Comments on Calculus

The dispute between Sir Isaac Newton and Gottfried Leibniz over who invented calculus.

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  • Max Wallis

    Patricia Fara responded to my Comment (below) that she'd addressed the issue in her book 'Newton: The making of genius'.I didn't think she does cover the actual questions (I replied):# Did the feud hold back mathematics?nor whether > Newton's belief in an interventionist god, > attacks on his scientist contemporaries, > and converting the Royal Society into a personal enterprisewere harmful to the progress of science.I referred to a Review of her book that concludes:There is a problem with the well-worn postmodernist point that Newton is what we have made him: Why have we made him anything at all? The answer to that question must surely involve science and ideas, as well as scientists and their images. After all, the cantankerous and elderly Newton, who has always vied for public attention alongside the promising youth with his apple, fought his intellectual battles with Leibniz and others to prove that he had actually been right, not just to seem that way to generations astonished by his skulduggery. [[Scott Mandelbrote, History, Peterhouse, University of Cambridge]].That helps us understand why modern historians of science tend to duck questions re. progressing science. So Melvyn needs to press the point on behalf of those who believe in scientific progress.

  • Newton and Leibniz

    Dear Melvyn, I very much enjoyed the "In Our Time" discussion of some of the principal mathematical ideas of Newton and Leibniz. I would like to add support for your own astute observation, confirmed by your guests, that Calculus was/is an idea born purely of the creative human imagination. That is to say natural philosophy as an approach of pure (objective) observation of the natural world, could never alone have led to its invention. Calculus was not a "discovery" it was an invention of mind giving rise to the ‘tools’ of differentiation and integration. What was the simple idea? The idea began with the idea (shared by many at the time) that to calculate the area beneath a curve you needed to divide it up into infinitesimally small slithers and then add them up. Hence the root of the word calculus (which I don’t think was noted in the program) is related to small stones or pebbles used for counting.Progress in human understanding is a multi-layered affair involving ideas building upon) earlier ideas/observations/discoveries. Newton was sufficiently humble to observe that in the formulation of his own ideas he had been "standing on the shoulders" of his forebears. The ‘evolution’ of human ideas towards a progressive understanding of universal phenomena has been a ‘sedimentary’ affair. Thus the invention of the conceptual ‘tools’ of mathematics and algebra were necessary to the eventual evolution of the ‘tool’ of calculus. At root this apparently intellectual creation was born of humans (philosophers?) seeking enlightenment as to "how things worked". If we go further back in history to the beginning of this path we find Archimedes who became fascinated with the practical question of the displacement of water. His self striking revelation was that the up-thrust on a material body, completely of partially immersed in a fluid, is equal to the weight of fluid displaced. This of course was the awakening of observational thought to the relevance of volume mass and density to ‘displacement’. Volume, mass and density were therefore continuous variables of central importance to Newton’s own laws of motion as applied to the paths/trajectories followed by objects under the force of gravity. Thus the time was ripe for the development of a mathematics of curvilinear relationships.The key problem for Newton, Leibniz (and others) was to minimise the inaccuracy inherent in using polygons (using the point gradient/tangent as one of its sides) to represent continuous rather than ‘blocky’ change. The actually leap in imagination that made the difference was the idea of introducing infinitesimally small changes (an unknown changes "dx" for a given change ‘dy’) which when squared in the resulting summation could be left out. That is to say the square (or cube in the case of volume) of an infinitesimally small fraction approaches zero and thus can be effectively taken out of the equation i.e. a fraction multiplied by itself gets smaller. Essentially it was the idea of introducing unknown infinitesimally small changes and then later removing them for the sake of simplification of the equation that was revolutionary. In this way a point gradient could then be simply estimated with the tool of differentiation and area could be estimated easily with integration.Seeking to resolve the question of who actually had this leap in imagination in the first place (perhaps, more a question of vanity) is less important than the question of why and when it was it invented. Essentially it was born out of the necessity of the time to solve a real world problem; a problem that had, one might say, "come of age". Throughout human history "necessity" of course has often been the true ‘mother’ of invention … but why had this question become so important? Why were philosopher/mathematicians so dedicated to resolving this question? I believe apart from the practical benefits of simplifying the mathematics of continuous change there were core philosophical reasons to do with Newton’s understanding of matter (as inert substance) and his equations of moti

  • Ian Banks - Leibniz vs Newton

    I think IOT is an excellent programme. Thank heavens for podcasts. I would miss it without them.I enjoyed the Leibniz v Newton programme this week but felt it meandered around the subject a bit too much.I wanted to learn more so I visited the IOT website for the first time hoping to find references to suitable next step material in print and on the web. I was disappointed not to see anything.This would be easy for your guests to provide and a big improvement to the usefulness of the website.

  • David Wolfe - Leibniz and Newton

    From an aborted emailDear Lord BraggAs someone deeply interested in my field of physics, and especially its history and philosophy, I was most interested in Thursday's programme. I was puzzled about the reference to Leonhard Euler as a Russian matehmatician (sorry to worry about such trivia). Euler was, of course, a brilliant Swiss and his portrait was on the Swiss 10 franc note for years. He was unable to find a decent job in Switzerland due to the monopoly on such positions by the Bernoulli family (deeply involved with the Leibniz-Newton controversy of course). Therefore, he went to St Petersburg where he spent much of his working life.On a personal note, I was a bit frustrated at the programme as I have written a 50,000 word monograph on Newton and his enmity with Hooke, Flamsteed and Leibniz and must admit to a small feeling of jealousy. "Too much high-powered competition" in the book world to get it published I'm told. But not important.On a much more important note, I admire your weekly programme immensely and am an avid listener. I cannot imagine a more pleasureable career than to spend time in this most Renaissance manner. Well done and please continue.Best wishesDavid WolfeDavid WolfeProfessor of Physics EmeritusUniversity of New Mexico

  • John - Leibnitz vs Newton

    We’ve had the measurement problem in physics,this was about the publication problem in mathematics. Newton was secretive about his invention of ‘fluxions’(calculus) as his method was still clumsy and obscure.He wrote in his letters and papers about calculus but he was not confident enough to publish in 1660s or the 1670s due to fear of criticism. Leibnitz independently invented calculus in 1674-5.Coming out of the 30 years war Leibnitz was motivated to invent a powerful universal tool of mathematics expressed in a language of symbols.As your speaker said they came to calculus from different approaches: Newton’s focus is Time as the absolute measure,temporal change,variation in time,rates of variation in time.We are not in the world of the Greek circle,timeless,perfect,but in the world of planetary ellipses from Kepler’s discoveries.How to represent the motion of a body in an instant,the time through which a planet has been orbiting.The new mathematics defined and operated with instantaneous motion and the infinitesimal step.Newton looked to the real world to evolve calculus:rates of change and movement and the slope of a curve.Leibnitz’s emphasis is on changing Space:he is finding areas under curves by dividing areas up into thin slices and thenadding them all together(integration). Newton attacked Leibnitz by saying infinitesimally small things don’t exist,but he was glossing over what he did and Leibnitz made explicit. Leibnitz is showing how quantities vary with each other and to capturetheir representation with the superposition of areas(the organization of spacesand the relationship between spaces) in the geometry of space. His inventionof the symbolic calculus notation was a great imaginative act and as a philosopher of genius he developed analytical logic in a world based on reason not power.He invented the word ‘dynamics’ for science, and mathematics becomes a dynamic modeof thought.He gave a rigorous meaning to the concept of an infinitesimal step.He was a mathematician and a mathematical physicist of genius. Newton’s campaign against Leibnitz(using the whole armoury of the Royal society) for plagiarism was effective and damaging.The differential and integral calculus was invented independently by Newton and Leibnitz.It is Newton who emerged with credit.He was given a state funeral, Leibnitz died dishonoured and unknown. However,a careful examination of the papers of Leibniz and Newton shows that they arrived at their results independently, with Leibniz starting first with integration and Newton with differentiation.We must view them as two monads coming independentlyto the same invention. Leibnitz’s contribution was to provide a clear set of rules for manipulating infinitesimal quantities, also his emphasis on formalism-– he often spent days determining appropriate symbols for concepts.His notation survived andthe name’calculus’ carrying the history of mathematics on its shoulders.Therefore we honour Leibnitz.

  • Jane - Leibniz and Newton

    Can't remember it very well, but I once read an aphorism to do with the power of 'an idea whose time has come'. I've noticed that it's often several people who 'have' the same idea somewhat simultaneously and wonder how much the idea comes 'from' and how much 'to' the individual. Sometimes it's almost as though part of the process is a drawing together or coinciding of particular elements or aspects within the general framework until a point of certain mass is reached and the receptive or prepared mind can grasp it.....necessity often, though not always, being the mother of invention. When it's so distantly posthumous, is all of this "who was first" stuff so important? There may be somebody totally unknown to mainstream history who was truly "the first".... and probably thousands have remained 'unsung' throughout the course of centuries. I'm not being disrespectful, I take a real delight in society's genuinely 'bright sparks', past and present, but find it hard to put them on high pedestals (with the single exception of the wonderful Alan Bennett). I suppose I have a rather pleasurable "Wow, thanks mate" feeling - a warm gratitude for their contribution to the scheme of things. Adulation I find rather unhealthy. Anyway, it's a tribute to (the also wonderful) Melvyn that he brought interest to a subject which, in the course of our general education, drained the life force and confidence out of far too many of us....."Thanks mate!" Best wishes as always.

  • Max Wallis - Leibniz & Newton

    Too much time spent on preliminaries to the feud, but we did learn that# the English all sided with Newton# Newton re-wrote history – like the apple myth, which accorded himself priority over gravitation, despite Robert Hooke’s lecture to the Royal Society (1666) on gravitation being "one of the most Universall Active Principles in the World"# when Leibniz appealed to the Royal Society (1708), Newton wrote the unsigned report judging himself in the right.Did the feud hold back mathematics? Jackie Stedall’s denial did not convince, given that English maths stuck with Newton’s clumsy formulation, while the Leibniz school’s symbolic algebraic approach roared ahead on the continent, especially with Euler (1736).Melvyn Bragg’s Newsletter admits to Newton being "secretive, mean-minded, perhaps a bit of a plagiarist" but loyally declares "none of this takes away at all from his prime place in the history of science".High time, surely, that the English admitted Newton’s belief in an interventionist god, attacks on his scientist contemporaries, and converting the Royal Society into a personal endeavour were all harmful to the progress of science.I wrote challenging Bragg in "People & Science of last March, but still await his response www.britishscienceassociation.org/web/News/ReportsandPublications/Magazine/MagazineArchive/index.htm

  • James Baring - Newton and Leibnitz

    I agree about the 'nugget' referenced by Peter Household, but it worries me that the 'best of all possible worlds' argument is not better understood by all. We must by definition live in the best of all possible worlds to the extent that the problems in it are either of our own making or dependent on us for their removal or resolution. That includes the possible loss of an eye, leg or ear, as the alternative would be to never have any of the aforementioned in the first place. The emphasis is on the word 'possible', and we should take into account the last 14 billion years of change. Leibnitz was certainly right about Voltaire, and in turn only approximately right when he said the only really worthwhile occupation is the cultivation of the vine.

  • Berrada.M.Ali

    the debate of this week on the history of calculus was good and instructive, however, much could be done if the invited Professors have dedicated much time to the explanation of the historical backgroung about the mathematical concepts used before the invention of calculus , for example the concept of function

  • Boris - Leibniz vs Newton

    The bias towards Newton in Britain is irrational and overwhelmingly nationalistic....which is hardly scientific. Your programme reflected this! For some reason the English can't think kindly of us Europeans.

  • "The System of the World"

    I'd just like to recommend Neal Stephenson's wonderful novel "The System of the World" as further reading on the subject of Leibniz and Newton and their times. He is particularly good on Leibniz and the Hanovarian princesses.One thing still puzzles me: Leibniz came at calculus by thinking about dividing space (integral calculus) while Newton was more interested in time (differential calculus) so it seems odd that Newton's followers primarily worked with geometry while Leibniz's school went in an algebraic direction. The old idea that differential calculus is "easier" that integral was also trotted out, and certainly this is the order it is usually taught in. But I certainly found it the other way round - I think it depends on whether you are primarily a visual or abstract thinker. Unfortunately most mathematicians and maths teachers are the latter and assume everyone else is the same!Jim (below): You must have been listening to a different program! It was very clearly brought out that Newton had control of the Royal Society and virtually dictated it's report on the controversy.

  • Kevan Martin - Leibniz & Newton

    Isn't it interesting that the two greatest English scientists - Darwin and Newton - both got into a tangle about priority of publication? Darwin handled his relations with Alfred Wallace rather better that Newton did with Leibniz. The problem of assigning priority was nominally solved by the founding in 1665 of the first scientific journals - Journal des Scavans, and the Philosophical Transactions of the Royal Society. The date of reception of the manuscript, or the date of publication then fixed the priority. With the advent of e-publication ahead of print, this mark in time has become blurred because the print version, which is the cited version, may only appear as late as a year after the e-publication. Tralaticious citations also blur greatly issues of priority in science, and now if the article cannot be easily downloaded, it is never read, let alone cited, even if it describes the original discovery.

  • Leibnitz and Newton:calculus

    Leibnitz invented calculus not knowingNewton had already done so,and he published it before Newton did:in fact it is his notation,not Newton's,that we use to this day.And he was among the greatest of philosophers.Leibnitz'swork was published in 1684,Newton's in 1687.The consequent dispute as to priority was unfortunate,and discreditable to all parties.Just thinkwhat would have happened if Watson published his theory of evolution before Darwin did,who would we remembertoday?

  • Muray Lamshed

    Yes, I noted it was said Euler was Russiam. Well, Catherine employed him and he is buried in St Petersburg but he was Swiss. Another great broadcast.

  • Peter Household - Leibniz & Newton

    If I were to single out one nugget, it would be this (around minute 29 of the podcast):- Jackie Stedall was explaining about infinitesimal quantities, which she said marked a clear break from Greek mathematics. Infinitesimal quantities may or may not exist, but it is the manipulation of them which led to the calculus. Melvyn asked: "It’s entirely imaginative in its origins then?" "Yes, it is." "So we’re talking about a work of the imagination?" "We are really, yes." She went on to say that there were great debates even in the 17th century about whether these infinitely small quantities are something or nothing. Either way you get a paradox. Both Leibniz and Newton knew this, and failed to resolve it. But the calculus was so powerful that no-one was going to give it up "because there a few arguments down at the foundations".

  • Themistocles, Leibniz vs Newton

    I think this was one of the better discussions. Two points of importance: Firstly there are good reasons to suggest that it was perfectly possible for both Leibniz and Newton to reach their conclusions independently. The key to this was overlooked although mentioned in the discussion. They both had classical Greek and Latin education as this is demonstrated through their mathematics and their philosophy. This is the common denominator. It is certain that they both were familiar with Euclid’s ‘Elements’ and hence with Book 10 which contains Eudoxus’s Method of Exhaustion (proposition 10.1). Eudoxus was a Greek mathematician from Cnidus that lived in the 4th century BC. According to this proposition if one subtracts from a quantity at least half of it and from the remainder at least its half and keep doing so FOR EVER, eventually there will remain something smaller than any pre-assigned quantity. Therefore, the idea of infinite regression was known to the Greeks and hence to both Newton and Leibniz to start with (it is true however that the Greeks did not have the instinct to develop it further.) This,together with Cavalieri's Geometria indivisibilibus (Italian mathematician of the 17th century) made the almost simultaneous discovery of calculus possible. Secondly, the philosophical differences between Leibniz and Newton are wonderfully reminiscent of the philosophical dichotomy that was developed in 4th century Athens through the defeat of democracy and the subsequent dominance of the undemocratic Platonic school of though (see Popper’s Open Society or the later work of Castoriades.) Newton with his belief in static matter and in an interventionist god that provides an exogenous rule to society strongly reminds of Plato. Leibniz on the other hand, with his theory of dynamics, his thesis of calculating out our differences and his belief in an non-interventionist god strongly reminds of the stoics and the philosophical movement that accompanied the birth and growth of democracy in Athens. Finally, the discussion totally missed the strangest fact in the history of Calculus: In 1906 a manuscript of Archimedes was found in which the mathematician discuses the validity of a method which seems to give correct results. He would cut an area or volume into infinitely many lines of parts, which (imaginary) he would place at one end of a lever as to balance the whole area or volume at the other end. This is how Leibniz started. However, neither Leibniz nor Newton knew of Archimedes' contribution. Why? His manuscript was washed away and then used by the Christian monks as raw materials for writing bibles. The original was written almost two thousand years before Newton. The big question for me is why having thought of so much, Archimedes never found the courage to go further?

  • Tom Milner-Gulland - Newton-and-Leibniz

    To my mind, the most impressive insight in Leibniz’ philosophy is in his argument that to speak of the difference between any two different spaces -- that is, worlds -- is to speak of a difference in the conditioning of their sensibilities. This is a superb insight, one the philosophical likes of which one doesn't find in the works of Newton. God creates beings to live in different universes (worlds), not by creating immense new volumes of space, but by manipulating the arrangement of their senses. But it begs the question of how *difference* is defined on an absolute scale, and one must surely be led to the conclusion that all differences are defined with reference only to time. Space is merely a vessel for temporal experience. So time is the conduit to the absolute, and space is our instrument for making sense of the that conduit.Just as space and time are incommensurable, so are curves and straight lines (gradients and tangents); so are number and infinity; so are experience and calculation. A metaphysics of incommensurability will embody all such distinctions as one. Incommensurability is the magic in the system that we call analysis. Further, insofar as *anything* is --- or is modelled to be -- *determinately* definable, so, in intellection, its ontology can be signified by an x-axis against which there must ipso facto be a y-axis. Just as God is not determinately definable, so one cannot plot the very being of God against time or anything else.

  • Anon Calculus

    "Truth is ever to be found in simplicity, and not in the multiplicity and confusion of things." ~ Newton "Politics is for the present, but an equation is for eternity." ~ Albert EinsteinUntil such Time, that we have a simple equation (foundation) for Calculus; we will neither be able to fathom, comprehend or understand the profundity of Newton’s legacy. Enjoyed the program

  • Leibniz vs Newton -who first calculated the calcul

    For some reason one of the guest this week said Euler was Russian. He was in fact Swiss.

  • PaulPic when first you practice to deceive…

    I wonder if the Calculus slight of hand when dealing with infinitesimals has anything to do with the current problems in reconciling relativity and quantum mechanics at very small scales. It just seems caustic to say the smallest things are nothing.

  • Jim - Newton-Leibniz

    Yet another splendid programme on a fascinating subject, although a couple of points did strike me. I was surprised that no mention was made of the fact that Newton was President of the Royal Society at the relevant time and was able to 'fix' the report. The discussion seemed to me to confirm the conclusion taught to me some fifty odd years ago that Newton was first but secretive with a convoluted and obscure notation while Leibnitz was second but with an understandable notation that we still use today. As I contemplate the programme I am beginning to understand the arguments that support these conclusions despite the passage of time and not really being a mathematician. Certainly one to download and listen to again

  • Aaron Sloman (University of Birmingham, UK) Calcul

    I think this must be one of the truly great radio discussions of all time. The subject is very deep, underlying a huge amount of science and engineering of the last three centuries, the subject matter should be part of everyone's education, the presenters and Melvyn made everything brilliantly clear using just words, with no spurious pictures or background noises, there are important philosophical (and even theological and political) issues involved, there's the drama of a personal battle of giants -- and all in less than 45 minutes.The replay should be advertised on all BBC web sites. It is one of the few things on the BBC that (almost) justify the use of that dreadful,patronising, word "unmissable".Aaron

  • Newton & Calculus

    Hard to believe "experts" when they get a basic fact wrong. Euler was Swiss and worked in Russia he was never Russian. He was without doubt the most productive of all mathematicians all his papers have yet to be published 226 years after his death. Yes he is on of my heroes

  • Thomas T

    During the recent show about Calculus, one of the speakers claimed - possibly as a joke, that both Newton and Liebnitz had biscuits named after them. This is only true in the case of Liebniz-keks. The Newton in Fig Newton's refers to the city of Newton, Massachusetts. Just thought I'd clear this up; biscuits are important.

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