The Rise of Science and the Mathematization of Reality: Competing Views

It [Mathematics] did not, as they supposed, correspond to an objective structure of reality; it was a method and not a body of truths; with its help we could plot regularities—the occurrence of phenomena in the external world—but not discover why they occurred as they did, or to what end.

– Isaiah Berlin, from an entry in Dictionary of the History of Ideas – The Counter-Enlightenment

Isaiah Berlin in his entry on what he termed the “counter-Enlightenment” tells us that opposition “…to the central ideas of the French Enlightenment, and of its allies and disciples in other European countries, is as old as the movement itself”. 1 The common elements that these reactionary writers opposed in the Enlightenment project were notions of autonomy of the individual, empiricism and scientific methodology, its rejection of authority and tradition, religion, and any transcendent notions of knowledge based on faith rather than Reason. Berlin himself places Giambattista Vico (1668-1744) and his Scienza nuova (1725; radically altered 1731) as playing a “decisive role in this counter-movement”. He specifically uses the term “counter-movement” rather than the appellation “counter-Enlightenment”.

I’ve been following – – blog Persistent Enlightenment, and one of the interesting threads or series of posts on his site deals with the concept of “Counter-Enlightenment,” a term coined by none other that Isaiah Berlin in the early 50’s (see his latest summation: here). I believe that he correct in his tracing of this concept and its history and use in scholarship. Yet, for myself, beyond tracing this notion through many different scholars I’ve begun rethinking some of the actual history of this period and of the different reactions to the Enlightenment project itself as well as the whole tradition of the sciences. One really needs to realize the Enlightenment itself is the culmination of a process that started centuries before with the emergence of the sciences.

Stephen Gaukroger’s encyclopedic assessment of the sciences and their impact on the shaping of modernity has been key in much of my own thinking concerning the history and emergence of the sciences as well as the understanding of the underpinnings of the mechanistic world view that informs it in this early period. One of the threads in that work is the battle between those traditionalist scholars of what we now term the “humanities” who seek to protect human learning – the study of ancient literature along with philosophy, history, poetry, oratory, etc. – as Gaukroger says, “as an intrinsic part of any form of knowledge of the world and our place in it” (1).1  He mentions Gibbon’s remark that during his time that the study of physics and mathematics has overtaken the study of belles lettres as the “pre-eminent form of learning” (1). In our own time this notion that philosophy and the humanities are non-essential to the needs of modern liberal democracies has taken on a slight edge as well.

Some like my friend R. Scott Bakker fantasy writer and independent thinker of Three-Pound Brain term this the “post-intentionalintentional view is the notion that we are missing information, that our is what limits our ability to cognize nature, and because of this even our ability to know what we are as conscious beings is severely limited in scope and power. As he states it: “If the lack of information is what prevents us from seeing our way past traditional conceits regarding the world, it makes sense to think it also prevents us from seeing our way past cherished traditional conceits regarding ourselves. If information privation plays any role in ignorance or misconception at all, we should assume that the grandiose edifice of traditional human self-understanding is about to founder in the ongoing informatic Flood…”. In private conversations and posts he has reiterated that most of our humanistic learning is becoming a relic of the past, a nice place to visit and enjoy, but that that it no longer offers a purview onto the world as such. Philosophy is dead. Of course there are many who see this elminativist and reductionary approach as erroneous, and the debates on this are endless. Scott’s own reaction is a “wait and see”: he is of that school that the sciences will prove him right in the end.

Yet, David Hume, was one of the first to reject such notions of reducing learning to one unified process based on the sciences alone. For Hume there was no such thing as a single form or enquiry, and he went on to conclude that even where such notions resided within different conceptions of the Enlightenment project of the eighteenth-century based as they were on ‘reason’, these notions left out the difference between propositional and non-propositional forms on knowledge. For Hume the sciences were based on propositional forms of knowledge, whereas natural history and other forms of knowledge were shaped by non-propositional forms. Now as Gaukroger attests Hume’s reading of natural history was not a ‘diachronic’ one, it went against the genealogical methodologies of the philosophes, and was based on what we term empirical or phenomenological versions of natural philosophy (445).

Ultimately the Humaen project took on a critique of the reductionary views of the philosophe’s of enquiry by limiting their systematic understanding based on propositional reason through an attack on the notion of innate ideas. Instead for Hume we needed to rephrase the question of enquiry by placing limits on propositional understanding, and emphasizing that the central problem is a conceptual issue about the forms of understanding available to us and in what circumstances it is appropriate to draw on them (445). It was out of the Humaen tradition that what we might term the “inferentialist” approach came about. In his essay ‘Of the Rise and Progress of the Arts and Sciences’, develops the notion that the relation between cause and effect is a matter of inference rather than direct perception, arguing that such connections are more easily established when changes in human conditions produce changes in large-scale human behavior, as in the case of the rise and progress of the arts and sciences, or that of the rise and progress of commerce, rather than on an individual level, and he distinguishes two forms of history corresponding respectively to cultural change and to individual actions (446).

Hume felt there should be a balance between competing views, rather than an oppositional method that placed these views against one another in what we now might term a binary mode of dialectical opposition. Against certain aspects of thinking in scientific thinking based on such oppositional notions Hume believed that  “forcing a choice of set over another, is not simply an issue about how to write history, but an integral aspect of his conception of what understanding the human condition consists in (446). Gaukroger summing up Hume’s method quoting another scholar Livingstone, says:

Hume’s philosophy is an attempt to show that what we would call rationality in science, morals, politics, and religion is the result of a long, gradual, and largely unreflective evolution of conventions, the end of which is the coordination and satisfaction of conflicting human needs and desires (446).

In some ways this notion of “unreflective evolution of conventions” goes to the heart of many debates, not just those about Enlightenment and Counter-Enlightenment, but in the sciences themselves and the our views of truth and reality. To go back to Berlin’s essay and the original quote at the beginning of this post in which Vico in his New Science questions the Rationalists telling us that the Cartesians were profoundly mistaken about the role of mathematics as the science of sciences; that mathematics was certain only because it was a human invention. It did not, as they supposed, correspond to an objective structure of reality; it was a method and not a body of truths; with its help we could plot reg- ularities—the occurrence of phenomena in the external world—but not discover why they occurred as they did, or to what end (Berlin). Here we see the juxtaposition of math as an invention, a language that does not describe the structure of reality, a method rather than a body of truths, as a tool of phenomenological enquiry not an ontological view onto the metaphysics of is and teleological finalities. 

For Hume it was more about what could help you get on with your work not about proving that one’s truth was the only valid truth. Against the dogmatists of reason he offered the empirical methodology of enquiry, and new that it was never final or fixed, but ever changing and evolving. That what we term truth is not some fixed category of unchanging substance, but was ever-changing bundle of propositional and non-propositional data to be sifted through the enquiring mind as part of the problem solving and decisional process. He knew this could never be concluded. One might say that the truth of history is that there are a histories of truth. His notion of striking a balance between competing views seems a good approach to current debates in the sciences and philosophy as well.

The notion of competing views of reality, and of whether math as a language can ultimately describe the structure of reality is at the heart of physical sciences and the debates in philosophy. The competing voices of materialist and idealist visions of reality have been playing themselves out for over two-hundred years or longer. Two hundred years ago Kant divided the world into two sets of objects. First is the “noumenon” –  in Platonic philosophy, the noumenal realm was equated with the world of ideas known to the philosophical mind, in contrast to the phenomenal realm, which was equated with the world of sensory reality, known to the uneducated mind. Much of modern philosophy has generally been skeptical of the possibility of knowledge independent of the senses, and Immanuel Kant gave this point of view its classical version, saying that the noumenal world may exist, but it is completely unknowable to humans. In Kantian philosophy the unknowable noumenon is often linked to the unknowable “thing-in-itself” (Ding an sich, which could also be rendered as “thing-as-such” or “thing per se“), although how to characterize the nature of the relationship is a question yet open to some controversy. The second is “phenonmenon” – Kant in his dissertation ‘On the Form and Principles of the Sensible and Intelligible World (1770)’, theorized that the human mind is restricted to the logical world and thus can only interpret and understand occurrences according to their physical appearances. He wrote that humans could infer only as much as their senses allowed, but not experience the actual object itself. Thus, the term phenomenon refers to any incident deserving of inquiry and investigation, especially events that are particularly unusual or of distinctive importance. The point here is that we are limited beings and because of our finitude we are limited to inferential knowledge that comes through our limited senses.

Those that affirm the notion that math is the language of reality, that it describes the structure of reality itself and that sooner or later it will build up a unified view of knowledge based on this conceptual framework. This notion can be traced back to the Pythagorean idea that whole numbers and harmonic (euphonic) sounds are intimately connected in music, must have been well known to lute-player and maker Vincenzo Galilei, father of Galileo Galilei. While possibly following Pythagorean modes of thinking, Vincenzo is known to have discovered a new mathematical relationship between string tension and pitch, thus suggesting a generalization of the idea that music and musical instruments can be mathematically quantified and described. This may have paved the way to his son’s crucial insight that all physical phenomena may be described quantitatively in mathematical language (as physical “laws”), thus beginning and defining the era of modern physics.

But why is this? How could math which seems to reside in our mind/brain also describe the structure of reality beyond our mind/brain? As a modern physicist Max Tegmark remarks why “has our physical world revealed such extreme mathematical regularity that astronomy superhero Galileo Galilei proclaimed nature to be “a book written in the language of mathematics ,” and Nobel Laureate Eugene Wigner stressed the “unreasonable effectiveness of mathematics in the physical sciences” as a mystery demanding an explanation?”3 As Tegmark tells us the aim of his scientific project is not only to show how math describes the structure of reality, but to further explicate what he terms a “crazy-sounding belief of mine that our physical world not only is described by mathematics, but that it is mathematics, making us self-aware parts of a giant mathematical object. We’ll see that this leads to a new and ultimate collection of parallel universes so vast and exotic that all the above mentioned bizarreness pales in comparison, forcing us to relinquish many of our most deeply ingrained notions of reality” (Tegmark, ibid.). This notion that we are components within a larger assemblage or mathematical object that is itself but one object in a pluralistic multiverse of parallel universes is part of the ongoing debates in physics.

In philosophy it is the work of Alian Badiou who tells us that the “mathematization of science eliminates at a stroke the tangled neo-Aristotelian interpretation of qualitative essences (hardness, lightness) smoothness, and so on) in favor of the anonymous, asignificant variables of quantity and motion” (Hallward, 8).4 Within the Newtonian framework this conception would lead mathematics and physics to essentially one and the same logic of explanation, a formal order of deductive thought directly articulated with the rational order of materiality itself.(Hallward, 8) As Hallward remarks what Badiou “has retained from the great movement of the scientific revolution and the subsequent Enlightenment campaign against parochial superstition is this conjunction of an isolated, self-grounding subjectivity and an indifferent mathematized rationality as the only fully adequate vehicle of truth” (8). Badiou situates his philosophy firmly within the mathematical rationality of Descartes speculations, saying, “Deep down, I am Cartesian…”(Hallward, 8). Badiou even aligns his Marxism within this mathematical reality, saying, “Marx accepted that there were formal similarities between the ambitions of emancipatory politics and the workings of capital. Because we can never go back on universalism. There is no earlier territoriality calling for protection or recovery…. We are rivals to capital, rather than merely reacting against it. It is a struggle of universalism against universalism, and not of particularism against universalism.”(Hallward, 9-10).

Instead of some reactionary opposition to the course of modern liberalism and capitalism per se, he sees Marxian thought as a rival economic system whose universal system might replace its rival through its purchase of truth and viability. Hallward describes Badiou’s guiding assumption which informs his notions of subjectivity and the event in the sense that “being of an individual or a situation is a matter of inconsistent multiplicity, an inconsistency that is accessible only once that individual has been subtracted from the regime of relations it has with other individuals…”(Hallward, 322). This eliminativist move of subtraction from all relations as allowing us to then inquire and understand the anomalous or ‘inconsistent multiplicity’ that is the individual or object seems rather eerily like Graham Harman’s notion of the “withdrawn object” disconnected from all relations. Not that we could equate the two views, for the one is based on a decisionary process of eliminating or subtracting the object from all relations to better understand its inconsistent multiplicity, while the other is based on a notion of substantial formalism that harkens back to the very Aristotelian of the medieval scholastics that Badiou himself rejects. Yet, there is still that connection between these two notions which is actually based on the concept of non-relation. How can something either be subtracted from all relations, or in itself withdraw from all relations? How is this possible? To answer these questions would take me too far afield. ( I will come back to this again!)

I have yet to explore those who would battle against this central notion of the mathematization of reality. I will need to break this into another post. I believe there is a kernel within this whole tradition will come down to the battle between two major competing views of science: one in which the structure of reality is reduced to mathematical language, which would entail that the neurosciences through their own elminativist processes and image technologies will someday find the mechanism(s) that unite mind with reality; and, second, those who oppose such unification of mind and reality in math, or those in philosophy that believe as Hume that neither math nor any form of propositional statement could ever describe the structure of reality, because reality in non-mathematical and non-propositional. This to me is the heart of the sciences and philosophy today in all its manifestations.

1. Stephen Gaukroger. The Emergence of a Scientific Culture (Oxford Press, 2006)
2. Isaiah Berlin, THE COUNTER-ENLIGHTENMENT. In Wiener, Philip P. Dictionary of the History of Ideas. electronic edition; based on the 1974 print edition by MacMillan Publishing Company, 1980.
3. Tegmark, Max (2014-01-07). Our Mathematical Universe: My Quest for the Ultimate Nature of Reality (Kindle Locations 145-148). Knopf Doubleday Publishing Group. Kindle Edition.
4. Peter Hallward. Badiou: A Subject To Truth. Kindle Edition.

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