Mathematical Reality?

    “To explain something new would mean to try to reduce it to the already known.” (Otte, 211) World history as we know it has taken huge leaps and bounds in the last hundred or so years. Things that we never thought possible like space travel and instantaneous international communication are yesterday’s news. The universe as we know it is rapidly expanding -- as is our knowledge of it. Having access to so much information helps us to take classic theories of antiquity to their full extent. We are able to put more pieces of the puzzle together now than ever before and see even more of how our world connects in a big picture. Its an exciting time to be alive. One such theory is on Mathematical Reality. There is a long-debated theory that there is a completely separate reality from our physical reality that is expressed using the language of mathematics. It seems that math is a pretty viable route to philosophy. People like to draw conclusions philosophically out of advanced mathematical phenomena that is as of yet unexplained in the physical world. Does mathematical reality exist? This is the classic debate in conversations on the subject. The two sides are Idealist and Empiricist. The Empiricist does not believe that this mathematical reality exists, they think that this physical world is all there is. The Idealist believes that mathematics express things that do not exist in our world, yet still hold true (Bouveresse, 55). The nature of math in conjunction with other phenomena lends credibility to the possibility of such another reality other than this physical world alone. I will be taking the side of the Idealist in this paper.

BASIC ROOTS/ HISTORY OF MATHEMATICS

    There are lots of interesting conversations that shoot off this subject and tie in well. Jacques Bouveresse wrote an article about Mathematic Platonism . The article talks about our discovery of math. Mathematics are not man made, or the invention of a genius. They are a discovery. Ratios and principles kept showing up all over the place in nature and people like the Pythagoreans used symbols (numbers) to express what they discovered in the language of mathematics (Clark,1989). “not only is Nature a reality independent of the physicist who could be tempted to study it, but physics itself is likewise a reality which would subsist even if there were no physicists” (Bouveresse, 65).
   

    Plato believed that every descriptive word or reference we use on earth at any time to anything from chairs to straight lines was derived from an ideal. We call something a “chair” because of its “chairness”. We call something a “cup” because it has cup-like qualities. These qualities are derived from their ideal. Somewhere there is the perfect wholeness of “chair”, exactly manifesting the completeness of what we now know as only shadows of a thing. Plato also could apply this same principle to mathematic or scientific principles. For example, if I was to tell you to draw a straight line, you’d probably get a ruler and trace along its edge. Then I could take that same straight line and hold it under a microscope and clearly demonstrate that it was not, in fact, a straight line. No matter how small you get or how perfect it looks, you would not succeed. Such a thing does not exist in our world. However, in mathematics, straight lines exist. Things that exist in mathematics are so much more than they appear to be. They are concrete where language and physical molecules are not. G.H. Hardy draws this distinction by saying,“A chair or a star is not in the least like what it seems to be ; the more we think of it, the fuzzier its outlines become in the haze of sensations which surrounds it ; but "2" or "317" has nothing to do with sensation, and its properties stand out the more clearly the more closely we scrutinize it. 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another,but *because it is so*, because mathematical reality is built that way."

    Although surprisingly deep and complex as a subject in its entirety, mathematics are primarily used today for functional purposes. Math can predict outcomes, solve problems, and communicate things through its symbolism that go much deeper than words do. Perhaps one of the greatest glories of mathematics is the interconnectedness of algebra and geometry. These two basic pillars of mathematics can describe the same things, but with completely different approaches. Geometry is about space, depth, and open-ended thinking. Algebra is more like rules and procedures. “. . . algebra is to the geometer what you might call the 'Faustian Offer'. You may recall, Faust in Goethe's story was offered whatever he wanted . . . by the devil in return for selling his soul. Algebra is the offer made by the devil to the mathematician. The devil says: 'I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvelous machine'. , , , the danger to our soul is there, because when you pass over into algebraic calculation, essentially you stop thinking; you stop thinking geometrically, you stop thinking about the meaning” (Atiyah, 2001).

ZERO, INFINITY, IMAGINARY NUMBERS & THE PHILOSOPHY OF MATHEMATICS

    The science and rationalism boom in the intellectual world over the last few hundred years has accomplished amazing things, but it seems that in the midst of it all, we may have sold out a little bit. In an article he wrote, Michael Otte has done a great job of taking relatively basic mathematical concepts and using them to reveal doorways that connect to some very unexpected places. He says that the problem with mathematics -- the reason why so many people brush it off as boring or too hard -- is not because its that difficult of a subject at all. The only problem is because people find it difficult to see the “development of ‘meaning’, of the ‘existence’ of mathematical objects” (Otte, 203). When most people think of math, they have flashbacks to some class they took--learning the quadratic equation or some other random thing they may never use again and probably don’t even come close to really understanding. They miss the meaning. None of the things their monotone teacher told them have any relevance to their lives. This is not how math works if you take a different kind of look at it. Ronald Glasberg addresses this issue similar to the way Otte did in his article “Mathematics and Spiritual Interpretation: a Bridge to Genuine Interdisciplinarity.” Glasberg, however takes a little bit more of a birds eye view of what he categorizes into three disciplines: The proper sciences, the humanities, and the spiritual pursuits. He says that these three are inseparably related and that mathematics is one of the best things to illustrate the connections. Glasberg proposes an intriguing connection because in our society today, the proper sciences including math are highly respected in the public and professional world. An appeal to science or math is much more credible for most people than spirituality. But spirituality has been around for much longer than math and science as a credible part of human history. Its history gives shape for the many mysteries of math and physics. It gives greater depth. “Thus, instead of our accepting a mathematically formalizable paradox as the best that can be achieved in our understanding of nature, those paradoxes might become the royal road to our understanding the world as an expression of the mind of God” (Glasberg, 18). And so, instead of seeing math as a big jumble of numbers that doesn’t mean much to the average non-genius we can see math as a connector between fact and feelings. Since math is such a different kind of discipline, we can find a very different kind of truth from the things it speaks of.

    The history of mathematics lends itself well to this line of thinking. For example, take “the number” zero. Zero is a pretty mysterious non-existence- “the nothing that is”(Kaplan). In the mathematical records of many ancient cultures before zero was “discovered” such as the Sumerians, Egyptians, Greeks, Indians and many others a simple symbol was used for the same purpose. It would show up either as a small-centered dot or a continuous even circle; almost like it appears today. They thought of it as a kind of place holder, a condition simply resulting from the absence of everything else. Its absence from greek texts and hot debate once revealed makes one wonder a little more deeply about this curious nothing. Plato writes of a man who is presenting a theory, talking about the concepts of “being” and “becoming”. In the middle of his discourse he found the subject expanded in his mind and he has to start over. There is another factor involved, like a receptacle. He names the three “being, space, and becoming”. Its nature is “like that of a container, waiting to be filled”. “...just like the base which the makers of scented ointments skillfully contrive to start with: they make the liquids that are to receive the scents as odorless as possible...”or “...space , which is everlasting, providing a situation for all things that come into Being, but itself apprehended without the senses by a sort of bastard reasoning...” (Kaplan, 64) So there it is: absolutely nothing. However, we have yet to see zero treated as an actual number. First, lets take a look at addition- zero plus any number is that number: 0 + 247 = 247. Now, on to subtraction- zero taken away from a number is also that number: 5 - 0 = 5. Or it could be negative- any number taken away from zero is the same number, but negative (unless it was already negative, then its positive): 0 - 5 = -5. Next, multiplication: any number multiplied by zero is zero. But what about division? How many times does zero go into forty? How many times can you take zero away from 8,945?

    Zero is not the only loophole in our empirical system of boxes and calculations. Next, lets take the imaginary number. The imaginary number is pretty much what it sounds like. It doesn’t exist. Any number times itself is called a square: 2 x 2 = 2^2 (two squared). To find the square root of a number is to find out what number, multiplied by itself is equal to your original number. For example: the square root of nine is three because 3 x 3 = 9 or 3^2= 9. The square root of nine is also negative three because -3 x -3= 9. A negative times itself is always a positive. So what’s the square root of negative nine? What number multiplied by itself is negative nine? Its imaginary. It doesn’t exist-not on this planet. John Wallis - a genius of many fields with a long list of credentials who lived in the 1600s- created a sort of theory on imaginary numbers with a geometrical approach (since they pretty clearly don’t compute in algebraic terms). “Wallis had stumbled on the idea that, in some sense, the geometrical manifestation of imaginary numbers is vertical movement in the plane. Wallis himself made no such statement...” Wallis got close, close enough to indirectly influence Einstein centuries later, but “ Even so, a close miss is still a miss...” (Nahin, 47)

    The imaginary does have a trace of connection to physics through complex equations that I won’t even try to understand for years. And so, when mathematics is viewed from a dry standpoint- as only a tool for technical use and nothing more, it may be dismissed with only a curious glance. They sure seem deeper though-“Think of irrational numbers, the guilty secret of the Pythagoreans, whose exposure shook Greek confidence to the core. Twenty-five hundred years later we can’t do without them, though the sense in which they exist is debated still. And imaginaries? Mathematicians, who love high-wire acts began thinking about the square roots of negative numbers as far back as Heron and Diophantus, but whenever these came up as solutions of equations they were called fictitious and the equations judged insoluble. Then in the Renaissance people began to calculate with them, fictitious though they were. In 1673 the great cryptographer John Wallis said they might be imaginary, but were no more impossible than the negative numbers; and now they sort with the reals in the street, drawing never a sidelong glance, although they still bear the caste-mark of their name.” (Kaplan, 70) Even irrational numbers- those which, when divided produce an infinite decimal that never resolves and never repeats- rocked Pythagoras’ world. You can plot irrationals on a two dimensional graph- sort of. We round it off, dismiss it as good enough. We know its probably somewhere in there, but can’t really get the cat back in the bag. Are we dismissing out of boredom the very thing we sold our soul in search of?

    The number pi is an irrational number. It is found by dividing the circumference of any circle by its diameter. The result of the division problem is: 3.1415926535897932384626433832795028841971693993751........ Pi goes on forever. We have not been able to precisely calculate it because the numbers continue to come. They are always random, never repeating, infinite decimals. “...a predestined yet unfathomable code.”( Witcombe, 1)

    Another equally mysterious number has almost the same name: it is called phi (or the golden ratio). The number phi is said to be the most beautiful number in existence. It is 1.618. (see first attached diagram)
This is a line which has been segmented into two parts- labeled above as “a” and “b”. The larger section “a” has a ration to section “b” of 1.618 to 1 . Not only that, but the ratio of section “a” to the length of the entire line is also 1.618 to 1. It occurs all over the place in nature.(Phi, 1)

    Infinity is an equally common yet mysterious concept. It is by definition incomprehensible. It is a number greater or smaller than any other number. The symbol for infinity is ∞. John Wallis and other famous mathematicians--noteably George Cantor--investigated infinity. They showed different principles it operates with. For example take a simple demonstration by George Cantor: (see second attached diagram)

    Counting by ones to infinity is the same as counting by twos to infinity. They both would yield an infinite progression until you get to infinity. What if we counted by tens, or hundreds, or millions? A very basic concept yet somehow still seems self-contradictory. Infinity is a word that we throw around casually in conversations--like we understand it. We think we know, but we have no idea. But what if infinity didn’t exist? What if everything was finite: able to be counted and measured on a precise scale. Infinity, like other things discussed here, ruffled plenty of feathers when discovered. But now that we have them--its practically impossible to imagine the world without them.
Math is clearly not an irrelevant hoop to be jumped through--as most would see it. It causes us to question where we would otherwise easily miss some of the deepest meaning in life. The same things that fit nicely in a box through any other discipline, will not stay put through mathematics.

MATHEMATICAL REALITY

    Carl Sagan is famous for saying “The cosmos is all there is.” This physicalist view of our world makes sense to some degree. If I can’t see, hear, feel, touch, or taste something, how can I be sure it’s real? Sounds reasonable right? Yeah, but something that’s equally as basic of a principle is Descartes’ famous line “I think, therefore I am”. This pulls on the same level of instinct that the physicalist view does- if not deeper. How can a person even be sure that they really exist except their thoughts? Is this all some dream? The mind is a powerful thing. Inside it are the thoughts that guide us through life- yet they are things that cannot be seen, felt, tasted, touched, or heard in and of themselves. Peter Bussey wrote an article called “Beyond Materialism: from the Medieval Scholars to Quantum Physics” in which he talks about the lack of physics’ ability to connect mind and matter. This adds to already credible reason to question Sagan and others who hold the empiricist position in our debate. He says “The boldest claim is that all existence is accounted for in terms of the science of physics, a postulate known as materialism or physicalism: the entire properties and behavior of any physical system arise, it is asserted, from the physics of its component elements. Now, human beings consider themselves marked by possessing qualities such as purpose, consciousness and moral responsibility, which in no clear way have anything to do with the atomic physics of a biological system. A rigorous belief in materialism threatens to eliminate these qualities, leaving us as physical and chemical automata. Such a view contradicts our deepest intuitions.” The facts of physics have taken us a long way. They’ve accomplished amazing things, no question about it. However, anyone who is human can tell you that’s only half of the picture at best. Physics has no way of connecting mind and matter. There is no empirical evidence that thoughts happen in the brain, or that a person’s soul or spirit exists. The only way we know those things (which are considered common knowledge) is because we just KNOW that we know. Nobody had to tell us about it because it’s built into being human. And so it is easy to see the dilemma we find ourselves in. There are two ultra-opposite, yet both reasonably acceptable theories at hand.

    A French philosopher of the Enlightenment named Denis Diderot, wrote of mathematicians that they “resemble those who gaze out from the tops of high mountains whose summits are lost in the clouds. Objects on the plain below have disappeared from view; they are left with only the spectacle of their own thoughts and the consciousness of the height to which they have risen and where perhaps it is not possible for everyone to follow and breathe”(Nahin, vii). It is not surprising that- though based on math, our debate is also left a piece of its heart in philosophy. The earliest record of these two opinions on the existence of mathematical reality that I was able to find was between Plato and Aristotle. In Plato’s opinion, the mathematical world was a completely independent world from the physical. It would have its own laws, systems, and be superior to the physical world. The existence of mathematical reality would also, therefore be independent of any human thought or any activity outside of itself. In Aristotle’s opinion, the mathematical world doesn’t actually exist. It is an idea that we speak of only in reference to abstracted ideas creatively constructed from human experience. He considers the actual mathematical facts to be the only reality. The abstract projections of these things that Plato believes refer to some other place apart from the known world are merely doxa-- idealistic appearances.(Bouveresse, 56)

CONCLUSION

    Albert Einsten said “As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality”(Harris). Does mathematical reality exist? Mathematics express things that do not exist in our world, yet still hold true. It appears to me that mathematical reality does not have a factual existence. Now, to be certain, I am no trained expert in this field - but I don’t think this is meant to be a cut and dried case. That would contradict the very nature of the conclusion. An absolutely airtight argument for the unquestionable existence of Mathematical Reality could very possibly prevent someone from understanding the very concept it exists for. My goal in taking the idealist position in the debate was not to prove the existence of things formerly mentioned. That would technically make me an empiricist. What I do think is important is that I allow this discussion to push me forward to question the boxes. The point is that you can’t prove it doesn’t exist. Try as we might, we cannot put fences or safety nets on the world. Its too big. It is very clearly beyond any of us to grasp, that’s why these things are so intriguing. Einstein also said, “The most beautiful thing we can experience is the mysterious. It is the source of all true art and all science. He to whom this emotion is a stranger, who can no longer pause to wonder and stand rapt in awe, is as good as dead: his eyes are closed.”(Harris)

Works Cited

Atiyah, M. F, Mathematics in the Twentieth Century. Am. Math. Monthly (2001), 108,no. 7, 654-666.
Bouveresse, Jacques. “III- On the Meaning of the Word ‘Platonism’ in the Expression ‘Mathematical platonism’” Proceedings of the Arostotelian Society. (paperback). Vol. 105 Issue 1, p55-79. Jan 2005.
Bussey, Peter. “Beyond Materialism: from the Medieval Scholars to Quantum Physics” Science & Christian Belief. Vol. 16 Issue 2, p157-178. Oct 2004.
Clark, Gillian. Iamblichus: On the Pythagorean Life . Liverpool University Press. Liverpool. 1989.
Crystal Reference Encyclopedia. Infinity. © Crystal Reference Systems Limited 2006. 28 May, 2006. <http://www.reference.com/browse/crystal/16423>.
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Hardy, G.H. A Mathematician’s Apology (paperback reprint edition). Cambridge University Press. Cambridge. January, 1992.
Harel, Guershon; Sowder, Larry. “Advanced Mathematical-Thinking at Any Age: Its Nature and Its Development” Mathematical Thinking & Learning. Vol. 7 Issue 1, p27-50. 2005.
Harris, Kevin. Einstein Quotes. 1995. May 28, 2006. <http://rescomp.stanford.edu/~cheshire/EinsteinQuotes.html>
Hart, W.D. The Philosophy of Mathematics . Oxford University Press. New York. 1996.
Kaplan, Robert. The Nothing that Is. Oxford University Press. New York. 1999.
Kau, Arron. Personal interview. Calculus teacher/ OSU bachelor’s of English. 25 May, 2006.
Nahin, Paul J. An Imaginary Tale. Princeton University Press. Princeton, NJ. 1998.
Otte, Michael. “Complimentarity, sets and numbers”. Educational Studies in Mathematics. Vol. 53 Issue 3. p. 203. 2003.
Phi in 1D. 24, May 2006. <http://www.beautyanalysis.com/mba_phiin1dBOTTOM_page.htm>.
Wikipedia. Philosophy of mathematics. 29 April 2006. Wikipedia.org.1 May 2006. <http://en.wikipedia.org/wiki/Philosophy_of_mathematics>.
Witcombe, Chris. Notes on Pi. “Profiles: The Mountains of Pi”. The New Yorker. 2, March 1992. Sweet Briar College.24, May 2006. <http://witcombe.sbc.edu/earthmysteries/EMPi.html>.

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