PLATO
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Is PLATO’s mathematical World real?
This was an extraordinary idea for its time and it has turned out tobe a very powerful one. But does the platonic mathematical World actually exist in any meaningful sense?
Many people including philosophers might regard such a World as a complete fiction – a product merely of our unrestrained imaginations.
Yet the platonic viewpoint is indeed an immensely valuable one. It tells us to be careful to distinguish the precise mathematical entities from the approximations that we see around us in the World of physical things.
Moreover it provides us with the blueprint according to which modern science has proceeded ever since. Scientists will put forward models of the World or rather of certain aspects of the World and these models may be tested against previous observation and against the results of carefully designed experiment.
The models are deemed to appropriate if they survive such rigorous examination and if in addition they are internally consistent sructures.
The important point about these models for our present discussion is that they are basically purely abstract mathematical models. The very question of the internal consistency of a scientific model in particular is one that requires that the model be precsely specified.
The required precision demands that the model be a mathematical one for otherwise one can not be sure that these questions have well-defined answers.
If the model itself is to be assigned any kind of existence then this existence is located within the Platonic World of mathematical forms. Of course one might take a contrary viewpoit :namely that the model is itself to have an existence only within our various mindsrather than to take Plato’s World to be in any sense absolute and real.
Yet there is something important to be gained in regarding mathematical structures as having a reality of their own. For our individual minds are notoriously imprecise unreliable and inconsistent in their judgements.
The precision reliability and consistency that are required by our scientific theories demand something beyond any one of our individual minds.
Aslında maddi destek istememizin nedeni çok basit: Çünkü Evrim Ağacı, bizim tek mesleğimiz, tek gelir kaynağımız. Birçoklarının aksine bizler, sosyal medyada gördüğünüz makale ve videolarımızı hobi olarak, mesleğimizden arta kalan zamanlarda yapmıyoruz. Dolayısıyla bu işi sürdürebilmek için gelir elde etmemiz gerekiyor.
Bunda elbette ki hiçbir sakınca yok; kimin, ne şartlar altında yayın yapmayı seçtiği büyük oranda bir tercih meselesi. Ne var ki biz, eğer ana mesleklerimizi icra edecek olursak (yani kendi mesleğimiz doğrultusunda bir iş sahibi olursak) Evrim Ağacı'na zaman ayıramayacağımızı, ayakta tutamayacağımızı biliyoruz. Çünkü az sonra detaylarını vereceğimiz üzere, Evrim Ağacı sosyal medyada denk geldiğiniz makale ve videolardan çok daha büyük, kapsamlı ve aşırı zaman alan bir bilim platformu projesi. Bu nedenle bizler, meslek olarak Evrim Ağacı'nı seçtik.
Eğer hem Evrim Ağacı'ndan hayatımızı idame ettirecek, mesleklerimizi bırakmayı en azından kısmen meşrulaştıracak ve mantıklı kılacak kadar bir gelir kaynağı elde edemezsek, mecburen Evrim Ağacı'nı bırakıp, kendi mesleklerimize döneceğiz. Ama bunu istemiyoruz ve bu nedenle didiniyoruz.
In mahematics we find a far greater robustness than can be located in any particular mind. Does this not point to something outside ourselves, with a reality that lies beyond what each individual can achieve?
Nevertheless one might sill take the alternative view that the mathematial World has no independent existence and consists merely of certain ideas which have been distilled from our various minds and which have been found to be totally trustworthy anda re agreed by all.
Yet even this viewpoint seems to leave us far short of what is required.
There seems to be a danger of circularity here fort o judge whether or not someone is in his or her right mind requires some external standard.
So also does the meaning of authoritative unless some standard of an unscientific nature such as majority opinion were to be adopted (and it should be made clear that majority opinion no matter how important it may be for democratic government should in no way be used as the criterion for scientific acceptability)
Mathematics itself indeed seems to have a robustness that goes far beyond what any inividual mathematician is capable of perceiving.
Those who work in this subject whether they are actively engaged in research or just using results that have been obtained by others usually feel that they are merely explorers in a World that lies far beyond themselves - a World which possesses an objectivity that transcends mere opinion be that opinion their own or the surmise of others no matter how expert those others might be.
İt may be helpful if i put the case for the actual existence of the Platonic wold in a different form.
What i mean by this existence is really just the objectivity of mathematical truth.
Platonic existence as i see it refers to the existence of an objective external standard that is not dependent upon our individual opinions nor upon our particular culture.
Such existence could also refer to things other than mathematics such as to morality or aesthetics but i am here concerned just with mathematical objectivity which seems to be a much clearer issue.
Let me illustrate this issue b y considering one famous example of a mathematical truth and relate it to the question of objectivity.
In 1637 Pierre de fermat made his famous assertion known as fermat’s last theorem which he wrote down in the margin of his copy o the Aritmetica , a book written by the third century Greek mathematician Diaphontos.
In this margin fermat also noted:
İ have discovered a proof of this which this margin is too narrow to contain
Fermats assertion remained unconfirmed for over 350 years despite concerted efforts by numerous outstanding mathematicians.
A proof was finally published in 1995 by Andrew wiles and this proof has now been accepted as a valid argument by the mathematical community.
Now do we take the view that fermat’s assertion was always true long before fermat actually made it or is its validity a purely cultural matter, dependent upon whatever might be the subjectie standarts of the community of human mathematicians?
Let us try to suppose that the validity of the Fermat assertion is in fact a subjective matter. Then it would not be an absurdity for some other mathematician X to have come up with an actual and specific counter- example to the Fermat assertion so long as X had done this before the date of 1995.
In such circumstance the mathematical community would have to accept the correctness of X’s counter-example. From then on any effort on the part of Wiles to prove the Fermat assertion would have to be fruitless fort he reason that X had got this argument i first and as a result the fermat assertion would now be false!
Moreover we could ask the further question as to whether consequent upon the correctness of X’s fortcoming counter-example fermat himself would necessarily have been mistaken in believing in th soundness of his truly proof at the tie that he wrote his note.
On the subjective view of mathematical truth possibly have been the case that fermat had a valid proof and that it was his secretiveness that allowed the possibility of X later obtaining a counter-example!
I think that virtually all mathematicians irrespective of their professed attitudes to Platonism would regard such possibilities as patently absurd.
Of course it might still be the case that Wiley’s argument in fact contains an error and that the fermat assertion is indeed false. Or there could be a fundamental error in wile’s argument but the fermat assertion is true nevertheless.
Or it might be that wile’s argument is correct in its Essentials while containing non-rigorous steps that wuld not be up to the standard of some future rules of mathematical acceptability.
Bu these issues do not adress the point that i am getting at here.
The issue is the objectivity of the fermat assertion itself not whether anyone’s particular demonstration of it might happen to be convincing to the mathematical community of any particular time.
İt should perhaps be mentioned that from the point of view of mathematical logic the fermat assertion is actually a mathematical statement of a particularly simple kind whose objectivity is especially apparent.
Only a tiny minority of mathematicians would regard the truth of such assertions as being in any way subjective although there might be some subjectivity about the types of the argument that would be regarded as being convincing.
However there are other kinds of mathematical assertion whose truth could plausibly be regarded as being a matter of opinion.
Perhaps the best known of such assertions is the axiom of choice.
Most mathematicians would probably regard the axiom of choice as true while others may regard it as a somewhat questionable assertion which might even be false .
Still others would take it as an assertion whose truth is a mere matter of opinion or rather as something which can be taken one way or the other depending upon which system of axioms and rules of a procedure one chooses to adhere to.
Mathematicians who support this final viewpoint would be relatively weak Platonists.
Those who adhere to objectivity with regard to the truth of the axiom of choice would be stronger Platonists.
I shall come back to the axom of choice since it has some relevance to the mathematics underlying the behaviour of the physical World despite the fact that it is not addressed much in physical theory.
For the moment it will be appropriate not to worry overly about this issue.
If the axiom of choice can be settled in one way or the other by some appropriate form of unassailable mathematical reasoning then its truth is indeed an entirely objective matter and it either belongs to the platonic World or its negation does in the sense that i am interpreting this em Platonic World .
İf the axiom of choice is on the other hand a mere matter of opinion or of arbitrary decision then the platonic World of absolute mathematical forms contains neither the axiom o choice nor its negation .
The mathematical assertions that can belong to Plato’s World are precisely those that are objectively true.Indeed i would regard mathematical objectivity as really what mathematical plaatonism is all about.
To say that some mathematical assertion has a Platonic existence is merely to say that it is true in an objectie sense.
A similiar comment applies to mathematical notions- such as the concept of the number 7 for example or the rule of multiplication of integers or the idea that some set contains infinitely many elements- all of which have a platonic existence because they are objective notions.
To my way of thinking platonic existence is simply a matter of objectivity and accordingly should certainly not be viewed as something mystical or unscientific despite the fact that some people regard it that way.
As with the axiom of choice however questions as to whether some particular proposal for a mathematical entity is or is not to be regarded as having objective existence can be delicate and sometimes technical . despite this we certainly need not be mathematicians to appreciate the general robustness of many mathematical concepts.
The mandelbrot set has an extraordinarly elaborate structure but it is not of any human design.
Remarkably this structure is defined by a mathematical rule of particular simplicity. We shall come to this explicitly but it would distract us from our present purposes if i were to try to provide this rule in detail now.
The pont that i wish to make is that no one not even mandelbrot himself when he first caught sight of the incredible complications in the fine details of the set had any real preconception of the set’s extraordinary richness .
The mandelbrot set was certainly no invention of any human mind. The set is just objectively there in the mathematics itself. If it has meaning to assign an actual existence to the mandelbrot set then that existence is not within our minds for no one can fully comprehend the set’s endless variety and unlimited complication.
Nor can its existence lie within the multitude of computer printouts that begin to capture some of its incredible sophistication and detail for at best those printouts capture but a shadow of an approximation to the set itself. Yet it has a robustness that is beyond any doubt fort he same structure is revealed in all its perceivable details to greater and greater fineness the more closely it is examined-independently of the mathematician or computer that examines it.
İt’s existence can only be within the Platonic World of matheatical forms.i am aware that there will still be many readers who find diffuculty with assigning any kind of actual existence to mathematical structures.
Let me make the request of such readers that they merely broaden their Notion of what the term existence can mean to them. The mathematical forms of plato’s World clearly do not have the same kind of existence as do ordinary physical objects such as tables and chairs.
They do not have spatial locations;nor do they exist in time.
Objective mahematical notions must be thought of as timeless entities anda re not to be regarded as being conjured into existence at the moment that they are first humanly perceived.
THREE WORLDS AND THREE DEEP MYSTERIES
THUS mathematical existence is different not only from physical existence but also from an existence that is assigned by our mental perceptions. Yet there is a deep and mysterious connection with each of those other of existence –the physical, mental, the Platonic –as entities belonging to three seperate worlds .the mysterious connections between the worlds are indicated where in drawing the diagram i have imposed upon the reader some of my beliefs or prejudices concerning these mysteries.
It may be noted with regard to the first of these mysteries –relating the platonic mathematical World to the physical World that i am allowing that only a small part of the World of mathematics need have relevance to the workings of the physical World. It is certainly the case that the vast preponderance of the activities of püre mathematicians today has no obvious connection with physics nor with any other science although w emay be frequently surprised by unexpeted important applications .
Likewise in relation to the second mystery whereby mentality comes about in association with certain physical structures i am not insisting that the majority of physical structures need induce mentality.
While the brain of a cat may indeed evoke mental qualities i am not requiring the sam e for a rock .
Finally for the third mystery i regard it as self-evident that only a small fraction of our mental activity need be concerned with absolute mathematical truth!
These three facts are represented in the smallness of the base of the connection of each orld with the next the worlds being taken in a clockwise sense in the diagram.
However it is in the encompassing of each entire World within the scope of its connection with the World preceding it that i am revealing my prejudices.
The entire physical World is depicted as being governed according to mathematical laws.
Everything in the physical universe is indeed governed in completely precise detail by mathematical principles perhaps by equations such as those we shall be learning about in chapters to follow or perhaps by some future mathematical notions fundamentally different from those which we would today label by the term equations .
If this is right then even our own physical actions would be entirely subject to such ultimate mathematical control where control might still allow
For some random behaviour governed by the strict probabilistic principles.
Three worlds and three deep mysteries
Thus,mathematical existence is different not only from physical existence that is assigned by our mental perceptions. Yet there is a deep and mysterious connection with each of those other two forms of existence:
The physical and the mental.
The mysterious connections between the worlds are also indicated where in drawing the diagram i have imposed upon the reader some of my beliefs or prejudices concerning these mysterious.
It may be noted with regard to the first of these mysterious relating the Platonic mathematical World to the physical World that i am allowing that only a small part of the World of mathematics need have relevance to the workings of the physical World.
It is certainly the case that the vast preponderance of the activities of pure mathematicians today has no obvious connection with physics nor with any other science although we may be frequently surprised by unexpected important applications.
Likewise in relation to the second mystery whereby mentality comes about in association with certain physical structures , i am not insisting that the majority of physical structures need induce mentality.
While the brain of a cat may indeed evoke mental qualities i am not requiring the same for a rock. Finally for the third mystery i regard it as self-evident that only a small fraction of our mental activity need be concerned with absolute mathematical truth!
These three facts are represented in the smallness of the base of the connection of each World with the next , the worlds being taken in a clockwise sense in the diagram.
However it is in the encompassing of each entire World within the scope of its connection with the World preceding it that i am revealing my prejudices.
Thus according to the figure the entire physical World is depicted as being governed according to mathematical laws.
We shall be seeing in later chapters that there is powerful evidence in support of this contention.
On this view, everything in the physical universe is indeed governed in completely precise detail by mathematical principles perhaps by equations such as those we shall be learning about in chapters to follow or perhaps by some future mathematical notions fundamentally different from those which we would today label by the term equations.
if this is right then even our own physical actions would be entirely subject to such ultimate mathematical control where control might still allow for some random behaviour governed by strict probabilistic principles.
Many people feel uncomfortable with contentions of this kind and i must confess to having some unease with it myself.
Nonetheless my personal prejudices are indeed to favour a viewpoint of this general nature, since it is hard to see how any line can be drawn to seperate physical actions under mathematical from those which might lie beyond it.
In my own view the unease that many readers may share with me on this issue partly arises from a very limited Notion of what mathematical might entail.
Part of the purpose of this book is to touch upon and to reveal to the reader some of the extraordinary richness power and beauty that can spring forth once the right mathematical notions are hit upon.
In the Mandelbrot set alone we can begin to catch a glimpse of the scope and beauty inherent in such things.
But even these structures inhabit a very limited corner of mathematics as a whole, where behaviour is governed by strict computational control.
Beyond this corner is an incredible potential richness.
How do i really feel about the possibility that all my actions and those of my friends are ultimately governed by mathematical principles of this kind?
I can live with that.i would indeed prefer to have these actions controlled by something residing in some such aspect of Plato’s World than to have them to be the subject to the kind of simplistic base motives such as pleasure seeking personal greed or aggressive violence that many would argue to be the implications of a strictly scientific standpoint.
Yet i can well imagine that a good many readers will still have diffuculty in accepting that all actions in the universe could be entirely subject to the scientific laws.
Likewise many might object to two other prejudices of mine that are implicit i the figure.
They might feel for example that i am taking too hard-boiled a scientific attitude by drawing my diagram in a way that implies that all of mentality has its roots in physicality.
This is indeed a prejudice for while it is true that we have no reasonable scientific evidence for the existence of minds that do not have a physical basis
We can not be completely sure.
Moreover many of a religious persuasion would argue strongly for the possibility of physically independent minds and might appeal to what they regard as powerful evidence of a different kind from that which is revealed by ordinary science.
A further prejudice of mine is reflected in the fact that im the figure have represented the entire Platonic World to be within the compass of mentality.
This is intended to indicate that –at least in principle- there are no mathematical
Truths that are beyond the scope of the reason.
Of course there are mathematical statements that are so vastly complicated that no one could have the mental fortitude to carry out the necassary reasoning.
However such things would be potentially within the scope of mentality and would be consistent with the meaning of the figure as i have intended to represent it.
One must nevertheless consider that there might be other statements that the outside even the potential compass of the reason and these would violate the intention behind the figure.
In figure as a concession to those who do not share all my personal prejudices on these matters , i have redrawn the connections between the three worlds in order to allow for all three of these possible violations of my prejudices.
Accordingly,the possibility of physical action beyond the scope of mathematical control is now taken into account.
The diagram also allows for the belief that there might be mentality that is not rooted in physical structures.
Finally it permits the existence of true mathematical assertions whose truth is in principle inaccessible to reason and insight.
This extended Picture presents further potential mysteries that lie even beyond those which i have allowed for in my own preferred Picture of the World.
In my opinion the more tightly organized scientific viewpoint of the figure has mysteries enough.
These mysteries are not remoted by passing to the more relaxed scheme of the figure.
For it remains a deep puzzle why laws should apply to the World with such phenomenal precision.
Moreover it is not just the precision but also the subtle sophistication and mahematical beauty of these successful theories that is profoundly mysterious.
There is also an undoubted deep mystery i how it can come to pass that appropriately organized physical material-and here i refer specifically to livig human brains- can somehow conjure up the mental quality of conscious awareness.
Finally there is also a mystery
About how it is that we perceive scientific truth.
It is not just that our brains are programmed to calculate in reliable ways.
There is something much more profound than that in the insights that even the hmblest among us possess when we appreciate for example the actual meanings of the terms zero one two three four, etc.
Some of the issues that arise in connection with this third mystery will be our concern in the next chapter in relation to the Notion of mathematical proof.
But the main thrust of this book has to do with the first of these mysteries :the remarkable relationship between mathematics and the actual behaviour of the physical World.
No proper appreciation of the extraordinary power of modern science can be ideas.
No doubt many readers may find themselves daunted by the prospect of having to come to terms with such mathematics in order to arrive at this appreciation.
Yet i have the optimistic belief that they may not find all these things to be so bad as they fear.
Moreover i hope that i may persuade many readeersthat despite what she or he may have previously perceived science can be fun!
I shall not be especially concerned here with the second of the mysteries depicted in figure namely the issue of how it is that mentality – most particularly conscious awareness – can come about in association with appropriate physical structures .
There will be enough to keep us busy in exploring the physical universe and its associated laws.
In addition the issues concerning mentality are profoundly contentious and it would distract from the purpose of this book if we were to get embroiled in them.
Perhaps one comment will not be amiss here however .
This is that in my own opinion there is a little chance that any deep understanding of the nature of the mind can come about without our first learning much more about the very basis of physical reality.
As will become clear from the discussions that will be presented in later chapters, i believe that major revolutions are required in our physical understanding.
Until these revolutions have come to pass it is in my view greatly optimistic to expect that much real progress can be made in understanding the actual nature of mental processes.
The good, the true, and the beatiful
In relation to this there is a further set of issues raised by figures i have taken Plato’s notion of a World of ideal forms only in the limited sense of mathematical forms.
Mathematics is crucially concerned with the particular ideal of truth.
Plato himself would have insisted that there are two other fundamental absolute ideals namely that the beatiful and of and to allowing the platonic World to be extended so as to contain absolutes of this nature.
Indeed we shall later be encountering some of the remarkable interrelations between truth and beauty that both illuminate and confuse the issues of the discovery and acceptance of physical theories .
Moreover quiete apart from the undoubted role of beauty fort he mathematics underlying the workings of the physical World,aesthetic criteria are fundamental to the development of mathematical ideas for their own sake providing both the drive towards discovery and a powerful guide to truth.
I would even surmise that an important element in the mathematician’s common conviction that an external platonic World actually has an existence independent of ourselves comes from the extraordinary unexpected hidden beauty that the ideas themselves so frequently reveal.
Of less obvious relevance here –but of clear importance in the broader context-is the question of an absolute ideal of morality:
What is good and what is bad and how do our minds perceive these values?
Morality has a profound connection with the mental World since it is so intimately related to the values assigned by conscious beings and more importantly to the very presence of consciousness itself.
It is hard to see what morality might mean in the absence of sentient beings.
As science and technlogy progress an understanding of the physical circumstances under which mentality is manifested becomes more and more relevant.
I believe that it is more important than ever in today’s technological culture that scientific questions should not be divorced from their moral implications.
But these issues would take us too far a field from the immediate scope of this book.
We need to address the question of seperating true from false before we can adequately attempt to apply such understanding to seperate good from bad.
There is finally a further mystery concerning the figure which i have left to the last.
I have deliberately drawn the figure so as to illustrate a paradox.
How can it be that in accordance with my own prejudices each World appears to encompass the next one in its entirety?
I do not regard this issue as a reason for abandoning my prejudices but merely for demonstrating the presence of an even deeper mystery that transcends those which i have been pointing to above.
There may be a sense in which the three worlds are not separate at all,but merely reflect individually aspects of a deeper truth about the World as a whole of which we have little conception at the present time.we have a long way to go before such matters can be properly illuminated.
I have allowed myself to stray too much from the issues that will concern us here.
The main purpose of this chapter has been to emphasize the central importance that mathematics has in science both ancient and modern.
Let us now take a glimpse into plato’s World at least into a relatively small but important part of the World of particular relevance to the nature of physical reality.
An ancient theorem and a modern question
Let us consider the issue of geometry.what indeed are the different kinds of geometry that were alluded to in the last chapter.
To lead up to this issue we shall return to our encounter with pythgoras and consider that famous theorem that bears his name:
For any right-angled triangle the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
What reasons do we hav efor believing that this assertion is true?
How indeed do we prove the theorem?
Many arguments are known.
I wish to consider two such chosen for their particular transparency each of which has a different emphasis.
For the first consider the pattern illustrated in the figure.It is composed entirely of squares of two different sizes.It may be ergarded as obvious that this pattern can be continued indefinetely and that the entire plane is thereby covered in this regular repeating way,without gaps or overlaps by squares of these two sizes.
The repeating nature of this pattern is made manifest by the fact that if we mark the centres of the larger squares they form the vertices of another system of squares of a somewhat greater size than either but tilted at an angle to the original ones and which alone will cover the entire plane.
Each of these tilted squares is marked in exactly the same way so that the markings on these squares fit together to form the original two-square pattern.
The same would apply if,instead of taking the centres of the larger of the two squares of the original pattern we chose any other point together with its set of corresponding points throughout the pattern.
The new pattern of tilted squares is just the same as before but moved along without rotation.
It should be clear that the area of the tilted square must be equal to the sum of the areas of the two smaller squares.
Euclıdes postulates
In building up his Notion of geometry euclid took considerable care to see what assumptions his demonstration depended upon.
In particular he was careful to distinguish certain assertions called axioms-which were taken as self-evidently true these being basically definitions of what he meant by points,lines,etc.
From the five postulates which were assumptions whose validity seemed less certain yet which appeared to be true of the gemetry of our World.the final one of these assumptions referred to as Euclid’s fifth postulate was considered to be less obvious than the others and it was felt for many centuries that it ought to be possible to find a way of proving it from the other more evident postulates.
Euclides fifth postulate is commonly referred to as the parallel postulate and i shall follow this practice here.
Before discussing the parallel postulate it is worth pointing out the nature of other four of euclid’s postulates.
The postulates are concerned with the geometry of the plane though Euclid also considered three-dimensional space later in his Works.
The basic elements of his plane geometry are points lines and circles.
Here i shall consider a line to be indefinitely extended in both directions;otherwise i refer to a line segment.
Euclid’s first postulate effectively asserts that there is a straight line segment connecting any two points.
His second postulate asserts the unlimited extendibility of any straight line segment.
His third postulate asserts the existence of a circle with any centre and with any value for its Radius.
Finally his fourth postulate asserts the equality of all right angles.
From a modern perspective some of these postulates appear a little strange particularly the fourth but we must bear in mind the origin of the ideas underlying Euclid’s geometry.
Basically he was concerned with the movement of idealized rigid bodies and the Notion of congruence which was signalled when one such idealized rigid body was moved into coincidence with another.
The equality of a right angle on one body with that on another had to do with the possibility of moving the one so that the lines forming its right angle would lie along the lines forming the right angle of homogeneity of space so that a figure in one place could have the same geometrical shape as a figure in some other place.
The second and third postulates express the idea that space is indefinitely extendible and without gaps in it whereas the first expresses the basic nature of a straight line segment.
Although Euclid’s way of looking at geometry was rather different from the way that we look at it today his first four postulates basically encapsulated our present-day
Notion of a metric space with complete homogeneity and isotropy and infinite in extent.
In fact such a Picture seems to be in close accordance with the very large-scale spatial nature of the actual universe according to modern cosmology.
Kinds of number in the physical World
Let us now return to the issue of proof by contradiction the very principle that Saccheri
Tried hard to use in his attempted proof of euclid’s fifth postulate.
There are many instances in mathematics where the principle has been successfully applied.
One of the most famous of these dates back to the Pythagoreans and it settled a mathematical issue in a way which greatly troubled them.
Can one find a rational number whose square is 2?
The real number system
Thus it was that the greeks were forced into the realization that rational numbers are not enough if the ideas of geometry are to be properly developed.
Nowadays we do not worry unduly if a certain geometrical quantity can not be measured simply in terms of rational numbers alone.
This is because the Notion of a real number is very familiar to us.
Although our pocket calculators express numbers in terms of only a finite number of digits we readily accept that this is an approximation forced upon us by the fact that the calculator is a finite object.
We are prepared to allow that the ideal mathematical number could certainly require that the decimal expansion continues indefinitely.
This applies of course even to the decimal representation of most fractions such as
1/3=0.3333
29/12=2.416666
9/7=1.285714285714285714..
237/148=1.601351351351….
For a fraction the decimal expansion is always ultimately periodic which is to say that after a certain point the infinite sequence of digits consists of some finite sequence repeated indefinitely.
In the above examples the repeated sequences are respectively 3,6,285714 and 135.
Decimal expansions were not available to the ancient greeks but they had their own ways of coming to terms with irrational numbers.
In effect what they adopted was a system of representing numbers in terms of what are now called continued fractions.
There is no need to go into this in full detail here but some brief comments are appropriate.
A continued fraction is a finite or infinite expression a+(b+(c+(d+….)-1 )-1)-1
Where a,b,c,d are positive integers.
Any rational number larger than 1 can be written as a terminating such expression :
52/9= 5+(1+(3+(2)-1 )-1)-1
And to represent a positive rational less than 1 we just allow the first integer in the expression to be zero.
To express a real number which is not rational, we simply allow the continued fraction expression to run on forever some examples being
√2=1+(2+(2+(2+….)-1 )-1)-1
7-√3=
5+
(3+(1+(2+(1+(2+(1+(2…)-1 )-1)-1)-1)-1)-1)-1
π=3+(7+(15+(1+(292+(1+(1+(1+(2+…)-1 )-1)-1)-1)-1)-1)-1)-1
In the first two of these infinite examples the sequences of natural numbers that appear namely1,2,2,2,2,…in the first case and 5,3,1,2,1,2,1,2,…in the second have the property that they are ultimately periodic .
Recall that as already noted above in the familiar decimal notation it is the rational numbers that have ultimately periodic expressions.
We may regard it as a strength of the greek continued fraction representation on the other hand that the rational numbers now always have a finite description.
A natural question to ask in this context is:which numbers have an ultimately periodic continued fraction representation?
It is a remarkable theorem first proved to our knowledge by the great mathematician Lagrange that the numbers whose representation in terms of continued fractions are ultimately periodic are what are called quadratic irrationals.
What is a quadratic irrational and what is its importance for greek geometry?
It is a number that can be written in the form ,a+√b where a and b are fractions and where b is not a perfect square .
Such numbers are important in geometry because they are the most immediate rational numbers that are encountered in ruler and compass constructions.
Particular examples of quadratic irrationals are those cases where a=0 and b is a natural number or rational greater than 1 .
The continued fraction representation of such a number is particularly striking.
The sequence of natural numbers that defines it as a continued fraction has a curious characteristic property.
It starts with with some number A then it is immediately followed by a palindromic sequence B,C,D,…,D,C,B, followed by 2A after which the sequence B,C,D,..,D,C,B,2A repeats itself indefinitely.
The number squareroot(14) is a good example for which the sequence is
3,1,2,1,6,1,2,1,6,1,2,1,6,1,2,1,6,…
Here A=3 and the sequence B,C,D,…,D,C,B is just the three term sequence 1,2,1.
Howmuch of this was known to the ancient greeks?
It seems very likely that they knew quiete alot- very possible all the things that i have described above, although they may well have lacked rigorous proofs for everything.
Plato’s contemporary theaetetos seems to have established much of this.
There appears even to be some evidence of this knowledge revealed in plato’s dialectics.
Although incorporating the quadratic irrationals gets us some way towards numbers adequate for euclidean geometry it does not do all that is needed.
In the tenth and most diffucult book of euclid numbers like √(a+√b)
Are considered.
These are not generally quadratic irrationals but they occur nevertheless, in ruler and compass constructions.
Numbers sufficient for such geometric constructions would be those that can be built up from natural numbers by repeated use of the operations of addition subtraction multiplication division and the taking of square roots.
But operating exclusively with such numbers gets extremely complicated and these numbers are still too limited for considerations of euclidean geometry that go beyond ruler and compass constructions.
It is much more satisfactory to take the bold step and how bold a step this actually is will be indicated in 9.5 of allowing infinite continued-fraction expressions that are completely general.
This provided the greeks with a way of describing numbers that do turn out to be adequate for euclidean geometry.
These numbers are indeed in modern terminology the so-called real numbers.
Although a fully satisfactory definition of such numbers is not regarded as having been found until the 19 th century the great ancient astronomer Eudoxos who had been one of plato’s students had obtained the essential ideas already in the 4th century.
A few words about Eudoxos ‘s ideas are appropriate here.
First we note that the numbers in euclidean geometry can be expressed in terms of ratios of lengths rather than directly in terms of lengths.
In this way no specific unit of length was needed.
Moreover with ratios of lengths there would be no restriction as to how many such ratios might be multiplied together .
The first step in the Eudoxan theory was to supply a criterion as to when a length ratio a:b would be greater than another such ratio c:d
This criterion is that some positive integers M and N exist such that the length a added to itself M times exceeds b added to itself N times while also d added to itself N times exceeds c added to itself M times.
A corresponding criterion holds expressing the condition that the ratio a:b be less than the ratio c:d
The condition for equality of these ratios would be that neither of these criteria hold.
With this ingenious Notion of equality of such ratios,eudoxos had,in effect an abstract concept of a real number in terms of length ratios.
He also provided rules for the sum and product of such real numbers.
There was a basic difference in viewpoint,however between the greek Notion f real number and the modern one because the greeks regardedthe umber system as basiclly given to us in terms oof Notion of distance in physicqal space so the problem was to try to ascertain how these distance measures actually behaved.
For space may well have had the appearence e of being itselfa Platonic absolute even though actual physical objects existing in this space would inevitably fall short of the Platonic ideal.
A pical object such as a square drawn in the sand or a cube hewn from marble might have been regarded by the ancient greeks as a reasonable or sometimes an excellent approximations to the Platonic to the Platonic geometrical ideal.
Yet any object would nevertheless provide mere approximation
Lying behind such approximation to the Platonic form –so it would have appeared-would be space than itself.
:an entity of such abstract or n otional existence that it could well have been regarded as a direct realization of a Platonic reality.
The measure of distance in this ideal geometry would be something to ascertain accordingly it would be appropriate to try to extract this idel Notion of real number from a geometrical of euclidean space that was assusmed to be given.
In effect this is what eudoxos succeeded in doing.
By the 19th and 20th centuries however the view Had emerged that the mathematical Notion of number should stand separately from the nature of physical space.
Since mathematically consistent geometries other than that of Euclid had been shown to exist this rendered it inappropriate to insist that the mathematical Notion of geometry should be necessity extracted from the supposed nature of actual physical space.
Moreover it could be very diffucult if not impossile to ascertain the detailed nature of this supposed underlying platonic physical geometry in terms of the behaviour of imperfect physical objects.
In order to know the nature of the numbers according to which geometrical distance is to be defined for example it woul be necessary to know what happens both at indefinitely tiny and indefinitely large distances.
Even today these questions are without clearcut resolution.
Thus it was far more appropriate to develop the nature of a number in a way that does not directl refer to physical measures.
Accordingly Richard DEDEKIND and georg CANTOR developed their ideas of what real numbers are by use of notions that do not directly refer to geometry.
DEDEKIND’s definition of a real number is in terms
Of infinite sets of rational numbers.
Basically we think of the rational numbers both positive and negative and zer oto be arranged in order of size.
We can imagine that this ordering takes place from left to right where we think of the negative rationals as being displayed going off indefinitely to the left with zero in the middle and the positive rationals displayed going off indefinitely to the right.
Dedekind imagines a cut which divides this display neatly in two with those to the left of the cut being all smaller than those to the right.
When the knife-edge of the cut does not hit an actual rational number but falls between them, we say that it defines an irrational real number.
More correctly this occurs when those to the left have no actual largest member and those to the right no actual smallest one.
When the system of irrationals as defined in terms of such cuts is adjoined to the system of rational numbers that we already have then the complete family of real numbers is obtained.
Dedekind’s procedure leads by means of simple definitions directly to the laws of addition subtraction multiplication and division for real numbers.
Moreover it enables one to get further and define limits whereby such things as the infinite continued fraction that we saw before 1/1-1/3+1/5-1/7+1/9-….
May be assigned real number meanings.
In fact the first gives us pi/4.
The ability to take limits is fundamental for many mathematical notions and it is this that gives the real numbers their particular strengths.
Real numbers in the physical World
There is a profound issue that is being touched upon here.
In the development of mathematical ideas one important initial driving force has always been to find mathematical structures that accurately mirror the behaviour of the physical World .
But it is normally not possible to examine the physical World itself in such precise detail that appropriately clear-cut mathematical notions can be abstracted directly from it.
Instead progress is made because notions tend to have a momentum of their own that appears to spring almost entirely from within the subject itself.
Mathematical ideas develop and various kinds of problem seem to arise naturally.
Some of these can lead to an essential extension of the original concepts in terms of which the problem had been formulated.
Such extensions may seem to be forced upon us or they may arise in ways that appear to be matters of convenience consistency or mathematical elegance.
Accordingly the development of mathematics may seem to diverge from what it had been set up to achieve namely simply to reflect physical behaviour.
Yet in many instances this drive for mathematical consistency and elegance takes us to mathematical structures and concepts which turno ut to mirror the physical World in a much deeper and more broad-ranging way than those that we started with.
It is as though nature herself is guided by the same kind of criteria of consistency and elegance as those that guide human mathematical thought.
An example of this is the real number system itself.
We have no direct evidence from nature that there is a physical Notion of distance.
That extends to arbitrarily large scales ;still less is there evidence that such a Notion can be applied on the indefinitely tiny level.
Indeed there is no evidence that points in pace actually exist in accordane with a geometry that precisely makes use of real number distances.
In euclid’s day there was scant evidence to support even the contention that such euclidean distances extended outwards beyond say about 10^12 meters or inwards to as little as 10^-5 metres.
Yet having been driven mathematically by the consistency and elegance of the real number system all of our exception still clung to this ancient of real number.
Although there might appear to have been little justification for doing this from the evidence that was available in euclid’s day our faith in the eral numbers system appears to have been rewarded.
For our successful modern theories of cosmology now allow us to extend the range of our real number distances out to about 10^26 metres or more while the accuracy of our theories of particle physics extends this range inwards to 10^-17 metres or might come about is some 18 orders of magnitude smaller even than that namely 10^^-35 metres which is the Planck scale of quantum gravity that will feature strongly in some of our later discussions
It may be regarded as a remarkable justification of our use of mathematical idealizations that the range of validity of the real number system has extended from the total of about 10^17 from the samllest to the largest that seemed appropriate in euclid’s day to at least the 10 ^^43 that our theories directly employ today this increase by a factor of some 10^^26.
There is a good deal
Nec hıv virüsünü mutasyona uratacak, aşıyık ı sadece Yahudilere verecek , onu durdurmak içiğn ana bilgisayarıb n kodlarını çözmeye çalışacaklar.
There is aa good deal more to the physical vaalidityt of the real number system than this.
In the first place we must consider that areas and volumes are also quantities for which real numbetr measures are accurately appropriate.
A volüme measure is the cube of a distance measure
Accordingly in the case of volüme w emay consideer that it isd the cube of a ddistance measure
Accordingly in the case ooff volumerss w emay cconnsiider that it is the cube of the above range that is rerelevant.
For euclid’s time this would give us a range of about 10^^ 51 fıor todays theeories at leastt 10^^129
Moreover thhere are other physical measures that require real-number descriptions , according to our presently successful theories.
The most noteworthy of these is time.according to relatiivity theory this needs to be adjointed to space to provide us with spacetime
Spacetime volumes arere 4-dimensional and it might well be consideredd that the temporall range should also be imncorporatedb into our considerations, giving a total of something like at least 10^^172
We shall see some far larger real numbers even than this coming into our later considerations., although it is not really clear in some cases that the use of real numbers is essential.
More importantly for physical theory from archimedes throughh Galileo and Newton to maxwell Einstein schrödinger dirac and the rest a crucial role fort he real number system that has been it providezs a necessary framework fro the standartt formulation of the calculus for their formulations.
Mnow the conventional approach to calculs requires the infinitesimal natuıre of the reals to be what it is.
That is to say on the small end of the scale it is the entire range of the real numbers that is in principle being made use of.
Tjhe ideas of calculus underlie other physicall notions such as velocity momentum and energy .
Consequently the real number system enters our successful physical theories in a fundamental way for our description of all cthese quantities also.
Here as mentioned earlier in coınnection with areas the infinite simal limöit of small scale strucre of the reall number system is being called upon.
Yet w emay ask whether the real number system is really correct fort he description of physical reality at is deepest olevels.,
When quantum mechanical ideas were beginning rto be introduced early in the 20th centtury there was feeling that perhaps we were now beginning to be introduced to witness a discerte or granular nature to the physicalo World at its smallest scalesç.
Energy could aperently exist only in discrete bundles or qwuanta and th ephysical quantities of action and spin seemed to occurt only in discrete multiples of a fundamental unit fort he classical *-+ concept o action and for its quantum counterpart
Accordingly various physicists attempted to builds up an alternative Picture of the World in wghich discrete processes governed all actions at the tiniest levels.
However as we now understand quantum mec hanics tha theory does not force us to the view that there is a discrete or granular nature to space time or energy at its tiniest levels.
Nevertheless, the idea has remained with us that there may indeed be at root such a fundamental discreteness to nature despite the fact that quantum mechanics in its standart formulation certainly does not imply this.
For example erwin schrödinger was among the first to propose that a change to some form of fundamental spatial discreteness might actually be necessary.
He related this proposal to some early greek thinking concerning the discreteness of nature.
Einstein also suggested in his last published words that a discretely based theory might be the way forward fort he future physics.
Others also have pursued ideas of this kind.
In the late 1950s i myself tried this sort of thing coming up with a scheme that i referred to as the theory of spin networks in which the discrete nature of quantum mechanical spin is taken as the funental building block for a combinatorial approach to physics.
Although my own ideas along this particular direction did not develop to a comprehensive theory , the theory of spin networks has now been imported by others into one of the major programmes for attacking the fundamental problem of quantum gravity.
I shall give brief descriptions of these ideas in chapter 12.
Nevertheless as tried and tested physical theory stands today-as it has for the past 24 centuries – real numbers still form a fundamental ingredient of our understanding of the physical World.
Do natural numbers need the physical World?
It is not a diffucult step from the integers to the rationals ; rationals are just ratios of integers.
What about the integers themselves then?
Are these rooted in physical ideas?
The discrete approaches to pysics that we were referred to in the previous two paragraphs certainly depend upon our Notion of natural number and its extension by the inclusion of the negative numbers to the integers.
Negative numbers were not considered by the greeks to be actual numbers sol et us continue our considerations by first asking about the physical status of the natural numbers themselves.
The natural numbers are the quantities that we now denote by 0,1,2,3,4…
They are the non-negative
Whole numbers.
The role of natural numbers is clear.
They are indeed the most elementary counting numbers
Which have a basic role whatever the laws of geometry or physics might be.
Natural numbers are subject to certain familiar operations most particularly the operations of addition and multiplication which enable pairs of natural numbers to be combined together to produce new natural numbers.
These operations are independent of the nature of the geometry of the World.
We can however raise the question of whether the natural numbers themselves have a meaning or indeed existence independent of the actual nature of the physical World.
Perhaps our Notion of natural numbers depends upon there being in our universe reasonably well-defined discrete objects that persist in time.
Natural numbers initially arise when we wish to count things after all.
But this seems to depend upon there actually being persistent distinguishable things in the universe which are available to be counted.
Suppose on the other hand our universe were such that numbers of objects had a tendency to keep changing.
Would natural numbers actually be natural concepts in such a universe?
Moreover perhaps the universe actually contains only a finite number of things in which case the natural numbers might themselves come to an end at some point!
We can even envisage a universe which consists only of an amorphous featureless substance for which the very Notion of numerical quantification might seem intrinsically inappropriate.
Would the Notion of natural number be at all relevant fort he description of the universe of this kind?
Even though it might well be the case that inhabitants of such a universe would find our present mathematical concept of a natural number diffucult to come upon it is hard to imagine that there would not still be an important role for such fundamental entities.
There are various ways in which natural numbers can be introduced in pure mathematics and these do not seem to depend upon the actual nature of the physical universe at all.
Basically it is the Notion of a set which needs to be brought into play this being an abstraction that does not appear to be concerned in any essential way with the specific structure of the physical universe.
In fact there are certain definite subtleties concerning this question and i shall return to that issue later.
For the moment it will be convenient to ignore such subtleties.
Let us consider one way that natural numbers can be introduced merely using the abstract Notion of set.
It also leads on to what are called ordinal numbers
The simplest set of all is referred to as the null set and it is characterized by the fact that it contains no members whatever!
Let us associate the empty set zero we can now proceed further and define a set with only one element and so on.
This may not be how we usually think of natural numbers as a matter of definition but it is one of the ways that mathematicians can come to the concept.
Moreover it shows us at least that things like the natural numbers can be conjured literally out of nothing merely by employing the abstract Notion of set.
We get an infinite sequence of abstract entities –sets containing respectively zero,one,two,…elements
one set for each of the natural numbers quiete independently of the actual physical nature of the universe.
Dedekind’s construction moreover shows how this purely mental kind of procedure can be carried further enabling us to construct the entire system of real numbers still without an reference to the actual physical nature of the World.
Yet as indicated above real numbers indeed seem to have a direct relevance to the real structure of the World.
Discrete numbers in the physical World
But i am getting slightly ahead of myself.
We may recall that dedekind’s construction really made use of sets of rational numbers not of natural numbers directly.
As indicated above it is not hard to define what we mean by a rational number once we have the Notion of natural number.
But as an intermediate step it is appropriate to define the Notion of an integer which is a natural number or the negative of a natural number.
In a formal sense there is no diffuculty in giving a mathematical definition of negative:
Rougly speaking we just attach a sign written as – to each natural number and define all the arithmetical rules of addition subtraction multiplication and division consistently.
This does not address the question of the physical meaning of a negative number however.
What might it mean to say that there are minus three cows in a field for example?
I think it is clear that unlke the natural numbers themselves there is no evident physical content to the Notion of a negative number of physical objects.
Negative integers certainly have an extremely valuable organizational role such as with bank balances and other financial transactions.
But do they have direct relevance to the physical World?
When i say direct relevance here i am not referring to the circumstances where it would appear that it is negative real numbers that are the relevant measures such as when a distance measured in one direction counts as positive while that measured in the opposite direction would count as negative.
I am referring instead to the numbers that are scalar quantities in the sense that there is no directional aspect to the quantity in question.
In these circumstances it appears to be the case that it is the system of integers both positive and negative that has direct physical relevance.
It is a remarkable fact only in about the last hundred years has it become apparent that the system of integers does indeed seem to have such direct physical relevance.
The first example of a physical quantity which seems to be appropriately quantified by integers is the electric charge.
As far as is known the electric charge of any discrete isolated body is indeed quantified in terms of integral mutiples positive negative or zero of one particular value namely the charge on the proton
It is now believed that protons are composite objects built up in a sense from smaller entities referred to as quarks .
There are three quarks to each proton the quarks having electric charges with respective values 2/3,2/3,-1/3.
These constituent charges add up to give the total value 1 for the proton.
If quarks are the fundamental entities then the basc charge unit is one third of that which we seemed to have before.
Nevertheless,it is still true that the electric charge is measured in terms of integers but now it is integer multiples of one third of a proton charge.
Electric charge is just one instance of what is called an additive quantum number.
Quantum nubers are quantities that servet o characterize the particles of nature.
Such a quantum number
Which i shall here take to be a real number of some kind is additive if in order to derive its value for a composite entity we simply add up the individual values for the constituent particles –taking due account of signs of course as with the above mentioned case of the proton and its constituent quarks.
It is a very striking fact that according to the state of our present physical knowledge that all known additive quantum numbers are indeed quantified in terms of the system of integers not general real numbers and not simply natural numbers- so that the negative values actually do occur.
In fact according to 20th century physics there is now a certain sense in which it is meaningful to refer to a negative number of physical entities.
Paul dirac put forward in 1929 his theory of anti-particles according to which for each type of particle there is also a corresponding anti-particle for which each additive quantum number has precisely the negative of the value that it has for the original particle.
Thus the system of integers does indeed appear to have a clear relevance to the physical universe-a physical relevance that has become apparent only in the 20th century despite those many centuries for which integers have found great value in mathematics commerce and many other human activities.
One important qualification should be made at this juncture however .
Although it is true that in a sense an antiproton is a negative proton it is not really minus one proton.
The reason is that the sign reversal refers only to the additive quantum numbers whereas the Notion of mass is not additive in modern physical theory.
This issue will be explained in a bit more detail in 1.7
Minus one proton would have to be an anti-proton whose mass is the negative of the mass value of an ordinary proton.
But the mass of an actual physical particle is not allowed to be negative.
An anti-proton has the same mass as an ordinary proton, which is a positive mass.
We shall be seeing later that according to the ideas of quantum field theory there ae things called virtual particles for which the mass can be negative.
Minus one proton would really be a virtual antiproton.
But a virtual particle does not have an independent existence as an actual particle.
Let us now ask the corresponding question about the rational numbers.
Has this system of numbers found any direct relevance to the physical universe?
As far as is known this does not appear to be the case at least as far as conentional theory is concerned.
There are some physical curiosities in which the family of rational numbers does play its part but it would be hard to maintain that these reveal any fundamental physical role for rational numbers.
On the other hand it may be that there is a particular role for the rationals in fundamental quantum-mechanical probabilities.
This kind of thing plays a role in the theory of spin networks as will be described in 2.9
Yet there are other kinds of number which according to accepted theory do appear to play a fundamental role in the workings of the universe.
The most important and striking of these are the complex numbers in which the seemingly mystical quantity sqrt(-1) usually denoted
By i is introduced and adjoined to the real number system.
First encountered in 16th century but treated for hundreds of years with distrust the mathematical utility of complex numbers gradually impressed the mathematical community to a greater and greater degree until complex numbers became an indispensable even magical ingredient of our mathematical thinking.
Yet we now find that they are fundamental not just to mathematics:
These strange numbers also play an extraordinary and very basic role in the operation of the physical universe at its tiniest scales.
This is a cause for wonder and it is even more striking instance of the convergence between mathematical ideas and the deeper workings of the universe than is the system of real numbers that we have been considering in this section.
The magic number i
How is it that -1 can have a square root?
The square of a positive number is always positive and the square of a negative number is again positive .
It seems impossible that we can find a number whose square is negative.
Yet this is the kind of the situation that we have seen before when we ascertained that 2 has no square root within the system of rational numbers.
In that case we resolved the situation by extending our system of numbers from the rationals to a larger system and we settled on the system of reals.
Perhaps the same trick will work again.
Indeed it will.
In fact that we have to do is something much easier and far less drastic than the passage from the rationals to the reals.
All we need do is introduce a single quantity called i and adjoin it to the system of reals allowing combinations of i with real numbers to form expressions such as a+ib, where a and b are arbitrary real numbers.
Any such combination is called a complex number.
When we get used to playing with complex numbers we cease to think of a+ib as a pair of things namely the two real numbers a and b , but we think of a+ib ,as an entire thing on its own and we could use a single letter say z to denote the complex number z=a+ib
It may be checked that all normal rules of algebra are satisfied by complex numbers.
In fact all this is a good deal more straightforward than checking everything for real numbers.
From this point of view it seems rather extraordinary that all complex numbers were viewed with suspicion for so long whereas the much more complicated extension from the rationals to the reals had after ancient greek times been generally accepted without question.
Presumably this suspicion arose because people could not see the complex numbers as being presented to them in any obvious way by the physical World.
In the case of the real numbers it had seemed that distances times and other physical quantities were providing the reality that such numbers required ;yet the complex numbers had appeared to be the merely invented entities called forth from the imaginations of mathematicians who desired numbers with a greater scope than the ones that they had known before.
But we should recall from 3.3
That the connection the mathematical real numbers have with those physical concepts of length or time is not as clear as we had imagined it to be.
We can not directly see the minute details of a dedekind cut nor it is clear that arbitrarily great or arbitrarily tiny time sor lengths actually exist in nature.
One could say that the so-called real numbers are as much a product of mathematicians’ imaginations as are the complex numbers.
Yet we shall find that complex numbers as much as reals and perhaps even more find a unity with nature that is truly remarkable.
It is as though nature herself is as impressed by the scope and consistency of the complex number system as we are ourselves and has entrusted to these numbers the precise shall be seeing in detail how this Works.
Moreover to refer just to the scope and to the consistency of complex numbers does not do justice to this system.
There is something more which in my view can only be referred to as magic.
In the remainder of this chapter and in the next i shall endeavour to convey to the reader something of the flavour of this magic.
Over the four centuries that complex numbers have been known a great many magical qualities have been gradually revealed.
Yet this is a magic that had been perceived to lie within mathematics and it indeed provided a utility and a depth of mathematical insight that could not be achieved by use of the reals alone.
There had not been any reason to expect that the physical World should be concerned with it.
And for some 350 years from the time that these numbers were introduced through the Works of cardano and bombelli it was purely through their mathematical role that the magic of the complex number system was perceived.
It would no doubt have come
As a great surprise to all those who had voiced their suspicion of complex numbers to find that according to the physics of the latter three quarters of the 20th century the laws governing the behaviour of the World at its tiniest scales is fundamentally governed by the complex number system.
For the moment let us concentrate on some of the magic of complex numbers leaving their physical magic until later.
Recall that all we have done is to demand that -1 have a square root together with demanding that the normal laws of arithmetic be retained and we have ascertained that these demands can be satisfied consistently.
This seems like a fairly simple thing to have done.
But now for the magic!
Solving equations with complex numbers
In what follows i shall find it necessary to introduce somewhat more notation than previously.
I apologize for this.
However it is hardly possible to convey serious ideas without the use of a certain amount of notation.
I appreciate that there will be many readers who are uncomfortable with these things.
My advice to such readers is basically just to read the words and not to bother too much about trying to understand the equations.
At least just skim over the various formulae and press on.
There will, indeed be a quiete number of serious mathematical expressions scattered about this book particularly in some of the later chapters.
My guess is that certain aspects of understanding will eventually begin to come through even if you make little atempt to understand what all the expressions actually mean in detail.
I hope so beceause the magic of complex numbers is a miracle well worth appreciating.
If you can cope with the notation then so much the better.
First of all we may ask whether other numbers have square roots.
What about the general complex number a +ib ?
We find that the complex number sqrt1/2(a+sqrt(aa+bb))+i.
Sqrt(1/2(-a+sqrt(aa+bb))
Squares to a+ib.
Thus we see that even though we only adjoined a square root for a single quantity we find that every number in the resulting system now automatically has a square root!
This is quiete different from what happened in the passage from the rationals to the reals.
In that case the mere introduction of the quantity sqrt(2) into the system of rationals would have got us almost nowhere.
But this is just the very beginning.
We can ask about cube roots fifth roots or even i-th roots.
We find that whether complex root we choose and whatever complex number we apply it to,there is always a solution to this problem.
We are still barely scratching the surface of complex number magic.
What i have just asserted above is a really quiete simple to establish.
Somewhat more remarkable is the so-called fundamental theorem of algebra which in effect asserts that any polynomial equation such as
Z^4=z-1
Must have complex number solutions.
More explicitly there will always be a solution to any equation of the form a0+a1z+a2z^2+…+anz^n=0
For comparison w emay recall that i was introduced in effect simply to provide a solution to the equation
1+zz=0
It is worth mentioning the problem that cardano had been concerned with from around 1539 when he first encountered complex numbers and caught a hint of another aspect of their attendant magical properties.
This problem was in effect to find an expression fort he general solution of a real cubic equation.
Cardano found that the general cubic could be reduced to the form
X^3=3px+2q
By a simple transformation.
Here p and q are to be real numbers and i have reverted to the use of x in the equation rather than z to indicate that we are now concerned with real number solutions rather than complex ones.
Cardano’s complete solution seems to have been developed from an earlier partial solution that he had learnt in 1539 from tartaglia although this partial solution had been found earlier by del ferro.
The del-ferro cardano solution was essentially the following:
X=(q+w)^1/3+(q-w)^1/3
Where
W=(qq-ppp)^1/2
Convergence of power series
Despite these remarkable facts we have still not got very far into complex number magic.
There is much more to come!
For example one area where complex numbers are invaluable is in providing an understanding of the behaviour of what are called power series.
A power series is an infinite sum of the form a0x+a1x+..+anx^^n+…
Because this sum involves an infinite number of terms it may be the case that the series diverges which is to say that it does not settle down to a particular finite value as we add up more and more of its terms.
Consider the series
1+x^^2+x^^4+x^^6+…..
If we put x;=1 then adding the terms we get
1,2,3,4,5,..
And we see that the series has no chance of settling down to a particular finite value that is it is divergent.
Things are even worse if we try x;;=2 since now the individual terms are getting bigger and adding terms successively we get
1,
1+4=5
1+4+16=21,
1+4+16+64=85,
Which clearly diverges.
On the other hand if we put x=1/2
Then we get
1,
1+1/4=5/4,
1+1/4+1/16=21/16,
1+1/4+1/16+1/64=85/64,
And it turns out that these numbers become closer and closer to the limiting value 4/3
So the series is now convergent.
With this series it is not hard to appreciate in a sense an underlying reason why the series can not help nut diverge for x=1 and x=2 while converging for x=1/2
To give the answer 4/3
For we can explicitly write down the answer to the sum of the entire series finding
1+x^^2+x^^4+x^^6+…..
1/1-xx
When we substitute x=1 we find that this answer is zero which is infinity and this provides us with an understanding of why the series has to diverge for that value of x.
When x=1/2 the answer is 4/3
And the series actually converges to this value as stated above.
This all seems very sensible.
But what about x=2?
Now there is an answer given by the explicit formula namely -1/3 although we do not seem to get this value simply by adding up the terms of the series.
We could hardly get this answer because we are just adding together positive quantities whereas -1/3 is negative.
The reason that the series diverges is that when x=2 eac term is bigger than the corresponding term was when x=1 so that divergence for x=2 follows, logically from the divergence for x=1
In the case of x=2 it is not the answer is really infinite but that we can not reach this answer by attempting tos um of the series successively up to 4 terms together with the answer 1/1-xx
And we can see that provided x lies strictly between the values -1 and +1 the curves depicting these partial sums do indeed converge on this answer namely 1/1-xx
As we expect.
But outside this range the series simply diverges and does not actually reach any finite value at all.
As a slight digression it will be helpful to address a certain issue here that will be of importance to us later.
Let us ask the following question :
Does the equation that we obtain by putting x=2 in the above expression namely
1+22+…=-1/3
Actually make any sense?
The great 18th century mathematician Leonhard Euler often wrote down equations like this and it has become fashionable to poke gentle fun at him for holding to such absurdities while one might excuse him on the grounds that in those early days nothing was properly understood about matters of convergence of series and the like.
Indeed, it is true that the rigorous mathematical treatment of series did not come about until the late 18th and early 19th century through the work of Cauchy and others.
Moreover, according to this rigorous treatment the above question would be officially classified as nonsense.
Yet, i think that it is important to appreciate that in the appropriate sense euler really knew what he was doing when he wrote down apparent absurdities of this nature and that there are senses according to which the above equation must be regarded as correct.
In mathematics it is indeed imperative to be absolutely clear that one’s equations make strict and accurate sense.
However it is equally important not to be insensitive to things going on behind the scenes which may ultimately lead to deeper insights.
It is easy to lose sight of such things by adhering too rigidly to what appears to be strictly logical, such as the fact that the sum of positive terms 1+4+16+64+256+…
Can not possibly be -1/3.
For a pertinent example let us recall the logical absurdity of finding a real solution to the equation x2+1=0.
There is no solution;yet if we leave it at that we miss all the profound insights provided by the introduction of complex numbers.
A similiar remark applies to the absurdity of a rational solution to xx=2
In fact it is perfectly possible to give a mathematical sense to the answer -1/3 to the above infinite series but one must be careful about the rules telling us what is allowed and what is not allowed.
It is not my purpose to discuss such matters in detail here but it may be pointed out that in modern physics particularly in the area of quantum field theory divergent series of this nature are frequently encountered.
It is very delicate matter to decide whether the answers that are obtained in this way are actually meaningful and moreover actually correct.
Sometimes extremely accurate answers are indeed obtained by manipulating such divergent expressions and are occasionally strikingly confirmed by comparison with actual physical experiment.
On the other hand one is often not so lucky.
These delicate issues have important roles to play in current physical theories anda re very relevant for our attempts to assess them.
The point of immediate relevance to us here is that the sense that one may be
Able to attribute to such apparently meaningless expressions frequently depends in an essential way upon the properties of complex numbers.
Let us now return to the issue of the convergence of series and try to see how complex numbers fit into the Picture.
For this let us consider a function just slightly different from 1/(1-xx)
Namely 1/(1+xx)
And try to see whether it has a sensible power series expansion.
There would seem to be a better chance of complete convergence now, because 1/1+xx remains smooth and finite over the entire range of real numbers.
There is indeed a simple-looking power series for 1/1+xx
Only slightly different from the one that we had before namely
1-x2+x4-x6+x8-…=1/1+x2
The difference being merely a change of sign in alternate terms.
At x=1:
1,0,1,0,1,0,1,…
At x=2:
1,-3,13,-51,205,-819,…
At x=1/2:
1,3/4,13/16,51/64,205/256,819/1024
We note that the divergence in the first case is simply a failure of the partial sums of the series ever to settle down although they do not actually diverge to infinity.
Thus, in terms of real numbers alone, there is a puzzling discrepancy between actually summing the series and passing directly to the answer that the sum to infinity of the series is supposed to represent.
The partial sums simply take off just at the same places as where trouble arose in the previous case, although now the supposed answer to the infinite sum namely 1/1+xx does not exhibit any noticeable feature at these places at all.
The resolution of the mystery is to be found if we examine complex values of this function rather than restricting our attention to real ones.
Caspar’s complex plane
In order to see what is going on here it will be important to use the now standard geometrical representation of complex numbers.
Caspar wessel in 1797 ,Jean Robert argand in 1806, john warren in 1828, and carl friedrich gauss well before 1831 all independently came up with the idea of complex plane in which they gave clear geometrical interpretations of the operations of addition and multiplication of complex numbers.
We are now to think of a real number x as a particular case of the complex number z=x+i.y
Where y=0.
Thus we are thinking of the x-axis as real line.
The complex plane therefore gives us a direct pictorial representation of how the system of real numbers extends outwards to become the entire system of complex numbers.
This real line is frequently referred to as the real axis in the complex plane.
The y-axis is correspondingly referred to as imaginary axis .
Let us now return to our functions that we have been trying to represent in terms of power series.
We took these as functions of the real variable x, namely 1/1-xx
And
1/1+xx
But now we are going to extend these functions so that they apply to a complex variable z.
Some relations to modern particle physics
Numbers such as these have interest in modern particle physics providing the possible cases of a multiplicative quantum number.
I commented on the fact that the additive quantum numbers of particle physics are invariably quantified as far as is known,by integers.
There are also a few examples of multiplicative quantum numbers and these seem to be quantified in terms of nth rooths of unity.
I only know of a few examples of such quantities in conventional particle physics and in most of these the situation is the comparatively uninteresting case n=2.
There is one clear case where n=3 and possibly a case for which n=4.
Unfortunately in most cases the quantum number is not universal that is it can not consistently be applied to all particles.
In such situations ,i shall refer to the quantum number as being only approximate.
The quantity called parity is a multiplicative quantum number with n=2.
The Notion of parity for a composite system is built up from those of its basic constituent particles.
For such a constituent particle its parity can be even in which case the mirror reflection is what is called its antiparticle.
Since the Notion of mirror reflection or of taking the antiparticle is something that squares to unity, the quantum number –let us call e -
Has to have the property e.e=1
So it must be an nth root of unity with n=2
This Notion is only approximate because the parity is not a conserved quantity with respect to what are called weak interactions and indeed there may not be a well-defined parity for certain particles because of this.
Moreover the Notion of parity applies in normal descriptions only to the family of particles known as bosons.
The remaining particles belong to another family and are known as fermions.
The distinction between bosons and fermions is a very important but somewhat sophisticated one .
There is a sense in which “two fermions make a boson”
And “two bosons also make a boson”
Whereas
“a boson and a fermion make a fermion”
Thus we can assign the multiplicative number -1 to a fermion and +1 to a boson to describe its fermion-boson nature and we have another multiplicative quantum number with n=2.
As far as is known this quantity is an exact multiplicative quantum number.
It seems to me that there is also a parity Notion that can be applied to fermions although this does not seem to be a conventional terminology.
This must be combined with the fermion-boson quantum number to give a combined multiplicative quantum number with n=4.
For a fermion the parity value would have to be +i or –i and its double mirror reflection would have the effect of a 2pi rotation.
For a boson the parity value would be +1,-1 as before.
The multiplicative quantum number with n=3 that i have referred to is what i shall call quarkiness.
I referred to the modern viewpoint that the strongly interacting particles known as hadrons are taken to be composed of quarks .
These quarks have values for their electric charge which are not integer multiples of the electron’s charge but which are integer multiples of the electron’s charge but which are integer multiples of one-third of this charge.
However quarks can not be exist as seperate individual particles and their composites can exist as seperate individuals only if their combined charges add up to an integer in units of the electron’s charge.
Let q be the value of the electric charge measured in negative units of that of the electron.
For quarks we have q=2/3 or -1/3 for anti-quarks q=1/3 or -2/3.
Thus if we take for the quarkiness the multiplicative quantum number exp(-2qpi.i)
We find that it takes values 1,w,w2
For a quark the quarkiness is w
And for an anti-quark it is w2
A particle that can exist separately on its own only if its quarkiness is 1.
I have exhibited some of the aspects of the magic of the complex numbers and have hinted at just a very few of their applications.
But i have not yet mentioned those aspects of complex numbers that i myself found to be the most magical of all when i learned about them as a mathematics undergraduate.
These things however depend upon certain basic notions of the calculus so in order to convey something of this magic to the reader it will be necessary first to say something about these basic notions.
Real number calculus
Calculus-or according to its more sophisticated name mathematical analysis-is built from two basic ingredients :
Differentiation and integration.
Differentiation is concerned with velocities accelerations the slopes and curvature of curves and surfaces and the like.
These are rates at which things change and they are quantities defined locally in terms of structure or behaviour in the tiniest neighbourhoods of single points.
Integration on the other hand is concerned with the other hands with areas and volumes with centres of gravity and with many other things of that general nature.
These are things which involve measures of totality in one form or another and they are not defined merely by what is going on in the local or infinitesimal neighbourhoods of individual points.
The remarkable fact referred to as the fundamental theorem of calculus is that each one of these ingredients is essentially just the inverse of the other.
It is largely this fact that enables these two important domains of study to ocmbine together and to provide a powerful body of understanding and of calculational technique.
This subject of analysis as it was originated in the 17th century by fermat Newton and leibniz with ideas that hark back to archimedes in about the 3rd century is called calculus because it indeed provides such a body of calculational technique whereby problems that would otherwise be conceptually diffucult to tackle can frequently be solved automatically merely by following of a few relatively simple rules that can often be applied without the exertion of a great deal of penatrating thought.
Yet there is a striking contrast between the operations of differentiation and integration.
In this calculus with regard to
Which is easy one and which is the diffucult one.
When it is a matter of applying the operations to explicit formulae involving known functions it is differentiation which is easy and integration diffucult and in many cases the latter may not be possible to carry out at all in an explicit way.
On the other hand when functions are not given in terms of formulae but are provided in the form of tabulated lists of numerical data then it is integration which is easy and differentiation diffucult.
The idea of a riemann surface
There is a way of understanding what is going on with this analytic continuation of the logarithm function –or of any other many-valued function in terms
Of what are called surfaces.
Riemann’s idea was to think of such functions as being defined on a domain which is not simply a subset of the complex plane, but as a many-sheeted region.
In the case of logz we can Picture this as kind of spiral ramp flattened down vertically to the complex plane.
I have tried to indicate this in figure.
The logarithm function is
Single-valued on this winding many-sheeted version of the complex plane because each time we go around the origin
And 2pi.i has to be added to the logarithm we find ourselves on another sheet of the domain.
There is no conflict between the different values of the logarithm now because its domain is this more extended winding space .
Riemann who introduced this idea was one of the very greatest of mathematicians he put forward a multitude of ideas that have profoundly altered the course of mathematical thought on this planet.
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