The Human Discovery of Software:
Turing and von Neumann as Biologists
Chapter Three excerpted from
Proving Darwin
Making Biology Mathematical
Gregory Chaitin
Pantheon Books, New York (2012)
In this chapter we present a revisionist history of the discovery of software
and of the early days of molecular biology, as seen from the vantage point
of metabiology; the present is always rewriting the past in order to justify itself.
As Jorge Luis Borges point out, one creates one’s predecessors!
As an example of this history rewriting process,
consider the atheist modern scientists
mechanical worldview image of Newton created by Voltaire.
In his essay «Newton, the Man,»
John Maynard Keynes describes Newton
as «the last of the magicians, the last of the Babylonians and Sumerians,»
not at all the first modern scientist, and closer to Doctor Faustus than to Copernicus.
Newton spent more of his time on alchemy and theology
than on mathematics and physics;
he assembled a remarkable collection of medieval alchemy books.
My friend Stephen Wolfram has three-hundred-year-old books on theology
by Newton and Leibniz (the originals, not copies) next to each other on a shelf;
Newton’s Principia is unaffordable, but not that many people are interested in theology.
In fact the past is not only occasionally rewritten,
it has to be rewritten in order to remain comprehensible to the present.
On the topic of how one should write the history of science,
see Helge Kragh’s brilliant An Introduction to the Historiography of Science.
Let’s now re-examine the early history of computing theory and molecular biology
in the light of metabiology, and see the threads that led -or that should have led,
if scientific progress were entirely rational- to metabiology.
Our story is full of surprises, starting with questions
about philosophy and the foundations of mathematics,
and including the creation of one-trillion-dollar technology,
which has already happened-and this will may well soon be followed
by the creation of a second such game-changing technology.
There are even connections with medieval magic.
Would you like to know more? Read on!
Few people remember Turing’s work
on pattern formation in biology (morphogenesis),
but Turing famous 1936 paper «On Computable Numbers»
exerted an immense influence on the birth of molecular biology indirectly,
through the work of John von Neumann on self reproducing automata,
which influenced Sidney Brenner who in turn influenced Francis Crick,
the Crick of Watson and Crick, the discoverers of the molecular structure of DNA.
Furthermore, von Neumann’s application of Turing’s ideas to biology
is beautifully supported by recent work on evo-devo (evolutionary developmental biology).
The crucial idea: DNA is multibillion-year-old software,
but we could not recognize it as such before Turing’s 1936 paper,
which according to von Neumann creates the idea of computer hardware and software.
We discussed this crucial idea in the previous chapter;
perhaps before proceeding we should summarize what was covered there:
Hardware Physics Dead Rigid Closed Mechanical
Software Biology Alive Plastic Open Creative
Natural Software DNA 3-4 x 10ˆ9 years old
Artificial Software Computer Programs 50 or 60 years old
Newtonian Math Continuous Math, Differential Equations For Physics
Postmodern Math Discrete, Combinatorial, Algorithmic For Biology
Definition of life (John Maynard Smith, The Problems of Biology, 1986).
Mathematical proof that something exists satisfying the definition (2010)
Remember Molière’s bourgeois gentilhomme who discovered
to his amazement that all his life he had been speaking in prose?
We are that gentleman.
Our bodies are full of software, they always had been,
but before we could recognize natural DNA software as such,
we had to invent artificial software, human computer programming languages.
There was software all around us, in every cell, ancient software,
but we couldn’t see that until we invented software ourselves!
Furthermore, as evo-devo shows, organisms contain their own history,
as per Shubin’s Your Inner Fish, your inner sponge, your inner amphibian…
Biology is just a strange kind of archeology, software archeology.
And the purpose of human love-making is to integrate software
from the male (in sperm) with software from the female (in the egg).
That is why people fall in love, because they want to combine their subroutines.
So, in a sense, von Neumann discovered why people fall in love.
Furthermore, the origin of life, which is still deeply mysterious,
is the origin of software-natural software, not artificial software.
But help is on the way. Stephen Wolfram’s A New Kind of Science
can be reinterpreted as a book about the origin of life.
One of Stephen’s main points is that it is very easy to get
a combinatorial symbolic system to be a universal Turing machine
or a general-purpose computer.
It is very easy to build a computer out of almost any discrete math components.
He refers to this, I believe, as the ubiquity of universality.
At a philosophical level, then, this means that the origin of life is not in general
that surprising, but perhaps in the particular implementation here on Earth it is.
Anyway, thank you, Stephen!
Enough generalities!
Now let me tell you in more detail how the computer was invented by Alan Turing
(and also simultaneously by Emil Post) to help clarify a question about the foundation of mathematics
-Nature, which invented the computer and hardware/software first,
doesn’t care a fig about the foundations of math, but it does care about plasticity.
First let’s talk about an old dream, certain knowledge.
Or, as the amazing polymath Leibniz put it, mechanical knowledge,
reasoning as certain as arithmetic, truths as obvious as 2 + 2 = 4.
No disputes anymore, said Leibniz: Gentlemen, let us compute
and determine who is correct! What a beautiful dream!
Leibniz did not do too much work developing what today
is called symbolic logic or mathematical logic,
but he stated the goal extremely clearly and forcefully,
and for this reason he is considered
the father, or the grandfather, of modern logic.
Leibniz could not devote too much time to any one topic;
he was omnivorous, he was interested in everything.
For example, he also invented binary arithmetic,
and calculating machines that could multiply
-Pascal’s original calculating machine,
the Pascaline, could only add and substract.
Through the years many logicians worked on Leibniz’s dream
of certain knowledge of mechanical reasoning: de Morgan,
Boole, Peano, Frege, Russell, Hilbert, Gödel, Turing, Post…
But it didn’t work, it turned out it could not work.
Instead of certain, mechanical knowledge,
Gödel found incompleteness
and Turing found incomputability.
But in the process Turing found
complete/universal programming languages,
hardware, software and universal machines.
One milestone was the German mathematician
David Hilbert’s improved version of Leibniz’s dream.
Hilbert wanted a formal axiomatic theory for all of mathematics,
a mechanized version of Euclid’s Elements
that would cover all of math, not just geometry.
The key point was that it should be possible to check mechanically
whether or not a proof is correct, whether or not the reasoning follows all the rules.
To do this one would need to invent a meticulous artificial language sufficiently powerful
to express all possible mathematical reasoning, all possible mathematical proofs.
In 1931, Gödel showed this is impossible: no such mechanical universal language
for reasoning can ever be found, can ever enable us to prove all mathematical truths.
This is called incompleteness.
But then in 1936 Turing showed thate there are in fact complete
or universal mechanical languages for performing mathematical calculations
instead of expressing mathematical proofs.
And the rest is history: the modern computer was born!
How did Gödel refute Hilbert and Leibniz?
By constructing an arithmetical mathematical assertion that asserts
its own unprovability: «I am unprovable,» which is provable if and only if it is false.
Turing’s technique was different, less tricky, deeper.
He studied what machines could compute,
and observed that most real numbers are uncomputable
and therefore have numerical values that cannot be determined
via formal proof, for otherwise one could mechanically run
through all possible proofs to systematically calculate
the value of these uncomputable real numbers.
This is actually my preferred version of Turing’s basic result.
The usual way it is explained is in terms of the famous halting problem.
Turing shows that there is no systematic way, no mechanical procedure, no formal axiomatic theory,
for deciding whether or not a self-contained computer program will eventually halt.
You can start it running and calculate step by step, but to decide if it is going
to go forever or not is in the general case quite impossible.
So we get computer programming languages, universal ones,
ones that are powerful enough to write any algorithm.
But we lose certainty, we lose mechanical reasoning.
That dream is gone forever.
Not to worry!
According to Emil Post-who is not as well known as Gödel and Turing
but was at their level (he came up with Turing machines too,
and also with an incompleteness theorem that remained unpublished for years)
-the axiomatic method, and especialy Hilbert’s formal axiomatics,
was just a terrible mistake, just a confused misunderstanding.
According to Post, math cannot provide certainty because it is
not closed, mechanical, it is creative, plastic, open! Sound familiar?
You bet, we have been talking about biological creativity all through the previous chapters,
and now we find something like it in pure math too!
So math is creative, not mechanical, math is biological, not a machine!
I told you that mathematical and biological creativity are not that different
-we’ll see that in even more detail in the next chapter.
The point is particularly well made in the titles of two of philosopher
Paul Feyerabend’s books: Against Method and Farewell to Reason.
Feyerabend advocates creativity and imagination-in a word, anarchy-
in science based on his reading of the history of science
and without ever mentioning Gödel or Turing.
But in my opinion Against Method would be the best title for a book
on the unsolvability of Turing’s halting problem. and Farewell to Reason
would be the best title for a book on Gödel’s incompleteness theorem.
What Feyerabend believes to be the case in the world of science
for purely philosophical reasons, in the world of mathematics
are actually mathematical theorems-there provably
are no general methods for solving all mathematical problems.
Nevertheless, as the mathematician Gian-Carlo Rota has observed in his essay
«The Pernicious Influence of Mathematics upon Philosophy,» philosophy
is actually the art of finding bad reasons for what one believes instinctively,
somewhere deep in the gut.
Because of subconscious childhood emotional cravings, in fact.
Rota made few friends in the philosophy community
with clever remarks like this, but I love philosophy
and I also that Rota has a point:
Philosophy should not try to imitate mathematics too much,
especially not the formal axiomatic method
that Hilbert championed.
For as Rota observes, if we could define our terms precisely,
that would be the end of philosophy.
Formal axiomatic is not creative.
There are, in fact, different kinds of mathematical creativity.
Some people, the majority, are interested in creativity
within a formal mathematical axiomatic theory
= find a proof but stay within the current paradigm
= normal science (Kuhn).
I myself am more interested in “savage creativity” (Deleuze)
= changing the formal theory
= new axioms, new concepts
= paradigm shift (Kuhn)
= against method (Feyerabend)!
And now for von Neumann’s self-reproducing automata.
Von Neumann, 1951,
takes from Gödel the idea
of having a description
of the organism within the organism
= instructions for constructing the organism
= hereditary information
= digital software
= DNA.
First you follow the instructions in the DNA
to build a new copy of the organism,
then you copy the DNA
and insert it in the new organism,
then you start the new organism running.
No infinite regress, no homunculus in the sperm!
For the details,
see the crucial part of von Neumann's paper
on self-reproducing automata in the first appendix.
This was the paper that inspired Sidney Brenner
to go to Cambridge to work with Francis Crick.
Crick needed someone to work with him,
for Watson had returned to the States
after publishing their famous paper in Nature.
First Crick shared an office with Watson,
then with Brenner; he could not work alone,
he needed to kick ideas around,
he needed to have someone to talk to all day long.
Francis Crick was the supreme theoretician,
the strategist, the person who masterminded
the creation of molecular biology.
And half of Crick was Brenner.
You see, after Watson and Crick
discovered the molecular structure of DNA,
the 4-base alphabet A, C, G, T,
it still was not clear how DNA worked.
But Crick, following Brenner and von Neumann,
somewhere in the back of his mind had the idea
of DNA as instructions, as software.
And he knew
that what counted,
what was important,
was how this information
flowed around the cell,
and how it turned into protein...
How did Brenner hear about
von Neumann's Hixon Symposium paper?
Back in South Africa,
before he went to Oxford
and then to Cambridge,
Brenner had as classmate
the computer scientist Seymour Papert,
who later worked with Marvin Minsky at MIT.
And it was Papert who alerted Brenner
to von Neumann's paper.
(I never met Papert, but I do know Marvin.)
Von Neumann's work
on self-reproduction
didn't stop in 1951 with
his so-called kinematical model,
essentially the plan for a physical device.
Following a suggestion by Stanislaw Ulam
(whom I had the privilege of meeting at Los Alamos),
von Neumann's model of self-reproduction
then moved from the physical world
to a toy graph paper two-dimensional plane
divided into identical squares,
each a finite-state machine,
a so-called cellular automata (CA).
This was published posthumously.
The cellular automata world used by von Neumann
is a homogeneous, uniform, totally plastic world,
a world where everything is software, information,
not hardware, a world where magic applies:
the right magic spell will create anything.
In it there is a universal constructor
as well as a universal computer.
That sounds good,
but in this cellular automata world
there are some problems too:
it is easier for an organism to reproduce itself
than it is for it to move (translate itself).
It has taken longer thant it did
with Turing’s work on the universal machine,
but von Neumann’s work on self-reproduction
and universal constructors is starting
to have technological applications:
namely printers for objects,
three-dimensional printers,
new flexible manufacturing technology,
3D printers that can print themselves (http://reprap.org),
ultimately perhaps, the universal factory that can build anything!
So you see, Turing’s paper
has had a tremendous impact,
and may have more.
The computer is not just a tremendously useful technology,
it is a revolutionary new kind of mathematics
with profound philosophical consequences.
It reveals a new world.
I have devoted
most of my life to exploring
one little aspect of that new world:
using the size of computer programs
as a complexity measure,
defining randomness
as irreproducible complexity,
as algorithmic incompressibility,
and studying a number
I call the halting probability Ω.
Ω compactly tells us
about individual instances
of Turing’s halting problem.
If we could know
the numerical value of Ω
with N bits of precision,
that would enable us
to answer the halting problem
for all programs up to N bits in size.
Ω is jam-packed
with logically and computationally
irreducible mathematical information.
The halting probability
of a program generated by coin tossing
is a paradoxical real number:
In spite of Ω’s simple definition,
it’s numerical value
is maximally uncomputable,
maximally unkowable,
and shows that pure mathematics
contains infinite irreducible complexity.
Ω can be interpreted pessimistically,
as indicating there are limits to human knowledge.
The optimistic interpretation,
which I prefer, is that Ω shows
that one cannot do mathematics mechanically
and that intuition and creativity are essential.
Indeed, in a sense Ω is
the crystallized, concentrated
essence of mathematical creativity
-Emil Prost all over again.
Furthermore, as I stated for the first time
in an article in the Bulletin of the European Association
for Theoretical Computer Science (EACTS)
in February 2007, the infinite irreducible complexity
of the halting probability Ω shows that pure mathematics
is even more biological than biology,
which is very complicated but only has finite complexity.
Pure mathematics has infinite complexity.
It was this clue that led me to try
to create metabiology, which I announced
in the EACTS Bulletin in February 2009,
only two years later.
So you see, Gödel, Turing, Post and von Neumann
opened the door from math to biology;
they gave us the necessary conceptual tool-kit.
We need postmodern discrete
algorithmic math to understand biology,
not Newtonian differential equations,
not old math, not analysis.
On pages 35-37 is a timeline and some references
to the work we have discussed in this chapter.
In the next chapter we’ll get down
to the nitty-gritty and see how to build a mathematical theory about randomly mutating software.
Following John Maynard Smith’s and Sidney Brenner’s advice, we shall ignore bodies and metabolism and energy and consider purely software organisms.
I’ll tell you how I go about building a mathematical theory, which I’ve succeded in doing once before (algorithmic information theory), and hopefully now once again (metabiology). And as you will see, the criteria for success are mostly aesthetic; math is an art form.
We’ll be able to prove that Darwinian evolution works in our toy model, and amazingly enough, the organisms that evolve by natural selection are better and better lower bounds on the halting probability Ω, which I did not at all foresee.
A pleasant surprise indeed.
But in retrospect inevitable, not surprising, as we will see in Chapter 5.
____________________________________________________________
Kurt Gödel, 1931 *
Self-reference: «This statement is unprovable!»
Incompleteness of formal systems for mathematical reasoning.
____________________________________________________________
Alan Turing, 1936
On computable numbers,
with an application to the Entscheidungs-problem
Completeness of formal systems for mathematical computations
= Universal Programming Languages, Universal Turing Machines
= general purpose computers
Theoretical Philosophy of Math Paper creates idea of software,
trillion dollar computer industry [von Neumann]!
____________________________________________________________
Turing’s late work on biology
Morphogenesis = Newtonian Math = Partial Differential Equations = Obsolete!
It was von Neumann who saw how Turing’s work applied to biology, not Turing himself!
Von Neumann starts mathematical biology!
Beginning of metabiology
____________________________________________________________
John von Neumann, late 1940s, early 1950s
Theory of self-reproducing automata
(self-reference becomes self-reproduction) **
Universal Constructors, Printers for Objects, 3D Printers,
Flexible Manufacturing, Self-Reproducing Printers!
Universal Factory = Future Trillion Dollar Business?!
____________________________________________________________
Von Neumann Died Young (53)
Died 1957; Watson, Crick paper on DNA = 1953
____________________________________________________________
Sydney Brenner, late 1950s, 1960s
Biologist inspired by von Neumann,
not by Schrödinger’s What Is Life?***
Influenced Crick, Shared office,
All that matters is information
= instructions for building things, doing things
= discrete algorithmic software
Nobel Prize winner
____________________________________________________________
Stanislaw Ulam ****
Cellular Automata World, 29 stages, 4 enighbors
Totally Plastic World, Everything is Software,
Magic Incantations, Information!
World as Idea!
Down with Materialism
____________________________________________________________
* Martin Davis, The Undecidable:
Basic Papers on Undecidable Propositions,
Unsolvable Problems and Computable Functions, Dover, 2004.
** John von Neumann,
«The General and Logical Theory of Automata,»
delivered 1948 at the Hixon Symposium
on Cerebral Mechanisms in Behavior, Pasadena, California,
and published in Lloyd Jeffries, Cerebral Mechanisms in Behavior:
The Hixon Symposium, John Wiley and Sons, New York, 1951, pp.1-41;
John Kemeny, «Man Viewed as a Machine,»
Scientific American 192 (April 1955), pp. 58-67
(describes von Neumann’s self-reproducing automata);
E.F. Moore, «Artificial Living Plants,»
Scientific American 195 (October 1956), pp. 118-126
(another reaction to von Neumann);
John von Neumann, Theory of Self-Reproducing Automata,
University of Illinois Press, Urbana, 1966
(posthumous; edited and completed by A.W. Burks).
*** Sydney Brenner, My Life in Science,
Biomed Central Ltd., 2001;
Matt Ridley, Francis Crick,
Eminent Lives, 2006.
****Stanislaw Ulam,
Adventures of a Mathematician, 2nd edition,
University of California Press, Berkeley 1991 (1st edition 1983).
See also Konrad Zuse, Rechnender Raum (Calculating Space),
Friedrich Vieweg & Sohn, Braunschweig, 1969).
An English translation od Rechnender Raum
will appear in Hector Zenil, A Computable Universe,
World Scientific, Singapore, 2012.
On the plasticity of the world see Freeman Dyson’s vision
of a totally green technology The Sun and the Genome,
and the Internet, Oxford University Press, 2000;
seeds that grow into houses instead of trees,
children performing genetic engineering
to design new flowers, etc.
See also Allison Coudert, Leibniz and the Kabbalah,
and Umberto Eco, The Search for the Perfect Language,
on the Adamic language used by God to create the world
and whose structure directly reflects
the fundamental inner structure of the world.
Knowledge of this language would give us
God-like powers (as in Jorge Luis Borges’s tale
«The Rose of Paracelsus»).
[¡Cuidado con la soberbia! ¿Seréis como dioses?
Querer suplantar a Dios es el camino de la ruina.]
LA ROSA DE PARACELSO
Jorge Luis Borges
publicado en La Memoria de Shakespeare
Ediciones Dos Amigos (Buenos Aires, 1982).
Emecé Editores (Buenos Aires, 2004).
En su taller que abarcaba las dos habitaciones del sótano, Paracelso pidió a su Dios, a su indeterminado Dios, a cualquier Dios, que le enviara un discípulo. Atardecía. El escaso fuego de la chimenea arrojaba sombras irregulares. Levantarse para encender la lampara de hierro era demasiado trabajo. Paracelso, distraído por la fatiga, olvidó su plegaria. La noche había borrado los polvorientos alambiques y el atanor cuando golpearon la puerta. El hombre, soñoliento, se levantó, ascendió la breve escalera de caracol y abrió una de las hojas. Entró un desconocido. También estaba muy cansado. Paracelso le indicó un banco; el otro se sentó y esperó. Durante un tiempo no cambiaron una palabra.
El maestro fue el primero que habló:
- Recuerdo caras del Occidente y caras del Oriente – dijo no sin cierta pompa. No recuerdo la tuya. ¿Quién eres y qué deseas de mí?
- Mi nombre es lo de menos -replicó el otro -. Tres días y tres noches he caminado para entrar en tu casa. Quiero ser tu discípulo. Te traigo todos mis haberes.
Sacó un talego y lo volcó sobre la mesa. Las monedas eran muchas y de oro. Lo hizo con la mano derecha. Paracelso le había dado la espalda para encender la lampara. Cuando se dio vuelta advirtió que la mano izquierda sostenía una rosa. La rosa lo inquietó.
Se recostó, juntó la punta de los dedos y dijo:
- Me crees capaz de elaborar la piedra que trueca todos los elementos en oro y me ofreces oro. No es oro lo que busco, y si el oro te importa, no serás nunca mi discípulo.
- El oro no me importa- respondió el otro.
- Estas monedas no son más que una parte de mi voluntad de trabajo. Quiero que me enseñes el Arte. Quiero recorrer el camino que conduce a la Piedra.
Paracelso dijo con lentitud:
- El camino es la Piedra. El punto de partida es la Piedra. Si no entiendes estas palabras, no has empezado aún a entender. Cada paso que darás es la meta.
El otro miró con recelo. Dijo con voz distinta:
- Pero.. ¿hay una meta?
Paracelso se rió.
- Mis detractores, que no son menos numerosos que estúpidos dicen que no, y me llaman un impostor. No les doy la razón, pero no es imposible que sea un iluso. Sé que “hay” un Camino.
Hubo un silencio, y dijo el otro:
- Estoy listo a recorrerlo contigo, aunque debamos caminar muchos años. Déjame cruzar el desierto. Déjame divisar siquiera de lejos la Tierra Prometida, aunque los astros no me dejen pisarla. Quiero una prueba antes de emprender el camino.
- ¿Cuándo?- preguntó con inquietud Paracelso.
- Ahora mismo - contestó con brusca decisión el discípulo.
Habían empezado hablando en latín; ahora, en alemán. El muchacho elevó en el aire la rosa.
- Es fama -dijo - que puedes quemar una rosa y hacerla resurgir de la ceniza, por obra de tu arte. Déjame ser testigo de ese prodigio. Eso te pido, y te daré después mi vida entera.
- Eres muy crédulo- dijo el maestro-. No he menester de la credulidad; exijo la fe.
El otro insistió.
- Precisamente porque no soy crédulo quiero ver con mis ojos la aniquilación y la resurrección de la Rosa.
Paracelso la había tomado, y al hablar jugaba con ella.
- Eres crédulo - dijo-. ¿Dices que soy capaz de destruirla?
- Nadie es incapaz de destruirla - dijo el discípulo.
- Estás equivocado. ¿Crees, por ventura, que algo puede ser devuelto a la nada? ¿Crees que el primer Adán en el Paraíso pudo haber destruido una sola flor o una brizna de hierba?
- No estamos en el Paraíso - habló tercamente el muchacho; - aquí, bajo la luna, todo es mortal.
Paracelso se había puesto de pie e inquirió:
- ¿En qué otro sitio estamos? ¿Crees que la divinidad puede crear un sitio que no sea el Paraíso? ¿Crees que la Caída es otra cosa que ignorar que estamos en el Paraíso?
- Una rosa puede quemarse- desafió el discípulo.
-Aún queda el fuego en la chimenea. Si arrojamos esta rosa a las brasas, creerías que ha sido consumida y que la ceniza es verdadera. Te digo que la rosa es eterna y que solo su apariencia puede cambiar. Me bastaría una palabra para que la vieras de nuevo.
- ¿Una palabra?- dijo con extrañeza el discípulo-. El atanor está apagado y están llenos de polvos los alambiques. ¿Qué harías para que resurgiera?
Paracelso lo miró con tristeza.
- El atanor esta apagado – repitió – y están llenos de polvo los alambiques. En este tramo de mi larga jornada uso de otros instrumentos.
- No me atrevo a preguntar cuáles son - dijo el otro con astucia o con humildad.
- Hablo del que usó la divinidad para crear los cielos y la tierra y el invisible Paraíso en que estamos, y que el pecado original nos oculta. Hablo de la Palabra que nos enseña la ciencia de la Kabalah.
El discípulo dijo con frialdad:
- Te pido la merced de mostrarme la desaparición y aparición de la rosa. No me importa que operes con alquitaras o con el Verbo.
Paracelso reflexionó. Al cabo, dijo:
- Si yo lo hiciera, dirías que se trata de una apariencia impuesta por la magia de tus ojos. El prodigio no te daría la fe que buscas: Deja, pues, la rosa.
El joven lo miró, siempre receloso. El maestro alzó la voz y le dijo:
- Además, ¿quién eres tú para entrar en la casa de un maestro y exigirle un prodigio? ¿Qué has hecho para merecer semejante don?
El otro replicó, tembloroso:
- Ya sé que no he hecho nada. Te pido en nombre de los muchos años que estudiaré a tu sombra que me dejes ver la ceniza y después la rosa. No te pediré nada más. Creeré en el testimonio de mis ojos.
Tomó con brusquedad la rosa encarnada que Paracelso había dejado sobre el pupitre y la arrojó a las llamas. El color se perdió y solo quedó un poco de ceniza.
Durante un instante infinito esperó las palabras y el milagro.
Paracelso no se había inmutado. Dijo con curiosa llaneza:
- Todos los médicos y todos los boticarios de Basilea afirman que soy un embaucador. Quizá están en lo cierto. Ahí está la ceniza que fue la rosa y que no lo será.
El muchacho sintió vergüenza. Paracelso era un charlatán o un mero visionario y él, un intruso, había franqueado su puerta y lo obligaba ahora a confesar que sus famosas artes mágicas eran vanas.
Se arrodilló, y le dijo:
- He obrado imperdonablemente. Me ha faltado la fe, que el Señor exigía de los creyentes. Deja que siga viendo la ceniza. Volveré cuando sea más fuerte y seré tu discípulo, y al cabo del Camino veré la rosa.
Hablaba con genuina pasión, pero esa pasión era la piedad que le inspiraba el viejo maestro, tan venerado, tan agredido, tan insigne y por ende tan hueco. ¿Quién era él, Johannes Grisebach, para descubrir con mano sacrílega que detrás de la máscara no había nadie?
Dejarle las monedas de oro sería una limosna. Las retomó al salir. Paracelso lo acompaño hasta el pie de la escalera y le dijo que en esa casa siempre sería bienvenido. Ambos sabían que no volverían a verse.
Paracelso se quedó solo. Antes de apagar la lámpara y de sentarse en el fatigado sillón, volcó el tenue puñado de ceniza en la mano cóncava y dijo una palabra en voz baja.
Y la rosa resurgió. (*)
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