FINITE CALCULUS

If we want to get real smart, we will have to let no reason unturned. Foundations of calculus have been debated for 23 centuries (from Archimedes to the 1960s’ Non Standard Analysis). I cut the Gordian knot in a way never seen before. Nietzsche claimed he “made philosophy with a hammer”, I prefer the sword. Watch me apply it to calculus.

I read in the recent (2013) MIT book “The Outer Limits Of Reason” published by a research mathematician that “all of calculus is based on the modern notions of infinity” (Yanofsky, p 66). That’s a widely held opinion among mathematicians.

Yet, this essay demonstrates that this opinion is silly.

Instead, calculus can be made, just as well, in finite mathematics.

This is not surprising: Fermat invented calculus around 1630 CE, while Cantor made a theory of infinity only 260 years later. That means calculus made sense without infinity. (Newton used this geometric calculus, which is reasonable… with any reasonable function; it’s rendered fully rigorous for all functions by what’s below… roll over Weierstrass… You all, people, were too smart by half!)

If one uses the notion of Greatest Number, all computations of calculus have to become finite (as there is only a finite number of numbers, hey!).

The switch to finitude changes much of mathematics, physics and philosophy. Yet, it has strictly no effect on computation with machines, which, de facto, already operate in a finite universe.

In the first part, generalities on calculus, for those who don’t know much; can be skipped by mathematicians. Second part: original contribution to calculus (using high school math!).

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WHAT’S CALCULUS?

Calculus is a non trivial, but intuitive notion. It started in Antiquity by measuring fancy (but symmetric) volumes. This is what Archimedes was doing.

In the Middle Ages, it became more serious. Shortly after the roasting of Johanne’ d’Arc, southern French engineers invented field guns (this movable artillery, plus the annihilation of the long bow archers, is what turned the fortunes of the South against the London-Paris polity, and extended the so called “100 year war” by another 400 years). Computing trajectories became of the essence. Gunners could see that Buridan had been right, and Aristotle’s physics was wrong.

Calculus allowed to measure the trajectory of a canon ball from its initial speed and orientation (speed varies from speed varying air resistance, so it’s tricky). Another thing calculus could do was to measure the surface below a curve, and relate curve and surface. The point? Sometimes one is known, and not the other. Higher dimensional versions exist (then one relates with volumes).

Thanks to the philosopher and captain Descartes, inventor of algebraic geometry, all this could be put into algebraic expressions.

Example: the shape of a sphere is known (by its definition), calculus allows to compute its volume. Or one can compute where the maximum, or an inflection point of a curve is, etc.

Archimedes made the first computations for simple cases like the sphere, with slices. He sliced up the object he wanted, and approximated its shape by easy-to-compute slices, some bigger, some smaller than the object itself (now they are called Riemann sums, from the 19C mathematician, but they ought to be called after Archimedes, who truly invented them, 22 centuries earlier). As he let the thickness of the slices go to zero, Archimedes got the volume of the shape he wanted.

As the slices got thinner and thinner, there were more and more of them. From that came the idea that calculus NEEDED the infinite to work (and by a sort of infection, all of mathematics and logic was viewed as having to do with infinity). As I will show, that’s not true.

Calculus also allows to introduce differential equations, in which a process is computed from what drives its evolution.

Fermat demonstrated the fundamental theorem of calculus: the integral was the surface below a curve, differentiating that integral gives the curve back; otherwise said, differentiating and integrating are inverse operations of each other (up to constants).

Arrived then Newton and Leibnitz. Newton went on with the informal, intuitive Archimedes-Fermat approach, what one should call the GEOMETRIC CALCULUS. It’s clearly rigorous enough (the twisted examples one devised in the nineteenth century became an entire industry, and graduate students in math have to learn them. Fermat, Leibnitz and Newton, though, would have pretty much shrugged them off, by saying the spirit of calculus was violated by this hair splitting!)

Leibnitz tried to introduce “infinitesimals”. Bishop Berkeley was delighted to point out that these made no sense. It would take “Model Theory”, a discipline from mathematical logic, to make the “infinitesimals” logically consistent. However the top mathematician Alain Connes is spiteful of infinitesimals, stressing that nobody could point one out. However… I have the same objection for… irrational numbers. Point at pi for me, Alain… Well, you can’t. My point entirely, making your point irrelevant.

***

FINITUDE

Yes, Alain Connes, infinitesimals cannot be pointed at. Actually, there are no points in the universe: so says Quantum physics. The Quantum says: all dynamics is waves, and waves point only vaguely.

However, Alain, I have the same objection with most numbers used in present day mathematics. (Actually  the set of numbers I believe exist has measure zero relative to the set of so called “real” numbers, which are anything but real… from my point of view!).

As I have explained in GREATEST NUMBER, the finite amount of energy at our disposal within our spacetime horizon reduces the number of symbols we can use to a finite number. Once we have used the last symbol, there is nothing anymore we can say. At some point, the equation N + 1 cannot be written. Let’s symbolize by # the largest number. Then 1/# is the smallest number. (Actually (# – 1)/# is the fraction with the largest components.)

Thus, there are only so many symbols one can actually use in the usual computation of a derivative (as computers know well).  Archimedes could have used only so many slices. (The whole infinity thing started with Zeno and his turtle, and the ever thinner slices of Archimedes; the Quantum changes the whole thing.)

Let’s go concrete: computing the derivative of x -> xx. it’s obtained by taking what the mathematician Cauchy, circa 1820, called the “limit” of the ratio: ((x + h) (x + h) – xx)/h. Geometrically this is the slope of the line through the point (x, xx) and (x + h, (x + h) (x + h)) of the x -> xx curve. That’s (2x + h). Then Cauchy said: “Let h tend to zero, in the limit h is zero, so we find 2x.”  In my case, h can only take a number of values, increasingly smaller, but they stop. So ultimately, the slope is 2x + 1/#. (Not, as Cauchy had it, 2x.)

Of course, the computer making the computation itself occupies some spacetime energy, and thus can never get to 1/# (as it monopolizes some of the matter used for the symbols). In other words, as far as any machine is concerned, 1/# = 0! In other words, 1/# is… infinitesimal.

This generalizes to all of calculus. Thus calculus is left intact by finitude.

***

Patrice Ayme

***

Note: Cauchy, a prolific and major mathematician, but also an upright fanatic Catholic, who refused to take an oath to the government, for decades, condemning his career, would have found natural to believe in infinity… the latter being the very definition of god.

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9 Responses to “FINITE CALCULUS”

  1. pshakkottai Says:

    Dear Patrice: In India, the mathematician-astronomer Aryabhata in 499 used infinitesimals and expressed an astronomical problem in the form of a basic differential equation. Manjula in the 10th century elaborated on this differential equation in a commentary. This equation eventually led Bhaskara (1114-1185) to conceive of differential calculus and a number of ideas that are foundational to the development of modern calculus, including the earliest use of the “derivative” and differential coefficient, and the first statement of the idea now known as “Rolle’s theorem”. This theorem is important as a special case of the mean value theorem, which is one of the most important theorems in modern analysis. Using these concepts, he solved Aryabhata’s differential equation to find the differential of the sine function, as well as the Earth’s velocity in successive positions of its elliptical orbit around the Sun.

    The 14th century Indian mathematician Madhava of Sangamagrama, along with other mathematicians of the Kerala School, studied mathematical analysis, infinite series, power series, Taylor series, trigonometric series, convergence, differentiation, integration, term by term integration, numerical integration by means of infinite series, iterative methods for solutions of non-linear equations, tests of convergence, the concept that the area under a curve is its integral, and the mean value theorem, which was later essential in proving the fundamental theorem of calculus and remains the most important result in differential calculus. Jyestadeva of the Kerala School wrote the first differential calculus text, the Yuktibhasa, which also includes discoveries of integral calculus, and explores methods and ideas of calculus that were later repeated, probably independently , in Europe during the 17th, 18th and 19th centuries. These contributions likely laid the groundwork for contributions by Descartes and Fermat, which in turn led to the developments of Newton and Leibniz. Some historians have suggested that the contributions of the Kerala School to calculus were transmitted to Europe, but this is not known for certain.

    In 1835, Charles Whish published an article in the Transactions of the Royal Asiatic Society of Great Britain and Ireland, in which he claimed that the work of the Kerala school “laid the foundation for a complete system of fluxions.” It was not until the 1940s however, that historians of mathematics verified Whish’s claims, proving that the Kerala school developed much of differential calculus well before Newton or Leibniz.

    In 17th century Europe, Isaac Barrow, Pierre de Fermat, Blaise Pascal, John Wallis and others are said to have discussed the idea of a derivative. René Descartes introduced the foundation for the methods of analytic geometry in 1637, providing the foundation for calculus later introduced by Isaac Newton and Gottfried Leibniz, independently of each other. Fermat, among other things, is credited with an ingenious trick for evaluating the integral of any power function directly, thus providing a valuable clue to Newton and Leibniz in their development of the fundamental theorems of calculus. James Gregory was able to prove a restricted version of the second fundamental theorem of calculus.

    Newton and Leibniz are usually credited with the invention, independently of one another in the late 1600s, of differential and integral calculus as we know it today. Their most important contributions were the development of the fundamental theorem of calculus. Also, Leibniz did a great deal of work with developing consistent and useful notation and concepts. Newton was the first to organize the field into one consistent subject, and also provided some of the first and most important applications, especially of
    integral calculus. Of course, important contributions were also made by Barrow, Descartes, de Fermat, Huygens, Wallis and many others.

    (from http://ibmathematics.tripod.com/id5.html
    Partha

    • Patrice Ayme Says:

      Dear Partha: I confess that I know nothing about India and the calculus. Sounds hyper impressive. One thing I know is that the development of calculus in Europe was internal (as we have it all). Fermat had the main theorem of calculus. The rest is details. I also know that claims are often made on account of Middle Earth os “Muslim” (often Jews on closer inspection!) scientists, that are not born by evidence.

      Krugman himself has done that about psychomathematics. establishing precedence of some Muslims by ignoring Polybius, more than16 centuries earlier.

      Anglo-Saxons have loved to put much under Newton’s banner, like the gravitation law (some French priest establishe it as Newton himself said and I documented) or the first law (word for word in Buridan, 350 years before0).

      One of my acquaintances is an engineer from Kerala, I will try to ask him… No wonder that Kerala has a reputation of education… Aand a communist party in power (no?)
      PA

  2. pshakkottai Says:

    Dear Patrice: I find it hyper impressive too the fact “Jyestadeva of the Kerala School wrote the first differential calculus text, the Yuktibhasa, which also includes….” I will try to see if it is accessible too. Yes, the communist party is in power.
    Partha

    • Patrice Ayme Says:

      That would explain it. Being so long brainy, Kerala (as far from Europe as possible) was bound to vote for people dedicated to equality!
      I am very interested by this. But, having been burned by wildly exaggerated claims about Chinese and “Muslim” science, a bit skeptical.
      India, though, is different. It’s truly part of the Middle Earth, and it perfected the Greek number system (introducing, or not, the zero), the so called “Arabic” numbers (truly the Greco-Indian numbers).
      PA

  3. Alexi Helligar Says:

    Patrice! This article is exciting. The ideas on infinity resonate strongly with me. In a universe of particulars, there can be no infinity and all slicing of real entities stops at the Planck length.

    However, there seems to be no limit to how arbitrarily big an entity can expand, whereas, there does seem to be a limit to how small an entity contract. What happens below the Planck length is (like Love itself) a matter that physics cannot speak to.

    • Patrice Ayme Says:

      Thanks Alexi, I love it! Sorry to be so slow, lots to do. Well, things can still be said, and I said some of them. For example quantum non separability implies non causal effects…. “at a distance!”… I have studied infinity for decades, before switching to my present hard line. As I discovered we really don’t need any of it. Zilch.
      PA

  4. de Foucaud Says:

    Just divide any number per zero and you are going inbound infinite. Yours.

    Envoyé de mon iPhone

    • Patrice Ayme Says:

      Dear Paul: Say x = m/0. Then x.0 = m (it’s actually what “divides” means!). So then YOU assume x = infinity. However, in Peano’s Arithmetic (PA), x.0 = 0 always… It’s an axiom! Peano’s arithmetic is the standard arithmetic everybody uses, even standard metamathematicians…

      (I’m being a bit impudent, as my system is non PA, but that’s an irrelevant point…I just put my conventional hat to answer you!)

      Tricky, tricky… Keep on trying, you are more than welcome.
      … And welcome to NON ARISTOTELIAN LOGIC!
      PA

    • Patrice Ayme Says:

      The Peano axioms can be augmented with the operations of addition and multiplication and the usual total (linear) ordering on N. The respective functions and relations are constructed in second-order logic, and are shown to be unique using the Peano axioms.

      Addition is the function + : N × N → N (written in the usual infix notation, mapping two elements of N to another element of N), defined recursively as:
      a + 0 = 0
      a + S(b) = S(a + b)
      For example,
      a + 1 = a + S(0) = S(a + 0) = S(a).
      The structure (N, +) is a commutative semigroup with identity element 0. (N, +) is also a cancellative magma, and thus embeddable in a group. The smallest group embedding N is the integers.

      Multiplication. Given addition, multiplication is the function · : N × N → N defined recursively as:
      a.0 = 0
      a. S(b) = a + (a.b)

      It is easy to see that setting b equal to 0 yields the multiplicative identity:
      a · 1 = a · S(0) = a + (a · 0) = a + 0 = a
      Moreover, multiplication distributes over addition:
      a · (b + c) = (a · b) + (a · c).
      Thus, (N, +, 0, ·, 1) is a commutative semiring.
      PA (hahaha!)

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