Monday, October 31, 2011

The Language of Science and the Tower of Babel

And God said: Behold one people with one language for them all ... and now nothing that they venture will be kept from them. ... [And] there God mixed up the language of all the land. (Genesis, 11:6-9)

"Philosophy is written in this grand book the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and to read the alphabet in which it is composed. It is written in the language of mathematics." Galileo Galilei

Language is power over the unknown. 

Mathematics is the language of science, and computation is the modern voice in which this language is spoken. Scientists and engineers explore the book of nature with computer simulations of swirling galaxies and colliding atoms, crashing cars and wind-swept buildings. The wonders of nature and the powers of technological innovation are displayed on computer screens, "continually open to our gaze." The language of science empowers us to dispel confusion and uncertainty, but only with great effort do we change the babble of sounds and symbols into useful, meaningful and reliable communication. How we do that depends on the type of uncertainty against which the language struggles.

Mathematical equations encode our understanding of nature, and Galileo exhorts us to learn this code. One challenge here is that a single equation represents an infinity of situations. For instance, the equation describing a flowing liquid captures water gushing from a pipe, blood coursing in our veins, and a droplet splashing from a puddle. Gazing at the equation is not at all like gazing at the droplet. Understanding grows by exposure to pictures and examples. Computations provide numerical examples of equations that can be realized as pictures. Computations can simulate nature, allowing us to explore at our leisure.

Two questions face the user of computations: Are we calculating the correct equations? Are we calculating the equations correctly? The first question expresses the scientist's ignorance - or at least uncertainty - about how the world works. The second question reflects the programmer's ignorance or uncertainty about the faithfulness of the computer program to the equations. Both questions deal with the fidelity between two entities. However, the entities involved are very different and the uncertainties are very different as well.

The scientist's uncertainty is reduced by the ingenuity of the experimenter. Equations make predictions that can be tested by experiment. For instance, Galileo predicted that small and large balls will fall at the same rate, as he is reported to have tested from the tower of Pisa. Equations are rejected or modified when their predictions don't match the experimenter's observation. The scientist's uncertainty and ignorance are whittled away by testing equations against observation of the real world. Experiments may be extraordinarily subtle or difficult or costly because nature's unknown is so endlessly rich in possibilities. Nonetheless, observation of nature remorselessly cuts false equations from the body of scientific doctrine. God speaks through nature, as it were, and "the Eternal of Israel does not deceive or console." (1 Samuel, 15:29). When this observational cutting and chopping is (temporarily) halted, the remaining equations are said to be "validated" (but they remain on the chopping block for further testing).

The programmer's life is, in one sense, more difficult than the experimenter's. Imagine a huge computer program containing millions of lines of code, the accumulated fruit of thousands of hours of effort by many people. How do we verify that this computation faithfully reflects the equations that have ostensibly been programmed? Of course they've been checked again and again for typos or logical faults or syntactic errors. Very clever methods are available for code verification. Nonetheless, programmers are only human, and some infidelity may slip through. What remorseless knife does the programmer have with which to verify that the equations are correctly calculated? Testing computation against observation does not allow us to distinguish between errors in the equations, errors in the program, and compensatory errors in both.

The experimenter compares an equation's prediction against an observation of nature. Like the experimenter, the programmer compares the computation against something. However, for the programmer, the sharp knife of nature is not available. In special cases the programmer can compare against a known answer. More frequently the programmer must compare against other computations which have already been verified (by some earlier comparison). The verification of a computation - as distinct from the validation of an equation - can only use other high-level human-made results. The programmer's comparisons can only be traced back to other comparisons. It is true that the experimenter's tests are intermediated by human artifacts like calipers or cyclotrons. Nonetheless, bedrock for the experimenter is the "reality out there". The experimenter's tests can be traced back to observations of elementary real events. The programmer does not have that recourse. One might say that God speaks to the experimenter through nature, but the programmer has no such Voice upon which to rely.

The tower built of old would have reached the heavens because of the power of language. That tower was never completed because God turned talk into babble and dispersed the people across the land. Scholars have argued whether the story prescribes a moral norm, or simply describes the way things are, but the power of language has never been disputed.

The tower was never completed, just as science, it seems, has a long way to go. Genius, said Edison, is 1 percent inspiration and 99 percent perspiration. A good part of the sweat comes from getting the language right, whether mathematical equations or computer programs.

Part of the challenge is finding order in nature's bubbling variety. Each equation captures a glimpse of that order, adding one block to the structure of science. Furthermore, equations must be validated, which is only a stop-gap. All blocks crumble eventually, and all equations are fallible and likely to be falsified.

Another challenge in science and engineering is grasping the myriad implications that are distilled into an equation. An equation compresses and summarizes, while computer simulations go the other way, restoring detail and specificity. The fidelity of a simulation to the equation is usually verified by comparing against other simulations. This is like the dictionary paradox: using words to define words.

It is by inventing and exploiting symbols that humans have constructed an orderly world out of the confusing tumult of experience. With symbols, like with blocks in the tower, the sky is the limit.


  1. "Lingua Franca" was a term to describe a universal language (not restricted to science). It used to be ancient Greek after Alexander and well into the roman Empire (where the eastern roman empire spoke a simplified common ancient Greek language, and even some Roman Emperors (like Marcus Aurelius) wrote their thoughts and memoirs in Greek). Then it was Latin, until well after the Renaissance, and recently it was replaced by... Broken English.

    That's the nice thing about Math and Music, their language is already universal, and that's why in Sci Fi books and movies with contact with other planets, we communicate with some kind of musical signal or geometric drawing or math equation or series.

    One problem the "Babelians" had was that they assumed there was some fixed ceiling (aka heaven) not far away from the ground that they thought they could reach with a tall enough tower. The project may have failed because they spoke many different languages, but even if they all spoke the same language, (not even dialects allowed), would they do any better?

    The blog reminded me of Angelos T., a very gifted student in my high school, one year older than me, who must have known 100+ languages (or at least the basics thereof). People were always surprised when he would greet them in their own language, whatever that was. from Serbo-Croatian to Tagalog.

    Surprisingly, he was also a big fan of Esperanto, the artificial universal language, and wore a button or pin with its symbol or flag, that included some kind of a star. A chemistry instructor suspiciously asked him (this was the late 60s, early 70s) if that pin was not the... Viet Cong flag.(!)

    In case you are curious, that top student continued to Princeton for his UG and Harvard for his PhD (which, he told me with a lot of pleasure when I met him again at MIT in 77, was in "Pure Math"), but for various reasons he never finished the Phd, at some point was back to his old high school as a lowly teacher, but eventually his linguistic talents (or maybe the family connections?) got him a job as a EC top level interpreter, where, as he said, he is "ashamed to admit" how high his salary is.

    BTW Angelos' father was an author of crime novels.

    Again, sorry for the lengthy comments

  2. We live our lives sensing what we sense, including what we sense we know. That is a belief, as much as believing that we know that we sense is a belief. So from the earliest of human times on, we have developed that knowledge, making sure we could believe it, but there is always a boundary beyond which we cannot validate our knowledge or make sure it is reliable and reproducible.

    What this boils down to, in my humble opinion, in circumstances of radical uncertainty, is pure sensing, using mechanical instruments with or without our human capacities, of justified true belief or Truth in the sense described above. So it is not what we say we know but what we believe we know, that we can sense and make sense of.

    If in radical uncertainty our best instruments are simply ourselves, including the depths of our being, to find this Truth all we must do is criticize ourselves and each other, looking for what cannot be rejected but only can be confirmed, independently. One independent can question assumptions of a whole group and when it cannot justifiably reject that, it is not Truth and NO critical goal for which we must seek responses immune to surprise.

  3. When we can only work with vague ideas and do not possess hard empirical evidence for probability calculations, the robustness of our ideas put together into a theory, philosophy or belief, depend on how well these ideas can be translated from one background (language) into another. I personally like to philosophize in psychology, taking it from the best education I could possibly have, I believe, and then speculating or producing alternative hypotheses for the things I see happening and how they are justified by seemingly unquestioned assumptions.

    Soon all sorts of small hypotheses start to emerge, and I capture them (write them down), until they mysteriously transform into other hypotheses, or are pushed aside by them. The better hypotheses get, the more likely is it, that one translates without residues into the other. Not surprisingly, after finishing my Masters and adding an extra year especially dedicated to theoretical psychology, discussing and criticizing with other students and the professor the latter's latest book in preprint, there was one moment in my life, in 1982, that I could see clearly and far.

    It must have been the confidence in my thoughts and ideas, reflecting reality without residues and only growing in strength at each translation from one vague idea, via the assumptions I could no longer question, into another, that produced a robustness transfixing my mind and urging me to write it all down as soon as I could and I did, for two weeks, in a state of feverish trance. It has always been my cornerstone ever since and 25 years later, something similar began, or repeated itself. This time much more developed, deeper and wider and applicable. Still there were times that robustness made room for doubt and vulnerability, but then it healed again.

    Isn't this how anyone or any group of people studying the strong and weak points of a reflection of reality (a model or a representation), takes steps towards greatest robustness, translating one take into another and improving the model or representation along the way, until everything fits beautifully, flawlessly and without residues lost in translation, across the tower of babel as the ship in Werner Herzog's Fitzcarraldo? That would imply that robustness and a high level of functional structure are the same.