jesuve

Unfortunately some of the symbols did not translate into this post but for the most part everything is here. Down with totalization!!!!

HEGEL, GÖDEL, AND BECOMING: A Hegelian Interpretation of a Mathematical Dialectic

This paper will, on the whole, investigate Hegel’s notion of the true and spurious infinity, specifically with its relation to Totalization. In short, it will explore Hegel’s critique of the mathematical infinite. Here, Hegel critiques the separation of the form of the mathematical symbol from its content, maintaining that the form is supposed to be the embodiment of content. This separation, according to Hegel, leaves mathematics hollow, lifeless, and stone-like; it cannot in anyway, then, describe the pregnant, burgeoning, and living movement of the Concept. Consequently, this critique has meaning far beyond mere mathematics. Not only is Hegel critiquing mathematics in general but language itself. If the form and content of our language is like that of mathematics, one will never be able to reach the Concept; our thought will, likewise, be a lifeless, a still-born thought. Still-born in the sense that our thought will not develop, since it is not caught up in the movement of the Concept, this is what Hegel has essentially in mathematical language termed the spurious infinity. Or perhaps a better way of describing this spurious infinity is when our thought could be described as caught in a never ending neurotic-like circle, or better yet gyre, speaking the same thing over and over again ad infinitum, if not ad nauseam—what Hegel dubbed the infinite line.

What is important to begin with in mention with regard to Hegelian dialectic and its movement is that it begins with a content that is indeterminate, and as such the form is immediate and universal. Through the movement of the Concept the form obtains for itself a determinacy which is its content. The content of any idea compels itself outside of itself, it in itself presupposes and points to further ‘information’, via the internal dynamic nature of the Concept. That which is immediate, the Concept, is mediated; i.e., contradiction is endemic to the movement. All of what again occurs because Being is self-related.

Furthermore as self-related, this mediation sublates itself into an immediacy which is essence. If Hegel is correct and form and content are intimately connected, language will not only truly be moved along through the power of the living Concept, but because it is moved along with the Concept; the Concept is known. Instead of an infinite line, Hegel illustrates the movement of the Concept as returning upon itself, through itself, like a circle. Moreover, this circle, the true infinity, is not a neurosis, but rather a circle of circles, a development that returns upon itself more wise and enriched. As the thought returns upon itself, it has incorporated what it has ‘learned’ from its past experiences, and in so doing has thrown aside that which is incomplete and unneeded, accomplishing a more complete and truer ‘picture’ not only of the development of the Concept, but of the Concept itself. So much so can this ‘picture’ become more complete that in fact, for Hegel, the Concept completes itself as the Absolute Idea. Moreover, unless the Concept can be fulfilled in itself, i.e., unless Form can return to Content fully, incoherent babbling is only possible since one could not even know of which one is speaking. Thus, totalization is not only possible for Hegel, but necessary.

I say that the paper is ‘on the whole’ concerned with Hegel’s concept of the spurious infinity because much of the paper will also be concerned with the mathematician Gödel, who showed, at least through my unique interpretation, that the movement of the Concept could be captured through mathematical symbolism. However, perhaps because of his reliance upon mathematical symbolism, Gödel’s interpretation of the Concept is that it cannot complete itself, that in fact it is incomplete. The question that this paper will be considering is whether Gödel’s Incompleteness Theorems do in fact capture the movement of the Concept and avoid the spurious infinity. In short, the paper will be exploring whether or not a different way of dialectical thinking, one that does not fall victim to the spurious infinity and yet nevertheless does not end in Totalization is -possible.

The inspiration for such an inquiry is the result of a rather long footnote in Jean Hyppolite’s book, Logic and Existence . Here, he speaks of the Hegelian notion of the autonomous development of the Concept as “consciousness being lost in its object”, and then states that perhaps such an internal and autonomous development is not as foreign to mathematics as Hegel believed. Although, Hyppolite admits that this is a very curious paradox indeed, he references, in this footnote, a work by Jean Caveillès, On Logic and the Theory of Science , who had tried to show the limits of mathematics and that such an autonomous development may be possible in mathematics itself. As in Hegelian dialectic, there is therefore an internal progression from singular content to singular content. ‘There is no consciousness which generates its products or is simply immanent to them. In each instance it dwells in the immediacy of the idea, lost in it and losing itself with it, binding itself to other consciousnesses (which one would be tempted to call other moments of consciousness) only through the internal bonds of the ideas to which these belong. The progress is material or between singular essences, and its driving force is the need to surpass each of them. It is not a philosophy of consciousness but a philosophy of the Concept which can provide a theory of science. The generating necessity is not the necessity of an activity, but the necessity of a dialectic’ [my emphasis].

Hyppolite, then asks the question if Caveillès is correct where would such a dialectic fit in a Logic like that of Hegel’s. As if answering his own question, Hyppolite writes that perhaps Caveillès’ Self is not as immanent to the content than it is in Hegel; but then immediately does write that it is interesting to note that Caveillès does speak of a mathematics in Hegelian terms, suggesting that such a project ought to be delved into with some profundity. While I cannot promise the depth of profundity that Hyppolite would be expecting, and I will be exploring the possibility of a dialectical, autonomous mathematical movement, using an esoteric interpretation of Gödel’s Incompleteness Theorems, specifically relying heavily upon the second theorem. This interpretation will show prima facie, that the immediacy of the idea will be lost in the autonomous movement, “lost in it and losing itself with it, binding itself to other consciousnesses (which one would be tempted to call other moments of consciousness).” And while the answer maybe ambiguous at best at the end—reminding us of the “curious paradox”—a demonstration of knowledge of both Hegel’s notion of the infinite and of Gödel’s Incompleteness Theorems will, at least, be demonstrated.

The natural problem of how to explore such a topic raises its ugly head. In keeping with Hegel’s emphasis upon history, a history lesson of sorts must first be given in order to set the stage further. This will set up the worldview under which both Hegel and Gödel were working and perhaps explain why both took the positions they did in reference to the Concept. But we must not lose our way in the history either, we must keep in mind the underlying question of the paper, can mathematics be thought in a dialectical manner without falling victim to the spurious infinity. One may ask whether or not we must view all mathematical symbolism as merely a quantitative relation. What if it was possible for a mathematics to be described as dialectical, i.e., as recounting of the movement of Concept returning into itself? To best explore this, a strict investigation of Hegel’s account of the remark entitled ‘Mathematical Infinite’ will be given, with the help of some commentators. Then, we will turn to the subjective side of the movement, relying somewhat upon Hyppolite’s interpretation of the Logic. This will give us a foothold into the less abstract account of the obscure mathematical-speak in the Mathematical Infinite remark. Next, we will approach Gödel’s theorems in the same manner; first nearing it qua mathematical and then stepping back, as it were, and examining it through the subjective side of its movement, with the help of Caveillès’ quotation. Finally, a recap of the argument and a comparison between the two thinkers will be given, and we will ask the question again, can a mathematical dialectic exist without falling victim to the spurious infinity.

During the 19th century great advances were made in both the expansion of mathematical systems and the solution of many earlier problems. One of the more significant developments was a solution 2000 years in the making. Euclid in systematizing the rules for geometry postulated an axiom that was ‘self-evident’ to ancients, but which was not so evident to the modern mind; this axiom is the ‘parallel axiom’. Mathematicians sought for years to derive this axiom from the other axioms in Euclid’s system, but to no avail. Not until the work of Gauss and others was the impossibility of such a project proven. This result is of the utmost importance, since through it proof for the impossibility of proving certain ‘truths’ through a given particular system was established. Moreover, mathematics was now recognized to be concerned primarily with drawing conclusions that are logically implied by the set of axioms in that particular system alone. Consequently, mathematics was understood to be more abstract and more formal than was thought before: more abstract because statements of mathematics can, in principle, be interpreted to be about anything whatsoever and not just to an objectively confined set of objects or statements; and more formal since the validity of demonstration is to be founded in the structure of the statements and not in the subject matter itself. So in a sense, mathematics was thought to be concerned not with whether the axioms assumed and consequences are true per se, but whether they necessarily follow logically from the assumptions.

At this time, mathematical formulization reached pandemic heights. It is obvious that ‘meaning’ was no longer considered important in pure mathematics; instead, ‘meaning’ is thought to be had merely from the logical relationships in which the axioms and the demonstrations find themselves. Form and content are now completely divorced. It was thought that this separation was of the utmost importance, since now unfettered by the constraints of traditional interpretations and limitations, one can investigate whether the bare relations between the terms hold logical necessity. However this ability depends on whether the foundation laid down by the axioms is internally consistent, i.e., no contradictory statements may be made from the axioms. This mission to find whether the axioms are inherently consistent was found to be of Herculean difficulty. Earlier attempts to derive the consistency of a system relied upon the consistency of a second, outside system, and this second upon a third and so on. For example, to prove that the rules of Geometry are consistent, Hilbert translated the axioms into algebraic truths—essentially using Cartesian coordinates to define point, line, etc. This, however, resulted in an infinite regress, for how can one be sure that this outside system is consistent without checking this against another system outside of this second system, and again this latter with a third and so on.

To remedy the deficiency of an infinite regress, Hilbert began constructing ‘absolute proofs’. Absolute proofs are constructed through complete formalization. All meaning is drained from the expressions occurring within the system. They are simply empty signs, which can be combined and manipulated through a series of pre-established rules of procedure. This absolute formalization was believed to have reached its pinnacle with the Principia Mathematica, constructed by Alfred North Whitehead and Bertrand Russell. By ridding Mathematics of meaning and allowing a mathematical system to make statements concerning itself, the problem of consistency was reduced to a problem of formal logic itself. If the axioms used in the system are derivable from the theorems of that system then the question whether those axioms are consistent is asking whether fundamental axioms of logic are consistent. The Principia Mathematica appeared to be the answer to the problem of consistency, since the problem had been reduced to that of the consistency of the formal logic itself.

However, a little known Austrian logician would shake the world of mathematics and logic to its core, overthrowing the Principia Mathematica’s claim to completeness. The reverberations begin in Gödel’s demonstration that even within any given system there will always be statements that cannot be proven by that system alone. Moreover, the way I see it, the reverberations do not stop there, in fact they are only further increased when Gödel reveals that the claims made in a system may not be able to be proven within that system, but they can be proven within a stronger, more complex system. I will take this one step further by arguing that this stronger system is in fact a priori derivable from the weaker system through the claims that the weaker makes concerning itself. In the end, this will be shown to be a Hegelian interpretation of a Mathematical system!

And yet, long before the Principia Mathematica (PM) and all related systems, the question of the usefulness of formalization had been posited. Leibniz had been working on a system of formalization by which all truth statements could be derived, much like the PM. Leibniz dreamed of a system by which symbolic writing could be used to examine the universal characteristics of truth claims. As such, Leibniz wished to abstract all form from content. He wished to rid argumentation from the fluidity of language, and instead focus on the fixity of an invariant signification of, ultimately arbitrarily chosen, symbols. In so doing so, Leibniz believed that he could indeed bypass all human contingencies and idiosyncrasies and look upon the ‘Truth’ that under-girds the conditional content of language by separating the content of language from the symbols of language. Thus, Leibniz was looking for a method by which Truth could be found via a truth-machine.

Like Gödel after him—but perhaps for different reasons—Hegel found the notion of formalization and its claim to find Truth to be absurd. Hegel understood the manipulation of pure symbols to be an external process, one that is not concerned with the Concept itself. The purpose of the Science of Logic is to give a systematic account of the Concept’s movement—the Concept being the movement of subjectivity which articulates the structure of objective reality. In this manner, Consciousness is the Concept itself. The Logic’s purpose, then, is to give a science, a cohesive and total body of knowledge which demonstrates the logical interlacing of the Concept in its movement; a kind of Science of Sciences, as it were.

As dealing with fixed symbols, the content of Mathematics is abstracted, external to the process of the autonomous movement of the Concept. As such, the relationship between the symbols is likewise an external relationship as well, and consequently an indifferent relation. For Hegel, mathematics is concerned only with quantity, the relation of indifferent externality among quanta. Consequently, the two quanta cannot be subsumed under one another. Their relation is forever external, resulting in what Hegel dubbed the spurious or bad infinity [die schlechte Unendlichkeit], an infinite line, instead of the true infinity [das wahrhafte Unendliche] where the Concept is able to return to itself through the related terms; a circle of circles, as it were.

In a very useful remark, entitled “Remark one: The Specific Nature of the Notion of the Mathematical Infinite,” Hegel explores the topic of the spurious infinity. Although this particular remark is dense in nature, in this paper I will select only those passages that are most applicable to this particular thesis. Hegel points out that mathematicians have not been able to justify the use of the infinite within mathematics in general. The correctness of the results, therefore, must rest within another system. The justification of the infinite lays outside of itself, in another. Presumably we could say the justification of the justification finds its justification likewise in another justification as well. “Ultimately, the justifications are based on the correctness of the results obtained with the aid of the said infinite, which correctness if proved on quite other ground: but the justifications are not based on the clarity of the subject matter and on the operation through which the results are obtained, for it is even admitted that the operation itself is incorrect.” Following this critique Hegel writes that the mathematical procedure is unscientific mostly because it is unaware that it has not “mastered the metaphysics and critique of the infinite,” as a result, much like we saw in the history section above Hegel points out that “Mathematics has to consider not what is true in itself but what is true in its own domain.” Taking what he found in Kant’s philosophy Hegel believes, mathematics relies upon the power of the understanding, which he distinguishes from activity of reason. Yet, Hegel takes this found distinction and gives it new meaning. The understanding and reason are not distinct powers, but rather levels of discourse of the same process, called thought; reason is not exterior to the understanding. Mathematics as understood as an activity of the understanding demarcates the limits it has set for itself through the observance of its own activity. Let’s look at the notion of the understanding more carefully.

The limit of the understanding, the law of identity, is its very power. Its function is to determine and differentiate thought. Its power, it seems, lies in its ability to make the different moments of thought possible, to separate the immediate from the mediate, the universal from the particular, and the particular from the singular. So the understanding is vital to thought and the movement of the Concept; yet, the law of identity is its limit. The understanding handles each of its categories as though they are as stable moments, each identical unto itself and isolated from each other. The power of the understanding, through itself, is abstract and finite. This is exactly what Hegel seems to be alluding to in his discussion since he states that the mathematical infinity is a magnitude of which there is no greater or lesser; this leads to the spurious infinity. In mathematics we are told magnitude is that which is defined as that which has the possibility for an indefinite amount of increase or decrease and, as such, it is an indifferent limit. To this extent, as infinite quantity, a mathematical infinite, is subject to a greater or lesser quantity. The absolute infinite, then, is something that cannot be completed. The static absoluteness cannot be what it claims to be, precisely because what it truly is always lies outside of itself; it is finite. This differs from the true, or metaphysical infinity because this type of infinite is not a quantum, it finds itself in the other, whereas a quantum is indifferent to its changes in magnitude, each lays outside of the next. As such the mathematical infinity is “burdened with a beyond which cannot be sublated, because to express as an amount that which rests on a qualitative determination is a lasting contradiction.” The dialectical movement, the immanent overcoming, of the moments of which the understanding demarcates and places into opposition to each other, cannot occur; their negative relation to one another cannot be realized, since they are understood to be isolated moments, never to be able to pass over into one another.

Yet, to be finite, the essence of finitude is to be able to sublate itself; hence “the truth of being is essence.” What is it to be? To be is to be measure, that is, it is to be a proportionality of quality and quantity; yet, at the same time, measure will reveal itself to be inherently indeterminate, through the understanding’s self-limitation; a substrate that is indifferent to its own self will be posited through itself—the indifference is endemic to its being. Nevertheless, to be a substrate requires that the substrate be self-related, and not indifferent. Consequently, the substrate must point beyond itself—positing some other ground—moving into essence, which will allow the substrate to be measured. Thus, essence allows being to explicitly be what it claims to be implicitly, i.e., to be a measured unity. Yet, mathematics will always see, according to Hegel, the substrate as isolated, the infinite is posited as something beyond the finite. Because the two moments’ unity is only implicit, each of the two determinations can alter to a certain extent without affecting the other. Quantitative determination can increase or decrease without the quality altering with it. And yet, because there is an implicit interdependency between these two aspects, quality and quantity limit each other. Quantity limits quality in how much a quantum can vary in magnitude before it undergoes a qualitative change, or a change in state. In this respect, quantity oversteps its own bounds and must change its quality. Likewise, quality limits quantity’s otherwise uninterrupted continuity, the continuous increase or decrease in the scale of magnitude. Any amount given to express the absolutely numerically infinite will always lack something; it must always posit something further, more complex than itself. An infinite series is expressed, and all because mathematics contains within itself a metaphysics that is always already beyond itself; consequently, mathematics can say nothing concerning what ought to qualify as a quantitative exteriority of any degree. Mathematics is infinite in the sense of being incomplete and therefore spurious. Hegel’s concept of the infinite, then, cannot be a potential infinite, which is endemically incomplete, but is rather an infinity that is complete in itself, our famous circle of circles, which is opposed to an infinite line. We now arrive at Hegel’s conception of the quantitative nature of number, or any symbolic language where the Concept lies outside of the symbols; it lacks the self-referential features that Hegelian logic demands. However I will now approach this in a more concrete manner.

We have seen above that Hegel wishes to say that the truth of being is essence; but what does this mean? The short version is as follows: Being is the most abstract class of thought, and as such there is no meaning without determination. We must be able to speak of it as a ‘this’ or as a ‘that’. This ability, however, requires that a limit be placed upon Being; we must be able to speak of what it is not, separating itself out from the rest of the world. We must be able to qualify what type of thing it is. We will see that the quality of a thing cannot change without changing likewise in its quantity, passing over into being-other and negating itself. There is a built in limit, a finitude in that which at first appears to be infinite. To illustrate this relationship between quality and quantity, let us use the example of water being heated or cooled. The alteration of the temperature of water is indifferent up to a certain point in its liquid state. There is, however, only so much of an increase or decrease in temperature that liquid water can withstand without changing its qualitative determination. Increase the temperature too much and liquid turns to steam, or if decreased too greatly, to ice. In this manner, the qualitative cohesion of the liquid water is shattered by the alteration in its quantitative determination; it is not an enduring substrate, unless we can understand it as changing into its other.

Hegel conceptualizes this reciprocal limiting of qualitative and quantitative determination in terms of a nodal line. This can be conceived of as a line with intermittent knots, the line representing the gradual quantitative continuity, while the knots signify qualitative “leaps,” interrupting the gradual quantitative continuity. The nodal line presupposes a substrate, a unity that is undergoing the qualitative and quantitative changes. A substrate is that which extends over the continuous and gradual quantitative change, while at the same time undergoes qualitative changes—and this must be so for if being is to be determinate, then, there must be a substrate which is unified in its alternating states of change. Quantity does not capriciously follow from quality; in its very self there is the autonomous urge to go out beyond itself. However, this urge does not come about from an external source, like the beyond of the Mathematical Infinite. Instead, the urge, the movement is autonomous, it arrives from the very nature of being finite. It must be other than it is.

As well as articulating the rules of Mathematics and the failings thereof, the Logic will also articulate the science of thinking, of its determinations and laws. Hegel believes this to be a comprehensive enunciation of the movement. This movement is pictorially represented as a circle of circles; each individual circle being the different ‘sciences’, or disciples, or movements that interlace to form the circle of the Concept itself. What is remarkable, among other things, about this image is that each of the sciences oversteps its limitations, moving into the other, what it is not, finding its completion in that into which it resolves itself. This is a useful representation of the movement of the Concept in general and what the Science of Logic is trying to convey. To define a concept, a science, or disciple, is to fix a meaning upon it, yet Hegel does not want to rest at fixity. Rather, he wishes to show the dialectical movement of these ‘fixed’ meanings. That is to say, Hegel wishes to take the ‘fixed’ meaning and move beyond it by thinking the meaning’s negation, its contrary nature. Moreover, Hegel, then, through reflectivity, shows how the whole movement can be grasped as a totality; the two moments, the fixed meaning and its negation, as conjoined, creating a third moment. Even though others have thought dialectically before Hegel, he believes that these thinkers have thought each moment as isolated unto itself, whereas Hegel views them as three moments of Thinking. Thinking, consequently, is fluid. The definition of anything will, through its own necessity, reveal its negation. This necessity moves us to see the totality within the system. If each moment necessarily moves us to its negation and through this negation to a third moment, then every thing must be able to be resolved into everything else—hence, totalization.

Let us briefly look deeper into the notion of totalization and how this is related to the different types of infinities. Dasein is the process of becoming, it is that which underlies and remains in the persistent dialectical flow. Dasein has being but is not complete being, it has deficiencies, and it, therefore, contains negation within itself. In containing negation endemically, Dasein will be seen to be inherently self-contradictory. This will force the concept of Dasein to point beyond itself, to necessarily posit another. This is Hegel’s way of trying to think of the Absolute as an internal reflection, it is the movement of the finite into the infinite, through an internal and endemic progression. We must be careful, however, in how we are to interpret this movement to infinity. It is not an endless progression, a progression that results in an incomplete enunciation of the Concept. We cannot hold the finite on one side and the infinite on the other side of an insurmountable divide. To do so would, in fact, be to make the infinite finite, abstracting the finite out of the infinite. Infinity is not beyond the finite. We have seen that the finite negates itself, through its own endemic movement, resulting in the infinite, and vice versa. This will allow the understanding access to solving contradiction. Every individual concept will posit its other, however in doing so it is pass over into what it truly is, i.e., its other. This is the movement of the true or genuine infinity.

To put it still another way. Dasein has a deficiency, a limit; and it is on account of this limit that determines what it is. However, Dasein excludes its deficiency from itself, that is to say, it demarcates itself from other things around it; Dasein becomes Something. Something is distinct from what it is not. Something has an identity in and through this relation with the other. This suggests that negation plays a constitutive role in something’s identity. Something, then, is mediated through negation. Something points beyond itself, in its limit it points to the other. Through a limit something is what it is. Something has its Dasein in its limit. Since the limit has a constitutive role in something, the limit acts as an ought; it is now a limitation. The ought, then, is part of something’s ontological structure. As such, something must move beyond its limitation, it ought to become other. Something or, in keeping our eyes to the greater project, the Concept limits itself. Its truth lies outside of itself, therefore, sublating its own limitation. The ought is built out of the relational structure. However, if the limit defines something and the limit is found through the other, the other defines something. There is no positive self for something; it is only a negative relation with the other. Something is itself in the other and it must overcome this otherness to become what it is. To be finite, then, is to be dynamic, it ought to be more than what it is. Something is always already other than it is. To think in terms of finitude is to think of becoming. The moment that a finite something comes into being it passes away, something is infected with negation from within. Therefore, in the relationship between something and the other, the other changes into something in an intrinsic manner through the something’s ought-to-be. Since it is intrinsic, this is self-relation with otherness.

We can relate this to language and cognition in the following manner, which will help to demonstrate Totality. The truth of the Hegelian Concept, as conceptualized in the syllogism, is three moments: Universality, Particularity, and Singularity. These moments are brought up in the second book of the Science of Logic, the ‘subjective logic’, within the chapter on the syllogism. Here in the subjective logic, how the content and the form are held together is described and laid out—that is to say, the movement of the Concept qua consciousness is given. We have seen that the system is not static or ossified. We must briefly consider judgment and the deficiency of the copula to make sense of the syllogism, and why the universal concrete is the true ‘word’ of the Concept.

Judging is the positing of the concepts; it will bring the content of concepts into objective existence, it will give them form. The copula’s deficiency lies in its functioning much like the understanding. The copula holds the subject and the predicate as distinctive things that bare only some extrinsic relation. For example, ‘The ball is red’. ‘Redness’ and ‘ballness’ can be understood as distinct entities holding some relation in that one is merely predicated of the other. The relation is external and not necessary. In judging and in the syllogism, there is the immanent judging of the Concept by the Concept itself. Judging will be seen not to be completely subjective in the sense of being related only to the individual, abstracted from the content, since there will be the relating of two concepts together through the middle term, which will take the place of the copula.

Due to its abstract nature, however, judgment must be replaced by the syllogism and its middle term, which will replace the function of the abstract copula. The middle term is mediation itself. The syllogism of necessity uses universality as its middle term; through it, particularity and individuality are thought together. This universality, however, must be distinctive and objective and neither qualitative nor quantitative in character. Qualitative universality has the property of inherence, a set of objects share a property. Each of the objects is distinct and yet contingently share some common property, something like a red ball, a red crayon, a red car, and a big red dog. None of these objects have anything real in common other than each is red, ‘red’ is a property among many others that just happens to be common among them all; it is contingently among them. Whereas in quantitative universality we do not have inherence but rather an essence itself, each of the distinct objects are subsumed under the weight of this type of universality. There is no particularity, and as such one cannot reach the movement of the Concept. This is what happens in the symbolization of language. We are left only with individualization. There is only one individual, representing all other possible particulars. Yet these particulars are only possibilities, since they have been subsumed under a universal that is abstract, since no particular is expressed under it. There is a ridged fixity. There is only one individual, one invariant significant term; the Concept’s movement is rendered impotent. The concrete universal, however, is able to hold the particular and the individual together, thinking them within a movement. A movement occurs since the identity between particularity and individuality is not dead, one understands how the particular is expressed within the individual object while each individual is nevertheless thought under a universal. The universal is thought as a concrete universal, since it is expressed as an individual object. We are able to have discourse concerning a particular thought.

What is important in the above discussion for our purpose in this paper is to point out that the above movement makes possible knowledge and the discourse of that knowledge. And not only knowledge of a limited sort, but a knowledge of the Concept in its entirety; hence, truly giving us something like a science of sciences, or a ‘Science of Logic’. Hegel believes that he is able to accomplish this, in part, because of the movement of the finite into the infinite. In the Phenomenology of Spirit, Hegel reveals the movement of the finite into the infinite in terms of human consciousness becoming aware of not only itself, but when it is able to have a philosophical discourse on Itself—when human consciousness is able to understand itself through the mediation of the other. In so doing, human consciousness moves to knowledge of the Absolute Concept, the Concept in its entirety. Yet, if this is to occur there can be no ineffable piece of knowledge, no complete Otherness, no absolutely beyond, as this would make impotent the dialectical movement of self into otherness. In a very real sense Hegel is championing discourse. By positing an ineffable beyond, one cuts short the possibility of a totalized knowledge. One would have to take recourse in Faith: Knowledge would not be able to overcome the structure of experience as it is considered by the understanding and which is already implicit reflection. But, thanks to explicit reflection, knowledge discovers its own finitude. It is therefore only capable of negating itself and of allowing faith to overcome this knowledge. The Absolute then is the object of faith and not of a knowledge.

Knowledge, thought of in the manner of faith, in and of itself can only be formal, grasping no content. It can describe how ‘ought-statements’ are put together, but what the statements mean is entirely out of its realm. Yet, faith allows for an immediate apprehension of a content that human consciousness cannot comprehend. Consequently, faith forces one into silence, into a world that lies outside the communication of mere knowledge. Faith compels us into a world that is ‘higher’ than discourse, and therefore ‘higher’ than knowledge. And since knowledge requires that the immediate is mediated by its other, knowledge cannot arise, and there is an unreachable beyond, an ought-to-be which the immediate can never live up.

What does this talk of language and the self-relatedness of the Concept and subjectivity have to do with our present topic of mathematics and dialectics? Everything as we will shortly see. The purpose of philosophical language is to trace the movement of Concept. We saw that for subjectivity to move along with itself it unfolds into itself, i.e., into objective reality. Language is the medium through which the dialectic becomes visible. Language is the signification by which the movement of dialectics is shown. The particular signs that are used for the language are intimately connected with the logic of the dialectic. We can say, then, that the logical dialectic is the dialectic of Being itself. However, if language itself is merely a sign or it is composed of contentless symbols, language negates the sensible, the content that is being expressed via the sign of language. Language, according to this way of thought, is form through and through. And if we ignore the interrelatedness of the sign with content, or rather return to this stage, form and content are completely divorced: The return to a symbolic writing, like Leibniz’s dream of a universal characteristic, is not only utopian but also absurd according to Hegel, because the progress of thought continually changes the nature and the relation of the objects of thought. One would constantly need new symbols corresponding to new discoveries and to new relations of thought…It pushes the negation of the sensible to the limit and takes into consideration only the expression of thought in language, as if signification could be an interiority without exteriority.

Hegelian logic is diametrically opposed to such a formalism, to a ‘mathematizing’ of movement of the logic. Quantity by its very nature is external, or indifferent to its object. The merit of symbolism is that it can avoid the confusion and ambiguity that everyday language can bring with it. The drawback, however, lies in the ‘I’ of the understanding forming the signification of the symbols itself. This ‘I’ finds the symbols dead and lifeless, isolated moments, something that would not occur if the ‘I’ already found the content present in the symbols, lying endemically there. The ‘meaning’ given to the concepts are merely external, they do not reveal the Concept in its movement. The ‘I’ isolates itself from the Concept, forming its own world, unable to communicate with the real Concept. The ‘I’ is unable to form the concrete universal that is the true word of the Hegelian discourse, that which shows the Concept in its totality.

The discussion of language and judgment is important in that it not only demonstrates that the movement of the Concept is autonomous, but also, and consequently upon this autonomy, a conscious-being must be present for the unfolding of the movement, i.e., Consciousness itself. We will see that one of Paterson’s critiques of Gödel’s incompleteness theorems is essentially that all mathematical dialectic can be checked through a computer-like machine, a branch of symbol sequences (well-formed-formulas [wff’s]). This is understandable since if the symbols of language truly are inherently empty and devoid of meaning then a computer could in fact run through all of the symbols checking simply for external consistency. In a sense the program could only check what is already present to see if it is consistent with what it already states. The human-mind would be reducible to a computer. But Hegel argues for a much more robust conception of Consciousness, it is living, breathing, capable of dialectic. Without the intuitive, autonomous movement of the Concept, the understanding would not, as we have seen, know what to even formalize in the first place. Unless meaning is inherent within the symbols themselves, the understanding can only view each moment as isolated and lifeless. However, I will show that where Paterson critiques Gödel is precisely where Gödel shines through and reveals the living, breathing, autonomous movement of his Concept.

Is it true, then, that symbolization cannot attain the movement of the Concept? Are we doomed to abstract speech when we use symbols, in the words of Russell’s epigram, “Pure mathematics is the subject in which we do not know what we are talking about, or whether what we are saying is true”? We are certainly doomed to this scenario if symbolism does not and cannot capture the movement of the Concept.

Gödel, like Hegel before him, understood that there were problems with formalization and attempted to reveal the shortcomings of it via thinking formalization to its conclusion. Both great thinkers understood that through strict formalization one could not glimpse the totality of the Concept as the system claims; Gödel’s thought, however, differs from Hegel’s in that Gödel does not believe that Totality can be thought at all—thus Hegel would believe the former to fall victim to the spurious infinity—although through my interpretation the movement of the Concept is nevertheless expressed. We have just seen that Hegel’s critique of symbolism rests in the fact that it does not capture the movement of the Concept but is instead an external and stagnant reflection of such a movement. This occurs essentially since mathematics and philosophy are thought to be fundamentally different projects. Mathematics functions through the use of axioms, or assumptions that are already thought to be true. Mathematics qua mathematics is not concerned with the Truth of these assumptions per se, but rather what is the consequence of these assumptions whether true or not, hence the explosion of theoretical mathematics systems. Philosophy, on the other hand, judges these assumptions; it tries to justify the assumptions, discovering their truth-value. While it has been mentioned in literature that Gödel was reflecting philosophically on mathematical principles, thus judging the truth values of them, through “self-referential character of its concepts,” it has also been noted that Gödel is unable to investigate the logic of the system. So, according to this view, while Gödel does at least reflect back upon the system judging it, the judging cannot be anything but external and alien to the Concept. In the following remaining pages, I will now present an interpretation of Gödel’s Incompleteness Theorems, which I hope to show, at least prima facie, that a dialectical mathematics is at least possible, a mathematics that will investigate the logic of the system. Through an entirely unique interpretation of the second Incompleteness Theorem, I will endeavor to reveal that an internal and inherent dialectic is possible within mathematics—that is to say, that Gödel truly thought of the movement of the Concept; there is being, nothing, becoming, through mathematics.

Gödel discovered an ingenious method of coding the rules of procedures for a formal system, which is demonstrated in the following note. Now, after discovering this coding scheme, Gödel then constructed a sentence out of these symbols. This sentence is as follows, G= ¬(y)Dem[y, sub(f, 11, f,)], stating of itself that “there is no Peano proof-number, the exponent of the largest prime factor of which is the number-G.” The number-G is an increasingly expanding number, choose the largest number one can and then add one more and to this number add one and so on and so on. In other words, there is no schema upon which we can map the number-G, outstripping itself. Hence, what the Gödel sentence states of itself is in effect, “Nothing proves this sentence.” And thus, we arrive at the paradoxical and ambiguous status of P.

G is said to be “undecidable,” i.e., neither its truth nor its contrary can be proven in a finite number of steps. And yet, even though G is undecidable, the sentence which it names must nevertheless be true. For if it is false, G would say that there is a proof. But if there is a proof of G we could prove a false statement within our system, giving rise to absurdity and incoherency. For this reason, G must be true, but cannot be proven so! Certainly at the immediate beginning, we have here the spurious infinite. The Gödel sentence must, by the way it has defined itself, posit another outside of itself, above and beyond itself, which it should not be able to do if form and content are indeed separate. Gödel seems to have painted himself in a corner from the very start. The genuine infinity cannot be expressed, as we have seen, in an endless series of meaningless symbols, nor statements made up of these symbols. Let us not be too hasty in judging Gödel though, this is the immediate stage; it is like beginning with immediate Being at the start of the Logic; we must wait to see what unfolds from its own autonomous movement.

Alan L. T. Paterson, the only commentator that I could find who explicitly links Hegel with Gödel, states that one of the problems with Mathematics, as conceived by Hegel, is that it ultimately says nothing interesting. “Why, in mathematics, should a certain result be seen as of fundamental importance while another is not. What makes one mathematical result (such as that there are infinitely many primes) interesting and even thrilling, while another—e.g., a huge tautology—is boring?” There is a kind of arbitrariness ascribed to mathematical discourse in that mathematics cannot say what result is philosophically more interesting, or more correct. There is no value laden insight available for the mathematician. When given two results a mathematician will not be able to judge which is ontologically more correct, only that both result from the axioms provided from the start; this can be shown primarily in the interest in theoretical mathematics. There should be an infinite number of theoretical geometries; for example, different axiomatic systems can be given for the same mathematical system, or the undecidable nature of the Gödel sentence. However, mathematics qua mathematics, according to Hegel and Paterson, will not be able to reveal to the mathematician which is more true, only that both result from the axioms. There is no intrinsic return within the system to its beginnings; there is no real self-referentiality—whereas in Hegel there is implicit within it, a self-referential character.

Neither Paterson nor Hegel should be chastised for such a statement, especially since it would seem that Gödel, himself, cannot make even the simplest of claims, since G is undecidable. We seem to have fallen victim not only to the spurious infinity, but also to the Banality of Inconsistency! There is no end to the ‘why game’. Mathematics, according to Paterson and Hegel, cannot give us the positive dialectic of Hegel’s, but only the negative sort; it cannot tell us that being and nothing are not identical. Becoming is left out of the equation, as it were. Where Paterson sees this inadequacy in Gödel is precisely where I see Gödel’s possibility of a discussion of a real movement of the Concept; this lies in the second incompleteness theorem.

Let us now look to the theorems themselves, particularly to the second where we will find the most fruitful place for discussion. The first Incompleteness Theorems states that the Concept is such that it cannot be proven directly—indeed if one attempts to prove it so one can only be lead into a contradictory position. Notice in the following note that a direct proof of G will yield an immediate contradiction; by directly demonstrating G we must end in proving ¬G. By attempting to directly prove the infinite nature of G we are forced to undo that very infinite nature. Through a direct proof, i.e., through a priori deduction, one must already assume that one has a complete understand of that concept. To claim that P entails G, this ever expanding number, we must be committed to the belief that our system of rule, P, which, if it is to be meaningful, has to be finite in relation to finite beings. By stating that G is derivable from P, then, is to say that our system of rules contains the absolutely infinite, and meaningfully so. In other words, we must be committed to stating that finite-cognitive-beings have created this absolutely infinite system. We turn the infinite upside-down; we make the infinite something that is dependent upon finite beings.

With these conclusions, Gödel has accomplished to set the limit. What Gödel has accomplished the setting of the limit and the nature of human reason. He has revealed, among many other interesting facts, that a finite mind cannot make statements that are applicable to the entirety of Mathematics. Not only is Peano incomplete with reference to a finite-cognitive-being, but it is even more unique, for it is incompleteable by us; it is what has been dubbed ‘essentially incomplete’. To have an idea of G, one must admit there will always be something that is improvable within the present system. Nagel and Newman do a wonderful job of illustrating why G is profoundly incomplete on page 103. Even if one were to take G as a new axiom, it still be impossible to claim that our new amplified system will prove all truth claims, since by taking G as a new axiom it will yield Dem'. Now, Dem' would need its own conclusion number satisfied by a proof number yielding P', giving us the choice between G' and ¬G' building the system upwards ad infinitum. And although our system is becoming more complex with every step we take, the system can, nevertheless, be augmented further. There can be, then, no possible structure or boundary in which we are able to place G, and as such we are not able, as finite beings, to place arbitrary boundaries upon it.

It is obvious that Hegel would claim that this is just dogmatism wrapped up in a nice packaging. How are we to know that such an infinite exists if we cannot ‘prove’ it. Gödel seems guilty of positing a transcendent world over and beyond the world that we are familiar with here and know most immediately. Being unable to ‘prove’ such a standard, or transcendent world, naturally leads one into a spurious infinity, since it forces us into silence due to faith. Within an implicit self-referential character, the dialectic can only be negative, telling us what we do not know, but never revealing what we do. Paterson writes that this is the result of the second Incompleteness Theorem.

Individuals are tempted to think of the consequences of Gödel’s Theorems as purely negative, revealing only the limitations of any formal system. We are lulled into a descent of madness, believing that inconsistency is ontologically real, rambling neurotically. Yet, Gödel did not design these theorems for such a destructive purpose. And I claim that these Theorems can be used to ascend to greatness and to further knowledge and wisdom. The secret lies in the second half of the first Incompleteness Theorem. However, before we arrive at this destination, let us look more closely at what is being claimed concerning mathematics and its inability to justify its own claims. Mathematics fails precisely because the intuitive and ‘living’ Concept—the movement—is separated by an impenetrable gap from the purely external and mechanical ‘pushing’ around meaningless symbols, the correctness of which could be checked by a machine. The Hegelian dialectic certainly cannot be checked by a machine, what is expressed in Judgment does not allow for an external reflection, but rather an understanding and insight into the movement of the Concept. A computer, in other words, could not follow Hegelian dialectic since it is concerned only with meaningless symbols; without the intuition involved in philosophical discussion, one would not know what to formalize let alone know the meaning of that which one does formalize, but only whether the formalized symbols bare some external relation.

It is no wonder then that a bad infinity would be thought to ensue in Gödel. Reflecting back upon itself the system cannot judge itself through itself, since it is essentially meaningless. Paterson takes this in reference to Gödel’s second Incompleteness Theorem. This theorem states that no formal system has the ability to prove its own consistency, that is, within the realm of its own resources but requires another through which it has no legitimacy to posit. In the following note, the second Incompleteness Theorem is presented. Here it is revealed that neither the consistency of Peano [Con(P)] nor its inconsistency [¬Con(P)] are decidable within Peano. With this proof, then, it would seem that we could choose whichever branch we want to set down upon with equal validity. Hence, we enslave Reason to our whims. The Banality of Inconsistency is seemingly thrown upon us with great force. It does indeed seem that we can decide what to believe and when to believe it. One could claim that because G and Con(P) are undecidable we could arguably interpret G as representing an ambiguity in our conception of mathematics, and since each branch is seemingly as viable an option as the other we are at liberty to dictate in which direction to move. Mathematics seems impotent as a means of directing us towards the ‘true’, meaningful branch of inquiry to tell us which branch is the ‘true’, meaningful branch.

However, it can be noted that even though G is an ambiguity and while neither Con(P) nor ¬Con(P) is the ‘true’ branch, one can look to the theorems and say, as Nagel and Newman do: “To have an idea of G, one must admit there will always be something that is improvable within the present system. For example, even if one were to take G as a new axiom, it still be impossible to claim that our new amplified system will prove all truth claims, since by taking G as a new axiom it will yield Dem'. Now, Dem' would need its own conclusion number satisfied by a proof number yielding P', giving us the choice between G' and ¬G' building the system upwards ad infinitum.” And although our system is becoming more complex with every step we take, ‘proving’ the previous step, the system can be, nevertheless, further augmented?. And because the system is embedded within itself, a spurious infinity could be interpreted. As Paterson writes, The bad infinite shows itself through the theorem inasmuch as we might try to deal with the problem by embedding the original formal system F within a larger one F2 relative to which we could determine the consistency of F….But then, of course, this only pushes the consistency issue over to F2 which in turn requires a larger formal system F3 to establish its consistency, and this is a bad infinity in the sense of Hegel, the sequence of formal systems never stopping.

While it is a true feature of the Gödelian system to continue on into infinity, it is not true, however, that a computer, or mindless zombie could correctly chose and calculate the next step in the dialectic; nor is it true that the ambiguity of which branch to take cannot be solved—undercutting both of Paterson’s critiques. And if we can show that the claims are undercut, it might be demonstrated that Gödel does indeed reach the movement of the Concept. I have focused primarily on Paterson’s claim that a system such as Gödel’s cannot tell us which branch to take since if the opposite can be demonstrated the breathing, living nature of Gödel’s system, at least prima fascia, can be revealed. If one can show that Gödel’s system is not ossified and yet does indeed end in an infinity—but one which will have to be different than the die schlechte Unendlichkeit—it will at least point us in the direction and lay a foundation for a fuller explication of a dialectical mathematics.

I will now attempt to layout a scaffolding as to what such system would look like. One could claim that because G and Con(P) are undecidable we could arguably interpret G as representing an ambiguity in our conception of mathematics, and since each branch is seemingly as viable an option as the other we are at liberty to dictate in which direction to move. However, there is a feature brought about through the second half of the first Incompleteness Theorem that will help us to determine which branch of reasoning to take. Here, the possibility of G’s non-existence is examined. Moreover, it ends in the weird and wonderful world of the ω-inconsistent. The ω-inconsistent is philosophically interesting in that it states its own inconsistency and yet acts as if it were true, confounding those who try to find its inner workings. If the reader will draw his attention to the applicable footnote, the beginning of the proof for ¬G states that there is a number of contradiction, i.e., that there is a proof-number for G; in other words, that G can be placed inside of a proof. Let us return now to the ambiguity of G. We are left with the choice of both P+Con(P), ‘P is consistent’, and P+¬Con(P), ‘P is inconsistent’, what will be dubbed respectively, P+ and P-. To decide which path to take we must keep in mind that while G is not provable within P, it is provable within P+. But why choose P+ over P-? Can Paterson be correct in arguing that Mathematics cannot give us an interesting, meaningful answer?

Imagine we are given a system for establishing mathematical statements, P. P is viewed as “ambiguous” given its P+ and P- extensions. G is to be understood as an “ambiguity” in our notions of mathematics (addition and multiplication) over which we are supposed to have dominion to choose which direction P+ or P- to take, since, by hypothesis, either is equally consistent provided that P is consistent. Further imagine that we are to accept that the rules of procedure for P are true and trustworthy in the sense that if we should perform a deduction within P, P will provide us with irrefutable demonstrations of these statements; for example whether P+ or P- is true. (It must also be noted that P+ and P- are explicitly represented as lying uphill, informationally speaking, from P alone, and while G is not provable in P it is provable in P+). Note also what ¬Con (P) says: (Ex) Dem(x, c), further note that we can perform an EI to introduce a new constant ‘k’ for this ‘x’: Dem(k, c). There is a strange peculiarity we must examine next. P- seems to state, that is, it seems to assert that, “P is true, yet inconsistent at the same time;” also if P itself is formally consistent so is P-…strange indeed (which would be strange indeed! Additionally, if ‘k’ were to name an ordinary number, i.e., a finite natural number, that number would decode in a finite proof of contradiction within P itself. Thus, we must, without any reservation, assert the unavoidable conclusion that in any model of P- the object named by ‘k’ cannot be an ordinary finite, nameable number—not 0, 1, 2, 3, 4,…n, as this number would have to decode into a finite proof of contradiction within P—but rather some kind of strange ‘infinite-number’. To put it another way, while it is possible for P- to be true—thus the ambiguity arises between the P+ and P- branches—P- can only be true under a non-standard interpretation. We, thus, arrive at the unavoidable conclusion that any object named by P- cannot be an ordinary nameable number and that the proof it names must be a non-standard proof. The proof cannot be ‘true’ in the standard sense of the word. The ambiguity of G thus arises simply because we are deceived by the imitation that P- represents. Therefore, we do have reason, a principled and a priori reason, for choosing P+ over P- as the correct theory of N; namely that the latter is only true under non-standard models, and hence the standard model must lie up along the P+ branch.

To further explicate this difficult discussion, P- is something I will dub a ‘meta-inconsistency,’ or what could be understood as a ‘pseudo-inconsistency.’ A meta-inconsistency is not formally inconsistent, but is inconsistent when understood in terms of something that it transcendentally presupposes, i.e., within the background context under which and within which it makes sense, e.g., the assumption that P+ is consistent. While any statement that is P- seems to be coherent, it must only be understood to be so as coherently in a non-standard manner. In other words, while P- is indeed consistent within itself, it is inconsistent with something that it transcendentally presupposes, within the background context under which and within which it makes sense, i.e., P+. We do, then, indeed have a priori grounds for choosing Con(P) over ¬Con(P). While P- may indeed be consistent, it can only be true in a non-standard manner, i.e., what passes for a real finite proof. ¬Con(P), then, has a pseudo-consistency. So, unless one can appreciate the model’s non-standard nature, through a reductio argument, one will forever remain trapped in the infinitely standing pairs of contradictories. The only way to appreciate their beguiling essence, though, is to admit of a standard model that will act as a standard of truth does exist, e.g., P+.

However, to even assume that P+ is true still presents us with an apparent ambiguity, by theorem one; even if we take P+ to be true we are presented with both P++ and P+-. Oh the humanity! We must find a way to shortcut the meta-inconsistency found in the P+- branch. Thus to prohibit the occurrence of the meta-inconsistency, we must assume a stronger form of consistency, what Gödel termed ω-consistency. For a system to be ω-consistent it cannot contain any statement such that it is both provable in Con(P) and ¬Con(P), that is to say it cannot be inconsistent. Hence, the conjunction of an ω-consistent system with the provability of ¬Con(P) will very much yield a contradiction, destroying our system. Thus if P is ω-consistent, all that is equivalent to the ¬Con(P) branch must be improvable within P. Our a priori rationale for choosing the consistent branch in lieu of the ω-inconsistent can now be justified. That is to say, we can now reason that P is consistent and nowhere ω-inconsistent, and therefore nowhere inconsistent. Consider the following: If P was inconsistent (and all that is inconsistent is meta-inconsistent) all statements made within P would be provable. Yet, if all statements were provable, the conditions for the ω-consistent could not arise, since ¬Con(P) is improvable within an ω-consistent system. Thus, any system that is ω-consistent is consistent in a cognitively finite meaningful manner, that is to say, it is describing objects that are real given our finite-cognitive abilities. We must, therefore, admit that we are held to the standard of the ω-consistent. If we are not willing to consent to this truth, we must admit that nothing is cognitively meaningful, since all would be ambiguous at best and we could only discuss objects that are out of our cognitive reach. We would be stuck in the spurious infinity, but we are not since we have a very real a priori reason to choose P+. Nonetheless, we must still keep in mind that while at every step along the way to the ‘plus-branch’ there will always be the correct one, even P+++++++…, no matter how far we proceed upwards it is incomplete. We move upwards in a dialectical sequence into systems that are forever increasing in complexity.

Why all of this is important and relevant to the present critique of Hegel’s notion of the Mathematical Infinite and whether Totality is possible will be shown next. Given Paterson’s and Lacroix’s assessment both what Hegel is doing and, in the case of Paterson what Gödel is doing, and if my interpretation of the transcendental argument in which Gödel is engaging is correct, we must see that a Mathematical Infinity can and does occur without being spurious in the sense of neurotically stating the same thing over and over again, but rather in fact does incorporate and sublate [aufgehoben] the next step ‘upwards’ in the dialectic; we have being, nothing, and becoming. To help illustrate this, let’s again recap what is occurring in Gödel’s dialectic and try to use Hegelian language to explain it. The best way to approach this in the present context is to examine it in light of the mathematical critique. It was said that the spurious infinity arises because there is no inherent limit within mathematics. This is because the understanding, being the moment of differentiation in thought, is constrained by the principle of identity. This differentiation leads thought via the understanding to consider each of the entities set before it as isolated moments—islands unto themselves. Thus the any two entities set before the understand’s gaze cannot be reconciled, brought back into a unity; there is no aufgehoben. Thus there is no immanent overcoming, no dialectical movement. I take this to be the equivalent to the inability, or unwillingness to appreciate the differences between the apparent consistencies in the meta-inconsistent branch and the ω-consistent branch. Without this appreciation any movement is a running in place, a mere stuttering of the same sounds, on and on, illustrated by Hegel as an infinite line. We are not only caught as to which branch to take, but we cannot even move upwards, we are simply caught uttering the same thing, “consistent, inconsistent.”

But if we have appreciation for the ω-consistent, and its ramifications, then there is an endemic self-negation within Gödel’s system. The understanding must realize that the two branches negate each other and inherently move us upwards toward a stronger system. There is an inherent contradiction within the two entities; thus instead of the understanding viewing them as two distinct constituent moments, the understanding is able to see the two as representing each in the other; quality in quantity and quantity in quality, they are determined by each other. There certainly seems to be in Gödel’s mathematics the spurious infinity. There is always an ungraspable ‘ought-to-be’, the theorems describe Incompleteness after all. But that is not entirely the question at hand, however. The question at hand is whether contradiction is an inherent, internal, autonomous, pregnant, burgeoning piece of the movement described in Gödel’s dialectic. Is there aufgehoben? At least how I have presented it, it would certainly seem to be the case that there is such an autonomous contradiction. Under Paterson’s critique that Gödel cannot tell us which branch of G to take, either P+ or P- is an understandable misconception given what Gödel gives us to work with. However, as can be seen above, mathematics, at least under my interpretation of Gödel’s theorems, does have the ability inherent within itself to reveal what is philosophically more interesting, as it were, e.g., P+ is correct given P-‘s non-standard nature. If ‘telling’ us which branch is more philosophically interesting is the key to an autonomous movement, which it must be if the form and content are inherently linked and the language is guided by the Concept, then Gödel’s dialectic contains within it an intrinsic and autonomous contradiction. Moreover, ‘choosing’ P+ over P- is a priori, meaning that Consciousness itself is at work within the movement of this Concept and not just an arbitrary preference on one’s own part; one is drawn by the movement to the next step.

Yet, one might still argue that a true dialect does not exist as there is only asymmetry between the two branches. The P- seems to depend only upon the P+, there does not seem to be the dependency upon the two that Hegel demands for the dialectic to occur. However, if we look closely we will notice that there is a kind of strange symmetry; for unless we knew that the P- was consistent only under a non-standard interpretations could we be made aware of the standard and true interpretation of the P+ branch. Thus, we understand how we move from one moment to the next in Gödel’s dialectic. We begin with P, which is ambiguous between its two extensions and G is the ambiguity, and we must look to both branches in order to decide that the P+ is the ‘correct’ choice. So we have the immediate, the mediated and becoming. Moreover, all of this is presented to us strictly through mathematical symbolization. The symbols themselves compel one to make the choices between the branches, content is necessarily contained within them.

What has been offered in this paper is a presentation of Caveillès’ idea that a mathematical dialectic exists: “There is no consciousness which generates its products or is simply immanent to them. In each instance it dwells in the immediacy of the idea, lost in it and losing itself with it, binding itself to other consciousnesses (which one would be tempted to call other moments of consciousness) only through the internal bonds of the ideas to which these belong. The progress is material or between singular essences, and its driving force is the need to surpass each of them.”

As I have presented Gödel’s system, Consciousness is certainly lost within itself, within other moments of consciousness that force one to surpass the present level of consciousness with which one is confronted. Yet, we must grant Hyppolite’s critique that such a system is paradoxical, but does this suggest that we ought to discount it? No, we must persevere through the paradox and try to think through what it means to have a dialectic that describes the movement of the Concept, and yet at the same time is incomplete but not subject to Hegel’s spurious infinity. This paper is merely the tip of the iceberg in the grand scheme of thinking through this paradox. Bibliography Hegel, G. W. F. The Encyclopaedia Logic. Trans. T. F. Geraets, W. A. Suchting, and H. S. Harris. Indianapolis and Cambridge: Hackett Publishing, 1991.

_____. The Science of Logic. Trans. A. V. Miller. Ed. H. D. Lewis. New York: Humanity Books, 1969.

Hyppolite, Jean. Logic and Existence. Trans. Leonard Lawlor and Amit Sen. New York: State University of New York Press, 1997.

Lacroix, Alain. The Mathematical Infinite in Gödel. New York: Baruch College of the City University of New York. The Philosophical Forum, vol. XXXI, No. 3-4, Fall-Winter 2000, pp. 296-325.

Nagel, Ernest and Newman, James R. Gödel’s Proof. New York and London: New York University Press, 2001.

Paterson, Alan L. T. Does Hegel Have Anything to Say to Modern Mathematical Philosophy? Massachusetts: Clark University. Idealistic Studies, vol. 32, No. 2, Summer 2002, pp. 143-158.

_____. Towards a Hegelian Philosophy of Mathematics. Massachusetts: Clark University. Idealistic Studies, vol. 27, No. 1-2, Winter-Spring 1997, pp. 1-10.