De 15 ianuarie
Binary logic is deeply rooted in the functionality of human reasoning. Some of the oldest examples in this respect are night and day, good and evil, life and death, front and back, left and right. Several of these examples fall under the general concept, later taken to be of utmost importance in science, of symmetry. Cultural models throughout history have dwelled in such binary logic. “Good deities” versus “bad deities “are one example, applied in religious models of which the Christian paradigm is one of obvious examples. Most interpretations of the world, many of them inherently anthropocentric, indeed correlate the concept of choice with the very existence of the world (regardless of whose choice we are speaking of – see the concept of “free will” in philosophy). Cosmological views would often describe a moment prior to the existence of the world as we know it, as a moment without choice (a feature that we do not seek, in this text, to establish as singular to this particular episode). One example is the Biblical phrase “In the beginning God created the heaven and the earth”,[i] illustrating that following the prior existence of a singular concept (God, or, as later described in John 1:1, “the Word”), the first act towards establishing the world as we know it was an act of duality, creating two clearly different entities. Furthermore, the following acts described in the same text also refer to two-fold divisions: light from the dark, and later water from the land. Romanian poet Mihai Eminescu, cited by critics to draw inspiration for cosmological models from rather cosmopolitan sources, likewise expresses this vision of dual choice as a pre-requisite of the world as we know it, in his poem “A Dacian’s prayer”:[ii]
“When death did not exist, nor yet eternity,
Before the seed of life had first set living free,
When yesterday was nothing, and time had not begun,
And one included all things, and all was less than one,
When sun and moon and sky, the stars, the spinning earth
Were still part of the things that had not come to birth,
And You quite lonely stood... I ask myself with awe,
Who is this mighty God we bow ourselves before.”
The binary logic does however have its limitations, and has in this respect found to be insufficient or even frustrating in its inability to solve more complex problems, or in the fact that it is unavoidably linked to the notion of conflict. Herder prize-winner Marin Sorescu offers an interesting illustration in his poem “Symmetry”:[iii]
“I was walking like that,
When all of a sudden in front of me
Opened two roads:
One to the right
And one to the left,
According to all rules of symmetry.
I took the one on the right
(Exactly the one that I shouldn’t have,
As was later proven)
I walked on it as I did,
Needless to give any details here.
And then, in front of me opened two
One to the right
And another one to the left.
I jumped into the one on the left,
Without even blinking, without even trying to gain impetus,
Down I went, the lot of me, into the one on the left,
Which, alas, was not the one covered in fluff.
And then in front of me opened two roads.
“I’ll show you!” – I said to myself –
And I took the one on the left again,
Just out of spite.
Wrong, very wrong, the one on the right was
The real, the truthful, the great road, supposedly.
So at the first crossroads
I gave myself entirely
To the one on the right. Again,
The other one was the correct choice, the other one…
And now in front of me again I find open
One to the right
And one to the left.”
These basic examples of cultural implications of symmetry are indeed mirrored in our scientific understanding of the world. In chemistry, substances are traditionally divided into two classes: organic and inorganic, depending on whether they belong to the type found in living systems (i.e., based on the chemical element carbon) or not. Specific to these two classes of substances are two kinds of bonding between atoms: in organic substances, two atoms bind to each other by sharing their electrons (“covalent bonding”) whereas in many inorganic substances (especially in those classified as “salts”) the bonding relies on attraction between opposite charges, without any electrons being shared between the two atoms (“ionic bonding”).[iv] In fact, the notion of bonding in chemistry is often defined as implying two atoms.
Going even deeper into the structure of matter, one finds that all atoms are formed from two parts of opposing charge: the nucleus (positive charge) and the electrons (negative charge); based on the balance between these two components, atoms and molecules can themselves become ions, of two kinds – positive and negative.
A further example of binary logic is offered by the notion of chirality. In everyday life, one example of chirality are the left and the right hand: although they appear similar, and are indeed mirror images of one another, they are not identical, insofar as they are not superimposable. Pairs of molecules are known, which possess this feature, of being related to each other by a plane of symmetry (“mirror”); these two mirror-related molecules are referred to as “chiral isomers”. In fact, one of the mysteries yet unsolved by chemists and biologists alike, is the reason why in certain cases – those found in living organism - one of the two isomers of a chiral pair is found in nature in higher concentration than the other isomer: there is indeed apparently nothing in their chemical and physical properties that should recommend one isomer to be more stable than the other. Yet, living organisms (or indeed their creator) have chosen one isomer over the other, in a blatant example of binary logic. Whereas for many such chiral cases (such as the proteins, sugars, or DNA) it is clear today that our organisms can only function with those isomers already inside our bodies, it is unclear exactly how and why the first ever chiral isomer of a compound was able to be produced in a higher concentration than its other isomer, and exactly what the road was from this one first chiral isomer to the complex set of chiral isomers needed to sustain even the simplest forms of life known to us. One may note that the choice between two chiral isomers, made, to the best of our understanding, at the very origins of life, mirrors in a remarkable way the binary logic applied in cosmological views cited earlier in this text (e.g., “In the beginning, God created the heaven and the earth”).
Two final examples in this line of reasoning are offered by modern theories on the structure of matter; these examples, however, also pave the way for elaborating beyond binary logic. Heisenberg’s uncertainty principle states that there are two features of any entity that describe it and that cannot be both known with exactitude at the same time: position and momentum (we shall purposefully avoid the Popper critique of this topic, as it is not directly related to our discussion). De Broglie likewise stated that all matter has dual nature and is thus at the same time describable as particle and as wave. True, for objects with very large mass, the wave character may in principle be discounted, and it is convenient to speak only of their particle-like features, as is done in classical physics. It is, nevertheless, useful to note that in this way classical physics operate with an approximation (albeit a very useful and accurate one). We cannot conclude this paragraph without noting that de Broglie’s and Heisenberg’s principles, while speaking of duality at the very core of our understanding of matter, do also imply reasoning beyond this duality. Indeed, in both cases the duality appears as an artificial, incomplete manner of attempting to divide something which is otherwise undividable (e.g., matter is at the same time particle and wave, and not either particle or wave).
We will now move to describe limitations of the binary logic within the chemical world, based on the very examples that we have provided above. Whereas in traditional chemistry the chemical combinations of carbon (“organic compounds”) and metals (“inorganic compounds”) would constitute two well-separated territories, the 20th century saw a new branch of chemistry establish, named organometallic chemistry,[iv] and which dwells on bonding between carbon and metal atoms. Notably, such organometallic interactions were always around us, as they are involved in compounds as simple and pervasive as vitamin B12. Organometallic chemistry further prompts us to more carefully investigate the covalent bond/ionic bond duality. Indeed, as most inorganic and organometallic chemists will acknowledge, bonding between metal atoms and other atoms (e.g., carbon) is best described as somewhere “in between” ionic and covalent. We can speak, thus, of a continuum of combinations and types of bonding, rather than of two discrete options – covalent or ionic. Furthermore, the very notion of “bonding” between atoms and molecules has recently been entangled with the advent of another recent branch of chemistry, “supramolecular chemistry”.[v] The recognition, and detailed study, of forces connecting molecules in complex manners, and an inevitable comparison in terms of strength and length between classical chemical bonds and the newly discovered interactions of supramolecular chemistry, has forced chemists to recognize that there is indeed a continuum in terms of the interactions found between atoms in manner, and that many examples can be found where atoms are such positioned with respect to each other that they cannot be properly described as chemically bonded, and yet are too close or too strongly connected to say that they are not chemically bonded.[vi] Furthermore, such bonding systems are nowadays well established in chemistry, as “three-center bonds” – thus lifting in one more way the binary logic from the notion of chemical bond.
We have cited above how atoms appear formed from two kinds of components with opposite electrical charge. This view is, however, incomplete. If we wish to speak of elementary particles, such as those constituting the atom, we can no longer today limit ourselves to the proton and electron that were deemed indivisible not so long ago. In fact, even accepting the simple proton(positive)-electron(negative) view, we would still have had to mention other elementary particles, such as the neutron and the photon, hence pointing us towards a world where elementary particles are not necessarily submitting to binary logic. Current views of the family of basic, indivisible elementary particles are still tinkering on the edge of the non-binary in logic: they include several (and not only two) particles – but these particles are indeed generally grouped in two classes (fermions and bosons). Furthermore, the concept of antimatter again places a mirror-type dual logic in the structure of matter.
A development worth noting in the mathematical construct allowing us to put forth models of the world has been fuzzy logic, which by definition denies binary logic by extending the range of choices beyond two. Conceptually (in our opinion) related to the fuzzy approach, although not directly connected mathematically, is the string theory, seeking to construct models of matter non-tributary to classical concepts. However, even in string theory the term “duality” is of the essence.
The case of chirality, discussed above as an important example of symmetry, may also be deemed to be taken as just a particular case within a range of symmetry operations of scopes greater than binary. Thus, whereas chiral molecules exhibit a certain element of two-fold symmetry (e.g., a mirror, or an inversion point) these elements are just some of the simplest cases of symmetry in a very long list a list whose elements are not necessarily directly derivable from two-fold elements. Thus, a three-fold axis of rotation does not necessarily appear as immediately derivable from the apparently simpler two-fold axis of rotation. In this respect, the elements of ‘symmetry’ present in molecules (and indeed anywhere in matter) are rather unlimited (if we were to only consider rotation axes based on all possible prime numbers), as opposed to binary – in a manner much reminiscent of the fuzzy logic mentioned above.
The very way our brains work, may be an interesting point to consider in terms of binary logic. Indeed, nervous impulses in neurons are transmitted by way of releasing atoms and signal molecules from one side of a membrane to another, or from one neuron to another. This clearly offers the premise for a binary approach (i.e., one either has, or has not liberated/transmitted the ion/molecule). Nevertheless, the exact amount of molecules/ions transmitted is still of the essence, and the magnitude of the signal is modulated by it. One can thus speak more of quantized response (again reminiscent of fuzzy logic), than of a simple yes-no logic. Complex, not necessarily binary, connections between neurons are indeed at the heart of the thinking process inside our brains. Notably, in this respect, one stereotypical metaphor expressing a weak intellect in colloquial non-academic speech involves the expression “he only has one neuron” – which incidentally breaks not only binary logic but also fuzzy logic. Nevertheless, such breaking of binary logic is not always culturally associated with weakness; - e.g. “hominem unius libri timeo” (I fear the man of a single book), a quote ascribed by some to St. Thomas of Aquino.
To conclude, we have explored here some basic considerations on binary logic in chemistry and beyond; the authors do express their hope that the reader will consider these ideas to be neither correct, nor incorrect.
[i] “The Holy Bible”, King James version, Genesis 1:1.
[ii] Mihai Eminescu, “A Dacian’s prayer”, 1879, Translated by Corneliu M. Popescu, http://www.romanianvoice.com/poezii/poezii_tr/dacian.php.
[iii] Marin Sorescu, “Simetrie”, in “Ceramica”, Fundatia Marin Sorescu, Bucuresti, 2004, 190 pp; translated by RSD.
[iv] I. Haiduc, J.J. Zuckerman, “Basic Organometallic Chemistry”, Walter de Gruyter Publ. Co., Berlin, New York, 1985, 488 pp.
[v] Ionel Haiduc, Frank T. Edelmann, “Supramolecular Organometallic Chemistry”, Wiley-VCH, Weinheim, New York, 1999, 470 pp
[vi] Ioan Silaghi-Dumitrescu, Dragos Horvath, “Mecanica Moleculara”, Editura Universitatii Babes-Bolyai 1996, 227 pp