THE TRANSMIGRATION OF THE IDEAL – ILYENKOV AND INTUITIONISM

Filipe Felizardo

Nov 8

Filipe Felizardo – International Friends of Ilyenkov Symposium 2022 – UCL PESGB

[here's the first attempt at tackling two subjects that met me recently - Ilyenkov's philosophy and intuitionism. The first, through Kyrill Potapov,and the latter, through a seminar by AA Cavia at the New Centre, Computation and the Real, sided with their book, Logiciel. I'm going to present this at the International Friends of Ilyenkov Symposium 2022 in London by the end of this week - which is why this essay is around 2K words,in order to be read under 20min. I plan to review it and expand it extensively, mainly to remove its personal, too oral vibe (i'm too green to improv on something by heart) and to elaborate its more compressed parts - which is all of it. You'll sense that the math part is way longer than the Ilyenkov. This is because i was advised to be clear with regard to former, given most of the audience isn't probably too familiar with it. There are two prices to pay: for readers of this blog, is that the Ilyenkov section is too compressed, for i'm cashing in on the fact that most of the audience is more familiar with him; for me, because readers familiar with the math will probably catch me erroneously belabouring some point, and possibly hatcheting the Ilyenkov into unintelligibility or plain mangling. That said, i also apologize for the lack of references, but that will be tackled when the text gets 10x its current size, and possibly fused with my previous essay on Socratic Death and Paideia, along with some others - The Unreality of the Human, and... we'll see! The fact is that i'm now arriving at the penultimate step in the construction of my final project for my certificate in philosophy at the New Centre. Thanks for reading!]

THE TRANSMIGRATION OF THE IDEAL – ILYENKOV AND INTUITIONISM

I picked mathematics for this presentation because i'm currently learning it from scratch. I figured that for the purpose of addressing the symposium's theme – Futures & Ideals – it would be fruitful to see how mathematical thought maps on to Ilyenkov's thought on thought – or his Dialectical Logic. What you'll hear is a case study for my understanding of both. I'll focus, on the one hand, in discussions in philosophy of mathematics regarding proof demonstration versus construction, and on the other hand, on Ilyenkov's notions of labour, activity, and of the ideal and its place in the dialectic. I hope that these are the beginning steps for the development of a pedagogical (and philosophical!) method that warrants futurity to learning, by activating it as an open ended process.

I will start with the mathematics. In order to be able to oppose the notions of demonstration and of construction, we need to understand what they refer to. And that is the concept of proof. A mathematical proof is an inferential web underlying a mathematical statement – a theorem - whose elaboration allows us to consider the statement's assumptions as logically sound or 'true'. It is usually agreed that this strategy was developed concurrently with land measurement and its science, geometry. In the early 3^rd century BCE, Euclid was the first to come up with a method for proofs as we know them today. The gist of it is an interplay between primitive notions – imported from intuition, such as 'point', 'line' – and axioms, statements about such primitive notions which are held to be inherently true. With this in our minds and hands, we can use logical deduction to check if a new statement or theorem is true. In this way, we demonstrate the proof of a theorem's truth.

Euclid's geometry contained 5 axioms. The 5^th is the most famous, by virtue of being the only for which no proof could be successfully deducted. This historical fact is essential for what we're dealing with here today. A very simplified account tells us that the 5^th axiom postulates that two non-parallel lines will always cross in the same plane if extended indefinitely. I chose this phrasing because this way I can say that its negation is 'two parallel lines will never cross in the same plane'. And this is where the proofs for the 5^th axiom always tripped, because most people working on it stopped short of being creative.

It was not unusual to try to prove a theorem by working on its negation. If its negation is false, then it can be deducted that the theorem is true. The issue lied in the very concrete fact that the negation of the so-called parallel axiom produced valid geometrical results – although completely new and inconsistent with Euclid's geometry. It is not an exagerattion to say that the contradiction was the birth of new geometries, stems off the main branch. In the late 19^th century, new, non-Euclidian geometries were produced by Riemann, Gauss, Lobaschevsky and Bolyai. What I want to stress is that from a contradiction – from going beyond ideological 'no-go' zones – it was possible to elaborate - to construct - new intelligibilities; to go beyond demonstration, which starts to appear, epistemically, as mere empirical redundancy – which is not bad in itself, but it is not sufficient. With contradiction, we can at least make future knowledge indefinite, instead of making present knowledge infinite. According to Jean-Yves Girard, "logic does not ensure termination, but absence of deadlock."

This promethean moment is very important for mathematics in general, but also for a humanist account of thought and logic, its science. In the early 20^th century, L.E.J. Brouwer, a dutch mathematician and philosopher of math, addressed the issue with great fervour.

In 1912, Brouwer wrote that he considered the birth of non-euclidian geometry the greatest blow to Kantian intuition. To put it briefly: we knew that Kant's account deemed space and time as a priori constituents of a transcendental logic. For Kant, our intuition of space was aprioristically conditioned by Euclidean geometry. We can immediately see that if someone brings up a non-euclidian geometry to the collective playground of the human intellect, Kant's account of intuitive space explodes. The blow's result is the liberation of the transcendental logic from axiomaticity.

For Brouwer, this entails that the mathematical intuition of space is not constrained by the logical principle of the excluded middle (PEM) – which states that for every statement, either itself or its negation are true, with no possible third option. We saw above that if the parallel axiom was true, its negation would be false. But we saw that if its negation was proved as true, we could get intelligible results – although the positive axiom would be deemed false. The knot that the branch created in the deductive tree thus created a contradiction – a third possibility beyond either/or.

Brouwer's new intuitionism keeps time, though, by considering "the falling apart of moments of life into qualitatively different parts, to be reunited only while remaining separated by time, as the fundamental phenomenon of the human intellect; passing by abstracting from its emotional content into the fundamental phenomenon of mathematical thinking, the intuition of the bare twoity." Brouwer, for other reasons, did not mind being called a mystic. But this notion of twoity is nothing of the sort. Although it compresses a beautiful dialectical movement in a strange term, it denotes the intuition of ordering the two, the many, from the one. The material incompatibility of two moments generates the possibility of further intelligibility. The one becomes itself by being 'not all other things', which gives us its twoity.

Iterating this movement allows us to construct the whole of arithmetic up to the smallest infinite ordinal number, and unites "the connected and the separate, the continuous and the discrete, gives rise to the intuition of the linear continuum, i.e., of the between", and thus, to space. Let's keep this notion of 'the between' for later. In sum, Brouwer is saying that from twoity and iteration, from time, we can construct the synthetic a priori, not only for arithmetic, but for geometry, and further dimensional geometries. In a sense, intuition is the labour of the intellect into activating new vistas for itself.

It may be the moment to cryptically quote S. Mareev, who says that "labour is a purposeful activity, aimed at a certain goal, and that goal is the knowledge of things that are non-existent as yet."

Now, we can see that this is strongly at odds with a philosophy which holds the PEM as eternally valid. Its application forecloses construction. For the intuitionist, assertions from the PEM are only true if realized – that is, they're merely local, and cannot be ampliated wholesale into the infinite. With this in mind, Brouwer goes further and states that logic is not the source of truth. Most remarkably for the debate of his era, and for our purposes, he also affirmed that an ideal logic is not the foundation of mathematics.

To finish my illustration of the import of intuitionism for this talk, I would like to end this section by stressing the following: realization plays a structural role in such an account of thought, truth, and meaning. For the intuitionist, there only exists what is concretely, actively constructed by the mathematician. This is not meant in a gratuitous nominalistic or solipsist manner; rather that existence and truth are not matters of end-all, be-all formalization, but the results of transformation; of reality realizing itself. Further developments in paraconsistent logic in the 20^th century even show us that there are ways of accomodating contradiction in systems that can see it as potentially informative. To stick to the PEM and to classical demonstration seems to amount to reification.

Cavaillès, french philosopher of math, put it quite well: "to speak of mathematics is to remake mathematics." In a way, to participate in mathematics is to activate its history, instead of atavistically fulfilling it.

I opt to be briefer in my dealings with Ilyenkov, having assumed that you are more familiar with his work – certainly more than I am. I'll give a summary of Ilyenkov's thought up to where it meets with the conflict between demonstration and construction. Afterwards, I'll propose a pedagogical position with regard to mathematical activity.

So, what does Ilyenkov think of thought? In Dialectical Logic, he invites us to "understand thought (thinking) as the ideal component of the real activity of social man transforming both external nature and himself by his labour." I mean, it's all in here. I'll take the liberty of paraphrasing 'real activity' as material activity, though. Although other Soviet philosophers have a different stance on the foundational problem of activity versus labour, and it might be tempting to take activity as an absolute under which labour is subsumed, we will abide by Ilyenkov and engage in the dialectical movement enabled by taking labour as its motor.

The key here is to understand that by virtue of being a concrete universal - something in which all humans actively participate in ever distinct ways - labour precedes the ideal. To quote him again: labour is [...]just the process – beginning and continuing completely independent of thought – within which the ideal is engendered and functions as its metamorphosis, idealisation of reality, nature, and social relations is completed (the metamorphosis and idealisation), and the language of symbols is born as the external body of the ideal image of the external world."

As we know, Ilyenkov agreed with Hegel with respect to thought taking more than simple linguistic shape. It is recorded in the human transformation of nature. But, as a follower of Spinoza, instead of going into absolute idealism, Ilyenkov posited – i'm paraprashing - that the thinking body actively constructs its relation to other bodies.

I understand this through the lens of the notion of material incompatibility. We've seen a version of it in Brouwer's account of the twoity in time. The body, through labour, idealizes meaning through the activity of itself in relation to another body, towards which it is materially incompatible. Through the ideal, the now thinking body compatibilizes, or better, synthetizes, the materially incompatible. This does not stop here, but it suggests an explication: to construct is to actively synthetize.

We now see that labour engenders the possibility of realizing, of transforming, through the ideal. In its turn, the ideal is the process of meaning-attribution, materialized in words and deeds which are reinjected into material activity, into ever different creative labours.

In order to understand the universality of this movement, I'd like to remember Brouwer's notion of 'the between', whose elaboration I now present: "the between is not exhaustible by the interposition of new units and therefore can never be thought of as a mere collection of units." This rhymes quite remarkably with Ilyenkov's presentation of the universal: it is "above all the regular connection of two or more particular individuals that converts them into moments of one and the same concrete, real unity. (…) This unity is an aggregate of different, separate moments rather than an indefinite plurality of units indifferent to one another."

In this way, we can understand concrete universality of labour as the construction – the active synthesis – of a universe through understanding. Such an spiralled universe is built on recognition of differences in extension, something only achievable through theoretical practice or activity in its concrete/abstract/concrete movement – in dialectical logic. Universality thus semantically realizes social, historical, scientific, artistic facts.

To bridge this with mathematics, I want to illustrate what happens if, against Ilyenkov, the Ideal is posited as a precursor. To do so, simply put, is to turn labour into an abstraction. If the human material activity masquerades as an abstraction, humans cannot but be alienated from it. Forbidden from actively transforming the ideal, the thinking body loses its ability to be active; the body is concretely alienated from thought; words are mere reifications of abstract universality; the ideal loses its normative historicity; activity is shun out of reality, and dehumanization begins.

To be frank, such a dim outlook strongly motivates my friendship with intuitionism. I hope it can be quite transparent that demonstration – against construction – maps quite well with this unfortunate relocation of the ideal.

In a way that is not dissimilar from Ilyenkov's account of the historical division of manual and intellectual labour – in which the hands of the scientist have to pay intellectual rent to a capitalized Science whose truths are absolute - , we have seen that proof by demonstration beholds the learner to timeless, idealized axioms. Simply put, it forces alienated labour. When faced with a contradiction, must the learner reject it as not holding up to the canon? Or, as Ilyenkov would suggest, should learners accept that they constructed a difference in the universe and thus, the possibility of realizing new meaning? Brouwer would certainly side with Ilyenkov in saying that "practical activity is the third thing on which all mutually contradictory systems come together on common soil."

Thought which does not conceive of contradiction as a local phenomenon that can be transformed into another facet of the universal is alienated thought. Dogmatism can only be broken by activity. Ilyenkov puts it clearly in his comments on Schelling: "here, in the field of intuition, he discovered dialectics as the true schema of the productive, actively subjective capacity of man to understand and alter the world of the images and concepts of science."

I want to end by suggesting that the classroom of mathematics is not a prefabricated space of 'self-evident', 'inherently true' axiomatics disguised as reason. It is rather the creative construction of time between the learner's and the pedagogue's intuitions. Freed from timeless dogma, logic is not a set of dehumanizing principles for a reified mathematics; it is thought actively knowing itself through its labour of construction, thus forming - activating - what it can become. This is the universalizing activity.

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