The best mind altering drug is truth

                                                                 -Lily Tomlin
 

 Why I wrote this paper

As a teacher of mathematics, I have often felt that the heavy emphasis on teaching computational methodology and its applications leaves students with the feeling that mathematics is a rather boring, dry and seemingly uninspired subject.  Additionally, students typically feel that mathematics has little to do with their lives.  It is my intention to present aspects of mathematics in this paper, which will serve to show a more "human side" to the subject.  This presentation will include morsels from the rich history behind modern mathematics, which has roots in a great diversity of cultures.  It seems to be a virtual trademark of the modern western viewpoint that academic thought can easily be divided into distinct subject areas.  I will, however, show that many of the fundamental ideas and modes of thought in mathematics cannot be well appreciated without regard to the underlying philosophical beliefs held by mathematicians1 and by the cultures in which the ideas were developed.  Mathematical thought seems to be nearly as old as human existence itself and much of its development is shrouded in religious belief and mysticism.  Indeed, one of the primary reasons why many ancient peoples first contemplated questions of math and science was an attempt to verify to themselves in explicit detail the perfection of God's creation.  The very human need to want to discover beauty and logic in nature persists today at the very highest levels of
scientific inquiry.

Modern mathematics, like most of modern scientific thought, is marked by discoveries that show many profound interconnections between apparently unrelated ideas2.  Indeed, when mathematicians speak about beauty in mathematics, they are often referring to a sense of inner harmony that is felt on the deepest emotional level when difficult mental connections are revealed through abstract mathematical notation, often scrawled out in a state of deep meditative concentration.  For the mathematician, the process of mathematical discovery is far from being cold or sterile.  Many have described the process of mathematical research as something akin to a religious experience.  Although it is relatively infrequent that one is actually able to attain such deep insight, most mathematicians do view mathematical systems as beautiful.  This appears to be quite similar to the way that many people have a high sense of aesthetic appreciation for music or poetry.  In music, completely different sounds can be blended together to create a delicate and unified whole, which often has an interpretation (to the listener) as far more aesthetic than the way in which any of the sounds would be perceived individually3.  In mathematics, the careful construction of logical systems (arithmetic would be an example of a logical system, geometry another) may lead to both profound "truths" as well as a certain kind of "self discovery of the mind".  It is my hope that in reading this paper, you will view math in a somewhat broader way than you would have otherwise, and that you will perhaps discover  a sense in which you may be more "mathematically curious" than you previously suspected.  Sit back, relax, and enjoy the ride!
 

 What this paper is about

There is a myriad of interesting elementary topics in the history of mathematics that are all worthy of extensive discussion.  I will, however, narrow the focus of the current presentation to one central theme-formal logical systems (also known as formal axiomatic systems).  I will say precisely what is meant by an axiomatic system in the next section.  You do, however, already know about several examples of such systems.  Any classical board game may be viewed as a logical system.  The pieces in, say chess, may be moved only according to a finite list of rules.  The same holds for other board games as well, even though the specific rules in two different games may be entirely different.  In any of these games, one starts with some initial configuration of the pieces and moves through a sequence of different configurations (usually by taking turns), the pieces being moved according to the rules of the game.  Anyone violating the rules is considered a cheater (or at least ignorant of the game).  The game ends when some pre-specified type of pattern is obtained by one of the players (e.g., checkmate in chess).  Arithmetic can be viewed as another example of an axiomatic system.  The "pieces" are numbers and symbols.  The "rules of the game" are properties of numbers and the way in which the operations (e.g., addition, subtraction, multiplication and division are operations) are legitimately performed.  So, for instance, we can start with an initial "configuration of the pieces", say, 12+13.  We then may apply the "rules" of arithmetic to arrive at the resulting configuration which, as you know, is 25.  Of course many calculations involve many more and complex steps, but the analogy to board games is still the same.  Perhaps human language could also be viewed as yet another example of a logical system.  The words are the "pieces", and the rules of grammar describe the "allowable moves" (that is the allowable ways of building sentences and phrases, etc.).  You might try to think about the important ways in which language truly is similar to a board game as well as the ways in which it may be different.  Linguists are still in a state of hot debate over such ideas.

After a more involved discussion of logical systems, I will describe some of the most important thinking of our current century on the topic of formal axiomatic systems.  Specifically, I will give a brief overview of the life and ideas of the Austrian born mathematician Kurt Gödel.  Not only did Gödel's discoveries about logical systems have implications in all areas of mathematics, but they seem to have potential implications regarding certain aspects of human consciousness itself.  It is the very fact that there seems to be an intimate connection between abstract mathematical reasoning and human philosophy/psychology that captured my interest in mathematics.
 

 Logical (axiomatic) systems defined

Let's take a careful look at elementary algebra as a system.  The elements of the system are of several sorts:

The objects of the system.  Some of the objects we call numbers.  Other ones we call the operation symbols representing addition, subtraction, multiplication, and division.
The relation symbols are the symbols used to compare the numbers.  Examples are =, <, >.
 

The axioms of the system serve to spell out explicitly what ground rules we are assuming (just as in chess, the written rules of the game specify exactly which kinds of moves are possible).  You have already been told about some of these axioms explicitly. To remind  you briefly, they are:
 

I.     a+b = b+a    (commutative property of addition)
       ab=ba        (commutative property of multiplication)

II.    (a+b)+c = a+(b+c) (associative propert of addition)
       (ab)c = a(bc)     (associative property of multiplication)

III.   a(b+c)=ab+ac       (distributive properties)
       (b+c)a=ba+ca

IV.    0+a=a+0 = a     (additive identity property )
       1a=a1=a              (mult. identity property)

V.     (1/a)a=a(1/a)=1, for a not zero.   (mult. inverse property)
       a+-a=-a+a=0     (additive inverse property).

Additionally, there are a good number of other axioms that were discussed, but not given special names.  An example would be the property that states we can always add the same number to both sides of any given equation.  This axiom is actually of a very special type called a rule of deduction.  It tells you a way in which you can start with one equation and produce another "legitimate" one from it.  Can you think of another very similar rule of deduction that we have been using repeatedly in the course?  Another axiom is called the axiom of closure.  It states that the sum or product of any two real numbers is always again another real number.  It also states that the opposite (negative) of any real is another real number.  "Obvious" as it may sound, many proofs would be incomplete without explicit reference to it.

Now once the axioms ("the rules") are specified, it would seem that in principle we should be able to decide whether any algebraic statement is true or false.  How would we do that ?  Well, how would you decide whether some random arrangement of chess pieces could actually arise in the course of some chess game4 ?  Certainly one way would be to actually start with a "legitimate" chess configuration (for example the initial configuration of any chess game ) and strategically apply the rules of chess to actually produce the configuration in question.  One of several things will happen.  If you are clever enough, you might actually succeed in producing the desired configuration after some finite number of moves (assuming, of course, that it can even be done at all).  Another possibility is that you keep trying to produce the desired configuration, but you die before this happens (so you will never know if it was possible).  Of course another possibility is that you simply get frustrated and give up (in which case you will also never know if your goal was actually achievable).  The same idea applies to establishing the truth or falsity of algebraic statements.  One can simply start with a known true statement (maybe itself an axiom!) and apply other axioms to it until one obtains the desired statement.  Of course, if one tries to prove a false statement in this way, one of two things will happen.  Either we will, at some point, notice that it is false, because it is found to contradict an already established truth, or we will die, never having discovered the its falsity.  Let's take a look at an actual example of a proof.  Any statement that a mathematician claims to have proved is called a theorem.  Let's prove the following theorem.

Theorem:  If X denotes any real number, then 0=0·X.
(note that despite the intuitive "obviousness" of this theorem, we can not be sure that it really does follow from the above axioms, until we have used the axioms to construct a proof).

Proof:  0·X is a real number (closure property of multiplication)
  0·X=(0+0)·X  (by the additive identity property)
         0·X=0·X+0·X  (by the distributive property)
  -(0·X) is a real number (by the closure property of opposites)
         -(0·X)+0·X=-(0·X)+[0·X+0·X]  (by adding the same number to both sides)
         0=-(0·X)+[0·X+0·X]  (by the additive inverse property)
         0=[-(0·X)+0·X]+0·X   (by the associative property)
         0=0+0·X  (by the additive inverse property)
         0=0·X    (by the additive identity property)

You should try to follow the above proof, even though it will certainly take a bit of serious concentration.  You should read it one line at a time, not moving on to the next until you have understood how the present one is justified by the axiom quoted in parentheses.

Whether or not you feel that you have a flawless understanding of the above proof, you should at least have a decent feel for what is meant by the notion of a formal axiomatic system.
 

 Why the need for formal axiomatic systems ?
 
Now you probably feel that the above proof is pretty abstract and not very "illuminating", even if you have managed to follow each of the steps.  You might feel that any mathematical statement should, somehow, have closer ties to "reality" (at least your personal sense of reality) than being the abstract symbol pushing exercise displayed above.  Thus you might endeavor to construct your own proof that 0=0·X in a way more similar to the following argument:
"multiplication by any number, say the number X, means to add the number to itself until
the number occurs X times in the sum.  So, 0·X must mean 0+0+....+0 (there are X zeros).  and since 0 represents "nothing", so does 0+0+...+0. So 0=0·X.  You would certainly not be alone in your feeling that such a proof "based on reality" is both easier to understand and perfectly acceptable. In fact, most schools of mathematical thought (until the current century) reflected this same type of thinking.  That is to say properties of numbers as well as other types of mathematical relationships were viewed as being correct if they seemed to coincide with the physical world.  Thus numbers themselves generally were regarded as being meaningful only when they represented physical quantities (e.g., numbers of stars, sheep, apples,grains of sand5, distance, temperature, etc.).  Properties of numbers were regarded more as properties of physical reality.  Indeed much impressive work in mathematics was done by people who held this viewpoint.  There was relatively little obsession with clearly stating the axioms (the base level assumptions) in explicit terms.  Great mathematicians thus avoided making too many serious fundamental errors in reasoning because they had extraordinary intuitive insight into many aspects of physical reality6.  It should be noted, however, that mistakes were made, even by many of the superb thinkers of the times.  Following the development of the Calculus in the seventeenth and eighteenth centuries, many mathematicians became more bothered than ever over certain seemingly paradoxical notions.  Many of these paradoxes revolved around notions of "indefinitely small positive quantities" and "arbitrarily large" quantities.  Although I will not give the details of how these paradoxes were ultimately resolved in modern math (the details are rather involved and technical), I would like to give several examples of them.  The first one can be easily stated in terms described by Zeno, an ancient Greek mathematician and philosopher, over two thousand years ago.  In describing the paradox he claims to assert that "a person in a room will never be able to leave that room".  In order to leave the room, argues Zeno, a person must first go halfway to the door.  He then says that even if by some miracle he should succeed in accomplishing this feat, he would still have to go half of the remaining distance, after which he would still have to go half the remaining distance, and so on-the process never ending.  Clearly a person would have to traverse some distance an infinite number of times, reasoned Zeno, so all these distances added together (to give the total distance to be travelled)  would give an "infinite distance", which no mortal could travel in our merely finite world.  The gist of Zeno's argument is most easily understood by studying the picture labeled Zeno's paradox.  Of course, being one of the great thinkers of his times, Zeno did realize that people really did leave rooms.  He could, however, not reconcile the apparent logic of his argument with the obvious reality.  The modern resolution of Zeno's paradox is rooted in a very careful and sharply defined notion of the infinite, which eluded mathematicians until the past century.  Indeed, this more sophisticated way of regarding infinite processes came about in large part when mathematicians recognized the need to put mathematics on more solid logical ground as a deliberate attempt to free mathematics from such bothersome paradoxes, thus making math more "pure".

Long after the time of Zeno, mathematics experienced a heyday in the seventeenth century when mathematicians made great strides in developing the area of math known today as Calculus (The great English scientist and mathematician Sir Isaac Newton then referred to his work as "the method of fluxions", and the great philosopher, mathematician, and scientist Gottfried Wilhelm Leibniz referred to his work as the "method of infinitesimals".).  Both Newton and Leibniz seem to have developed many of the same methods independent of each other, and a bitter dispute over priority of discovery, rooted largely in national pride, broke out.  But that's a paper in itself.
Like their predecessors, both Newton and Leibniz were not so much concerned with providing a rigorous logical foundation for their methods, but employed new methodology freely so long as it was consistent with their intuition.  Both men were, after all, largely concerned with questions of physics and viewed math as a tool for reasoning about physical phenomena.  So long as their methods seemed to work as a reliable tool for their science, neither man felt any particular need to agonize over isolated and infrequent "quirks" in the fine details.  There was, however a growing number of mathematicians and philosophers who continued to be bothered by these quirks or logical gaps.  To get a feel for the kind of "logical gap" in their methodology to which I am referring, I will very briefly describe the "big bold idea" of a major portion of calculus-that part known as integral calculus (calculus is divided into two parts in modern textbooks: integral and differential calculus).  The idea is that of an infinitesimal.  Consider how one might try to find the area of some irregular shape (see picture) such as the one shown.  Note this is a harder problem than finding the area of a "simple" figure like a triangle or rectangle, for which there exist nice area formulas from ancient geometry.  In fact the problem is much harder, since there are no "straight" or "regular" sides to the figure.  Well, you might first start by finding the approximate area by covering the figure by a grid of small squares.  You could then compute the total area occupied by the squares, which comes close to the area of the strangely shaped figure itself (see picture)-not bad!  But the nagging fact does remain that your answer is still only an approximation to the actual area of the figure, and not an exact answer.  So, you say, how about trying to fit in more squares (thereby leaving less of an "error region") by using smaller squares? So far, so good-your answer is now closer to the actual area-but still only an approximation.  You might repeat the process ten billion times (or even more), but each time you would still only have an approximation, albeit better ones each time.  To get the exact area, what Newton and Leibniz did was to assume the existence of infinitely many "indefinitely small" little squares (called infinitesimals) and devise a method for summing (adding up) these infinitely many pieces.  As you can well imagine, such methodology received wide criticism from those types bothered by Zeno's paradox and the like.  Their contention was, essentially, how do we really know the methods to be valid if Zeno's reasoning produced the wrong answer.  Could it be that these new methods of Newton and Leibniz might also be wrong, even if their incorrectness is harder to "see" ?  Bishop George Berkeley, the famous English philosopher and theologian, mirrored these sentiments in 1734 when he published a satirical work entitled The Analyst, Or a Discourse Addressed to an Infidel Mathematician.  In referring to the infinitesimals used by both Newton and Leibniz he says:  "May we not call them ghosts of departed
quantities ?"  It was not until nearly a century after Newton's death in 1727 that great strides would be made in dealing with "the problem of infinitesimals".  Ultimately, the imprecision in Newton's logic was to be ameliorated by mathematicians such as the French born Louis Augustine Cauchy.  Although Cauchy's resolution of the problem of infinitesimals involved a kind of ingenious reformulation of an infinite process in finite terms, there has been work done in more recent times which gives a valid formulation of the infinitesimal concept, deemed to be on rather shaky ground in the time of Newton and Leibniz.  In fact, this relatively recent work is the basis for an entire branch of modern mathematics called non-standard analysis.  Essentially, the way in which the modern approach to infinitesimals "preserves the logical integrity" of infinitesimals is by creating a whole new formal axiomatic system in which the infinitesimals are viewed as a "new sort of number".  As in any formal axiomatic system, it is spelled out in most explicit terms what types of manipulations one can make with these "numbers".  The "correctness" of the system lies in its logical consistency and rigorous methodology.
 

Ultimately the focus on axiomatic formalization of mathematics would prove to be the catalyst for the development of entirely new branches of mathematics.  People began to consider formal systems of axioms for their own sake, rather than always being focused on issues of practical utility7. Thus, much of mathematics began to be regarded (and still is today, even more so than ever) as an elaborate "game", where "game strategies" were studied for their inherent interest8.  One of the consequences of this view was the creation of a "new kind" of geometry.  It was the hungarian Janos Bolyai (1793-1856), the German Karl Gauss (1777-1855, born in Brunswick, Germany) and the Russian Nicolai Ivanovitch Labachevsky (1802-1860) who pioneered the beginnings of non-Euclidean geometry9.  I would like to describe the most basic idea behind the development of non-Euclidean geometry10.  Many of the axioms of Euclidean geometry simply describe the basic notions of point, line and plane.  One comes quickly to an axiom known as Euclid's fifth axiom or the parallel postulate (axiom=postulate).  This axiom has played a special role in the history of geometry, so I will state it here.

The Parallel Postulate:

Through any point P, not lying on a given line L, there is exactly one line through P which is parallel to line L (see the picture labeled 'the parallel postulate').

In any case, for centuries mathematicians have had the feeling that the parallel postulate could actually be proved from the other axioms (in which case it need not be taken to be an axiom at all; it would simply be proved from the remaining axioms as a theorem11).  However, nobody had actually ever succeeded in supplying such a proof (many claimed to have had proofs, but serious errors in reasoning had been found in all of them).  It was the genius in Karl Friedrich Gauss that began to suspect that not only was it necessary to take the parallel postulate as an axiom after all (as the ancient Greeks seem to have suspected all along), but that it was, in fact, independent from it.  By this he meant that if one were to completely replace the parallel postulate with an alternative assumption (such as there being two  or more lines through point P that are parallel to L), this would not conflict with the other axioms (that is to say it would not produce any logical inconsistencies in the resulting system of geometry)-truly amazing!  One could either choose to accept the parallel postulate or one could choose to replace it with an entirely different assumption and either way would not result in any logical inconsistencies-even though the two systems would be different!  Neither system would be any more "correct" (logically consistent) than the other-the two would just be different.  For instance, if we assume that through P, there are at least two lines parallel to L, then it can be proved that the sum of the interior angles in any triangle is always less than 180 degrees.  There are many other seemingly strange consequences that result as well.  The collection of all such results together with their proofs is known as hyperbolic geometry.  By changing the parallel postulate in various other ways, modern mathematicians have developed entirely new varieties of geometry that are wildly different than Euclidean geometry, which was taken as "absolute truth" for several thousands of years.  In such geometries the fundamental relationships between points, lines and planes have virtually nothing to do with the "geometric (visual) intuition" that we all develop as mere consequences of observing, and otherwise perceiving the physical world through our senses.  By constructing geometries in which all lines had only finite length, the German mathematician George Bernhard Riemann (1826-1866) provided physicists such as Albert Einstein (1879-1955) with the basic mathematical tools necessary for the precise expression of his general theory of relativity.  In Riemann's geometry, space itself could have variable curvature (in contrast to Euclidean geometry in which space is always thought of as uniform-that is, "the same everywhere").  In Einstein's theory, light rays act as "straight lines" that are curved in the direction of the fabric of space-time12 in the presence of massive bodies.
 

 Old Paradoxes Revisited

Having made considerable progress in repairing certain logical gaps in mathematics and having developed whole new mathematical systems, mathematicians now turned their attention to yet another kind of paradox in the early part of our century.  The sort of paradox to which I am referring can be demonstrated by the statement S below, and is called the Liar's Paradox.

                 (S) 'This statement is false.'

Well, is statement S true or not ?  If you think it is true, then you must accept its assertion-which immediately contradicts your supposition that it is true.  On the other hand, if you say that statement S must be false, then you must reject its assertion (of falsity)-which then immediately forces you to accept it as true.  Either way leads to a sort of "self contradiction". Another similar classic paradox is the following statement:

 'The barber of Seville shaves exactly those people who can't shave themselves.'

The question then is; who shaves the barber (he does need a shave)?  If you say he should shave himself, then you only need glance at the above "rule" which rules that out (do you see why ?).  On the other hand, if the barber can't shave himself, then he is the type of person mentioned in the second half of the sentence (those who can't shave themselves), so the sentence says he is one of the people he shaves! AAAARRGGGHHH!!!

You might think this is all intriguing, but wonder what it has to do with mathematics.  The answer is that if such seemingly self-contradictory statements can arise so easily from "everyday language", then might not a similar phenomena occur within the framework of formal axiomatic systems?  These worries were dispelled when mathematicians, like Bertrand Russel and Alfred Whitehead, working in the areas of logic and set theory, made clear that it was important to spell out in exact terms what is even meant by the notion of a well-defined sentence in math.  They built axioms into mathematical logic which would have the effect of ruling out such aberrations as S above or the barber sentence on the grounds that it didn't fit conditions stipulated by the axioms.  Statements such as S were, therefore, deemed not to be meaningful sentences at all, so that notions of truth or falsity simply didn't apply (much as it makes no sense to ask whether the utterance 'tree!' is true or false).
 
 
 
 
 

 Kurt Gödel

In this final section I will describe a result that many deem one of the deepest and most profound achievements of twentieth century mathematics.  The result is known to mathematicians as Gödel's Incompleteness Theorem.  In order to begin to understand the result, it is necessary to first make a distinction between formal axiomatic systems (which have already been discussed at length in previous portions of this paper.) and what often goes by the name of metamathematics.  Simply stated, a metemathematical statement would be any statement in a human language (e.g., English, German, Greek, Chinese, etc.) about mathematics.  A metamathematical statement is not a statement within a formal system, but rather a statement outside the formal system.  For example, the statement 'there are infinitely many integers' is an example of a metamathematical statement since it is expressed in human language, not in the formal symbols of an axiomatic mathematical system.  On the other hand a statement like '2+3=5' is a formal mathematical statement, not a metamathematical one, since it is expressed using only the symbols of mathematics.  Writing the sentence 'two added to three is the same as five' is a metamathematical statement as it is expressed in human language.  You might say that the last two sentences express the "the same truth."  You say this because you have interpreted the formal statement.  There are a great many simple metamathematical statements that you would have a great deal of difficulty expressing in formal mathematics because human language is easier for humans than the awkward formalism of axiomatic systems.  This is why in practice mathematicians mix formality and metamathematics in describing their work.  Its just much easier to understand.

O.K. Now lets briefly discuss the notion of truth in a formal axiomatic system.  We have already discussed the idea of logical deduction from axioms, otherwise known as proof (much earlier towards the beginning of the paper).  Based on this discussion, you might say that the true statements in a formal system are the ones for which a proof exists within the system.  This has nothing to do with whether a given person happens to be able to discover a proof.  Once the system is defined (the axioms specified), either there does or there doesn't exist a proof (or maybe more than one) "waiting to be discovered."  What Gödel proved in 1930 was the following:

Gödel's Incompleteness Theorem:
In any formal axiomatic system (which has at least the complexity of the system of formal arithmetic) it is possible to construct a sentence (call it G for Gödel's sentence) that is neither provable nor refutable from the axioms.

At first this might not strike you as so incredible sounding, for you might be thinking that Gödel meant that he (or any other living person) could not find a proof or refutation of the sentence G (I still haven't told you anything about what G actually says.  I will, be patient) within formal mathematics.  However this would be a misunderstanding of Gödel's theorem.  What the Incompleteness theorem does mean is that he could formally prove that a proof (of the truth of G) doesn't exist!  In other words he showed that formal axiomatic systems do not posses the "logical strength" to validate themselves.  This implies the notion of "unknowable truths" within axiomatic systems.

In case you want to get a feeling for the kind of sentence G is, here it the idea.  Essentially G says:

(*)   'This statement of arithmetic has no proof in formal arithmetic.'

Does this remind you of the liar's paradox ('this sentence is false') ?  It should!  It should be noted, however, that the sentence marked (*) is not really exactly Gödel's sentence G.  After all, the precise statement of G is a formal statement in mathematics, not a metamathematical statement in English as (*) expresses it.  So what Gödel had to do was to find a way to "translate"  this sentence into the language of formal mathematics.  I will not give the details of how he did this; its very technical.  The essential idea, however, is that he developed an ingenious way of coding statements by numbers.  This coding procedure is called Gödel numbering today.  Anyway, in this coding scheme, each statement can be coded by numbers.  For instance the statement 'five is greater than two' might get coded as the number 90732.  The sentence 'there are infinitely many prime numbers' might just happen to get coded as 3097744527.  The statement marked (*) would have its own particular Gödel number associated with it.  In coding statements in metamathematics as numbers within a precise formal axiomatic system, Gödel was, in effect, "making mathematics talk about itself".  This was to most mathematicians in 1930 such a bizarre and unanticipated result that most of them simply chose to ignore it for two reason's.  One was that because the ideas involved seemed so novel and strange that few wanted to stake their professional academic reputation on it.  Secondly, it was, to many, an emotional and disturbing result.  The reason for this was that Gödel's theorem implied that one of the major goals of twentieth century mathematics of the time could never be achieved.  This meant that mathematicians who felt that they "were within range" of formally establishing the absolute consistency of certain axiomatic systems (and had taken most of their adult life trying) were doomed to failure.  Their professional life's ambitions had just been suddenly wiped out in the course of a twenty minute talk delivered by the young Kurt Gödel who had just barely completed his doctoral dissertation.

Since Gödel first proved his incompleteness theorem, others such as the English mathematician Alan Turing have established other results that seem to have a flavor similar to Gödel's theorem.  Turing's halting problem has fascinating implications for theoretical computer science and has relevance in discussing the question of whether humans could ever hope to construct a machine so complex that it could "think" (no one knows the answer-we are probably light years from it.  The real issue is trying to understand what "thinking" even means13).

It is my hope that you have enjoyed the paper and found some of the ideas discussed intriguing.  Perhaps it has even changed your previously held belief about the kind of subject that mathematics is.  My real intention was to peak your curiosity a bit.  I will consider myself to have succeeded if I have accomplished that.
 
 

You can only find truth in logic if you have first found truth without logic

                                                                            -G.K. Chesterfield
 
 

 
 Notes

[1] I am using the word mathematician here in a broad sense.  By it I mean anyone who is  intensely interested in mathematics and spends a great deal of time with it, whether or  not it be their profession.

[2] For two excellent accounts of this phenomena in physics, see The Tao of Physics by  Fritjof Capra and the film Mindwalk, available on video.

[3] Although this phenomena occurs in most kinds of music, one of the most striking  examples occurs in much of the music of J.S. Bach.  Bach's music has been quite well  studied from a mathematical point of view and has been found to contain intricate  complex mathematical patterns.  The great German mathematician Gottfried Leibniz once  said that "music is the pleasure the human soul experiences from counting without being  aware that it is counting."  If there is any truth to this thought, then not only must Bach  have been counting, but he was indulging in an orgy of complex calculations.

[4] You might actually try an intriguing problem that deals with a modified version of chess.   On an ordinary empty chess board, try to put down eight pieces (pretending that each one  is a queen) in such a way so that no queen is in a position to take any other queen.  Can  this be done ?

[5] There is a classical work by the immortal Greek mathematician, scientist and philosopher  Archimedes called the Sand Reckoner addressed to King Gelon in which the author  endeavors to estimate the number of grains of sand on all beaches on earth.  Although  the answer obtained is probably in great error (due to such a limited knowledge of our  own planet at the time), the work is nonetheless considered a jewel in the history of math  and science.  In the process of carrying out his reasoning, Archimedes proceeds to invent  a whole new notation for representing large numbers and performing complex  calculations with them.  He uses methods of reasoning unmatched for his time.

[6]  Until about 130 years ago, many areas of math and science were not regarded as distinct  areas of study.  There was tremendous overlap between areas of physics, astronomy and  math.  Thus, many of the pioneers of modern math were just as concerned, if not more  so, with questions in the physical sciences.

[7] It should be pointed out that the ancient Greeks were actually quite concerned with  developing their geometry as a formal axiomatic system, and made great strides in doing  so.  However, following the "golden era of Greek geometry", mathematics experienced  an overall decline in interest for such rigorous methodology until the eighteenth century.   Mathematical giants such as Euler, Cauchy, Weirstrauss, Lagrange, Legendre helped to  resurrect and further develop this formal approach to mathematics.

[8] Most mathematicians would not have used such words as "game" to describe their own  work.  They would be more likely to call it research.  For a hilarious spoof on  mathematicians who take themselves too seriously, listen to the song 'Lobachevsky' by  the comedian Tom Lehrer (available on CD and tape) who was popular in the 1960's.
 

[9] Euclidean geometry (also known as plane geometry) was the geometry studied by the  ancient Greeks.  This remained the only kind of geometry studied for several thousand  years.  It still accounts for most of the geometry that high school students study today  around the world.
 

[10] I know, you are thinking that this is going to be rough going, since you probably don't  even remember the basics of your high school geometry (or maybe you don't remember  being awake in geometry....).  Don't panic!  You will need to remember almost nothing.   All you need to recall is that Euclidean geometry was used to establish certain facts,  mostly about plane figures (e.g., squares, rectangles, parallelograms, etc.).  The exact  methods and formulas are not important here.  The facts (which were called theorems  and corollaries) concerned such notions as angle, area, and perimeter.  For instance one  of them states that the sum of all the interior angles in any triangle is always 180 degrees.   Another one states that the area of any triangle may always be calculated by multiplying  the base by the corresponding altitude (height) and dividing this result by 2.  There were  dozens upon dozens of other theorems, but recounting them here would serve no  purpose.
 

[11] Remember that axioms in a logical system are not "derived" or proved.  They are the  fundamental assumptions.
 

[12] The term space-time refers to a type of inseparability of the notions of space and time.   Einstein represented this interconnection mathematically as a four dimensional  mathematical entity.  While the notion of a "four dimensional reality" may have a certain  aesthetic appeal to science fiction fans, it should be noted that some of the leading  physicists today consider theories that depend on even higher-dimensional mathematical  representations.  Is reality really more bizarre than fiction?  Probably so.  In the words  of Leo Rosten:  "truth is stranger than fiction, because fiction has to make sense."

[13]   See the book The Emperor's New Mind for an interesting discussion on the nature of  thinking and human consciousness.
 
 
 

Bibliography
 

[1] Asimov, Isaac.  ASIMOV ON NUMBERS.  New York, Pocket Books, 1977.

[2] Barrow, John D.  PI IN THE SKY.  Cambridge: Oxford University, 1922.

[3] Hofstadter, Douglass.  GODEL, ESCHER, BACH: AN ETERNAL GOLDEN BRAID.   New York, Basic Books, Inc., 1979.

[4] Maor, Eli.  TO INFINITY AND BEYOND.  New Jersey, Princeton University Press,  1991.