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.
Hegel Was Right_nodrm
From an 'it is not' no one can extract an 'it cannot be' (ex non facto ad non posse non valet illation), and the senses do not even deliver us the 'it is not', since as we already see, 'no' is not an empirical data.
It would be superfluous to stop in nai? ve formulations of the law, which boast they do not need an 'all' and say: an acid reacts with a base forming salt. It is obvious that if the 'an' has the intention of a singular, the formulation in question is not a law; and if it has a universal inten- tion, the formulators wanted us to understand it as an 'all' and their attempt of concealing this was in vain.
To tell the truth, we do not need to lengthen our journey. The example that we have mentioned shows that, if a formulation of the law really becomes a law, it would necessarily be the logical equivalent of any other formulation of the law. Now, if we have demonstrated that one of them is unempirical, that also applies to the others, because they are the logical equivalent of the first one. This is why all the attempts in the future of formulating empirically a law are doomed to failure. The reader sees immediately that 'an acid cannot' means the same as 'always that acid. . . ' or that 'every acid. . . ' etcetera.
10. neceSSary
The only formulation that deserves a closer look is this one: 'an acid necessarily reacts with a base forming salt'.
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Science and Literature 135
The reader notices immediately that it is the logical equivalent of 'an acid cannot no. . . ' Therefore, it is as unempirical as this and all the former ones.
A symptom of intellectual despair has been the critical acclaim that Saul Kripke's attempt at defining 'necessary' and 'necessarily' obtained. It first defines 'possible', for he thinks that what all possible worlds have in common will be a good definition of 'necessary'. He says that an M world is possible if it is logically compatible with the natural laws that govern our real natural world M. Here again the 'if' particle reap- pears --a particle which we have seen is unempirical. That very thing frustrates the whole enterprise. Afterwards, Kripke considers all pos- sible worlds M1, M2, M3. . . and triumphantly exclaims: what they have as a common denominator is the content of the word 'necessary'.
This illustrious theorem has so many deficiencies that one can hardly believe that it has been taken seriously at all. We have already pointed out its gratuitous 'if'. But it also assumes an 'all', whose unem- piricy we have already proved. If it is not the common denominator of all possible words, a common denominator cannot be called necessary, for some world could be possible without it. Besides, logical compatibil- ity also is not an empirical data. It is evident that Kripke forgot that necessity --that by means of which he tried to define 'law'-- needed to be defined. Kripke wants to define the necessary going through the possible, but he defines the possible as that which is compatible with the law. And there is the law again, which was just the origin of the entire problem. The circularity is manifest, and at the end of the day nothing has been defined.
I do not know why one would want to deceive someone by concealing the fact that when some processes started to be called 'material', this was in contrast with what is 'free'. Men would have not come up with the idea of calling these processes 'necessary' if it were not by contrast- ing this with his freedom, which is continuously experimented in self- consciousness This was done precisely with the purpose of denying that stones, rivers and stars are free. The only meaning of 'necessary' is 'not free'. In fact, we saw here (III, 2) and also in Kant that self-determi- nation is the most intelligible concept that exists. But that is, necessary means not free; this implies not only that freedom is more intelligible that this necessity that scientists look for, but also that this big neces- sity lacks all intelligible content whatsoever and can only be alluded negatively without us understanding what kind of fixed point this is.
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 136 Hegel was right
Determinism --as the thesis that all that exists is necessary-- is one of those funny statements that in order to have meaning need to be false, for the concept of freedom could only have originated in self- consciousness; i. e. , in the direct knowledge we have from a being that is truly free; otherwise, it would be unexplainable the fact that one day the concept of freedom started to be. The thesis according to which everything is necessary lacks meaning if the concept of freedom does not exist; therefore, in order to have meaning, this theses actually needs that some beings are free, that is to say, it requires that the thesis itself to be false. The same happens with the thesis which says that everything is inexistent.
And I am not only referring to the philosophical and thematic deter- minism of Laplace or Holbach, but also to the 'methodic' determinism which many scientists feel obliged to profess within their disciplines. As we have said, determinism refutes itself in each and every one of its forms. I do not see why physics had to wait for Heisenberg, Bohr and von Neuman in order to bury determinism deep in earth. It was obvi- ous even from before that it was not only a gratuitous and unverifiable thesis, but also that the word necessity itself lacks empirical meaning and hence has nothing to do within Physics.
It is very important to notice, however, that the true concept of ne- cessity --not that fictional necessity without content which scientists have pursued in vain--, certainly has meaning. Not an empirical one, naturally; we have seen that there is no way to express the necessity of a law in empirical terms. The meaning is something that must be and that has to be. To be sure, it was obscurely uttered before by frustrated formulations, such as: 'it can not be not being'. But if no empirical con- tent could have given origin to this meaning, the origin had to be self- consciousness. Now, what we know by self-consciousness is freedom. Therefore, the concept of necessity cannot be different from the con- cept of freedom.
The identity of necessity and freedom is probably the most remark- able feature of the moral imperative, since the moral imperative (III, 7) addresses freedom and, in doing that, constitutes freedom for the very first time. The moral imperative makes the subject free by drawing him to the responsibility of a necessary behavior. As Kant says in precise terms, 'I am free because I ought'. Before being responsible I am not free; I am free in virtue of the necessity that imposes to me called impera- tive: "Neither freedom on its own, as subjective and separated from
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Science and Literature 137
necessity, is something absolutely real, nor necessity on its own, as isolated, can be called real" (A? sth I 165). "What is real is the unity of (those) contrasts, and we must say that the spirit is free in its necessity; only in necessity it is free; conversely, its necessity consists in its free- dom" (EGP 116); "the absolute necessity contains in itself its freedom" (PR II, II 28).
Only due to an absolute misunderstanding of Hegel's thought one can explain the fact that the identity of necessity and freedom has been interpreted as 'known and accepted necessity', granting necessity thereby with a meaning of natural law and determinism. Such neces- sity does not exist. The necessity of the natural laws "is itself only a sham, false necessity". (PG 118). "But if the law does not have its truth in the Notion, it is a contingency, not a necessity, not, in fact a law. " (PG 189)
No one knows what the necessity that has been praised by physics and other scientists --including deterministic psychologists and phi- losophers of history-- means. The only possible meaning of the word necessity is known by introspection, or better said, by self-consciousness, where its content is freedom itself.
"One must not understand by necessity the exterior, but rather the irresistible, the divine, which is an end in and of itself, in relation to freedom" (VG 263).
We saw (III, 9) that this is the moral imperative which, as we saw, is of a divine nature. That is the only possible meaning of the word neces- sity. As we will see (V, I), this imperative makes man free, for one can- not speak of the autonomy of the self when the course of life is decided by impulses and instincts which were not introduced by the self and appeared miraculously. Only in my positive response to the imperative am I autonomous and free. "The necessity deepens into the concept. And this, which is freedom, is the true of necessity" (PR II, II, 199).
It is shameful to watch the spectacle given by those who distrust of our knowledge and perception of the moral imperative because, they say, it lacks the necessary character that the empirical science and its laws have. Such science would want that their knowledge had a neces- sary character, but they do not even know what it would want, and in fact, such necessity lacks all kind of meaning whatsoever, weather it is empirical or not. Thus, those who disdain the imperative because they prefer necessary knowledge, are going, in fact, after a revelry which does not have meaning nor can it have it. They disregard what is truly
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 138 Hegel was right
known to us and constitutes the only possible meaning of the word 'necessary'.
We mentioned (III 7) the group of 'roles' and 'expectations' that constitute a society, and we showed that its moral character could not be concealed. This system of intersubjectivity --which, as we have ex- plained, is free and necessary-- Hegel calls it 'free mechanism' in con- trast with the laws of mechanic (the Newtonian mechanics) which in fact lack all definable necessity and are not thus laws. "Only the free mechanism has a law, the proper determination of pure individuality, that means to say, the concept that exists on its own; as distinction in itself, this law is a restless source of movement that lightens itself up and is free necessity" (WL II 375).
Here we have the 'origin of movement' Aristotle longed for in his Physics. It is the soul that Plato defined as "the movement capable of moving itself" (Laws X 896A). What mechanics blindly looked for with its laws was to explain what happens in the world, but it lacked true causality and the production of more being.
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 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
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 140 Hegel was right
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).
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Infinite and Distinction 141
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,
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 142 Hegel was right
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,
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Infinite and Distinction 143
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
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 144 Hegel was right
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
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Infinite and Distinction 145
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,
? ? ? ? ? ?
It would be superfluous to stop in nai? ve formulations of the law, which boast they do not need an 'all' and say: an acid reacts with a base forming salt. It is obvious that if the 'an' has the intention of a singular, the formulation in question is not a law; and if it has a universal inten- tion, the formulators wanted us to understand it as an 'all' and their attempt of concealing this was in vain.
To tell the truth, we do not need to lengthen our journey. The example that we have mentioned shows that, if a formulation of the law really becomes a law, it would necessarily be the logical equivalent of any other formulation of the law. Now, if we have demonstrated that one of them is unempirical, that also applies to the others, because they are the logical equivalent of the first one. This is why all the attempts in the future of formulating empirically a law are doomed to failure. The reader sees immediately that 'an acid cannot' means the same as 'always that acid. . . ' or that 'every acid. . . ' etcetera.
10. neceSSary
The only formulation that deserves a closer look is this one: 'an acid necessarily reacts with a base forming salt'.
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Science and Literature 135
The reader notices immediately that it is the logical equivalent of 'an acid cannot no. . . ' Therefore, it is as unempirical as this and all the former ones.
A symptom of intellectual despair has been the critical acclaim that Saul Kripke's attempt at defining 'necessary' and 'necessarily' obtained. It first defines 'possible', for he thinks that what all possible worlds have in common will be a good definition of 'necessary'. He says that an M world is possible if it is logically compatible with the natural laws that govern our real natural world M. Here again the 'if' particle reap- pears --a particle which we have seen is unempirical. That very thing frustrates the whole enterprise. Afterwards, Kripke considers all pos- sible worlds M1, M2, M3. . . and triumphantly exclaims: what they have as a common denominator is the content of the word 'necessary'.
This illustrious theorem has so many deficiencies that one can hardly believe that it has been taken seriously at all. We have already pointed out its gratuitous 'if'. But it also assumes an 'all', whose unem- piricy we have already proved. If it is not the common denominator of all possible words, a common denominator cannot be called necessary, for some world could be possible without it. Besides, logical compatibil- ity also is not an empirical data. It is evident that Kripke forgot that necessity --that by means of which he tried to define 'law'-- needed to be defined. Kripke wants to define the necessary going through the possible, but he defines the possible as that which is compatible with the law. And there is the law again, which was just the origin of the entire problem. The circularity is manifest, and at the end of the day nothing has been defined.
I do not know why one would want to deceive someone by concealing the fact that when some processes started to be called 'material', this was in contrast with what is 'free'. Men would have not come up with the idea of calling these processes 'necessary' if it were not by contrast- ing this with his freedom, which is continuously experimented in self- consciousness This was done precisely with the purpose of denying that stones, rivers and stars are free. The only meaning of 'necessary' is 'not free'. In fact, we saw here (III, 2) and also in Kant that self-determi- nation is the most intelligible concept that exists. But that is, necessary means not free; this implies not only that freedom is more intelligible that this necessity that scientists look for, but also that this big neces- sity lacks all intelligible content whatsoever and can only be alluded negatively without us understanding what kind of fixed point this is.
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 136 Hegel was right
Determinism --as the thesis that all that exists is necessary-- is one of those funny statements that in order to have meaning need to be false, for the concept of freedom could only have originated in self- consciousness; i. e. , in the direct knowledge we have from a being that is truly free; otherwise, it would be unexplainable the fact that one day the concept of freedom started to be. The thesis according to which everything is necessary lacks meaning if the concept of freedom does not exist; therefore, in order to have meaning, this theses actually needs that some beings are free, that is to say, it requires that the thesis itself to be false. The same happens with the thesis which says that everything is inexistent.
And I am not only referring to the philosophical and thematic deter- minism of Laplace or Holbach, but also to the 'methodic' determinism which many scientists feel obliged to profess within their disciplines. As we have said, determinism refutes itself in each and every one of its forms. I do not see why physics had to wait for Heisenberg, Bohr and von Neuman in order to bury determinism deep in earth. It was obvi- ous even from before that it was not only a gratuitous and unverifiable thesis, but also that the word necessity itself lacks empirical meaning and hence has nothing to do within Physics.
It is very important to notice, however, that the true concept of ne- cessity --not that fictional necessity without content which scientists have pursued in vain--, certainly has meaning. Not an empirical one, naturally; we have seen that there is no way to express the necessity of a law in empirical terms. The meaning is something that must be and that has to be. To be sure, it was obscurely uttered before by frustrated formulations, such as: 'it can not be not being'. But if no empirical con- tent could have given origin to this meaning, the origin had to be self- consciousness. Now, what we know by self-consciousness is freedom. Therefore, the concept of necessity cannot be different from the con- cept of freedom.
The identity of necessity and freedom is probably the most remark- able feature of the moral imperative, since the moral imperative (III, 7) addresses freedom and, in doing that, constitutes freedom for the very first time. The moral imperative makes the subject free by drawing him to the responsibility of a necessary behavior. As Kant says in precise terms, 'I am free because I ought'. Before being responsible I am not free; I am free in virtue of the necessity that imposes to me called impera- tive: "Neither freedom on its own, as subjective and separated from
? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Science and Literature 137
necessity, is something absolutely real, nor necessity on its own, as isolated, can be called real" (A? sth I 165). "What is real is the unity of (those) contrasts, and we must say that the spirit is free in its necessity; only in necessity it is free; conversely, its necessity consists in its free- dom" (EGP 116); "the absolute necessity contains in itself its freedom" (PR II, II 28).
Only due to an absolute misunderstanding of Hegel's thought one can explain the fact that the identity of necessity and freedom has been interpreted as 'known and accepted necessity', granting necessity thereby with a meaning of natural law and determinism. Such neces- sity does not exist. The necessity of the natural laws "is itself only a sham, false necessity". (PG 118). "But if the law does not have its truth in the Notion, it is a contingency, not a necessity, not, in fact a law. " (PG 189)
No one knows what the necessity that has been praised by physics and other scientists --including deterministic psychologists and phi- losophers of history-- means. The only possible meaning of the word necessity is known by introspection, or better said, by self-consciousness, where its content is freedom itself.
"One must not understand by necessity the exterior, but rather the irresistible, the divine, which is an end in and of itself, in relation to freedom" (VG 263).
We saw (III, 9) that this is the moral imperative which, as we saw, is of a divine nature. That is the only possible meaning of the word neces- sity. As we will see (V, I), this imperative makes man free, for one can- not speak of the autonomy of the self when the course of life is decided by impulses and instincts which were not introduced by the self and appeared miraculously. Only in my positive response to the imperative am I autonomous and free. "The necessity deepens into the concept. And this, which is freedom, is the true of necessity" (PR II, II, 199).
It is shameful to watch the spectacle given by those who distrust of our knowledge and perception of the moral imperative because, they say, it lacks the necessary character that the empirical science and its laws have. Such science would want that their knowledge had a neces- sary character, but they do not even know what it would want, and in fact, such necessity lacks all kind of meaning whatsoever, weather it is empirical or not. Thus, those who disdain the imperative because they prefer necessary knowledge, are going, in fact, after a revelry which does not have meaning nor can it have it. They disregard what is truly
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known to us and constitutes the only possible meaning of the word 'necessary'.
We mentioned (III 7) the group of 'roles' and 'expectations' that constitute a society, and we showed that its moral character could not be concealed. This system of intersubjectivity --which, as we have ex- plained, is free and necessary-- Hegel calls it 'free mechanism' in con- trast with the laws of mechanic (the Newtonian mechanics) which in fact lack all definable necessity and are not thus laws. "Only the free mechanism has a law, the proper determination of pure individuality, that means to say, the concept that exists on its own; as distinction in itself, this law is a restless source of movement that lightens itself up and is free necessity" (WL II 375).
Here we have the 'origin of movement' Aristotle longed for in his Physics. It is the soul that Plato defined as "the movement capable of moving itself" (Laws X 896A). What mechanics blindly looked for with its laws was to explain what happens in the world, but it lacked true causality and the production of more being.
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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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