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The Beauty and Mystery of Mathematics
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BarkellWH
Posts: 3458
Joined: Jul. 12 2009
From: Washington, DC
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 The Beauty and Mystery of Mathematics
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quote:
It took centuries to develop and refine the concepts of modern mathematics. There was quite a flowering of this effort in the 20th century. The teachability of many mathematical concepts indicates that these concepts are well suited to our aptitudes and abilities.
This is an interesting topic in its own right and probably should be removed from the "Cataract Question" thread. I have taken the liberty of creating a new thread for this discussion and have left Richard's quote above as a point of departure.
The British mathematician Godfrey H. Hardy, who specialized in number theory and mathematical analysis, in his classic 1940 work "A Mathematician's Apology," extolled the aesthetic beauty of pure mathematics. Hardy declared that beauty, not usefulness, is the true justification for mathematics. And Bertrand Russell in his autobiography wrote that it was his desire to know more of mathematics that kept him from committing suicide. Russell wrote that the beauty of mathematics was "cold and austere, like that of a sculpture...sublimely pure and capable of a stern perfection."
There is an interesting concept that mathematics has a reality that transcends the human mind; that mathematical structures are not the product of human thought, but that they exist timelessly, apart from human thought, in what might be called a Platonic realm of their own, waiting to be discovered by mathematicians. Einstein once asked, "How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably appropriate to the objects of reality?" Great mathematicians such as Robert Langlands, Kurt Godel, Sir Roger Penrose, and others would reply that mathematical structures are not just "admirably appropriate to the objects of reality," they ARE objects of reality.
Cheers,
Bill
_____________________________
And the end of the fight is a tombstone white,
With the name of the late deceased,
And the epitaph drear, "A fool lies here,
Who tried to hustle the East."
--Rudyard Kipling
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Date Nov. 19 2013 16:46:59
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BarkellWH
Posts: 3458
Joined: Jul. 12 2009
From: Washington, DC
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RE: The Beauty and Mystery of Mathem... (in reply to guitarbuddha)
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quote:
Good to see you back Bill. I hope you aren't hoping to trap Richard into acknowledging that he is reversing his previously stated position on the determinist nature of the 'laws' of physics. You naughty naughty Bill.
Never left, GuitarBuddha. I just reserve my comments for topics in which I have some degree of knowledge and interest. In this case, it has nothing to do with Richard per se; rather, I find the intersection of mathematics and philosophy to be very interesting, and I have taken your and Richard's comments on mathematics in the "Cataract Question" thread and tried to steer it in a more philosophical direction under a new topic and thread.
Cheers,
Bill
_____________________________
And the end of the fight is a tombstone white,
With the name of the late deceased,
And the epitaph drear, "A fool lies here,
Who tried to hustle the East."
--Rudyard Kipling
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Date Nov. 19 2013 19:24:18
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Ruphus
Posts: 3782
Joined: Nov. 18 2010
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RE: The Beauty and Mystery of Mathem... (in reply to BarkellWH)
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When you are situated in a discrete urban environment of the west that appears as if everyone was aware about the phenomenon of methexis, you might feel it only consequential to ponder about further playgrounds of entity and perception.
But when getting an idea of world view and lacking level of sophisticataion in vast spheres of the planet, where people live with standards off any natural science, and proceed accordingly all the time, with all the sick and destructive custom that yields of it ...
Then you wished the academies turned around to first ensure a catch up of the worlds population to at least basic knowledge, before resuming exploration.
The daily inferno caused among fellow men, children, animals and environment sure does ask for a making up leeway.
For anyone doubting the urgency, come to the east and be for example ...
of another religion / custom.
just a moved out bachelor girl ( if you dare).
a tortured animal.
a desinformed and string-led kid in an orthodox family.
grossly robed by crafty routines and then neglected by ambience that shrugs off perfidy as common behaviour.
& & &
It really is so sheer unbelievable how the vast of people live and deem still in modern times, indulging themselves in blatant injust over and over again. Constantly since so many centuries.
And that is only the most crass of superstition and desinformation, which the insanity of the west has yet to be added to.
Like those 1,5 mio square kms of primare wood that have been depletet from the year 2000 to 2012 alone.
Where will we file our newest scientific insights when all what is left be burned soil?
Which famous living mathematician will calculate the current stupidity and insanity?
How will most basic info be getting around finally?
... At least fundamentals like grasp about the difference of believing and knowing? Finally? Within this world maybe?
Ruphus
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Date Nov. 19 2013 20:07:27
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BarkellWH
Posts: 3458
Joined: Jul. 12 2009
From: Washington, DC
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RE: The Beauty and Mystery of Mathem... (in reply to Ruphus)
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quote:
When you are situated in a discrete urban environment of the west that appears as if everyone was aware about the phenomenon of methexis, you might feel it only consequential to ponder about further playgrounds of entity and perception. But when getting an idea of world view and lacking level of sophisticataion in vast spheres of the planet, where people live with standards off any natural science, and proceed accordingly all the time, with all the sick and destructive custom that yields of it ...Then you wished the academies turned around to first ensure a catch up of the worlds population to at least basic knowledge, before resuming exploration.
As one who has lived in both sophisticated, Western, urban environments and underdeveloped areas of the world lacking in the sophistication you mentioned, I can assure you that the pursuit of the phenomenon of methexis (the relation of a particular to a Platonic form) in mathematics or any other endeavor, and working toward societal advancement in underdeveloped societies are not mutually exclusive. One need not be sacrificed in order to pursue the other. Both can be, and are, pursued simultaneously in this world. There are pure mathematicians, theoretical physicists, and philosophers working on their questions, while education and development specialists work on raising the educational level and basic standard of living of those in the least-developed areas of the world.
Cheers,
Bill
_____________________________
And the end of the fight is a tombstone white,
With the name of the late deceased,
And the epitaph drear, "A fool lies here,
Who tried to hustle the East."
--Rudyard Kipling
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Date Nov. 19 2013 20:36:51
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Richard Jernigan
Posts: 3430
Joined: Jan. 20 2004
From: Austin, Texas USA
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RE: The Beauty and Mystery of Mathem... (in reply to BarkellWH)
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It's been a while since I last read Hardy's autobiography. I don't remember whether Hardy was a declared Platonist or not. Certainly Platonism was more prevalent among Hardy's generation than among more recent ones.
During Hardy's youth belief in the supernatural was far more prevalent than nowadays. For instance, in the biography of R.L. Moore commissioned and published by the Mathematical Association of America, Moore's tactic while an undergraduate for becoming better acquainted with a young woman was to invite her to attend church with him. Moore was a young undergraduate, having matriculated at 16.
Some time later, by his early twenties Moore experienced a crisis of faith, as evidenced in correspondence with his earliest mathematical mentor, Halsted. I suspect Moore may have undergone a similar experience with Platonism.
During my time as a student with Moore he certainly behaved as though mathematical concepts partook of "reality", though he was careful to skirt the issue. Indeed he changed from referring to the Axiom of Choice as "that fact" to carefully pointing out in the second edition of his American Mathematical Society Colloquium Publication that it was an axiom, not a fact.
The Axion of Choice occupies a position similar to Euclid's parallel postulate. There are systems of geometry which satisfy all of Euclid's axioms except the parallel postulate. Similarly there are systems of logic which obey all the Zermelo-Frankel axioms of set theory, some incorporating the Axiom of Choice, others negating it.
The Classical Greeks were pretty much Platonists in geometry, mistaking the intuitive appeal of Euclid's axioms for reality. At present the Non-Euclidean Minkowskian geometry of General Relativity is taken as a better representation of reality.
The discovery of non-Euclidean geometries in the mid-nineteenth century led to a weakening of the Platonist position. The problem was that any geometry that obeyed Euclid's axioms necessarily harbored within itself a model for non-Euclidean geometry. Which of the geometries was "reality"?
Two strong supports for Platonism are the "universality" of mathematical concepts, and the intense feeling of discovery occasioned by the extension of these concepts to new realms, or the discovery of the proof of a difficult proposition, or the solution of a hard problem.
The "universality" argument is undermined by the unresolved controversies on the foundations of mathematics. Russell and Whitehead tried to develop all mathematics from set theory, an effort effectively dynamited by Gödel. The Intuitionists counseled a radical retreat, and so on. For those interested, Morris Kline's "Mathematics, the Loss of Certainty" is an informative read.
The emotional feeling of discovery brings forth an analogy on my part. During Newton's lifetime, the great majority of educated Europeans believed the Bible story of language. It was a gift from God at the Creation. Human hubris at the attempted construction of the Tower of Babel annoyed their Creator, who confounded the universal language of mankind up until then, resulting in the polyglot world.
The modern story of language is that it is an innate, genetically determined aptitude of humans, produced by a long running course of evolution. Toddlers learn with striking efficiency to speak and understand because their brains are built to do so. The diversity of languages was produced, and continues to be produced by the evolution that brought it into being in the first place. Details of the aptitude and its development are the subject of a minor industry of debate, but few linguists publicly adhere to the supernatural origin of language.
Personally, I find believable a similarly modern story for the human aptitude for mathematics, and the ready adoption of its concepts by students. To me the relationship of Newtonian mechanics to the ability of a dedicated adolescent to become a pool shark is quite clear. Newtonian mechanics is the mathematization of universal intuitive concepts of motion. The rapid development of Newtonian mechanics and classical electromagnetic theory ironically led to the ability to detect their inadequacies.
The mathematical developments that enabled the elaboration of classical physics formed the basis for newer, more accurate theories.
Counting and measurement are absent from only a very few studied societies. The application of measurement to motion was a ball buster for centuries. Much of the difficulty was overcome by Newton and Leibnitz, finally tidied up by Cauchy, Dedekind et al in the 19th century. Mathematical analysis is a centuries long conscious work of extending the ideas of counting and measurement and their application.
I find it both hard to conceive of, and unproductive to believe in the supernatural existence of the Platonic entities.
This doesn't diminish the esthetic and emotional pleasures, the sense of discovery and wonderment in mathematics, and indeed in all of science. I was raised a Christian. I fell away from major components of belief at the age of 18. Still the drill, ceremonies and rhetoric of the Anglican liturgy have a strong emotional resonance for me. It speaks to important human issues.
I don't need to divorce myself from the pleasures of Platonism in order to doubt its supernatural claims.
RNJ
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Date Nov. 20 2013 1:18:52
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guitarbuddha
Posts: 2970
Joined: Jan. 4 2007
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RE: The Beauty and Mystery of Mathem... (in reply to Richard Jernigan)
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quote:
ORIGINAL: Richard Jernigan
The modern story of language is that it is an innate, genetically determined aptitude of humans, produced by a long running course of evolution. Toddlers learn with striking efficiency to speak and understand because their brains are built to do so.
Newtonian mechanics is the mathematization of universal intuitive concepts of motion. The rapid development of Newtonian mechanics and classical electromagnetic theory ironically led to the ability to detect their inadequacies.
I don't need to divorce myself from the pleasures of Platonism in order to doubt its supernatural claims.
RNJ
I remember when I was first introduced aged six or so to the concept of syllables. I am not sure what the preferred definition is but I like to think of a syllable as a vowel sound which may be more less separated. My name is Freel my teacher asked me how many syllables it had. I suggested two Free-el. I was corrected. This surprised my as any attempt to say it in one robs it of its natural and correct pronunciation and this offended me somewhat. Of course the problem was simple. Rather than an analysis of language as spoken syllables were actually being applied to spelling where a written vowel sound is articulated on the page by an intervening consonant. I guess I was thinking of phonemes. It is odd to note that phoneme is a newer word. So easily the grammer which we choose to describe a thing can become confused for the thing itself.
I am sure you are aware of this but it is important to note that in a very real sense a child actually BUILDS ITSELF a brain which is capable of making sense of the sounds of the as yet unrecognised individuals around it. Now it is true that there is a basic template and that at birth sensitivity to some of the possible sounds of human speedh which are hardwired pre birth are damped down as the brain becomes more specialised to the Aesthetic of the particular and somewhat arbitrary music of the language to which the child is exposed. However I like to believe that this specialisation is a consequence of the problem solving of random instigation followed by a natural selection and reinforcement of neural pathways which allow enhanced access to resources. These conditions for which are present in each fledgling brain) which with each birth begins,to a certain extent, anew. Every child will cry naturally and without an agenda but very soon the crying assumes modes to manipulate the objects(parents) who are it's primary instruments to gain resources (food, warmth and later affection and cars).
The grammar of mathematics which is part of a continuum from Euclid to Newton and beyond has certainly been designed to mirror a small range of the innate intuitions to which all humans are privy and is certainly very able for those tasks. But is this seed all that is available ? The angler fish can predict the path of a jet of water passing through one medium to another to topple a fly which is a rather complicated optical and mechanical problem. Thus it equals our teenage pool player effortlessly. Thus might a fish seed all of our mathematical grammar ?
Well maybe so....But even a teenager (separated from me in age in precisely the same proportions as I am from you Richard) is capable of thoughts which would be very difficult to express with accuracy by any known means (least of all a conversation).
Now the basic components of a computer were as far as I aware inspired by symbolic logic, as you alluded. The same logic that was such a terrible failure with regard to finding a basis for mathematics. I am certain that a computer model, hidebound by its physical basis of such a small part of human potential, will be seriously limited in describing or efficiently modelling even the identifiably logical functions that constitue thought.
And may that is the temptation of the platonic entities. We look for truth in their completeness and indeed knowability because when we look at the true power of the human mind and it's emotional nature we are always somewhat at sea. We can know a platonic solid in its entirety because of its fascinating simplicity. We simply cannot know a fellow human or even ourselves with such satisfying objectivity.
I wholeheartedly agree with your last point. I find it childish and shallow that the claim was even once made that understanding unweaves the rainbow. When we close the desk drawer and look out the window it will still be there and if we take the time we can enjoy it in the moment fleeting and imperfect and laugh at ourselves for imagining that the model we built in our heads was the thing itself. And of course continue to cherish both.
The loss of beauty is suffered by those who hide behind the claim that analysis robs events of beauty. And of course the temptation to make this claim is all the harder when the analysis has proven troublesome in the past. Don't think more just decide that the people who understand are you aesthetic inferior.
Of course that last paragraph dovetails neatly into the field of music which is a collection of variously related and discrete styles of language. The analysis of which can lead to, as I tried to point out in the opening paragraph, some pretty unguarded assumptions when one confuses the grammar of analysis with the thing itself.
First always, the thing itself.
D.
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Date Nov. 20 2013 2:36:55
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BarkellWH
Posts: 3458
Joined: Jul. 12 2009
From: Washington, DC
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RE: The Beauty and Mystery of Mathem... (in reply to Ruphus)
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quote:
Displayed in your example of incredible neglection of future damage, or likewise in the US foreign concept of past century in Middle East to play with and awake the fading backwardedness. To this day I wonder whether the planners of that policy could have been really that stupid to not estimate what would be coming out of such, and coming back to bite their own offspring into the butt. And much more than that.
Much of United States Middle East policy has been counter-productive; ironically, not least with regard to the US national interest. Nevertheless, the regrettable condition of the Middle East is primarily a function of its own culture and religion, which makes no distinction between the secular and sacred realms. As a result, the Middle East finds it difficult coming to terms with modernity in all its manifestations, including governance. It has always been my opinion that what the Middle East desperately needs is an equivalent of the Western Eighteenth Century Enlightenment, which to our great good fortune separated the world of faith from the world of rational thought. Turkey was the only country to separate the two, and Kemal Attaturk had the strength to enforce it. Others have not even attempted it to their detriment.
Cheers,
Bill
_____________________________
And the end of the fight is a tombstone white,
With the name of the late deceased,
And the epitaph drear, "A fool lies here,
Who tried to hustle the East."
--Rudyard Kipling
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Date Nov. 20 2013 14:39:17
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BarkellWH
Posts: 3458
Joined: Jul. 12 2009
From: Washington, DC
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RE: The Beauty and Mystery of Mathem... (in reply to Richard Jernigan)
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quote:
I find it both hard to conceive of, and unproductive to believe in the supernatural existence of the Platonic entities.
I do too, and I mention mathematical structures as potentially the result of "Platonic Forms" because that is how the mathematicians I mentioned thought of (or described) them. (I don't know if they actually believed in Platonism and its "Forms," or if they were simply using Platonism as a way of describing their belief that mathematical structures exist independently of human thought.) Nevertheless, I do think that mathematical structures exist as a reality that transcends the human mind.
Cheers,
Bill
_____________________________
And the end of the fight is a tombstone white,
With the name of the late deceased,
And the epitaph drear, "A fool lies here,
Who tried to hustle the East."
--Rudyard Kipling
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Date Nov. 20 2013 14:51:01
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BarkellWH
Posts: 3458
Joined: Jul. 12 2009
From: Washington, DC
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RE: The Beauty and Mystery of Mathem... (in reply to guitarbuddha)
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quote:
As do paintings and recordings ?
In some sense, yes. But obviously without physical properties. Nevertheless, I think mathematical structures transcend, and are independent of, the human mind (forget Platonic Forms). When mathematicians make a "discovery," or solve a problem (Fermat's Last Theorem), they are bringing forth something that existed prior to their "discovery"; something that was just waiting for a human mind (or minds) to reach the level of sophistication required to "crack the code," so to speak.
By the way, I sent you a PM, GuitarBuddha, but it didn't go through as your In-Box was full.
Cheers,
Bill
_____________________________
And the end of the fight is a tombstone white,
With the name of the late deceased,
And the epitaph drear, "A fool lies here,
Who tried to hustle the East."
--Rudyard Kipling
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Date Nov. 20 2013 15:08:45
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Ruphus
Posts: 3782
Joined: Nov. 18 2010
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RE: The Beauty and Mystery of Mathem... (in reply to BarkellWH)
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quote:
ORIGINAL: BarkellWH
... the regrettable condition of the Middle East is primarily a function of its own culture and religion, which makes no distinction between the secular and sacred realms. As a result, the Middle East finds it difficult coming to terms with modernity in all its manifestations, including governance. It has always been my opinion that what the Middle East desperately needs is an equivalent of the Western Eighteenth Century Enlightenment, which to our great good fortune separated the world of faith from the world of rational thought. Turkey was the only country to separate the two, and Kemal Attaturk had the strength to enforce it. Others have not even attempted it to their detriment.
Cheers,
Bill
Agreed.
However, the degree of status quo has been threadened by foreign policies.
Until the Seventies of last century a broad tendency had come up in Middle East of acknowledging western achievements and admiring them. ( James Dean, Fred Astair and fashion from Paris had their high times in that region; just to name symbols.) Simultaneously recognizing the backwardedness of own traditions.
At that time superstition was on decline and mainly preserved by elderly.
That trend was countered and turned to the opposite by foreign strategies, who by-passed oriental intellectuals ( and even surrendered them to dictators, or instigated / helped to remove or kill them) and flattered the retarded.
Secondly, with the same time of specifically reviving fading Middle Age, technological amnety was massively exported, reducing once more recognition of traditional mental standard.
Today tyrants who call for re-establishing stone age, come along in western cloth and limousines, check the time per watches and appoint their sessions per mobile phone, whilst at home heating systems, AC, panel-TVs, fridges and microwaves round up the comfort. Only supporting a historical mind-set of repressing objects causality and origin, that engages all the amneties without the providing civil foundation.
Without those policies superstion in the Orient today should be rudimentary and there would certainly be in place intellectual and democratic progress of a whole other category.
Ruphus
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Date Nov. 20 2013 16:09:34
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BarkellWH
Posts: 3458
Joined: Jul. 12 2009
From: Washington, DC
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RE: The Beauty and Mystery of Mathem... (in reply to Ruphus)
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quote:
Until the Seventies of last century a broad tendency had come up in Middle East of acknowledging western achievements and admiring them. ( James Dean, Fred Astair and fashion from Paris had their high times in that region; just to name symbols.) simultaneously recognizing the backwardedness of own traditions. At that time superstition was on decline and mainly preserved by elderly.
I agree that certain Western (both American and European) policies in the Middle East have been misguided, but I do not think the policies themselves have prevented the countries and cultures of the area from embracing modernity and developing rational, mature, political, economic, and social systems. You correctly call James Dean, Fred Astaire, and fashions from Paris just "symbols," and that's what they are. They represent a very superficial view of the West. It is not Western foreign policy that creates the Middle Eastern mindset. And it is not just superstition either.
The problem with societies and cultures in the Middle East goes much deeper and is very much a function of a certain religio-cultural outlook that makes no distinction between the secular and sacred realms. Religion is paramount in all aspects of life, and it places a bind on Middle Eastern cultures in ways that make it difficult to accept modernity. In fact, one can say that religion is culture. And it pervades their entire existence in everything from their overall belief system to the laws that govern inheritance and court cases (in which a female inherits only half of what a male sibling inherits; and in court, where the testimony of two women is required to equal that of one man). The examples above are just small elements of a much larger problem that has nothing to do with Western foreign policy. Rather, it has everything to do with elements inherent in the religio-cultural milieu in which they exist.
Cheers,
Bill
_____________________________
And the end of the fight is a tombstone white,
With the name of the late deceased,
And the epitaph drear, "A fool lies here,
Who tried to hustle the East."
--Rudyard Kipling
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Date Nov. 20 2013 17:13:10
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BarkellWH
Posts: 3458
Joined: Jul. 12 2009
From: Washington, DC
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RE: The Beauty and Mystery of Mathem... (in reply to Ruphus)
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quote:
There would undoubtedly be other development under educated and sincere minds like say of a Mossadegh, compared to the regency of Taliban alikes.
I take your point, Ruphus, but I'm not sure that in the long run things would have turned out differently (in terms of a modern mindset) under Mossadegh. Just look at Egypt. When Mubarak was deposed and an election was held, Mohammed Morsi and the Muslim Brotherhood won. What did Morsi do? He attempted to undermine the very democratic process that brought him to power. The so-called "Arab Spring" has turned out to be more a "winter." Egypt, Libya, Tunisia? And in Syria Bashar al-Assad may be an authoritarian thug, but look at what might take his place if he is deposed.
When I speak of the religio-cultural milieu that makes no distinction between the secular and the sacred in the Middle East, I am not talking about the Taliban or terrorists, Ruphus. I am talking about the majority of the population living in these societies. That same religio-cultural mindset exists whether it is Saudi Arabia or Morocco. That is why Kemal Attaturk's enforcement of secularism was so important in Turkey. For decades Turkey was the one secular country in the Middle East, and it prospered. Women did not have to wear scarves and veils, and they could become professionals: doctors, lawyers, business executives. Now, sadly, that seems to be changing under Erdogan.
I do not like any society that places religion in the public square. I am disgusted that so-called Christian "Creationists" and "Intelligent Designers" have such a loud voice in the U.S. They even have significant influence in what is taught in secondary schools in some parts of the country. So we have some students who "learn" (if that's what you call it!) that it is just as valid to consider the age of the Earth at 6,000 years as it is to consider it's true age at 4.5 billion years. And that it is equally valid to believe or disbelieve in evolution as a fact. Absolute ignorance! Nevertheless, for the most part, we have managed to escape these religious constraints on free enquiry. The Middle East has not managed it, and, in my humble opinion, it is because they have never experienced the equivalent of the West's Eighteenth Century Enlightenment that separated the secular world from the religious world.
Cheers,
Bill
_____________________________
And the end of the fight is a tombstone white,
With the name of the late deceased,
And the epitaph drear, "A fool lies here,
Who tried to hustle the East."
--Rudyard Kipling
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Date Nov. 20 2013 21:24:36
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