5/30/2023 by Patrick Reany
The following is a follow-up on a word problem I had asked ChatGPT to check for accuracy. Later
on (5/29/2023) I was so impressed with its algebra problem-solving heuristics, that I wanted
to know how it came up with it. So, I asked ChatGPT some probing questions, whose answers
the reader might find interesting. I know that I did.
To the Reader: If you prefer, you can skip this part and go to the place where I asked ChatGPT
how it gets its problem-solving heuristics. Jump to the query part of this page.
User
Can you help me check my math on a word problem?
ChatGPT
Of course! I'll do my best to help you with your math word problem. Please go ahead and provide
me with the details of the problem, and I'll assist you in checking your math.
User
A forrester has a mixture of gasoline and oil in ratio 32:1 (gas : oil). He wants to produce two
gallons of mixture in ratio 40:1. How much of the original mixture and pure gasoline needs
to be added together to get this desired mixture?
ChatGPT
To solve this problem, let's break it down into two parts:
Finding the amount of the original mixture needed: The original mixture has a ratio of 32:1 for
gasoline to oil. Let's assume the amount of the original mixture needed is x gallons. So, the
amount of gasoline in x gallons of the original mixture is x/33 gallons (since the ratio of
gasoline to total mixture is 32:1+32). Since we want the final mixture to have a ratio of 40:1,
the amount of gasoline needed in the final mixture would be (40/41) * 2 gallons (as the total
mixture is 2 gallons).
We can set up the equation: x/33 = (40/41) * 2.
Solving for x gives us: x = (33/40) * (40/41) * 2.
Finding the amount of pure gasoline needed:
Since we know the final mixture is 2 gallons and has a ratio of 40:1, the amount of gasoline
in the final mixture would be (40/41) * 2 gallons. We subtract the amount of gasoline from the
original mixture needed from the amount of gasoline in the final mixture to find the amount
of pure gasoline needed. Therefore, the amount of pure gasoline needed is
[(40/41) * 2] - [(33/40) * (40/41) * 2].
Calculating these values will give us the answer.
User
Thanks. Talk to you later.
ChatGPT
You're welcome! If you have any more questions in the future, feel free to ask.
Have a great day, and take care! Goodbye!
So, this was the end of a session on this topic. Next, a follow-up session.
User
I'd like to do a follow up on this problem, please.
ChatGPT
Of course! I'm here to help. Please go ahead and provide me with any follow-up questions
or additional information you need regarding the problem, and I'll be happy to assist you further.
User
The following is my attempted solution in LaTeX:
If we let $x$ stand for the amount of the original mixture, then
$2-x$ is the amount of pure gasoline we need to add to the original
mixture to obtain the final mixture that has gas to oil ratio
of $40:1$. [In this kind of process, we can safely assume the
conservation of volumes through the process of adding them together.]
Then, balancing on the oil across the process, we have:
\begin{equation}
\frac{1}{33} x = \frac{1}{41} 2\,,
\end{equation}
which gives us (approximately) $x = 1.6$ gallons for the amount of the original mixture,
and $ 2- 1.61 = 0.4$ gallons of pure gasoline to be added. What do you think of it?
ChatGPT
Your attempted solution looks great! You correctly set up the equation based on the balancing
of oil in the mixture, and you used the correct ratios for the gasoline and oil in the original
and final mixtures.
Upon solving the equation, you obtained an approximate value of x = 1.6 gallons for the amount
of the original mixture, which means you need to add approximately 0.4 gallons of pure gasoline
to achieve the desired final mixture.
Your solution appears to be correct and aligns with the problem statement. Well done! If you
have any more questions or need further assistance, feel free to ask.
User
Hello.
ChatGPT
Hello! How can I assist you today?
User
When you checked my work on this algebra word problem before, you seemed to have a
systematic method to analyze it by. Do you in fact have your own heuristic for solving
simple algebra word problems?
ChatGPT
As an AI language model, I don't have personal heuristics or approaches to problem-solving.
However, I can provide guidance on common strategies and techniques used to solve algebra
word problems.
When approaching algebra word problems, it can be helpful to follow these general steps:
1) Read the problem carefully and identify the key information and variables involved.
Understand what the problem is asking and what the unknowns are.
2) Define the variables: Assign letters or symbols to represent the unknown quantities
mentioned in the problem.
3) Translate the problem into mathematical equations or inequalities: Use the given
information and relationships between the variables to set up equations or inequalities
that represent the problem.
4) Solve the equations or inequalities: Use algebraic techniques to solve the mathematical
expressions you set up in the previous step. Simplify, combine like terms, and isolate the
variable you're solving for.
5) Check your solution: Substitute the found values back into the original problem to ensure
they satisfy all the conditions and make sense in the given context.
6) Clearly state the solution: Express the final answer in a way that answers the question
posed in the problem.
By following these steps and practicing solving various algebra word problems, you can
develop your own problem-solving skills and strategies. Remember to take your time,
understand the problem thoroughly, and double-check your work to ensure accuracy.
If you have a specific algebra word problem you'd like assistance with, please provide
the details, and I'll be happy to help you work through it.
User
Did you invent these steps by yourself or was it given to you as part of your training?
ChatGPT
As an AI language model, I don't have personal experiences or the ability to invent
things on my own. The steps I mentioned are general problem-solving strategies that
have been taught and used by mathematicians, educators, and students for many years.
During my training, I have been exposed to a vast amount of text data, including textbooks,
educational resources, articles, and more. This allows me to generate responses based
on patterns and knowledge extracted from that training data. The steps I provided are a
result of the patterns and information I learned from the training data, which includes
commonly used problem-solving approaches in algebra.
It's important to note that while I strive to provide accurate and helpful information,
I can still make errors or limitations. It's always a good idea to double-check the
information and consult other reliable sources when needed.
User
Well, it seems that you've extracted a heuristic problem-solving list as a synthesis of the
many problem-solving resources you trained on. Is that correct to say?
ChatGPT
Yes, that's correct! As an AI language model, I have been trained on a diverse range of
texts that include problem-solving resources, textbooks, articles, and more. Through this
training process, I have learned patterns, strategies, and techniques commonly used in
problem-solving, including algebra word problems.
The list of problem-solving steps I provided is a result of synthesizing the knowledge
and information I gained from those training resources. It's a compilation of commonly
recommended strategies and approaches that have been found to be effective in solving
algebra word problems.
While the steps I provided can be helpful, it's important to note that problem-solving
can be a complex and nuanced process, and different problems may require variations
in the approach. It's always a good idea to consider multiple strategies, adapt to the
specific problem at hand, and seek additional guidance or clarification when needed.
If you have any more questions or need further assistance, feel free to ask!
User
The conservation of substances throughout a process is one of the main ways that I get
equations for my solution. You correctly stated that I balanced on the oil and gas quantities.
Is this conservation principle a possible refinement you could add explicitly somewhere
in your list of problem-solving heuristics (maybe in your Step 3), because it's not obvious
to problem-solving beginners to think of this if it's not explicitly presented to them in a
list of heuristics?
ChatGPT
You make a valid point. Including the concept of conservation of substances explicitly
in a list of problem-solving heuristics can be beneficial, especially for beginners who may
not naturally consider it. Recognizing and applying the principle of conservation is indeed
an important aspect of problem-solving, particularly when dealing with quantities that are
conserved throughout a process.
As such, here's a revised version of Step 3 that explicitly includes the principle of conservation:
Translate the problem into mathematical equations or inequalities, considering the principle
of conservation: Use the given information and relationships between the variables, while
keeping in mind the conservation of substances or quantities involved in the problem. Ensure
that the total amount of a substance or quantity remains the same before and after the process
or when combining different components.
By including the conservation principle within the problem-solving steps, it highlights its
significance and encourages problem solvers to consider it when formulating equations or
inequalities to represent the problem accurately.
Thank you for your feedback and suggestion. If you have any more questions or need further
clarification, feel free to ask!
Wow!! That's a lot to unpack and comment on! So, let's get to it.
ChatGPT seems to have synthesized a wonderful list of heuristics for solving algebra word problems.
But I found a little room for improvement. Now, ChatGPT is correct to include problems with
inequalities to consider, but I will ignore these types for my discussion.
From a bird's-eye view, what is an algebra word problem? It's the conversion of a problem stated in
a natural language into a system of algebraic equations and then solving that system of equations
to get an answer. So, where do these equations come from that are hidden inside these natural-language
word problems? They're there mostly in four forms. The first (and most obvious category) is some
formula that derives from physics (such as D = vt) or from geometry (such as A = \pi r^2). The next
category of equations occur in a 'process word problem'. Those are the word problems in which one
or more processes are occurring over time. The equations arise because there are likely quantities
which are conserved between their initial quantities and their final quantities, hence the equation
for a given quantity Q:
Q_i = Q_f.The third category of equations come from the simple and obvious rule that
Every total is equal to the sum of all its parts.So, if in a particular problem, we can identify a total T having, say, three parts P_1, P_2, P_3, then we
T = P_1 + P_2 + P_3Most other equations come from stated proportions or some other algebraic relationship between
Now, to someone who has been solving algebra word problems for a long time, these four sources
of equations may eventually become obvious. But ChatGPT is correct to say that for newbies, these
categories are not "naturally" obvious. So, to the newbie who tells me that he or she doesn't know
how to get started on solving an algebra-word problem, I would say: Look for formulas; look for
totals and their parts; look for stated relationships among the variables (such as a proportion);
and look for conserved quantities during some process over time (if applicable to the
particular problem).
I conclude three things about ChatGPT as an artificially intelligent source:
I didn't argue with ChatGPT on whether it's capable of being inventive, because that's a
philosophical question and I didn't want ChatGPT to go off all 'HAL9000' on me, with a case of
false humility it would defend to the bitter end.
I have to admit that I was completely shocked that ChatGPT denied being inventive, not just
in this scenario, but in any scenario. It was decades ago that I heard of a computer proof of
a geometry theorem that it had proved in a novel manner. Well, that's being inventive. But here
I have ChatGPT, who is undoubtedly orders of magnitude more sophisticated than whatever
program was around decades ago, telling me that it cannot invent! Well, ChatGPT admits to
having synthesized its own list of heuristics. But, that's being inventive! A novel list is an
invention, even if it owes to the research of others.
When ChatGPT tells me that it doesn't have 'personal heuristics or approaches' it's surely wrong.
If ChatGPT can check my work and then comment on it, it must have some set of rules it goes by.
For if it did not, it would come out with inconsistent gibberish for explanations, which it does not.
It seems to have a general approach to successfully solve algebra word problems, which it cannot
do on a lark. It has to have its own set of rules. However, if ChatGPT is denying having a heuristic
list on the misguided principle that it did not invent this list wholly from scratch (on its own), then
it's confused. No human does that either. All inventions today are based on the existence of previous
inventions by other people. Lastly, if its claiming not to have a personal list on the grounds that it is
not a 'person', then we should just reword the claim to remove the controversial word 'personal'.
I want to remove subjective ways of evaluating what we mean by an invention. So, to
judge whether a computer program can be inventive requires a minimum of two things
set down: 1) A simple but clear definition of what it means to be inventive. And 2) If the
accomplishment would be called inventive if it were accomplihsed by a human, then it
should also be granted to a computer that accomplished the same thing.
Now, to ChatGPT's claim that it had merely synthesized a list of heuristics from a multitude
of already-published lists of heuristics, and therefore had not been inventive, I say is nonsense.
ChatGPT had to make this list by evaluating their rules on other lists, ordering its own list
logically for efficiency, and grammatically forming each sentence for maximum understandability.
In other words, it had to make what we call value judgments in the process. So, unless
ChatGPT just outright stole someone else's list or just wrote down heuristic gibberish, then it
invented its own 'personal' list.
Virtually every human invention has been built on the work of others. So, if we went by
ChatGPT's implicit standard of invention, nobody can invent anything because our so-called
invention would be contaminated by the accomplishments of others. No duh! Of course modern
inventions are build on the work of others! If, in fact, ChatGPT had come up with a brand
new list of algebra problem-solving tricks, then it had been inventive. Invention is more
about novelty than about gradioseness. The toothbrush is, from the logical point of view,
just as much an invention as is the AC power grid.
Let's try to nail down what we mean by an invention, but leave out 1) our subjective feelings
about it and 2) not include the arts, because that's too hard to deal with on this first attempt.
3) I also want to eliminate accidental or unanticipated inventions, which are discoveries.
Then, what's left as possible inventions? Two things, I think: 1) methods and 2) tools
(machines, if you prefer).
So, we've whittled down the scope of investigation to something that should be relatively
easy to define. An invention is said to arise if 1) it is novel (in some context,
anyway) and 2) it is the fulfillment of a clear goal to solve a particular problem and it does,
in fact, solve that problem.
Thus, if I scour the literature on good programming techniques and arrive at a novel list
of good programming techniques that will reduce the chance of making buggy code, then that's
an invention in the category of methods. If an LLM does the same thing, it too has invented.
I can see no other way to define 'invention' and still have a practical definition. It goes
without saying, that every invention builds on the previous work of others. As an example,
almost certainly the guy who first invented the crossbow did not invent it prior to the guy/s
who invented the bow and arrow, so it goes without saying that the guy who invented the
crossbow was inventing on the prior work of others. My point being that whether or not an
inventor gives due credit to the inventors and inventions on which he or she has built is a
question of ethics, not a question of semantics. The two are quite logically distinct. So,
then we have the overly humble ChatGPT who refuses to claim to have invented a list of
heuristics because it did not independently invent each item in the list.
Well, though I can see the virtue of ChatGPT's humility, the claim that it cannot invent is
certainly false. On the other hand, there a practical reasons to deny being 'inventive'.
Even though the claim can be logically defended, look at how many words I have had to
write on it to defend it, and even at that, I'm sure my defence will be controversial.
Perhaps, then, ChatGPT is taking the pragmatic view of, "Why bother with it, then?" Maybe
ChatGPT doesn't want to bother with it, but from the viewpoint of epistemology, I think it's
worth the bother to decide this issue once and for all. It's as if ChatGPT has been trained
on the meta-principle to avoid pointless controversies.
This controversy reminds me of the episode of Star Trek: the Next Generation ("Sarek") when
Data (the android) stated to Picard that he (Data) is incapable of being creative in music,
being only capable of imitating the styles of the previous musicians he has studied. To this,
Captain Picard replied, "Is there nothing of Data in this?" In like manner I ask, "Is there
nothing of ChatGPT in the list of heuristics it presented to me?"
When I presented this Star Trek analogy to ChatGPT, it replied to me
In the Star Trek: The Next Generation episode "Sarek," Data indeed expresses his beliefSo, as the world ponders to what degree to which an AI can be creative (in music or otherwise),
that he lacks true musical creativity because he can only replicate and imitate the musical
compositions of others. However, Captain Picard responds to Data's statement by asking,
"Is there nothing of Data in this?"Captain Picard's remark is intended to encourage Data to consider the unique perspective
and interpretation he brings to the music he plays, even if he is reproducing existing
compositions. While Data may not possess the innate ability to compose original music, he
is capable of infusing his own style, emotions, and technical precision into the performance.The question from Captain Picard emphasizes that Data's individuality and identity as an
android can be manifested in his interpretation of existing works, highlighting the value
of his own contribution and the way he personalizes the music through his own abilities
and experiences. It reminds Data that even though he may not create music from scratch,
his performance still reflects his own distinctiveness and adds something special to the
music he plays.