The conclusion is more or less what I believe about mathematics. The practicality of mathematics is often vastly overstated, and yet there is something that I find fulfilling about it. Over Spring Break I read A Mathematician's Apology, by G.H. Hardy, and since then, I have come to agree more and more that mathematics is a field concerned mostly with aesthetics. When dealing with messy fractions involving square roots and the like, one often calls the result ugly; certain proofs are called elegant almost without fail. I have a textbook which compares spectral theorem to a symphony. The language of beauty is tied into mathematics more deeply than most realize.
If you asked an average person to describe mathematics, they would likely respond by saying that mathematics is the study of numbers. Some might remember to mention geometry, a topic most don’t deal with beyond one class in high school. Although this is a limited view of mathematics, it’s not wrong. And how hard can numbers be? Or plane geometry, for that matter? While some may say that they are not mathematically inclined, most people will tell you that they know what a number is, and they certainly know how shapes work. So 22/7, or the square root of 2? Definitely numbers. And two straight lines side by side will obviously never meet. Then what about the square root of -1? Is that a number? And what about hyperbolic geometry? Is it useful, or simply a warped version of what is real? Mathematical constructs like these raise questions about the very nature of mathematics.
The square root of -1, more commonly referred to as i, gives one pause. The term “square root” highlights the difficulty. The square root of 2 is the length of a side of a square with area 2. The square root of -1, then, is the length of a side of a square with area -1. Such a square makes absolutely no physical sense, which is why i is often called the imaginary number.
Is mathematics concerned with physical reality, though? This discussion goes back quite a ways. Plato, in his Republic, discussed mathematics at some length. At one point, Socrates says in reference to geometers (the mathematicians of their day), “They talk of squaring, applying, adding, and the like; whereas, in fact, the entire subject is practiced for the sake of acquiring knowledge.” (Book VII, 527a-b) So the physical impossibility of i is of no concern to mathematicians. Earlier in the Republic, Socrates takes the geometers to task for using physical objects (specifically, “the animals around us, every plant, and the whole class of manufactured things” (Book VI, 510a)) as images on which they base their studies. But no such physical object exists for i. Instead, one might say that when dealing with i, mathematicians are using other numbers as images. In the same way that geometers make arguments about squares and diagonals, drawing inspiration from the imperfect squares and lines that they see in nature, a mathematician might make an argument for the existence of i by looking at more conventional numbers. -1 is really not that different from 2. They are both integers. So if 2 has a square root, than why wouldn’t -1? This is how the square root of 2 acts as an image for i. When dealing with i, one must work completely through reason, and although still falling short of dialectic, one is still operating in the third portion of Socrates’ divided line, referred to as thought, well into the intelligible section of the line[1]. It is ironic that such numbers are called imaginary, when it is the lowly first section of the line that is referred to as imagination.
Although the concept of i has been around for centuries, it was ignored and disparaged for quite some time. Leonard Euler put in a lot of work to make i respectable. He receives the credit for introducing the notation of i for the square root of -1, and the famous equation eiπ+1=0 bears his name. Starting in the 19th century, mathematicians began to discover a number of surprising properties of functions involving i. This was part of a general movement in mathematics, mainly spurred by developments in geometry (which I will return to shortly), toward increased rigor and examining the foundations of mathematics. The question inevitably arose: What is mathematics? Kant classified mathematics as a synthesis of a priori knowledge with knowledge from the content of specific experience. In response, Dedekind, Frege, Russell, and Whitehead attempted to show that mathematics (in Dedekind’s case, just arithmetic) is entirely based on a priori knowledge: that is, that it is entirely analytic and concerned only with pure logic. This school of thought is known as logicism and is, at least initially, an attractive explanation of what mathematics is and with what it is concerned.
Of course, i fits into all of this. If anything, the existence and study of i in mathematics would seem to provide an excellent counterexample to Kant’s argument that mathematics is synthetic a priori, for there is no specific experience from which one might get the concept of i. As stated earlier, no squares have area -1. How could a field of study that deals with such a concept be synthetic? Then again, as also stated earlier, one might draw the concept of i from previous experience with other square roots by simply generalizing to the non-physical. I have studied no Kant (unfortunately), but I imagine something known through this type of mental experience is still considered a posteriori knowledge.
Still, the logicists have a trick or two up their sleeves to talk about i in terms of pure logic. It requires only a rather simple argument to show that all numbers involving i (called complex numbers) can be represented by 2 real numbers, and any operations you can perform with complex numbers are the same as certain operations on those 2 real numbers. So if the real numbers can be said to be purely logical construction, it follows trivially that i, and indeed all complex numbers, are also purely logical constructions. But there’s the rub; it has turned out to be subtly and maddeningly difficult to give a logical formulation of the real numbers. By the time one has arrived at such a construction, its definition has enough conditions on what a number is that it seems far from a priori knowledge. And Gödel’s Incompleteness Theorem brings into question whether looking at mathematics only in terms of logic is even useful. After all, one could derive a statement whose truth is undecidable from its axioms. This is a grave limitation that naturally leads to the assertion that math is more than just logic, for good or ill. So although i has risen to be as respected as the other numbers, it turns out that this is only so far up. It would seem that even as unreal a concept as i must be taken to be known only through experience, albeit experience of a very particular sort.
All this focus on number has led us away from the other branch of mathematics, dealing with shape. Geometry has been intensely studied at least since the time of the Greeks. It is geometers that Plato alternately criticized for basing their studies on unshakeable, physically-based hypotheses, and lauded for working by means of pure thought when making their arguments. Later, probably at least in part as a response to Plato, Euclid would attempt to lay down a set of axioms and postulates from which the important geometric theorems of his day could be logically derived. Many of these were uncontroversial, perhaps because they fell into the trap of using the perceived behavior of physical objects as their basis. It seems clear from experience that “a straight line may be drawn from any one point to any other point” (Euclid’s Elements, Book 1, Postulate 1), or “The whole is greater than its part” (Book 1, Axiom 9). But one axiom stood out as being different from the others, and caused a great deal more concern among mathematicians. Euclid’s parallel line postulate can be stated in many different ways, most commonly as, “Given a point and a line, there is exactly one line through that point that will never intersect the given line,” although this is not the form found in the Elements. Though this postulate was fiercely debated, and to some seemed that it should be somehow derivable from the other axioms, it was shown much later to be indispensable when dealing with Euclidean geometry. In fact, if one removed the parallel postulate, he or she would end up with a completely alien system of geometry, one in which nearly everything diverges from everything else, given enough time. This system is called hyperbolic geometry, and given a point and a line, there are multiple lines through that point that will never meet the given line. To many, it sounds ridiculous. However, it has been shown that all of Euclid’s other axioms work just fine in hyperbolic geometry. There is even a simple model of hyperbolic geometry, a circular plane with its border at infinity, and straight lines drawn as circular arcs that cross the border at a right angle. But as simple as this model is, it seems absurd, and a little lonely. Everything ends up infinitely far away from everything else.
It was inquiry into alternate geometries such as these that helped lead to serious discussion of the nature of mathematics. In dealing with such geometries, it was shown that any theorem that could be deduced from its axioms should hold in any accurate model of those axioms. This lent itself to a logicist view of mathematics, because so long as the axioms of geometry could be considered pure logic, all of its theorems were deduced through pure logic, and so geometry itself could be considered purely logical. This ran into the same problem as the logicist view of numbers, namely, the Incompleteness Theorem. Other schools of thought arose in the late 19th and early 20th centuries as well. One, formalism, is in many ways similar to logicism, and so will not be discussed in this paper. But the third major school of thought, intuitionism, has serious differences from the other two. It seems to be a complete restructuring of mathematics that leaves behind some of the most important tools in mathematics to arrive at new and novel ideas.
Intuitionists believe that mathematics deals solely with mental constructions based on intuition. Based on this belief, they reject the Law of the Excluded Middle, which holds that if something is not not true, then it is true. An intuitionist would respond that by showing that something is not not true, one has not shown any construction in which it is true, so how can one draw such a conclusion? This thinking led to a radical reconstruction of analysis and logic, albeit one that appears in many ways limited and, ironically, counter-intuitive. What would an intuitionist make of the hyperbolic plane? A better question might be, what would an intuitionist make of the parallel postulate? It is doubtful that they would hold it in high regard. Although it is simple to construct two lines with a common perpendicular, an intuitionist would likely scoff at the claim that all the perpendiculars between those two lines are now common. Such a thing is not able to be shown, because it deals with an infinite number of perpendiculars. Or when dealing with Euclid’s original statement of the parallel postulate, one can certainly show where two lines that meet on a straight line at less than two right angles meet using trigonometric functions. But an intuitionist would challenge the thought that you can tell whether or not lines meet at a right angle. If one measures the angle between the two to be 90.0000o, it still cannot be said with certainty that it’s a right angle, because it could really be 90.000010. So an intuitionist would not hesitate to throw out the parallel postulate, opening the door for hyperbolic geometry. It seems likely, however, that an intuitionist would also not hesitate to throw out quite a bit more. Any theorem in hyperbolic geometry proven by contradiction would be thrown out. Any proof that uses such a theorem would be thrown out. Precious little remains of geometry after an intuitionist has his or her way.
The hyperbolic plane and the number i both turn our focus to the unbelievable in mathematics. Their existence seems nothing more than useless mathematic abstraction, separated from reality. An intuitionist would say that mathematics is wholly based on our intuitions about reality, and although intuitionism is largely dissatisfactory to many mathematicians, if the intuitionists are wrong, then what is mathematics if it is not concerned with reality? Some would say it is concerned only with logic, but this view does not hold up to close scrutiny. Others reduce mathematics further, to simple operations on strings of symbols, but this seems to ignore the power of mathematics. Plato described mathematics as reasoning from assumptions, and few would argue with that. It is still a good description, and it is by working logically from simple assumptions that surprising results like the properties of i and non-Euclidean geometries have been discovered. But what good is such reasoning? Although Plato found the assumptions mathematicians made appalling, he still recognized that mathematicians worked through pure reason, and in doing so, turn their minds toward what he termed the Good, by which all truth and beauty is illuminated. Ultimately, mathematics is the pursuit of the Good through logic, with help from a few extra-logical assumptions. From as simple a set as possible of first principles, mathematicians work toward beauty and truth. By working with concepts like i and the hyperbolic plane which are far removed from the physical world, mathematicians come to deal more completely in pure thought, moving ever closer to that goal.
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