Questions on the History of Mathematics

Arthur Miller
Einstein and Schrödinger never fully accepted the highly abstract nature of Heisenberg's quantum mechanics, says Miller. They agreed with Galileo's assertion that "the book of nature is written in mathematics", but they also realized the power of using visual imagery to represent mathematical symbols.

For most people I am sure it is of little interest that such an abstract language could have ever amounted to anything,since we might have been circumscribed to the natural living that is required that we could do without it. But really,  can we?

 Paul Dirac 

When one is doing mathematical work, there are essentially two different ways of thinking about the subject: the algebraic way, and the geometric way. With the algebraic way, one is all the time writing down equations and following rules of deduction, and interpreting these equations to get more equations. With the geometric way, one is thinking in terms of pictures; pictures which one imagines in space in some way, and one just tries to get a feeling for the relationships between the quantities occurring in those pictures. Now, a good mathematician has to be a master of both ways of those ways of thinking, but even so, he will have a preference for one or the other; I don't think he can avoid it. In my own case, my own preference is especially for the geometrical way.

So of course one appreciates those who start the conversation to help raise the questions in ones own mind. Might it be a shared response to something existing deeper in our society that it would warrant descriptions that we might be lacking in. Ways in which to describe something about nature. There is something definitely to be said about the geometer that can visualize the spaces within which they are working. It has to make sense. It has to describe something? Why then not just plain English(whatever language you choose)

String theory's mathematical tools were designed to unlock the most profound secrets of the cosmos, but they could have a far less esoteric purpose: to tease out the properties of some of the most complex yet useful types of material here on Earth.What Good are Mathematics in the Real World?

Do you know how many mathematical expressions are needed in order to describe the theory?

 The language of physics is mathematics. In order to study physics seriously, one needs to learn mathematics that took generations of brilliant people centuries to work out. Algebra, for example, was cutting-edge mathematics when it was being developed in Baghdad in the 9th century. But today it's just the first step along the journey.Guide to math needed to study physics

Conversations on Mind, Matter, and Mathematics

How mathematics arose from cognitive realizations. Ex. Newton and Calculus. The branches of mathematics. Who are it's developers and what did they develop and why?

 It may be as important as the history in relation to how one may perceive the history and development of mathematics. These were important insights into the way one might of asked how did emergence exist if such things could have been imagined in the mind of the beholder. To attempt to describe nature in the way that one might do by invention? So are these mathematical things discovered or are they invented? Why the history is important?

 This is the basis of the question of what already exists in terms of information has always existed and we are only getting a preview of a much more complicated system. It does not have to be a question of what a MBT exemplifies in itself, but raises the questions about what already exists, exists as part of what always existed. Where do ideas and mathematics come from?

This is a foundation stance that is taken right throughout science? If it exists in the universe, it exists in you? How does one connect?

See Also:

WHAT IS YOUR FAVORITE DEEP, ELEGANT, OR BEAUTIFUL EXPLANATION? 

See Also: Some Educational links to look at then.

•  Wolfram\alpha
-What is Wolfram\Alpha
• Khan Academy

Artifacts in the Exploration of Geometry

Ashmolean Museum, Oxford, UK

It should not be lost on individuals who have followed this blog, that there is a range of connection to Platonic Forms idealization, that such an artifact in Ashmolean Museum although modeled to represent a reality and constituent forming basis, it is by this choice,  that I exercised a" foundational attitude"  about what I can use to push my own perspective forward in science. What others were using.

"The Artist and his Museum"

The first public showing of the mastodon (also known as the "Mammoth", the American incognitum and the "animal de l'Ohio") took place next door to Independence Hall, the building in which both the Declaration of Independence and Constitution were finalized. The venue, known variously as Peale's Museum, the American Museum or simply as The Museum, was the remarkable product of a resourceful, versatile and passionate artist and showman, Charles Wilson Peale.

Peale (1741-1827) was born and raised in Maryland. A vocal opponent of the Stamp Act, he was effectively driven from his first trade, saddle making, when loyalist merchants cut off his credit. He turned to a traveling life of a self-taught, itinerant portrait painter. After a short apprenticeship with Benjamin West in London, Peale returned to Maryland in 1769 to paint wealthy patrons throughout the Chesapeake region.
In 1776 he moved to the largest city of the colonies, Philadelphia, in the hopes of further developing his career. Through his contacts made while serving as a captain of the Continental Army, Peale painted a remarkable assemblage of Revolutionary War figures, including the most comprehensive portrait series ever painted of George Washington. See:Charles Willson Peale's Museum

After doing quite a bit of reading over the years it is surprising what one can come across as they look at the historical perspective with artifacts which sat on shelves to curious onlookers as they examine these items.

Shown here are the models in the mathematical wunderkammer located in the Department of Mathematics at the University of Arizona. Like those in most modern mathematics departments, the collection is a combination of locally-made student and faculty projects together with a variety of commercially produced models. Sadly, a century since their Golden Age, many of the models are in disrepair and much of their documentation has been lost. However, some recent detective work, with the help of the Smithsonian Institution in Washington, has helped the department identify models by the American educators W. W. Ross and R. P. Baker in the collection.

Also see here for further thoughts on this




So you have in fact "forerunners of museums today" revealed in pursuits by individuals to catalog items according to the range of professions and undertakings. In this case, I was interested on geometrical forms as it was some interest to me that we could move our minds around in abstract spaces . I followed the surfaces of "dynamic movement"  issued forth by theoretical application. These would be,  modular forms or Genus figures of string theory, that raised my interest about the space we are working in.

Sylvester's models lay hidden away for a long time, but recently the Mathematical Institute received a donation to rescue some of them. Four of these were carefully restored by Catherine Kimber of the Ashmolean Museum and now sit in an illuminated glass cabinet in the Institute Common Room.

Now you must know that I do not have the education of the universities but this did not stop me from trying to understand what these artifacts in geometry actually represented. Where they were placed by theoreticians to represent the figurative evolution of what actual begins in this universe, from beyond time and space and arrived to a direction of expressions unfolding in the arrow of time. This was a recognition of the times in microseconds that had been "used in minutes" of Steven Weinberg.

A giddy craze was sweeping across Europe at the turn of the 17th century. The wealthy and the well-connected were hoarding things—strange things—into obsessive personal collections. Starfish, forked carrots, monkey teeth, alligator skins, phosphorescent minerals, Indian canoes, and unicorn tails were acquired eagerly and indiscriminately. Associations among these objects, if they were made at all, often reflected a collector's personal vision of an underlying natural "order". Critical taxonomy was rarely in evidence.

So this historical perspective of the artifacts moved my perspective to today and what is going on in mathematical abstraction. What are these shapes actually representing in reality? Is there such a thing once perception has been granted of the close correlative function of the description of that microscopic reality?

It would be that the mind has become capable of moving into the realm of the microscopic, that by measure of energy used, details the plethora of particle and constituents of that energy, that each artifact is leading toward ever finer issues of what began in the formation of the matter, to allow us to see it's constitutions as they are revealed today macroscopically.

IN Search of Mandelstam's Holy Grail

There are two posts that reflect the purpose of this post today. One is Clifford's linked through Lee Smolin's comment and the other, at Backreaction. Good Physics is Conflict

A lot of you may never understand the significance of the mystery that follows the thinking of the Holy Grail. Yet is it more the knowledge that can be gained from all soul's day, that on this occasion we may have called it Halloween.

We celebrated the past, in the living of today? You philosophize, while you become the thoughts of models created by science leaders shared? I do not think any have a "personality disorder" like I do:)

Lee Smolin:

Here is an example of the kind of question I found I needed a book to explore: what to think of the problems that arise from the need for higher dimensions in string theory, such as the problem of moduli stabilization and the vast number of static solutions. To approach this I read books on the early history of GR and unified field theories and learned that higher dimensional compactifications were explored many times between 1914 and 1984 and that close to the beginning these problems were appreciated and discussed by Einstein and others. I weave this story into my book because I find it useful when trying to judge how serious the present issues in string theory are to know how Einstein and many others struggled with the same issues over decades.

So of course when we think of the persons of science who walked before us (shoulders of giants), what are their whole stories, but what is evidenced to us as we read those words? So you compile your data accordingly, and from it, we say at certain spots, how are we to react to the challenge now facing us?



Stanley Mandelstam, Professor Emeritus, Particle Theory

My present research concerns the problem of topology changing in string theory. It is currently believed that one has to sum over all string backgrounds and all topologies in doing the functional integral. I suspect that certain singular string backgrounds may be equivalent to topology changes, and that it is consequently only necessary to sum over string backgrounds. As a start I am investigating topology changes in two-dimensional target spaces. I am also interested in Seiberg-Witten invariants. Although much has been learned, some basic questions remain, and I hope to be able at least to understand the simpler of these questionsStanley Mandelstam-Professor Emeritus Particle Theory

As a lay person watching the debate it is difficult for me to discern the basis of these arguments. But I strive to go past what you think is surface in conduct in science's response, as some may show of themself in a reactionary pose. Should we all be so perfect, that the human condition is not also the example by which we shall progress in science?

Dealing in the Abstract



A sphere with three handles (and three holes), i.e., a genus-3 torus.

Of course the thinking may seem so detached from reality that one asks for some reason with which to believe anything. It required, that the history of this approached dust off models in glass cabinets, that were our early descendants of the museum today.

Sylvester's models lay hidden away for a long time, but recently the Mathematical Institute received a donation to rescue some of them. Four of these were carefully restored by Catherine Kimber of the Ashmolean Museum and now sit in an illuminated glass cabinet in the Institute Common Room.

How many of you know how to work in such abstract spaces, and know that what you are talking about has it's relationships in the physics of today? Or that, what satellites we use in measure of, have some correlation to how one may have seen "UV coordinates supplied by Gauss?"

Artifacts of the Geometrical WunderKammern

As one visits the mathematical puzzles and conjectures, what value these insights to the physics or our universe if we did not see things in this way? As artifacts of some other kind of geometrical thinking that we could then apply it to how we see the micro-perspective and macro-perspective working within the Quantum or cosmological realms?

So abstract and foreign to our eyes that we let it escape our attention while we talk about all our theoretical points of view and divergences from what is symmetrical?

In the past, new scientific discoveries, strange finds, and striking pieces of original artwork were greeted with awe and wonder. It became popular during the Renaissance to build a "cabinet of curiosities" to display a private collection of art and natural objects of which the owner was extremely proud. These groups of objects were at first housed in an actual cabinet or ornate piece of furniture, known as Wunderkammern or Wunderkabinetts. They are simultaneously pieces of furniture and the collections of items within them.

In the exhibits of these early Wunderkammern, owners might display strange, beautiful, mysterious, and precious marvels like starfish, monkey teeth, alligator skins, phosphorescent minerals, Indian canoes, Egyptian figurines, and “unicorn tails.” Rich art patrons would display their new art acquisitions in the intimate backdrop of a prized spot in an ornate carved cabinet. At Kensington castle, Sir Walter Cope is said to have displayed, “holy relics from a Spanish ship; earthen pitchers and porcelain from China; a Madonna made of feathers; a back-scratcher; a Javanese costume, Arabian coats; the horn and tail of a rhinoceros; the baubles and bells of Henry VIII's fool; and a Turkish emperor's golden seal.” The collections demonstrated manmade wonders and the diversity of God’s creations as well as a fascination with new scientific approaches to the study of natural phenomena. Each collection’s commitment to miscellany dependended on the idiosyncratic interests of the collector.

So it was again important to bring people back to the ways of geometry working in spaces, that although seemingly detached from our reality is the underlying basis of the physics involved.

Mine is a layman's perspective so I cannot say for certain that all I write here will be of value. It is up to you whether you think it important or not.

Figure 2. Clebsch's Diagonal Surface: Wonderful.

Sylvester's models lay hidden away for a long time, but recently the Mathematical Institute received a donation to rescue some of them. Four of these were carefully restored by Catherine Kimber of the Ashmolean Museum and now sit in an illuminated glass cabinet in the Institute Common Room.

The Museum when thought in context of "Platonic solids" was some what of a contention when I showed it's location in early historical context as a artifact.

Shown here are the models in the mathematical wunderkammer located in the Department of Mathematics at the University of Arizona. Like those in most modern mathematics departments, the collection is a combination of locally-made student and faculty projects together with a variety of commercially produced models. Sadly, a century since their Golden Age, many of the models are in disrepair and much of their documentation has been lost. However, some recent detective work, with the help of the Smithsonian Institution in Washington, has helped the department identify models by the American educators W. W. Ross and R. P. Baker in the collection.

Also see here for further thoughts on this

I thought it important to quickly post this so that people understand that "a glass case" can hold many things for inspection, but in this case ,I was referring to the geometrical forms. If any of these form were to show symmetry in action which of these would do so? Sylvestor surfaces?

An attempt in Latex to map these functions from a layman's perspective. What use if I cannot understand the mathematical language as it is written, yet, I can see "acrobatically" the way geometry works in space?

Visual Abstraction to Equations

Sylvester's models lay hidden away for a long time, but recently the Mathematical Institute received a donation to rescue some of them. Four of these were carefully restored by Catherine Kimber of the Ashmolean Museum and now sit in an illuminated glass cabinet in the Institute Common Room.

Some of you might have noticed the reference to the Ashmolean Museum?


Photo by Graham Challifour. Reproduced from Critchlow, 1979, p. 132.

It seems only the good scientist John Baez had epitomes the construction of the Platonic solids? A revision then, of the "time line of history" and the correction he himself had to make? Let's not be to arrogant to know that once we understand more and look at "the anomalies" it forces us to revise our assessments.

The Art form

I relayed this image and quote below on Clifford's site to encourage the thinking of young people into an art form that is truly amazing to me. Yes I get excited about it after having learnt of Gauss and Reimann's exceptional abilities to move into the non euclidean world.

Some think me a crackpot here? If you did not follow the history then how would you know to also include the "physics of approach," as well? Also, some might ask what use "this ability to see the visual abstraction" and I think this art form is in a way destined, to what was kept in glass cabinets and such, even while the glass cabinet in analogy is held in the brain/space of them) who have developed such artistic abilities.

It's as if you move past the layers of the evolution of the human being(brain casings) and it evolution and the field that surrounds them. Having accomplished the intellect( your equations and such), has now moved into the world of imagery. Closet to this is the emotive field which circumvents our perspective on the greater potential of the world in the amazing thought forms of imagery. This move outward, varies for each of us from time to time. Some who are focused in which ever area can move beyond them. This paragraph just written is what would be considered crackpot(I dislike that word)because of the long years of research I had gone through to arrive at this point.

Of course, those views above are different.

Mapping



Is it illusionary or delusional, and having looked at the Clebsch's Diagonal Surface below, how is it that "abstraction" written?

The enthusiasm that characterized such collections was captured by Francis Bacon [1, p. 247], who ironically advised "learned gentlemen" of the era to assemble within "a small compass a model of the universal made private", building

... a goodly, huge cabinet, wherein whatsoever the hand of man by exquisite art or engine has made rare in stuff, form or motion; whatsoever singularity, chance, and the shuffle of things hath produced; whatsoever Nature has wrought in things that want life and may be kept; shall be sorted and included.

There is no doubt that the long road to understanding science is the prerequisite to mapping the images from an equation's signs and symbols. While not sitting in the classroom of the teachers it was necessary to try and move into the fifth dimensional referencing of our computer screen to see what is being extolled here not just in image development, but of what the physics is doing in relation.

In 1849 already, the British mathematicians Salmon ([Sal49]) and Cayley ([Cay49]) published the results of their correspondence on the number of straight lines on a smooth cubic surface. In a letter, Cayley had told Salmon, that their could only exist a finite number - and Salmon answered, that the number should be exactly 27

So of course to be the historical journey was established like most things, Mandelstam current and what is happening there as an interlude, as well as helping to establish some understanding of the abstractions that had been developed.



But yes, before moving to current day imagery and abstraction, I had to understand how these developments were being tackled in today's theoretical sciences.

Megalithic carved stone balls from Scotland

With the discovery of sound waves in the CMB, we have entered a new era of precision cosmology in which we can begin to talk with certainty about the origin of structure and the content of matter and energy in the universe-Wayne Hu

I mean what influence might we have gained from looking at such ancient pieces?

The balls were located at the Ashmolean Museum, Oxford, UK at one time, so I do not know if they are still there.


Photo by Graham Challifour. Reproduced from Critchlow, 1979, p. 132.

Sylvester's Surfaces

Figure 2. Clebsch's Diagonal Surface: Wonderful.

We are told that "mathematics is that study which knows nothing of observation..." I think no statement could have been more opposite to the undoubted facts of the case; that mathematical analysis is constantly invoking the aid of new principles, new ideas and new methods, not capable of being defined by any form of words, but springing direct from the inherent powers and activity of the human mind, and from continually renewed introspection of that inner world of thought of which the phenomena are as varied and require as close attention to discern as those of the outer physical world, ...that it is unceasingly calling forth the faculties of observation and comparison, that one of its principal weapons is induction, that it has frequent recourse to experimental trial and verification, and that it affords a boundless scope for the exercise of the highest efforts of imagination and invention. ...Were it not unbecoming to dilate on one's personal experience, I could tell a story of almost romantic interest about my own latest researches in a field where Geometry, Algebra, and the Theory of Numbers melt in a surprising manner into one another.

I had been looking for the link written by Nigel Hitchin, as this work was important to me, in how Dynkin drawings were demonstrated. Although I have yet to study these, I wanted to find this link and infomration about James Sylvester, because of the way we might see in higher dimensional worlds.His model seem important to me from this perspective.

Sylvester's models lay hidden away for a long time, but recently the Mathematical Institute received a donation to rescue some of them. Four of these were carefully restored by Catherine Kimber of the Ashmolean Museum and now sit in an illuminated glass cabinet in the Institute Common Room.

The reason for this post is th ework dfirst demonstrated by Lubos Motl and th etalk he linked by Nigel Hitchin. The B-field, which seems to no longer exist, or maybe I am not seeing it in his posts archived?

In 1849 already, the British mathematicians Salmon ([Sal49]) and Cayley ([Cay49]) published the results of their correspondence on the number of straight lines on a smooth cubic surface. In a letter, Cayley had told Salmon, that their could only exist a finite number - and Salmon answered, that the number should be exactly 27

.

There had to be a simplification of this process, so in gathering information I hope to complete this, and gain in understanding.

James Joseph Sylvester (September 3, 1814 - March 15, 1897) was an English mathematician and lawyer.

Now as to the reason why this is important comes from the context of geometrical forms, that has intrigued me and held mathematicians minds. Sometimes it is not just the model that is being spoken too, but something about the natural world that needs some way in which to be explained. Again, I have no teachers, so I hope to lead into this in a most appropriate way, and hopefully the likes of those involved, in matrix beginnings, have followed the same process?

The 'Cubics With Double Points' Gallery




f(x,y,z) = x2+y2+z2-42 = 0,

i.e. the set of all complex x,y,z satisfying the equation. What happends at the complex point (x,i*y,i*z) for some real (x,y,z)?

f(i*x,y,z) = x2+(i*y)2+(i*z)2-42
= x2+i2*y2+i2*z2-42
= x2-y2-z2-42.

Has it become possible that you have become lost in this complex scenario? Well what keeps me sane is the fact that this issue(complex surfaces) needs to be sought after in terms of real images in the natural world. Now, I had said, the B-field, and what this is, is the reference to the magnetic field. How we would look at it in it's diverse lines? Since on the surface, in a flat world, this would be very hard to make sense of, when moved to the three coordinates, these have now become six?

fancier way of saying that is that in general, it's okay to model the space around us using the Euclidean metric. But the Euclidean model stops working when gravity becomes strong, as we'll see later.

Now what has happened below, is that what happens in quark to quark distances, somehow in my mind is translated to the values I see, as if in the metric world and moved to recognition of Gaussian curves and such, to decribe this unique perspective of the dynamics of Riemann, lead through geometrical comprehension ad expression. No less then the joining of gravity to Maxwells world.

Like the magnetic field we know, the lines of force represent a dynamcial image, and so too, how we might see this higher dimensional world. Again I don't remember how I got here, so I am trying hard to pave this road to comprehension.



"Of course, if this third dimension were infinite in size, as it is in our world, then the flatlanders would see a 1/r2 force law between the charges rather than the 1/r law that they would predict for electromagnetism confined to a plane. If, on the other hand, the extra third spatial dimension is of finite size, say a circle of radius R, then for distances greater than R the flux lines are unable to spread out any more in the third dimension and the force law tends asymptotically to what a flatlander physicist would expect: 1/r.

However, the initial spreading of the flux lines into the third dimension does have a significant effect: the force appears weaker to a flatlander than is fundamentally the case, just as gravity appears weak to us.

Turning back to gravity, the extra-dimensions model stems from theoretical research into (mem)brane theories, the multidimensional successors to string theories (April 1999 p13). One remarkable property of these models is that they show that it is quite natural and consistent for electromagnetism, the weak force and the inter-quark force to be confined to a brane while gravity acts in a larger number of spatial dimensions."

Now here to again, we are exercising our brane function(I mean brain)in order to move analogies to instill views of the higher dimensional world. The missing energy had to go somewhere and I am looking for it?:) So ideas like "hitting metal sheets with a hammer", or "billiards balls colliding", and more appropriately so, reveal sound as a manifestation of better things to come in our visions?

See:

Unity of Mathematics

The Sound of the Landscape

Ashmolean Museum, Oxford, UK

As you know my name is Plato (The School of Athens by Raphael:)I have lived on for many years now, in the ideas that are presented in the ideas of R Buckminister Fuller, and with the helping hands of dyes, have demonstrated, the basis of these sounds in balloon configuration worth wondering, as simplice's of these higher dimensional realizations.

A Chladni plate consist of a flat sheet of metal, usually circular or square, mounted on a central stalk to a sturdy base. When the plate is oscillating in a particular mode of vibration, the nodes and antinodes set up form a complex but symmetrical pattern over its surface. The positions of these nodes and antinodes can be seen by sprinkling sand upon the plates;

Now you know from the previous post, that I have taken the technical aspects of string theory, and the mathematical formulations, and moved them into a encapsulated state of existance, much as brane theory has done.

I look at this point(3 sphere derivation from euclid point line plane), on the brane and I wonder indeed, how 1R radius of this point becomes a circle. Indeed, we find this "idea" leaving the brane into a bulk manifestation of information, that we little specks on earth look for in signs of, through our large interferometers called LIGO's



John Baez:

Ever make a cube out of paper? You draw six square on the paper in a cross-shaped pattern, cut the whole thing out, and then fold it up.... To do this, we take advantage of the fact that the interior angles of 3 squares don't quite add up to 360 degrees: they only add up to 270 degrees. So if we try to tile the plane with squares in such a way that only 3 meet at each vertex, the pattern naturally "curls up" into the 3rd dimension - and becomes a cube!

The same idea applies to all the other Platonic solids. And we can understand the 4d regular polytopes in the same way!

The Hills of M Theory

The hills are alive with the sound of music
With songs they have sung for a thousand years.
The hills fill my heart with the sound of music
My heart wants to sing every song it hears....

It's a wonder indeed that we could talk about the spacetime fabric and the higher dimensions that settle themselves into cohesive structures(my solids) for our satisfaction? What nodal points, do we have to wonder about when a string vibrates, and one does not have to wonder to much about the measure of the Q<->Q distance, as something more then the metric field resonates for us?

This higher dimensional value seen in this distance would speak loudly to its possiblites of shape, but it is not easily accepted that we find lattice structures could have ever settled themselves into mass configurations of my solids.



Lenny Susskind must be very pround of this landscape interpretation, as it is shown in the picture above. But the question is, if the spacetime fabric is the place where all these higher dimensions will reveal themselves, then what structure would have been defined in this expression from it's orignation, to what we see today?

Alas, I am taken to the principles of," Spacetime in String Theory," by Gary T. Horowitz

If one quantizes a free relativistic (super) string in flat spacetimeone finds a infinite tower of modes of increasing mass. Let us assume the string is closed,i.e., topologically a circle