Where is the Derivation, Dude?


All over the internet and even in the graffiti of underpasses, we are seeing more of science’s and mathematics’ iconic formulas advertised as never before.  On the surface they seem comforting to aficionados, but are they really making the public more likely to delve into the fields?

Anyone can use a hammer, so to a society, the knowledge and skill needed in making one is more important than the skill involved in using it. Similarly anyone— even computers– can plug numbers into mathematical or scientific formulas, so how to derive them is key. Yet schools struggle just to get students to acquire the simple mechanics and end-products of math and science. Increasingly most high schools, perhaps due to preconceived notions, do not emphasize or even cover derivations, acting as if the formulas were brought down on two stone tablets by Moses.

As a result, you can argue that the crux of math and science are not really taught to those who don’t continue to specialise in math and science. Could it be partly why in the United States, evolution is not understood and why there is a large fraction of climate change deniers in the population? Many other countries’ educational systems are not that far behind in misguiding their youth.

In the same vein,  due to oversimplified models of matter we carry around a bricks-and-mortar-simplification in our heads every day. When some say, “we are nothing but an assembly of molecules”, they don’t realise how much they are shortchanging our understanding of chemistry, physics and biology.

Even if we don’t buy arguments along the ideas of those of Philip Goff, who suspects there are aspects of matter itself that play a key role determining the nature of consciousness, there is simply the fact that atoms and molecules are not solid lego blocks. On the contrary they are something quite special, and the sophisticated models we have arrived at , are just that, models. We have no conclusive, final understanding of what they are.

More precisely, the atom’s constituents: protons, neutrons and electrons are not like anything we experience at the macroscopic level. Sometimes they behave like particles, but in other experiments they also show wave-like properties. (Light also has a dual nature, but it’s distinct from particles because light has no mass.) When atoms bond into molecules, the energy levels of electrons change significantly. And the relationship between molecules and their energetic environment is a very dynamic one, leading to an assortment of effects, ranging from colour, synthesis and decomposition.

Several storms of Jupiter caught by NASA’s Juno

And things get far more complicated when molecules are part of a living entity. It’s even far more complex than having to account for the weather of both Earth and Jupiter. If the reference seems outlandish, here’s what I mean. On Earth, storms are caused by the uneven solar heating of our surface and the ensuing pressure differences. On Jupiter, storms result from the uneven heating from within the Jovian planet. In living systems, molecules not only respond to external energy and to molecules from outside of cells, they respond  to the energy provided from ingested molecules and to the information provided by genetic material. The latter is not static itself–its molecules are also subject to the environment.

And it continues. At every level that molecules enter, new realities emerge. There are beautiful mutual relationships that occur between organisms: trees and orchids whose roots need ectomycorrhizal fungi to absorb adequate ions; microbiota in our gut that increase our ability to harvest otherwise inaccessible nutrients; parents who cling to their children from birth to adulthood to nourish them with molecules, habits, morals, skills and knowledge.

Isn’t it then an intellectual crime to proclaim “we are nothing but an assembly of molecules” ? The concepts of math and science are too intricate for parents to handle by themselves. But schools, books and the internet have to do a better job. Simple advertising, mere use of formulas or articulation of technical vocabulary alone will leave children’s minds unnourished.

N.B. And in case you were curious about the origin of eπi + 1 =0, here it is:

Start with the MacLaurin series for e, and use xi as its power. After simplifying the powers of i, select-terms turn out to be the MacLaurin series for cos x, and ,after factoring out i, the series for sin x. Then if we let x = π radians and evaluate the trigonometric terms, we end up with our equation.


Certain aspects of mathematics are a form of poetry of the rational mind. Another consequence of the relationship exi = cos x + i sin x is that if we let x= π/2, we realize that ln(i) = πi/2. From that, we can realize that ii = e-π/2. Where’s the poetry? It surfaces when you put it into words. The irrational number e, raised to an irrational power, is equivalent to an imaginary number raised to the power of itself. 🙂

3 thoughts on “Where is the Derivation, Dude?

Add yours

  1. Hey Enrico,
    It’s been a while. I like your derivation of Euler’s Identity, and also many of the other articles on your site the few that had a look at were very interesting and reminded me of high school!
    I stumbled upon your website because I was trying to contact you but I’m not sure how…
    I wanted to talk to you about a possible collaboration between your writing and a non-profit I founded a few years ago which attempts to reduce food waste in Montreal.
    I say attempts because it is very very difficult, however we have a few initiative which allow us to make a small impact with hopes of a snowball effect. If you’d like to check it out we have a website it’s http://www.fondationlafourmiverte.org.
    Anyways, I’d really like to exchange emails with you or meet eventually if you’d be interested in writing a blurb in our member newsletter that we send out twice a year to all our members. If possible it would be best to reach me either through the contact us page on the website or at my person email at andrewliberio@hotmail.com since I’m very unfamiliar with WordPress.
    I hope you’re doing well, and that you’re enjoying your “retirement”.
    I hope to hear from you soon,
    Andrew Liberio

    Liked by 1 person

  2. Hi — I’m creating a presentation on the hidden meaning behind Euler’s ID, and searching for images I found this brilliant one of you (presume it’s you?) with the graffiti.
    Can I have your permission to use it? It will be ultimately be filmed and posted online, all being well.
    Let me know?


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