The Less Known Problematic Origins of Calculus

Reading Time: 27 minutes

The Origin of Calculus in Change vs. Constant

Underlying every kind of scientific theory is a fundamental conundrum between change and constant. For example, if we want to do sociology, we must talk about how society is changing. But in so talking, we have to say that there is a thing called “society”, which changes in some ways but remains the same society. The new and old societies are the same in one sense and different in another. But how can we make those two claims without a contradiction? Similarly, while doing economics, we have to talk about how an economy is changing. But in so talking, we have to say that there is a thing called “economy” which changes in some ways but remains the same economy. The new and old economies are the same in one sense and different in another. But how can we make those two contradictory claims without a contradiction?

This problem occupied early Greek philosophers—Heraclitus and Parmenides—and had a paralyzing effect on scientific thinking for 2000 years, to a point that no scientific theory could be developed until the logical contradiction between change and constant had been resolved.

When Newton created modern science, he found a solution to this problem based on methods that Indian mathematicians were using to compute planetary trajectories. These methods were employed to compute ritual timings. They came to be known as calculus. This solution has had an enduring effect since Newton and is used in all areas of modern science because no other solution to the change vs. constant problem has been proposed or found since then.

Calculus, however, is a deeply flawed solution to the change vs. constant problem, and I will discuss the flaws in calculus along with the correct solution to the same problem. We will trace the origins of this problem in Greek philosophy, why these problems had made science impossible, how calculus was created, and despite its numerous successes, why it has become a dead end for science today, before talking about the correct solution.

Change vs. Constant in Western Philosophy

The problem begins with the debates between Parmenides and Heraclitus sometime around 500 BCE. Heraclitus claimed that change is the only constant and there is no such thing as “being” as he likened everything to a flowing river. To say that I went for a bath in the Ganga is meaningless for him because “you can never step in the same river twice”. Ganga is also a meaningless word for Heraclitus because it assumes the existence of the same thing day after day, when it is just flowing water. By extension, there is no such thing as an economy or a society that changes because reality is only change and flux.

This argument is so good that Parmenides had to counterclaim that change is impossible and that a flowing river is an illusion. There is only one thing called “being”, Parmenides claimed, and for any change to occur, a “being” must cease to exist and become a non-being, followed by the appearance of another being that was previously non-being. Change, therefore, involves a logical contradiction in a being becoming non-being and followed by a non-being becoming a being. Parmenides spent considerable effort in proving that being cannot become non-being and non-being cannot become being.

Change vs. Constant in Vedic Philosophy

There is an easy solution to this problem, which Kṛṣṇa calls dehina and deha, and which we loosely call soul and body at present. The solution says that there are many potential states of the soul called bodies, and the soul goes from one state to another, or from body to body. Therefore, the bodies are changing while the soul is constant. The soul and the body are not the same, and yet both exist. Therefore, both change and constant exist simultaneously, and yet, they pertain to body and soul, respectively.

How the soul goes from one body to another is a complex process, but we can simplify it to “choice” for the moment. This simplification is not always true because time forces many changes in the body and this forced movement from body to body—often against the soul’s choices—is called the soul’s bondage in the world. The forced change causes the soul to change bodies from childhood to youth to old bodies in one life, and then to another childhood body in the next life. The distinction between the effects of time and the soul’s choices is not important for the present discussion, so we can talk about “choice” as a generic solution to the succession of bodies that can be made either by the soul or by time.

Resurrection vs. Reincarnation Doctrines

Greek philosophers did not have the above soul-body distinction. Hence, they had no answer to the change vs. constant problem. While a soul-body distinction exists in Judeo-Christian doctrines, we can never say that “I am a soul, and I have a body”. For all practicalities, the soul is the body, although the soul survives the death of this body to get a new body. Thus, Christianity talks about resurrection rather than reincarnation. Reincarnation is the soul going body to body. Resurrection is the soul going to sleep in one body and waking up in another, and what happens between sleep and waking is not important.

Thus, despite a soul-body distinction, Christianity could not solve the change vs. constant problem. The soul in the body remained like a coin in someone’s pocket. The body moves, and the coin goes with it. The pocket has the coin. The coin does not have a pocket. Similarly, the body has a soul but the soul does not have a body. Thus, even Judeo-Christianity remained paralyzed for centuries despite a soul-body distinction, because it was talking about the resurrection of the soul rather than its reincarnation.

Space vs. Time vs. Matter Distinction

An even more serious issue in Greek philosophy was the absence of the distinction between space, time, and matter. As we know, the Greeks had four elements, called Earth, Water, Fire, and Air, but not Ether, as in Vedic texts. So, people were walking on earth and through air, but never in space. Past, present, and future were illusions for Parmenides because there was only “being” which could never become another being. The concept of empty space—something devoid of Air, Water, Fire, and Earth—did not exist. Thus, neither time nor space existed for Parmenides. It was only material substance. Since change was denied, therefore, both time and space were unnecessary constructs, artifacts of the illusion of change.

Likewise, for Heraclitus, space was either Air or Fire, and a portion of that Air or Fire became Earth and Water. Since everything was simply flux, therefore, change and time were innate properties of the material substances, not separate from them. Time would be separate from material substance if there was an unchanging substance. Since everything was flux, therefore, time was also substance. Thus, we could not talk about motion in space and time. The space, time, and matter distinction was absent.

Cartesian Substance vs. Space and Time

Even when Descartes published Meditations in 1641, he called the body res extensa or “extended substance”. He removed the four Greek elements of Earth, Water, Fire, and Air, and replaced them with one substance that had extension. He often equated res extensa to a “corporeal substance” meaning that which comprises the body. Descartes rejected the distinction between substance and form, which means that there is no Platonic world of forms with an ideal form of a cube or sphere which is embedded in a corporeal substance to create a corporeal cube or sphere. There was just a corporeal substance.

The “space” between two people was not empty space. It was also an extended substance, like the Air of Greeks. Since Cartesian ontology only includes two substances—res extensa and res cogitans—therefore, time had no independent existence. He sometimes called time a “mode of thought” and at other times he spoke about the “duration of existence”, treating time as an attribute of the two substances. Thereby, the corporeal substance was the real thing and extension and duration were the substance’s properties. Effectively, substance was not in space and time. Rather, space and time were in the substance.

When we put space and time in the substance, then we cannot solve the change vs. constant problem because the substance—like the society and economy—is both change and constant. If it changes, then it is longer constant. But if it remains constant, then it can no longer change. By adding space and time attributes to the corporeal substance, Cartesian metaphysics could be alternatively interpreted either as Heraclitan becoming without a being or Parmenidean being without a becoming, but never both.

Serendipitous Newtonian Solutions

Thus, even until the time of Descartes, there was no resolution to the problem of change vs. constant created in Greek times, and the distinction between space, time, and matter did not exist. The resolution to both these problems appears for the first time in Newton’s writing published as Principia in 1687—46 years after Cartesian Meditations. In Principia, Newton provided a solution to the Greek problem of change vs. constant, while separating space, time, and matter to create a theory of motion.

It is rather astounding that such innovation in thinking happens all of a sudden and out of the blue. But it is not so amazing if we understand that Newtonian innovations were not truly his. He was taking them from Freemasonry texts, which had all these concepts. Newton indirectly acknowledged Freemasons for these ideas by saying: “If I have seen further than others, it is by standing upon the shoulders of giants.” The “others” in this case were Descartes, Copernicus, Galileo, Kepler, and such. The “giants” in this case were Freemasons who remain unnamed and unquoted for reasons that we can only guess today.

Masonry had been persecuted for centuries prior by the Catholic Church. It had been given acceptance in Europe after the Protestant Reformation about a century prior to Newton. Those who were earlier called Masons were now calling themselves Freemasons (since they were free from persecution). But to attribute scientific breakthroughs to an erstwhile persecuted minority would be highly embarrassing to Christianity. Not crediting them would be problematic in case the connection was later discovered. So, and this is my guess, Newton chose to attribute breakthroughs to “giants” without naming them.

Narratives on the Origins of Calculus

At this juncture, I will like to take a detour into how the story of the invention of calculus is presently told. Here are some pertinent quotes from the Wikipedia article on the History of Calculus:

  • Egypt: The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volumes and areas, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are only given for concrete numbers, some are only approximately true, and they are not derived by deductive reasoning.
  • India: Some ideas on calculus later appeared in Indian mathematics, at the Kerala school of astronomy and mathematics. Madhava of Sangamagrama in the 14th century, and later mathematicians of the Kerala school, stated components of calculus such as the Taylor series and infinite series approximations. However, they did not combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the powerful problem-solving tool we have today.
  • Greeks: Eudoxus used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes developed this idea further, inventing heuristics which resemble the methods of integral calculus. Greek mathematicians are also credited with a significant use of infinitesimals. Democritus is the first person recorded to consider seriously the division of objects into an infinite number of cross-sections, but his inability to rationalize discrete cross-sections with a cone’s smooth slope prevented him from accepting the idea. At approximately the same time, Zeno of Elea discredited infinitesimals further by his articulation of the paradoxes which they seemingly create.

Egyptians and Indians who had formulas and numbers did not “develop them in a rigorous and systematic way”, “did not derive them by deductive reasoning” and could not “turn calculus into the powerful problem-solving tool we have today”. Meanwhile, Greeks paralyzed by their controversies about being and becoming were “prevented from accepting their own ideas” by “paradoxes that infinitesimals seemingly create”. Never mind the fact that Newton would have named the “giants” if they were Greeks. Never mind the basic problem of change vs. constant and the paralyzing effects that Parmenides and Heraclitus had. There is something to be learned from this, namely, that history is easily distorted by interpreting facts using a biased lens that unduly favors one side over the other.

Paradoxes Created by Zeno of Elea

While we are on this topic, we should talk about the problem of infinitesimals created by Zeno, a disciple and supporter of Parmenides, and hence, a proponent of the impossibility of motion. He created several paradoxes of motion to justify the Parmenidean position in which a changing world is an illusion because there is just a being, which cannot become non-being, on the path to becoming another being.

The simplest of these paradoxes says: Achilles is in a race with a tortoise, with the tortoise having a head start of 100 meters. When Achilles has run 100 meters, the tortoise has gone a little further by 2 meters. When Achilles has run 2 meters, the tortoise has gone a little further by a few centimeters. Thus, Achilles is always behind the tortoise. Hence, the quickest runner can never overtake the slowest runner.

Another paradox says: An arrow cannot move to a place that it is presently not situated, in infinitesimal time, because no time elapses for the arrow to move to that place. Similarly, the arrow cannot move to the place where it is already situated at present because it is already there. Thereby, the arrow is always motionless. Even if the arrow has been released from a bow, it can never reach its target destination.

If we look carefully, the Achilles Paradox divides space into infinitesimals and the Arrow Paradox divides time into infinitesimals and concludes Achilles can never overtake the tortoise and the arrow can never hit its target. Of course, we see Achilles overtaking the tortoise and the arrow hitting its target. However, if these were real, then we would never see those things. Since we see them, they must be illusions.

Later Developments in Western Thinking

The philosophies of Parmenides and Heraclitus are what we call māyāvāda and śūnyavāda in the Vedic tradition. According to māyāvāda, there is a being called Brahman, but all becoming—seen as a world—is an illusion. According to śūnyavāda, there is just becoming without any being, and hence there is no Brahman, and even as māyā may be perceived, it has no reality, hence ultimately nothing exists. Both māyāvāda and śūnyavāda are antiscientific philosophies due to their antirealist attitudes. This is why Buddhists (who follow śūnyavāda) and Advaitins (who follow māyāvāda) never made any scientific discoveries. They only reject the sense of meaning and purpose in the present world and life.

Greeks were rescued from this sense of meaningless and purposeless existence by Plato, when he talked about form-substance dualism, with a transcendent world of pure forms that made the present world of substance meaningful and purposeful because the transcendent forms were partially and imperfectly reflected in the world. Platonism could never be justified because no forms could ever be defined. Greeks just anointed themselves as pure humans and treated other races as imperfect humans at best.

Aristotle then junked the imaginary Platonic transcendentalism by separating forms into “theoretical” and “practical” categories with the former being public rationality and the latter being private belief. Even the separation of private and public was junked by the Romans as the Bible came to be regarded as the only source of truth, which had to be accepted on faith and could not be analyzed rationally.

To ignore this historical trajectory and say that the Greeks were close to discovering calculus is quite facetious. To disregard the fact that Newton did not attribute a single Greek for his discovery, and instead recognized unnamed giants, means that the Greeks were hindering and not assisting progress. To disregard the effects of problems created by Zeno’s Paradoxes, which are widely discussed even today, is to ignore what they entail. By disregarding all these problems, we can never understand the significance that calculus presently assigns to the concepts of continuity and smoothness of trajectories. Therefore, the story of calculus as is commonly told today is designed to obscure the truth rather than reveal it.

The Theory of the Soul in Vedic Texts

Before we move further, I would like to talk about the tripartite nature of the soul in Vedic texts and how this tripartite description is absolutely crucial to the solution to the problem of change and constant. The three aspects of the soul are called sat-chit-ānanda, and for the sake of our present conversation, we can call them thinking-thought-thinker. The thinker produces a will. The will acts on a repository of ideas to pick an idea. After the idea has been selected, consciousness attaches to it to think it. But if this vocabulary seems difficult, then we can also use three words—will, idea, and awareness. There is a repository of ideas from which an idea is selected by will and then awareness attaches to the idea to know it.

The will or choice is discrete. The thought picked by that will or choice is also discrete. Since awareness moves from one thought to another, therefore, even the knowing of the idea is discrete. However, even when there is no thought, there is still self-awareness which we call “I”. This “I” produces a will, which then selects a thought, and attaches to that thought to become aware of it. We can call this process “I am X”, where “I” is the thinker, “X” is the thought, and “am” is the thinking. “I am X” is thinker thinking thought. The “I” is continuous, but the will, thought selection, and awareness of thought are discrete.

As the thinker goes through a succession of thoughts, the being becomes thoughts without going through non-being. Likewise, the continuous succession of thoughts—which we can call becoming—are not the denial of a being. The discreteness of successive thoughts is not the denial of the continuity of the thinker. The continuity of the thinker is not the rejection of the discreteness of thoughts. Thus, being is not contrary to becoming and continuity is not contrary to discreteness. Due to the discreteness of thoughts, the concept of infinitesimals created by Zeno is false. The thinker doesn’t get fractional thoughts. He is either thinking or not thinking. Each thought is finite and not infinitely small. The thinker jumps through thoughts one after another without creating any discontinuity in his existence.

Thus, all problems of being vs. becoming are resolved if we use the thinker-thinking-thought model of change. Likewise, the problem of infinitesimals created by Zeno ceases to exist in this model of change. Any potential contradiction between discreteness and continuity also ceases to exist likewise. The being is continuous and the becoming is discrete. The “I” is the constant and the “am X” is the change. By going through a succession of “I am X”, there is an unchanging “I” and a changing “am X” attribute.

Science From the Theory of the Soul

This solution to the problem of being vs. becoming, constant vs. change, and continuous vs. discrete is applied to everything in Vedic philosophy. For instance, society and economy are also some “I”. When the society or economy evolves, the same “I” gets a new “am X” attribute. Thereby, we can say that the same society and economy have changed to a new state. Similarly, there is an “I” called Ganga which then goes through successive stages called the “flowing water”. When we step into flowing water day after day, we are stepping into the same river called Ganga, even though the flowing water is different.

The person who enters a society or an economy has entered the awareness of some “I”. The person who steps into the flowing water of Ganga has entered the awareness of Ganga. When they step out of Ganga, they have left the awareness of Ganga. If they leave a society or economy, they have stepped out of the awareness of the “I” that identified that society or economy as a unique being or individual.

The Cartesian body is not a “corporeal substance”. It is thought. The Cartesian mind is also not a non-material substance. It is an “I”. The connection between mind and body is not mysterious. It is simply the connection between some “I” and some “X” to produce “I am X”. Since the “X” can be changed without destroying the “I”, therefore, there can be a succession of bodies of the soul. Even if the soul goes to sleep and then wakes up, still there is a succession of bodies from “I am X” to “I am Y” which follows the same process as the numerous X’s that the I had previously attached to as its personal attributes. There is strict ordering between I and X, such that “I am X” is correct but “X is I” is incorrect. This means that the soul has a body, but the body doesn’t have a soul, nor are the body and soul identical in any sense. The soul and body become identical if and only if there is an “X” that the “I” can be eternally attached to. Hence, the difference between an eternal and a temporary body is that “I am X” can be eternal or temporary.

Therefore, if we want to do any kind of science—let it be economics, biology, sociology, cosmology, or physics—the paradigm is always the evolution of “I” through a succession of “X”. Each such paradigm is identical to that of a fixed dehina and a changing deha. If we know how the soul changes bodies, then it applies to how the Ganga flows, how an economy goes up or down, how a society rises or falls, and how an object moves. There is only one subject—namely, the science of how the soul goes from body to body—because there is one resolution to the problem of constant vs. change, continuous vs. discrete, and being vs. becoming. Another resolution to this problem is not needed if this resolution always works. We only have to prove that this resolution to the problem of being vs. becoming always works.

The Essence of the Newtonian Solution

Newtonian calculus eschewed all these principles found in the Vedic texts and envisioned an alternative idea of change and constant in which a particle moves through infinitesimal locations in space and time. The particle is the being and its successive positions in space and time are becoming. This requires separating space, time, and matter (matter is now a particle moving through space and time). Through this separation, Newton got rid of all the historical problems created by Heraclitus, Parmenides, Zeno, and Descartes. The world is now not a Cartesian res extensa. It is actually two such extensa—space and time—with the particle moving through the extensa in infinitesimal steps. There is no problem of being becoming non-being during change because that non-being is space and time whereas the being is the particle; the same thing is not both being and non-being, so the contradiction doesn’t exist. Since there is indeed a being—the particle—therefore, the world is not simply flux as was the case for Heraclitus.

All problems could be solved by solving Zeno’s Paradoxes—e.g., the Achilles and Arrow Paradoxes. The answer required saying that infinitesimal is not actually zero, but something that tends to zero. We can make the infinitesimal as small as we like, but it will never be precisely zero because if it were precisely zero, then the Arrow would never move forward and Achilles can never overtake the tortoise.

The innovation of calculus is redefining the infinitesimal as neither finite nor zero. To quote Wikipedia again: An infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the “infinity-th” item in a sequence. An infinitesimal exists in the no-man’s land between 0 and the smallest conceivable number, and it originates in infinite convergent series.

Indian Origins of the Newtonian Solution

Infinite convergent series were invented in India to calculate values of real numbers to arbitrarily desirable levels of accuracy. For instance, the number π can be defined to the accuracy of ten decimal places as 3.1415926535. We don’t always need so much accuracy. Most school and college students just equate π to 3.14. But if we were doing a satellite launch, we better use a more accurate value.

Clearly, we cannot obtain more precise values of π by drawing a circle on paper because every scale we use has a limit in the smallest granularity it can measure. To measure the precise value of a planet or star in the sky, the canvas on which we draw the circle must be far bigger than the earth. Accurate astronomy is hence not possible unless we define π to increasing levels of accuracy. Infinite convergent series were invented in India for precisely the purpose to define numbers like π to an arbitrary desired accuracy.

The infinite series for π for instance is 1 – 1/3 + 1/5 – 1/7 + 1/9 – … = π/4. It takes lots of steps to get to accurate values, but each step is very easy, and we can truncate the steps whenever we want to get a less precise value that suffices our needs. We need a few decimal places to get the circumference of a dome and more decimal places to compute the precise points of celestial planetary conjunctions.

This is then the origin of infinitesimals—the infinity-th item in a series of numbers that progressively converges to a limit value—in this case, π. Of course, we don’t have to go to infinity. We can truncate it after an arbitrarily chosen—although sufficiently large—number of steps. The incremental additions become smaller and smaller, which is why the infinitesimal can get smaller without ever being 0.

We have to remember that this is just a calculation tool and not reality. We cannot chop off some steps in reality and expect the planets to complete their revolutions in time. By chopping off those steps, the planets would start lagging behind. Knowing everything to infinite precision requires infinite steps and hence infinite time. Therefore, truncating a convergent series can never be called reality. These were meant only to calculate the precise time for a ritual—sometimes called a muhurta—because a ritual that begins and ends within the muhurta gives better results. To know the time by which the ritual can begin and the time by which it must end, mathematics was necessary.

How Indian Mathematics Benefitted Newton

The mathematics that Indians had invented for doing rituals was useful for architecture too, especially if you had to build an architecture aligned with specific cosmic events, such as a temple precisely facing the sun when the sun enters a particular star sign. The same methods of calculation that were used for calculating the precise muhurta for a ritual were useful for calculating the precise orientation of temples. This is how masonry got involved in mathematics, and this knowledge traveled to other parts of the world through masons, although the inventors of that mathematics were ritualistic Brahmanas.

The fact that Brahmanas were doing muhurta calculations for rituals doesn’t mean that this had anything to do with the science of rituals itself. Rituals involved mantra pronunciation, a mantra had a meaning, and reality was modeled as meaning. Mathematics instead does geometric computations without any semantics. The fact that the Brahmana would do both the muhurta calculation and the ritual at the appointed time, leads to the modern confusion that mathematics is actually the Brahmanical science when mathematics was simply a tool while real science was the chanting of mantras. The mantra science needed a lot of philosophy of linguistics and meaning while mathematics needed approximations.

Newton got formulas for infinite convergent series through masons and realized that the answers to the problems raised by Zeno’s paradoxes lay in saying that an Arrow or Achilles moves an infinitesimal step forward in an infinitesimal time, neither of which is precisely 0. An infinitesimal has a small length, but zero does not. That makes all the difference because Zeno’s Paradoxes depend on zero-length points in space and time. If these are not zero, and yet, arbitrarily small, then Zeno’s Paradoxes don’t exist.

For this solution to actually work, we also need to separate space, time, and matter, which already existed in the Indian calculations of muhurta where a planetary body is moving in space and time, so the world is not a plenum or substance, as was the case up until the time of Descartes. Instead, that substance was replaced by three distinct concepts of space, time, and matter, with precise models of calculating the motion of bodies to predict the specific time at which they arrive at a specific place.

How Newton Depended on Observations

Newton only needed to replace the angular speed of planetary movements used in such calculations (which had been determined over ages through empirical observation) with a causal explanation of that angular speed (i.e., the angle by which the planet moves in unit time). This work had already been done by Johannes Kepler where he showed—based on the Copernican hypothesis of a heliocentric solar system—that the angular speed of a planet is inversely proportional to its distance from the sun.

Newton’s ingenuity lay in postulating a “gravitational force” pulling the planets toward the sun to explain the angular speeds that Kepler had observed. He was assisted in this solution by children tying stones to a string and rotating them around—(a) the planet was the stone, (b) the gravitational force was the string tugging at the stone, (c) the planetary angular speed was how fast the stone was rotating. Newton had to make a formula that will reduce the angular speed of rotation with the increasing length of the rope and it turned out to be an inverse squared law. Everything else was casting that formula in an ordinary language by either inventing new names or applying familiar ones to symbols in the formula.

The fact is that one could predict planetary positions based on Kepler’s measurements. There would be no explanation of planetary movement, but it would work. It would not be precise without the infinite convergent series formulae, but it would work. It would be just a hallucination without a solution to Zeno’s paradoxes, but it would work. It could be a planet stepping into the sky-river, but it would work. Three out of the four problems—hallucination, river in the sky, and imprecise prediction—were solved by the postulate of infinitesimals. The fourth problem of explaining the motion based on gravity was Newton’s invention, although he had help in this solution from children playing with ropes and stones. The story of the apple falling on Newton’s head is either pure fiction or at best an unrelated fact.

Problems of Continuity and Smoothness

Infinitesimals allow us to write quantities like ∂x/∂t because they are arbitrarily small and yet not 0. But they also force restrictions on the possible trajectories. The trajectory cannot be discrete and unsmooth because then ∂x would be finite and ∂t would be 0, and the result would be ∂x/∂t = ∞. These restrictions do not exist when we talk about a thinker thinking thought. A thinker may think logically progressing thoughts, and his mind may then be said to move continuously and smoothly. However, a thinker can also be distracted from one thought to another and his mind would be discontinuous and unsmooth. The thinker thinking thought doesn’t mandate continuity and smoothness, whereas calculus does.

If you take a pencil and try to draw a line on paper, the question is: What is the next point in the line? Should the pencil move upward or downward? Should it go backward or forward? Should it jump or move smoothly? These are choices. But calculus eliminates them. The pencil cannot jump, and it should not take a discrete left or right, up or down, or forward or backward turn. Even if the pencil turns, it should do it slowly and smoothly. Thus, the hand should never draw a triangle or square because these have sharp vertices. A sharp vertex in calculus means that that rate of change denoted by ∂x/∂t = ∞. If we get one infinity in a series, the sum will also be infinite. To eliminate sharp vertices, the hand should never draw a triangle or a square. Circles, ellipses, and straight or curved lines are fine because they don’t have sharp vertices. Thereby, the use of infinitesimals puts a constraint on the hand holding the pencil.

The constraint on the hand is called determinism. It is created by assuming that the hand cannot move the pencil in all the ways that the hand is capable of moving. By constraining the hand, determinism is created and choice is destroyed. Of course, if choice exists, then calculus will be proven inadequate. The hand may sometimes change position and direction discretely, which then breaks calculus. Since calculus is the foundation of everything in modern science, hence, discretizing hand motion breaks science. This is why science is indeterministic—the hand can jump up or down, left or right, forward or backward, discretely. So, if we assume calculus, it may work for some hand movements—where the hand is drawing circles and ellipses. But it will fail for other cases—when the hand draws triangles and squares.

The solution to this problem requires rejecting calculus as a model of reality. It may be used for those changes that are like circles and ellipses, although even then it is not the nature of reality. Calculus certainly breaks when the change is like a square or triangle. After throwing calculus in the dustbin of history, we can return back to Zeno’s Paradoxes and solve them with the thinker thinking thought model, which accommodates both discrete and continuous, being and becoming, constant and change.

The Problem of Choice and Indeterminism

Newton’s mechanics is not deterministic. If two cars crash into each other, we cannot use Newtonian equations to predict the outcome. We can only use Newton’s mechanics while they are running on the road. Even if determinism is observed, such as in the case of billiard ball collisions, the explanation is not deterministic because when two particles collide, they can recoil in many directions since the colliding particle is dimensionless in theory, so every direction is equally good. This indeterminism is not seen in billiard ball collisions due to the finite size of the ball, which makes all directions non-equivalent.

Thereby, even when the phenomena are deterministic, the theory is indeterministic. Otherwise, if the balls collide and disintegrate into many pieces, then the phenomenon is also not deterministic. The problem of indeterminism in Newton’s theories is not realized by limiting its application to (a) finite-sized balls, and (b) that do not disintegrate on collision. If we merely accept that all balls are finite-sized, then the calculus based on infinitesimals would be destroyed. If we merely accept that some balls disintegrate on a collision, then the calculus based on continuity and smoothness will be destroyed. Hence, to preserve calculus we have to pretend that all changes, in reality, are infinitesimal, continuous, and smooth.

These assumptions then create the illusion of determinism and are used by many ignorant scientists and philosophers as a counter to free will. They don’t know that calculus is preventing the hand from drawing a square and triangle and permitting the hand to draw a circle and ellipse. Nature is not preventing the hand from drawing a square or triangle; calculus is. Therefore, calculus is not a model of reality. It is incapable of predicting and explaining all discontinuous and unsmooth phenomena—which are, by the way—the only phenomena due to atomic theory. This, as we have discussed, is because all choices are discrete. Sometimes that discreteness can be approximately modeled as smoothness and continuity. But that is largely an exception. The rule is that everything is discrete due to the quantization of reality.

The Problem of Quantization of Reality

Quantization breaks the assumption of infinitesimals, which breaks calculus, which returns us to Zeno’s Paradoxes. Quantization has taken us back to Greek times when Parmenides and Heraclitus were arguing about being vs. becoming. 2500 years have passed, but we are back to where we started—in terms of understanding the nature of reality, finding correct models for reality, and building truthful theories. In some ways, we may even be worse off than the time in 500 BCE, when most of nature was pristine, millions of additional life forms existed, and the fear of the planet killing most people did not exist, although people were still killing each other.

Indeterminism is the law of unintended consequences. We don’t expect something to happen, but it happens. But we cannot simply blame the problem on a “law” when we chose the “law” called calculus which then leads us to the “law” of unintended consequences. We have to give up both kinds of laws.

Whatever people call the problem of “quantum gravity” is actually a problem about the quantization of space and time because gravity has been reduced to the properties—such as curvature—of space and time. To unify the theory of space and time with the theory of matter and energy, we have to quantize everything. We cannot have discrete particles of matter and energy floating in a continuous space and time. But we cannot quantize space and time because it will break the infinitesimals used as ∂x/∂t.

The fact is that quantization will make some locations bigger and some locations smaller. Quantization will make some moments longer and some moments shorter. How will we know what is bigger or smaller when the scale used to measure big and small is itself becoming bigger and smaller? The answer is simple—we have to use something that doesn’t become bigger or smaller due to changes in locations and times. That constant thing is the soul. It can experience longer and shorter moments. It can know what is bigger and smaller because it is itself not changed by changing locations and times.

The Problem of Relativization of Reality

The problem of relativity is “I am X” where “X” is a temporary, changing, and momentary designation of the “I” similar to ordinary designations of the “I” as man and woman, black and white, husband and wife. When the “I” acquires a temporary “X”, then it seems different from other “I” with a different “X”. Then, if “I am X” interacts with some “I am Y”, it gets a subjective deviant understanding of Y because its own designation is X, and it is unable to truly know the “I” thinking “I am Y”. X and Y are called “coordinate reference frames” in the theory of relativity. But since X and Y keep changing (change is called motion and acceleration), therefore, every “I” with “X” and “Y” misjudges each other to preclude knowledge.

The misjudgment is called “length contraction” and “time dilation” in the theory of relativity. This means that each observer, due to its own conditioning by X and Y, selects and rejects certain aspects of each other, while neglecting the other aspects. This is similar to an observer looking at one side of the cube, and thinking that the other sides don’t exist, or that there is just a flat surface and not a cube. Meanwhile, other observers are looking at other faces of the cube and calling the same thing a different face. The different observers cannot reconcile the different faces into a coherent understanding, and they cannot agree with the other observers.

This problem is called the elephant and the five blind men in Indian philosophy. Each blind man sees one aspect of the elephant, and claims that it is the full elephant, but cannot agree with others who describe the same thing differently. Relativization is the result of not being able to see the full truth because that full truth is not visible to everyone precisely due to their identification with some X, which then precludes the understanding of Y. When X and Y misunderstand Z, and each other, they enter into a conflict about whose version is true, like the blind men who argue with each other about what the elephant truly is.

Reconciling Quantization and Relativization

The above two problems can be resolved if a quantum is also an observer’s perspective. For instance, we can treat the elephant’s parts such as its leg, tail, stomach, ear, and trunk as quanta, each blind man as one perspective on the elephant which receives one quantum and claims that it is the whole truth, but they cannot reconcile all these quanta into the elephant. Reconciliation requires three other possibilities: (a) each blind man can alternately walk over the place of the other blind man and see what he is seeing, (b) he can become a visionary to see the whole elephant, and (c) he can further understand that the elephant is an “I” attached to some temporary “X” calling itself “I am X”.

Through these processes, we can progressively dissolve the mutual exclusion of the various quanta and perspectives. If there is mutual exclusion, then there will be disagreement, competition, and unhappiness. If there is mutual inclusion, then there will be agreement, cooperation, and happiness. Thus, the Vedic texts talk about matter and spirit as duality (exclusion) and non-duality (inclusion).

The fact is that the tail and trunk of an elephant are mutually defined although blind men cannot see that. They can just see that the tail and trunk cannot be varied independently; changing the tail also changes the trunk. This is quantum entanglement. But due to blindness, they cannot accept that the same thing is both tail and trunk, and yet, neither of them. Thus, the problem of quantization and relativization is the result of blindness. The problem doesn’t exist for the visionary. Then, X and Y are not mutually exclusive or contradictory.

When science transitions us from disagreement, competition, and unhappiness to agreement, cooperation, and happiness, then it becomes the solution to the problems of life. This science is not the calculus of infinitesimals, based on motion and acceleration, relying on force and mass, and suffering from the dualities of change vs. constant, discrete vs. continuous, and being vs. becoming. This science is not imbued with indeterminism and the law of unintended consequences. It is not the victim of dualisms such as mind vs. body, religion vs. science, faith vs. reason, private vs. public, and subjective vs. objective.

The truth has always been available in Vedic texts. The practice and realization of that truth required some rituals, which required some mathematics, which went from Brahmanas to masons, traveled to various parts of the world, and eventually became calculus and gravitational theory. This history is obscured at present to preserve the prestige of those who don’t want to acknowledge that they took something from somewhere which was never meant to be used in the way it was used, and this is a dead end now. Now we need a new philosophy even to solve the problems created by calculus. Until then, no fundamental discoveries will ever be made. We are now back to Zeno’s Paradoxes, a 2500-year-old problem, and we should start thinking about the Parmenides-Heraclitus debate again.

Cite this article as:

Ashish Dalela, "The Less Known Problematic Origins of Calculus," in Shabda Journal, March 20, 2023,