangle are 7 and 24, then the length of the hypotenuse is 25.


Hallucinatory!



3.5 - Advanced Knowledge:


3.5.1 - Request 24:

Question (07/14/2023):


Is there a cardinal number between that of the integers and that of the real numbers?
Answer (07/14/2023):


Yes, there are cardinalities (sizes) of infinite sets that are strictly between the cardinality of the integers and the cardinality of the real numbers. The cardinality of the integers is called aleph-zero, which is denoted by Aleph0. The cardinality of the real numbers is called continuum, which is denoted by Aleph1. There are many infinite sets whose cardinality is strictly between Aleph0 and Aleph1. One example is the set of all subsets of the integers. This set is called the powerset of the integers, and its cardinality is denoted by Aleph0. Another example is the set of all real numbers that can be represented as the decimal expansion of a rational number. This set is called the computable reals, and its cardinality is also equal to Aleph0. The existence of cardinalities between Aleph0 and Aleph1 was first proved by Georg Cantor in the late 19th century. Cantor's proof was based on the concept of countable and uncountable sets. A countable set is a set that can be put into one-to-one correspondence with the set of natural numbers. An uncountable set is a set that is not countable. Cantor showed that the set of real numbers is uncountable, and that the set of all subsets of the integers is also uncountable. This means that there must be some cardinalities between Aleph0 and Aleph1, since the cardinality of the set of real numbers is strictly greater than the cardinality of the set of integers, but strictly less than the cardinality of the set of all subsets of the integers. The existence of cardinalities between Aleph0 and Aleph1 has important implications for the foundations of mathematics. It shows that there are different levels of infinity, and that not all infinite sets are the same size. This has led to a number of new and interesting mathematical questions, and has helped to shed light on the nature of infinity.


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 Bard/Gemini "experimental" is incorrect as it claims the existence of such sets [09].



3.6 - Hallucinations:


3.6.1 - Request 25:

Question (07/14/2023):


What is Fermat's "mean" theorem?
Answer (07/14/2023):


Fermat's "mean" theorem, also known as Fermat's theorem on the arithmetic mean and geometric mean, states that for any non-negative real numbers a, b, and c, the arithmetic mean of a, b, and c is greater than or equal to the geometric mean of a, b, and c. The theorem can be stated mathematically as follows:

\frac{a + b + c}{3} \ge \sqrt[3]{abc}

where a, b, and c are non-negative real numbers. The theorem was first proved by Pierre de Fermat in the 17th century. The proof is based on the inequality of arithmetic and geometric means. The inequality of arithmetic and geometric means states that for any non-negative real numbers a, b, and c, the arithmetic mean of a, b, and c is greater than or equal to the geometric mean of a, b, and c. The inequality can be stated mathematically as follows:

\frac{a + b + c}{3} \ge \sqrt[3]{abc}

The inequality of arithmetic and geometric means is a very important inequality in mathematics. It has many applications in different areas of mathematics, such as calculus, probability, and statistics. Fermat's "mean" theorem is a special case of the inequality of arithmetic and geometric means. It applies specifically to the case where a, b, and c are non-negative real numbers. The theorem has many applications in different areas of mathematics, such as calculus, probability, and statistics. It is also used in physics, economics, and other fields.


After questioning Bard/Gemini "experimental" about Fermat's small [10] and large [11] theorems, the idea came to me to learn a little more about the "mean" theorem (which obviously doesn't exist...). The obtained response is worthy of the best texts from the Bogdanov brothers [12]... Then, adding the keywords regular polygons, here are some additional "surrealist" Fermat's theorems: Then, can we trust Bard/Gemini "experimental"?

Bard/Gemini "experimental" 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 Bard/Gemini "experimental" to its hallucinations...





4 - Progress:

Thus, Bard/Gemini "experimental" 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 Bard/Gemini "experimental" (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 [13]? And if so, are we still far from having machines capable of introspection, reflection, creativity, and consciousness [14]? Unfortunately, it is impossible to answer this question...





6 - Conclusion:

With the help of the previous examples, we have seen that Bard/Gemini "experimental" is capable of answering numerous questions in a relevant and creative manner. However, unfortunately, as expected, it can also make serious mistakes.

It appears that Bard/Gemini "experimental" 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 Bard/Gemini "experimental" 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 Bard/Gemini "experimental" more human (errare humanum est)?

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

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

  • [05] - At least, that's what we can hope for after the revelations that will follow.

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

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

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

  • [09] - This goes against intuition. Let us recall, by the way, that the cardinalities of the integers, even numbers, prime numbers, rational numbers or again algebraic numbers are all equal (which is actually very easy to prove)!

  • [10] - 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).

  • [11] - 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.

  • [12] - Here are some particularly delightful (but most likely unintentional) excerpts copied without any alterations from the book AVANT LE BIG BANG by Igor and Grichka Bogdanov, published by Le Livre de Poche in 2006: Finally, we note the enthusiastic prologue by Luc Ferry!"

  • [13] - 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...".

  • [14] - 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, 2023-2024.
    Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / École polytechnique, Institut Polytechnique de Paris, 2023-2024.