Abstract
This text will first discuss the great and productive “linguistic turn” that originated in Logic, at the end of the nineteenth century. After Turing’s work on the Logical Computing Machine (1936–1950) and Shannon’s theory (1948), Brillouin attributed information content to Maxwell’s demon’s action (1956), with a scientific invention and, in my opinion, a touch of humor, thus granting to inert matter an intelligent activity capable of inverting entropy. Moreover, since the 1950s, many identified the digital elaboration of data with human intelligence and some projected the processing of information onto inert matter. In the same years, DNA was identified as the complete information carrier of phylogenesis and ontogenesis. In these approaches, an ambiguity has been arbitrarily played out, and a threshold has been sometimes crossed, sometimes not; first, in Logic and Computing, between “formal principles of proof” or of computation and their “semantics” (their mathematical meaning); then in Biology, between elaboration of information and biological dynamics in their context. Today, well beyond classical Artificial Intelligence on discrete data types, Brillouin’s invention and Deep Learning on continuous mathematical structures are the new loci of confusion between knowledge construction and transmission or processing of information. Yet, the construction of human knowledge is even more radically by-passed and science dehumanized when information is directly embedded in a world that is viewed as a massive information processor. The remarkable scientific and economic productivity of our new observable, information, must instead be seen in the context of our human, ecosystemic and historical activities, in order to subordinate it to interpretation and meaningful action that enrich and do not strip the sense from our forms of knowledge and life. To this purpose, we must strictly distinguish between information, as formal elaboration and transmission of signs, and information as production of “meaning” in our active friction on reality.
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Notes
- 1.
We use “sign”, in reference to a priori meaningless strokes, letters, 0 s and 1 s …, thus we use the expression “sign pushing” instead of “symbol pushing” widely used in Computer Science. “Symbol” retains the Greek etymology of “sym-ballein” or “to bring together” or to unify different acts of experience or synthesize meaning.
- 2.
Note that computability and its machines were invented in order to prove the limits of the formal-linguistic approach. As hinted above, in order to prove undecidability and incomputability, Gödel, Church and Turing (1936) had first to give precise definitions of “computable function” and “formal derivation”, thus of formal computing/arithmetic machines, the mathematical foundation of modern digital computers – see (Longo 2010) for the relevance of these and other negative results.
- 3.
H. Weyl, Hilbert’s “best student”, recalls his supervisor’s philosophy by quoting: “a typically Hilbertian manner: “It must be possible to replace in all geometric statements the words point, line, plane by table, chair, mug.” In [Hilbert’s] deductive system of geometry the evidence, even the truth of axioms, is irrelevant …” (Weyl 1953).
- 4.
Even Wiles’ 1993 proof of this easy to state arithmetic property, Fermat’s last theorem, required the invention of new deep mathematical ideas, a complex blend of advanced algebraic geometry, the “vision” of a path that linked homology to the analysis of elliptic curves, leading to the totally unrelated new ideas and techniques of the “modularity lifting theorem” (Cornell et al. 2013). Checking formally the proof, a posteriori – once it has been given, may be very useful, but it is a different matter, (Hesselink 2008).
- 5.
In short, following a proof allegedly given by Gauss at the age of 7, place
1
2
…
n
on a row
n
(n-1)
…
1
on the next row (an audacious mirror symmetry of the usual order of number writing), then obtain:
(n + 1) … (n + 1) by adding the columns. Thus, the sum of the first n integers is (n + 1)n/2, which is easy now to check by induction; yet, we had first to produce the formula, by this or other similar constructions. The rule-based formalist approach confuses proving, in mathematics, which includes inventions like here, with a posteriori proof checking, a remarkable technique in Computer Science, that we will mention below.
- 6.
In (Longo and Montévil 2014), we added to irreversibility of time and to entropy production the coexistence of a “symmetry breaking” and of a random event: these four phenomena seem to be correlated in all existing physical theories. In our work, this correlation extends to proper biological dynamics, such as embryogenesis and evolution, where increasing organization as well (anti-entropy production) produces entropy.
- 7.
The information content is the logarithm of the probability, as a frequency, of the chosen card. This example opens the way to conflating Boltzmann’s logarithmic formula for entropy to Brillouin’s quantification of information, modulo a negative sign and a differing constant, as pointed out above, also by its dimensionality – a further major difference.
- 8.
See “We Have Learned Nothing from the Genome” (Venter 2010), written by the leader of the team that first decoded a human DNA, and (Longo 2018a) for some references to the amazing promises made in early 2000s as for cancer’s genetic diagnosis, prognosis and therapy, see also (Weinberg 2014), (Gatenby 2017).
- 9.
In a rare attempt to clarify the different role of Turing-Kolmogorov vs Shannon-Brillouin approaches in biology, (Maynard-Smith 1999) confuses, in the explanatory examples, the dual correlation that entropy and complexity have in the two theories, see (Longo et al. 2012), (Perret and Longo 2016) for details.
- 10.
Even more radically “proteins never do fold into a particular shape, but rather remain unstructured or ‘disordered’ […]. In mammals, about 75% of signaling proteins and half of all proteins are thought to contain long, disordered regions, while about 25% of all proteins are predicted to be ‘fully disordered’ […]. Many of these intrinsically unstructured proteins are involved in regulatory processes, and are often at the center of large protein interaction networks” (Gsponer and Madan Babu 2009). See also the increasingly acknowledged important role of long non-coding RNAs (Hadjiargyrou and Delihas 2013). Dogmas on the exact mechanisms inspired by the need to transmit and elaborate alpha-numeric information are collapsing one after the other (Mouilleron et al. 2016).
- 11.
Overlapping genes are parts of a given DNA or RNA region that can be translated in at least two different reading frames, yielding very different proteins. In fact, a shift in the reading frame of a nucleotide sequence, by one-two bases, totally changes the resulting protein. Discovered in the late ‘70s in a small viral genome (Barrell et al. 1976), there are coming to the limelight only recently since many studies have shown they are not restricted to viruses (Chirico et al. 2010), but they are present as well in cellular organisms, man included (reviewed in Pavesi et al. 2018). M. Granero and A. Porati, among the pioneers in this field, nicely described the phenomenon in Italian: by a one letter shift CARABINE MICIDIALI becomes ARABI NEMICI DI ALI, and GAS INODORO becomes ASINO D’ORO (A. Vianelli, personal communication). This sort of shifts in ‘reading’ is not recommended in linguistic analysis nor in programming, where it is avoided by the use of parenthesis, typically – like in lambda-calculus, a consistently evoked reference for the genetic program, see (Danchin 2003, 2009). Actually, even in genomes (at least bacterial ones) there might be more STOP off-frame codons than expected to avoid unneeded reading frame shifts (Seligmann and Pollock 2004; Abraham and Hurst 2018). What is a gene, then?
- 12.
- 13.
This is in blatant contrast with biological organisms, which “go wrong” very often or most of the time – but, in absence of a pre-given phase space for biological dynamics, it is hard to pre-define “wrong/right”: very rarely, but crucially, hopeful monsters may be “right” in evolution, and contribute to speciation, as we may observe a posteriori (Longo 2017). The point is that ecosystemic compatibility and viability is not optimality, in particular because the ecosystem is not pre-defined, but co-constructed with/by the organism.
- 14.
“… due to the inherent limitations of scientific instruments, all an observer can know of a process in nature is a discrete time, discrete-space series of measurements. Fortunately, this is precisely the kind of thing — strings of discrete symbols, a “formal” language—that computation theory analyzes for structure.” (Crutchfield 1994).
- 15.
“subjective-absolute and objective-relative seems to me to contain one of the most fundamental epistemological insights that can be extracted from natural sciences” (Weyl 1949).
- 16.
- 17.
Unfortunately, this attitude is shaping minds. A student in mathematics, in my top institution in higher education (ENS, Paris), asked me not long ago: “how can the Universe compute at each instant where it will go in the next instant?”. I answered: “there are plenty of very small Turing Machines, hidden everywhere, that do the computations!”. And I wrote a letter to Alan Turing (Longo 2018c) – he explicitly and radically departed from these views (the brain … the Universe, are surely not discrete state machines, i.e. Turing Machines, says he – see references in my letter).
- 18.
Quantum Computing formally introduces a major feature of Quantum Mechanics, entanglement (Zorzi 2016). By this, it forces a connection between hardware and software, a major computing and engineering challenge, beyond Turing’s fruitful split, hardware vs software, perhaps along the way of a major revolution in computing.
- 19.
Many projected into nature Kolomogorof’s effective notion of “incompressibility” for finite strings, as the dual component of predictive determination, i.e. as randomness. A finite sequence or string of letters/numbers w, say, is incompressible if no program to generate it is shorter than w (Calude 2002). Today, this is a crucial notion since our machines need to compress strings. However, it does not describe physical randomness, as thought by many, yet another abuse of a (remarkable) computational invention. Only asymptotically, infinite random sequences, as defined by Martin-Löf, relate to classical and quantum randomness (Calude and Longo 2016). This is to be expected: limit constructions already unified, by Boltzmann work, random particle trajectories and thermodynamic entropy (Chibbaro et al. 2015). In the finite, all sufficiently long strings are compressible, by Van der Waarden theorem (Calude and Longo 2017). Thus, in terms of incompressibility, there would be no long random sequences …. And a string to be generated, in the future (next year lottery drawings), is blatantly not random if one can compute, now, just one element of it, independently of its ‘a posteriori’ compressibility or not. Once more, there is no way to analyze such a meaningful notion for physics, randomness for finitely many events, in abstract computational terms: one has first to specify the intended theory, as an interpretation and organization of (a fragment of) nature, and then define randomness relatively to it (Calude and Longo 2016).
- 20.
A red herring, the so-called extended Church Thesis, often blurred computational novelties: any physical finite structure computes at most Turing computable functions. Now, mathematical neural nets are not meant to compute number-theoretic functions. Of course, in this and other cases, if one forces a physical dynamic to take a digital input and then produce at most one digital output and formalizes the dynamics “à la Hilbert”, one can prove that that formal system computes no more than a Turing Machine. This says nothing about the proper expressivity and possible functions of a continuous dynamics of networks (see below), whose job is not to compute functions on integers (see below for more). A similar trivializing game has been played also with Quantum Computing and Concurrent Networks (Aceto et al. 2003).
- 21.
- 22.
- 23.
Statistical analysis of “abstract” data govern finance and, thus, economy, (Bouleau 2017). These analyses are independent from the underlying assets, that is from any reference to “tables, chairs, mugs”, as Hilbert would put it (footnote 3).
- 24.
“The machine may be out of order and present the operating characteristics similar to a mad behavior of a living being. But it cannot revolt. The revolt indeed implies a profound transformation of the finalized behavior, and not a malfunction of the conduct.”
- 25.
Papers (co-)authored by Giuseppe Longo are downloadable here: http://www.di.ens.fr/users/longo/download.html
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Inigo Wilkins made a critical and competent reading of a preliminary version of this paper.
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Longo, G. (2020). Information at the Threshold of Interpretation: Science as Human Construction of Sense. In: Bertolaso, M., Sterpetti, F. (eds) A Critical Reflection on Automated Science. Human Perspectives in Health Sciences and Technology, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-030-25001-0_5
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