No More Learning

? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? chapter iv
Infinite and Distinction
? ? We just showed that the so much praised necessary character of the 'objective universe' as opposed and confronted to the subject, is a deceit because it lacks all kind of meaning. Now, this overwhelming intention and effect has affirmed the infinite charac- ter of the physical world, and I believe that when Pascal was frightened because of 'the eternal silence of these infinite spaces', his fear was de- rived principally from the infinite.
The great public is still shocked by the idea of a space which has been declared for more than three thousand years as infinite. Although the majority of actual physics says that it does not believe any more in the spatial infinity of the cosmos, infinitude is reintroduced in their dis- course under the form of the infinitesimal (infinitely small). In addition, must physics still believe in the temporal infinitude of the universe: in other words, they believe that time extends to the past and to the future infinitely, which amounts to keep extolling that the space is infinite for that is the concept of time that physics have. They do this as if the word infinite could have any meaning at all in its application the physical.
As it is obvious, we must add that every particularity involves limi- tation and consequently carries negation to universality. The subject of the infinitude is identical with the subject of universality. No sci- ence can do without this subject, let alone logic. It is unsettling and
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outrageous how one believes that everybody understands what the word 'universal' means. Another disconcerting belief is that it is very easy to give meaning to it by empirical means.
Following Hegel we will show here that, if it is not in terms of self- determination and morality of the spirit, the words 'infinite' and 'uni- versal' lack all kind of meaning.
On the other hand, a true definition of infinite brings a capital prob- lem to philosophy and theology, namely, the distinction between the infinite and the other beings. The illusion that consists in believing that the word "infinite" has some physical meaning besieges theologians too, since they try to distinguish God from the human spirit by means of this term --God and man are identical at least in regard of their spiritual- ity--, as if infinitude consisted in something different than spirituality itself. The definitive clarification that Hegel provided has not been un- derstood by his commentators, despite that it is one of the most revolu- tionary and irrefutable contributions that a thinker has ever made.
1. pSeudoinfinite
"The infinitely great and infinitely small are therefore pictorial concep- tions which, when Lockheed at more closely, turn out to be nebulous shadowy nullities. " (WL. I 236).
These expressions are breakouts towards the irrational. One can be- gin to see this with those who define infinite as 'bigger as anything that we could possible think'. That means, we cannot think; one grants us permission to abandon rationality in the wings of imagination; we are authorized to proceed without concept.
It is time to proclaim to the entire world that the infinite has nothing to do with a big size. In general, it does not have to do with magnitude either. Something is infinitely big when it cannot be bigger, but the definition of magnitude is: that which can be bigger or smaller. There- fore, it is a contradiction to talk about an infinite magnitude. If it were infinite, it would be no magnitude.
Likewise, something infinitely small (= infinitesimal) is something that cannot be smaller. But by definitions a magnitude is what can be bigger or smaller. Therefore, an infinitely small magnitude is a contra- diction. It is nothing more than fog and shadows. Hegel warns us: "this is a necessary and direct consequence" (WL. I 242).
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At this point, mathematicians make a drastic turn and define infini- tesimal quantities as quantities that can be always smaller. But this is mere mambo jumbo. In fact, a quantity is what it is and it cannot be either bigger or smaller. What they mean actually is a quantity that can be substituted or replaced by another smaller quantity. They are imagining that they substitute it for a smaller quantity: Not always, (cfr III 9) though; and this is what is decisive. An 'always' is not a sensible data, and consequently, it is not imaginable. Not even the imaginary representation fulfills the definition they are giving of in- finitesimal. And the same goes to every definition of theirs that defines big as a quantity that can be always bigger.
When the mathematician or the empiricist says that he imagines himself repeating the operation indefinitely, he is only imagining the beginning of it (five or six times). But the beginning is not infinite. In addition: with the adverb 'indefinitely' the definiendum reappears, there is circularity and they have defined nothing. In regard of the concepts, they do not manage to define 'always' nor 'indefinitely'; in regard of imagination, they may 'repeat' but they do not this "indefi- nitely". Without this last element the definition leaves aside what is essential. According to them, this is what would make the quantity infinite. "This is the bad infinite: when one says, and so it goes in infi- nitum" (GP III 171). "It has been exposed that the indefinite progress be- longs completely to the reflection which lacks the concept" (WL II 500); "supposing and determining never fulfill the goal" (JS 27).
What we must reproach to the indefinite process is that it is not understood; it only appears to be so; we never get through it to the con- cept. It is useless to say that Mathematics and Physics understand by infinite an indefinite process, for there is nothing to understand there. What we have there is an imaginary construction that is not infinite. It is an abuse of language to call something a thing that it is not. "The in- finity of the infinite progress remains burdened with the finite as such is thereby limited and is itself finite. " (WL I 131).
If it was not an inveterate self-deceit, we would not insist upon it. Perhaps somebody thinks that the concept of infinite could have an empirical or operational meaning in the following way: I could add a stretch to a straight line, and I could add to the result another stretch, and so on.
But this 'so on' is not empirical; the first four or five times perhaps they are, but they do not suffice to build up an infinite; on the contrary,
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the expression 'and so on' would need to be defined, and its definition would be precisely the definition of quantitative infinite that we were looking for. If they tell us that 'in principle' the operation of adding a stretch is indefinitely repeatable, they are not delivering us any empiri- cal data or observable operation, but only a statement whose meaning is not understood so long the adverb 'indefinitely' is not defined, which was actually the only thing at stake here. Furthermore: a given straight line is perhaps an empirical data, but 'any' straight line is not. 'Any straight line' is the same as 'all straight lines' and we have showed (III 9) that 'every' is neither an empirical nor an operational data. Besides, they said that I 'can' add a stretch to the line. The experience that origi- nates the concept of 'can' is not an empirical but a selfconscious one: I can give myself other determinations; I can choose either this or that. The concept of this concept is freedom and self-determination.
It is obvious even for the most superficial reflection that sensibility does not bring us any data at all that could be called infinite and that the origin of the concept of infinite is not empirical. On the other hand, the said concept is evidently not constructed by the negation of empiri- cal finite data, for negation does not add any content whatsoever and, consequently, we would still be missing the characteristic content of infinite, since it was not given by any of the empirical data. Further- more, the empirical data do not have a label that indicate it is finite or infinite; if they seem to us infinite (and they actually are) it is by con- trast to the idea of infinite that preexisted in the mind. Descartes had already explained it. That color is green and period; the green does not tell us anything about finite or infinite things. The sound is sharp and period; the sharp does not say anything about finite or infinite things. We would have never come up with the idea that things are finite if the concept of the spirit did not preexist in the spirit.
"It is sheer irresponsibility no to see the fact that we call finite or limited something contains the demonstration of the real presence of the infinite and unlimited, that the awareness of the limits can only exist insofar that the unlimited exists there in consciousness" (EPW 60 A).
The preexistence of the infinite in consciousness is the condition of possibility of all the attempts to grant the word infinite some quantita- tive, empirical, imaginary or operational meaning. The only bizarre thing is that the authors of such attempts do not realize what is it that guides them, for it is obvious that no element of the above mentioned kinds brings up to their minds the idea of infinite; on the contrary,
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their mind has this initiative has taken their minds and it costs them a lot to invent artificial combinations of such elements that bear some resemblance with the idea of infinite. Besides, the similarity is always false and the attempts fail. Their definitions are either circular or they use in the definer a word or sign which meaning is not as the men- tioned ones.
For instance, the rule for the formation of mathematical series (which are the modern version of the indefinite process), includes the letter n, which means 'any', and we already pointed out that 'any' is not an empirical or an operational element.
In addition, one works with 'sets', probably with the belief that a set is an empirical data. Despite the popularity of this trend, it is imperative to realize that empirical things are not constituted as sets nor do they form them. The oblique objects that exist in the world are separated from each other by thousands of objects that are not oblique; our intel- lectual consideration is what delineates the set of oblique objects. We could say exactly the same thing about the organisms called mammals and about the metals called bodies. Therefore, a set as such is not an empirical data. In order to build up a set, it is necessary that it is con- stituted by all oblique objects, but we have exposed (III 9) that 'all' is neither an empirical nor an imaginary data.
Despite the importance of the contribution he made to mathematics, Georg Cantor --supported by Richard Dedekind-- did not say any- thing new in regard to the fundamental problem we are dealing with. It is true that unlike his predecessors Cantor does not attribute infini- tude to the last addend of the series, but rather to the entire series: he attributes infinitude to the sum of all finite elements. No one could deny the sagacity and agility of this radical turn that made Kroenecker fly into rage. But, evidently, no one can speak of a sum or a set without implying the concept 'all', whose meaning is not empirical or imagi- nary because only the infinite is universal. As a definition of infinite Cantor's great construction is circular: it presupposes that we under- stand the meaning of the word 'all', which can only be understood in function of the infinite. Of course, he does not ever say a word about the non-existence of the series: it does not exist either in reality, imagina- tion, paper or blackboards. Once again, we are dealing with the indefi- nite process, which we could define as the search of a meaning with the condition of never finding it --in other words, with the condition of never reaching the concept. The reason of not calling the last element
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of the series infinite is that the last element does not exist. And the same thing goes to the entire series: it does not exist.
Further, all these attempts work with numbers. But a number is not a sensible data. It is an invention of the understanding. Animals do not know how to count; Thomas Aquinas and Kant already said this: sensibility does not perceive the number. Hegel said that too: "Number is a non-sensuous object" (WL I 212).
In order to be convinced of this, we must bear into account that we are able to count the most heterogeneous things, without them having any sensible relation whatsoever between them. A flower, an emotion, a tempest, a flavor, a soul, a bull and a thought sum up seven. Perhaps one could say that the common denominator is that all of them are beings. Now, this would be enough to demonstrate that the number is not an empirical data, for we have exposed (II 7) that empirical data do not apprehend the being. However, we could add nothingness as another numeral and then we would have eight.
Whenever we have the whim of counting stuff, we do not depend at all on the empiri- cal: numbers have nothing to do with sensibility. The example we just mentioned teaches us that --even when the countable elements are ob- jects that can be perceived empirically-- the idea of number does not enter through the senses but is rather an initiative on account of the intellect. Besides, in order to constitute a number, the elements need to be summed up between them; otherwise, each element would exist independently and we would not have the total number. But empirical impressions --as they originally come to us-- do not come this way: each of them is what it is and knows nothing about sums. The idea of summing them up must come from someplace else. Not even the fact that five empirical impressions could come simultaneously to our minds would make us count them; let alone the case of successive impressions, for the best thing that mechanical memory could do is to present them at the same time.
In a word, the efforts to give meaning to the word 'infinite' which are not based on self-consciousness employ the number, but for this very reason they fail since the number is not a sensible data. Furthermore, what they build up is neither the infinite nor anything that resembles it. Naturally, they can always employ the arbitrary recourse: we under- stand this by infinite. But we already said that whoever speaks like that remains only with this and renounces thus to the infinite; he prefers this and quits thereby the search for the infinite. Not even he knows
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why he takes this and not that, for he renounces to look up for the meaning of infinite. He could say I understand 'abracadabra' or 'X' by this, but whoever hears him does not know what he is talking about: he does not even know it himself.
We would now have to proceed positively and disentangle the true meaning of the infinite, but since it is also the meaning of the universal, it would be more convenient to dissipate the common illusion of those who believe that the universal has a meaning which is not intersubjec- tivity as such.
2. pSeudouniverSaL
First and foremost, one should notice how inadmissible it would be to build up the universal as the negation of the particular and the indi- vidual. First, because negation does not provide any content at all and thus the only positive content would be singularity, which is precisely what we cannot employ in order to construct the universal. Second, because the individual would be defined through the negation of the universal, the process would be circular and nothing would have been defined.
What we have just said refers to the alleged singular that we sup- posedly perceive by means of sensation. In fact, the empirical data says nothing about individuality or universality, the same way it says noth- ing with regard to finitude or infinitude. All these considerations are provided by the mind: they are not contributions of the sensibility. On the level of individuality, the fact that the individual must be mentioned in the definition of the universal and vice versa is very illustrative; this demonstrates that, if we leave aside the fixed and unintelligible abstrac- tions of the abstract intellect, the concrete and real individual is univer- sal in itself, and it would be the more universal the more individual it is. But let us not rush too much.
It is important to remind (II, 6) the reader how frivolous the theory of abstraction is. According to this theory, the origin of the universal is empirical data through a mysterious process of generalization. Before making a generalization in order to get a concrete universal, the mind needs to know which are the pertinent data, among the innumerable ones we have in the world, because just from them, not from all the existent ones, could the mind abstract the universal in question. Now,
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in order to know which objects we need and which aspects of these ob- jects are relevant, the mind needs the universal in question as a previous guide; otherwise, it would be trapped by a total despair, it would not know where to begin or where to draw its attention. Far from being the product of empirical impressions and the result of the said process of generalization, the universal is the condition of possibility of the for- mer and precedes the latter.
The aforementioned theory of abstraction is an unforgivable mis- placing, because Plato had already showed --in the decisive way we just evoked-- that the universal necessarily preexists.
Some people asked: where is it?
The key point is what we have to say. The question is imagining the universal (what is imaginary is spatial), not understanding it, because only in regard to a spatial object can one say things like location, posi- tion, etcetera. And so the same happens as in the case of the infinite: the infinite of imagination is not the infinite, and a universal of imagination is not universal. Neither the infinite nor the universal are imaginable.
By definition, the universal is what is fulfilled in many, in the belong- ing particulars. But an imagined and spatialized universal falls beyond the singulars; it is distinct and distant from them; it is not in them, it is not fulfilled in them and hence it is not universal; it is in the imagination and the particulars remain outside. The abstract intellect --whose mo- dus operandi is to follow imagination quietly-- believes that the fulfill- ment of the universal in the particulars consist in its mere considering that the universals are fulfilled in the particulars. But that would be a fiction of universality, not true universality. Nothing can be legitimate- ly called if it is not effectively fulfilled in the singulars. All the farce would be reduced to say that intellect has the whim of calling universal something that is not universal; but a whim does not legitimize the use of the term in the opposite sense contained in its definition.
Of course, it is useless to reply that one can define universal as that which is predicated of many. To predicate is to affirm that something is fulfilled in the thing that is being predicated. This is the only meaning of the operation called predicating. All the attempts to define predication by empirical means are doomed to failure. For instance, to say that one predicates something when one has the verb 'to be' is entirely misguid- ed, for all propositions predicates something about something, and the immense majority of propositions are built up without that verb. To sustain that there is predication when the attribute 'goes along' the
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subject in question is also completely mistaken, because the hyper- baton or other grammatical device can separate them to the opposite extremes of a sentence and yet the predication remains. No empirical data corresponds to the meaning of the predication. Its meaning is to affirm that something is fulfilled in that which is being predicted. And here we are, exactly where we were before this way out came to light.
One should notice that despite the pretensions of the abstract intellect of considering the universal within the particulars, the fact is that this does not happen as tall. What he is considering universal is either an imaginary confusing representation as indeterminate as pos- sible (for all determination would wipe out its alleged universality) or a word that supposedly consists in a set of sounds in a blob of ink. Any of these alleged universals is an entity on its own, which is completely different and distant from the singulars being referred. It is false that the intellect considers or even imagines it fulfilled in the singulars, for it considers or imagines it separately from the objects. It is not even a fictitious universal; it is not universal even for the deceiving intellect that calls it universal.
3. infinite and univerSaL
Infinite is the spirit. Every spirit, "since there cannot be two classes of reason or two classes of spirit" (PR I 43).
This is a necessary consequence of what we have said: since the con- cept of infinite cannot be empirical or imaginary, its origin cannot be sensibility or imagination; therefore, its origin is self-consciousness. But what we know by self-consciousness is the spirit. It follows that the meaning of the said concept is the spirit.
When it comes to the spirit, the point of departure has to be --as Hegel affirms emphatically-- the immensely meritorious contribution of Spinoza: every determination is negation; every peculiarity is limi- tation. Nevertheless, Spinoza did not understand that, if the limit is negation, the infinite is negation of negation.
When I published my previous work I had not yet realized that fini- tum and determinatum are passive past participles. If we say that some- one is infinite, what we are actually doing is denying that this person has been determined, for he is giving himself determinations, that is to say, he is self-determining himself. Infinitude is self-determination.
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"The spirit is not natural; it is only that which he makes of itself" (GP II 494).
Infinite is what is not determined by nature but what gives itself its own determinations which he can wipe out and replace with other ones. "Freedom consists precisely in the indetermination of the will, that is to say, the will does have in itself any natural determination. This is why it is in itself a universal will" (NH 224). "This absolute unity of individuality and universality called the self" (JLMN 163).
"The infinitude is the self-determination which refers to itself, the posing of an immanent and own determination, which is truly some- thing indifferent for he who poses it" (NH 87s).
"The practical spirit (moral) is first and foremost free will in the sense in which the self can abstract from any determination in which he is, and in any determination he remains undetermined and identical with himself" (NH 57).
We showed (III 7) that self-determination consists precisely in moral; the appearances are only natural determinations; self-determination consists precisely in not letting them to determine the self; this task is only for the self to do.
"The awareness of freedom contains the understanding that the sub- ject has of himself as a person, that means to say, as a universal in his individuality, as something which is capable of making abstraction of everything particular and of laying it down as an infinite" (VG 175).
The last thing we said should not be understood as if we were talking as potential infinitude, that is to say, as if infinitude consisted in the spirit's capacity of having all the determinations. It is the other way around: Determination is limit, negation. The infinitude is negation of that negation. The spirit is infinite because it does not have determi- nations "he is only what he makes himself being" (GP II 494).
In itself, the spirit does not have limits, and that is the definition of infinite: that which has no limits. As Hegel says, the limits and the de- terminations that he gives to himself are indifferent to it. In fact, they do not limit him, because he can do without them whenever he wants. Even in the act of giving itself certain determination or peculiarity, the spirit manifests its infinite and universal nature, for he could give itself this one instead of that one; these determinations do not deter- mine the spirit; it holds the dominion over his own acts. "By virtue of this original unity it follows, in the first place, that the first negative, or the determination, is not a limitation for the universal which, on the
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contrary, maintains itself therein and is positively identical with itself. " (WL II 241)
It is useless to object that such thing is not what is commonly under- stood by infinite. By infinite one can only understand that which has no limits. The only sense in which the definiendum can be real is the one we have been referring to. We have showed that people commonly imagine something that is not infinite, despite the fact that mathematicians and physics have the whim of calling it like that. The concept of infinite that we have showed can only be rejected by someone who has anoth- er concept of infinite, but neither philosophers nor physics have such concept or can have it. In particular, we said (IV, 1) that one should not confuse the infinite with a big magnitude. Further, one should not be tempted to say that the infinite is a 'magnitude without limits'! By definition, every magnitude is a limit, so to talk about a magnitude without limit equals to speak about a limit without limit. "What has all determinations" is (implicitly) contradictory, because there are de- terminations that are contradictory between themselves. Besides, if all determination is limitation, this statement would be useless to define what is unlimited and infinite.
Let us do without the resistance of those who are not willing to admit that the human spirit is infinite. Evidently, they think that the human spirit is 'here' or 'there', in a given place and not in the an- tipodes; therefore, they conclude that it is not infinite. The ship of this objection is so feeble that one big wave will be enough to sink it. On the one hand, they suppose that the spirit is something material and spa- tial, for that would be the only way they could speak of location and place. On the other hand, they believe that infinitude is spatial; some- thing so big that is everywhere. We have said enough in this regard. Another recourse the objectors could employ is to say that the infinite is that which has all determinations, denying thereby that the human spirit is not in that case. But we pointed out that this could not be the definition of infinite:
In a word, finitude consists in having a limit, which means to say, that it puts its not-being in that point; in other words, in that point this thing ceases to be and makes reference to some other thing. On the other hand, the infinite reflection consists in that I do not refer to anything else but my- self and I am an object to myself. This pure reference to myself is the self, the root of the infinite Being itself. The leave aside completely, everything
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that is finite. The self as such does not have any content given by nature (immediate), but rather he has only himself as content. This pure form is at the same time his content.