PHYSICAL SCIENCES

Mathematics


MATHEMATICS DEFINED

Mathematics(1) is the study of quantity, structure, space, and change. Mathematicians seek out patterns,[2][3] formulate new conjectures, and establish truth by rigorous deduction from appropriately chosen axioms and definitions.[4)  Through the use of abstraction and logical reasoning, mathematics evolved from counting, calculation, measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Mathematics continued to develop, for example in China in 300 BC, in India in AD 100, and in the Muslim world in AD 800, until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that continues to the present day.[5]

Benjamin Peirce (1809–1880) called mathematics "the science that draws necessary conclusions". Benjamin Peirce(1809-1880) was an American mathematician who taught at Harvard University for approximately 50 years. He made contributions to celestial mechanics, statistics, number theory, algebra, and the philosophy of mathematics. He was the son of Benjamin Peirce (1778–1831), later librarian of Harvard, and Lydia Ropes Nichols Peirce (1781–1868). After graduating from Harvard, he remained as a tutor (1829), and was subsequently appointed professor of mathematics in 1831. He added astronomy to his portfolio in 1842, & remained as Harvard professor until his death. In addition, he was instrumental in the development of Harvard's science curriculum, served as the college librarian, and was director of the U.S. Coast Survey from 1867 to 1874.

David Hilbert said of mathematics: "We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise."  David Hilbert (German:(1862-1943) was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th & early 20th centuries. Hilbert discovered & developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of geometry. He also formulated the theory of Hilbert spaces, one of the foundations of functional analysis.

Albert Einstein (1879–1955) stated that "as far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." Mathematics is essential in many fields, including natural science, engineering, medicine, finance & the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics & game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure & applied mathematics, and practical applications for what began as pure mathematics are often discovered. More?https://en.wikipedia.org/wiki/Mathematics


MY STUDY AND TEACHING OF MATHEMATICS

By 1971, after 4 years of teaching and 13 years study of mathematics, I was 27. Other subjects occupied my attention until my retirement from FT & PT work in 1999. This webpage has become the focus for my continued interest from the age of 55 to 71.  
Secondary school mathematics consists of mathematics typically taught in middle schools (a.k.a., junior high schools) and high schools (or secondary schools) — that is, roughly ages 11–17. It is preceded by primary school mathematics and followed by university level mathematics.

Note that the usage of the term secondary education is not consistent around the world, but it is convenient to separate the mathematics typically studied before adolescence from that studied during and after this period, since the latter is generally of a qualitatively different nature (i.e., more abstract). University mathematics, on the other hand, typically has a breadth (and depth) lacking in the high school curriculum, and so deserves its own classification. Major areas of mathematics that I studied at the secondary level included: Pre-algebra, Elementary algebra, intermediate algebra, geometry, and trigonometry. Precalculus, calculus, and elementary probability and statistics (usually non-calculus based) have been added since I studied and taught maths from the 1950s to the early 70s.  More: http://math.wikia.com/wiki/Secondary_school_mathematics

In contemporary education, mathematics education is the practice of teaching & learning mathematics, as well as associated scholarly research. Researchers in mathematics education are primarily concerned with the tools, methods & approaches that facilitate the practice or the study of practice; however, mathematics education research, known on the continent of Europe as the didactics or pedagogy of mathematics, has developed into an extensive field of study, with its own concepts, theories, methods, national and international organisations, conferences and literature. This article describes some of the history, influences and recent controversies. More:https://en.wikipedia.org/wiki/Mathematics_education

Elementary mathematics consists of mathematics topics frequently taught at the primary or secondary school levels. The most basic topics in elementary mathematics are arithmetic & geometry.  Beginning in the last decades of the 20th century, there has been an increased emphasis on problem solving. Elementary mathematics is used in everyday life in such activities as making change, cooking, buying and selling stock, and gambling. It is also an essential first step on the path to understanding science. In secondary school, the main topics in elementary mathematics are algebra and trigonometry. Calculus, even though it is often taught to advanced secondary school students, is usually considered college level mathematics. For more: https://en.wikipedia.org/wiki/Elementary_mathematics


MATHEMATICS AND REALITY

There is debate over whether mathematical objects such as numbers and points exist naturally or are human creations.  Mathematics is used throughout the world as an essential tool in many fields, including natural science, engineering, medicine, & the social sciences. Applied mathematics, the branch of mathematics concerned with application of mathematical knowledge to other fields, inspires and makes use of new mathematical discoveries and sometimes leads to the development of entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind, although practical applications for what began as pure mathematics are often discovered.[8]  Bertrand Russell once said that 'Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture.'

-----------------------------------------FOOTNOTES-----------------------------------

1. Wikipedia
2. L. A. Steen, April 29, 1988. The Science of Patterns Science, 240: 611–616. and summarized at Association for Supervision and Curriculum Development, ascd.org
3. Keith Devlin, Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe, Scientific American Paperback Library, 1996.
4. Philip Bertrand Jourdain(1879-1919), The Nature of Mathematics, 1913.

5. Howard Eves, An Introduction To The History Of Mathematics., 1964.
6.
Charles S. Peirce, The New Elements of Mathematics
7. The quote is Einstein's answer to the question: "how can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality?"
8.
  Ivars Peterson is an award-winning mathematics writer.

JOURNALS

A. Printed Journals with Mathematics and Internet sites have burgeoned with the arrival of the Internet. Over 650 websites of printed journals. They contain tables of contents, abstracts, information about submissions and subscriptions, and in some cases, electronic versions of papers as of 25 September 2008. Many journal links include redirects, as a result of mergers and transfers among publishing corporations and their sites, and can be slow to respond. Go to:http://www.mathontheweb.org/mathweb/mi-journals5.html and https://en.wikipedia.org/wiki/List_of_mathematics_journals

One Example: Annals of Global Analysis and Geometry

Examines global problems of geometry and analysis
Looks at interactions between differential geometry and global analysis and their application to problems of theoretical physics
Contributes to an enlargement of the international exchange of research results in the field
Comprehensive coverage, from complex manifolds and related results from complex analysis and algebraic geometry to Lie transformation groups and harmonic analysis
100% of authors who answered a survey reported that they would definitely publish or probably publish in the journal again

B. This journal examines global problems of geometry and analysis as well as the interactions between these fields and their application to problems of theoretical physics. It contributes to an enlargement of the international exchange of research results in the field. The areas covered in Annals of Global Analysis and Geometry include: global analysis, differential geometry, complex manifolds and related results from complex analysis and algebraic geometry, Lie groups, Lie transformation groups and harmonic analysis, variational calculus, applications of differential geometry and global analysis to problems of theoretical physics. Related subjects » Algebra - Analysis - Business, Economics & Finance - Geometry & Topology -Theoretical, Mathematical & Computational Physics

ABSTRACTED/INDEXED IN

Science Citation Index Expanded (SciSearch), Journal Citation Reports/Science Edition, SCOPUS, Zentralblatt Math, Google Scholar, EBSCO, CSA, ProQuest, Academic OneFile, Academic Search, CSA Environmental Sciences, Gale, Mathematical Reviews, OCLC, Referativnyi Zhurnal (VINITI), SCImago, STMA-Z, Summon by ProQuest http://www.springer.com/mathematics/analysis/journal/10455/PSE

MATHEMATICS: A SCIENCE OR AN ART?

Many philosophers believe that mathematics is not experimentally falsifiable, and thus not a science according to the definition of Karl Popper. However, in the 1930s Gödel's incompleteness theorems convinced many mathematicians that mathematics cannot be reduced to logic alone, and Karl Popper concluded that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently." Other thinkers, notably Imre Lakatos, have applied a version of falsificationism to mathematics itself.

An alternative view is that certain scientific fields (such as theoretical physics) are mathematics with axioms that are intended to correspond to reality. The theoretical physicist J.M. Ziman proposed that science is public knowledge, and thus includes mathematics. Mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics.

The opinions of mathematicians on this matter are varied. Many mathematicians feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts; others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). It is common to see universities divided into sections that include a division of Science and Mathematics, indicating that the fields are seen as being allied but that they do not coincide. In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics.

 mathematical logic and set theory


In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics; set theory is the branch of mathematics that studies sets or collections of objects. Category theory, which deals in an abstract way with mathematical structures and relationships between them, is still in development. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930.[50] Some disagreement about the foundations of mathematics continues to the present day. The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theoryand the Brouwer–Hilbert controversy.

Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. As such, it is home to Gödel's incompleteness theorems which (informally) imply that any effective formal system that contains basic arithmetic, if sound (meaning that all theorems that can be proven are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system). Whatever finite collection of number-theoretical axioms is taken as a foundation, Gödel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Therefore, no formal system is a complete axiomatization of full number theory. Modern logic is divided into recursion theory, model theory, and proof theory, & is closely linked to theoretical computer science,[citation needed] as well as to category theory.

Theoretical computer science includes computability theory, computational complexity theory, & information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most well-known model – the Turing machine. Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with the rapid advancement of computer hardware. A famous problem is the "P = NP?" problem, one of the Millennium Prize Problems.[51] Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as compression and entropy. For more: https://en.wikipedia.org/wiki/Mathematics#Inspiration.2C_pure_and_applied_mathematics.2C_and_aesthetics


U-TUBE VIDEOS

There are now dozens of maths videos. An exploration of mathematics, including where it comes from and why it explains the physical world; and whether it’s a human invention or a hidden language of the universe. Aired on 15 Apr 2015, on PBS network's NOVA program series, this is a collector's item! Start here:https://www.youtube.com/channel/UCT4-UAcRfvBtO76gX2vexpA and https://www.youtube.com/watch?v=mJbChZrXDJE


HOUSEHOLD FINANCE and MATHEMATICS

Back in the 90s, a young teacher known for his knowledge in Maths, developed a comprehensive Maths program based on the curriculum of the time. He personally had great success with the program, which he dubbed “Household Maths”, and with his support several teachers he worked with also followed the program enthusiastically. With little co-ordination or entrepreneurial skills, he even managed to sell a few copies of it.  He was drawn away from using the program for many years, even though he still had a strong belief in its purpose and results. Now with a renewed push for purposeful Maths, he wanted to bring “Household Maths” back again. The basic premise of the program and the majority of its content and curriculum base is still sound, 20 years after he first created it.  

Back in the 1990s I was not teaching maths and so, what follows is the original introduction and program summary(with some comments about how he would integrate new technologies).  During my years involved in education, whether as a student or teacher, many teachers have made Maths such a boring subject. In turn, their classes have responded by being bored. Sheets and sheets of repetitive sums have done nothing other than keep the bright child occupied and the struggler frustrated. I have been guilty of this myself. The struggler learnt to hate Maths and the bright child just did the sums because they were easy.

I have always looked for programs that made Maths interesting for the children when I was a maths teacher.  Many books and programs have been released under the heading of “Real Maths”. Too many, though, are just a book of activities that are not related to each other and could be dealt with in a single session, are part of a program that still had too many worksheets filled with monotonous equations or aren’t that real to the children, anyway. Finally he decided to do something himself. He thought to himself: When is Maths most useful and meaningful? The simple answer was in daily life at home. Maths is all around us in our house. Paying bills, going shopping, looking for bargains, building a house, developing the garden, planning holidays –  all of these tasks are Maths at work.

He wanted more than a book of activities to keep the children busy once or twice a week, though. He wanted his entire Maths program for the year to be a rewarding, interesting and entertaining learning experience based on Maths at home. Children love pretending to be adults. So this program was going to treat them like adults. The key to it all was always going to be making it interesting and fun. When faced with a policy that says Maths must be taught for one hour a day, so many teachers decide to make a worksheet of equations with as many sums as they can fit on it to keep the children working for the hour. Of course what happens is that the bright children barely have to think and finish within twenty minutes while the strugglers get stuck on the first sum for twenty minutes and just know they’ll never finish in time. This only builds up their frustration and hatred towards maths while the bright sparks just confirm what they already know – they’re good calculators. But can they think? Have they been taught to think? More:http://mgleeson.edublogs.org/2012/05/26/662/

Have you ever dreaded going to your letterbox because you don't have the money to pay the bills you'll receive? Are you struggling to reduce your credit card debt? Are you sick of never being able to build up your savings enough to go on that trip you dream of or buy the home you really want. This section of the website aims to get you in control of your money to help you achieve your goals.

Budgeting: wife in charge(WIC)
Saving:WIC
Banking: $6000 in the bank and in stock.
Managing debts: WIC
Insurance: WIC
Donating: $1200
Income tax: WIC

I could write in detail about the financial aspects of my life; I will do so at a future time to cover the period: 1956 to 2016. For now start here: http://www.curriculumsupport.education.nsw.gov.au/secondary/mathematics/years11_12/teaching/fs4.htm   As I head through my 70s, I will summarize my financial state. In those 60 years my finances changed significantly withe the years.

TWO BBC DOCOS ON MATHEMATICS

A. THE CODE


A.1 The Code is a mathematics-based documentary for BBC Two presented by Marcus du Sautoy, beginning on 27 July 2011. Marcus Peter Francis du Sautoy OBE (London, 1965) is the Simonyi Professor for the Public Understanding of Science and a Professor of Mathematics at the University of Oxford. Formerly a Fellow of All Souls College, and Wadham College, he is now a Fellow of New College. He is currently an Engineering and Physical Sciences Research Council Senior Media Fellow. He was previously a Royal Society University Research Fellow. His academic work concerns mainly group theory and number theory. In October 2008, he was appointed to the Simonyi Professorship for the Public Understanding of Science, succeeding the inaugural holder Richard Dawkins.

I saw this 3-part series, The Code, in Australia in the summer, in February, of 2012.(1)  Each episode covered a different branch of mathematics. Du Sautoy reveals a hidden numerical code that underpins all nature. A code that has the power to explain everything, from the numbers and shapes we see all around us to the rules that govern our own lives. In the first episode, Du Sautoy revealed how significant numbers appear throughout the natural world. They're part of a hidden mathematical universe that contains the rules that govern everything on our planet and beyond. (1) SBSONE TV, 12/2 to 26/2/2-12, 8:35-9:40 p.m. For more details on Du Sautoy go to: http://en.wikipedia.org/wiki/Marcus_du_Sautoy  For more on this TV series go to:http://en.wikipedia.org/wiki/The_Code_%28UK_TV_series%29

A.2 For a person like myself who only studied mathetics from 1949 to 1963, as part of my primary and secondary school curriculum in Ontario, from the age of 5 to 18, and for the most part more than 50 years ago, the study of mathematics has been far out on the periphery of my life. Except for its use, of course, in its practical functions of daily life, a use whose importance is taken-for-granted, I did not teach or study mathematics from 1972 to 2002.  I took a course at teachers' college in 1966/7 on teaching maths to primary children. I also taught primary school maths from 1968 to 1971. After 2002, I occasionally came to study this field as it arose somewhat serendipitously during the years of my retirment from jobs. The above TV series, entitled The Code, helped to bring this important, and for most people essentially practical, subject into my world of fascination and mystery, the world of the hidden mathematical patterns in nature, the geometric rules of the universe and the laws of shape. My understanding of the field of maths was immensely enriched by this series.

B. TO INFINITY AND BEYOND


B.1 To Infinity and Beyond is a series exploring topical scientific issues. By my third year, in 1947 while living in RR#1 Burlingtonm I was able to count; so I am told by students of human development in psychology. Once a person knows how to count, it seems as if there would be nothing to stop them counting forever. Of course, such a person would need more familiarity with arithmetic; let's say, then, by the last year of childhood, the age of 12, such a person could theoretically keep counting forever. But, while infinity might seem like an perfectly innocent idea, keep counting and you enter a paradoxical world where nothing is as it seems.

Mathematicians have discovered there are infinitely many infinities, each one infinitely bigger than the last. And if the universe goes on forever, the consequences are even more bizarre. In an infinite universe, there are infinitely many copies of the Earth and infinitely many copies of you. Older than time, bigger than the universe and stranger than fiction. 
Mathemathics states nothing is beyond infinity, for infinity is endless. It does, however, depend on your own personal definition of infinity. Here are 3 different definitions of infinity: 1. Something that truly goes on forever and cannot be overpowered or outlasted by any force imaginable in this universe or any other; 2. Too large or powerfull to ever be understood fully by the one using the term; for example, the universe goes on till infinity, nobody can live that long and thus to humans using the term like this, the universe will seem like infinity even though it has a limit beyond reach; and 3. A number so high that it cannot be measured; for example, there is an infinity of molecules in the universe. There aren't, but the number is too high to measure.

B.2 Most people use the 1st definition, but some use the 2nd term, especially religious people. They say God is infinite, but they just don't understand him. I'm guessing God understands Himself and thus He is the 2nd definition of infinite. Nothing in all of reality can come anywhere near the first definition of infinity. This doco presents viewers with what I'm sure for many are new terms like: (i) google, and googolplex; the latter term is the name given to the large number 10 to the 10th to the 100th, or 10 to the google; and (ii)
Graham's number, named after Ronald Graham, which is a large number that is an upper bound on the solution to a certain problem in a mathematical theory known as Ramsey theory. Graham's number gained a degree of popular attention when Martin Gardner described it in the "Mathematical Games" section of Scientific American in November 1977. He wrote that, "In an unpublished proof, Graham has recently established ... a bound so vast that it holds the record for the largest number ever used in a serious mathematical proof." The 1980 Guinness Book of World Records repeated Gardner's claim, adding to the popular interest in this number. Go to this link for more:http://en.wikipedia.org/wiki/Graham%27s_number This doco presents the story of infinity and for more on this topic go to:http://docuwiki.net/index.php?title=To_Infinity_and_Beyond

MORE ON DEFINITIONS

Mathematics from Greek máthēma, “knowledge, study, learning” is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics.  is a conceptual system possessing internal necessity that can only be so  by no means otherwise." Mathematics is essential in many fields, including natural science, engineering, medicine, finance & the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics and game theory. Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered.

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement,architecture and later astronomy; today, all sciences suggest problems studied by mathematicians, & many problems arise within mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning & physical insight, & today's string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics.

Some mathematics is relevant only in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, & joins the general stock of mathematical concepts. A distinction is often made between pure mathematics and applied mathematics. However pure mathematics topics often turn out to have applications, e.g. number theory in cryptography. This remarkable fact, that even the "purest" mathematics often turns out to have practical applications, is what Eugene Wigner has called "the unreasonable effectiveness of mathematics". As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages. Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.

For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the elegance of mathematics, its intrinsicaesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple & elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. G.H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic. Mathematicians often strive to find proofs that are particularly elegant, proofs from "The Book" of God according to Paul Erdős. The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions.

Notation, language, and rigor

Most of the mathematical notation in use today was not invented until the 16th century. Before that, mathematics was written out in words, limiting mathematical discovery. Euler (1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. It is compressed: a few symbols contain a great deal of information. Like musical notation, modern mathematical notation has a strict syntax and encodes information that would be difficult to write in any other way.

Mathematical language can be difficult to understand for beginners. Common words such as or and only have more precise meanings than in everyday speech. Moreover, words such as open and field have specialized mathematical meanings. Technical terms such as homeomorphism  and integrable have precise meanings in mathematics. Additionally, shorthand phrases such as iff for "if and only if" belong to mathematical jargon. There is a reason for special notation & technical vocabulary: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".

Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject. The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verify, such proofs may not be sufficiently rigorous.

Axioms in traditional thought were "self-evident truths", but that conception is problematic. At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas; and so a final axiomatization of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory. For more: https://en.wikipedia.org/wiki/Mathematics#Inspiration.2C_pure_and_applied_mathematics.2C_and_aesthetics

ENGLISH MATHEMATICS WRITERS

The following is a review in the London Review of Books(8/10/15). The review is by Adam Smyth of two books: (i) ‘Brief Lives’ with ‘An Apparatus for the Lives of Our English Mathematical Writers’ by John Aubrey, edited by Kate Bennett(Chatto, 520 pages); & (ii) John Aubrey: My Own Life by Ruth Scurr(Oxford, 1968). Adam Smyth has edited A History of English Autobiography which will be published by CUP early in 2016. He co-hosts the literary podcast litbits.co.uk. Smyth writes:

"A friend who teaches in New York told me that the historian Peter Lake told him that J.G.A. Pocock told him that Conrad Russell told him that Bertrand Russell told him that Lord John Russell told him that his father, the sixth Duke of Bedford, told him that he had heard William Pitt the Younger speak in Parliament during the Napoleonic Wars, and that Pitt had this curious way of talking, a particular mannerism that the sixth Duke of Bedford had imitated to Lord John Russell who imitated it to Bertrand Russell who imitated it to Conrad Russell who imitated it to J.G.A. Pocock, who could not imitate it to Peter Lake and so my friend never heard it. But all the way down to Pocock was a chain of people who in some sense had actually heard William Pitt the Younger’s voice." For another review go to: http://www.tandfonline.com/doi/abs/10.1080/0268117X.2015.1060672?journalCode=rsev20

MATHEMATICS OF LOVE

Part 1:

From algorithms used on dating sites to a range of equations, formulas and patterns: mathematics could help you find your perfect match. Phillip Adams, Austrralian writer and intellectual, speaks with Hannah Fry, Lecturer in Mathematics of Cities at the University College London.  To listen to this interview go to: http://www.abc.net.au/radionational/programs/latenightlive/the-mathematics-of-love/6089506 To listen to a talk by Hannah Fry go to: https://www.ted.com/talks/hannah_fry_the_mathematics_of_love  Maria Popova has written a review of Hannah Fry's new book at the internet site: brain pickings. Popova's article is entitled: "What Mathematics Reveals About the Secret of Lasting Relationships & the Myth of Compromise."  Popova begins with some questions: "Why 37% is the magic number, what alien civilizations have to do with your soul mate, and how to master the “negativity threshold” ideal for Happily Ever After."

Popova continues:  "In his sublime definition of love, playwright Tom Stoppard painted the grand achievement of our emotional lives as “knowledge of each other, not of the flesh but through the flesh, knowledge of self, the real him, the real her, in extremis, the mask slipped from the face.” But only in fairy tales and Hollywood movies does the mask slip off to reveal a perfect other. So how do we learn to discern between a love that is imperfect, as all meaningful real relationships are, & one that is insufficient, the price of which is repeated disappointment & inevitable heartbreak? Making this distinction is one of the greatest and most difficult arts of the human experience — and, it turns out, it can be greatly enhanced with a little bit of science."

Part 2:

"That’s what mathematician Hannah Fry suggests in The Mathematics of Love: Patterns, Proofs, and the Search for the Ultimate Equation (public library) — a slim but potent volume from TED Books, featuring gorgeous illustrations by German artist Christine Rösch. From the odds of finding your soul mate to how game theory reveals the best strategy for picking up a stranger in a bar to the equation that explains the conversation patterns of lasting relationships, Fry combines a humanist’s sensitivity to this universal longing with a scientist’s rigor to shed light, with neither sap nor cynicism, on the complex dynamics of romance and the besotting beauty of math itself.

One of the criticisms of Hannah Fry and her work has come in from a writer who does mathematics for a living. He has written that "the applications of mathematics like thos of Hannah Fry are frivolous. This is because mathematical results are very narrow and always require very stringent assumptions. These assumptions are not realised in any of the "love" situations described by Fry. Of course, in a classroom it is fun to make a passing comparison between a daily life situation and a real physics or maths problem in order to make the latter easier to understand.  I have done so on occasion in my teaching, but over time I have learned that this can be misleading to students. Mathematics in Western Culture by Morris Kline is a well-written, non-technical book, that friends with various backgrounds have greatly appreciated. There are others. As usual, if you say "sex" or "love" then you get a lot of attention. The attention Fry gets is clearly undeserved in this case."


MORRIS KLINE: a famous teacher

Morris Kline(1908-1992) was a Professor of Mathematics, a writer on the history, philosophy, and teaching of mathematics, and also a popularizer of mathematical subjects. Kline grew up in Brooklyn & in Jamaica. After graduating from Boys High School in Brooklyn, he studied mathematics at New York University, earning a bachelor's degree in 1930, a master's degree in 1932, and a doctorate in 1936. He continued at NYU as an instructor until 1942. During World War II, Kline was posted to the Signal Corps (United States Army) stationed at Belmar, New Jersey. Designated a physicist, he worked in the engineering lab where radar was developed. After the war he continued investigating electromagnetism, and from 1946 to 1966 was director of the division for electromagnetic research at the Courant Institute of Mathematical Sciences.

In 1977 Kline turned to undergraduate university education; he took on the academic mathematics establishment with his Why the Professor Can’t Teach: the dilemma of university education. Kline argues that requiring professors  to produce original mathematics and do research distracts them too much from the broad knowledge that is necessary if they are to be successful teachers.  He lauds mathematical scholarship as expressed in the form of expository writing or reviews of the original work of others. When Kline was 48, in 1956, he called the mathematics community to action. But once that community was  mobilized he turned into a critic. Skilled expositor that he was, editors frequently felt his expressions were best tempered with rebuttal. Kline recalls E. H. Moore’s recommendation to combine science & mathematics at the high school level in his Why Johnny Can’t Add (p. 147).  Closer reading of Kline's life and work shows that he saw mathematics as a "part of man’s efforts to understand and master his world". He also saw the role of the mathematician within a broad spectrum of sciences. For more on Klein go to: http://en.wikipedia.org/wiki/Morris_Kline

NUMERACY AND SUSTAINABILITY

Part 1:

"Numeracy and sustainability" is an article by John Cairns, Jr. of the Department of Biology, in the Virginia Polytechnic Institute and State University in Blacksburg, Virginia. The article appears in the online journal Ethics in Science and Environmental Politics in 2003. The article begins as follows:  "Sustainable use of the planet is based on the assumption that humankind has the right to alter the planet so that human life can inhabit Earth indefinitely. In doing so, environmental conditions of the planet may be shifted so that they are optimal for one species, but not necessarily for all species or even a majority of species now alive. Clearly, humankind does not value all life equally. Sustainability is based on the assumption that acceptable environmental conditions can be maintained. The assumption has not been validated, nor is it likely to be for centuries, if ever."

It continues: 

"Numeracy is the ability to use or understand numerical techniques of mathematics (a useful introduction is available in Bartlett 1994). However, the important decisions humankind makes should not be based on numbers, even economic numbers, but rather on eco- and sustainability ethics, which provide a values framework that indicates how the numbers should be used & interpreted. The emphasis on severely limited numbers is a major weakness of the United Nations Commission on Environment and Development (1987) report, which focused on development (commonly regarded as synonymous with growth). Development is just one metric valued by one species. Sustainability involves a variety of metrics for a complex, multivariate living system called the interdependent web of life."

Part 2:

Sustainability is the study of patterns involving all forms of life and the conditions necessary for them to flourish as a community. One can place an infinite value on human life and on other life forms as well. One can place an infinite value on one’s own life but be willing to sacrifice it to protect one’s offspring. This example illustrates that there can be more than one value for infinity. Sustainability involves a similar situation—humankind must place infinite value on personal life, on the lives of its future generations, and on those of other life forms. Balancing these seemingly incompatible values will never be fully achieved since life consists of dynamic and stochastic events that will frequently alter the factors affecting this precarious balance. Sustainability policies may be developed for individual components; for example, agriculture, energy, transportation, communities, and fisheries, but achieving sustainability will not be possible unless the policies are integrated into a master policy and plan that does not adversely affect other components. Numeracy will be helpful in establishing component balancing. For more go to: http://www.int-res.com/articles/esep/2003/E40.pdf


THE BEAUTY OF MATHEMATICS

For those who have learned something of higher mathematics, nothing could be more natural than to use the word “beautiful” in connection with it. Mathematical beauty, like the beauty of, say, a late Beethoven quartet, arises from a combination of strangeness and inevitability. Simply defined abstractions disclose hidden quirks and complexities. Seemingly unrelated structures turn out to have mysterious correspondences. Uncanny patterns emerge, and they remain uncanny even after being underwritten by the rigor of logic.  So powerful are these aesthetic impressions that one great mathematician, G.H. Hardy, declared that beauty, not usefulness, is the true justification for mathematics. To Hardy, mathematics was first and foremost a creative art. “The mathematician’s patterns, like the painter’s or the poet’s, must be beautiful,” he wrote in his classic 1940 book, A Mathematician’s Apology. “Beauty is the first test: there is no permanent place in the world for ugly mathematics.”

And what is the appropriate reaction when one is confronted by mathematical beauty? Pleasure, certainly; awe, perhaps. Thomas Jefferson wrote in his seventy-sixth year that contemplating the truths of mathematics helped him to “beguile the wearisomeness of declining life.” To Bertrand Russell—who rather melodramatically claimed, in his autobiography, that it was his desire to know more of mathematics that kept him from committing suicide—the beauty of mathematics was “cold and austere, like that of sculpture…sublimely pure, and capable of a stern perfection.” For others, mathematical beauty may evoke a distinctly warmer sensation. They might take their cue from Plato’s Symposium. In that dialogue, Socrates tells the guests assembled at a banquet how a priestess named Diotima initiated him into the mysteries of Eros—the Greek name for desire in all its forms. For more on this theme and a review of the 300 page 2013 book Love and Math: The Heart of Hidden Reality by Edward Frenkel go to:  http://www.nybooks.com/articles/archives/2013/dec/05/mathematical-romance/

MANDELBROT and THE FRACTAL

Benoit Mandelbrot, the brilliant Polish-French-American mathematician who died in 2010, had a poet’s taste for complexity and strangeness. His genius for noticing deep links among far-flung phenomena led him to create a new branch of geometry, one that has deepened our understanding of both natural forms and patterns of human behavior. The key to it is a simple yet elusive idea, that of self-similarity.

To see what self-similarity means, consider a homely example: the cauliflower. Take a head of this vegetable and observe its form—the way it is composed of florets. Pull off one of those florets. What does it look like? It looks like a little head of cauliflower, with its own subflorets. Now pull off one of those subflorets. What does that look like? A still tinier cauliflower. If you continue this process—and you may soon need a magnifying glass—you’ll find that the smaller and smaller pieces all resemble the head you started with. The cauliflower is thus said to be self-similar. Each of its parts echoes the whole.

Other self-similar phenomena, each with its distinctive form, include clouds, coastlines, bolts of lightning, clusters of galaxies, the network of blood vessels in our bodies, and, quite possibly, the pattern of ups and downs in financial markets. Only in the last few decades has a mathematics of roughness emerged, one that can get a grip on self-similarity and kindred matters like turbulence, noise, clustering, and chaos. And Mandelbrot was the prime mover behind it. Go to this link for a commentary on the book The Fractalist: Memoir of a Scientific Maverick by Benoit B. Mandelbrot:http://www.nybooks.com/articles/archives/2013/may/23/mandlebrot-mathematics-of-roughness/                   

UNPROVABLE TRUTHS IN MATHEMATICS

In Solomon Feferman's 2006 review in the London Review of Books, "Provenly Unprovable" Feferman examines the book Incompleteness: The Proof and Paradox of Kurt Gödel by Rebecca Goldstein. Like Heisenberg’s uncertainty principle, Gödel’s incompleteness theorem has captured the public imagination, supposedly demonstrating that there are absolute limits to what can be known. More specifically, it is thought to tell us that there are mathematical truths which can never be proved. These are among the misconceptions that proliferate around Gödel’s theorem and its consequences. Incompleteness has been held to show, for example, that there cannot be a Theory of Everything, the so-called holy grail of modern physics. Some philosophers and mathematicians say it proves that minds can’t be modelled by machines, while others argue that they can be modelled but that Gödel’s theorem shows we can’t know this.

On the face of it, Goldstein would appear an ideal choice to present Gödel’s work: she has taught philosophy of science and philosophy of mind at several universities and has also written successful novels and short stories, including The Mind-Body Problem and Properties of Light: A Novel of Love, Betrayal and Quantum Physics. What she does very well is to provide a vivid biographical picture of Gödel, beginning midstream with his touching relationship with Einstein at the Institute for Advanced Study in Princeton, where, over the 15 years before Einstein’s death in 1955, they were often seen together. 

Postmodernists have claimed to find support in Godel's theorem for the view that objective truth is chimerical. And in a book with the surprising title: Bibliography of Christianity and Mathematics, it is asserted that ‘theologians can be comforted in their failure to systematise revealed truth because mathematicians cannot grasp all mathematical truths in their systems either.’ The incompleteness theorem is also held to imply the existence of God, since only He can decide all truths. For more on this topic go to:http://www.lrb.co.uk/v28/n03/solomon-feferman/provenly-unprovable

THE MAN WHO LOVED ONLY NUMBERS

Part 1:

Being affectionate with numbers, endlessly wondering about them, loving them, is, though impersonal and bloodless, no more strange perhaps than being possessed by the endless ramifications of cricket or trout fishing. Being consumed by numbers to the exclusion of all else, sounds deranged. The Hungarian mathematician, Paul Erdös, number theorist and combinatorialist extraordinary, eccentric, socially dysfunctional, obsessive, childishly egocentric, helplessly dependent on fellow number freaks to feed him, transport him, put him up and put up with him, was certainly outside the normal range, but not insanely so.

Like most mathematicians, Erdös had a deep need to be ordered and structured, so requiring long immersion inside mathematical abstractions. He thought numbers more interesting and comforting than anything else in this world and was able to spend most of his waking life in contact with them. When he died in 1996, at the age of 83, he had worked on more problems, made more conjectures, proved more theorems, collaborated with more people, and written and co-written more mathematical papers (over 1500) than any mathematician in history.

Part 2:

If one had to pick a mathematical genius at the furthest pole from Erdös, it might well be John Forbes Nash, 1994 Nobel Laureate in Economics, the subject of Sylvia Nasar’s biography: A Beautiful Mind. Where Erdös was impish, kind, open to all and monkishly pure, Nash was overbearing, secretive and abrasive, with a stormy marriage, an illegitimate son and several complicated liaisons with men. Erdös was a problem-solver delighted with progress, given to clever formalisms, inching towards the truth in many ways at once: Nash, a self-proclaimed conqueror, iconoclast and revolutionary who wrestled with ultra-difficult problems in the depths. Erdös all speed, lightness, constant movement: Nash heaviness, depth and endurance – carrying the same problem around with him in his head, for months on end.

This review in the London Review of Books back in September 1998 appeared in the last months before I retired after a 50 year student-and-working life.  It is a review of three books:(i) The Man who Loved Only Numbers: The Story of Paul Erdös and the Search for Mathematical Truth by Paul Hoffman(300 pages, 1998), (ii) Proofs from the Book by Martin Aigner and Günter Ziegler(200 pages, 1998), and (iii)  A Beautiful Mind: Genius and Schizophrenia in the Life of John Nash by Sylvia Nasar(500 pages, 1998). For more of this review go to:   http://www.lrb.co.uk/v20/n18/brian-rotman/fortress-mathematica

THE STEM SUBJECTS


The STEM subjects in the UK are: science, technology, engineering and mathematics. STEM fields is an acronym for the fields of study in these categories. The acronym is in use regarding access to United States work visas for immigrants who are skilled in these fields. The initiative began to address the perceived lack of qualified candidates for high-tech jobs. It also addresses concern that the subjects are often taught in isolation, instead of as an integrated curriculum. Maintaining a citizenry that is well versed in the STEM fields is a key portion of the public education agenda of the United States.

ALAN TURING: Part 1

The following paragraphs are written as a tribute to Alan Turing.  Alan Turing(1912-1941) is considered by many to be the father or godfather of computer science and modern computers. He was a mathematician, logician, wartime codebreaker and victim of prejudice. In 1999 Time Magazine named Turing as one of the 100 Most Important People of the 20th Century for his role in the creation of the modern computer, and stated: "The fact remains that everyone who taps at a keyboard, opening a spreadsheet or a word-processing program, is working on an incarnation of a Turing machine."

In August 2009 John Graham-Cumming started a petition urging the British Government to posthumously apologise to Alan Turing for prosecuting him as a homosexual. The petition received thousands of signatures. Prime Minister Gordon Brown acknowledged the petition, releasing a statement on 10 September 2009 apologising and describing Turing's treatment as "appalling": Thousands of people have come together to demand justice for Alan Turing and recognition of the appalling way he was treated. While Turing was dealt with under the law of the time and we can't put the clock back, his treatment was of course utterly unfair and I am pleased to have the chance to say how deeply sorry I and we all are for what happened to him ... So on behalf of the British government, and all those who live freely thanks to Alan's work I am very proud to say: we're sorry, you deserved so much better.


From 1952 until his death in 1954 Turing worked on mathematical biology, specifically morphogenesis. He published one paper on the subject called The Chemical Basis of Morphogenesis in 1952. This paper put forth the Turing hypothesis of pattern formation.  His central interest in the field was understanding Fibonacci phyllotaxis, the existence of Fibonacci numbers in plant structures. He used reaction–diffusion equations which are central to the field of pattern formation. Later papers went unpublished until 1992 when Collected Works of A.M. Turing was published. His contribution is considered a seminal piece of work in this field. While I do not claim to understand this field, I draw on some of its concepts, its history, its leading figures and its recent developments in my poetry.

ALAN TURING: Part 2

The Chemical Basis of Morphogenesis describes the way in which non-uniformity, as found in stripes, spots, and spirals, etc., may arise naturally out of a homogeneous, uniform state. The theory, which can be called a reaction–diffusion theory of morphogenesis, has served as a basic model in theoretical biology, and is seen by some as the very beginning of chaos theory. Chaos theory is a field of study in applied mathematics, with applications in several disciplines including physics, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions: an effect which is popularly referred to as the butterfly effect. Small differences in initial conditions, such as those due to rounding errors in numerical computation,  yield widely diverging outcomes for chaotic systems, rendering long-term prediction impossible in general.While Newtonian physics dominated the physical sciences until the twentieth century, chaos theory among other developments like the work of Einstein, altered the paradigms in which phsics worked.

Long-term prediction is often impossible even though the systems in which the prediction takes place are deterministi. This means that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos.  Chaotic behavior can be observed in many natural systems, such as the weather.  Explanation of such behavior may be sought through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps.
  For anyone interested, the  6 You-tubes of the BBC documentary on Alan Turing, chaos theory and the above are at: http://www.iceinspace.com.au/forum/showthread

Part #1: http://www.youtube.com/watch?v=HACkykFlIus
Part #2: http://www.youtube.com/watch?v=wHu8iaLs9i4&NR=1
Part #3: http://www.youtube.com/watch?v=Oj--pxcFUjg&NR=1
Part #4: http://www.youtube.com/watch?v=AMbua0BGfFE&NR=1
Part #5: http://www.youtube.com/watch?v=wc0IYJm5-mE&NR=1
Part #6: http://www.youtube.com/watch?v=-1x-7ZLKhjw&NR=1


ALAN TURING: Part 3

The last physical and biological sciences as well as mathematics that I studied formally in school were back in high school in the years 1961 to 1963. In the five decades, 1963 to 2013, these academic disciplines were still far out on the periphery of my knowledge universe. Many other disciplines occupied my attention during those 50 years, years in which I continued my studies as a student(until 1988), as a teacher(until 2005), and finally during my leisure-time, the years of my retirement from FT, PT and volunteer teaching.  In the last decade, I have begun to make up for this dearth, this pittance, in these areas of my knowledge base. Occasionally I am able to pick up some of the content of these fields. On 5 April 2011, in the first weeks of autumn in Australia, I had the opportunity to view a BBC Documentary: "The Secret Life of Chaos." (1) –Ron Price with thanks to (1)SBSONE TV, 5 April, 8:30-9:30 p.m.

I had heard of you Alan Turing
due to my life with computers,
my knowledge of chaos-theory,
and the interdisciplinary nature
of my study over those 40 years.

But I really hardly knew you at all,
Alan, until this evening when your
life was given a context of meaning,
your birth in 1912 to your death in
1954: mirabile dictu---second most (1)
distinguished alumnus at Princeton
University-its Institute for Advanced
Study was your home at the start of
the Baha’i teaching Plan in the years
1936 to 1938. I wondered to myself...

if you even heard of this newest of the
Abrahamic religions, this new Force in
the world of being, Alan...its so simple
rules, unpredictable outcomes, & the
patterns which emerged from its self-
organization. I take my hat off to your
genius, Alan, and I trust you are now
discovering new patterns in retreats
where new friends are found in new
neighbourhoods of Heavens(2), the
world of lights where the Beauty of
Existence is finally unveiled to the
eyes of us humans in a final hour.(3)

1 Latin phrase meaning "marvellous to relate"
2 Abdul-Baha, Memorials of the Faithful, Baha’i Pub. Trust, Wilmette, p.144.
3 ibid., 75.

Ron Price
6 April 2011 to 27/7/'12


Some of my internet posts below in relation to physics and mathematics:

http://www.iceinspace.com.au/forum/


http://www.sciencechatforum.com/viewtopic


MATHEMATICS AND LEIBNITZ

I could write much more about the history of mathematics and I will at a later date at this sub-section of my website.  For now I will add Gottfried Wilhelm von Leibniz(1646-1716), a German mathematician and philosopher. He occupies a prominent place in the history of mathematics and the history of philosophy. Leibniz developed the infinitesimal calculus independently of Isaac Newton, and Leibniz's mathematical notation has been widely used ever since it was published. His visionary Law of Continuity and Transcendental Law of Homogeneity only found mathematical implementation in the 20th century. He became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685 and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is at the foundation of virtually all digital computers. For more on Leibnitz goi to:http://en.wikipedia.org/wiki/Gottfried_Wilhelm_Leibniz

MATHEMATICS AND SOCIOLOGY


Part 1:

I stopped studying mathematics in 1963. In 1963, in grade 13 in Ontario, I took algebra, geometry and trigonometry. I got a 2nd and two 3rds in these 3 subjects, and entered university in September of that year. Since I only studied one of the sciences, chemistry, I was not able to go into the fields of law or medicine or any of the sciences. I needed physics and, after studying it for three months in grade 13, I realized that I would not pass it in the final exam and so switched to history. In that same year, in September, I entered an arts program, and began to study sociology, among other subjects.

Thomas J. Fararo’s review of theoretical sociology, a field I took a special interest as both a teacher and a student for over several decades, reveals the extraordinary complications in theoretical sociology in the twentieth century.  I stopped teaching theoretical sociology in 1999 and took an early retirement at the age of 55. I have continued my study of this field in the last dozen years of my retirement.  So it was that I took an interest in Fararo's article in this online electronic journal. Since this subject is, at the very least, esoteric, I will not dwell on it extensively here, but simply refer to this journal article.

Part 2:

The complications that occur in theoretical sociology and its mathematical side are largely concerned with differences in mathematical approaches and mathematical units of analysis. Why should this be?  The two scientific fields that have progressed the most in mathematical approaches in the last 100 or so years are physics & economics.  Also in the field of mathematics itself & its branches, there is a notable difference from mathematical sociology and mathematical psychology. That notable difference is the openness of the former fields to non-data-based exploratory analysis in the preliminary phases of investigation as a general rule. In sociology, sociological theories are statements of how and why particular facts about the social world are related. They range in scope from concise descriptions of a single social process to paradigms for analysis and interpretation.

Some sociological theories explain aspects of the social world & enable prediction about future events, while others function as broad perspectives which guide further sociological analyses. In sociology, sociological theories are statements of how and why particular facts about the social world are related. They range in scope from concise descriptions of a single social process to paradigms for analysis and interpretation. Some sociological theories explain aspects of the social world and enable prediction about future events, while others function as broad perspectives which guide further sociological analyses.https://en.wikipedia.org/wiki/Sociological_theory

Mathematical sociology and mathematical psychology take great pains to insure that nothing gets into the exploratory stage which is not thoroughly sifted and filtered as belonging to some “proper” existing theoretical school and even theoretical unit of analysis like a group or an individual respectively. This helps protect the discipline’s “integrity”, but to what extent does a discipline need to be protected beyond what science and logic already do? For more on this article by Osher Doctorow
entitled "Two New Tools for Mathematical Sociology with Applications," in The Electronic Journal of Sociology, 2005--go to:http://www.sociology.org/archive.html

MATHEMATICAL SOCIOLOGY

Mathematical sociology is the use of mathematics to construct social theories. Mathematical sociology aims to take sociological theory, which is strong in intuitive content but weak from a formal point of view, and to express it in formal terms. The benefits of this approach include increased clarity & the ability to use mathematics to derive implications of a theory that cannot be arrived at intuitively. In mathematical sociology, the preferred style is encapsulated in the phrase "constructing a mathematical model." This means making specified assumptions about some social phenomenon, expressing them in formal mathematics, and providing an empirical interpretation for the ideas. It also means deducing properties of the model and comparing these with relevant empirical data. Social network analysis is the best-known contribution of this subfield to sociology as a whole and to the scientific community at large. The models typically used in mathematical sociology allow sociologists to understand how predictable local interactions and are often able to elicit global patterns of social structure.

Mathematical sociology remains a small subfield within the discipline, but it has succeeded in spawning a number of other subfields which share its goals of formally modeling social life. The foremost of these fields is social network analysis, which has become amongst the fastest growing areas of sociology in the 21st century. The other major development in the field is the rise of computational sociology, which expands the mathematical toolkit with the use of computer simulations, artificial intelligence and advanced statistical methods. The latter subfield also makes use of the vast new data sets on social activity generated by social interaction on the internet. My study of this aspect of sociology has been limited. More:    https://en.wikipedia.org/wiki/Mathematical_sociology 

MATHEMATICAL psychology

In my many years of teaching psychology my exposure to its mathematical aspects was limited. Mathematical psychology is an approach to psychological research that is based on mathematical modeling of perceptual, cognitive and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characteristics with quantifiable behavior. The mathematical approach is used with the goal of deriving hypotheses that are more exact & thus yield stricter empirical validations. Quantifiable behavior is in practice often constituted by task performance.

As quantification of behavior is fundamental in this endeavor, the theory of measurement is a central topic in mathematical psychology. Mathematical psychology is therefore closely related to psychometrics. However, where psychometrics is concerned with individual differences (or population structure) in mostly static variables, mathematical psychology focuses on process models of perceptual, cognitive and motor processes as inferred from the 'average individual'. Furthermore, where psychometrics investigates the stochastic dependence structure between variables as observed in the population, mathematical psychology almost exclusively focuses on the modeling of data obtained from experimental paradigms & is therefore even more closely related to experimental psychology/cognitive psychology/psychonomics. Like computational neuroscience and econometrics, mathematical psychology theory often uses statistical optimality as a guiding principle, assuming that the human brain has evolved to solve problems in an optimized way. Central themes from cognitive psychology; limited vs. unlimited processing capacity, serial vs. parallel processing, etc., and their implications, are central in rigorous analysis in mathematical psychology.

Mathematical psychologists are active in many fields of psychology, especially in psychophysics, sensation and perception, problem solving, decision-making, learning, memory, and language, collectively known as cognitive psychology, and the quantitative analysis of behavior but also, e.g., in clinical psychology, social psychology, andpsychology of music.More:https://en.wikipedia.org/wiki/Mathematical_psychology


THE HISTORY OF MATHEMATICS

The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments have come to light only in a few locales. The most ancient mathematical texts available are Plimpton 322 (Babylonian mathematics c. 1900 BC), the Rhind Mathematical Papyrus (Egyptian mathematics c. 2000-1800 BC) and the Moscow Mathematical Papyrus (Egyptian mathematics c. 1890 BC). All of these texts concern the so-called Pythagorean theorem, which seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. For more on the history of mathematics go to this link:http://en.wikipedia.org/wiki/History_of_mathematics


ISLAM AND MATHEMATICS

In the history of mathematics, mathematics in medieval Islam, often termed Islamic mathematics or Arabic mathematics, covers the body of mathematics preserved and developed under the Islamic civilization between circa 622 and 1600.[1] Islamic science and mathematics flourished under the Islamic caliphate established across the Middle East, extending from the Iberian Peninsula in the west to the Indus in the east and to the Almoravid Dynasty and Mali Empire in the south.(1) 

The history of mathematics during the Golden Age of Islam, especially during the 9th and 10th centuries, building on Greek mathematics (Euclid, Archimedes, Apollonius) and Indian mathematics (Aryabhata, Brahmagupta), saw important developments, such as the full development of the decimal place-value system to include decimal fractions, the first systematised study of algebra (named for The Compendious Book on Calculation by Completion and Balancing by scholar Al-Kwarizmi), and advances in geometry and trigonometry.[1] Arabic works also played an important role in the transmission of mathematics to Europe during the 10th to 12th centuries.(2)[2] https://en.wikipedia.org/wiki/Mathematics_in_medieval_Islam & (1) go to:http://en.wikipedia.org/wiki/Mathematics_in_medieval_Islam


DETAILS ON THE HISTORY OF MATHEMATICS

The following link takes you to a detailed history of mathematics
:http://www-history.mcs.st-and.ac.uk/index.html


The Enhanced Developments in Mathematics (DEVM):

This DEVM book series is devoted to publishing well-written texts within the broad spectrum of pure and applied mathematics. DEVM volumes range from research monographs to texts which may be used in a classroom setting or for self-study as a reference. Ideally, each text should be self-contained and fairly comprehensive in treating a particular subject. Topics in the forefront of mathematical research that present new results and/or a unique and engaging approach with a potential relationship to other fields are most welcome. High quality edited volumes conveying current state-of-the-art research will occasionally also be considered for publication. The DEVM series appeals to a variety of audiences including researchers, postdocs, and advanced graduate students. For more on this subject go to:http://www.springer.com/series/5834