Author Archives: ineedmath

Illiterate Math or ∫ ? what? ≠ :)

In class the other day, a student wondered, “Why don’t they just use symbols? Didn’t Euclid or Eudoxus notice how much easier it would be to use symbols and pictures? All of this language is so difficult.”

equation1

I believe no one would deny that mathematics is its own language. In fact, I think most people would categorize modern mathematics as a very foreign language. Symbols, operations, variables, etc. are mashed together in such a way that only a mathematician can understand what it is expressed or being questioned. This is absolutely necessary on some level. Mathematicians, and other users of mathematics, need an efficient and effective way to represent operations and the language of computation. On the other hand, it is quite sad to me that a “universal” language is so foreign. Indeed, if mathematics is a universal language, why is it understood by so few, and even despised by so many?

Mathematics didn’t start out being a foreign language. As described in Boyer and Merzbach, the earliest algebra descriptions did not use symbols. It was referred to as rhetorical algebra and an equation like x+2-z=3 would be written similar to: a thing plus two and minus a thing is equal to three. Sure, this description seems a little weird and clunky, but for most people, I argue it is easier to figure out and less “scary” or off-putting than the typical symbolic equation given above.

In Book I, proposition 41 from Euclid’s Elements, it describes something in language that all people could attempt to understand.

Thus, if a parallelogram has the same base as a triangle, and is between the same parallels, then the parallelogram is double (the area) of the triangle.

The above sentence may be difficult to understand, but it is approachable by anyone. All of the words can be defined or looked up. Representing mathematics with words allows everyone to be mathematically literate. I believe the loss of language in mathematics is a hindrance to it accessibility, popularity, and literateness.

Why was Euclid’s Elements so popular? Everyone seems to have read it, from President’s to peasants, for at least 1000 years. Part of the reason for this may be that it was actually readable. Anyone who could read, could read Euclid’s Elements. It is hard to imagine my calculus text being written in such a way, but it is important to ask, is there value in writing it in that way? Would it be a worthwhile undertaking to attempt to translate mathematics into words? Would that make it more accessible, more enjoyable, and more relevant to everyone? Would non-mathematicians be able to pick up a book and attempt to understand it? I know very few people who could pick up this and understand it:

equation2

However, most people might understand or could at least attempt to understand: write out one-half, eight times. Between each one-half write a plus symbol. Now, starting at zero and continuing until seven, raise each iteration of one-half to a power. The first one-half is raised to zero and the last one-half is raised to 7. Compute each iteration, and then add it all together. This is called summing a sequence of numbers from zero to seven. If I read this description when I first saw the above notation, I would not have been so afraid of it. I also might have tried to do it. Instead, seeing this foreign language scared me off. It just seemed too complicated to understand and figure out.

Most of my favorite math books do just that (Zero: The Biography of a Dangerous Idea; 100 Essential Things you didn’t know you didn’t know). They are books about math that contain numerous verbal descriptions of operations, ideas, and calculations. Sure, they also contain real formulas, but that is a supplement to the written description of the idea. Finding ways to use language to write about and describe math is important. Math should be accessible to all even if it is not computable by all. Wouldn’t approaching math in this way allow people to be at least as literate in mathematics as they are in English?

The Thirteen Books of Euclid’s Elements; translated from the text of Heiberg with introduction and commentary by Sir Thomas L. Heath, K.C.B., K.C.V.O., F.R.S., SC.D. Camb., Hon. D.SC.
Oxford, Volume III, Books X-XIII and Appendix: https://archive.org/details/JL_Heiberg___EUCLIDS_ELEMENTS_OF_GEOMETRY

A History of Mathematics, Jan 11, 2011 by Carl B. Boyer and Uta C. Merzbach

equation taken from http://www.math.utah.edu/grad/exam/DiffEquatF2014.pdf

My infinity is bigger than your infinity

When I was a child, I purposely found something to think about to help me fall asleep. Usually I picked cartoons or super powers, but sometimes things just came into my head, like it or not. What was the worst? Thinking about heaven. At first, heaven seems all right. There is a lot to do, gold everywhere (though no purpose for it), people are nice (it’s a prerequisite), you get to see most of your family, and there is plenty to eat (though no one is ever hungry). Anyway, I start thinking about FOREVER.

pic1cropAt first, it is just a sensation; a weird sensation like tingling and falling and nothingness. It is not a sensation that I can make sense of really because forever doesn’t really make sense, at least not to a 10 year old. I try to get away from forever but forever is a huge part of the definition of heaven. Then, the opening credits of the Twilight Zone, with the music, and starry sky, usually appear. Fade to myself standing, looking at heaven, in the dressing room mirrors of infinity. You know, when dressing rooms have those three mirrors that are angled just perfectly so the images are smaller and smaller replicas of one another, on and on, into infinity. This picture, and thoughts of the foreverness of heaven, kept me up at night as a child.

I am glad to say that forever no longer keeps me up at night. While I still find no comfort in the foreverness of heaven, the lack of a middle ground between forever and my time on earth is what usually keeps me up at night now. However; I still can’t stand it when mirrors are angled that way. It creeps me out, and I can’t help but wonder if there is an end, or if I can find a flaw from one image to the next. In my opinion, we are not meant to look into infinity like that, squarely.

When beginning to pursue mathematics, I thought math might clarify, or in some way define, forever (or as adults call it, infinity). On the contrary, Math has actually made it stranger. Theories in math have shown numerous types of infinity, and infinities within infinities, and sizes of infinities, and calculations of infinity. None of this brings me any comfort, except to say that we obviously don’t have this figured out yet because that is just not possible. Infinity is infinity, and it is very large, incalculable and non-denumerable, and there is only one kind; it is called forever. Heaven can only exist in one, all-encompassing infinity.

pic2crop

When reading A History of Mathematics, I read about Zeno’s paradox. That led to an internet search, and then to Numberphile. I watched the video, accepted the idea, and left it alone. The solution seemed reasonable enough. Later in the semester, I was required to do a research project. By some unknown scheme, we picked Georg Cantor, whom I had never heard of. If you haven’t either, he is the creator of set theory but also perhaps the mathematical or scientific father of infinity. You just can’t shake things off in life. They follow you.

My research for that project led me to question the mathematical view of infinity. Let me start by saying, I know very little of Math’s view of infinity. It seems to be an infinite topic. This is where I am in my understanding – so please comment, post, reply, educate me, and critique my understanding. Calculus one is a prerequisite for the course, and being a rule follower, I have that. So, I had experience computing limits to infinity. That is relatively easy. BUT, those are just numbers. They aren’t real things. Numbers aren’t real. So, of course I could compute the infinity of something that isn’t really real. What numbers represent is real; like Zeno’s paradox. Zeno’s paradox applies numbers to something real – something actually happening in the world (theoretically). In other words, when I take the limit of a sequence that goes to infinity, it has no relation to time or space. It is just numbers. But, if I were taking the limit of Zeno’s paradox to see how far Zeno actually travels, or to find the time it takes to travel, or to see if he can ever catch the turtle, I would have to do so in relation to time and space. When I do that, the exact opposite answer occurs. Zeno will never catch the turtle. That mathematics isn’t computing real infinity or perhaps all of infinity is perhaps echoed by the Numberphile narrator when he asks, “What I want to ask a physicist is, can you divide space and time infinitely many times?” Similarly, Kelly MacCarthur wonders in the Calculus 2 video used for online math courses, “Can I take infinitely many steps?”

However, if all of space and time existed at one instant, forever, then Math has it right. It could calculate the infinite because it occurs all at once. There is no sequence, event after event – in essence, no time or space really because it is all at once, everywhere. Yes, there are scientific theories, philosophies, and religions which believe this is the case. Of course, this idea is contrary to most people’s understanding of infinity. Whenever math instructors talk about infinity, they always say, “Infinity is only a concept. It is not a number.” Yes, it is only a concept but is it also something real? If it is only a concept then why are we computing something real that is a concept? Why would we bother to compute a concept? It seems like Math is walking a funny line here.

Math has worked something out though. I’m just not sure what it is. Math is summing an infinite process (as if infinity happened to end). Obviously, Math’s understanding of infinity has proven useful in mathematical calculations and many practical applications. To paraphrase others before Cantor, “It works. So, no need to define it. It works.” So, Math has worked something out about infinity but what has Math worked out, and is it really infinity?

pic3cropMathematicians always like to joke about engineers rounding numbers to 3 or 4 places because it doesn’t really matter to engineering after that, but is mathematics rounding off infinity or at least only capturing some aspect of infinity? After all, how can there be different types of infinity? My preferred illustration for the existence of multiple infinities is from Galileo. Galileo used a thoughtful but intuitive approach to understand infinity. He drew a circle. Then, he drew an infinite number of rays from the center of the circle. These rays filled up the space inside the circle. But then, he drew a larger circle around the smaller one and extended those rays to the larger circle. Though he drew as many rays as possible (an infinite number perhaps), the infinite number of rays did not fill up the larger circle; there were spaces between the rays. This led him to believe that first infinity was not large enough for the second circle; not even close. He would need another size of infinity to fill up the larger circle. [BAM! PHH! Did your mind just explode?] It is important to note that intuitively, his illustration makes sense. However, with today’s current understanding of infinity and better ability to calculate infinity, we now know that the infinity in the smaller circle leaves no space between the rays when extending to a larger circle. But, I liked his intuitive approach. Though intuition seems to be severely lacking when it comes to infinity.

References:

Dangerous Knowledge: http://topdocumentaryfilms.com/dangerous-knowledge/
Georg Cantor His Mathematics and Philosophy of the Infinite by Joseph Warren Dauben
TML: The Infinities In Between (1 of 2): http://www.youtube.com/watch?v=WihXin5Oxq8
TML: The Infinities In Between (2 of 2): http://www.youtube.com/watch?v=KhgNiqI-bt0
Infinite Series: http://stream.utah.edu/m/dp/frame.php?f=f55f900bec01a3106121
Zeno’s Paradox – Numberphile: http://www.youtube.com/watch?v=u7Z9UnWOJNY

My new bumper sticker.

pic4crop

Incommensurability

Incommensurable

I had not heard of the term incommensurable prior to reading it in the History of Mathematics (Merzbach and Boyer). According to the online dictionary (free dictionary), it means these numbers are “impossible to measure,” or in mathematics specifically, “having an irrational ratio.” This term illustrates the practicality of early mathematics. Merzbach and Boyer assert that early mathematics were used for practical things like measuring. So, instead of saying a number is irrational, early mathematicians more accurately described these numbers as un-measurable.

What fascinated me about the term incommensurable, and the ratios or numbers that fit the definition, is how mysterious and troubling incommensurables must have been to these earlier cultures. I imagine early mensurers, or as we call them today, surveyors, were befuddled that nearly every object had a length or a side that was not measurable or re-creatable. To paraphrase Alex Bellos (Numberphile, Pi), pi “is the simplest possible ratio (circumference to diameter) of the simplest possible shape (a circle) and yet it is this ugly, complicated number,” and it is everywhere. Sure, the length of an incommensurable is exactly equal to a ratio, but that doesn’t help someone who is actually measuring or creating an object.

I imagine one of the King’s workers saying, “Yes, that circular wall on the side of your castle will be about 3 units. That’s as close as I can estimate.” Now obviously, estimating pi to about 3 worked, and worked very well, but it does seem funny none-the-less. Early cultures came up with close approximations like “about 3” or square root of 10, and these were surely close enough for most tasks required of them. Archimedes’ approximation of Pi as 3 + 10/71 was used for more than 1000 years (Numberphile – PI).

The square root of 2 as the hypotenuse of a right triangle. Image: public domain, via Wikimedia Commons.

The square root of 2 as the hypotenuse of a right triangle. Image: public domain, via Wikimedia Commons.

That some early mathematicians avoided incommensurables is no surprise to me. Apparently, Pythagoras was afraid of the incommensurables. Merzbach and Boyer describe the Pythogorean belief as possessing the tenet, “that the essence of all things, in geometry as well as in the practical and theoretical affairs of man, is explainable in terms of arithmos, or intrinsic properties of whole numbers and their ratios.” We now know that whole numbers cannot even describe something as simple as the diagonal of square. I imagine the Pythagoreans believed they simply had not discovered or understood something about numbers, shapes, and math. To the Pythagoreans, incommensurables probably made the universe seem poorly planned or unorganized. However, I think the opposite view could also be taken. The universe was so delicately planned or created or happened in such a way that every object is made up of an unmeasurable number. How planned and organized and amazing is that? That is way more amazing than a universe based on whole numbers.

I am awestruck by the fact that we live in a universe where nearly every object has some incommensurable aspect to it? We can exactly represent the incommensurables or irrationals in numbers, thanks to Math, but we cannot measure them. This may be a leap, but to me, that means the universe is unmeasurable. There is something satisfying about living in an unmeasurable universe. Of course, if I were a god or the master of all creation, I think I would do the same thing. I would create objects that could never be recreated exactly the way I created them, and every object would have this small mystery to it. I love a universe that is unknowable. A Pythagoreans universe would be boring and unpleasant.

In the end, however the universe came into being, the existence of incommensurables is one convincing reason to believe that humans did not create math. If humans created math, I doubt they could have ever thought to have such numbers or for them to be so pervasive. Score one point, for math existing, not being created.

Also, on your next date, remember to tell the person you are with that they are incommensurable. It is not synonymous with irrational, which would be an insult. Being incommensurable means they are mysterious, unmeasurable, and a perfect ratio that is undefinable any other way.

References:

History of Mathematics, 3 ed., Uta C. Merzbach and Carl B. Boyer

http://www.thefreedictionary.com/incommensurable
incommensurable. (n.d.) The American Heritage® Dictionary of the English Language, Fourth Edition. (2003). Retrieved September 14 2014 from http://www.thefreedictionary.com/incommensurable

Pi – Numberphile

Wikipedia: http://en.wikipedia.org/wiki/Mathematics
http://en.wikipedia.org/wiki/Mathematics#cite_note-Waltershausen-13

Bellos, A. Here’s Looking At Euclid.

tags: incommensurable, irrational, transcendental, pi, sqrt(2)