Concept Formation and Spiritual Aspiration
The View from Oregon — 360: Friday 26 September 2025
In newsletters 266, 280, and 296 I discussed something that I first wrote in the summer of 2022, to which I find myself returning again: “There is a fundamental intellectual difference between those who get their rigor primarily from mathematics, and those who get their rigor primarily from logic.” I have a larger framework, to a certain extent derived from Husserl, in which I would eventually like to place this, but I haven’t yet taken the time to write out the necessary exposition. Although my own thought is naturalistic, there are aspects of Husserl’s non-naturalistic philosophy of science that I find congenial, and which I think can be naturalized within a naturalistic framework, so to speak (one could interpret Marvin Farber’s work as being in this vein). In the meantime, whenever the idea recurs to me, I see it in a new light, and this new light picks out details that I hadn’t previously noticed.
Mathematics constitutes a distinctive kind of intuitive formalism. Arithmetic (especially with Hindu-Arabic numerals) and geometry are intuitive on their own account; they don’t need to be made intuitive. They possess a certain conceptual naturalness, despite exemplifying formal concepts. Logic in its many forms constitutes a much less intuitive formalism; logic doesn’t come naturally to us — we need to force ourselves to think logically. I’m assuming that intuition is here a continuum, so that given concepts can be more or less intuitive — it’s not a case of all-or-nothing where intuitiveness is concerned, but a matter of degree. Also, I’m quite sympathetic to the reader who doesn’t follow me on these claims. As with my background idea of how to develop these ideas in an Husserlian framework, I haven’t written out the exposition of this. It’s been the background of my thought for many years, and I just keep working on it in isolation like Nietzsche writing about the philosophical “secret garden” he had cultivated, and which culminated in On the Genealogy of Morals. So I make no claim to here stating the obvious or representing my ideas as a consensus position: my own secret garden is as idiosyncratic as Nietzsche’s.
In any case, granted the presuppositions that are the manure in my secret garden, it seems to me that when we conceive of anything by way of an intuitive formalism — e.g., laws of nature expressed in mathematical terms — the continued extrapolation of this formalism, even its infinitistic extrapolation, feels as conceptually natural as the original statement of the law of nature itself. This is the mathematical route to formalization, and since it takes place in, by, and through the intuitive formalisms of mathematics, and the mathematical path to rigor obeys its own internal logic. The connection to Husserl is his critique of mathematicized natural science, especially in his posthumously published The Crisis of European Sciences. For Husserl, the naturalistic mathematization of what he called Galilean science skirted the necessity of some procedure like phenomenology as a means to clarifying the foundations of formal thought. I don’t think the means of conceptual clarification must be phenomenological, but I agree in principle with Husserl that this neglect of clarification is the original sin of human thought, and this original sin has metastasized like a cancer through science, until extreme means like constructivism come to be suggested as a way to get right with the world again. But I’m not a constructivist either; constructivism is a symptom, not a cure.
On the other hand, when we first conceive of anything intuitively (and here I am excluding the intuitive formalisms of mathematics) and then only later, after having intuitively conceptualized it, seek a formal expression to capture the meaning of the intuition, the extrapolation of the subsequent formal concept (which attempts to capture an intuitive concept) becomes a test as to whether the formal capture of the intuitive concept is successful, or, rather, a test of the degree to which it was successful. If the extrapolation of this formalized intuitive concept yields contradictions, then our formal capture was a little off the mark, and we have to try again. This way of thinking is more typical of logic, though it’s also evident in set theory, which is kind of a half-way house between mathematics and logic.
With intuitive concepts that we seek to capture in an adequate formalism, progress in formal conception formation takes the form of a sequence of increasingly adequate formal surrogates for the intuitive concept. Ultimately, all we’re doing is trying to do justice to our original intuition, and usually finding that we come up short, either because we discover that our formal surrogate leads to a contradiction, or it fails to capture something essential in our initial intuition, and so is inadequate to what we were trying to do. We might limp along with the inadequate formal concept, but we’ll never be satisfied as long as there’s a glimmer of the original intuitive concept that we once believed we grasped, and we will never entirely give up trying to capture this original intuition in a formal concept that can be assimilated to the larger structure of formal knowledge.
Get Nick Nielsen’s stories in your inbox
Join Medium for free to get updates from this writer.
Another form of progress in concept formation is the purification of a concept as it develops from its initial particularistic form to the universalistic form of the concept that is the presumptive telos of the particularistic concept. The intuition here is that different minds may differently conceive of one and the same concept, and the reason that these different minds do so is that there remain some particularistic accretions on the concept, and these need to be purged away to get at the true concept that lies hidden somewhere under the obscuring particularistic artifacts of a given mind. We can assimilate this purgative conception of concept formation (or, rather, concept development) to the formal capture conception if our successive attempts at the formal capture of an intuitive concept incrementally strip away particularistic layers to the concept as we seek the true, universal concept beneath these layers. But it’s not always like this. We may regard our initial intuition, not yet adequately captured by any formal concept, as the true version of the concept, and the subsequent attempts at formal capture as obscuring with formal machinery the pristine concept within. In this case, the formal capture conception is antithetical to the purgation conception of formalization.
It’s entirely possible that all of these forms of concept formation and development obtain at some time or another, and more and better besides. We have no a priori assurance that there is one and only one mechanism of concept formation, and by focusing on the mathematical and logical forms of concept formation in the formal sciences I’m not suggesting that this distinction is exhaustive. But the distinction is useful, and it reminds me of something Wittgenstein wrote in the Foreword of his Philosophical Remarks:
“This book is written for such men as are in sympathy with its spirit. This Spirit is different from the one which informs the vast stream of European and American civilization in which all of us stand. That spirit expresses itself in an onwards movement, in building ever larger and more complicated Structures; the other in striving after clarity and perspicuity in no matter what structure. The first tries to grasp the world by way of its periphery — in its variety; the second at its centre — in its essence. And so the first adds one construction to another, moving on and up, as it were, from one stage to the next, while the other remains where it is and what it tries to grasp is always the same.”
The two methods of attempting to grasp the world that Wittgenstein expressses here correspond to concept formation through intuitive formalism, which I said naturally lend themselves to extrapolation, and concept formation through formal capture of an intuitive concept. Formal capture is, as Wittenstein says, always trying to grasp the same thing, while the other movement of thought is to extrapolate structures, always moving onward and upward. In a sense, this latter movement of thought is evolutionary: our elaboration tends to diverge indefinitely from the original concept, so that this conceptual development takes us farther afield, farther from our point of origin. Insofar as this is an evolutionary movement of thought, it is naturalistic. The other approach to concept development, always trying to grasp the same thing, might then be thought of as non-naturalistic, and perhaps this betrays its proximity to Husserl’s non-naturalistic conception of natural science, which I mentioned earlier.
We can recognize a kind of spiritual striving in both of these forms of concept formation and development: the one is Faustian, always striving toward the infinite, to invoke a familiar Spenglerian association, but the other… the other isn’t exactly Apollonian or Magian, which are the two other categories to which Spengler always recurs. Apollonian thought gave us the original conceptions of formalism that we still employ today, and insofar as the Apollonian is finitistic, we could assimilate the repeated attempts to grasp an elusive intuition to the Apollonian, but the “fit” isn’t quite right.
However, it does bear some resemblance to traditional forms of Western mysticism, and especially to asceticism. And we find lots of hints of this in Wittgenstein as well. When Russell met Wittgenstein again after their years of separation during the First World War, he was surprised to find that Wittgenstein had, in the meantime, become a mystic. Russell was flummoxed, but he shouldn’t have been, since he had his own form of mysticism, but that must be a story for another time.
