“Ideas came with explosive immediacy, like an instant birth. Human thought is like a monstrous pendulum; it keeps swinging from one extreme to the other.”
--Eugene Field
“Mistakes are at the very base of human thought, feeding the structure like root nodules. If we were not provided with the knack of being wrong, we could never get anything useful done.”
--Lewis Thomas
Defays continued his discussion on Numbo and how it functions and raises some interesting points. He reviewed how Numbo relates to human thought, which is something that I have rarely seen stated in learning about specific computer programs. He made some claims, most of which I agree with wholeheartedly, about how Numbo resembles human performance.
He stated that, "Ideas are not systematically explored, and in fact are often abandoned before having been fully examined"(pp.150). This is true not just to solving math problems but to nearly every aspect of human life. It seems ideas in all fields are subject, such as, but not limited to, social & economic reform, music & movie reviews, and even something so simple as deciding what to have for dinner. Humans just aren't always systematic and often give up on ideas before considering them fully, two characteristics which I often consider flaws.
Defays continued with, "The combinations created are not always strongly goal-driven, so that unmotivated-appearing avenues are occasionally embarked upon"(pp.150). This relation to human thought processes seemed so apparently obvious to me. The first thing that came to mind was, 'Numbo daydreams too?' and as silly as that sounds, the analogy made sense to me and helped clarify how the temperature and the urgency systems in Numbo operated. It also made me consider that not all 'unmotivated-appearing avenues' are irrelevant and sometimes Numbo's and our wandering avenues can lead to solutions.
A third observation Defays made was that, "Solutions are often found that involve the chaining of several arithmetical operations in a seemingly logical way"(pp.151). While I didn't have many vivid analogies at hand when reading this, I couldn't deny its truth. In his detailing of Numbo's processing, Defays has shown this and I know humans do it too. Not just in math, of course, humans come to solutions of problems by way of linking different thoughts together, such as by using analogies. For example: analogy to better clarify a misunderstanding.
Although I do agree with these three similarities, I do not agree with his fourth, "Obvious solutions are found at once"(pp.150). Which is kind of funny because neither does he. He states earlier, on the same page, that due to the randomness of Numbo, "...it is sometimes possible to miss an obvious solution." And this argument, I believe, could have made a better comparison because occasionally humans do miss an obvious solution to a problem.
Overall, I give much credit to Defays' work on the Numbo system, I found it to be extremely interesting. I also greatly appreciate his discussion on how the program is similar and different to human thought. In the computer science courses I've taken for my cognitive science major it is not often discussed how the programs we use and write mirror, or don't mirror, human thought.
Something Like That
“And therefore education at the University mostly worked by the age-old method of putting a lot of young people in the vicinity of a lot of books and hoping that something would pass from one to the other, while the actual young people put themselves in the vicinity of inns and taverns for exactly the same reason.”
--Terry Pratchett
Daniel Defays' Numbo system is interesting but not quite as interesting, to me, as his rationalizing of why it operates the way it does. He claims that humans effortlessly use three types of knowledge to solve mathematical problems such as "rote small-number arithmetic (e.g., 6+1=7), ...knowledge of approximate sizes of numbers(e.g., multiplying 20 by 6 should bring us into the vicinity of 114), ...[and] procedural arithmetical knowledge (e.g., we can multiply 6 by 19)" (pp.135).
The third one mentioned somewhat surprised me because when doing math problems, such as crypto problems, I overlook that procedural knowledge I'm using because it is so basic. The second knowledge he mentions however, excited me in a nerdy way because it was something that I do often, not just in mathematics. "Just tell me how to get to NYC, we'll see if we can't find the hotel from there," is not an uncommon way to think about a problem for humans but I had never really considered it as a way to program a computer. This is just one way in which computer science has helped clarify my own ways of thinking. It amazes me how these simple thought patterns that humans use every day can turn into a year long programming project for a Belgian mathematician/psychologist.
Although Numbo is a wonderful work of art, it understandably still lacks many features which a human has when solving problems. One such feature is not having all learned processes always readily available to utilize when confronted with a problem. Another, which Defays specifically tried to correct, was that of association resulting in blindly-spreading activation leading to "uncontrolled, chaotic behaviors of the network"(pp.138). I'm not saying humans don't have some degree of control over their mind running through associations but instead that perhaps in some program, it should be embraced to better mimic humans. This happens all the time in humans, people losing focus or wandering off on some tangent. For example a typical student might be faced with a crypto problem, "Well, you can multiply 5 and 5, which is 25, which is like a quarter, four quarters in a dollar, that's equal to 100, I wonder how many licks does it really take to get to the center of a tootsie-pop, what was I doing?" Although something like this in a program would most likely never serve a purpose, other than to demonstrate association, I'm just saying that something error-like in a program doesn't always have to be rejected, especially when attempting to mimic human thought.
--Terry Pratchett
Daniel Defays' Numbo system is interesting but not quite as interesting, to me, as his rationalizing of why it operates the way it does. He claims that humans effortlessly use three types of knowledge to solve mathematical problems such as "rote small-number arithmetic (e.g., 6+1=7), ...knowledge of approximate sizes of numbers(e.g., multiplying 20 by 6 should bring us into the vicinity of 114), ...[and] procedural arithmetical knowledge (e.g., we can multiply 6 by 19)" (pp.135).
The third one mentioned somewhat surprised me because when doing math problems, such as crypto problems, I overlook that procedural knowledge I'm using because it is so basic. The second knowledge he mentions however, excited me in a nerdy way because it was something that I do often, not just in mathematics. "Just tell me how to get to NYC, we'll see if we can't find the hotel from there," is not an uncommon way to think about a problem for humans but I had never really considered it as a way to program a computer. This is just one way in which computer science has helped clarify my own ways of thinking. It amazes me how these simple thought patterns that humans use every day can turn into a year long programming project for a Belgian mathematician/psychologist.
Although Numbo is a wonderful work of art, it understandably still lacks many features which a human has when solving problems. One such feature is not having all learned processes always readily available to utilize when confronted with a problem. Another, which Defays specifically tried to correct, was that of association resulting in blindly-spreading activation leading to "uncontrolled, chaotic behaviors of the network"(pp.138). I'm not saying humans don't have some degree of control over their mind running through associations but instead that perhaps in some program, it should be embraced to better mimic humans. This happens all the time in humans, people losing focus or wandering off on some tangent. For example a typical student might be faced with a crypto problem, "Well, you can multiply 5 and 5, which is 25, which is like a quarter, four quarters in a dollar, that's equal to 100, I wonder how many licks does it really take to get to the center of a tootsie-pop, what was I doing?" Although something like this in a program would most likely never serve a purpose, other than to demonstrate association, I'm just saying that something error-like in a program doesn't always have to be rejected, especially when attempting to mimic human thought.
Intelligence: A Chance Occurrence?
“To dare every day to be irreverent and bold. To dare to preserve the randomness of mind which in children produces strange and wonderful new thoughts and forms. To continually scramble the familiar and bring the old into new juxtaposition.”
-- Gordon Webber
-- Eric Hoffer
Hofstadter talks about the intricacies of Jumbo and how it is a self-driven system, in that it has a flow of operation which is determined by factors of varying urgency and random numbers. He also claims that because of the great deal of randomness in his system that it may appear less 'intelligent' than it actually is. He does defend Jumbo by explaining that much of what we do is also random. Such things as deciding where to sit in a room, what to have for lunch, or what to wear on any given day (this is the case, at least, for most of the guys I know) have many weighted factors that may help you decide but these decisions can also be heavy with randomness. It is not in most people's best interest to take the time to rationalize out what their most viable option is for every decision they make. Just because we may not always be aware or willing to admit that many of our decisions are randomly driven, does not detract from our intelligence and could be an argument through simile for possible machine intelligence in Jumbo, or AI programs with randomness.
I enjoy that his defense of an intelligence that Jumbo may have is followed by a description of its epiphenomenal intelligence. He states that its intelligence, if any, was not intentional from its creators and emerged from the way in which the "small program-fragments interct with each other." I believe that our intelligence arises the same way, which could further link human and machine intelligence. I'm an advocate against "thinkodynamics", as Hofstadter puts it, or "laws governing thoughts at their own level." I am a staunch believer that our intelligence and our consciousness, if they differ, arose from largely random interactions between seperate parts, or functionings, of our brain, which themselves arose from more basic interactions among neurons.
So I do not believe that randomness in a system must deny that system of intelligence but perhaps that any system that is truly intelligent does have randomness in it. I believe that Hofstadter's views on entropy in machine and human systems and their relation to possible intelligence are ones that should be more widely considered by students and other computer scientists.
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