My answer is very simple; yes, I do think that the main mathematicians, scientists or philosophers recognized as such by the international community, had left their mark in the most relevant historical events registered until today and somehow are responsible for them.

Yes. Those great people have a magical historical touch!

I add people like Leibniz, Carnot, Clausius, Boltzmann. Mendel and last but not least Poincaré. However sometimes their influence can really be destructive. A cocktail of Hegel's theory of totality with a misunderstanding of Nietzsche's conception of morality has indeed influenced the former century. but in a wrong way.

I think the conversation should move to people among us today. We all know about all the great minds of times past. but what about the now. I find often that we are not recognizing the genius thinkers of today and I give you an example at the following link: http://www.sciencedirect.com/science/article/pii/S1364815213000224

We do have among us some of the greatest minds of all time, should we not celebrate them while they are among us?

All the names that have been mentioned were mostly not recognized during their time and a few were prosecuted (Galileo anyone) so should we not strive to change this. I have found that most of the time the fault lies with us (the scientific community) for not celebrating big enough the achievements of our peers, that why people love Edison, but no one knows about Tesla. Lets make a concerted effort to change, what do you say friends.

Dear Professor Esfahani,

Your suggestion is excellent. In addition, to giving attention to contemporaries in mathematics, science and philosophy, I suggest that a form of integration. That is, in considering contemporary scholars, we continue to measure the influence of the great persons of the past.

Please comment on the contributions of Tesia.

Dear Prof James and friends: the biggest fan of Tesla is Prof Ljubomir. Just view his thread about his favorite scientist :))

'Nikola Tesla was a Serbian American inventor, electrical engineer, mechanical engineer, physicist, and futurist best known for his contributions to the design of the modern alternating current electricity supply system.'

Ibn sina (Avesina) who was born in 980 A.D., was very influencing. He was a famous philosopher of the time. He also contributed to mathematics, physics, music and other fields.

Brunelleschi excelled not only as an architect but a brilliant mathematician who invented the principles of linear perspective with the use of mirrors around 1415. The use of Linear perspective became widespread in Western Europe along with Italy and soon it was a common practice all over the artistic world. Brunelleschi invention came into being when he constructed two paneled painting demonstrating geometrical optical linear perspective in 1400. http://www.famous-mathematicians.com/filippo-brunelleschi/

Dear Miranda,

Many thanks for the information about Nikola Tesla, truly a remarkable scientist, inventer and philosopher. The attached pdf file contains an overview of Tesla's contributions on the occasion of the Centennial of the birth of Nikola Tesla:

H. Pratt. Nikola Tesla. 1856-1943, Proceedings of the IRE, vol. 44, 1956, 1106-1108.

Briefly, Tesla is known for his discovery of the rotating electric field that made it possible to have a commutatorless, nonsynchronous, polyphase induction motor which required alternating current and eliminated the need for costly brushes and communtators necessary for direct current usage.

Yordon,

Many thanks for pointing out the contributions of Saint Clement of Ohrid. An iconic painting depicting Saint Clement is shown in the attached image. The readers of this thread will probably be interested in the St. Kliment Ohridski library.

The National and University Library "St. Kliment Ohridski" in Skopje was one of the first national institutions to be formed by the Parliament (ASNOM) of the young Macedonian state on the 23rd November, 1944.

The rich library tradition on the soil of the Republic of Macedonia whose roots stretch back to the deeds of the pan-Slavonic educators St. Cyril (826-869) and St. Methodius (820-885) is the same foundation on which the Library began to build up and develop its activity. Its patron saint, Clement of Ohrid (830-916), established the first monastic library in Ohrid in the Monastery of St. Panteleimon, and he is thus the founder of librarianship in these parts.

The initial fund of books in the National and University Library "St. Kliment Ohridski" at its foundation in 1944 amounted to some 150,000 library items. The majority of these were inherited from the Central Library of the pre-war Faculty of Philosophy in Skopje, founded in 1920 in Skopje.

Besides the holdigns, the National and University Library "St. Kliment Ohridski" also inherited the modest premises of the Central Library of the Faculty of Philosophy, located in the central city area, on the left bank of the River Vardar. The Library was open with 50 places for readers and 12 employees.

The documents which are of significance in confirming the status of the National Library of Macedonia were the ASNOM Decision of 18th January 1945 concerning the statutory depositing of an obligatory copy of all books published in Macedonia and the Decision of the National Committee for the Liberation of Yugoslavia (8th February 1945) concerning the statutory depositing of a copy of all books published in Yugoslavia. Thus the National and Univesrity Library became national deposit library and one of the eight Yugoslav deposit libraries.

Yordon,

Benjamin Pierce was truly a polymath! I just now found a copy of his biography published in the second volume of the American Mathematical Monthly, 1895 (see the attached pdf file).

His work on linear and associative algebra appeared in "little" lithographed volume in which he describes 70 or 80 different kinds of simple calculus.

Here is a link about Tesla and how we are still discovering things he was ridiculed for.

http://sciencepanorama.com/teslastory/

I think we should also mention some Persian Mathematician from the golden age of Islamic period, such as the great polymath Omar Khayyam whose calender is used in Iran currently and is proved to more accurate than the Gregorian calender and the great mathematician Jamshid Kashani who we owe algebra to. Truly, without the work of all the scientists in the middle east during Europe's dark ages, all the great works of the Greeks would have been lost to time.

Akbar,

Many thanks for the suggestion. Here is what I found.

Al-Kashi was born in Kashan which lies in a desert at the eastern foot of the Central Iranian Range. At the time that al-Kashi was growing up Timur (often known as Tamburlaine) was conquering large regions. He had proclaimed himself sovereign and restorer of the Mongol empire at Samarkand in 1370 and, in 1383, Timur began his conquests in Persia with the capture of Herat. Timur died in 1405 and his empire was divided between his two sons, one of whom was Shah Rokh.

While Timur was undertaking his military campaigns, conditions were very difficult with widespread poverty. al-Kashi lived in poverty, like so many others at this time, and devoted himself to astronomy and mathematics while moving from town to town. Conditions improved markedly when Shah Rokh took over after his father's death. He brought economic prosperity to the region and strongly supported artistic and intellectual life. With the changing atmosphere, al-Kashi's life also improved markedly. The first event in al-Kashi's life which we can date accurately is his observation of an eclipse of the moon which he made in Kashan on 2 June 1406.

It is reasonable to assume that al-Kashi remained in Kashan where he worked on astronomical texts. He was certainly in his home town on 1 March 1407 when he completed Sullam Al-sama the text of which has survived. The full title of the work means The Stairway of Heaven, on Resolution of Difficulties Met by Predecessors in the Determination of Distances and Sizes (of the heavenly bodies).

Yordan,

I agree that there are quite a number of ideologues who changed (influenced) people's view of the world. You mention religion. Religion does fit within the scope of this thread, provided those who had an impact on religion were either mathematicians or scientists or philosophers.

The polymath Omar Khayyam, mathematician Jamshid Kashani (Al-Kashi), Saint Clement, ames Watson and Francis Crick (among others) are excellent examples of influential scholars that we should consider.

It is truly amazing how many bright lights (truly significant scholars) have had an impact on the history of mankind.

Another influential poet, philosopher and mathematician is Viktor Vladimirovich Khlebnikov, better known by the pen name Velimir Khlebnikov (Russian: Велими́р Хле́бников, IPA: [vʲɪlʲɪˈmʲir ˈxlʲebnʲɪkəf]; 9 November [O.S. 28 October] 1885 – 28 June 1922), was a poet and playwright, a central part of the Russian Futurist movement, but his work and influence stretch far beyond it.

In his work, Khlebnikov experimented with the Russian language, drawing upon its roots to invent huge numbers of neologisms, and finding significance in the shapes and sounds of individual letters of Cyrillic. Along with Kruchenykh, he originated zaum.

He wrote futurological essays about such things as the possible evolution of mass communication ("The Radio of the Future") and transportation and housing ("Ourselves and Our Buildings"). He described a world in which people live and travel about in mobile glass cubicles that can attach themselves to skyscraper-like frameworks, and in which all human knowledge can be disseminated to the world by radio and displayed automatically on giant book-like displays at streetcorners.

In his last years, Khlebnikov became fascinated by Slavic mythology and Pythagorean numerology, and drew up long "Tables of Destiny" decomposing historical intervals and dates into functions of the numbers 2 and 3.

For more about Khlebnikov's life and work, see

http://en.wikipedia.org/wiki/Velimir_Khlebnikov

Many thanks to Alexander Yurkin for pointing out the importance of Viktor Vladimirovich Khlebnikov. To read about Khlebnikov's radio of the future, see

http://acousmata.com/post/90347897/the-radio-of-the-future

English translations of Khlebnikov's poems are available at

http://www.albany.edu/offcourse/issue41/cigale_translations2.html#khlebnikov

Khlebnikov's self portrait is attached to this post.

Following up on what Kamal Bani-Hani has written,

One of the major discoveries comes from James Watson and Francis Crick.

In mid-March 1953, using experimental data collected by Rosalind Franklin and Maurice Wilkins, Watson and Crick deduced the double helix structure of DNA.Sir Lawrence Bragg, the director of the Cavendish Laboratory (where Watson and Crick worked), made the original announcement of the discovery at a Solvay conference on proteins in Belgium on April 8, 1953; it went unreported by the press. Watson and Crick submitted a paper to the scientific journal Nature, which was published on April 25, 1953. This has been described by some other biologists and Nobel laureates as the most important scientific discovery of the 20th century. For more about this, see

http://en.wikipedia.org/wiki/James_Watson

The attached image shows the DNA structure for four common deoxyribonucleotides are connected by phosphodiester bonds to form a single-strand.

Clearly all major scientific figures such as Nobel winners etc correspond well to this description yet not any achievement in science change in a deep way the manner that we think about the universe or about ourselves. Only few people in history made such a deep change in our views. One name that is dear to me is Imanuel Kant that made us understand that knowledge is created from the combination of sensory data with our own inner representations and change the focus from outside "objectivity" to the way we percieve the world via our own filters.

Certainly Nicolaus Copernicus is the one who changed our view of the world around us. Of course, he was not the one such person.

I hate to say that, but if you mention philosophers then the name of Marx comes to my mind. The everyday life of millions of people is still dictated by his ideas as well as by his cruel followers.

Historian of medicine Pedro Laín Entralgo theorized that in every historic period medical scientists affect the way we Westerners view the human body in all spheres of culture. From Greek Antiquity to the end of the European Middle Ages, the Hippocratics' and Galen´s vision of the human body as a piece of the natural order impacted on medicine, art, architecture, and worship. In the Renaissance, Vesalius´ notion of the human body as a work of architecture established a new bodily paradigm for the rest of culture. This paradigm became dynamic in the Baroque. Mechanicism and organicism took over in the later Baroque and early Enlightenment. The view of the body as a State of cells in revolution marked the age of political revolutions in 1848, Darwinian evolutionism influenced the view of the body in all culture in the mid to late nineteenth. The vision of the body as a resume of all earlier evolutionary phases (visible in the embryo) became paramount at the end of the nineteenth. The notion of the body as a factory dominated culture in the early twentieth. And the idea of the body as a synthesis of all earlier paradigms ruled the rest of Western culture throughout the rest of the 20th century.

@Nir Sochen: Clearly all major scientific figures such as Nobel winners etc correspond well to this description yet not any achievement in science change in a deep way the manner that we think about the universe or about ourselves. Only few people in history made such a deep change in our views. One name that is dear to me is Imanuel Kant that made us understand that knowledge is created from the combination of sensory data with our own inner representations and change the focus from outside "objectivity" to the way we percieve the world via our own filters.

I agree with you comment about Kant and would like to return to your comment a bit later. First, there is the issue of the Nobel laureates and their impact on the world.

Consider, for example Hermann Muller. Thanks to the work of Hermann Muller, who won the 1946 Nobel Prize for Physiology or Medicine, people realized the importance of tempering our knowledge with safety and care.

Muller won his prize for proving that X-rays cause mutations (called X-ray mutagenesis) in the human body. In the mid-1920s, he'd gathered significant evidence that exposing Drosophila flies to X-rays caused genetic mutations that shortened their lifespans. He was certain that the same kind of damage would occur in humans.

Although he'd been trying to publicize his work for around 20 years, it took the World-War II atomic bombings of Japan to underscore the dangers of radiation, X-rays and nuclear fallout. It was then that the Nobel committee finally recognized his research.

Muller's discoveries, as well as his anti-nuclear weapons politics, made him an invaluable counterweight to the world-changing technological advances of the Atomic Age. For more about this, see the attached paper.

Other Nobel laureates to consider are Watson and Crick (1962), Marie Curie (1935), Albert Einstein (1921), Sir Alexander Fleming, along with Sir Ernst Boris Chain and Sir Howard Florey (1945), and others.

If I understood correctly, a good answer to your question (James Peters) should contain: name of the scientist, mathematician or philosopher, what that person did, and what was the historical consequence of his/her work, well beyond rational understanding. For example (taking some of the figures mentioned, or being inspired by the responses already given):

1) Albert Einstein, his work on the transformation of matter into energy (or viceversa), his formula E=mc2. Although Einstein did not work directly on the atomic bomb, his ideas did inspire those scientists and engineers who made it possible. Einstein (with Leo Szilard) wrote to USA president Roosevelt, urging USA to make a head start for developing the bomb, believing USA was a much better promoter of personal freedom than (say) Germany (at the time).

2)Mathematicians Karl Weierstrass or David Hilbert, allowed furthering the development of female rights, in admitting to their tutoring Sophia Kowaleskaya (the former) or championing Emmy Noether for an academic position (the later).

3) al-Khwārizmī. Just his name evolved into our modern word "algorithm", and the name of one of his books (Al-Kitāb al-mukhtaṣar fī ḥisāb al-jabr wa-l-muqābala) contains the expression al-jabr, from which derived our modern word "algebra".

4) The ethnologist Thor Heyerdahl. His travels on the Kontiki made clear to many orthodox academicians that travels which they thought "impossible" to the technology of primitive people, were actually feasible.

5) Sigmund Freud. The vocabulary used in his work has permeated down to current language, and has been used, abused and misused, like "unconscious", "subconscious" and "projection".

6) Ada Lovelace's work rendered the first (computer) programs, executed by Charles Babagge's machines. Although mathematical initially, programs have evolved into an entire abstract realm dubbed "cyberspace".

7) Aristotle influenced deeply the theology of Jewish and Christians during the middle ages, for good or for evil. Of course, since Aristotle thought he could have an opinion on whatever he deemed possible, regardless on the actual knowledge he could bear on the subject, he did influence a vast amount of aspects of science, mathematics and philosophy, if, for no other thing, to trigger someone to challenge his views.

8) The sociologist Carlos Castaneda ("The teachings of Don Juan", among his books), in my humble view, has instilled some modern questioning about our relationship of external (physical) and internal (psychological) realities, figuring them closer to each other and more intermingled that we wouldn't know. At any rate his work has inspired serious scientific work in neurology and psychiatry, like that performed by Stephen LaBerge, on lucid dreaming. He has also inspired many artists, and teachers of art, chiefly Betty Edwards with her "teaching with the right side of the brain".

9) Jacques Derrida work on deconstructionism, denying intrinsic, thus stable meaning to (theoretically) any concept. Thus, the moment we embark on defining something, one must cloth it within a meaningful context, otherwise loosing the perspective of the significance intended by the original user of the word, towards a possible listener.

10) Adam Smith, thinker on moral philosophy and what is now known as political economy, has influenced economical theory into situations that in mathematics are described as zero-sum games (the earnings of one are the losses of the other). Until the mathematician John Forbes Nash described the possibility of cooperative games, with a long term mathematical stability.

Dear Arturo,

In all fairness to Aristotle, he invented the scientific treatise. Without him, we researchers would still be using Platonic dialogues. I am not certain that he was as skeptical as his teacher Plato. So he can be forgiven for writing treatises on all imaginable scientific knowledge of his own time. His most prolific disciple Saint Thomas Aquinas did the same, influencing all Medieval thought in his treatises. Just as Maimonides Judaized Aristotle, so Aquinas Christianized him.

Arturo and Nelson,

Thomas Acquinas referred to Aristotle as The Philosopher in his Summa Theologica, some of which I studied in Latin and actually wrote an essay in Latin on Acquinas's De Ente et Essentia (quite a remarkable work) in a graduate course on Acquinas.

You might be interested in the fact that it was Aristotle who first gave the philosophy of nature the status of a specific discipline, "organizing a large number of empirical observations within a coherent body of philosophical reflections" (page 1 of the attached paper).

In Aristotle’s philosophy, the concept of nature involves both a metaphysical and a physical outlook. For Acquinas,

nature is nothing other than a principle of motion and rest in that in which it is

primarily and in virtue of itself (per se) and not accidentally (per accidens). See page 6 of the attached paper.

Dear Arturo Ortiz Tapia,

Your post is excellent and points in the direction of many interesting and important scholars. There are many points to consider, further. Consider, for example, the work of Al-Kwarizmi.

Al-Khwarizmi is best known for his work entitled Hisab al-jabr w'al-muqabal. Although there are various different translations of this title, scholars agree that the term ‘al-jabr’ refers in some way to an equation and it is the origin of our word ‘algebra’. It was a text that was designed to be practical for those working in professions such as law, medicine, construction and land ownership that needed to use various sorts of numerical computation (from page 1 of the attached paper).

The important thing to notice here is that Al-Khwawrizmi included a chapter on the solution of linear and quadratic equations in his al-jabr book. From this, it is obvious that this scholar had an enormous influence on the subsequent history of Algebra and its later extensions.

Charles Darwin: his theory of Natural Selection, (Survival of the fittest), was interpreted by many people in a different way. One of them lead to social darwinism.

Social Darwinism arose in the US around the year 1906, starting the Eugenics Movement, with its "better baby contests", "fitter family competitions", and finally the eugenic laws with its compulsory sterilizations which lasted up till the 1970´s. It seems that Darwin unconciously influenced the creation of a social theory (which already existed in the past), or its rebirth, which gave way to dividing people into "fit" or "unfit" categories, being that "geneticaly", "socially" or "economically" fit or unfit to survive.

Euthanasia was also proposed to get rid of the unfit. A 1911 Carnegie Institute report mentioned euthanasia as one of its recommended "solutions" to the problem of cleansing society of unfit genetic attributes. The most commonly suggested method was to set up local gas chambers. All of this long before Hitler got to power, and started the exact same programs proposed decades before in the US. Hitler even seems to have copied the US propaganda to promote euthanasia, because in the 1930s, there was a wave of portrayals of eugenic "mercy killings" in American film, newspapers, and magazines.

It seems that California eugenicists began producing literature promoting eugenics and sterilization and sending it overseas to German scientists and medical professionals in the 1930´s, probably influencing the Nazi movement in how to efectivally execute their own eugenic programs. In Nazi germany the elimination of the feebleminded was done for eugenical reasons (biological), but the elimination of jews and gypsies was done for social reasons. In Hitlers theory (or his interpretation of darwin´s theory), they degenerated society, and to have a fit society, they had to go.

But Social Darwinism, and Eugenic laws didn´t only prosper in the US, it also prospered in Scandinavia. in all four scandinavian countries, Norway, Sweden, Denmark and Finland, compulsory sterilizations were implemented from 1930´s till 1975, long after Hitler´s eugenics program got out in the open. It was propagandized, once again, as a means to prevent the degeneration of the race by "helping" the feebleminded against propagating their own weak genes. It is a subject which most of today's Nordic folk would rather keep in decent obscurity.

In all countries where Social Darwinism prospered, it was not only a biological or social eugenical movement, it was also based on economics. In the US, it gave way to Rockefeller capitalism, while in the USSR it gave way to communism. Karl Marx was very impressed by Darwins theory, and wrote in 1860: "Although it is developed in the crude English style, this is a book which contains the basis of natural history for our views." The Soviets tried to create the super-human by socially engineering him, but to accomplish this, many "sub-humans" not "fit" to "evolve" into capitalists slaves, so they could suffer and then become revolutionary, to then become communist, just had to be eliminated too. Here a quote from Friedrich Engels:

"Among all the nations and sub-nations of Austria, only three standard-bearers of progress took an active part in history, and are still capable of life -- the Germans, the Poles and the Magyars. Hence they are now revolutionary. All the other large and small nationalities and peoples are destined to perish before long in the revolutionary holocaust. For that reason they are now counter-revolutionary. ...these residual fragments of peoples always become fanatical standard-bearers of counter-revolution and remain so until their complete extirpation or loss of their national character ... [A general war will] wipe out all these racial trash down to their very names. The next world war will result in the disappearance from the face of the earth not only of reactionary classes and dynasties, but also of entire reactionary peoples. And that, too, is a step forward."

- Friedrich Engels, "The Magyar Struggle," Neue Rheinische Zeitung, January 13, 1849

See how he used "racial trash", and "a step forward"... Maybe that is why the Soviet Union (or Stalin, whichever you prefer) exterminated millions of ukranians in the 1930´s by using a controlled famine.

Whatever way, although Darwin´s theory was interpreted in ways he maybe never would have pretended, it must also be said that many of his relatives became eugenicists themselves, maybe again, influenced by good old Charles Darwin himself, and his theory. They all seemed to have intermarried between a couple of high-class families, maybe to conserve their "superior" traits. This endogamy is very common with royalty and high-upper-class aristocracy.

Francis Galton (his cousin) was a fierce defender of Eugenics, and some say he even coined the term "Eugenics" himself. He became chairman of the National Eugenics Society, and Darwin’s son Leonard replaced him in 1911. In the same year an offshoot of the society was formed in Cambridge. Among its leading members were three more of Charles Darwin’s sons, Horace, Francis and George. And lets not forget about Charles Galton Darwin (Charles Darwin´s grandson), involved in the British Eugenics Society for over 30 years, and president for over 6 years. A good book to read is his "Next Million Years". Here is a link to a biography: (http://www.galtoninstitute.org.uk/Newsletters/GINL0412/chief_sea_lion.htm)

So, to answer your question: yes, I am pretty sure that Charles Darwin´s work influenced historical events in a mayor way. An important extra point may be that sometimes the work from a scientist or philospher may influence historical events based on a wrong interpretation.

(I have left almost no links nor notes because all of this is very general information, available for the main public)

Dear Arturo, the work of Bernard Riemann contributed to biology and research methodology. He applied the long forgotten (epistemologically diluted) method of analysis and synthesis to explore the mechanism of the ear that are to date still influential. The method goes back to ancient geometers such as Pappus and philosophers such as Aristotle. The method itself has been revisited by George Polya and it makes an enormous contribution to mathematics and epistemology. It also provides insights into the design process in design science in current times.

Arturo and Ricardo,

You may find the attached article on Poincare interesting.

Starting on page 16 of the attached article, Poincare and Riemann are mentioned. Poincare became interested in Riemann surfaces and considered ways to extend Riemann's ideas to 3 dimensions.

Fernando,

You may find the attached paper on Wittgenstein interesting.

Briefly, Ludwig Josef Johann Wittenstein was born in Vienna in 26 April 1889 (just two days form now will be Wittgenstein's birthday!). During the summer of 1908, he experimented with kites at the Kite Flying Upper Atomosphere Station near Glossop, Derbyshire. That same year, he became a research student in the department of engineering at Manchester University. After his kite flying experiments, he worked on the design of a jet reaction propeller for aircraft. After his initial fascination with the engineering design of the propeller, he then became interested in the mathematics associated with the propeller design.

What's about Pythagoras of Samos- Greek philosopher, mathematician, and founder of the religious movement called Pythagoreanism

Golam,

I agree with you that Pythagoras--as philosopher and mathematician and founder of Pythagoreanism--had a huge influence on others.

Surprisingly, very little is known about Pythagorus (580-500 B.C.). He was born in Samos on the western coast of what is now Turkey. He met Thales, who recommended that he travel to Egypt. It is conjectured that Pythagoras learned about what is known as the Pythagorean theorem from the Egyptians. The school that he founded was as much a religion as a school of mathematics.

Bertrand Russell succinctly summarises Pythagorus's contribution: "It is to this gentleman that we owe pure mathematics. The comtemplative ideal -- since it led to pure mathematics -- was the source of a useful activity. This increased its prestige and gave it a success in theology, in ethics, and in philosophy."

For more about this, see

http://www.math.tamu.edu/~dallen/history/pythag/pythag.html

My favourites:

Spinoza, Gould, and Husserl were looking for the 'geometry of mind' that I found, partly thanks to their previous work, partly thanks to a little known work of Newton, on biblical prophets' texts. This 'geometry of mind' is actually also a geometry of physiological sensation, and I have modeled it using a simple form of geometric (without the 'metric') topology. It is of tremendous use for human physiological states of illness and 'bad' behaviours (related to 0 and 1, in the other RG question)

Broaden a bit:

Sure, these thinkers influenced 'events' and technology and gave us wonderful tools. But far more, their misunderstood work (as mental modelling) gave rise to distorted linear or globalist, often logically stupid human ideas (and detoured use of the tools). See for example Gould on the Tree of Evolution and the big logical errors of concluding that humans are the summum of Creation... and THIS does influence nature! including the drift in human health, which statistics dutifully re-hide regularly.

Not mentioning the name of Archimedes until this moment may have only one cause: it is obvious. In my opinion he was the greatest scientist of all times, active and successful both in theory and practice. Wonderful example to follow.

Marek,

Yes, I agree with you that Archimedes was a truly great scientist (and mathematician).

Perhaps you will find the attached overview of Archimedes's work on the Sphere and Cylnder interesting. Unlike Pythagoras, we know quite a bit about Archimedes thanks to a combination of public stories and his writings. Recall that Pythagoras (580-500 B.C.) lived almost 300 years earlier than Archimedes and we have no writings by Pythagoras to study. By contrast, Archimedes lived in the era up to 212 B.C. and he was a prolific writer of mathematics.

Marek,

Perhaps more intriguing in the study of Archimedes is connected with infinitesimals.

With Archimedes' method of exhaustion in estimating the area of a bounded region by inscribing a polygon with sides so small that the polygon (with known area) that the polygon is indistinguishable from the bounded region (with unknown area). See Section 3 of the attached paper.

Leibniz introduced a quantity that was infinitely small not as an absolute zero but as a relative zero. Sect. 5 of the attached paper.

Euler introduce a number 1/oo that is less than every positive number.

But the infinitesimal was later banned from mathematics with the introduction of the epsilon-definition of limit by Cauchy.

Later, Dedekind (1831-1916) and Georg Cantor (1845-1918) introduced the actual infinite (but not infinitesimals) in mathematics. Sect. 7 of the attached paper.

A rigourous foundation for infinitesimals was introduced by Abraham Robinson (1918-1974) with the introduction of Non Standard Analysis, North Holland, 1966.

James,

I wonder why you put forward your question. You seem to have a pretty large knowledge in this area, at least when we are talking about mathematicians or philosophers. Are you trying to increase your knowledge in other areas or just to check what are the living ideas in other parts of world, outside Manitoba?

I think both goals (if I guess correctly) are valuable. It is a pleasure to hear again about great figures born in Arab countries, not always recalled in western culture.

But great mathematicians were born in India, too. Will anybody recall the invention of the number zero?

Dear Marek,

Many thanks for your observations. The question for this thread came to mind because of rather long-term interest in the contributions of mathematicians, scientists and philosophers. I am continually amazed and grateful for the contributions of others to this thread.

The subject of infinitesimals and infinity alone are enough to occupy a multi-volume library. And enough to occupy me for quite some time. Thanks to this thread, I recently discovered that infinitesimals became an acceptable part of mathematics with the introduction of Non-Standard Analysis by Abraham Robinson in 1966:

http://www.cut-the-knot.org/WhatIs/Infinity/NSA.shtml

See the revised 1996 edition of the NSA book:

http://www.amazon.com/exec/obidos/ISBN=0691044902/ctksoftwareincA/

Of particular interest is the study of Internal Set Theory, with articles by

Edward Nelson, Internal set theory: A new approach to nonstandard analysis, Bulletin of the American Mathematical Society 83(6), 1977, 1165-1198.

For more about Nelson's approach to IST, see

https://web.math.princeton.edu/~nelson/books.html

John Nash's work on Game Theory surely influenced the progress of the cold war.

Nash is a good contribution to this thread but not for whatever his work did during the cold war, but for his contribution to game theory and more generally, to Rational Choice Theory. Rational choice theory dictates what we should take as rational in the wider sense of the term. The analysis of rational versus irrational behaviour has been crucial in many areas such as philosophy, economics, biology,... I'd include Nash along with von Nuemann and Morgenstern.

Michael and Alfonso,

There is an excellent biography for John Forbes Nash at

http://www-history.mcs.st-andrews.ac.uk/Biographies/Nash.html

You may also find the following book interesting:

Tom Siegfried, A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature, Academies Press, 2006

http://www.nap.edu/catalog.php?record_id=11631

It is often argued that Alan Turing's work influenced the Second World war.

Deep in my hart I should like to promote Boltzmann, Carnot and Clausius as people who influenced the progress of thermodynamics and particularly their research for efficient energy transformatiom from heat to mechanical energy. The Carnot cycle approves that only 41% of heat can properly transformed in useful energy. This principle is used in energy plants of any walk. Moreover besides the maximum transformation efficiency they gave evidence that heat is a form of molecular dynamics and not an elementary particle as longtime claimed in the 18th century. Perhaps they showed the way for falcification of the the existence of some other elementary particles

Dear Professor Kamal Bani-Hani,

Yes, definitely, it appears that "every scientist is standing on the shoulder of other fellow researchers"!

@Guido J. M. Verstraeten: about Carnot cycle.

Perhaps you will find the following paper interesting:

L. Wang, Carnot theory: Derivation and extension, Int. J. Engng Ed 14, 1998, no. 6, 426-430.

http://www.ijee.ie/articles/Vol14-6/ijee1031.pdf

The optimization theory is used to derive and

extend the classical Carnot theory.

Dear James, thank you

In my opinion following paper will have immense effect in future;

Thomas L. Saaty and H.J. Zoffer, Principles for implementing a potential solution to the middle east conflict, Notices of American Math Soc. 60(10) 2013, 13001321

DOI: http://dx.doi.org/10.1090/noti1053

Ismat,

That article in the Notices of the AMS has generated lots of controversy. I just did not expect the middle east conflict appearing as a topic in the Notices. Now that you have mentioned it, I will go back and read it again.

James,

Important thing is an attempt for a fair solution, not where and who did it. In my opinion a fair solution to middle east conflict will be an historical event.

There are scientists who you've never heard of who have influenced historical events in a major way; partly because their work was classified, and partly because those scientists were considered "the enemy" of the country where the significance of their work was finally realized. Giving credit to "enemy" scientists is something that countries, most countries, don't like doing.

"

The Russian theoretical scientist K. Tsiolkovsky began the space age.

D. I. Mendeleev opened the whole new field in chemistry by the discovery of the periodic table.

Ivan Pavlov has played a very important role in biology by proving the conditional reflex.

Ivan Pavlov had a very important contribution to biology by proving the conditional reflex.

Milutin Milankovic is one of the most influential Serbian scientist in the history! Very impressive, in many areas. He was mathematician, astronomer, climatologist, geophysicist, civil engineer, doctor of technology, university professor...!

http://earthobservatory.nasa.gov/Features/Milankovitch/

http://www.serbiaconsulatenyc.com/en/greatscientists.html#Milutin Milanković

@Natalia Duxbury: The Russian theoretical scientist K. Tsiolkovsky began the space age.

In 1911, Tsiolkovsky observed:

„Earth is the cradle of mankind, but one does not stay in the cradle forever.”

This is Tsiolkovsky’s Imperative.

For more about this, see

D.J. Tamm, The Reinvigoration of the West Through Outer Space Development, or

Tsiolkovsky’s Imperative in the 21st Century,

M.Sc. thesis, Jagiellonian University in Krokov, 2006:

http://www.hudsonfla.com/tsiolkovsky.pdf

@Natalia:

Perhaps you will find the following article interesting:

V.O. Samoilov, Ivan Petrovich Pavlov (1849-1936), Journal of the History of Neurosciences 16, 2007, 74-89.

http://www.google.ca/url?sa=t&rct=j&q=&esrc=s&source=web&cd=12&ved=0CCwQFjABOAo&url=http%3A%2F%2Fxa.yimg.com%2Fkq%2Fgroups%2F28073569%2F1746981659%2Fname%2F09%2B-%2BIvan%2BPetrovitch%2BPavlov%2B1849-1936.pdf&ei=mB14U6SxAYOayASS44DIBA&usg=AFQjCNEl3jcbT3BivPsybO03y9Fa3_p6jg&bvm=bv.66917471,d.aWw

Samoilov observes:

There are many names in the history of Physiology. It is hard to choose even ten outstanding names. However, during the construction of the new building of Physiology Laboratory of Leiden University — the cradle of modern physiology and medicine — sculptor Oswald Wenckebach6 carved four sculptures in the walls of the main entrance. There is W. Harvey with a heart in his hands, C. Bernard with a liver in his hands, A. Berthold holding a rooster to his chest, and there is I.P. Pavlov holding the brain in his hands (p. 89).

@Ljubomir Jacić :

Milutin Milankovic is one of the most influential Serbian scientist in the history!

The following Ph.D. thesis should be of interest:

P.K. Munneke, Snow, ice and solar radiation, thesis, 2009, Utrecht University.

This thesis carries forward Milankovic"s work on climate theory in his book published in 1930.

Thanks @James. Good observation speaking on importance of his work which is invariant to time! Have You noticed his work relation to NASA?

@Ljubomir,

I just now found a magnificent collection of abstracts from papers from a symposium dedicated to Milutin Milankovic:

Serbian Scientific Society, Nonlinear Dynamics Mulitin Milankovic. Multidisciplinary and Interdisciplinary Applications, Beograd, October 1-5, 2012:

http://www.mi.sanu.ac.rs/projects/booklet_of_abstracts.pdf

See, for example,

the mathematical model for aerial robots, p. 99.

digital library preserving the works of Serbian scientists, p. 42,

comments about NASA, p. 174.

Milutin Milankovic was not recognized adequately for his valuable work in different fields and many contributions to human mankind!

Thanks @James for fine link!

Ljubomir,

I suspect that there are many scientists who go unnoticed. Science itself is a quiet and somewhat isolated occupation. Little will be known about a scientist who works quietyly, without fanfare.

When I was working at Moscow State University, we've used Milankovic's cycles a lot to mathematically model subsurface permafrost reconstruction. With a due reference.

First, I will repeat your question: """Are there mathematicians, scientists or philosophers whose work you view as influencing historical events in a minor or in a major significant way? """ SO, THE POINT IS, THAT YOU SPEAK ABOUT HISTORICAL EVENTS. The most direct influence of a philosopher on history - and it was not necessarily a good one - was the influence of Karl Marx. Then there are different influences from philosophers generating fascism... OK; another point could be the that different scientific achievements influence historuical events directly. The wars of the 20th century would have been different without phones and without planes. The Moral is that "historical events", like revolutions and wars, are not necessary good things. History work better when it does continuous development, and not events - which are ruptures and discontinuities. All possible achivements, from Marxism to phone, Internet, airplane, play a very good role in peaceful continuous developments and mostly a bad role in case of wars and revolutions, which are disruptive events. So, please, better don't ask about historical events. We don't want events anymore.

Nice imput, Mihai, it is true that many philosophies, inventions and scientific theories led to horrible historical events, i mentioned Darwin's theory in an earlier comment, and how this led to social darwinism and eugenics. But it must also be said that many of these people never intended any harm (except Marx maybe), and on the other side of the story, mathematicians, physicists, nuclear scientists and inventors working for the military invented things to increase military power and advancement, but these inventions are now being used publicly, with the best non-violent intentions. The knife cuts both ways. Its maybe the interpretation and intention of the peoples that influences historical events. A knife can be a usefull tool, or a deadly weapon.

Right, Yarci. I think, in fact, that the use of the word "Event" was a bad choice in the question. Maybe he just asks about achievements, or influence, or impact. If we assimilate Jesus Christ with a philosopher, we get another example of unbelievable ambiguous long-lasting good and bad impact. Darwin is wonderful pointing out that humans are a particular case of animals. Good impact: we must love and respect nature and life. Bad impact: social (and economic!) darwinism, eugenics, and even creationism done only to combate Darwin, by obscurantists who believe to be injured by this only idea. Again, the problem does not come from the inventor, but from a lot of users. Well, it is not bad to discuss about the ethics of science. But I think that the Society in general needs more ethics. It is so easy to blame the scientists and to continue what somone did before, without any problem. Or to say (what I have heared a lot since the crisis begun) "we don't want to finance science so much, because they are not ethical". In contrary, politicians which use money from the Budget in their own interest - and I have thousends of examples - are very ethical...

@Mihai Prunescu:...I think, in fact, that the use of the word "Event" was a bad choice in the question.

Point well-made! I have revised the question, replacing "events" with the term "outcomes". Basically, it does seem to be important to consider those mathematicians, scientists and philosophers whose discoveries led to outcomes that have become part of the history of humankind.

The work by Charles Darwin definitely had a profound influence on subsequent work by others on biology, anthropology, archaeology, and philosophy.

@yarci francisco rodriguez santana:...i mentioned Darwin's theory in an earlier comment, and how this led to social darwinism and eugenics.

Yes, social darwinism and eugenics, can be traced to the work of Darwin on the Descent of Man. You may find the following article interesting:

D.J. Galton, C.J. Galton: Francis Galton: and eugenics today, J of Medical Ethics, 2014

http://jme.bmj.com/content/24/2/99.full.pdf

See page 100 and the discussion on the inheritance of intellectual ability.

James, did You hear for Mihajlo Petrovic-Alas, famous Serbian mathematician who did fine contributions in the area of differential equations, and later on, the invention of analogue computer. For me he is very important, as I am someone who is control engineer and particularly I like that he was a fisherman who had written a book on fishing, as it is my favourite hobby! :)

http://en.wikipedia.org/wiki/Mihailo_Petrovi%C4%87

On may 6th, Google devoted to him this page!

https://plus.google.com/+google/posts/EzxChM5D8X3

Ibn Sina (Persian ابن سینا or ابو علی‌ سینا ; August c. 980 – June 1037), commonly known as Ibn Sīnā, or in Arabic writing Abū ʿAlī al-Ḥusayn ibn ʿAbd Allāh ibn Al-Hasan ibn Ali ibn Sīnā[2] (Arabic أبو علي الحسين بن عبد الله بن سينا) or by his Latinized name Avicenna, was a Persian[3][4][5][6] polymath, who wrote almost 450 works on a wide range of subjects, of which around 240 have survived. In particular, 150 of his surviving works concentrate on philosophy and 40 of them concentrate on medicine.[7]

His most famous works are The Book of Healing, a vast philosophical and scientific encyclopedia, and The Canon of Medicine,[8] which was a standard medical text at many medieval universities.[9] The Canon of Medicine was used as a textbook in the universities of Montpellier and Leuven as late as 1650.[10] Ibn Sīnā's Canon of Medicine provides an overview of all aspects of medicine according to the principles of Galen (and Hippocrates).[11][12]

His corpus also includes writing on philosophy, astronomy, alchemy, geology, psychology, Islamic theology, logic, mathematics, physics, as well as poetry.[13] He is regarded as the most famous and influential polymath of the Islamic Golden Age.[14]

http://en.wikipedia.org/wiki/Avicenna

Thank you for the link, Peter. I already read this one some time ago. It made me buy David Galton's book : " In our own image ". It is true that Darwin didn't endorse Francis Galton's eugenics views at first, like mentioned on page 100 of the pdf, but he did get influenced by it later on (page 101-102).

In relation to your question, it seems to me that most of the works and theories that influenced historical outcomes can not be traced back to a single man. Evolution was already in the hindu myths (Dashāvatāras), and Charles Darwin's grandfather, Erasmus Darwin, introduced the idea of evolution in a chapter of one of his books (explaining it using differences in bird beaks!!). Then it seems that Alfred Russell Wallace send him his work, which made Charles Darwin get in a rush to publish "his" theory, making me wonder if he didn´t "borrow" some of Wallace´s own ideas and conclusions.

But it was really Thomas Malthus that influenced him mostly (though Malthus was proofed wrong in the long run). It seems Malthus and Erasmus Darwin influenced Charles Darwin's "Origin of Species", a book which then influenced Francis Galton, and Galton's eugenics views then, later on, influenced Charles' "On the Descent of Man", Which then all combined, influenced english, scandinavian, american and german biologists, creating the idea of superior races, superior genetics, and the horrible eugenic laws that followed. And let´s not forget, Eugenics is not a thing of the past. It is being put back on the discussion table, beeing dubbed: Modern Eugenics, which includes Transhumanism, Genetic enhancement, In-Vitro, etc., etc. Just read through some bioethics sites.

An example of a single persons investigation leading to changes in worldviews (as mentioned above by someones earlier comment), in my eyes at least, would be Thor Heyerdahl, and the conexions he made when studying and visiting Polynesia, from which he concluded that Polynesia got inhabited by pre-incan white, red-haired, bearded peoples who fled Peru when the Incas invaded, by small rafts. He then mounted the Kon-Tiki expedition and proofed it was at least possible, crossing the Pacific Ocean. He at first had virtually no help, and academics insisted he shouldn't even try it, as it would change orthodox ethnological theories.

Karl Heinrich Marx

@Ismat Beg:

Please consider adding your reflections on why you consider Karl Marx important. Which historical outcomes did Marx influence?

Karl Heinrich Marx: Due to his work in economics that laid the basis for the current understanding of labour and its relation to capital. It has also influenced much of subsequent economic thought. Marx attempted to separate key findings from ideological biases. Marx's ideas have had a profound impact on world politics and intellectual thought.

@Ismat Beg:

Another side of Karl Marx that may be of interest to followers of this thread is Marx's work in mathematics.

A collection of papers translated from Marx's Math. Rukopsii is given in

Mathematical Manuscripts of Karl Marx, 1983:

http://www.marxists.org/archive/marx/works/download/Marx_Mathematical_Manuscripts_1881.pdf

See, for example, the calculus of zeros, page 97 in the attached pdf file.

James, I'm completely surprised with the paper you made available for all of us. As a citizen born in a communist country, I was taught in school about Karl Marx and his achievements for the fate of "ordinary citizens belonging to the working class". And again, during my studies - here Marx was presented as a great philosopher. But the word "mathematics" never appeared in conjunction with his name. Contrary to his friend, Engels, who was considering the Moon's orbit (why it doesn't fall on the Earth) as an example of "struggle of classes". We, the physicists, were laughing. Well, perhaps my country (Poland) was not so communist as it was expected to be ...

Incredible and funny, isn't it?

Indeed, Marek, I was not aware of Marx's math either. James is an example of how to present a question, and then also provide excellent feedback with documents and fresh viewpoints. Excellent post. Thanks to all.

@Marek Wojciech Gutowski, @Ismat Beg:

I myself was surprised to find that that Karl Marx had an intense interest in mathematics.

From an economics perspective, this make sense, since economics has a solid foundation in mathematical analysis (and probability theory).

Slightly off-record: I think the world would be different (maybe better?) if James' opinion concerning economy and its relations to mathematical analysis and probability theory (and statistics) was indeed true. Economists' activity seems to be fueled not by mathematics but rather by an idea to make rich even richer. This is done rather by good deals, various agreements, and such, almost without any reference to math. They will never say "a derivative of a function" but "dynamics", they write minus sign *after* the number, and quite often confuse millions and billions during public presentations.

@Marek Wojciech Gutowski, J.F. Peters; I share James' opinion. Economists' are just giving tools like scientists. It depends how we use them. We need to use these theories carefully for the betterment of human being.

Returning to Karl Marx's interest in the calculus of zeros, notice an interest in zeros originated in the work of Leonard Euler on infinitely large and infinitely small numbers.

A good discussion about this can be found on page 5 in

E.I. Gordon, A.G. Kusraev, S.S.Kutateladze, Infinitesimal Analysis, Kluwer:

http://www.math.nsc.ru/LBRT/g2/english/ssk/infa_e.pdf

It was while Marx was writing his Capital that he became interested in mathematics. His mathematical studies were mainly related to the study of political economy. There is still the question concerning what enticed Marx to study the calculus of zeros.

Does anyone have any insights or thoughts about this?

For more about this, see p. 7 and elsewhere in

http://cfcul.fc.ul.pt/varios/Karl_Marx_small.pdf

@James F. Peters. Very interesting observations about Marx. Actually, in the second reference you gave us, it is mentioned that Marx got interested particularly in analytic functions, because he found them tractable "algebraically". Marx, it seems, was preoccupied about methodology. He wanted his ideas to have a coherence in a systematic way, and his usage of mathematics was adhered to that part which could be thought of as "Algorithmic". Thus, Marx was probably aiming at what today we would call numerical finding of roots, for two reasons: numerical analysis is among the most systematic, and algorithmic practically by definition (as Marx would have wanted). The second reason is that, Marx probably tried to understand how to find zeros (the roots of equations), but also he wanted to go further: to *systematically* find functions as solutions for a problem, given certain constraints, boundary and initial conditions. Marx probably glimpsed what Isaac Asimov later explicitly set up in his novel "I, Robot" (the final chapter), concerning the possibility that machines could take care of us, by balancing out any fluctuation, even of a sociological scale. All that within the reach of algorithmic mathematics.

The balance of equations concerns systematic modeling of processes: for example, in thermodynamics. The total energy of the system is equal to the energy used plus the energy wasted. You put everything in one side, and solve it, you get the "zeroes" of the equation or system of equations.

Come to think of it, if I am correct on my assumptions about Marx, he would not be the first to envisage such possibilities. In fact, it was Pierre-Simon, Marquis de Laplace, and I shamelessly quote from Wikipedia (Laplace's demon):

" We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

—Pierre Simon Laplace, A Philosophical Essay on Probabilities "

Naturally, for an intellect like Marx's, I can imagine that he would have found appealing to actually try and apply analysis to the problem of unbalances of sociological and economical nature. Instead of just thinking of "celestial bodies" and "atoms", and if really something (or someone) with sufficient information and systematic, algorithmic capabilities of analysis, then why not? society as a whole could be mathematically modeled, and figure out what could be the best for them.

Only mind you. Back at the time of Karl Marx, Chaos Theory was still considered "pathological functions", even for mathematicians like Henri Poincare. My point is, in balancing out equations, the balance is not "perfect", and then nonlinear (and recursive) effects tend to depart from assumed results, even for small differences in the initial conditions. Marx wouldn't have changed the core of his thinking, his ideals, but probably the way of implementation, and his way of explaining things about it.

Also notice that I didn't mention Kurt Godel (or others, like Church-Turing Halting problem), because, assuming you already have the correct algorithms, you only want to have systematic solutions, and one is not longer worried whether new "truths" could be found from the given axioms that make your mathematical algorithms correct.

Arturo,

Kurt Godel was a polymath, truly an amazing scholar.

Perhaps you already know that Godel was a Platonist (he considered mathematical objects to be part of the real world, not mere abstractions.

For more about this, see

C. Parsons, Platonism and mathematical intuition in Kurt Godel's world, The Bulletin of Symbolic Logic 1 (1), 1995, 44-74:

http://www.thatmarcusfamily.org/philosophy/Course_Websites/Intuitions_F09/Readings/Parsons_godel.pdf

Do you think that Godel view is correct?

Perhaps I forgot to mention that my commentary about Godel, was in connection to Karl Marx's interest in zeroes. Having algorithms you presume correct is aside from the possibility of finding new mathematics.

Now, as for Godel's Platonism, I think that at least as far as I have seen, mathematics is the same in *all* our observable Universe, hence, mathematics is something that can abstracted always one way or another from the real world, meaning that in many ways it has to be immanent to it.

Dear all, during a conference in the Netherlands one of the participants claimed the equivalence work produced by the Carnot heat engine and the theory of surplus value of Marx. All comment are very welcome

@Arturo Ortiz Tapia: ...Now, as for Godel's Platonism, I think that at least as far as I have seen, mathematics is the same in *all* our observable Universe, hence, mathematics is something that can abstracted always one way or another from the real world, meaning that in many ways it has to be immanent to it.

One problem with viewing mathematics as something that can be abstracted from the real world, is the tacit assumption that structures such as straight lines or unbounded sets (not found in nature) can be found in nature. Perhaps you will agree that it is more the case that line segments and bounded sets found in nature have been extended by mathematicians in an ideal world.

@ James F Peters. True, quite a few, if not most of mathematics is an extension from what is observed in Nature. The interesting thing is those same abstract extensions (like those you mentioned: The Straight Line without beginning or end, or unbounded sets) can be, and have been made by two independent people. Take Isaac Newton and Gottfried Leibniz, both having a claim on their own over calculus, one of the most remarkable abstractions ever made about Nature. My point being, I agree with you or whoever that there are mathematical concepts without apparent mirror in Nature. Having said that, how come two different people may arrive to the same abstraction? alright, both have the same entity in front (or inside) them: Nature. But once in the realm of insights and imagination, is it not amazing -and very telling on the "hidden" abstract part of Nature - that it is not the case that one arrives to one scenario, and the other person arrives to another, substantially different from the former?

Arturo,

It seems that the structures in nature that we perceive are very dependent our on perceptiveness. That is, those structures in nature that we perceive and that we conceptualise, generalise and extend in mathematics, depend on our perceptual acuity.

James, et al,

Studying math via problem solving reported by specific people personalizes math in important ways. Our US schools have depersonalized math and its history since the 1990s, a major error in judgment.

Our schools should be interested in the multiple math tool kits that have been created for specific purposes over the last 5000 years. Egyptian Eye of Horus binary math anticipated modern binary notation. Both were filled with round off errors. I learned modern computers love affair with fixed and floating point notations in the 1960s. Even IEEE comments on Eye of Horus connections to early computers.

Egypt solved its rational number round off problem 4000 years ago. Algorithms became questions for 3600 years:

How can an infinite series like 1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 add back missing 1/64 units by finding an exact finite series? Archimedes loved the approach so much that the first calculus found an exact finite series for the 1/4th geometric series

4A/3 A + A/4 + 1/16 + 1/64 + ... .. found 4A/3 = A + A/4 + A/12

By 800 AD Arabs replaced the Egyptian and Greek rational number system based on scaled rational numbers n/m byLCM m/m = mn/mp with the best divisors of mp used to find mn , that recorded concise unit fraction series. Arabs created a subtraction based rational number system that used an algorithm. Arabs and Fibonacci scaled

(n/p - 1/m ) =( mn - p)/mp with (mn - p) set to 1 to obtain 2-tern unit fraction series. When impossible like 4/13 a second LCM, in this case 1/18 found 3-term series

(4/13 = 1/4 + 1/18 + 1/468). Galileo used the notation in his square root method that would have been recognized by Fibonacci, Archimedes and Egyptians.

By 1600 AD, base 10 decimals, created by an algorithm and a form of the binomial theorem replaced Arab and medieval scaled rational numbers ... returning to Babylonian round off issues ... built into our early computers ... interesting how 4,000 year old solved problems return as unsolved problems ..

@James F Peters

So, the requirement for two people to arrive to the same abstraction is that they both have the same level of perceptiveness or acuity? Of course! I can go with that. Take Janos Bolyai and Lobachevsky (perhaps we could include in the lot Frederick Gauss). They studied Non-Euclidean geometries based on the same Euclidean postulate of parallel lines, and among them, they did not know the work of each other, so far as we know. Taking Gauss aside for a moment, we could say that Bolyai and Lobachevsky had comparable levels of perceptiveness (Gauss, for me IMHO, was amazingly far for perceptual than the former put together).