When I first opened up the readings for today I had a mild panic attack and painful flashbacks to general physics, but surprisingly I actually kind of enjoyed these readings (much more than the calculus readings anyways!). The Thorne readings especially did a really great job of explaining Newton’s and Einstein’s theories and the relationship between the two- at least as good of a job as my Physics 211/212 classes did! One of the first things that really struck me was Einstein’s realization that his theory of relativity and tidal gravity were actually the same theories worded in different ways. I was always under the impression that the physicists of the world had stumbled along for centuries using Newton’s laws until Einstein came along and was able to prove that Newton was wrong. In reality however, scientists using Newton’s laws alone were able to make incredibly accurate predictions!  According to Thorne, Einstein’s discovery did not prove that Newton was wrong, simply that he did not have “the whole picture.” I also found Einstein’s history and the process he went through over the years to develop his theory of relativity very interesting and amusing- especially his professors description of his as lazy. If only this was the kind of stuff we learned in gen. physics...

Another point that interested me was the distinction both authors made between physics and mathematics. Both articles stated several times that Einstein was a brilliant physicist but only an adequate mathematician at best. From my frame of reference (lol), physics and math are so similar that I cannot imagine someone being good at one but not the other. I always believed that to understand math was to understand physics and vice versa... Apparently Einstein is proof that I was wrong! Of course Einstein’s “bad at math” is not equivalent to my bad at math. If I was as good at math as Einstein was bad at it I could probably be a math major. 

The final thing that really stood out to me was Greene’s claim in the third article that eventually all matter in our universe would be converted to energy. I of course knew that there was a relationship between mass and energy, and that mass can be converted into usable energy, but it has never crossed my mind to emphasize the equation the way Einstein himself did- with the emphasis on the creation of mass from energy rather than the other way around. To my knowledge, we have not yet been able to do this in a lab, but I am very ignorant about physics so someone please correct me if I am wrong! The ability to create mass from energy such as heat or light would be a truly amazing thing! 

Einstein

I had never really thought about space and time before in as much depth as the book presented. It really amazed me how complex the whole spacetime four-dimensional fabric worked and made me realize how crucial Minknowski’s discovery was. The fact that gravity is produced by a curvature of spacetime’s fabric, making it a warpage of spacetime seems very strange. At first it made me wonder how can this could be so. I feel like a warpage of spacetime is something to be regarded as scary and mysterious like the black holes and wormholes mentioned, but gravity isn’t all that mysterious nowadays, maybe a little scary if you’re far above ground, but that’s it. In today’s society, our viewpoints have changed dramatically since BC times, but does his implications of humans not realizing spacetime in this reading still apply? Do we really not notice our spaces and times are relative and unified? I feel as if this would only be true if we did not have the ability to imagine ourselves in other places, including other organism’s lives, like the phrase of “putting yourself in someone else’s shoes”. But most of us have this ability, don’t we? Is this ability learned or innate? (This also reminds me of the tv series “24” which follows several characters in 24 hours of real life time, all at the same time.)

Gravitational time dilation really intrigued me, but I still don’t understand how it’s possible. The clock example made me really curious, but as I was reading I couldn’t understand what it was saying. Could someone possibly explain it to me how you understand the example to prove gravitational time dilation?

I had always wondered every time I boarded a plane, how the heck they stayed parallel to the earth’s surface. I noticed that most trips I took, the plane was substantially higher at the end of its journey than at the beginning (on board computers on overseas flights), but thinking back now that distance was far smaller than if the plane had gone linearly straight unrelative to the earth after first taking off. Up until class Tuesday, I had never realized parallel lines could be drawn on spherical objects or even intersect, but seeing it now makes everything make a lot more sense. Has anyone had a similar experience before? I was also wondering if anyone here has gone skydiving before? I was wondering if one would feel the vertical stretch and lateral squeeze of tidal gravity during those long seconds free falling, even just a tiny bit. 

I was also wondering what do you guys thought about the “enormous energetic wealth contained in mass itself.” I found this to be intriguing, but it’s hard to wrap my mind around it. Then, my biggest ‘a-ha!’ moment was when I read this: “two balls thrown into the air on precisely parallel trajectories, if able to pass unimpeded through the Earth, will collide at the Earth’s center” (p110). So, I wonder to you, as a though experiment, do you think that if we dropped only one ball, that it would stop at center of the Earth unretrievable, or do you think it may have enough energy to come back out on the other side?

this ain't your grandma's version of snake

We've all played the classic game of snake but it just gets old playing on flat surfaces. here's a wonderful game where you can play snake on what looks like a plane but the mechanics play like you're on the surface of a torus (doughnut or innertube shape) as well as some other options. You have to download but definately worth a look!

https://sites.google.com/site/tjgaffneymath/snakes-on-a-projective-plane

Non-Euclidean Response

After the readings on Non-Euclidean Geometry, a few things stuck out to me. The creation of non-Euclidean geometry came from trying to prove something the parallel axiom. How this mathematical advancement came out of actually trying to prove something to be correct, and ultimately not being able too. Again I find it interesting how humans react to problems, we immediately to try to solve them rather then bringing up the possibility that they may be wrong. Individuals spent their live trying to prove Euclidean’s parallel axiom was true; Hume was one of the only ones to express a little doubt. Everyone else was trying to prove that this axiom existed. Ptomey was the first, next Proclus; then another long list of individuals spent their lives trying to prove the parallel axiom. They were all using an inductive method, once they weren’t able to prove it that way, another group switched to try to prove it through deduction.  At the end, all of these individuals were using the previous knowledge that other had discovered and Gauss was labeled the creator of non-Euclidean geometry. I thought it was interesting that Gauss the name changed the name of the geometry three times until he chose non-Euclidean. I wonder about this, was there some reason behind this? Or am I simply focusing on a fact that isn’t important?

        The fact that we can label different creators was also brought up in the first article from chapter 36, defending that Gauss as the creator, because the most significant part was not needed for mathematical development. Stating how Lobatchevskey and Bolyai learned the ideas from their teachers who learned it from Gauss. In the other article there was opposing views, and stated that Lobatchevskey, Bolyai and Riemann were the creators of this geometry that had the greatest advancement since the Greeks. I find it interesting that they have opposing views on the creators, and important influences in this new geometry. Looking at the creation of non-Euclidean geometry do you agree that the most important fact is the development of the properties of physical space? Can there be one person who could have been the “liberator” of math? 

Non-Euclidean Geometry

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.

Chase's contribution for 9/20/2012

Can anyone explain to me why highly scholastic people always seem to be the most awkward?  That's one thing I found really interesting while reading these articles.  Newton not having friends or even to call someone his students, and the Bernoulli Brothers fighting over the discovery of their cycloid and the differential equations with the second and third degree.

Besides that, I find it interesting on how closely related all these Mathematicians were.  It seems like they all bounced, borrowed, or claimed ideas that were very similar, if not the same.  They always seemed build off each other's previous ideas and try to make it their own.  Like the instance with Newton and Leibniz fighting for who invented Calculous and when Newton, Leibniz and the Bernoulli Brothers all figured out the cycloid, the solution to the brachistochrone problem.  This just makes me wonder who really invented what in the mathematical world and who just borrowed ideas and called it their own?

Where would we be without calculus?

Wow! That was a lot of reading! In general, it was all about the beginnings, criticisms, and applications of calculus, as well as some information on some early mathematicians. Now what does this all mean? Well, the way I saw (or read) it, ever since Newton (and Leibniz) brought forth Calculus, the subject has been under much debate. In fact, one of the first criticisms that caught my attention from the readings was when Leibniz accused Newton of “implying an imperfect God” (Gleick, 171), and apparently Newton took the comment to heart as it was noted to have stung. In today’s society, we (or at least I) don’t acquaint God to mathematics. This alone shows me the type of thinking that went on during this time period. For some reason, the people of that time period believed that God and mathematics were related. What do you think?

Another thing that got my attention the Feynman article on the relationship between physics and mathematics. Now if I recall correctly, it has been said that Newton invented calculus in order to understand physics. Now, the article has been written many many many years after the lifetime of Newton, so it stands that Feynman has a different mindset than say, Newton and the Bernoulli brothers. Reading deeply into the article, it somewhat implied (at least to me) that in order to understand physics, once must know math. Of course this makes sense to me because in today’s society, calculus and physics is intertwined. However, this makes me want to know how physics was taught back in the Newton time period. What do you think? If what we know now is all thanks to the creation of calculus, what was the understanding of physics like back in the day? Clearly that had some concepts of basic physics like the concept of gravity, but I’m sure the understanding was different.

Some Light Reading on Calculus

Hmmm.... well this certainly was a nice, cut-and-dry reading - very straightforward and pretty much entirely based in "exact" math and history. Makes it somewhat difficult to think of much to say ethically/philosophically on the general reading as a whole. Yet I find that I would agree with this reading's other poster (Jason's "The Rate of Right and Wrong") in that an ethical spin on this very long reading of hard facts and proofs would most likely be focused on uncertainty - which makes perfect sense that our discussion last time ended with its brief introduction.
On that note, I would like to point out a psycho-philosophical observation I've made through the years: that we, as humans, feel a very strong necessity to cling to concrete, certain facts. Stemming from a terror of the unknown, we find some absolute "truth" that we rigorously cling to and, upon it, construct our entire understanding of the world. Sometimes that comes in the form of religion, where those who base their knowledge from the Bible claim it as absolute fact and, as a logical result, must therefore take its words literally. Most others (especially in modern times) prefer to base their concrete "certainties" on math and science, generally assumed to be disciplines of certainty itself. So it makes perfect sense that most would be terrified to think of the possibility that their entire understanding (and their perspectives on everything they know) is built on something that isn't absolutely certain.
And yet, for someone such as myself [someone else may conclude otherwise] who analytically questions everything they "know", I find that the only thing we can ever know for absolute certain is that there is nothing that we know with absolute certainty. Our knowledge of science is just the current (albeit tested) theories of observable trends; history often dwindles down to the perspectival interpretation of whoever recorded it; and math, being the archaic human method of explaining patterns/relationships of reality, is only fact insomuch as the situation's parameters define it as such.

This last becomes the major issue we see in the reading where the objections of others to either Newton's or Leibniz' approach to calculus seemed (as I understood the reading) to be concerned mainly with issues such as the logic behind the applicability of the concept of ratios with infinitesimal denominators; in short, with limits.
Though perhaps more of a curiosity than a substantive addition to this post, I cannot help but comment on the inability of us/myself as an observer in trying to objectively understand the entire context of this situation - including the historical context of the society (whose influential perspectives caused both Newton and Leibniz to refrain from publishing their "radical" ideas) as well as the sensibility of the math itself.
Using myself as an example, I once was in absolute concert with the objections set forth in the reading. Yet now, (having taken and understood multiple courses that explore and define the many aspects of calculus) I have become so mentally well-versed in the concepts of limits and infinitesimals that my method of logic itself has conformed to these ideas; accordingly, it is incredibly hard now to understand where these logical objections in the reading are coming from, and a "how does this not make sense to you" kind of attitude sets in, and I find it difficult if not impossible to be able to dissect their objections and truly prove the certainty of my current understanding against their arguments. As a result, having been unable (at least in terms of the modern state of such developed knowledge) to analyze the situation objectively, I often end up concluding teleologically that the ends justify the means, and that so long as that method calculus keeps giving us the observably "correct" results, that's all the proof we need. Then again, I'm not a true mathematician.

One final side observation I wanted to comment on was the personalities of each of these two brilliant pioneers as well as that of other proclaimed "geniuses" of history. Course I can't say this for everyone who has ever made a significant contribution to math, science, etc., but is it not curious that the most brilliant minds tend to be socially... unfit? For Newton (and probably the majority of other "geniuses" - Einstein comes to mind) of course this is obvious in his eccentricities and nervous irritability, while Leibniz' apparently profound philosophy had to be kept a secret in order for him to remain popular. And I mean this is by no means a novel idea - perhaps even an intuitive one since the only way evolutionary advancements can be made is when one abnormal ("mutated") individual is lucky enough to surpass the status quo; not to mention the mentally conducive environment that social aloofness provides for learning.
Even so, I just find it interesting that very nearly all aspects of social development encourage conforming to the standard norms (and necessarily so, in many cases), while progress and advancement rather clearly favors a highlight of the individual.

The Rate of Right and Wrong

We talked at the end of last class about the uncertainty of math. In the schools that I have grown up in, there persists a distinct contrast between the accuracy of math and science versus the humanities. In the former, there was "one right answer," while the latter is left up for interpretation. However as made evident by "The Calculus" reading, the fundamental concepts that "allow" calculus to work are not clear cut logic.

In both Newton's and Liebnitz's calculus, there is debate on the validity of finite limits expanding to infinity and infinitesimals (respectively). If we question a math that is so persistent in describing the world around us and in predicting systems, is there actually a black and white outcome? In other words, just because the math works, does that mean that the answer given is solely right or wrong? Or does the justification between right and wrong come from what the answer means to us, an interpretation? For me, I think (but am not sure) the vindication of the answer arises from the ability to predict what is going to happen next. This requires a faith in the answer you found, sort of like religion. Nevertheless, I think that math to us has no meaning without the interpretations we give it, requiring some subjectiveness.

One note: after further reading multiple sources seem to point to nonstandard analysis as an answer to the problem of infinitesimals, but debate in the validity of certain fundamental principles in math and science is not just present in calculus.

Sorry I posted this on Blueline2 instead of here and just realized now that it's 5 hours late.

Kline begins this chapter in his work on the history of mathematics by bridging a gap that is not often subject to questioning, the gap in question being the transition from Medieval thought to the ideas of the Renaissance.  Kline uses Jerome Cardan, who had a notable impact on the aforementioned progression, in two senses; Kline employs Cardan's personal life and philosophy as a sort of illustration of the change while also discussing Cardan's contributions.  After leaving the topic of Cardan, Kline takes a step back and approaches the forces behind the transition from a more universal perspective.  Some of the forces, or causes, of the change include the entrepreneurial motives of a developing middle class, ambitious states seeking power through warfare, and the Protestant Reformation, which put a crack in the monolith of thought-oppression known as the Catholic Church.  Despite the impact these forces had, none of them were the primary factors involved in the iconic and world-transforming renaissance men discussed afterward.  

Kline describes some of these major contributors and their contributions, including Newton, Descartes, Huygens, and Galileo.  The one that was particularly interesting to me was Descartes, as his transition from Euclidean Geometry to a knowledge of the world using space was an interesting development of the work of Euclid, who we'd already discussed in class.  "The essence of objects or matter is space, and objects are essentially chunks of space, space solidified, or geometry incarnate."  (Kline, 106)  This new understanding of the material world was truly revolutionary, and it also went well with his understanding of time, and how space and time were not only "the machinery" of the universe, but also the things that made nature knowable.  But the most interesting thing to me is the motivation behind why Descartes and his peers were uncovering the mysteries of the natural world using mathematics; their motive being a desire to know God.  

According to Kline, these renaissance men worked not for the glory of the Ancient Greek idea of a "hero-scholar," for the money and opportunities, or for the development of science and mathematics in order to know more about the world, but in order to know and glorify God.  The Islamic scholars in Lindberg's article are motivated by the idea of "completing the Greek project." (Lindberg, 176)  Do you think Kline is right when he attributes their efforts to a desire for the divine, or do you think that some of the other aforementioned forces played more significant roles than are otherwise indicated?  Does it matter to you personally "why" the progress was made so long as it was actually made?  

Mathematical Spirit and Western Science

In the chapter “Renewal of the Mathematical Spirit,” Kline discusses many aspects that lead to the rekindling of mathematical interest.  He mentions the influence of Greek thought and expanding trading interests as major factors leading to mathematical innovation. Lindberg further conveys the idea that the Greeks had a major impact on Islamic science.   This displays the importance of trade and exposure to other societies in the development of science and mathematics.  Another major aspect of scientific advancement Kline presents is battle or war.  Kline states “the needs of war have always aroused nations to put forth money and efforts unimaginable during times of peace (102).”  He also mentions the idea proposed by many scholars, including Francis Bacon and Rene Descartes, that the foremost intention of science is the domination of the natural world by man.  He presents the idea that the universe was “designed by God in accordance with mathematical laws” and it is man’s goal to use science and mathematical laws to maintain a certain level of supremacy over nature.  Does this imply a level of competition and a desire for dominance is required for the advancement of science?  I do not believe it is man’s role on earth to reign over nature as intellectuals such as Bacon and Descartes have suggested.  But the motivations of war and competition do imply that the desire for power/control may be a significant factor in the progression of science and mathematics. 

More Prime Numbers

Thought this article tied in well with the New York Times article about prime numbers we read:
http://au.news.yahoo.com/world/a/-/world/14831400/mathematician-proof-prime-numbers/

Do Dogs Understand Calculus?

Interesting study done by a professor and students at Hope College.

http://web.williams.edu/go/math/sjmiller/public_html/103/Pennings_DogsCalculus.pdf

Societal Aspects to Mathematical Development

As I went through the readings I was both impressed at the incredible developments of the Arab Mathematicians when Europe was in a period of mathematical stagnation during the Medieval period and I found myself considering repeatedly just what factors need to be present in a society for these kind of mathematical advancements to be made.

For instance, it's clear that a society that was only focused on the hereafter and spiritual understanding of a God stifled mathematical growth, but at the same time so to did a society focused solely on practicality like the Romans. I was particularly surprised that the Romans made no development in Mathematics (their running water and complex engineering would seem to counteract that to my mind), certainly they had the money and the leisure at hand to think about such things. So mathematical development is seems not to be solely a function of time and wealth, or even practicality.

The common aspects I see between between the Greek, Islamic, and Renaissance societies is that they 1)all encouraged curiosity, and for the Islamic and Renaissance societies 2) they had a tradition to build upon and 3) had centers of learning. Islamic societies were curious about the world around them because it was a method by which to learn about God, and they also had Madrasas--schools where one could learn and discuss--and a growing collection of Greek philosophical and mathematical literature gained through conquest by the growing Arab empires. Both Nobility and the Merchant class was willing to patron mathematics, giving scholars the ability to develop theories.

Similarly Renaissance thinkers had a theology that now taught them that to learn about nature and the world God had created was to learn about God, intellectualism was no longer sinful. A influx of greek(preserved by the arabs) and arab literature translated into latin gave the Renaissance men a tradition to grow upon. Finally there were colleges established to both teach and share these new traditions.

These are all aspects that I feel intuitively I knew, but I would have also thought that things like practicality and need, would drive invention. Furthermore, could a society have just a one or two of the qualities a I describe and still have mathematical developments like we see in the Islamic and Renaissance societies?

A day in the life...

As I was going through the readings, I gave my best effort to really picture myself living in each time period that was mentioned throughout the readings. In Kline's article especially, I found that it was difficult to imagine myself being a mathematician living in the medieval European times.The most interesting point that I found was the idea that all nature served man however man's life here on Earth was just a means to a final end goal of spiritual exaltation. So with man's main purpose here on earth being spiritual salvation, where does a physical life, health, science, literature, and philosophy lay on the spectrum of priorities in comparison? It's hard to think about living such a one dimensional life coming from the world we live in, especially with Creighton's extensive core, filled with the arts and philosophy's of the world. It really does make sense, however, that mathematics and abstract thinking took a back seat to the idea of saving one's own soul. Moving to other parts of the world where math made a bit more of an evolution, the Hindus and the Arabs, it makes one wonder what the connection between math and religion actually is. With such pivotal advancements being made in Hindu and Arab communities such as the notion of zero and negative numbers, what was the difference between medieval Europe and these nations? How did religion play its role in both of these nations? Why is there such a stark difference in the advancement of mathematical thinking?

It's even more interesting to move to the other end of the spectrum with the Romans, being unproductive in the advancement of mathematics because they were "too much concerned with practical results to see farther than [their] nose." The medieval world was too much preoccupied by the does and don't of everyday life in order to grant access into the spirit kingdom of God to worry about abstract thinking here on Earth. Mathematics can't bloom in either of these kinds of nations, so, what kind of conditions do you think would foster the highest mathematical advancement in the shortest amount for a given civilization?

Prime number visualization

Hey guys, I found this really neat visual representation for prime and composite numbers. Enjoy!

http://www.numbersimulation.site88.net/

Just a note: if the block appears by itself along the red line it represents a prime number. If other blocks appear along with it those are the factors of the number the block represents.

Perelman article

http://www.newyorker.com/archive/2006/08/28/060828fa_fact2

Some topic areas for papers

You need to come up with a topic next week and pitch it to Dr. Dugan and myself. Note that this meeting must be substantive, so do some research BEFORE you come to the meeting-you need to present your ideas to us. Here are some possible ideas that you can pursue further (don't limit yourself to these-there are a huge number of topics that have math/history/ethical content!):

Artificial intelligence
Cosmology
Insurance
Warfare modeling (e.g., collateral damage, or casualty modeling)
Climate modeling and climate change
Population modeling
Economic modeling
Behavior prediction
Genetics
Traffic modeling
Finance
Math and aesthetics
Evolutionary game theory
Political prediction/modeling
Polling
Testing

Please consult the syllabus for a description of the pitch process.

Ancient Mathematics (Yay!)

Hey everyone, just thought I'd link a site I thought about while doing yesterdays readings. It has examples of the numbering systems used not only in ancient Egypt, Babylon, and Greece, but also the Mayan numeral system and Quipa, the Incan counting system. Enjoy!

Heres the link: http://www.math.twsu.edu/history/topics/num-sys.html

Greek Mathematics

The reading “The Rise of Theoretical Mathematics in Ancient Greece” summarized the development of mathematics in ancient Greece.  Many people think of the ancient Greeks and think of their application of mathematics to various fields.  A lot of what the Greeks discovered (e.g. golden ratio which has applications to architecture, music, nature, etc.) and expanded upon appeals to various senses, and this reading (in addition to explaining the evolution of math) mentions that this was a large part of Greek culture.

And going along with the idea of mathematics influencing various fields, the New York Times article discusses prime numbers and the mystery behind them.  The infinite quantity of primes also has fueled various attempts to better understand this “mathematical anomaly.”  And out of curiosity, I went to the Prime Island site (http://yoyo.cc.monash.edu.au/~bunyip/primes/primeIsland.htm) and “listened to primes” (http://primes.utm.edu/programs/music/listen/).  It’s interesting to see numbers interpreted this way.

But I think the essence of mathematics is explained by the last sentence of “The Euclidian Synthesis”: “the attempt to create a deductive system, far from hindering imagination, actually stimulates it to create new ideas.”  Euclid’s work is one example.  Another is people who have tried to explain primes but have not been able to establish a predictive pattern.  Instead, they have established things such as Prime Island and musical interpretations, imaginatively creating something organized out of a random pattern.  Math is often used to model situations, and I think this statement about mathematical discoveries applies to other fields as well.  When someone says they have reached a breakthrough, their work is scrutinized by many parties.  And while the initial findings may or may not hold true, the inquiries by others result in tangential findings and subsequent breakthroughs.  Society as a whole is all about expanding knowledge bases, which is exactly what happens in this process.  And while we can sit back and amass information, it’s what we do with that knowledge, ethical or unethical, that defines us.

Multi-what?

While reading this article my thoughts seemed to drift quickly to how "unintuitive" the methods of solution for simple arithmetic and geometric problems appeared in Egyptian and Babylonian mathematics. But I quickly realized that it was not the case that these methods of solution were truly unintuitive but rather that they constructed at a time when the basic frameworks we all take for granted were in their infancy. As such, the choice of operations was relatively unrestricted, after all, there really didn't exist systems of multiplication, counting or division that were a part of the norm. So, counting in base 60, using only 'unit' fractions and multiplication based solely on doubling were chosen for their understandability within existing frameworks. This, I find, is an excellent example of how human understanding of concepts grows with their use over time. It is often the case that we may have a hunch that a certain state of affairs is true but lack the means to covey it in a formal way. As such, we can construct examples to build upon an intuitive grasp on a given concept, thus flushing out important details that are otherwise obscured from intuition. 

This article also made me consider how motivations for the evolution of mathematical though has changed since its' infancy. It seems that the primary motivation for the Egyptian and Babalyonians was to solve basic problems related to measurement and construction (essentially geometry and algebra). Nowadays, the study of mathematics need not have any particular set of applications in mind outside of explaining certain phenomena within a mathematical system. So it seems that the motivations underlaying the study of mathematics have drifted beyond mere utility. I am left to wonder if utility was the prime motivation for other human intellectual endeavors? Is the move away from solely utilitarian grounds reflect a shift from primal needs to intellectual ones? Or is such a shift towards abstraction natural and would occur regardless if primal needs still remained central to everyday life?

Egyptian and Babylonian Mathematics

After reading the document on Egyptian and Babylonian mathematics I found it extremely interesting that the started to deal with fractions but weren't able to get to the same sophistication that we have today. This seems to imply mathematics is an evolving discipline. It is also necessary to note that both civilizations used mathematics more as a way to relate to observed objects and not some abstract thought that current exists today. This seems to imply that humans used mathematics in the beginning as a relationship between objects in our observed reality and the names we give them (one, two, three,...) to make some understanding of the world around us. Once this understanding was reached did we start to play with the concepts as the Egyptians did with circles to find pi and the Babylonians did to find a relationship between squares and triangles to reach an early form of the Pythagorean theorem.

Looking at each civilization independently we see that our counting system relates to the Egyptian counting system, which is based in tens, but varies greatly from the Babylonian counting system, which is in base 60. Two civilizations separated from each other came to described the same world each observed by means of two different counting systems. This relates to the exercise we did in class where each group developed different counting systems. It seems that this implies that mathematics started off as a human construct, a tool if you will, created to understand and manipulate the world about us. But simple arithmetic seems to have come from observing our world and algebra and geometry came from relating object to each other and trying to manipulate them. The intrinsic number of stuff never changed, rather our ways to understand the objects did (base 10 vs base 60). In this case mathematics seems like a tool created by use based on some abstract relation between our understanding of the world and how it actually is. Or alternatively put, amounts of objects never change but our understanding of the relations will, and this is where mathematics seems to be evolving with us.

Any ethical implications would come from how to apply the algebra and geometry the ancient civilizations found by way of relationship. The reading gives one account of it being used for astronomy and astrology. Once a stable cycle of the stars has been mapped out the civilizations could plan events by the year, when to plant crop, harvest, and react to other environmental phenomena associated with the region (like winter and summer). This would give the kings and scribes great power to dictate the course of their civilizations well being and open up questions of ethics for conducting their orders.