idealization blindness
idealization mathematics modelstoday i’ll be writing about the common scientific deepity “everything is just x“. i previously approached this, but today i’ll give a clearer example.
idealization is a common habit of most knowledge based activities, best exemplified by the problem “describe the forces affecting a falling rock R on earth E”. to answer “F=mg” where g is the acceleration of gravity, we would have to idealize the problem and its constituents:
- g is a constant not depending on where R is relative to E;
- R can be modeled by its center of mass and therefore all its “parts” are equally affected by the force;
- there is no atmosphere on earth to cause air resistance, no wind to push the rock;
- the rock does not absorb, dissipate or produce any kind of heat or energy that might cause it to move;
- the rock does not lose mass, dissolve, evaporate;
- the planet does not suffer from any of the issues mentioned.
so our idealization of the problem will give us a working, real world approximation of the answer, which, for the most part, should be enough. it is important to understand this because this is where some people diverge when dealing with models.
our model of the rock was initially “F=mg” where m is the mass of the rock. but if we drop some of the idealizations and add wind resistance for example, it is now “F=mg – bv” where v is the velocity and b is a constant that depends on the density of the air and on the size of the object. so already idealizing our rock has prevented us from a more accurate solution. we can keep adding terms to this total to get a more accurate description of the problem, yet already with this one we have to deal with differential equations (remember g is acceleration and v is velocity, meaning v will change over time). if we then add the density of the air around the rock, which depends on atmospheric density, which changes in space and time, we will complicate our problem so much we won’t be able to solve it properly. then there is the heating of the rock and the likelihood of it breaking and/or losing mass and its shape. we can go on endlessly.
now, it is true that usually, the broader idealization has results good enough for every day precision, and precision can increase by “de-idealizing” the problem. the more we “de-idealize”, the more accurate (and complicated) our model will become. but there is no way of completely “de-idealizing” something, since we are always dependent on observer error. we can increase our precision to a certain extent, but our accuracy might be fundamentally biased.
so a model is just that, a model, with a certain accuracy and precision. so saying that when a rock falls all it has is gravity pulling it is wrong on many levels. but this is not the main problem of idealization, since what i described is what engineering deals with every day, and we don’t trust our bridges any less because of this.
the problem of idealization is when it is subsequently used for induction. for example, i assumed my rock was a certain idealized hypothetical rock, and then i use this idealized rock as an argument for another model and deduce the properties of the new system with the idealization as an undiscussed assumption. the deduction will have amplified my idealization error and might yield completely wrong results.
for example: there is no such thing as a square or a circle in nature (not as real objects, but they exist as physical ideas in brains). these are two examples of completely idealized shapes. now, multiplication, as it is defined, calculates the area of a rectangle a times b. so the area of a square is its side l times itself. but an idealized circle has no sides! how can we calculate an area if our multiplication operation assumes two sides of a rectangle? we conjure up a magical number, lets call it , that turns our circle with radius r into a rectangle with side and side b = r. our area is now , a very known formula. note that what i did was idealize a magic number that turns a circle into a rectangle (that number happens to be ). since this situation is completely disconnected from the “real world”, this number can “exist” in our minds. and it’s very surprising how we can manipulate these transcendental numbers as if they were real things, and how idealizing actually “works”, in the sense that it makes us be able to do calculations properly.
but this is where the blindness begins. instead of saying “there will always be error in calculating the area of a circle because the formula for area is based on multiplication, which is based on rectangles” we say “there will always be error in calculating the area of a circle because is an infinite number”. isn’t this confusing? is this number real, so much that it can be used like any other? the idealization has taken over the very definition, and with it, turned us away from the real problem itself, and allowed us, on one hand, to advance our abstractions, but also, to disconnect from reality.
my favorite example is when people say “everything is energy” or “everything is just atoms and molecules” or “reality is quantum physical”. following the example, it’s like saying “the area of a circle is “. these terms used are idealizations, superficially true, but deeply false (hence the deepity). if any of these sentences was true, we would be talking of a total and complete understanding of nature, which is not the case.
but an anecdote illustrates what i mean very simply. to a mathematician, everything is just numbers in very elaborate ways. and to deal with them, we can just approach reality using these abstractions instead of the real thing. but this would be like saying “the entire works of william shakespeare is just the collection of letters from a to z and some minor punctuation and line breaks, so we can just deal with the alphabet instead of reading the books”. sounds ridiculous, but it’s exactly what idealization blindness is. we must be aware that reality is beyond our subjective ideals, and that all we do must be tested by it, and our accuracy will always be limited. at least, that’s how i’ve idealized my own subjective experience.