Approximations to Reality

When I wrote my previous entry, I thought that the criticisms to rationality that I heard in a Philosophy conference a few weeks ago were just some extreme views by some professors operating on the fringes of academia. But I was wrong. After spending some time learning more about this topic and discussing it with other people, I realized that such views are moving now to the mainstream, while the case in favor of rationality is being pushed to the side, falling out of fashion.

Read my previous post on this topic on the following link: A Man tells you He Knows the Exact Truth.

Empirical sciences are under intense scrutiny and settling their role in our world, and in our lives, could be one of the most critical questions in this century. I always took for granted that the “scientific revolution” that started four centuries ago would keep accelerating, a perception that agrees with the current boom of scientific production and the fascination with STEM education.

But if more and more people are feeling disillusioned with the paradigm of knowledge built on rational inquiry and the scientific method, we may well be creating a society where the answers provided by science carry the same weight as the ones obtained with a non-scientific approach. And this should not be a problem, as long as the matters in which non-science has the last word are those that fall outside the boundaries of science. But, does anyone dare to propose a clear demarcation between the two? I would say that Religion, for example, is clearly outside the remit of sciences, but tell that to Richard Dawkins or Peter Atkins.

It seems that the “incompleteness” in the exact sciences frustrate some people, while the hubris of the scientific community irritates many more. I’m not ready yet to say anything about the latter, as I don’t identify arrogance like a badge only worn by scientists: Lawyers, architects, and historians display, in my view, the same sense of superiority as physicists, biologists or geologists. But reflecting about the other point, the incompleteness of sciences, I found a metaphor that can shed some light into this problem, a metaphor having its origins in one of my favorite results in Mathematics: Weierstrass Approximation Theorem.

Athens and Jerusalem

Some time ago, my friend Daniel Andres Diaz wrote an excellent post about the limits of knowledge obtained through reason, emphasizing the intrinsic deficiencies of inductive reasoning in reaching the full understanding of our Universe. For example, empirical sciences are never 100% reliable and rely on statistical methods and the language of probability to shape our confidence around the uncertainty. But the incompleteness problem goes beyond the limitation of our measurement devices: No matter how well calibrated is your equipment, Heisenberg’s uncertainty principle tells you that knowledge of the position of a particle precludes you from knowing its momentum. Always. Everywhere.

And these epistemological constraints are not even exclusive of the physical world: Gödel’s Incompleteness Theorems show with perfect logical rigor that it’s impossible to find a complete and consistent set of axioms for all Mathematics (a matter that deserves its own entry in this blog).

I might be getting this wrong, but the element that seems to unify these perceived shortcomings in science is the nature itself of the material we are using to reach that which is the Whole and covers Everything. From the finitude of our minds, and our reasoning, and our material universe, what hope can we have to answer questions about the infinite and that which goes beyond our Universe? This argument is critical, and one that Jorge Luis Borges entertains in his short story “On Exactitude of Science”, juxtaposing the inherent limitations of some silly maps to figure a representation of the Empire (that Empire which nihil maius cogitari possit, to make the quest truly hopeless).

Reconstructing Reality

One could identify that the insurmountable chasm between science and reality emerges from the nature of the former, being just a proper subset of the later. The tools and procedures that sciences give are contained in reality but clearly don’t correspond to the whole reality, and perhaps we need to go outside of science to make up for the difference. This deficiency is not exclusive of sciences, of course, and language, literature, and arts would face the same limitations. Perhaps the moral of the story is that you need the whole reality to have a chance of knowing the reality, and we are sentenced forever to incomplete knowledge.

I think something different. I think that just by using fragments of the Whole, one has the possibility of reconstructing the Whole; and that this synthetic reality is indistinguishable from reality. This notion has its roots in a fundamental question in mathematics, which deals with the capabilities of collections of subspaces, contained in a universe, to approximate any object in the Universe.

To illustrate this point, think for a moment about the Universe of all possible types of food: vegetables, beans, and legumes; lean meats and poultry and fish; eggs, tofu, nuts and seeds; cereals, milk, ice cream, and biscuits; beer, whiskey, and tequila. Think of the breakfast you have every day, and those exotic insects you ate on your last trip to Mexico. Think about sushi and pizza, chocolate and vanilla, haggis, and ajiaco.

Now, let’s pick one specific type of food, say, for example, fruits, and let’s ask the following questions: To what degree we can use fruits to replace other kinds of food? To what extent can we use apples, pears, bananas, berries, and all the “sweet and fleshy products of trees or other plants that contain seeds and can be eaten as food”, to recreate recipes that use a whole variety of other ingredients?

From our experience in the kitchen, the answers to these questions are pretty obvious, as it is quite intuitive that you won’t be able to replicate the taste of sirloin steak with a bunch of grapes and lemons. But, can you imagine how shocking it would be if someone claims that they have found that just by combining fruits, you could replicate any food? And to be more scandalous, that such replication is “to any degree of accuracy”, meaning that you could always modify a little bit more the fruits’ recipe to make the flavors, textures and nutritional values of your meal indistinguishable to the ones you would have prepared with the standard list of non-fruity ingredients.

In 1885, Karl Weierstrass did just that when he shocked the mathematical community by announcing that polynomials, a humble and delicious type of continuous function, that could well serve the role of the fruits of mathematics, could approximate with any degree of accuracy any continuous function, even the most pathological ones. Weierstrass result sent shockwaves in all directions and could be considered as the formal starting point of what is known as Approximation Theory, a well-established branch of Mathematical Analysis.

Even the Pathological Ones

Everybody should love polynomials. They are the nicest, most well-behaved functions you could think of. They are smoother than silk, extremely versatile, and very pleasant to work with. My cousins in Theoretical Mathematics use them to construct polynomial rings and algebraic varieties, but I prefer to use them for computational purposes, as the raw material for a myriad of beautiful algorithms. Weave sequences of polynomials by interlacing them at right angles, and you form families of orthogonal polynomials, exquisite but strong fabrics that have hypnotized me for a decade. Sadly, the only thing about polynomials that kids learn in school, and some adults can still remember, is that dreadful nursery rhyme: “minus b plus minus the square root of b squared minus four times a times c, all over two a”.

But reality is not the families of polynomials. The Universe (that Empire that extends in all directions), if anything, is the space of continuous functions, of which polynomials are just but one tiny example. The intuitive idea of a continuous function is that is “a function whose graph can be drawn without lifting your pencil from the paper”, a description that most surely was in the minds of XVI and XVII mathematicians, and nowadays in the minds of high school kids taking Calculus. But with the program to formalize mathematics, undertaken by Cauchy and his followers, this vague description was replaced by a technical characterization of what it meant to be continuous. With this more powerful and precise machinery, mathematicians saw in front of them a vast landscape of infinite possibilities, and launched themselves into the wild to explore every corner of the Universe of continuous functions.

I can’t say precisely the date when mathematicians made the first contact with the pathological, monster-like functions. Was it Bolzano in the 1830s? Or Riemann in 1860s? In any case, the creatures that they found (or perhaps more precisely, that they engineered), displayed the most extravagant of behaviors, making some doubt the wisdom in pursuing this path of intellectualism that opened the door to objects that defied the common sense and were stupidly useless. But it was in 1874 when our own hero, Karl Weierstrass, presented to the community the most bizarre specimen known up to that date: a function continuous everywhere but differentiable nowhere. This was a function that, in theory, you could draw without lifting the pencil from the paper, but one that at every point, makes a sharp, aggressive turn. And when I say “at every point” I really mean it: at every single point, of the “two to the aleph sub-zero” infinite number of points there are in the continuum, you need to change directions while keeping the contact with the paper. If you want to have a glimpse of what it would feel to try to do it, think for a second that you are Jack Nicholson in that mental institution of “One Flew Over the Cuckoo’s Nest”, and use a pencil to scratch a piece of paper as violently as you can.

Screen Shot 2019-06-30 at 13.58.01

Now you can appreciate better Weierstrass’ result of 1885. Defying intuition and common sense, Weierstrass showed that using the beloved, polite and gracious polynomials, one could approximate, to any degree of accuracy that you want, even the beast that he had created a decade earlier. Name the most demanding precision that you can, the smallest margin of error that not even the gods could see, the epsilon that could render any plausible gap into dust: you can find a polynomial that will be that close to these functions of madness.

A testament that Weierstrass result was an instant classic is that over the next thirty years, at least thirteen alternative proofs were put forward by the most prominent names in Mathematics of the turn of the 20th century. And although the theorem doesn’t apply to fruits and food, it certainly has been pushed in all possible directions within Mathematics: In 1937, Marshall Stone took Weierstrass result to new heights when he generalized it to compact Hausdorff spaces, and in 1951, after a tour-de-force, Sergei Mergelyan proved it for functions in the complex plane.

Reality Redux

Pathological functions might have looked like freaks back when they were discovered, but in the XX century, they became a pillar of science and mathematics. In my own work, I can think of at least three examples of continuous functions with point-wise erratic behavior: Brownian motion, fractals, and wavelets. I would not advise using polynomials to model them, as there exist more suitable tools to do so; however, that is not the point I want to make here. My argument is that Weierstrass theorem serves as a metaphor of a way to cope with that gap that exists between the cosmic scale of the Everything, and some scattered set of elements in it, which in all appearances look minuscule when compared to the Whole.

Our knowledge of ourselves and the Universe may look incomplete when we insist that reality is achieved only when the error is exactly zero, throwing up our hands in horror when facing the prospects of an alternative. But in fact, what we are doing (what we have been doing for centuries) is closing the abyss, moving one step at a time towards the Truth.

The incompleteness that might still exist after our slow progress does not bother me. That vacuum is just a fantasy, which we use to punish ourselves for not being perfect. But that clumsy understanding of the Universe, which we are creating through language, science, mathematics, and arts, is an approximation more valid and profound than any platonic ideal, forever beyond our grasp. The sequences of steps we are taking are giving us such a crisp and refined understanding of reality that only fear could prevent us to embracing them with determination.

It is perhaps more powerful than all that: The approximating reality is reality itself.

3 thoughts on “Approximations to Reality”

  1. Yes, badge of arrogance, especially by scientists is actually unscientific ! Since you’re a mathematician, you’d know the diff. between not proving and disproving.
    Love your blog !

    Like

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