In a recent article for this very magazine, Dimitris Moustakas used formal logic to confront Zeno’s paradoxes in a deconstruction of Marxian “dialectical logic.” It is unclear to me whether Moustakas views himself as an analytic, yet his attempted deconstruction fits into a series of attacks on Hegelian dialectics by analytic philosophers, most significantly by the veritable Bertrand Russell, himself a disappointed mystic in search of a formally clear ground for his assumptions:
The great mathematicians of the seventeenth century were optimistic and anxious for quick results; consequently they left the foundations of analytical geometry and the infinitesimal calculus insecure. Leibnitz believed in actual infinitesimals, but although this belief suited his metaphysics it has no sound basis in mathematics. Weierstrass, soon after the middle of the nineteenth century, showed how to establish the calculus without infinitesimals, and thus at last made it logically secure. Next came Georg Cantor, who developed the theory of continuity and infinite numbers. “Continuity” had been, until he defined it, a vague word, convenient for philosophers like Hegel, who wished to introduce metaphysical muddles into mathematics. Cantor gave a precise significance to the word, and showed that continuity, as he defined it, was the concept needed by mathematicians and physicists.1
Like Russell before him, Moustakas establishes himself as a representative of the “scientific” side of the debate as opposed to the dialecticians, Hegelian or Marxist, which, while often proclaiming themselves to be scientific as well, seem all too content to ignore any scientific developments not fitting into their paradigm as bourgeois distortions or regression. While the latter part might certainly be true, the former bears questioning and it is my contention that while perhaps following the flow of science a little longer than its counterpart Moustakas himself ignores many of the developments of current logic. To understand this, let us first examine Moustakas’ treatment of Zeno’s paradoxes. From a modern perspective, it can be said that Zeno’s concern is primarily continuity, and as such Moustakas places himself again in a line with Russell, who used Hegel’s treatment of continuity as the point of his attack. Moustakas writes:
…any declaration of a body being in motion or at rest at an instant is tied to its immediate past or future state. Therefore, it is impossible to infer a body’s state (either rest or motion) at an instant just by knowing its position at that instant, without considering its position before or after that moment. Zeno’s argument erroneously draws this conclusion, and therein lies its fallacy.
Thus, the argument goes: by considering the position of an arrow a at an instant t it is impossible to determine its velocity, one needs to know where it was and will be before and after t. However, it is also not the case that the position of the arrow at any given point in time t− before t is enough to determine its velocity, as this velocity can change by an arbitrary amount between t− and t. One might instead say that it is required to know the positions of a in an interval I of time around t to determine its velocity. However, any given interval I already contains more information than is needed and can (and needs to) be made smaller to determine determine a’s velocity. One might in fact define the velocity of a at t as the limit of the distance a passes through as I is made smaller and smaller. This is has been the foundation of calculus ever since Weierstrass provided one, as Russell hinted at.
However, while this is certainly a viable definition and still what is used in Calculus courses, it is not how any of the pioneers of calculus, neither Leibnitz nor Newton nor their immediate successors2 derived their results. This should not in itself be a call for us to abandon the modern treatment as it might only show the advancement of science since their time, though it should warrant consideration. However, what one finds when reaching even the lower spheres of differential geometry is that the modern treatment based on Weierstrass becomes very cumbersome very quickly, whereas thinking, as his precursors did, in terms of an infinitesimally small interval sidesteps the issues that plague Weierstrass’ formalization of Calculus. This was done even within a time when it was not yet clear that such considerations could be given formal meaning. Thus wrote Sophus Lie:
The reason why I have postponed for so long these investigations, which are basic to my other work in this field, is essentially the following. I found these theories originally by synthetic considerations. But I soon realized that, as expedient [zweckmässig] the synthetic method is for discovery, as difficult it is to give a clear exposition on synthetic investigations, which deal with objects that till now have almost exclusively been considered analytically. After long vacillations, I have decided to use a half synthetic, half analytic form. I hope my work will serve to bring justification to the synthetic method besides the analytical one.4
A side effect is that in such a setting the continuum can never be decomposed into two non-overlapping parts. With this in mind, it is at least vaguely ironic that the only time Hegel mentioned Zeno at least in Science of Logic is to attack the idea that the continuum could be decomposed:
Bayle, who in his Dictionary (Article, “Zenon”) finds Aristotle’s solution to the dialectic of Zeno “pitoyable,” does not understand what it means to say that matter is infinitely divisible only as possibility; his retort is that, if matter were divisible in infinitum, it would actually contain an infinite aggregate of parts and would be, therefore, not an infinite “en puissance,” but an infinite that exists really and actually. On the contrary, divisibility itself is already only a possibility, not a concrete existing of parts, and plurality in general is posited in continuity only as moment, as sublated.5
The law of the excluded middle had come under attack before and for entirely different reasons by the intuitionists and in particular Luitzen Egbertus Jan Brouwer in the great foundational disputes of the last century, giving rise to constructive logic, now one of the cornerstones of provability theory, and one finds, when one follows along the considerations of Lawvere and their extensions by Urs Schreiber, about which I wrote an exposition here, that the suspension of the law of the excluded middle gives rise to an astonishingly simple and extremely expressive geometric theory that even goes into deep territories of current-day physics.
So far so good, however, Moustakas’ attack was about the law of non-contradiction, that it cannot be that both a proposition and its negation can be true at the same time, and the intuitionists and even Lawvere left this law untouched. So should we perhaps assume that this law is some God-given truth according to which the universe works?
Well, for this we should first off note that the law of non-contradiction is, in a formally precise way, dual to the law of the excluded middle, and if the suspension of the law of the excluded middle has such interesting effects it would be of a disquieting asymmetry if similar could not be said about the law of non-contradiction. And it is in fact true that, while geometric logic, which can very roughly be said to be classical logic without the law of the excluded middle, has become one of the most interesting bridges between logic and geometry, there is an entire dual logic that has received less attention simply because it hasn’t come up as much yet. However, this is not to say that it will not in the future or is of less fundamental importance, after all geometric logic didn’t come up for thousands of years until only a few decades ago, and there are clear theoretical reasons for assuming it will come up sooner rather than later.
Moreover, there is the promising field of linear logic, in which the law of non-contradiction does not hold. This field is an interesting case study to address one of Moustakas’ objections, namely that in the realm of formal logic it follows from the assumption of a contradiction that any given statement would be true. This might be true of classical logic but is certainly not an an automatic truth in any logical formalism. Moustakas refers to this as the principle of explosion, but a better name is its classical one ex contradictione quot libet, from a contradiction follows arbitrariness. This logical proposition can be decomposed into two parts ex contradictione quot falsio, from contradiction follows falsehood, and ex falsio quot libet, from falsehood follows arbitrariness. What one finds now in linear logic is however a logical system that decomposes into two parts which are intertwined, one in which ex contradictione quot falsio holds and one in which ex falsio quot libet holds, and classical logic is precisely the domain in which the two logical systems overlap.
Why now is linear logic important? Well, most tantalizingly it is becoming increasingly clear that it is a logic suitable to describe quantum processes, a quantum logic6, which can be extended into dependent linear type theory, a system that is able to describe both the quantum realm and its interaction with the classical one, which is some of Urs Schreiber’s (and as well as David Jaz Meyers’ and Hisham Sati’s) most current and interesting work.7 Moustakas acknowledges the possibility of quantum systems complicating his argument but writes off considerations in that direction as “very dangerous and cranky”. Perhaps so, but only if reality itself is dangerous and cranky! In his refusal to consider quantum theory, Moustakas manages to catch up with scientific theory around the beginning of the last century, complete with all its trappings, as exemplified by Russell.
Yet, linear logic also has other applications, for instance to probability theory.8 And this brings me to my final point: the law of non-contradiction is an assumption about the syntax of logical systems, not their semantics, what they describe in the real world (or other interesting systems), however, what the negation of a proposition corresponds to is a question of semantics, not syntax. Thus, if the negation of a proposition is defined to correspond to the fact that the proposition is not factual, then the law of non-contradiction is tautologically true, however, there is no reason to assume that this is the only possible semantics for a logical syntax, and in fact the semantics of classical logic are woefully inadequate to cover reality.
To make this clearer, consider the following example: think of the sun at sunset, going down and being covered halfway by, say, a hill. Is the proposition “the sun in the sky” now true or untrue? Well, the sun is not completely in the sky, so the proposition is perhaps not completely true. Assuming both the law of the excluded middle and the law of non-contradiction we would be forced to conclude that the proposition is false. Yet, part of the sun is still in the sky. Colloquially we might say the proposition is half-true, which we can formalize if we suspend the law of the excluded middle and allow for states between complete truth or complete falsehood. Or, dually, we might say that it is both true that the sun is still in the sky and that it is not in the sky, after all, part of it is in the sky and part of it is not, which we can formalize by suspending the law of non-contradiction. Does it necessarily follow from this that the moon is made of cheese? Of course not! Thus, we see how the introduction of even the slightest geometry is enough to give rise to situations that classical logic is ill-equipped to deal with, and how our colloquial language is in some ways more advanced than its formalizations.
In summary, while dialectical Marxists or Hegelians certainly cannot afford to discard current-day science, neither can the analytics, and neither side can claim its mantle in this debate.
Alexander Prähauser
- Bertrand Russell. A history of western philosophy. 1967.
- Continuity and Infinitesimals, Stanford Encyclopedia of Philosophy. url: . https://plato.stanford.edu/entries/continuity/
- Sophus Lie. Allgemeine Theorie der partiellen Differentialgleichungen erster Ordnung. 1873.3
The issue is, vaguely speaking, that such a formally clear “synthetic” treatment of differential geometry requires an infinitesimal neighborhood D around each of real number x that is so small that each function would be linear on that interval, and thus continuous, yet if in such a neighborhood existed even one number y different from x, meaning (x=y), then a function could simply be defined such that there was a jump between x. However, it was found by William Lawvere, himself a Hegelian Marxist, using dialectical methods, that one could define such a theory as long as one assumed that numbers x and y could exist for which it neither held that the two were equal, x=y, nor that they were wholly different, (x=y), meaning, as long as one suspended the law of the excluded middle, that either a proposition is true or its negation is, one of the classical rules of Aristotelian logic or its successor Boolean logic. One can collapse such an “enriched” universe into a classical one and retain any propositions arrived at without use of the law of the excluded middle. As Lawvere was directly inspired by Hegel it should come as no surprise that this state of affairs is exquisitely described in his dictum on infinitesimals [3]:
These magnitudes have been defined as such that they are in their vanishing, not before their vanishing, for then they are finite magnitudes, or after their vanishing, for then they are nothing. Against this pre-notion it is objected and reiterated that such magnitudes are either something or nothing; that there is no intermediate state between being and non-being (’state’ is here an unsuitable, barbarous expression). Here too, the absolute separation of being and nothing is assumed. But against this it has been shown that being and nothing are, in fact, the same, or to use the same language as that just quoted, that there is nothing which is not an intermediate state between being and nothing. It is to the adoption of the said determination, which understanding opposes, that mathematics owes its most brilliant successes.3 Georg Wilhelm Friedrich Hegel. Wissenschaft der Logik. Ausgabe B. 1813.
- Ibid.
- Jean-Yves Girard. “Linear logic”. In: Theoretical Computer Science 50.1 (1987). doi: 10.1016/0304-3975(87)90045-4 . url: https://www.sciencedirect.com/science/article/pii/0304397587900454.
- David Jaz Myers, Hisham Sati, and Urs Schreiber. Topological Quantum Gates in Homotopy Type Theory. 2023. arXiv: 2303.02382.
- Michael Shulman. “Affine Logic for constructive Mathematics”. In: The Bulletin of Symbolic Logic 28.3 (June 2022), pp. 327–386. doi: 10.1017/bsl.2022.28. url: https://doi.org/10.1017%2Fbsl.2022.28.