Miseducation in STEM

Preface

There is nothing inherently evil about a traditional liberal arts education. In fact, what poses as a well rounded education today pales in comparison to the wisdom taught in ancient times. So what was the original outlook for this discipline of thought? In the times of classical Greece, it was quite apparent most men were born into a craft and were only technically proficient in one skill. Workers such as metalsmiths, potters, and farmers were capable of learning how to read and write, but never saw the need to. This wasted potential hindered expression of thought, limiting the extensibility of the human mind. Even if layman weren't interested in the abstractness of academics, it would at least be sensible to give them the choice over being locked into a single trade.

The original seven arts first require a mastery of grammar, logic, and rhetoric. The next four arts are more specialized which include: arithmetic, geometry, music, and astronomy. It doesn't matter that these skills were taught 2,000 years ago; they will always be relevant and are demonstrably not taught well enough in public education. The majority of students at big universities would find better purpose in their abilities by pursuing a vocational craft rather than fall for the debt traps of academia because self studying and a sharp mind are free.

But as high school students are propagandized to pursue higher education, the ones who are passionate enough about pursuing the truth see no other choice than to sign up for the innocent and objective fields such as Math or Engineering. We see this now as many college programs report an increase in STEM applicants. These youth want to learn a practical skill they'd imagine could help them in their professional careers only to realize it's all corporate grooming.

Academia, irrespective of field, is a tit for tat game. If you help a professor prove his research, he will be more likely to advertise and vouch for yours in peer review. There's also fiscal corruption where money laundering incentivises academics to look the other way when a new ill-formed concept is being pushed. These ideas are taught as if they were truth with a capital T.

Academia used to be a privilege for the gifted, but thanks to the standardization of everything, colleges are available to all walks of life, the difference this time is the rigor or the lack there of in curriculums and the moral shortcomings of those who teach them. The spells and deceptions of academia do not stop at gender studies or CRT. Not only are curricula in math, physics, and computer science filled with unhelpful repeated material but the way they are taught contests the original meaning of liberal in liberal arts: liber, free and unrestricted. In other words, STEM students are taught to depend on and appeal to academic institutions instead of learning how to learn on their own.

Pedagogical Practices

The flip side of what teachers are taught to do showcases what they ought to do: provide a grounding or a framework for students to stand upon to see a glimpse of Truth. Teachers must balance student's understanding of the material and their fluency in problem solving. Old school methods of rote memorization are great for cranking out problem after problem. For example if we had to rigorously determine simple arithmetic like ⅔ ÷ 49, nothing would ever be accomplished. Using muscle memory frees our minds to work on harder problems. That doesn't mean we should neglect to understand what the question is asking: how many 49'ths can be repeatedly subtracted from two thirds of the unit?

On the other hand, in common core there's too much emphasis put on the understanding side of things. Kids have to know 10 different methods of doing a simple calculation, only one of which is the accepted way on the exam. Everyone theoretically knows what they need to do, but it takes them to long to do it. Ideally you should know what fraction division is and be able to answer a question on it instantly. You need both the fluency with basic operations which comes with drills and the deep understanding of why the operations are valid in order to graduate on to the harder and more abstract problems or you'll be lost. The more calculations one does the greater their intuition becomes; the more intuition one has the better at calculations they become.

Learning Styles

A rather large amount of time used to be spent on training teachers to cater to learning styles. VARK (visual, auditory, reading/writing, and kinesthetic) are the four main alleged learning archetypes for each person. While it is true that not everyone learns the same way, when students are taught with their own style preference, exam scores do not indicate any advantage. In fact, when someone says they learn by seeing or doing they are just confirming their past biases and successes, when a teaching environment that uses all modes of learning could've been equally or even more effective.

Learning styles makes learning worse. It gives teachers unnecessary things to tend to, and makes students reluctant to engage with certain types of instructions. So not only is involving multiple ways of understanding the same concept important, but integrating many concepts to arrive at the correct approach to seeking knowledge is the fundamental skill.

STEM curricula will ingrain into student's mind, that episteme, in other words pure scientific knowledge, is the only legitimate kind. Techne, or technique (craftsmanship), is a completely separate yet valid body of knowledge, and there are many others. Wisdom (sophia), is a special kind of knowledge which can not be accounted for under the scientific method, but rather through philosophical thinking. As cliché as "thinking outside the box" sounds, it's a skill all of us need.

Platonism

Disclaimer: The actual identity and personhood of the man we know of as "Plato" is independent of the metaphysics he presented. Lets take a look at an article by the lecturer Balaguer from the Stanford Encyclopedia of Philosophy who does not take kind to his thinking. Note, many academics will bring up topics that are completely irrelevant to the argument as we'll see. First off, Platonism in Metaphysics is redundant. Platonism is metaphysics: the discussion of which concepts are independent of the human mind. Next, he defines it as:

Platonism is the view that there exist such things as abstract objects - where an abstract object does not exist in space or time and which is therefore entirely non-physical and non-mental.

While this definition is a somewhat amateur take, it's good enough to give anyone a basic grasp on the concept. Here's where it goes wrong: neither space nor time are relevant where abstract objects are concerned. In fact, time is meaningless without matter, so to include these in the definition in an encyclopedia shows how tethered the author is to the material world. A better definition would be: ... - where an abstract object is entirely non-physical, inanimate and independent of the human mind. Abstract objects being inanimate, have always existed and continue to exist, whether anyone chooses to think about them or not. These objects are referred to as perfect concepts or noumena (non-mental) coined by Kant, where phenomena are objects of the five senses and are used for observations in the scientific method.

Balaguer's first example, Consider the sentence '3 is prime' is objectively a bad choice to illustrate what it means to be an abstract object. Numbers are already abstract objects and primeness is a property of only certain kinds of numbers. It also assumes you know some math which by what I've been saying so far would not be likely as STEM students aren't supposed to go anywhere near philosophy. Other red herrings are creatures like unicorns or the flying spaghetti monster which borrow from the material world and are most certainly not abstract objects. A better example would be π as the length of the circumference compared to the length of the diameter. As described here circles are on the higher order of reality: a universal form, and so even an alien civilization would be able to derive an identical understanding of π.

Another strawman is as so:

Consider the property of being red. According to the platonist view of properties, the property of redness exists independently of any red thing.

Wrong! Properties of color do not apply to an abstract object because color presupposes a physical object and a light source. Abstract objects do not take on the same attributes of physical objects: it's the other way around. Known as reification or mimicry, physical symbols or objects are used to represent absolute concepts. There is no way to reify color because abstract objects cannot be seen.

So he goes on and on about redness, and finally gets to his argument: the Epistemological one, which goes:

1. Human beings exist entirely within spacetime.
2. If there exist any abstract mathematical objects, then they do not exist in spacetime.
3. This means human beings could not attain knowledge of said objects.
4. If mathematical Platonism is correct, then humans can't attain mathematical knowledge.
5. Humans have mathematical knowledge therefore, Platonism is not correct.

The last two claims are so fanatical they aren't even worth analyzing. Platonism is the only worthwhile philosophy and is how the Ancient Greeks came to realize all of their well formed mathematics; You cannot think properly without understanding it. Undoubtedly, without philosophy there would be no realization of any other knowledge or science. This is why it should be included in STEM curricula not as an elective, but as an actual course instead of long winded research paper classes. This would apply especially to math majors.

What Defines a Well Formed Concept?

Circularity

So before we go into what it means to be well defined, we should note that in philosophy, circularity or circular reasoning is acceptable. To understand why take a look here. Essentially math cannot prove itself, and so philosophy fills this epistemic gap by anchoring what it means to be true, and accounting for why we can construct arguments in the first place. Truth is the ultimate circular concept. It permits its own existence. We will be defining what it means to define a concept, so obviously we have to appeal to circularity.

Sometimes circular definitions are acceptable in math. For example, one is one. This is the law of identity, and does not need rigorous proof. Generally speaking, mathematicians do not care about circular definitions because it's not describing anything new. While not incorrect, it's generally bad form and creates tautologies. Circularity in science is strictly forbidden and is completely incorrect. This leads us to then ask why we care about well defined concepts in the first place.

STEM students are presented with dump trucks full of propositions and theories often without any in depth explaining. It is up to them to see why they are all supposedly true. Here is how one can systematically see for themselves if a concept is well defined:

Conditions

1. Must be reifiable either intangibly or tangibly.
2. Must be defined in terms of attributes which it possesses, not of which it lacks.
4. Must exist as a perfect concept, independent of the human mind.

For example, the concept of infinity is ill-formed because we cannot produce an instance of it. Not only does it fail this first step, but it also fails the next, since it's coined as the negation of being finite: defined by an attribute which it lacks.

Again, we might know in our hearts infinity exists but its best we leave philosophy and theology to tackle it as the STEM fields are not an appropriate paradigm.

As another example, let's analyze irrational numbers. Well we could compare the measure of the hypotenuse of any isosceles triangle to one of its legs and call it,  2 , so we pass step one. However, in step two, we realize irrational numbers are defined as being not rational, and are therefore defined by something they lack i.e. a ratio. Irrationals are incommensurable meaning impossible to measure, therefore we could never define them to be numbers, the official definition of which we will now move on to.

Origin of Numbers

number - coming to the realization that the notion of number is a difficult if not impossible idea to define precisely. - Edwin Clemenz, PhD

It's rather surprising math major's and even computer science majors aren't taught the formal definition of number, and even more outrageous the professors don't know it. When studying how numbers came to be, one must realize that ratios and magnitudes (short vs long, light vs heavy) came way before the discovery of numbers as an abstract object, just as division came way before multiplication.

One has to read Euclid's Elements, the second most printed piece of literature after the bible, to even start to understand the number concept. In Ancient Greek there was no word for abstract and it was assumed scholars should know math is a concept of the mind, so the direct translations can get hairy / circular. By book seven, a weaker form of geometry called algebra is introduced through the abstract unit. Why is algebra weaker? In geometry we can arbitrarily choose what our unit length shall be, so we can physically construct the length of something like π ×  2 . In algebra, our reference length is abstracted to a dimensionless unit, called 1 making it impossible to know the exact value of the aforementioned quantity. Before we can say what 1 or any number is for that matter, we need to discuss magnitudes.

Magnitudes

Magnitudes are measuring attributes such as length, weight or density. They are an exact measure of a physical or visual quantity with dimensions. In language we use them qualitatively such as, A tree is bigger than a sapling but this doesn't tell us by how much. For more exact language, we need to compare the magnitude of what we're measuring to something else for reference.

Ratios

A ratio is simply a comparison of two homogeneous magnitudes. By homogeneous, we mean the two lengths have a common measure. For example, _ : _ _ and _ _ _ : _ _ _ _ are both valid ratios because both their antecedent and consequent pieces are measured with the unit of size _. If they didn't have a common measure then they would not be considered numbers. In the case of π, the ratio of the length of the circumference compared to the length of the diameter is measureless. This is because there is no common measuring unit or building block which can compare them. This is why π cannot be written as a ratio or rational number and is consequently irrational (not a number). When p : q both have a common measure we can write p / q as the name of the measure, and the name is what we call a number.

Number

Number, [nombres in French: number. nombre in Spanish: name] is the name given to a measure which describes a magnitude. So, _ _ : _ is a ratio with a common measure of _, and so _ _ / _ is the name of this measure. In algebra we refer to this number as 2/1 or simply 2.

In algebra there is no distinction made between 12 and 24, but in geometry while they are proportional they aren't congruent since their measure's are different. Look at the case where we have two right isosceles triangles whose leg's are different magnitudes. In either triangle, we see the ratio, hypotenuse : leg =  2 . But which measure yields the real  2 ? Really neither, since there is no ideal unit in geometry. More importantly, we see that  2  ≠ hypotenuse / leg, because in algebra there is no such thing as length. So of course when we abstract out a definition which depends on a unit length (the leg) we will get an indescribable magnitude (1.414...).

This makes you wonder what other concepts throughout STEM are most likely left incomplete or inadvertently taught flat out wrong.

Calculus Without Limits

It is without doubt that many calculus students dread the first few weeks of class. Why? Because it is nothing but limit theory. The idea that an object can be equal to the limit of itself is not only incorrect but extremely confusing for new students. Dancing around the idea of zero divided by zero being indeterminate yet the basis of the fundamental theorem of calculus is silly. What if I told you all of calculus could be done with no knowledge of infinities, infinitesimals, or even knowing what a limit is?

Derivatives

Here is the mainstream single variable definition of a derivative which any calculus student knows by heart:

$$\frac{d}{{dx}}f\left( x \right) = \mathop {\lim }\limits_{\Delta \to 0} \frac{{f\left( {x + \Delta } \right) - f\left( x \right)}}{\Delta }$$

If you haven't taken calculus the geometric meaning of the derivative is as follows: given any smooth and continuous curve, the secant line (a line which touches the curve twice) eventually becomes the tangent line (touches the curve once) at the coordinate (x, f(x)), as Δ goes to zero. Here's a visual:

The definition certainly gives you the correct answers but why confuse students with the zero divided by zero nonsense? Also, they find it rather cumbersome to learn a different derivative rule for each type of function. Surely there is a systematic method which does not involve limits.

In this diagram we can observe by similar triangles that the slope of any tangent line is given by the slope of a parallel secant line. The secant line is given by connecting the coordinates (x - m, f(x - m)) and (x + n, f(x + n)), where m, n are the distances to the left and right of x, the spot we are trying to find the slope at. We will derive an auxiliary equation which gives us m with our arbitrary choice of n at any x in an example.

So from the total rise over run in the diagram, we can yield an alternate definition of the derivative without limits:

$$f'(x) = \frac{{f\left( {x + n } \right) - f\left( {x - m} \right)}}{m + n }$$

Different values for m and n are possible, but what isn't is m + n = 0 because m + n represents the distance of the leg of a triangle. If m were to equal n we wouldn't have a secant line. Two non-zero positive rational numbers cannot be combined to equal zero, meaning our denominator never equals zero. This demonstrates our definition can never be undefined. Now we have to work out how we get m and n, then we're done.

Let's find the general derivative of x3:

$$f'(x) = \frac{ (x + n)^3 - (x - m)^3}{m + n }$$ $$f'(x) = \frac{x^3 + 3x^2n + 3xn^2 + n^3 -x^3 +3x^2m - 3xm^2 +m^3}{m + n }$$ $$f'(x) = \frac{3x^2(m+n) +3x(n-m)(m+n)+(m^2-mn + n^2)(m + n)}{m + n }$$ $$f'(x) = 3x^2 + 3x(n-m) + (m^2-mn + n^2)$$

We can now hopefully see we have our answer in the form:

$$f'(x) + Q(x,m,n)$$

Q is the auxiliary function which always equals zero. Instead of throwing this extra information away like in mainstream calculus we can use it to find values of m or n. By completing the square, we can express m in terms x and n or vice versa.

$$m^2 - (3x +n)m + (3xn + n^2) = 0$$

$$m(x,n) = \frac{3x+n \pm \sqrt{(3x+n)^2 -4(3xn +n^2)}}{2}$$

Obviously if all you care about is the general derivative itself, the last part is not mandatory. For functions like sin(x) we'd plug in the Taylor series expansion. Keep in mind the discovery of the derivatives to the trigonometric functions had to come AFTER the discovery of their Taylor series expansions. Note that Newton worked out these Taylor series by trial and error in De Analysi, but Brook Taylor formally generalized polynomial expansions for all functions. To prove derivatives without first appealing to derivative rules, the function in mind needs to be in polynomial form.

Going back to differentiating sin(x), all the odd degree terms will cancel and we'd be left with the even middle terms, giving cos(x). This will work for any function which is smooth and continuous. Sure, memorizing the rule via drill might be easier, but if you wanted to gain a deep understanding without limits it is certainly possible and even more powerful.

Here is some C which uses the root approximation method to calculate n given x and m for sin(x):

#include <stdio.h>
#include <math.h>

#define e    2.7182818284590452353602874713527
#define pi   3.1415926535897932384626433832795

double f(double x, double m, double init_n) {
double r=init_n, t=0, t1=0, n;
do {n=r;
t = ( sin(x+n)-sin(x-m)-m*cos(x) ) / (n*cos(x)) - 1;
t1= (-(1/cos(x))*(sin(m-x)+sin(n+x))+m+n*cos(n+x)/cos(x))/(n*n);
r=n-t/t1;
} while  ( fabs(r-n) > 0.00001  );
return r;
}

int main() {
\\ Function is called with angle, m and an initial value for n
double res=f((double) (pi/3), (double) 0.1, (double) 0.1);
printf("\n%lf", res);    \\ res contains the value of n
res=(sin(pi/3+res)-sin(pi/3-0.1))/(0.1+res);
printf("\n%lf", res);     \\ res contains the value of cos pi/3.
return 0;
}

With integrals, something clever can be done which relates to this alternate definition of the derivative. It removes the need for differentials and infinitesimals, but it's best if we save that for another day. What one should take from this is calculus was discovered in a brute force fashion as physicists in the seventeenth century were searching for a better toolkit. And thus calculus was not founded in a purely rigorous mathematical environment but from an empirical one.

One last misconception in STEM that needs to be addressed is a concept which directly relates to calculus, notably in physics. So what specifically about time in Einsteinian physics does academia get so undoubtedly wrong?

Time as a Concept

In order for anything you say about time in physics to make sense, you must first define what time is.

Right off the bat: no, time is not the indefinite continued progress of existence and events in the past, present and future; time is not defined by the jittering of cesium isotopes; and no, time is not the mystical fourth dimension. It certainly does not exist merely within our minds and neither is it a grand illusion. We need to return to what makes a concept well defined. For time to be reified we need two conditions:

1. The existence of matter
2. Constant repeatable action/motion called events

Take a sandglass for example, we take some sand (matter) and create a glass tube which is constricted in its center. When ALL the sand passes through from one end to the other at a constant rate we call this an event.

Just like in geometry we can make our unit of measurement anything we like. For simplicities sake let's call the event just mentioned a cycle. Now we can measure how many cycles it takes to drive to the supermarket or for the sun to rise and set. This process determines time intervals in terms of cycles. When dealing with distances we can get a grasp of how fast we're going by forming the ratio distance : cycle.

One drawback is not everyone can reference our sandglass and know how long a cycle is. However, there is one event everyone becomes used to because it's always happening: it's the rotation of the cosmic heavens. The sun tells us the time of day, the moon tells the day of the month and the stars reflect the time of year or even the ages (eons).

If you thought astronomy being apart of the original seven liberal arts was foolish, think again. The ancients knew the intimate link between the motion of the heavens and the passage of time:

And God said: Let there be light made in the firmament of heaven, to divide the day and night, and let them be for signs, and for seasons, and for days and years. To shine in the firmament of heaven, and to give light upon the earth. And it was so done.

The book of Revelation even describes the end of time coinciding with stars falling out of the heavens or the pausing of their movement.

Living in large cities (worldliness), the light pollution and the tall buildings obstruct your view of the cosmos. This severely warps your sense of time.

So, we use the cosmic rotation event as the standard unit of time. We notice how one complete rotation consists of light and darkness. The fact that these actions or motions in the sky are constant and repeatable, we can verify that time can be reified.

After this, we define time as so: it is the name given to a distance that describes the motion of a certain body of matter from one position to another at a constant rate. Note the definition of time almost identically lines up with the concept of numbers. This is exactly Aristotle's view: time is a number of motion.

In our definition we could easily have focused on mass instead of distance. If we define a second as the number of times a cesium-133 atom performs 9,192,631,770 oscillations, we are STILL describing a mass, and a constant repeatable action, although this is the materialist's view on time. The definition would be: time is the name given to a mass that describes the decay of a certain radioactive material from one state to another, not as intuitive. Whichever scalar we choose, we more importantly should know the concept of time describes the occurrence of an event and distinguishes the event from all other events that do not occur simultaneously or concurrently.

Now that we have defined time in terms of what it possesses, we can check off step two from the well definedness criterion. For step three, lets assume the previous definition of the concept led to logical contradictions. Then it would follow that we would not be able to distinguish one event from any other, which if true, you would not be able to read this article. Reading requires distinguishing one event of processing a word or group of words from the next event. So obviously, the concept of time couldn't lead to a logical contradiction (paradoxes are not necessarily contradictions). Finally, we should confirm time is non-mental i.e. a noumena.

When we measure time we use a physical device like a clock or a sun dial. That does not mean time is a physical quantity. There is no particle made of time, and if there were you'd have to be outside of time to verify and manipulate it which by the first requirement of reification, would imply you are not made of matter. You'd be timeless, spaceless and all eternal. So just like numbers we know that time exists independently of the human mind. It's not a concept mankind made up to organize society; it's absolute. Therefore we have passed step four and have rigorously shown time is well formed. It isn't up for debate or a mystery that needs string theory to be explained. Einstein did not understand what time is, and consequently anyone who teaches his plagiarized guess work is also deceiving you.

Closing Remarks

If the original goal of the liberal arts was to free the student, in the sense that they could abstract on what they learned and take on vastly more skilled labor positions, then modern academia (especially in the sect of STEM) has utterly failed. Instead, students are coached to go to college with only one job description in mind.

Do not believe you're protected because you're enrolled in a STEM program. Anything which modern academia touches turns to filth. In no way should you disregard pursuing a degree but always remain skeptical of what is being taught. All it takes to commit an atrocity is to teach and instill one absurdity. Given that the left brain dominant occupy the corporate ladder, the pharmaceutical industry, the industrial military complex, space agencies and the intelligence community: STEM students have the utmost responsibility of not succumbing to their ideologies by challenging the convenient lies told to them in school and beyond.

These lies are used as a grooming ritual to render students useless on their own, especially in corporate environments. Thinking for oneself is only encouraged for menially niche tasks and often ostracizing when it isn't in the range of acceptable discourse. STEM's broad range of disciplines breeds busy bees, a destructive and containerized hive mind network, which is already here and will be the demise of what we know of as Western Culture. How so? We don't need more engineers, we need people who can fix stuff. The fear mongering that being a jack of all trades is risky, has produced worker bees who can only do one thing well. The stigma that trade skills and the arts are antiquated will inevitably turn people into useless-meaningless husks of nihilism when left to their own devices. White collar jobs are a spiritual dead end.

What can you do? Rise to the occasion. Challenge professors or other peers when you are presented with an ill-formed idea. See past deceptions by learning real philosophy. Learn a side trade / skill. Don't get tunnel vision. Start thinking about how you can gather multiple streams of income and focus on setting yourself up to build a family (otherwise known as success). God bless.