We all know how the discovery of calculus has changed the course of mathematics. There is hardly any branch of science that has not been effected by calculus. But what would have happened had calculus not been there? How would the world of science have proceeded in this kind of situation?
I find the question strange. As far as I can recall from books, it could have been discovered 1000 years before Newton and Leibniz (!) by Archimedes. He was very close to this discovery and was actually using infinitesimals in his methods.
Inevitably someone had discovered the calculus. The mathematical tools are like power tools: the problem to solve promotes their development. Perhaps a more interesting question would be what if someone other than Newton or Leibniz had developed calculus notation would be different then? Can you imagine the calculus with a different notation?
I find the question strange. As far as I can recall from books, it could have been discovered 1000 years before Newton and Leibniz (!) by Archimedes. He was very close to this discovery and was actually using infinitesimals in his methods.
Calculus is the mathematics of change, of calculating problems that are continually evolving. Now, mathematics is essentially the language of science and so science is not possible without mathematics. Change is a relative concept that can involve any pair of dimensions, time, force, mass, length, temperature, velocity, momentum etc. It is when our dimensions change that our study becomes a bit more complicated and calculus arises.
Can you think of anything that does not change in nature? I cannot imagine science and engineering as we know today without calculus.
Valery, Archimedes lived almost 2000 years before Newton and Leibniz.
Yes, of course. But a thousand years from this span was the dark ages, just plain mechanical clock time...
Of course, If Newton and Leibnitz would not have discovered Calculus, someone else would have discovered it anyway. As for Archimedes, he did find the area bounded by a parabola and the X-axis f0r X from 0 to a finite value. Yes, he was very near to discovering Calculus.
My question was actually aimed at making a list of things which would not have been possible had there been no calculus.
@Hemanta, I simply do not understand your question. I support Issam's view that calculus is the language to describe change, which is basic to nature. 'Things' are independent of their description. So, if calculus would not have been invented (I do not see discovery in math, I'm afraid), we still had natural processes. But maybe you mean 'ideas' instead of 'things'? Then you may find some ideas here
http://en.wikipedia.org/wiki/History_of_calculus#Applications
@Hemanta, your question may not be much interesting because had there wasn't calculus, it might have been something else, who knows?
There were four problems facing mathematicians in the sixteen hundreds. All four problems can thought of in terms of geometry, given the graph of a function. The first problem was that of finding the tangent line to a curve at a particular point, meaning a line that intersects the curve at a single point in the vicinity of the chosen point. The second problem involves finding the length of a curve from one point to another. The third problem was that of computing the area between the x-axis and the curve over a certain domain. The fourth problem involved finding the maximum and minimum values attained by a function for a specified domain.
These problems were solved with other methods (other than Calculus) such as Descartes' and Fermat's Methods for tangent line problems, Archimedes first method (method of exhaustion) for computing areas and so forth.
Moreover, fractional calculus came into being now and it has been found out that most of nowadays real world problems cannot be handled with the usual order calculus. Thus, one could asked, "What if FRACTIONAL calculus would not have been discovered?".
@Yusuf,
I would like to ask Hermanta's pardon for bringing up a question that is only loosely related to his original question: can you describe us a single important real world problem the solution of which requires fractional calculus?
Dear Ulrich,
Thank you. In another link, there was a discussion regarding fractional calculus. However, can it be more important than even Newton's calculus?
Dear Hemanta,
I read your answer saying that you would prefer not to see your topic abused with the fractional calculus issue since it was already discussed at another place.
If this is true, I ask you to rethink and change your mind. My reasons: we have here an opportunity that in a specialized question seldomly is addressed. Not to go into technical details but to put the whole thing into perspective by considering a single characteristic example. I'm sure that Yusuf has something to bring to the table to substantiate his dedicated assessment.
In mathematics as well as any other branch of science "one has to stand on the shoulder of giants" as did Newton and Leibniz. Their discovery was a just a step on a process that in principle was started by Euclid and he himself stood on top of other people before him.
While both newton and Leibniz reconceptualized what we know today as the Calculus, there was a lot of background done before them and it is most likely that in one form or another.
Dear Ulrich,
Okay then, let us wait for a reply from Yusuf to your question.
@Hemanta and Ulrich. What reply should I provide specifically? Thanks for the discussion.
Okay Ulrich.
1. Ultrasonic Wave Propagation in Human Cancellous Bone:
Reflection and transmission scattering operators are derived for a slab of cancellous bone in the elastic frame using Blot’s theory. Experimental results are compared with theoretical predictions for slow and fast waves transmitted through human cancellous bone samples.
2. Viscoelasticity:
Many works has been done such as a model suggested by G. W. Scott Blair, that for “intermediate” materials stress may be proportional to the stress derivative of “intermediate” (non-integer) order:
Where are material-dependent constant.
A. N. Gerasimov suggested a similar generalisation of the basic law of deformation using Caputo fractional derivative.
3. Neurophysiology of Eye Movements
Anastasio noticed some problems using classical integer-order models to describe the behaviour of premotor neurons. In order to overcome these difficulties, he proposed a fractional-order model in terms of the Laplace transform of the premotor neuron discharge rate and the Laplace transform of the angular velocity of the head in different form.
4. Anomalous diffusion and propagation problems...
Initial value problem involving the fractional diffusion equation. The solution involves the use of Caputo fractional derivatives which later reduced to one-parameter Mittag-Leffler function.
I forget to point out that the first application of fractional calculus was made by Abel(1802-1829) in the solution of an integral equation that arises in the formulation of the tautochronous problem. This problem deals with the determination of the shape of a frictionless plane curve through the origin in a vertical plane along which a particle of mass m can fall in a time that is independent of the starting position.
I have a lot to say about this @Ulrich, I hope those made some sense to you. Thanks.
Dear Yusuf, I hoped that you would make a decision and would select one particularly convincing application. If you, for instance, list 'Initial value problem involving the fractional diffusion equation' it is both logical and trivial that you need fractional calculus.
The discovery of calculus -- i.e., the fundamental theorem -- was an important milestone that combined two up til then separate pursuits, areas and tangents. The study of tangents becomes unavoidable as soon as one considers perspective geometry, which the Renaissance artists initiated. And once tangents and areas are both being studied independent of conic sections, the fundamental theorem will - and did -- pop up repeatedly in many contexts, one of which we call "the" discovery of calculus. So the more relevant question is "What if artists had never begun to use perspective?" and this is a question that actually has been explored in depth in non-scientific areas.
I would have had a higher GPA in college!
Seriously, this is like asking what would the world be like if we had not invented the wheel. I think the concept has always been entrenched in mathematics and physics, but it took a while to develop the rules and standardize the terminology to what we now know as calculus.
Since Jeff identifies 'calculus' with 'the fundamental theorem of calculus' I feel invited to tell a short story about the the fundamental theorem of calculus. That this theorem is an interesting and important matter is certainly not controversal among mathematicians. As we all know this, it is a theorem on functions of a real variable (to use old-fashioned terminology). The natural question is: what's about n-variables? That's not so easy, mathematicians say in unison. Finally they found a generalization of Stoke's theorem of vector calculus and declared this to be t h e generalization of the fundamental theorem to higher dimensions. I could hardly believe that it was not a fallacy when, in 2004, I discovered an absolutely verbatim generalization of the fundamental theorem from R^1 to R^n. It is among my articles that can be downloaded from RG and its title, of course, is: 'The Fundamental Theorem of Calculus in R^n'. There is a further article there: 'Higher-dimensional antiderivatives and the efficient computation of electrostatic potentials' which shows the practical background of this accidental discovery.
Mr Hemanth Baruah Good morning ! before this please think about what if birth of ones not happened by ones parents? your question reflects and arises so many doubts and so many base causes for which you can not answer some times beyond your level. And now coming to point CALCULUS means doing mathematics by finding difference and differentiating things and causes and functions so on like Brahma Vishnu Maheswara saying Srushti Sthithi Laya they are equivalent in nature there itself created calculus by Brahma it is not that finding out calculus now itself. It is very much essential and already exisitng one not discovered one after identifying they are saying that which was existed calling as discovered it is not a new one in mathematics calculus is part and parcel of mathematics. .. .N V Nagendram Asst. Prof. LBRCE,Mylavaram INDIA.
@Ulrich Your result (which I'll call FTC^n) is somewhat interesting and, apparently, occasionally useful, but you seem to be implying that it is a more interesting and fundamental generalization of FTC than the generalized Stokes's Theorem (GST), which is not the case. It is true that your result is more elementary than GST: FTC^n is nothing more than a repeated application of FTC to each coordinate of the function; however, there is no motivation for the introduction of the quantity you call f'(x), and no indication that f'(x) represents anything meaningful. Indeed, as your 'limit expression' shows, f'(x) is basis-dependent, so it will take different values if you change the coordinate system. This is not surprising, since the integral you call int_a^b f(x)dx represents the integral over the n-dimensional cuboid with diagonally-opposite vertices at a and b oriented with respect to the basis vectors, and is therefore also basis-dependent.
So FTC^n only gives the relation between a very particular differential operator and a very particular volume form, both of which are basis-dependent. It may well be a useful trick occasionally (particularly in physical applications where there is a natural basis such as current-time), but GST is very much the 'true' generalization of FTC to higher dimensions (and to other, stranger, spaces).
Forgive me if I've interpreted your post incorrectly, but you seem to be claiming that the generalization of FTC to GST by mathematicians is a somewhat unsatisfactory solution, that they only arrived at 'eventually', and that a simpler solution would somehow be more beneficial to mathematicians. That is not the case: mathematicians declare GST to be the natural generalization of FTC not because they can't come up with anything better, but because it takes the concepts behind the definitions in FTC, extends them in a natural way, defines new mathematical objects and yields a theorem which not only generalizes FTC but also shows that the equivalent formulation holds among the new objects. Thus, it gives us a deeper understanding of the result of FTC. FTC^n is a generalization of FTC (since taking n=1 yields FTC), but it gives no extra understanding, since it is just an inductino where the only non-trivial step is an application of FTC. Moreover, the result for n>1 is largely uninteresting to mathematicians.
On a brighter note, I don't think I'd seen that result before, so if I ever find myself needing to integrate a function f over an n-dimensional cuboid where f is the directional derivative of some other function in the direction (1,1,...,1) (divided by 2^n) then I'll know exactly what to do!
@John,
I read your contribution with pleasure and with most of your views I agree. Let me, nevertheless add a few observations.
Your statement
seems not well justified since I make some remarks on the occurence of this quantity in the literature and formulated my own judgement on its nature and particularity.
From your last paragraph it is evident that you misunderstood my one-limit formula for f'(x) to mean that it is a directional derivative. Of course, as an differential operator of order n it never (unless n=1) can be a directional derivative. Recalling the definition of the vertical bar will make this clear to you.
You may have not noticed that FTC^n provides a method to practically (i.e.in arbitraryly good approximation) integrate arbitrary functions f over arbitrary domains of R^n: Represent the domain as a union of cuboids, find an antiderivative F of f (symbolic or numeric) and evaluate F at the extremal points of the domain and add these suitably (an algorithm for this is given in the second article to which I referred).
Altough I formulated a bit provocative in my contribution here, my original articles hopefully don't convey the idea that GST would be superfluous if FTC^n would have been recognized earlier. Nevertheless I consider it a severe omission, at least from a pedagogic point of view, not to mention FTC^n when it comes to calulus in R^n.
@Ulrich,
First of all: sorry for not reading your paper correctly and for missing your definition of the vertical bar. I don't doubt that your method is useful sometimes and may even be useful as an algorithm for doing volume integrals.
I disagree that it is a pedagogic ommision not to mention FTC^n when it comes to calculus in R^n, since I feel it is a rather specialist result and might distract students away from learning about multidimensional derivatives and Stokes's theorem, which are of more fundamental importance.
In the end, I can't really think of any situation in which you might use FTC^n explicitly (unless, of course, you were trying to program a computer to do volume integrals). If I were given a volume integral over a cuboid, I would find it far easier to integrate it the standard way (one variable at a time) than I would to find an 'antiderivative' explicitly and then try to remember the definition of the vertical bar.
@John,
Try it with the field generated from a homogeneously charged cuboid. As I explain in both of the mentioned aricles (in the second in more detail) this is the problem for which I did both methods (explicit multiple integration, and usage of a antiderivative) and concluded that the latter was much more controllable and transparent (due to the fact that the anti-derivatives of the Coulomb potential are nice functions; they further, when found, can easily be shown by simple differentiation to be antiderivatives, whereas the huge expressions resulting from multiple integration are harder shown to be exactly right ).
No big deal. Folks would have worked in delta notation.
Actually, it is a rather informative exercise to think through what calculus actually does & try solving problems using delta notation (ie. before the limiting step taken in calculus).
Answers to some incredibly non-linear terms become reasonable.
An acquired taste - not for the faint-hearted... :)
@Hemanta,
Human ingenuity has always solved problems. Every old civilisation including India had developed its own methods to solve problems what is presently known as calculus and its invention/discovery attributed to Issac Newton. It is a pity that you have forgotten achievements our great Indian civilisation.
The truth is that Calculus was understood and practised in Malabar of India which had the School of Astronomy and Mathematics, even before Christ was born. It was used to express Vedic physical laws in mathematical terms and to calculate rates of change in Astronomy and Astrology. Aryabhatta (born in Kerala) in 2700 BC, was one of the early members of this school.
Dear Firoz Khan,
Indeed, I started this discussion to enumerate the uses of what we commonly know as calculus. I have not mentioned anything about Newton! In fact, I feel, the discovery should rather have been attributed to Leibnitz too, and not just to Newton! This is my personal feeling anyway; I may be wrong.
Coming to mathematics studied in ancient India, I would like to add the following two points. If you study Narada Purana, you will find that Narada, son of Lord Brahma, was taught how to find the area of a circle! The area thus found has been found to be correct up to two places of decimals! Later, much later, the area of a circle could be found using calculus. What I have just mentioned dates back much beyond 2700 BC. The mathematics that was used to find out the exact date and time of lunar and solar eclipses was in existence in India from time immemorial. There was no calculus at that time.
No, I have not forgotten the contributions of the Indians towards mathematics. In fact, I have studied Vedic Mathematics to some extent. However, I am not proficient in Sanskrit, and that is a hindrance towards learning Vedic Mathematics. Translations are of course available, but they are not quite sufficient perhaps.
I would like to repeat, I started this discussion to know how knowledge in modern times would have been affected had there been no calculus. I had no idea that this would hurt your feelings! Please forgive me if I have done so unknowingly.
@Hemanta Baruah,
I don't know what Purana tell, but I know with some conviction that to know calculation of area of a circle involves Pi and it was again Aryabhatta from Kerala’s (Malabar) school of Astronomy and Astrology ,who for the first time calculated its value correctly to four points of decimals.
@ Hemanta Baruah,
I understand your feelings and the purpose(ambition) of your question. If our country wouldn't come under the invasion of any foreign nationals, in particular if our education system would remain independent (i.e. free from the autocratic influences of bureaucrats ) from ancient stage, then perhaps another more efficient powerful mathematical engine would have been developed by using the methods , developments and techniques given by our Vedic Mathematics. For example, imagine the situation of computation of simple multiplication of two numbers having three or five digits; usually we immediately search for a calculator. But a normal student of fifth standard (from Saraswati Sisu Vidya Mandir in Orissa) can answer it orally by using the tricks given in Vedic Mathematics. (for details regarding the tricks, see the book : Quicker Mathematics by M. Tyra). Note, by the phrase "autocratic influences of bureaucrats", I mean that the persons devoted to mathematics would have been encouraged in the shape of physical, mental, psychological and financial form.
Pradhan,
Indeed in Vedic Mathematics, such short cut methods of multiplication and such other matters are there. However, more than that, in Vedic Mathematics, even the concept of infinity was discussed. 'Vedic Mathematics' was the mathematics of ancient India, done during the vedic period. Not just in the Vedas, in some of the 'puranas' too, there are certain mathematical matters included.
As for the other part of your post, I should like to add the following. In fact, during the ancient times, education was available for the people of high caste only! In addition, the Sanskrit language did not have its own script. In fact, during the reign of Emperor Harsha Bardhana, the Sanskrit scriptures were 'written' using the Devanagari script. Accordingly, earlier to that, the Vedas and the Puranas were available in unwritten form only to the people of high castes! The Vedas and the Puranas were just recited by the disciples of sages and gurus. That is why, Shruti was another name of the Vedas. Shruti means 'what you learn by hearing'.
This was the root reason why not all could study mathematics in those days. Much later, when the idea of zero was discovered in India, there was this trend already that studying was not exactly for all!
These are the basic reasons why Indian mathematics could not prosper much. The idea of zero was first taken out from India by the scholars who came with the Muslim invaders. The scholars in those countries developed 'Al zabarwal' which is currently known as school 'algebra'. The idea of algebra thereafter entered into Europe, and mathematics started to grow in Europe. Meanwhile, the Indians had already forgotten that they had an ancient tradition of mathematics already.
The fault was in India, in the caste system in India. Education was not available to the people of lower castes.
Accordingly, if mathematics have grown in Europe, the Europeans are not to be blamed! In Europe, there never was a caste system that debarred education!
Well, @Hemanta, we Europeans did not engage in castes, rather we had classes, e.g. the medieval cleric and the nobility (high level of education), the guilds (middle level of education, mostly vocational), and the farmers (low level of education). For the longest time of European history higher education was available only to clerics and nobility - and for men. In most countries women have been allowed to study at university only since the 19th century.
As a second thought: education for all is possible, but where it is not free, people face another obstacle accessing knowledge - and later contributing to science and other fields of study.
Dear All,
The question, as it looks, is too strange as each and every bits of information which finally grows into knowledge or wisdom, if it still really exists even now, is something divine; not to talk of the case of calculus alone. Whatever knowledge, that the human generation holds in their brains or stored somewhere have been given by a divine power and if calculus has to come on a date destined by the Almighty, He shall be doing so at his will and He did it and we had it and is still in good use till the last day.
Michael,
Right. In Europe, there was no caste system, although due to the class system education was not accessible to most of the people.
In India, the problem was indeed very different. The Sanskrit language did not have its script in the sense that it was a spoken language. The students in those old days had to learn by hearing and remembering the hymns.
The educational system in India during the ancient times was caste based. People of low castes were not allowed in those educational institutions. On top of that, the language through which education was available did not have script. Accordingly, the lower caste people were not in a position to get educated. Had Sanskrit been a written language, this problem would perhaps not have been there.
After the Sanskrit language owned the Devanagari script, the system was eased a bit. However, by that time it was too late. Already a culture had grown that education was not for all!
In Europe, the situation was different. The languages did have their own scripts. So in the form of books, knowledge was available. Therefore, in Europe knowledge grew more freely than in India.
In fact, even in the 21st century, the caste system is still there in our country. It is another matter that the caste system is no longer a hindrance in getting education. But the system is still there among the Hindus. You can imagine how strong the system had been in the old days.
@Filippo Salustri,
It is what I meant by in my earlier response, "Human ingenuity has always solved problems. Every old civilisation including India had developed its own method to solve problems what is presently known as calculus".
Dear Colleagues,
I have followed the interesting discussions on the history of calculus and its relation to and influence by the society of India as it was (and is still) organized in castes.
However, I would like to come back to the very beginning of this discussions, to the questions whether there were inventions which can be compared to the (independent and rather different) invention of calculus by Leibniz and Newton, beyond their quarrel to which any thing has been said; see, e.g., http://en.wikipedia.org/wiki/Leibniz%E2%80%93Newton_calculus_controversy. Here one also has to take into account Fermat who knew how to differentiate polynomials and to determine their extremas.
By the way, a similar discussion as on calculus could have been arised between Johann and Jacob Bernoulli and Newton concerning the invention of the calculus of variations, resp. the first problem of the calculus of variations, either the brachistochrone problem of Johann Bernoulli or the problem of minimum restistance of Newton.
I am particularly interested, whether there have been acchievements outside Europe concerning both calculus (differentiation and analytical integration). I do not mean here the determination of areas by geometrical means or, for example, the figures which contain the maximum area, resp. volumne, the circle, resp. the sphere, which has been proven by Zenodoros around 200 B.C. Since series were already known in India very early, there must have been the knowledge on limits.
Moreover, since I have learned that the rate of change of variables were also known, I would be very interested in some references.
Although I am not a historian in Mathematics I am very interested in these subjects; see my contributions on my web page http://www.ingmath.uni-bayreuth.de/FORSCHUNG/Pesch_veroeff.html: A15-17, B17, 37, 40.
Thanks in advance and best regards,
Hans Josef
This question is somewhat like the old joke about how, if electricity had not been invented, we would be watching television by the light of a candle.
The question is interesting -- albeit requiring counter-factual speculation, something historians avoid. Presumably we would all be solving problems by approximation, and declaring some problems too difficult to solve that way.
But Prof. Baruah raises interesting historical ruminations. The oral tradition in India was very, very important in that each word of religious mss. (all mss. were religious in early times) had to be committed to memory exactly. There was no room for paraphrasing or loose translation (think, mathematical proof). (See the article by Dalrymple in the New Yorker about finding a bard who could actually recite the Mahabharata exactly word for word over a series of nights.) Yes, this was affiliated with some form of the Varna or Jati (sub-caste) system. And yes, it excluded the vast majority of the populace, but the obverse was true - that it gave those on the "inside" license to concentrate on these oral mss. as also certain classes had access in the West. In addition, ancient India allowed the interplay of schools of philosophy with budding mathematical creativity. Zero and nirvana being identical conceptualizations. The multi-dimensionality of Hindu thought having direct correspondence in mathematics, etc.
Even in this case, someone else would have discovered it a couple of hundreds of years ago!
The different perspectives of this long list of comments is so enjoyable for me. I really thankful to researchgate.net for providing us this space, for me its so enriching....
@Prof. Dickason
Whatever you said is right and as a proud Indian national I adore ancient Indian achievements. Whatever, Hemanta is telling is the dark side of of our civilasation and we have to be reminded of it again and again so that fruits of development and progress should reach the historically downtrodden people and governance should be transparent in its approach to all people. In case of mathematics and science there is story retold by famous Indian historian and philosopher of science. According to his investigation when Al-Beruni with Mahmud of Ghazni came to India, Al-Beruni met a Brahmin both were impressed by knowledge of mathematics with each other, when Al-Berun asked him for leave the Brahmin advised him, you told everything so clearly with any ambiguity, never do it with other people, in public or with stranger. If you have to mean TWO, say say Moon and Sun so that your knowledge remains in a specific circle. This type of encryption in the form of mythology, rituals in teaching and leaning makes it decipher the ancient knowledge. It is a complete denial of knowledge outside the inner circle. It has made geometry of Hindu mythology so difficult that meaning and purpose conveyed is difficult to be comprehended by Sanskrit speaking Brahmins presently.
Firoz Khan,
Yes, Al-Beruni was the scholar concerned! Thank you for reminding this. It was because this great scholar accompanied Mahmud of Ghazni, Indian mathematics could spread everywhere.
All sciences are in nature and all nature is numbering. When Human needed counting, they do it first with their fingers and stones or branches. Later they discovered + and – operators, and by a pastime they come across * and / operators…. And …..Mathematics was discovered by the fantastic mental imagination. Later zero was discovered as it is not stated in fingers ….. and opens afterward doors to the numerical technology , computers and high tech… without calculus no theories, no technology, no development could happen in the human civilization
While all of the conversation about Indian mathematics is interesting, the question originally posed --"But what would have happened had calculus not been there? How would the world of science have proceeded in this kind of situation? " -- remains. As an historian of mathematics, I think a prior question needs to be answered first. What constitutes calculus? As various writers have observed, many aspects of what we now call calculus were known, often for a long time, before Newton and Leibniz. What was it that they did to make it calculus? Depending on the answer to that question, we can then ask if the questions posed by science could have been answered without calculus.
Prof. Sinjab reminds us calculus is the analysis/science of change. Prof. Rosenstein reminds us that we need to be clear about our terminology, in that calculus in various forms (calculi?) has/have existed for a long time.
I would like to add that (cognitive psychology scholars have identified this since about 2000) for human beings there is such a thing as "change blindness" -- as also other "blindnesses" such as "attention blindness" (perhaps better would be "inattention blindness" or "distraction blindness"), etc. As a species (and across cultures) we are not good at observing, assessing, and analyzing change(s). So the major contribution of mathematicians working on dimensions of calculus through its long history is they have filled in for the limitations in human consciousness -- a great advance for the species (and, ultimately hopefully, for other species and ecosystems too).
I find the question strange, too. Really, we can't talk about something that did not occur. It's impossible to know. As the professor Dickason said, "requiring counter-factual speculation, something historians avoid".
Indeed, this question is definitely strange. In my former counter-questions I have asked who else than Fermat, Leibniz, and Newton can be considered as contributers or inventors of Calculus (analytical, not geometrical differentiation and integration) outside Europe? However, nobody followed this quiestion. There were some specialists involved at the beginning of this endless story who might know the if there exist somebody in Asia.
Dear Hans,
As far as analytical matters of calculus are concerned, no, no one there was in Asia to have invented it.
However, what I meant was in which way science would have proceeded had there been no calculus.
Dear Nelson,
In Bayesian Statistics, they use a concept of finding out the probability of an event that had already happened in the past! Don't you find that peculiar? At least, the Bayesians find that fine enough!
In my case, I am asking what would have happened if a certain historical event had not taken place! At least, I am not trying to find out the probability of 'discovery of calculus'!
Since the 15th century, the art of solving algebraic equations also contained methods to compute fractional or decimal approximations of the roots. The procedure was to expand the previous approximation plus error term using the binomial theorem and then disregard everything of higher order than the linear. This should remind of Newton's method. The most comprehensive collection of instructive examples is attributed to Viete at the end of the 16th century. See Cajori, A History of (1919) Mathematics resp. (1928) Mathematical Notation.
Dear Hemanta,
Universe is there before Calculus invention. Calculus is a device (made by human) that explains and/or helping scientist to understand a tiny part of the Excellence of Universe. That is the changing behavior of anything. Using Mathematical induction (another tool) we now able to say that Universe and it characteristics if full of Organized principles operating with perfect harmony whether you understand it or not. Therefore, my point is - laws of Universe/Nature and its Excellence have been there before any Mathematical principals (Theory) developed. And it will continue to be there even scientists developed many other theories like calculus. This techniques (calculus etc. ) gives us to get a virtual identity of this Universe. For more details - I will definitely recommend you to read Quantum Principal (may help to understand a bit more about our limitation of Knowledge). If you feel - it is too tough to understand Quantum Theory - then please try to read consciousness & Hinduism. At least later will help you to understand the concept, motivation & design of this Universe and its components.
Dear Asit,
Incidentally, I would not like to accept that Quantum Mechanics, or any other field of mathematics, applied or pure, is 'too tough' to understand! In my eyes, mathematics is the best kind of poetry! But of course, it would be poetry only if you can find the rhythm in every mathematical result you come across!
It is another matter that some people may actually find even the English Grammar 'too tough' to understand, and end up writing horrible English! Should that mean that the English language is tough to write in?
As for ancient Indian scriptures, I have read most of them perhaps! Could you please cite a specific example of an ancient Indian scripture, other than the Gita perhaps, that describes consciousness in particular in detail?
I have found your answer quite interesting indeed, although it is hardly related to my question!
My recommendation to you to visit www.youtube.com and type "Amit Goswami - Quantum Mechanics". There is an International Forum on Quantum Physics, Consciousness also. You may Google it. Overall perception of that forum is "OM=mC2". You may find some input there too.
Reading a scriptures may not always give you the perception. Please read Swami Vivekannada's views in this regard. Unless you have the perception - you can't gain knowledge. I find that "Knowledge will be limited if ego comes in front of it".
Again I am saying that Universe is extended from infinity to infinity. Man's created mathematics extended within a finite domain. Due to induction Man thinks that their knowledge is well enough to understand the full extend of Universe - in true sense - definitely impossible only with mathematics, you need philosophical realization of The Universe.
This is what I realized. I did not find any theory and practice invented by Human civilization that can deals with infinity. How close our invention is - still from that point to infinity is infinitely many steps.
I appreciate your comments on my comments. And once again asked myself," what is my purpose?".
Appreciate your future comments.
@Hemanta & Asit
I find historically speaking Hemanta is to a great extent correct that there was no concept of Advaita Vedanta which states that there is no difference between Brahman and Atman, or “All there is...is Consciousness” in the sense of Asit used it prior to Gita.
Aryans were simple people with their living gods with them and a concept of One but inactive God, inactive in the sense It has nothing to do with the affairs of the world or humans, simply a force/energy. Latter, they developed their mythology woven around Hindu (indigenous) philosophy and accepting all deities of locals. Scholars believe that the Vedantic philosophy is as old as the Vedas, since the basic ideas of the Vedanta systems are derived from the Vedas and attempts were made to relate it to Rgveda. However, in spite of claims that there existed 99 canons of Advaita Vedanta-tradition prior to pre-Shankara doctrines and sayings are traced in the works of the later schools therefore this evidence does not credit existence of this school of philosophy prior to Adi or First Shankara who expanded it on the interpretation of Gita and teaching of his Guru. The fact remains that the key source texts for all schools of Vedanta are the Prasthanatrayi, the canonical texts consisting of the Upanishads, the Bhagvad Gita and the Brahma Sutras, of which they give a philosophical interpretation and elucidation.
Asit, if you perceive reality in QM it is your choice and you can become another Tao of Physics.
Mr. Khan,
Thanks for your beautiful comments. My knowledge did come from reading scriptures or book. It is simply - my realization though using mathematics in many system. Initially I had an idea that - Mathematics can solve everything in the Universe, Then in practice when I was applying it in various fields - I find system is not like what I thought. More, precisely, if the boundary of the system extended in a domain which is infinite then no mathematics can solve it. Universe is like that.
The other side of it - is to realize it from its dual (complementary) space. That domain is untouched. Mathematics does not work there. That is your true perception - knowledge of Infinity.
Dr. Sha,
For my infinity perception, please visit:
https://www.researchgate.net/post/Does_infinity_exist_or_it_is_a_product_of_human_mind
Calculus became possible and necessary with the scientific analysis of large scale dynamical processes. To do experiments with large scale dynamic processes, with noticeable effects in a short time, one needs a concentrated but finely controllable energy source. The historically first such source is gun powder, the first "scientific" experiments were precision shooting competitions of the waffenmeister in the 14-16th century. Loading, aiming and shooting a canon is indeed a rather precisely repeatable experiment.
A second ingredient to calculus, or the quantitative treatment of dynamical processes, is the time unit that stays constant over time and is infinitely divisible. In consequence the invention of the time line that allowed for the first time to solve the paradoxes of Zeno. It would not do to need in the morning a different theory of a dynamical process than at noon; or if theory had to change with the seasons. The introduction of this idea (measuring time by the swings of a pendulum of fixed length) into science is commonly attributed to Galilei. And unsurprisingly, one of the biggest achievements in physics of Galilei was the parabolic shape of the path followed by a cannon ball.
After these two achievements, the invention of calculus was "in the air", and was enthusiastically picked up once formulated.
Catapults and trebuchets would also qualify for "large scale dynamical process", however, calibration is more complicated than measuring a volume of gun powder. Were there precision shooting matches with trebuchets?
For some reason this question launches strange reflexions on conciouness and philosophies. But it also shows some confusion. If math is a common laguage for physics it is not physics though.
Can one really speak about the "invention" of calculus. Calculus is a synthesis of a very large corpus of mathematical techniques developed for centuries, starting with the Greeks reasonning on infinetesimals (like in the Zeno paradoxe) and certainly not finished nowadays. In between, one finds Newton and Leibnitz interested in the mouvement of planets, but also Archimedes and Pascal trying to calculate the area of complexe surfaces (no movement in that case). Progress in this field is related to the progress in the understanding of convergent series in the XIXth century*.
- Only the word calculus has been invented.
- If it had not, the world would be the same.
- The ways that have been followed by physicists could of course have been different as physics cannot be confused with the math that it uses as a language.
Think to the present situation: what is going to survive between, strings, branes or loops? Maybe none! But these corpus would remain nice math anyway.
*The Zeno paradoxe should not be taught anymore, because most poeple do not know it has been solved for long and they waste time philosophing around.
@Charles,
nevertheless 'quantum Zeno effect' is quite iteresting (and real!).
Hemanta,
Do you know - what you are! Slave of this Universe! Do you know what Calculus mean to you? One of your arms as a freedom fighter to get yourself free from the rules of the Nature/universe. Your Calculus is a toy to play with in your material life in a finite domain.
No human invention is permanent. Everyday, millions mathematical theory will come and go - but Universe will go with its own rule.
X, Y, Z axis you defined in your vision is your virtual reality - May be you add some more dimensions to it. It is still a virtual reality.
Before, Calculus, Human was there. Even animal does to know Calculus. They survived. If you observed an animal, for example, Bird. How civilized they are. They do not have savings account, checking account, retirement plan, stock market., capitalism , industrial revolutions etc. They do have one thing - that we do not have as Human - is Peace & Harmony.
"Calculus" gives you much pleasure in your virtual world.
@Ulrich. Yes, really, QM Zeno effect is interesting. I was only speaking of the Greek version that confuses non-physicists and make them loose their time. C4H
Asit, be careful with birds. They are just cleverly disguised and evolved velociraptors. One day the chicken will remember their heritage.... And I'm sure that the vultures will be grateful that you consider them civilized.
And 95% of humanity (if you take the ones that know what the words mean) would vehemently contradict you about "infinitesimal calculus gives much pleasure."
While one cannot and should not speculate too much on such ;what if' questions because small changes (no pun intended) can lead to large consequence in history, there is a possible answer, by demonstration. The most precise astronomical calculations of orbits, before Kepler's laws came along, were done by a small group of mathematicians in Keralam (southmost state of India) using approximations through infinitesimals. The had almost everything in hand to do calculus without limits. They had realized the 'harmonic' nature of the second differences etc. There are several studies now, some of which speculate that this knowledge could have traveled to Europe due to strong trade or through Jesuit priests. In any case, published texts of 'Ganita Yukthi Bhasha' in original (Malayalam) with interpretation and detailed commentary in English are now available.
Dear Lutz,
Nice comments on birds' life. Everything you see in this Universe (either living or non-living) has at least two qualities. One half of that quality may attract you and other half may not. When you accept something in NATURE you have to accept both of these qualities. Without this dualism nothing can be stabilized in the sense of Mathematics. I am not sure in the infinity whether this dualism is true or false.
Dear Asit,
You actually mean that no one can change the rules of the Universe. You are right.
However, without calculus, would it have been possible for us to study Nature? But of course, animals, as you have mentioned, do show certain traits of understanding Nature. They do not use calculus anyway!
Hemanta,
Your thought is good - I have no question about it. But there are many many spices in this Nature. They do not have mathematical formula with them - Look how beautifully they understand the NATURE/Universe without having any trouble!. In other words
I would say there are some alternate paths to understand the Universe without using our calculus and higher mathematical theories.
It seems same as one person asked: What had happened if your father was not there? Person replied, it makes more sense if my mother was not there? Who cares about father? Mom can find many!
Sarjinder Singh,
What branch of Science you talking about I am not sure. The topic says that if the calculus is not there.......? Your concern is "Operation" - if you operate two elements you may create and/or not create a third element. Rule of operation could be your choice!
How the third element will develop depending on your operation between the elements. This is auto catalytic process and that is local in time. Biggest concern is - what is the importance of calculus in Science and its purpose to unfold complexities of Nature/Universe. Your answers to these concerns will be much appreciated.
Thanks!
Dear Joseph,
You are right. However, deciphering such things is a problem. So an exact description of how much of those things resemble calculus is difficult.
Joseph, did You just switch the topic from "calculus" in its modern meaning of "inifinitesimal calculus" to a more general meaning including "doing arithmetic calculations"?
And, it has been my understanding that since the not so recent discovery of the Rosetta stone, much progress has been made in deciphering the ancient mediterranean and mesopotamian scripts. Surprisingly, most of the clay tablets from mesopotamia were found to be bookkeeping entries of storage and delivery of goods, very few military communications and even less pure prosa.
I imagine empirical achievements in the physical and social sciences would far less advanced. Technology today may have been more similar to that of the Roman Empire than the technology we are familiar with in 2013.
Dear Joseph,
It would be interesting to know more about those fragments.
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