Calculus

Derivatives

Derivates are an expression of the instantaneous change with respect to time. so they are expressed as dx/dt. Because they are attempting to measure the change at a given instant or moment, by definition they are already problematic, because change by definition does not occur at an instant, rather it is a measure of change over time, and hence the whole concept of “infinitesimal” times are instriduced, which is really an imaginary quantity that seeks to bridge this fundamental contradiction in our concept of change.

Plotting change for simple functions against time on the x axis enables us to see that the instantaneous change at any point on the resulting curve is given by the slope of the curve at that point. This is quite intuitive- a more steeply sloping change, the kind that gets everyone excited when watching the poll results, house prices or financial markets is indicative of a more rapid growth, whereas flatlining is indicative of a lethargic or dead market, which is good for personal weight records, cholesterol levels etc., you get the point. The slope at any point, in turn is the relationship between the two sides of the infinitesimally smal triangle at that point whose sides are given by dx and dt, and whose hypotenuse is itself the slope. So for example, where there is a high ratio between the height and breadth of the stairs, you would get a steeply sloping staircase.

The practical applications of this observation is measurements of quantities that involve such infinitesimal changes, like the area of a circle. The reason that the circle is such a beguilng figure is because it is a collection of dimensionless points which bend around each other by an immeasurable amount- if it were measurable, then the points would not be dimensionless. But at the same time we can have an idea of the overall result of such infinitesimal changes, which is a circle. But because our reprsentation of points and a circle are approximations, so also are our measurements of such objects, and this is where derivatives come in.

The Area of a Circle

A circle, as we have been saying is an example of necessary approximations in our measurements of objects. We think of quantities such as the length of a line or the area of a tetrahedron as definite quantities, although even here, we ignore infinitisemal quantities like the inherent curvature of surfaces on which that measurement is occuring. In order to specify the are of a circle, one should then be able to insert a definite amount of squares, or tetrahdrons into that circle, since after all, that’s how we express area, isn’t it?- as”square” metres. We’ve defined area as an approximation of the number of squares we can get into it.

Here lies the problem- no matter how tiny the squares near the circles edge, there will always be an area between the edges that remains unaccounted for, because the points being infinitesimally small and dimensionless, we can always “zoom in” an increase the resolution to expose the irregularities that might have become “smoothed over” in our eyesight. It’s more or less precisely the problem of fitting a square into a circular peg, it turns out we can do it- approximately. It turns out also that babies are not silly to try and have a go at it. Approximations are highly scientific.

So this is how derivatives deal with the problem: Rather than squares, imagine a set of rectangles wrapped around each other like rubber bands to obtain the circle, or like a child’s wooden toy with concentric rings, you know, all put together a central pegto make a big circle. Each of those bands has a length and height, being rectangular, and all the heights put together equal the radius of the circle. Each subsequent band is taller than the first, of course, like the child’s toy, so you can arrange them all neatly in a row along the x-axis, can’t you? That’s your area of the circle- its the area of that triangle. The tallest band on the end is one side, its length is 2piR, the circumference, while the other side is simply all the heights added together, which as we have said is the radius, R. The area of that triangle is 1/2 times 2piRsquared. which is piRsquared. Voila, dudes, easy. But that’s still an approximation, as we have seen.

So where is the approximation represented in the formula? I suspect it is in the nature pi, but I’m not sure. Well, it must b, because we never explained how pi got there in the first place. pi of course is the ratio of the circumference and radius of any circle. That we never manage to stop at a partucular decimal is reflective of the fact that as we increase the resolution to orders of ten, the uncertainty remains, infinitely.

Euler’s number and the derivative of exponents

An exponential function is essentially any number raised to an increasing power. When plotted this is a curve of increasing slope. The bigger the base number, the steeper the slope, because the rate of change depends on the number. If the number is 2, quantity is only doubled each time its multiplied by 2. But if it is 3, each time its tripled and if it’s say, 33, each time it’s mutiplied by itself, it’s “33’ed”. So if you had one farm, you got 33 farms. The next time you go crazy, you now got (wait a second while I do the um..mental math) 1089 farms. Yes, you’re just sat there with 1089 farms. You can’t even get around them in a day. The next time you got 35,937. In just three steps, now you can’t even count them in a day (just out of curiosity, the next steps are 1,185,921, 39,134,393, 1,291,424,969- ChatGPT took full 3 seconds to get that last one).

So an exponential always starts at “1” (anything to the power of 0), but the first step is the base number itself (raised to 1 is the number itself). The next step is as many copies of the base as the base itself. So the bigger the base, the more copies of it. You’ve added the number to itself by as many times as the number itself. So if you started with 2, you only get another set of 2. If 2, you now have three sets of three. That’s not a cube, its a square. if 33, then you now have 33 33’s, like the amazed farmer. Every time, you get a number of copies of whatever you had in the previous step equal to the base number- THAT’s the principle of the exponent. The increase at any particular point depends only on the base number.

So to give an example, if you got to a million through a base of 2 as well as through a base of 33 (albeit through differing amounts of steps), the instantaneous change from there on, only depends on the base (granted that this is a fictitious example): in the case of base 2 the next step is 2 million, in the case of base 33, you now have 33million. If you noticed, I was able to calculate the change even (albeit fictitiously) without even going through the initial calculations. The difference in increase was 31million (the first increased by 1million and the second by 32 million).

So you see that the increase is not “equal to” the base, rather it is merely proportional to it, that it, it depends on it in some respect, it is some multiplication of it, at any point on the graph. Now because this “increase” is a time dependent function, we get a specific increase if we introduce a specific time along the x-axis, but because we now fix this time, we get a fixed increase and thereby we fix the proportionality constants to each base. So for the base 2, this value is , for 3 it is , for 4 it is …. and so on. We are tempted to see an exact reflection of the number in the increase in the number, but this is not the case, because of the effect of time.

There is a number for which it is 1 and this number is e which is 2.71818, “Euler’s number”, in the case of which it is an exact reflection of the number itself in the infinitesimal time.

Understanding Gravity

We cannot really imagine cuboidal space bending. When you squash a cube, you’re really only distorting its two-dimensional faces (or envisage a stack of two-dimensional squares that are each being distorted). In order to really bend a cube you would have to make part of it invisible, since we cannot see the fourth dimension. You would need to send part of it into an invisible dimension. This is precisely why demonstrations of space-time bending are always made on two-dimensional membranes (reducing the number of dimensions by 1), else we could not visualize it at all.
To make it slightly more complicated, if we take the grid to be the closest representation of space, then in a two-dimensional space, a third line cannot intersect the plane traced out by any two, it can only intersect the lines- a further line can only intersect lines, not planes (in one dimension two lines cannot intersect period). So also in three-dimensional space, a fourth line cannot intersect the cube described by three. A plane can intersect other planes in those dimensions, it cannot intersect the space itself, it can only lie within the space. Should it be able to do so, that’s space-bending.
It’s one thing to hold that space-time as a whole is non-Euclidean in three dimensions, but for this to be true of localised distortions of space time is more difficult to accept, one feels that here a fourth dimension must be called upon, or that the dimension itself be gravity.
This is difficult to conceive, but if we take the lines as “directions along which the dimension can extend”, then there is no fourth direction along which three-dimensional space can extend. But apparently space has a fourth direction into which it can bend. We never “see” space bend, because its not being bent into any of the three dimensions. When a satellite orbits the Earth, its travelling along a gravitational well, yet there is not visible distortion in space, and the same for the Earth’s orbit around the Sun. Just like a two dimensional object can perform no act that can bend a two dimensional plane of which it were part, since all its acts would also be two dimensional in the same plane, so three dimensional objects like us can perform no action that will bend three dimensional space into a fourth dimension, since our acts are also three-dimensional. There is nothing for us to “push” space into, just like the two dimensional creature cannot push into a third dimension. That’s also the reason that when an object falls, it does so in a straight line, not curved. Curvature of space is felt by three dimensional objects as a force travelling in a straight line, as all forces do. They only orbit under special conditions like water in the sink. The Solar System was formed from a rotating cloud of gas and dust which spun around a newly forming star, our Sun, at its center.

The planets all formed from this spinning disk-shaped cloud, and continued this rotating course around the Sun after they were formed. if you pour a small amount of water into the sink it goes straight in, it does not swirl. But for reasons that I cannot fully explain here, which have to do with the manner in which angular momentum stabilizes systems, this “whirlpool” is a preferred configuration, as we can see it is the shape of the vast mount of galaxies in the Universe that are formed around gravitational sinks. The force the planets experience is directed straight at the Sun, not circularly around it. The sink is only the 4-D representation of what is a straight line in 3-D. So all those straight lines are pushed out into the 4th dimension like a sink. But obviously in 3-dimensions its flat as a pancake and straight as an arrow.
So gravity is basically a force felt in a straight line (which is the only way a force can be felt), caused by an extra-dimension that in the case of a spherical object is shaped as a cone. Is this dimension a spatial dimension? We’d like to say “yes” because after all a cone is a shape in space. However the truth is that gravity is the least well explained factor in physics, so we cannot give a certain answer here. We cannot illustrate or conceive of this bending except mathematically, because we only perceive things in three dimensions. Either there are at least four spatial extended dimensions or General Relativity is false.
In any case I have no reasons to doubt that my interpretation of contemporary physics is anything but true (I have no intention of presenting pseudo-science if I am shown to be wrong). But this: Did the Universe have only three spatial dimensions, one would not return upon oneself were one to travel to the end of it. One does not do so through travelling through a sphere, but only along the surface of it. The fabric of our space-time is curved as the surface of a sphere, in the 4th dimension.
Gravity is the equivalent of this spatial curvature. Thus although we cannot perceive it, we experience it, and in this sense I think it is right to state that our bodies perceive the 4th spatial dimension even though our eyes do not. Since that force is felt by all matter including even light, we are able to measure that curvature through the gravitational lensing of light. Gravitational lensing of light is at this point extremely well attested, and was first famously proven experimentally by Sir Arthur Eddington in confirmation of Einstein’s predictions of it from his theory of Relativity. It goes as thus- when light from a star reaches us, if it has travelled past a heavy body en route, then its path is bent and it is seen away from its true position as a result. Light curves around massive objects in the same manner that planets orbit them, as though drawn by an invisible force- the curving of spacetime in another dimension.

Now as it turns out, the equations of relativity do no include an extra spatial dimension, and so it is that the distortions in space occur within the three dimensions themselves. This is best visualized (IMHO) a concentration of space in some areas and rarefaction in others, rather than a bending into another dimension. There is a change in the metrics of space such that they are not uniform throughout. It’s like a cake which is badly mixed, or a 3-D cobweb-grid filling a box that is distorted by the presence of an insect within it. Space is not bent out into another dimension, rathe it just happens to be more space-y in some places moreso than others. So if space is a quantity (which is debatable) then there is more of it in some places than others and that is what the distortion consists in, that would be another manner of thinking about it.

Gravity and General Relativity

The “Equivalence Principle”

Einstein realised that the weighlessness of free fall is equivalent to the weightlessness of space. A person in free fall does not, contrary to what one might imagine, experience the gravitational force. He ceases to feel the force that we experience, which is the constant pressure on our weight bearing parts and instead and what he experiences instead is acceleration. This is equivalent to a rocket accelerating at 9,82 m/sec. If you wanted to feel nothing on Earth, like someone stationary i space, I think you would need rocket boosters to your feet that held you suspended in mid-air stationary, or perhaps even better, moving with constant velocity (or perhaps both).

The entire Universe, on the other hand, might or might not have an overall curvature. Our best measurements to this moment do not yield any sign of curvature, however like an ant on the globe, this might purely be because we are measuring too small a part of it. This means, for example that if the Universe is “positively” curved then it actually is like the surface of the Earth and you can walk around in it in circles.

Toroid shaped Universe

The universe shape that seems most in vogue is the toroid. A cylinder, which is NOT a sphere, rather a rolled up flat surface is also (shockingly!) a flat surface, equilateral triangles continue to be 180 degrees (trying drawing one and rolling it). When you join the ends of a cylinder therefore the donnut that you get is also flat- a toroid. Equivalent shapes are those that can be bent or moulded into each other as long as you don’t cut and join., so you can have many-shaped toroids, for eg a coffee mug is a toroid becase of its donut handle. The thing about a toroid is that although its flat, its stilll finite, you can alwasy come back to the initial position simply by walking in straight line, and there are a number of suuch possible straight lines, the circumference being only one of them, others are diagonals for eg. The 2-d version of this is the pac-man video game screen.