I think we can all agree that reading Dragon's Egg was a bit strange. I still can't completely picture what a cheela looks like or figure out why they call north and south the "hard directions". However, it was an interesting story, and its lesson is hard to miss: math helps us survive. As human beings, we must utililze multiple aspects of mathematical thinkings to help us overcome daily situations. While the story does a great job of illustrating just how great math can be, it fails in regards to another facet of this course: ethics. Now, I understand that the math told the leader of the tribe that they didn't have enough food for the entire tribe to make it, so she made the logical choice of leaving the elderly behind. Ethically, however, was that the right decision? When the elder made the long journey to deliver food for the infants of the tribe, did the tribal leader have to just take the food and send the starving hero back to die? I guess what I'm saying is that this story really told me two things: one, that mathematical thinking is and must be used to ensure the survival of our species, and two, that mathematical thinking devoid of anything else is a sure-fire way to make some horrible decisions.
If, as the second reading seems to suggest, mathemetical thinking is indeed an innate process, one that we are virtually born with, then it must be tempered with the other abilities of the human species. It is all well and good that humans can subitize - but so can chimps, pidgeons, and rats. Advanced mathematical thinking should not be the only thing separating us from our animal brethren. I agree that it is amazing that the brain - not just the human one - is capable of such abilties from birth. It suggests that mathematical thinking is a key component of how beings make sense of the world around them - a tool given to the creatures of creation to help them understand existence and stabilize their lives. But, since it is not a unique abilty of the human race, we must ensure that we allow all the various virtues and vices of our species to influence the way we make choices and take actions; if we do not, we'll be acting about as human as a rat.
You bring up a good point about both readings: math is a crucial component of our lives, but we need things other than math to help us. And I agree. While both readings are very much about math, the idea of what utilization of math makes us is an interesting angle to consider. You touch on it briefly with the ethics thought. I understand that sending the elder back was not necessarily the greatest decision, but in the given situation, it might have been necessary and unavoidable. I’m not saying this is what I would do, but it comes down to the dilemma of whether the lives of a few are worth sacrificing for the good of many. And this opens up a whole other Pandora’s Box of questions, one of which asks what makes someone a good leader. Is math (or logic) what allows us to make good decisions both in leadership positions and daily life?
ReplyDeleteIn crisis situations, I think math is used more often than other concepts because of the objectivity of math and logic. In daily life though, I think math is support or a foundation for our decisions, but not the only thing that helps with decisions making. People often justify their actions with logic. On the other hand, we’ve all heard of people who use logic too much and lack compassion or are too emotional and not objective enough. Simply put, it comes down to the, “Think with the head, or think with the heart?” idea. But I think using either solely would only work for so long before a balance needs to be attained. And there have been various iterations of the idea that there needs to be a balance. To bring in some literature/philosophy, Plato’s The Republic talks briefly about balance of the various parts of the soul. Achieving a balance of all components, according to Plato, was the greatest virtue. To do so, there needed to be a balance of reason, spiritedness, and appetite which can be analogous to the mind (logic), heart (emotions), and gut (impulses), respectively. Obviously, if one thinks solely with logic, emotions, or impulses, good decisions will not be made. So maybe it is our ability to apply math to situations, and not utilize it as the sole source of information, that helps us make decisions as we do.
**Oh wow... apparently I wrote too much so I'm going to post the rest of my comment as a reply to this one**
ReplyDeleteGood stuff, man. Love the quick focus on ethics, and I very much agree that our decisions (not just as humans but also) as living creatures cannot rely solely on the input of any one source - misguided logic can be just as dangerous as blind philosophy/ethics.
Unfortunately, when it comes to ethics itself, I find I must disagree with you [at least partly] in terms of the situation of the Dragon's Egg story. If we are to even begin trying to introduce the philosophy of right and wrong into this scenario, we must analyze it from its very definition. Now I suppose this is more or less of a personal opinion (because who among us can rightfully claim an objective definition?), but I have always figured that the definition of ethics is wholly relative and constantly evolving based on the circumstances in which one finds oneself. Oh sure, there are always ethical 'universals' like "It's wrong to end the life of another" and yet more than once there is a major dilemma of the story that puts that code to the test when the main character is forced to take actions that will inevitably kill (albeit indirectly) members of her own tribe [tribe? hive? community] in the hope of preserving the lives of the rest. And, in what appears to be a fairly primitive society, we have already developed a strict assumption that utilitarianism - not absolute morality - is necessary to ensure survival (which is also curious that these cheela seemed to have little to no concept of most math or arithmetic and yet would willingly submit to a utilitarian philosophy that incorporates the basic understanding that the needs of the many outweigh the needs of the few).
From there, I posit the very crude definition that, in its most basic form, the moral codes of ethics were designed/brought about chiefly for the survival of the community (and, to a lesser extent, the betterment of its individuals' quality of life). And, given the sheer variety of the term "community", such an understanding of ethics would furthermore center around situation and circumstance until one might conclude that the foundation of ethical differences is cultural and historical relativity - that is, what is morally "right"/appropriate for individuals of a certain community can be quite different from what is "right" for another society, or even the same society at a different period of time. And because of that variation in circumstance, it might be very generally said that each of those communities' moral codes/actions are comparatively just as relatively "right" as the morals of any other situation since they all set with the goal of the survival of the community - though I by NO means assume that this is true in all cases (lest I accidentally defend the horrors of slavery).
.....wow that took a lot longer to say than I had wanted... ANYway I suppose my point with all of that was to heavily stress the fact that ethics is so very much of an ambiguous and relative topic that, as I alluded to before, it must also be tempered by logic. Sure, it's easy for us as 21st centurions - pioneers of political correctness - living in the good ol USA to read this and feel appalled that the main cheela would calculate decisions such as veritably "aborting" her premature egg in order to win the duel for leadership or so "heartlessly" foregoing the lives of the elders in order to more plentifully distribute food for the rest of the colony. But are we really in any place to talk? These fictional characters have spent generations evolving under ludicrously harsh conditions and have only survived as a species in a manner similar to other insects [I totally think the cheela are insectoid - they have to be] - that is to say their evolutionary path is such that the individual holds far less relative strength and longevity (and therefore much less importance) than a human individual and that they simply cannot be judged by the same circumstantial ethical guidelines [although one might say something different regarding the human author who created this story].
DeleteIn other words, I mean to say that perhaps, if put in their shoes [keep in mind that also includes restricting our mental affinities for arithmetic and higher math/problem solving/critical thinking], perhaps we might very well also come to the very same tough ethical conclusions regarding the needs of the many.
BUT, of course, this class and these readings are about far more than just ethics (which is pretty much all I've commented on so far) so I will begin to wrap this up by mentioning that I think it is very curious how math works in to the relationship between these two stories. I agree very much with what you pointed out/mentioned [though not explicitly, I think this was where you were going] about how mathematical abilities may be innate (and relatively universal across species), but what separates us from the rest is our ability to incorporate that crude sense of math into all other aspects of our mental knowledge and through it form a much fuller comprehensive understanding of the many problems of life that plague every individual.
DeleteEven a mouse can count, but it only call upon those faculties in order to get food or avoid "punishment" - or, more generally put, to survive. It is only humans [as yet] that can take the concept of that number and link it to a billion other things, incorporate it into all aspects of their life, and through it form infinite connections to anything else even remotely related to that concept. Example? Three. How many things run through your mind when you just envision the concept of three? The arabic numeral 3, the 3 symbolic medals of recognition [gold/silver/bronze], the 3 most important little words ["I love you"], the ultimate pattern of 3 that exists in any dualist continuum of that we are forced to use to understand the world [positive, negative, and neutral], perhaps you're religious and also think of 3 persons in one God or the fact that Jesus was resurrected 3 days after his crucifixion.......THE POINT is that rather than being forcibly limited in our mental faculties to pigeon-holing the applications of our knowledge, we can instead take one tiny little concept and apply it to anything and everything and, through it, understand the workings of the world exponentially faster than any other creature.
And, to tie this up into the last thing we talked about in class (or at least the last thing I remember), the more we take each new concept we learn and apply it to everything else in our life, the more we understand of the relationships that form between all manner of things. Soon, the different relationships begin to resemble each other and before you know it we begin to formulate patterns by which each relationship occurs and, taking that even further, we observe patterns of patterns that lead to comprehensive wisdom and eventually even predicting the future.
DeleteThus, the key to truly knowing anything is to comprehensively understand everything.
For what ever reason, I chose to read the math as an innate character text before reading the dragon's egg text. After finishing the first reading, I was a bit skeptical to say the least. It's not that I didn't agree with what the authors said, I just wasn't convinced that the time a baby sat and stared at an object really correlated with some innate ability to count and do simple arithmetic.
ReplyDeleteWith these doubts in mind, I read the dragon's egg reading. I think it was really important for me to read the innate math reading first because I was able to better connect the dots in the dragon's egg story and realize how ingrained these simple concepts of counting and math really are in our everyday lives. It wasn't until I saw an example of someone who didn't have those skills that I realized how true the first article was.
From what I’ve seen I could easy see some logic to the possibility that humans might be born with some intrinsic form of arithmetic that is somehow coded into our subconscious. The arithmetic I am speaking of might manifest subconsciously and discovered through means of physical displays of simple understanding through reasoning and logic. Based on no facts whatsoever and only on my personal experience; I am of the opinion that even at young ages children can show that they are capable of understanding basic forms of arithmetic through ‘semi’ rational thought. If for example it you were to give two children candy in different quantities, say child A gets one piece of candy while child B gets 3 pieces of candy, for their good behavior at the supermarket. Child A, who does not receive the mathematical equivalent of candy will now use their inherent sense of basic reasoning and logic to become upset by this injustice and will demand that they be given the same equal shares candy. Even though child A has not yet been taught to consciously understand rudimental mathematics such as addition and subtraction they still project an inherent form of logical mental persuasion. In this scenario the candy represents a means for a child to project a mathematical argument through very basic cognitive thinking.
DeleteI think that even at very young ages we are capable of picking up from their conscious existence very rudimental arithmetic without being taught such concepts using our innate sense of fundamental logic and reasoning. The Question is, if we see these examples of modest mathematical thinking in babies, toddlers, and young children; are they learning through visual interpretation or an innate aptitude for metallization.
Brandon, I thought that you had a good point with the role that "mathematical thinkings" have on in our daily life. I would also say that my own opinions on the second reading closely match yours. However, like Matthew, I would have to disagree with the role mathematics plays in ethical decisions inside the text of Dragon's Egg.
ReplyDeleteI believe that the point that the author is trying to make is the question of ethics (was the decision to leave the elders behind good or bad) does not arise without an understanding of the numbers (math). Without understanding or realizing there is a problem (the symbolic representative of food left did not match the symbolic representative of cheela's still alive) the decision to leave the elders behind for the good of the group does not contain an ethical dilemma.
I think this relates to the Brain's Innate Arith. Reading in how basic math exists in our brain, which can help us understand how it started and why it has evolved. But as we understand the world around us using math, we introduce problems, some ethical.
One last thing: I think an interesting point is the understanding of time (clicks I believe) in the Dragon's Egg Reading. This takes some sort of math understanding. It also brings up how relative accomplishments can be dependent on the relativity of time. Imagine if our days were half as long?
I think the two readings really lead me to think about the most basic of building block in modern mathematics: definitions. In the Dragon's Egg short story, it was immediately obvious that these insectoidal creatures lacked even the most basic mathematical notions beyond counting to three. Despite these limitations, Great-Crack was able to define lingual representations for concepts of numbers greater than 3. Ignoring the prime inspiration for a moment, the notable thing here is that this situation illustrates something important about mathematics: it is constructed. That is to say, our heroin had to create a lingual means to convey conceptual knowledge concerning a particular mathematical idea like a number past three or the notion of a bijective correspondence (each seed "represents" a pod).
ReplyDeleteThe question then becomes, does our invented verbiage and symbolism convey constructive frameworks or is it merely descriptive, that is to say, a means of discussing preexisting truths merely stumbled upon by a series of hairless primates? I think a lot of answers to questions imposing two options is that the question poses a false dichotomy. Consider this: say you have a box full of legos but no manual. Now, you didn't make the box of legos, they were gifted to you by Uncle John who used to play with them back when "he was your age." Taking it a little further, lets say there are legos of every size, shape and color you could possibly imagine (no doubtedly a massive box of legos). One day, you realize you can build a big stack of legos using only one of the various types of legos (think a bunch of 2x2 blocks stacked). Well, back in the day your Uncle made that same design but never told you about it. Did you discover the arrangement of lego blocks or did you create it? If we take a linear, time-centric viewpoint we can reasonably conclude that one discovered a pre-existing but known configuration of the blocks. But, perhaps the world is stranger than it seems and we really live in some sort of deceptively linear in recollection, but highly nonlinear experience of time. In such a world, all bets are off to who was the first to create this particular arrangement of legos. The key here is that you both constructed something influenced only by the prime building blocks chosen. The point being, certain aspects of mathematical knowledge is discovered and other aspects are merely stumbled upon or are implications of certain properties. In particular, it seems that various properties are seemingly discovered either through idle inspiration or experiential matters but implications are dependent on the selection of 'discovered' properties and how they lock together to form a coherent statement or piece of deductive abstraction.
The idea behind Dragons Egg, as I understand it, was that relating mathematics to the real world and back again is an abstract idea. The application of mathematics, which is represented by associating seeds with their source of food, was used by one creature to show that they would not have enough food to make the long journey. While only one creature understood this over time many learned to use math and associate it with the real world, thus inferring that while association with mathematics is abstract mathematics is something that can be shared and learned. I also feel that the main character represented different thinking due to the fact that she kept collecting trinkets that she found during hunting party excursions. This seems to infer that some minds are more aligned towards thinking with math where others aren't. And as a final piece of the Dragons Egg, I feel that a major barrier for the sharing of mathematical abstract thinking (teaching others) came primarily from difficulties in language. The characters had to count using a set language system they created, meaning that their base for counting is not necessarily a base 10. The characters also had to teach each other which proved more difficult than imagined and this could be largely due to language barriers associated with trying to convey complex ideas.
ReplyDeleteThe second text, The Brain's Innate Arithmetic, was more interesting for me because I didn't know babies could intuitively count. But I am still skeptical on this part of the text. I thought that this was very well thought out but that it didn't necessarily show a innate knowledge of mathematics. It seemed that only after 3 to 4 days did the babies react to the change in expected numbers of puppets. My question is: what were the babies doing for the 3 to 4 days prior to this? My answer is observing the world. It seems that our brain is basically an organic super computer and that babies are no exception to this. This test could show that for the 3 to 4 days prior to the test the babies were observing our reality and "learned" some fundamental laws of conservation. While this does imply an abstract application of mathematics it doesn't rule out that babies learned this for observation rather than having an innate command for math. The other studies of the text pertaining to humans showed the different specializations of the brain for the ability to comprehend and command math while the animal sections showed that animals brains are a more basic form of organic computer than learn from observation of their surrounding world.
Brandon I think it's very interesting that you bring up the decision to leave the elders behind in Dragon's Egg as an ethical issue. The use of math in that decision supplied the leader with the knowledge that the entire clan would not make it so most reasonable solution for the majority of the clan was to leave the elders behind and lose a few instead of many. When reading this I immediately related it to economic decisions made everyday in the world that are morally questionable. One specific case I thought of was the case against Ford Motors over a faulty part in the Ford Pinto in the late '60's. They made the decision to distribute these cars despite this dangerous flaw because they calculated the cost of replacing the part to be more expensive than the cost of the law suits against them for injuries and death due to the defective part (http://www.engineering.com/Library/ArticlesPage/tabid/85/ArticleID/166/Ford-Pinto.aspx). This example instantly came to mind when reading Dragon's Egg. The decisions made based on math may seem ethically wrong or immoral at times, but they also may cause one to avoid larger consequences or repercussions that may overshadow the consequences of the math based decision. It is difficult to look at cases such as these and decide if the decision was morally right or wrong. However, logically leaving the elders behind most likely benefited the majority of the clan.
ReplyDeleteI at first was extremely perplexed about “The Brain’s Innate Math.” I was uncertain how this was proof of actual math occurring. I realize the child could tell the removal and presence of the objects but not how that correlated to mathematical skill. I agree with Cole. It didn’t make sense to me how this could be proof of basic arithmetic.
ReplyDeleteIn the “Dragon’s Egg” I thought it interesting how I didn’t quite grasp that the basic math and counting was a skill that not everyone had. I almost just assumed that these creatures had this skill and was confused at first how the main characters’ counting was so life shattering. It helped me realized just how much we rely on math in our daily lives and simple actives.
For a story that needed some serious help in its story writing, I thought the ideas in the "Dragon's Egg" were very integral in our lives, just as a lot of you mentioned. Math is very necessary for our survival. An example of this in real life is budgeting. If one does not budget, or not well enough, they could possibly not have enough to buy food at the end or the month. The chart that Great-Crack made reminded me of a multiplication table or an early abacus, the way she laid all of the stones and pods out. I also had to suspend belief for a minute when their number system stopped at a "dozen" and a dozen's dozen was a great (our gross). It's such an abstract concept and they could have easily stopped at 8 or 14 instead of 12, par se. What is so special about "12" that a second "society" would make it integral in their lives? Also, it really surprised me when she got so upset the elder was following them. I don't understand why she was so adamant that the elders not leave the old home at all; her dealing with the one seemed so harsh!
ReplyDeleteI understand the skepticism surrounding "The Brain's Innate Math" whether or not babies can count using the time they stare at objects; I have seen this experiment before in some of my classes. I think a more accurate conclusion from the study is that babies stare longer when they are not allowed to see a change that occurs, and their staring is like a double-take (they don't understand what happened). Even at this point in our lives, if something startles or surprises you, chances are high that you'll look again and/or stare to try to make sense of it. Also, in this reading, I thought it was very interesting that rats could do basic arithmetic. I knew about things such as Skinner's Box, but I did not realize arithmetic figured in so substantially. Instead, I previously thought it was more a sequential pattern they learned.
I think I reread "Dragon's Egg" three times and did a little extra looking around the internet about it. What I found out about it, almost confused me even more. Epic failure on my part. I kept looking for understanding of more than the mathematical aspect, like Brandon stated, he couldn't picture the "cheela", my brain was actively searching to put an image to the word. This happened a lot throughout this story, which was interesting to see how it affected my personal relationship with the story.
ReplyDeleteJoelle, I really like your example about budgeting. I think it's interesting to look at where math is actually necessary for our survival verses where math is in our society. Like math is needed for calculating money, food, water supplies and everything. Yet it is also see in items we enjoy and consume like clothing items with patterns and number of sizes, paintings, and any item that is created from a large factory. The time, materials and everything needed is calculated to ensure quality in a product.
In reading the second article I found it very interesting how they tied the time a baby stared at an object to correlate with knowing numbers. I have worked with young child, and was at preschool today and wondered how a student with special needs would fair in this experiment. The whole idea of conservation also popped into my head while reading this. If it was changed to an amount pour into an object instead of numbers would that make a difference. Children usually don't realize/know the law of conservation until six or seven years old.
Similar to Joelle also I remembered about skinner's Box but didn't realize that is was a specific number of times. I think it's interesting that we test animals math abilities, when it is something we humans created.
Alot of people above have talked about the ethics of Dragon's egg and how Great-Crack's dealing with the aged ones was too harsh, particularly with the "starving hero" mentioned by Brandon. I'm not one who loves playing devil's advocate, but I feel that to analyze the ethics of the situation you have to put yourself in Great-Crack's place; we should not assume that any of the ethical standards that we hold apply to them. For example, the idea of waiting to die didn't seem to trouble many of the aged ones that were being left behind. It appeared like a sort of duty, as if obeying the leader of the clan is of higher ethical value due to the maintaining of order as opposed to trying to survive. In my opinion the aged one that attempted to join the group was probably violating Great-Crack's word, and more importantly, her logic. Great-Crack had painstakingly decided that in order for many to survive, exactly a dozen must not eat. We have no way of knowing whether or not if letting the "starving hero" join the group would've affected the outcome, so, as far as I can tell, we must assume that they are using a different system of ethics than us. We're taught to give little value to "arguments from authority," but that was not the case in that particular clan of cheelas.
ReplyDeleteAlso, similar to many of the postings above, I was surprised and impressed at the information provided from the psychological studies concerning babies and math. The idea that people are looking at the brain's structure to understand how math works is interesting, and I've heard that Einstein's brain had a significantly higher lipid count, so I wonder if there ever could be a sort of scan that analyzes how "good" someone is at math. Overall, both readings were interesting.