Are Science and the Irrational Incompatible?

Abstract

Can science examine what does not yet present itself in a rational, measurable or stabilised form without abandoning its own standards? In mathematics, the term “irrational” has a precise technical meaning: it refers to numbers that cannot be expressed as the quotient of two integers. Philosophy, however, gradually gave the word a broader negative charge by opposing it to Reason, measure and order. This study revisits the role of Greek thought in that semantic shift, then traces several moments when objects first regarded as unproductive, impossible or disturbing found a place within the work of knowledge: approximation, complex numbers, fractals and quantum indeterminacy. The aim is not to praise the irrational, but to ask whether excluding it in advance may close off possible paths of inquiry. Science is not the same thing as the irrational; it can nevertheless investigate what still resists established forms of rationality, provided that such objects are subjected to examination, criticism and control.

Keywords — alogos – chaos – knowledge – semantic shift – repudiation of the irrational.

PART I — How the irrational became suspect

From the incommensurable to the unreasonable

To question the irrational is difficult because the word reaches us already burdened with judgement. In mathematics its meaning is precise. In philosophical and ordinary language it came to designate what escapes Reason, what cannot be measured, ordered or placed within a recognised category. Yet the passage from one meaning to the other is not neutral. It is precisely this shift that I wish to examine.

The Latin ratio may refer to calculation, proportion and measure, but also to Reason as the faculty of thought. Greek thought encountered a problem when it discovered magnitudes that could not be expressed as a ratio of integers. The discovery of incommensurability was experienced as a profound rupture. Árpád Szabó, in Anfänge der griechischen Mathematik, Munich, 1969, p. 119, describes it as a “scandal for Greek mathematical thought”.

The term alogos, applied to such numbers, already carries a judgement: these are numbers for which logos cannot account in the same way as it can for regular ratios. An arithmetical difficulty thereby becomes a broader intellectual difficulty. What cannot be reduced to a measurable proportion is gradually placed on the side of disorder, emotion, the paranormal or, more generally, whatever appears to resist the protocols of Reason.

The irrational as an object of knowledge

It would nevertheless be contrary to the philosopher’s vocation to stop at this condemnation. To examine the irrational is neither to defend it nor to restore some lost prestige. The question is whether what has been dismissed as unproductive may, in certain contexts, contain a possibility of knowledge. Our ignorance remains immense; it would therefore be imprudent to decree that objects which initially resist recognised categories deserve no examination.

Gilles Gaston Granger himself feels the need to warn the reader when he approaches this question in L’Irrationnel (1998): his purpose is not to praise the irrational, but to show that reflection on it is particularly instructive. My own argument adopts the same caution. I do not seek to celebrate the irrational. I question a repudiation that has been turned into an apparent self-evidence although it rests on historical and philosophical choices that remain open to discussion.

The privilege accorded to the rational

The rational has much in its favour. It reassures because it accords with method, precision, exactness and procedure. Alain states that the rational is proper to humanity and superior to the irrational. Hegel’s famous proposition declares that “what is rational is actual; and what is actual is rational”. Amartya Sen, in Rationality and Freedom (2002, p. 4), associates rationality with the discipline of subjecting one’s choices — actions, objectives, values and priorities — to reason. Habermas likewise stresses justification by reasons and arguments.

The attraction of this tradition is understandable: rationality promises order and the possibility of giving reasons for what one advances. Plato’s intelligible world offers stable and immutable Ideas; Greek mathematics likewise works with magnitudes and ratios capable of comparison. Measure and balance thus become guarantees.

The limits of Reason

Yet the usefulness of Reason does not establish its omnipotence. Amo, in De arte, associates contradiction with the absence of tranquillity of the soul: “Nulla tranquillitas animi in contradictione.” The search for tranquillity partly explains distrust of what appears indeterminate.

Herbert Simon’s notion of bounded rationality shows that decision is constrained by available information, cognitive capacities and the conditions under which choices must be made. Recognising these limits does not mean abandoning rationality. It simply prevents us from turning a method that is effective within a domain into a universal power. Nor does it confer truth upon the irrational. It prevents us from identifying the present limits of our instruments with the definitive limits of what can be known. The irrational then changes status: it is not necessarily what knowledge must exclude, but what knowledge must first determine whether it can investigate.

PART II — When Reason itself fixes the limits of what can be known

The absolutisation of Reason

I use “dogmatism linked to Reason” to describe the tendency to grant Reason a scope it does not possess. The problem is not Reason itself; it begins when we decide, without examination, that whatever exceeds its familiar forms must be useless, false or non-existent. Kant uses a striking image in his 1796 essay on perpetual peace in philosophy: dogmatism is “a cushion for falling asleep”. In the Critique of Pure Reason he also criticises the presumption of knowing things that lie beyond the limits of human knowledge. Laberthonnière similarly criticises forms of dogmatism that freeze thought and turn intellectual constructions into authorities protected from examination.

In my view, the repudiation of the irrational belongs to this difficulty. The Greeks judged it unfruitful because they did not immediately know what to do with a quotient that could not be reduced to a ratio of integers. Yet the later history of mathematics would provide examples of fruitfulness precisely where the rule initially seemed to prohibit any solution. The number i is one such example.

Presence of mind before the unknown

To avoid this dogmatism, one needs an intellectual disposition capable of keeping attention on the problem rather than retreating into habit. Tradition calls this “presence of mind”. It does not mean abandoning rules, but retaining enough lucidity to recognise when a new situation requires another route.

Bergson links consciousness to the preservation of the past and anticipation of the future: “All consciousness is therefore memory — conservation and accumulation of the past in the present. But all consciousness is anticipation of the future.” William James makes the pursuit of future ends and the choice of means for attaining them a criterion of mentality.

Amo gives this notion an important place. In De arte he uses praesentia animi, explicitly related to the French expression présence d’esprit, and defines it as promptness in thinking rightly and applying what has been thought on the basis of right reason. He also connects it with momentary deliberation adapted to circumstances and the end to be achieved. The notion matters here because the irrational does not initially provide the landmarks to which the philosopher is accustomed. To venture into it requires us not to confuse the absence of an immediate landmark with the definitive absence of meaning.

PART III — The scandal of irrational numbers

Pythagoras, Plato and the ideal of a measurable world

The Greek rejection of irrational numbers is rooted, in the interpretation defended here, in a representation of the world in which harmony, proportion and measure occupy a privileged place. Pythagoreanism sought regulated ratios in the cosmos. Within such a vision, a magnitude that cannot be reduced to a ratio of integers is more than a technical problem: it threatens a conception of order.

Plato inherits this environment. Mathematics provides him with conceptual instruments for thinking about cosmos and city. In the Statesman, 259e, he distinguishes the master builder from the manual worker. In Republic VIII, 546b-d, rational and irrational diagonals enter a political reflection; the failure of human calculation becomes an image of maladjustment capable of contributing to civic decline. Proclus likewise testifies to the important place of mathematics in the Platonic tradition, although precise ancient attributions concerning the golden ratio must be handled cautiously.

Aristotle and the limits of demonstration

Aristotle cannot simply be presented as ignoring or excluding irrational magnitudes. In Prior Analytics I, 23, he uses the incommensurability of the diagonal and side of the square as a classic example of proof by impossibility: assuming the diagonal commensurable leads to the consequence that odd numbers would be equal to even numbers. In I, 31, he again considers how one may demonstrate whether a diagonal is commensurable or incommensurable. Incommensurability therefore belongs fully to his mathematical and logical horizon.

The word alogos may nevertheless be heard as referring to what cannot be accounted for by the expected logos, what appears irregular or does not fit the anticipated model. Yet irregularity is not rare in nature. One of my claims is precisely that regularity cannot be turned into a prior condition of existence or knowledge.

√2: a magnitude that measure could not contain

The square provides the most striking example. If its side is 1, its diagonal is √2. The theorem attributed to Pythagoras gives 1² + 1² = 2; hence the diagonal is √2 = 1.41421356… Its decimal expansion is infinite and non-repeating, and it cannot be written as a fraction of two integers.

Late traditions reported by Pappus and Iamblichus associate the disclosure of incommensurable magnitudes with punishment or death by drowning. The versions differ and cannot be treated as established historical fact. Whether history or legend, however, these narratives show the cultural representation attached to the irrational: a dangerous secret, a rupture of order, a truth one might have preferred to keep at a distance.

PART IV — When the irrational enters science

From “non-fractional” to “unreasonable”

The Greek difficulty, according to my thesis, was not merely encountering a number that could not be written as a fraction. It lay in confusing two judgements: “non-fractional” and “useless”. From this confusion comes the semantic shift by which mathematical alogos moves towards the philosophical irrational, soon associated with the unreasonable, madness and chaos.

Einstein before indeterminacy

The preference for exactness did not disappear with Antiquity. Einstein expressed discomfort with the role of probability in quantum mechanics. In his letter to Max Born of 4 December 1926 he wrote that quantum mechanics was impressive, but that an inner voice told him it was not yet the real thing; the theory brought us scarcely nearer to the secret of “the Old One”, and he was convinced that “He does not play dice”.

This disagreement does not make quantum mechanics irrational. It shows rather that the rationality of one domain need not coincide with habits acquired in another. Quantum randomness obeys its own formalisms. We must therefore avoid making our older expectations the measure of every possible rationality.

The impossible number that becomes a tool

From complex numbers to fractals

This fruitfulness does not stop with complex numbers. Benoît Mandelbrot uses the complex plane in the study of fractals. Expressions such as z² + c reveal self-similar structures capable of describing irregular forms: contours, shorelines, branching patterns and textures. Irregularity becomes an object of mathematisation.

I am not saying that the irrational is science. I am saying that it can be raw material for science. Egypt built with approximations; modern mathematics constructed theories from numbers once judged impossible; Mandelbrot gave mathematical form to geometric irregularity. This history is enough to challenge the idea that the irrational is intrinsically sterile.

PART V — The unknown before it is put in order

Protagoras: knowing also means putting in order

If the irrational can be raw material, we must ask how knowledge is built from what initially appears formless. Protagoras’ formula, “man is the measure of all things”, has often been read, following Plato, as a profession of relativism. I propose another reading: generic humanity intervenes in a world that does not spontaneously deliver its categories and, through its work, gives form to what it seeks to know.

Plato reports the formula in the Theaetetus and opposes it in the Laws with the proposition that God is the measure of all things. Thomas Aquinas later summarises the relativist reading. Other interpreters — Heinrich Maier, Mario Untersteiner, G. B. Kerferd, Gilbert Romeyer-Dherbey and Barbara Cassin — have emphasised, in different ways, human activity in ordering, structuring or producing meaning. On this reading, humanity does not arbitrarily invent reality; it works to make it intelligible. Knowledge does not always receive its object already constituted: it contributes to organising it as a knowable object.

From chaos to the object of knowledge

Heinrich Rickert insists that reality in all its richness cannot be exhausted by concepts. Conceptualisation selects, divides and organises. Rudolf Carnap explicitly works on the passage from chaos to constructed order in Vom Chaos zur Wirklichkeit, an unpublished manuscript of July 1922 preserved in the Rudolf Carnap Papers at the University of Pittsburgh (RC 081-05-01). In this text, preparatory to the Aufbau, chaos is a starting point for reconstruction: not nothingness, but material not yet endowed with the order that knowledge must establish.

The irrational may therefore be understood as what precedes rationalisation rather than as what is absolutely alien to it.

The limits of the knowable

This question meets the problem of the limits of knowledge in Amo. In De arte he writes that knowledge extending beyond the limits or boundaries of the thing in itself is not possible. Kant, in the Critique of Pure Reason, likewise states that what objects may be in themselves, apart from the receptivity of our sensibility, remains entirely unknown to us.

The author advances the hypothesis that Kant takes up an idea already formulated by Amo without acknowledging its authorship. This belongs to a broader dossier concerning illnesses of the soul and the treatment of African predecessors by some European thinkers. Kant’s well-known 1764 statement about Africans is retained as evidence of his racial prejudice. Whether or not one accepts the proposed intellectual filiation, the issue rejoins the general question of the article: what realities do we discard because they do not fit the categories we have chosen to privilege?

PART VI — Making the irrational an object of inquiry

From number to human phenomena

The semantic shift is now visible. In mathematics, an irrational number is a magnitude that cannot be expressed as a ratio of integers. In philosophical language, the term comes to receive what appears without measure, rule or explanation: passions, affective states, unusual phenomena and experiences difficult to objectify. The move is not self-evident. An impossibility of expressing a number as a fraction becomes an impossibility of understanding; a limit of calculation becomes a judgement on reality.

Unamuno occupies an important place in reflection on the limits of reason and on what in existence cannot be wholly absorbed by it. Camus, in The Myth of Sisyphus, shifts the question towards the absurd: the world itself is not reasonable; the absurd lies in the confrontation between this irrationality and the human desire for clarity. To resist one form of rationality, however, is not to be absolutely devoid of rationality.

Quantum mechanics and the boundaries of the rational

Einstein remained committed to a realism according to which reality exists independently of the observer. Abraham Pais reports that, during a walk, Einstein asked him whether he really believed that the Moon exists only when one looks at it.

Quantum mechanics nevertheless requires rules that are not those of ordinary experience. Entanglement, probability, wave functions and the role of measurement long appeared strange. In a 1947 letter to Max Born, Einstein used the expression spukhafte Fernwirkung, usually rendered “spooky action at a distance”. The 1935 EPR argument sought to show that quantum theory was incomplete. Bell’s inequalities and later experiments, notably those of Alain Aspect, profoundly changed the debate.

Should we therefore call quantum mechanics irrational? No. It has formalisms, produces explanations and predictions, and its predictions have been confirmed with remarkable precision. It teaches rather that rationality need not be unique. A domain may possess its own rules provided they are coherent, testable and fruitful. What appears irrational in an older framework may become rational in a new paradigm.

What is not yet scientific

Science does not begin with objects already constituted as scientific. It begins with phenomena, observations and hypotheses, sometimes with myths or poorly defined objects, which it then subjects to procedures of control. The atom was speculative for centuries before theoretical and experimental developments gave it scientific status. Radioactivity likewise had to be constructed as an object of knowledge.

It would therefore be imprudent to confuse the non-scientific character of a starting point with the impossibility of knowledge about it. Scientific work consists precisely in transforming initial material into an object that can be described, tested and shared.

Can the paranormal be an object of inquiry?

Émile Kenmogné argues in Maladies paranormales et rationalités (2016) that no fact should escape the necessity of philosophical inquiry, and that some phenomena require multidisciplinary illumination. The paranormal is not thereby established as constituted knowledge. Making a phenomenon an object of inquiry does not amount to recognising its reality as described, still less to granting it scientific status. Inquiry may lead to error, illusion, an already known explanation or the impossibility of concluding. Its task is precisely not to decide the result before examination.

The Terentian formula — “I am human: I think nothing human alien to me” — expresses this openness. Jean Désy likewise invites reflection on the role of the irrational and intuition, while Jung, Simone Weil and Saint-Exupéry, each in a different way, explored areas of human experience not wholly enclosed by measurement.

The irrational as a deposit

I am not pleading for the rehabilitation of the irrational as a substitute for Reason. I ask that it cease to be treated as metaphysical waste. An object may be obscure, imprecise or unproductive in an initial state and become fruitful after conceptual work. This is why I prefer the image of a deposit or raw quarry.

Bachelard showed how representations can form epistemological obstacles. An obstacle, however, need not be a final term; it may become the point from which inquiry reorganises itself. Granger likewise insists on thinking the irrational as a problematic complement to rationality rather than its simple negation. Popper writes: “Science must begin with myths, and with the criticism of myths.” The second half is decisive. Science may start from myth, but it does not stop there: it subjects myth to criticism. That is the difference between openness and credulity.

Conclusion — From exclusion to examination

The repudiation of the irrational in the Western tradition is therefore not philosophically necessary. In the account proposed here, it results from a historical displacement: an arithmetical problem, that of incommensurable magnitudes, acquired a negative qualification that overflowed its original domain. According to the author’s thesis, the Greek rejection of certain numbers helped consolidate the idea that to escape measure is to escape logos.

The history of mathematics nevertheless invites us to distinguish what is devoid of reason from what merely resists the available forms of rationality. √2 is not an absurdity: it is a number. Egyptian approximation is not a surrender: it is a technique. The number i, born from an operation that ordinary rules seemed to forbid, became a major instrument. Fractals give mathematical form to irregularity. Quantum mechanics shows that a new rationality can unsettle the habits of an older one.

Science and the irrational are incompatible only if the irrational is given a definitive meaning: that which, by nature, would be withdrawn from every possible knowledge. That is not the meaning adopted here. The irrational may also designate what resists available forms of rationality, what has not yet been ordered, measured or explained. To exclude it in advance would be to ask science to concern itself only with what it already knows how to make scientific.

The boundary must be maintained elsewhere. The object of inquiry may be obscure, uncertain, irregular or reputed irrational; the method used to examine it cannot abandon criticism, control and the administration of evidence. The irrational may enter science as a problem. It does not enter as an established truth. Philosophy need not celebrate obscurity; it must prevent the door from being closed too early.

Questions for further reflection

• Was the passage from the mathematical to the philosophical meaning of the “irrational” necessary, or was it the result of a historical choice?
• Is difficulty of measurement enough to disqualify an object from knowledge?
• How can legitimate confidence in Reason be distinguished from its absolutisation?
• Can presence of mind resist intellectual habits without abandoning method?
• Why could incommensurability be experienced as a threat rather than simply as a discovery?
• Does approximation constitute a form of rationality different from the demand for exactness?
• Does the number i show that a rule may have to be exceeded for a new theory to emerge?
• Should “man is the measure of all things” be read as relativism or as a theory of the constitution of the knowable?
• How can one investigate an object without granting it scientific status in advance?
• What distinction must be maintained between philosophical openness and credulity?

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