Gödel, criticizing a paper by Turing, remarked on how 'concepts' are grasped by the mind in different ways, that certain concepts can become clearer, sharper and richer as time goes on. 'Concept' can be taken to mean one's conception of something (which can involve more than the 'psychological' as understood by Frege and Husserl) or it can mean the thing (the ideal unity of all adequate conceptions of the concept) of which one has a (possiby imperfect) conception of. Gödel's observation may not apply to all concepts, but only to some, for instance mathematical or metaphysical concepts. The clarification, sharpening and enriching of one's concept of a concept has to be carried out through logical and intuitive exercise (which will involve interaction with other concepts). This exercise will have a spiral structure, for one will return to the concept again and again but now in a slightly different light.
We have no idea how concepts, meanings, intentions, references are related or even what these things are. We have no idea how their mereology works or the nature of this relation.
Do we think of concepts or do we think through concepts ? And at a given moment can we be thinking of more than one concept or be thinking via more than one concept ? When I think of the concept of prime number am I also thinking of the concept of number ? (and is it not curious that we ask about the number of numbers satisfying a certain property ?).
What Frege got wrong was not knowing that the purity and objectivity he postulated in thought, meaning and reference is only an approximate ideal which depends on the exercise and training of the mind. How can we understand (think of) this pure thought in its activity ?
In other words there is vagueness and there is clarity and objectivity - and there is a path and exercise leading from one to the other. But ordinary conceptions and meanings of the ordinary mind - maybe these do not have any one definite clear objective counterpart. These are simulacra, pseudo-conceptions, shadows.
“Good Morning!” said Bilbo, and he meant it. The sun was shining, and the grass was very green. But Gandalf looked at him from under long bushy eyebrows that stuck out further than the brim of his shady hat.
“What do you mean?” he said. “Do you wish me a good morning, or mean that it is a good morning whether I want it or not; or that you feel good this morning; or that it is a morning to be good on?”
“All of them at once,” said Bilbo.
Here is a recapitulation of what we wrote about Gödelian-Platonic dialectics which is central to the process of concept refinement and enrichment and the progressive clarity and objectivization of concepts:
We recommend this essay by Tragasser and van Atten on Gödel, Brouwer and the Common Core thesis. Gödel's theory, as recounted by the authors, is of utmost significance. Gödel was promoting the restoration of the authentic meaning of Plato's dialectics and the role of mathematics expounded in the Republic and other texts. Perhaps Gödel has pointed out the best path (at once philosophical and self-developmental) to the absolute. Here is a relevant quotation from the Tragasser and Van Atten chapter p. 179:
Rudy Rucker (1983, 182–183) has reported on his conversations on mysticism with Gödel. Gödel’s philosophy of mathematics is called Platonism. He held that mathematical objects are part of an objective reality, and that what the mathematician has to do is perceive and describe them. Gödel once published some very brief remarks on how we have a perception of the abstract objects of mathematics in a way that is analogous to our perception of concrete objects (Gödel 1964). Rucker, seeking elucidation of these remarks, asked Gödel ‘how best to perceive pure abstract possibility’. Gödel says that, first, you have to close off the other senses, for instance, by lying down in a quiet place, and, second, you have to seek actively. Finally, The ultimate goal of such thought, and of all philosophy, is the perception of the Absolute. When Plato could fully perceive the Good, his philosophy ended. Therefore, according to Gödel, doing mathematics is one way to get into contact with that Absolute. Not so much studying mathematics as such, but studying it in a particular frame of mind. This is how we interpret Gödel’s remark about Plato. There is, then, no break between mathematical and mystical practice. The one is part of the other, and the good of mathematics is part of the Good. Gödel also talked about his interest in perceiving the Absolute with his Eckermann, Hao Wang.And here is what is remarkable about the Platonic-Gödelian method: the confluence between pure mathematical thought and introspective transformative philosophical psychology. But this project can be discerned in Husserl's Logical Investigations and Claire Ortiz Hill has written extensively about the objective, formal and logical aspect of this work, in particular the important connection to Hilbert's lesser known philosophical thought. However the psychological and phenomenological aspect is just as important, just not in the way of the later Husserl, rather in the Platonic-Gödelian and transformative philosophical psychological way.
The epokhê as Husserl outlined (in the Ideen) is not possible (and even less is the Heideggerian alternative valid), rather such a clarity and 'transcendental experience' is possible through the Platonic-Gödelian method.
For a good summary of the role of mathematics in Plato see Sir Thomas Heath, A History of Greek Mathematics, Vol.1, Ch. IX.
The Socratic method of abstraction must not be confused with that of mere generalization or induction. Rather the examples are skillfully chosen so that the mind is (re)awakened to the cognition of a certain idea (the applicability of the idea to different situations comes afterwards). A geometric analogy for Socratic abstraction might be that the examples are like points and the ideas are like the lines, planes or other figures determined by those points.
The Platonic idea is in itself the Fregean objective concept (the ideal pole) while its relational aspects are the conceptions, meanings and intensions that the mind has.
Platonic dialectic is like an ars inveniendi, a general method, a universal intelligence, capable of producing new results and solving problems in all subjects. Its interdisciplinary nature reflects it's being 'beyond hypotheses' yet 'using hypotheses as stepping-stones'. Dialectic is like a universal strategy and practice that is applicable to a great variety of games (we use 'game' in a non-pejorative sense and rather in the spirit in which in the Platonic dialogues dialectics is seen as a form of intellectual athleticism). It is also an exercise as well as a 'game'. There is a similarity to the Pyrrhonian epekhein (abstaining from views), however: the practice of mathematics is an essential preparation for Platonic dialectic and its aim is to produce illumination and authentic knowledge (the vision of the good or the absolute). Platonic dialectic also involves 'phenomenology' in the sense of including introspective psychology, but phenomenology as such is only a preliminary part and phenomena are seen in an entirely different light (as part of the 'friction' and spiral of the cognition) and subordinated to an entirely different purpose (which is neither mindfulness of the present moment, living in the moment, nor a rationalization and becoming-at-home-in the world of the cave).
From some perspective, an individual's conception of a concept can be seen as a point in a space of cognitive or representational states. However, many such conceptions that differ in detail, perspective, or level of development may correspond to the same underlying concept in its ideal form. This is analogous to the construction of a quotient space, where an equivalence relation partitions a larger space into classes of points that represent the same abstract entity. The concept itself, then, is the equivalence class; an ideal unity that transcends any single instantiation or representation.
ReplyDeleteThe clarification of a concept over time, through logical and intuitive effort, can be thought of as moving within the space of representatives, gradually homing in on the most canonical or "sharpest" representative of the class. This reflects Gödel’s idea that certain concepts become richer and more refined through mental discipline.
The analogy becomes even more vivid with the Hopf fibration. In this view, the concept corresponds to the base point (a structure in the lower-dimensional base space), while each conception or mental representation of the concept corresponds to a point on the fiber (a circle) lying above that base point. The various conceptions are related but differ in their local features, perspectives, or contexts. The full richness of the concept is not contained in any one conception, but in the entire fibration structure above the base point.
Moreover, the spiral nature of conceptual refinement (as mentioned in the original text) aligns beautifully with the twisting structure of the Hopf fibration. Returning to the same concept from different angles can be seen as moving around the fiber, each time arriving at a slightly shifted, though related, perspective.
Let's make it simple again. We have a point (“Good Morning!”) and a set of straight lines (“Do you wish me ‘, ’feel good this morning”,...) passing through that point. If you ask what line the point belongs to in response you will find out that it belongs to all of them. Instead of a point, you can take an event and for straight lines you can take a context . Or the point is one move on the chessboard and the lines will be the equivalent of sequences forming the whole combination.
DeleteOf course, Hopf fibration is more impressive
https://www.youtube.com/watch?v=jbvEzeVgBZU
than these straight lines :))))
Thank you for your interesting comment. But while the fibers of a fiber bundle have a certain homogeneity I view our different conceptions underlying the same objective (ideal) concept as a progressive hierarchy (or ascending spiral) which may exhibit qualitative discontinuities. I made an important addition to this post explaining how this idea of the refinement of concepts is connected to Platonic dialectics and Gödel's theory of the highest purpose of mathematics.
DeleteThank you for the thought-provoking elaboration. The idea of a progressive hierarchy of conceptual refinement, marked by qualitative leaps, indeed resonates with mathematical structures such as Grothendieck topoi or large cardinal hierarchies, where each level introduces new modes of "seeing" that recontextualise the levels below. One might also point to Turing degrees or the arithmetical hierarchy, where ascending strata unveil deeper orders of computability and definability.
DeleteGödel’s Platonic dialectic clearly transcends mere formalism. His view suggests that mathematical practice, rightly approached, becomes a kind of intellectual asceticism, purifying the intellect to allow glimpses of the Absolute. This is where the “hierarchy” turns spiral; not merely climbing but returning, at a higher turn, to the same conceptual nexus, now newly illuminated.
Gödel’s vision aligns less with the later Husserlian bracketing of the world and more with the anamnetic recovery of truth through intellectual intuition, echoing Plato’s own ladder in the Symposium. In this sense, mathematical ascent is not a rejection of the phenomenal but its transfiguration.
The convergence of mathematical clarity and mystical perception, what we might provocatively call the Gödelian eudaimonia, challenges the impoverished dichotomy of logic vs. life. There is, perhaps, no pure mathematics without transformation of the self.
That is an interesting remark about cardinals. I would say that as we ascend the structures become richer in intension but smaller in extension and yet having the potentiality of giving rise to higher extensions. Thus God would have the greatest (infinite) intension (the sum total of all positive attributes) but the extension of this intension would be a unique being. Yet if we consider the logical implications of this infinite intension and take there extensions then we determine ever greater multiplicities of beings. The increase in set theoretic cardinality would reflect rather than an ascent, a descent into the unlimited (apeiron) and into multiplicity and lack of form and limit (though Gödel did however consider the universal class V as a form of the Absolute). I think the neoplatonists debated whether Arithmetic (which is closer to finitary mathematics) should be placed on a higher plane than Geometry (though the Republic would suggest otherwise).
DeleteThe Arithmetical hierarchy is very interesting for it is defined without leaving the limit of countable infinity. And indeed I am writing about how computability can be defined in mathematical analysis, for instance by considering analytic functions having series expansions with computable coeficients.
It seems strange that nobody has asked the following question:
Given a polynomial equation of degree n can we expressed its solutions as a convergent series with computable coeficients and is there a way of determining this algorithm conceretely ? (I believe Wronski attempted to do something similar).
I think the Husserl of the Logical Investigations and the Husserl of the Ideen both practiced phenomenology however the method of the Logical Investigations was more than just a phenomenology it was a method that involved at once psychological introspection, analysis and the logical-axiomatic method (in one section Husserl even sets up an axiomatic system for mereology) - hence its affinity to Platonic dialectics. The Logical Investigation was a methodological synthesis of Fregean logical objectivism, Hilbertian axiomatic formalism, Brentanian psychological instrospection and the best of the philosophical psychology of the 19th century.
PS. I suppose that considering Newton's method might be a way of answering the question about polynomial equations.
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