Chapter 36 in Morris Kline’s “Mathematical Thought: From
Ancient To Modern Times” discusses the profound importance of non-Euclidean
geometry and the process behind its creation. I found it interesting how this creation so greatly
influenced the world of mathematics by prompting mathematicians to rethink the
nature of mathematics in relation to the physical world. Again we see, similar to other great
findings, it is difficult to credit a single individual with the creation of
non-Euclidean geometry as the work of Lambert, Taurinus, Gauss, Lobatchevshy,
and Bolyai all played a role in its creation. As noted in this chapter, the process of creating
non-Euclidean geometry was affected and delayed by social acceptance and
publication. Do ideas today face a
similar process or has the “spirit of the times” completely changed? In the modern world ideas can be shared
globally on the web, which can provide the security of anonymity for those who
may fear criticism. The “spirit of
the times” in today’s society is perhaps more accepting and excited about new
ideas.
Kline also points out the creation of non-Euclidean geometry
endorsed pure intellectual curiosity as sufficient justification for exploring
any mathematical idea. I believe
this statement advocates abstract thought in math and other subjects, which is
to suggest an idea with no use or potential benefit is not necessarily
useless.
In “Mathematics in Western Culture” Kline discusses the
various types of geometry as relative to our perspective as “earth-bound
mortals.” This brings to question
why Euclidean geometry was developed first? The creation of non-Euclidean geometry radically affected
science and in many ways was made possible by breaking the habit of thought at
the time. I find it surprising how
existing definitions and theorems in mathematics help us better understand a
subject matter while also promoting a particular style of thought. Does the study of math lack creativity,
which potentially narrows our view and holds us back from further creation or
discovery? I can hardly get
through my modern algebra homework let alone make a mathematical discovery, but
what aspects of our society might affect progress or better understanding of
mathematical ideas as seen in the creation of non-Euclidean geometry? Perhaps the perspective of mathematics
as absolute or unflawed leads people to believe everyway to think of something
differently has already been thought of.
hmm... this is an interesting topic and you ask a lot of questions (and my answers are characteristically long) so I'll try to just focus on your last paragraph here..
ReplyDeleteDoes the study of math lack creativity? Personally, I don't think so at all, but I do believe that [as you mentioned at the end] the popular view that mathematics is absolute or flawless does create a sort of toxic learning environment that gives the idea that there is in fact no creativity. And can you really blame it? Math today is so far advanced that a person spends the majority of their educational career simply memorizing equations and relations because it's been proven time and time again and why should we bother learning to play around and figure out others ways of doing it if we already have a method we know works? But in my experiences with (slightly) higher level math courses, I found that more and more complex problems almost require waves of mathematical ingenuity - being able to see physical/logical relations and figure out new ways of arithmetically describing the same thing, if the old ways don't work. SO perhaps my answer would really be that maybe the early study of math (insomuch as many people today undertake it simply so they can pass the class and never have to use it again) does indeed narrow the creativity of the subject; but the further discovery of math seems to beg for it.
In a way that kind of touches on all but the first question in your last paragraph, but I also wanted to add something about that. Now I've gone through this Kline reading a couple times and quite honestly I still don't quite understand non-Euclidean geometry. I know it's got something to do with being inapplicable to the real world and, based on their example of Lobatchevsky's formulas, looks like it's taking trigonometry and converting it into complex numbers (numbers with real + imaginary components), which is strongly reminiscent of Euler's formulas, which were made some decades prior to these discoveries. So clearly I must be missing something; nevertheless, if I think I understand your question at all, then it would seem to me that Euclidean geometry was developed first because the perspective as "earth-bound mortals" thought that really was the way of the world. No one had ever seen an edge of this seemingly flat earth, so it could easily be presumed that physically two parallel lines would indeed go on forever because, so far as they knew, the world was infinite. Even later on, when they knew this to be false and came around with non-Euclidean geometries, they still believed that if the earth was not infinite then surely the universe was. And sure enough, as we are learning in this modern age, that too is not entirely correct.
But it does spawn one last question from me (yes I'm sorry I wrote so much) and that is that.. well like I said I don't claim to understand all the logic behind these non-Euclidean readings, but if mathematicians were so shocked to learn that the earth was round and that flat Euclidean geometries did not technically apply to a spherical surface.... well I dunno it sounded like they were getting ready to discount everything they learned from Euclidean geometry just because they learned that it didn't apply to the earth exactly as they had thought. What then would they say to the realization that the universe is actually finite (albeit expanding infinitely) and that, technically, there's no such thing as a line that goes on forever outward?
Will, you bring up good questions about new ideas being influenced by social acceptance. Some people are fearful of criticism, others of sharing their ideas, and still others are fearful of both. Being able to share ideas anonymously does help mitigate some of those fears, but as we’ve seen in some of our readings, people rarely remain anonymous when coming forward with large breakthroughs like the ones we have been learning about. We as a society like to think that we are more open to new ideas and it’s hard to fairly judge if that is true because we often look back on history with hindsight bias. I think we would be just as stubborn in accepting or publishing someone’s work if it were an idea that could completely change how we look at the world. It is said that humans are creatures of habit and I think that applies to intellectual knowledge as well. We like advancements but not changes. Advancements mean we add to the knowledge we already have while changes also imply that we were wrong for the past 10, 50, 100, or 1000 years. I know very few people who like to admit that they are wrong, and I think no matter the time or the subject, people will always be reluctant to accept things as earth-shattering as non-Euclidian geometry was in its birth.
ReplyDeleteWill, you mentioned that you believe the "spirit of the times" today is more accepting of radical thoughts and ideas, and to a certain extent I agree with you- but mainly in the areas of politics, lifestyle, religion etc... However, I don't think this applies to the areas of math and science. The hurdles scientists have to jump through in order to get research published in a reputable scientific journal are daunting (maybe even more so than in the days of Gauss since there are so many more mathematicians and scientists today) and even after publication there is no guarantee that the research will be accepted (or even noticed) by the scientific community. In my opinion this is a good thing because it prevents us from accepting willy-nilly anything put in ink, even though it means the "truth" generally emerges very slowly. People generally are opposed to change that refutes long held assumptions and beliefs because it can feel very uncomfortable and strange- this is as true today as it was three hundred years ago. Just look at the way people reacted when astronomers decided Pluto was no longer a planet!
ReplyDeleteAs to your other question, I certainly believe that making groundbreaking mathematical discoveries requires a great deal of creativity! Math and science are typically thought of as rather cold and sterile- but in reality if Gauss and other hadn't had the creativity to take the concepts of Euclidian geometry and question them/apply them in innovative ways (ie. to the surface of a sphere) we would probably still be living in the "dark ages."
I think there are several factors playing key roles in the way that our society might affect progress or better understanding of mathematical ideas as seen in the creation of non Euclidean geometry. As a whole, our society has troubles breaking away from the common western thought process that we are so accustomed to that it becomes hard to really have any new ground breaking ideas. I don't know much about euclidean vs non euclidean math but from what the reading it seemed like the non euclidean geometry basically scrapped everything it new about the previous world and started from scratch. These concepts are so deeply ingrained in our society and culture that its going to take a truly brilliant or ignorant mind to really break through those ideas and come up with something that no one has seen before.
ReplyDeleteHowever, I think social media and the internet are a great tool is helping great mind collaborate on what they are working on. This may be a good thing or a bad thing depending on how you look at it. In someways, it might foster thinking with other combining ideas and coming together. On the other hand, it might make it more difficult to really break out of the axioms that we are so used to and try and come up with something new and different.
William, I also share your interest as to how the creation of geometry influenced mathematicians to trying to relate mathematics to that of the physical world. If anything, I believe that geometry was definitely a “stepping stone” for mathematicians to think differently when it came down to making other mathematical discoveries. For example, if it weren’t for geometry, then Calculus, as we know it today, would not exist.
ReplyDeleteGoing slightly off topic, but still somewhat relevant is my question of why, despite all other geometries, is Euclidean geometry still the geometry that most people are familiar with and use most frequently? For example, when doing Calculus, we still are using Euclidean geometry to solve problems, but why didn’t someone invent another version of calculus utilizing one of the Euclidean geometries? Or if something like this is indeed invented, then why doesn’t it see more uses? I know that as the first geometry, Euclidean is probably going to be the most popular, but if other geometries are just as consistent as Euclidean, then why do we still (in modern day) feel that at minimum, Euclidean geometry is the most that a person needs to know?