×àñòü 7.

demnach keine blosse Rhetorik, drueckt kein leeres Pathos aus, sondern eine dem Erfahrungsurteil berechtigterweise zustehende Emphase.

Willem van der Kuijlen

Infinite Judgment in Kant's

«Critique of Pure Reason»

1. Introduction

In the table of logical functions of understanding Kant introduced «infinite judgments» in addition to the common logical distinction between affirmative and negative judgments. Correspondingly, Kant included «limitation» in the table of categories in addition to the categories of «reality» and «negation» (KrV, B106). Kant's example of infinite judgment was the sentence, «The soul is non-mortal». Kant's division of judgments may come as a surprise to those familiar with ordinary logical distinctions, especially because, in general logic, infinite judgment is classed with affirmations (KrV, B96f.). In view of transcendental logic, therefore, Kant took care to distinguish it from both negative and affirmative judgments. It is not negative because it is a logical affirmation, since the copula is not negated. Unlike ordinary affirmative judgments, on the other hand, it contains a negative predicate. Hence, infinite judgment is a logical affirmation by using merely a negative predicate in order to say something about our total knowledge (KrV, B97) and the content of knowledge «überhaupt».

Notwithstanding this explanation, Kant's introduction of infinite judgment and the category of limitation has met with neglect and misunderstanding, if not downright rejection, ever since. Schopenhauer, as always, was clear and quite amusing in his disapproval. He called infinite judgment «einen spitzfindig erdachten Lückenbüßer, was nicht ein Mal einer Auseinandersetzung bedarf, ein blindes Fenster, wie er zu Gunsten seiner symmetrischen Architektonik deren viele angebracht hat.». According to Hegel it was not a judgment at all. Peirce also maintained that Kant added infinite judgments «because it rounded out his triad of categories of quality». As to the category of limitation, De Vleeschauwer added a philological reason: «Elle constitue donc une retouche de la dernière heure.». To Kemp Smith, Kant's distinctions reflected «a very artificial and somewhat arbitrary manner» in preparation «for the 'discovery' of the category of limitation».

Instead of blaming Kant for unclarity or architectonical preoccupations to account for misunderstandings on our own part, Kant's distinction itself should be taken more seriously. The following sections will examine Kant's notion of infinite judgment, its meaning and function in view of the evidence found throughout KrV instead of particular passages only. The methodological starting point of this analysis is the assumption that Kant must have had good reasons to introduce infinite judgment; otherwise he would not have mentioned it at all and he certainly would not have added his warning observation in KrV, B97f..

Kant's insistance on this form of judgment is not as far-fetched as his commentators would have it. In ordinary language it is not uncommon to use negative predicates and even double negations in a meaningful way. Especially cases of double negation may help us to understand the meaning and function of infinite judgment. Generally, double negation amounts to an affirmation of what is negated: duplex negatio affirmat. This may apply in the case of classical bivalent logic, which is the logic most philosophers tend to subscribe to, but if it is considered as a principle of logic (sc. the law of double negation) it is not appropriate to serve as a logical means to analyse occurrences of double negation in ordinary or non-formal language. For example, the use of figures of speech such as understatement and litotes, which may contain double negations, cannot be adequately understood if they are supposed to represent affirmations. In order to account properly for occurrences of double negation, we must assume that there are good reasons for using the long-winded expression instead of a shorter one. These reasons may be of a pragmatic, rhetoric, or political nature, but in each case we will have to take the context into consideration.

From the following example it will be clear that double negation does not amount to a simple affirmation. Sentence 1 is a statement in response to official claims that there was no point in arresting Bouterse in Brazil, because subsequent extradition to The Netherlands would be impossible.

1 «The arrest and extradition of Desi Bouterse by the Brazilian government is complicated, but not impossible.»

«Not impossible» in this case does not mean the same as «possible», at least that is not the exact meaning of the statement. The statement merely points to the fact that the opposite claim about the impossibility is not valid. This does not entail a true claim about the possibility of arrest and extradition.

For the sake of introduction it is assumed here that this example should be understood in terms of infinite judgment. Infinite judgment, then, seems to be a highly common means to express specific meanings in specific contexts for specific reasons. As we shall see, the topics related to this example (esp. double negation) will recur when infinite judgment in the Kantian sense is discussed.

In the next section we will have a closer look at infinite judgment in relation to transcendental knowledge. In section 3 we will assess historical and systematic objections raised against infinite judgment. Implications resulting from this evaluation are presented in section 4 in relation to Kant's conception of transcendental logic. Sections 5 and 6 will determine the meaning and function of infinite judgment in the transcendental dialectic and doctrine of methods of KrV.

2. Infinite judgment and transcendental knowledge

In his observation regarding infinite judgment (KrV, B97f.) Kant clearly distinguished it from affirmative and negative judgment. His example was «The soul is non-mortal.», in contrast with the negative judgment «The soul is not mortal.». We may rephrase the latter example as «It is not the case that the soul is mortal.» to show that negative judgment is the negation of an affirmative one. Although Kant did not give a corresponding example of affirmative judgment, it obviously would have been «The soul is mortal». These three examples, the three types of judgment and their formalisations, are represented as follows:

2 The soul is mortal - affirmative judgment - S is p

3 The soul is not mortal - negative judgment - ¬(S is p)

4 The soul is non-mortal - infinite judgment - S is ¬p

Kant did not mention an example of affirmative judgment because, in this case, it would have been false: the soul does not belong to the set of mortal things. This implies an important aspect of Kant's view of judgment. By «judgment» he meant «true judgment». This is confirmed by Kant's conception of the function of negative judgment. In KrV, B97 he stated that a negative judgment regarding the soul would have warded off error. Negative judgments prevent us from making mistakes (cf. KrV, B737). True infinite judgments, however, serve another purpose.

Kant also provided an explicit reason to introduce infinite judgment: only in transcendental logic is infinite judgment a separate member in the division of judgments. In general logic, however, it is rightly classed with affirmative judgments. Apparently, it is necessary to distinguish infinite judgment because it serves a particular function in transcendental logic. In order to understand this function we have to get a clear picture of how transcendental logic should be distinguished from general logic. The remainder of this section is devoted to the main aspects of this distinction.

General logic abstracts from all content of the predicate. It abstracts from all content of knowledge and deals with nothing but the mere form of thought (KrV, B78). Transcendental logic, however, also considers what may be the worth or content of a judgment (KrV, B97). Here, «content» should be taken in a very specific way, i.e. in the sense of «Gegenstände a priori» (KrV, B81) or «Erkenntnis a priori» (KrV, B80). Thus, in transcendental logic, content is limited to pure knowledge a priori (KrV, B171). The content matter of transcendental logic does not consist in objects (e.g. bodies) or properties (e.g. mortality) in the ordinary sense, but in knowledge. Transcendental logic expresses knowledge about knowledge.

If we take a closer look at Kant's explanation of the term «transcendental» this conception of logic is easier to understand. In his introduction to KrV Kant stated:

«Ich nenne alle Erkenntnis transzendental, die sich nicht so wohl mit Gegenständen, sondern mit unserer Erkenntnisart von Gegenständen, so fern diese a priori möglich sein soll, überhaupt beschäftigt» (KrV, B25).

Transcendental logic is a specific kind of transcendental knowledge. The latter concerns the a priori possibility of the mode of knowledge of objects, rather than one of the objects themselves. More specifically, this is stressed right at the beginning of «Die transzendentale Logik»:

«nämlich: daß nicht eine jede Erkenntnis a priori, sondern nur die, dadurch wir erkennen, daß und wie gewisse Vorstellungen (Anschauungen oder Begriffe) lediglich a priori angewandt werden, oder möglich sein, transzendental (...) heißen müsse. (...) Der Unterschied des Transzendentalen und Empirischen gehört also nur zur Kritik der Erkenntnisse, und betrifft nicht die Beziehung derselben auf ihren Gegenstand» (KrV, B80f.).

According to this passage transcendental knowledge belongs to the project of the critique of knowledge, i.e. the critique of pure reason as it is presented in KrV. From this perspective we could distinguish between various levels of knowledge. The most general level is that of critique. Transcendental knowledge, as distinguished from empirical knowledge, is a more specific kind of knowledge. It is not so much concerned with a priori knowledge tout court, but only with a priori knowledge about the a priori possibility or application of certain representations (intuitions and concepts), i.e. the mode of knowledge (as specified in the transcendental aesthetic and the transcendental logic respectively). The third level contains knowledge about what is ordinarily regarded as the content, or subject matter of statements. This knowledge is expressed in statements like «Bodies are heavy». The content referred to in this type of statement is possible experience, which would be yet another level, i.e. the «lowest» content level.

For our present purposes we have identified four levels: 1) objects and properties within the field of possible experience; 2) judgments about 1; 3) transcendental knowledge about the a priori conditions of 2 (i.e. all that Kant said about space and time, the categories, schematism and the principles); and 4) critique as presented in KrV. As Kant stated, and as has been pointed out above, transcendental logic considers what may be the worth or content of a logical affirmation in infinite judgment, and what is thereby achieved in the way of addition to our total knowledge or in respect of the content of knowledge in general («überhaupt»). Thus, infinite judgment as part of transcendental logic is not a logical distinction for the affirmation or negation of anything at levels 1 and 2. Rather, it is a means to express something about knowledge überhaupt, i.e. about a higher level of knowledge. Seen against this background, infinite judgment is probably the only form of judgment which is specifically fit to put critical knowledge into words. Infinite judgment is indispensable for the purpose and means of Kant's critique. In the following sections this claim will be substantiated.

3. Against infinite judgment

In his study of Kant's infinite judgment Albert Menne echoed Schopenhauer's reproach: Kant introduced infinite judgment to save the symmetry of the table of categories (Menne, l.c., 151f., 159). Menne denied the independent status of infinite judgment for historical and systematic reasons. Historically speaking, there is no justification for using the notion of infinite judgment, since, as Menne suggests, the use of the term «infinite» is based on an incorrect translation of «indefinite judgment» (enuntiatio infinitus), which is a judgment containing a negative or infinite term with an infinite extension. The misunderstanding of Aristotelian «indefinite judgment» probably lead Kant to disregard this kind of judgment as the fourth, quantitative form of judgment and to introduce a missing, third, qualitative form of judgment, thereby completing the trichotomy of the table.

However, within the historical context, Menne acknowledged that some logicians prior to Kant also spoke of «infinite judgment», for example, Scharffius, Micraelius and Jungius (l.c., 155-157), although the use of infinite judgment was not wide-spread and limited only to approaches that were not Aristotelian or scholastic. According to Menne, Kant used an unusual type of judgment in a very specific way as he attached literal meaning to «infinite», although he was ignorant of its proper historical and philosophical background. Besides, Menne also maintained that there were no systematic reasons to use it in the same literal way as Kant did.

Menne (l.c., 159-162) denied Kant's literal reading of infinite judgment systematic relevance, especially in so far as it is distinguished from negative judgment, because there is no difference between a negative and infinite judgment, if the relations between subject and predicate are represented according to set theory. In agreement with this view negative judgment 3 and infinite judgment 4 are set theoretically identical, if negation in either judgment is understood in terms of complements: «not being mortal» and «non-mortal», both formalised as «(mortal')», refer to the complements of the set of mortal things. The formalisation expressing this identity would run: {x | soul (x)} ? {x | mortal' (x)} (Menne gives diagrams). In Menne's view, therefore, there is no difference between negation of the copula and predicate negation. To support this, he also called upon the generally accepted laws of traditional logic, particularly Apuleius' laws of equipollence, which state that an affirmative judgment is equipollent to the corresponding negative judgment if the predicate is denied as well. Menne concluded that both kinds of negation should be equivalent in order to yield an affirmative result. Applied to 2 and 3 above this may be expressed as:

5 (S is p) ? ¬(S is ¬p)

Menne's argument is circular (claiming the equivalence and proving it by referring to a defining, common logical principle) and appears to be based on the principle of double negation: two negations (of a single term or within a single statement) equal the affirmation (of that term or statement):

6 A ? ¬ ¬A principle of double negation

Menne's unconvincing, or at least unsatisfactory rejection of infinite judgment is based on this principle and on the principle of excluded middle, as is clear from his conception of «non-p» in terms of the complement of «p». Thus he maintains that if the introduction of infinite judgment were to make any sense, it would have been necessary for Kant to show that these formal principles do not hold in the case of transcendental logic. According to Menne (l.c., 160) Kant failed to supply such arguments and only offered problematic speculation. I believe that these arguments, apart from the basic distinction between transcendental and general logic (see section 2), can be found quite explicitly in other parts of KrV. A more detailed discussion of infinite judgment in the next section will deal with the principles of double negation, excluded middle and, in addition, the closely related principle of contradiction.

4. Infinite judgment in transcendental logic

An explanation of infinite judgment should be based on an explanation of the applicability of the principles involved, as it does not make sense to introduce infinite judgment if these principles apply without any restrictions. Again, the starting point is Kant's conception of «transcendental». In his view, not all types of a priori knowledge are transcendental. Analytic judgments are a priori judgments, but they are not transcendental. Because analytic judgments only express the fact that the predicate is contained in the concept of the subject, which is exacly what makes them analytic (KrV, B10), it is not necessary to rely on experience in order to be able to make them (KrV, B11f.).

Reference to experience is what distinguishes analytic from synthetic judgments. Synthetic judgments express a relation between subject and predicate which cannot be derived from the relation between the concepts involved. Synthetic judgments a posteriori can only be derived from experience, e.g. «This body is heavy» (KrV, B12). However, in the case of synthetic judgments a priori (e.g. «Everything which happens has its cause.», KrV, B13) things grow more complicated. Experience can no longer serve to account for the synthetic character of these judgments, since they are a priori. Thus the key problem of the critique of pure reason can be defined as the search for an answer to the question «How are a priori synthetic judgments possible?» (KrV, B19, 73). The answer in terms of the a priori possibility and applicability of concepts and intuitions is what Kant called «transcendental knowledge».

Now we should determine in what transcendental way the principle of contradiction applies to analytic and synthetic knowledge. In KrV, B190 Kant defined the principle of contradiction as the highest principle of analytic knowledge:

7 «Keinem Dinge kommt ein Prädikat zu, welches ihm widerspricht»

In a sense, the principle of contradiction applies to analytic as well as synthetic judgments:

«ein synthetischer Satz kann allerdings nach dem Satz des Widerspruchs eingesehen werden, aber nur so, daß ein anderer synthetischer Satz vorausgesetzt wird, aus dem er gefolgert werden kann, niemals aber an sich selbst» (KrV, B14).

Thus, the principle can be applied to a synthetic judgment, but only in so far as yet another synthetic judgment is presupposed from which it follows. To the latter it cannot be applied (unless in so far as... etc., which would initiate an infinite regress). Again, this limited application of the principle of contradiction indicates a need to account for the possibility of a priori synthetic judgments.

In addition to the principle of contradiction, Kant also defined the highest principle for all synthetic judgments:

8 «ein jeder Gegenstand steht unter den nothwendigen Bedingungen der synthetischen Einheit des Mannigfaltigen der Anschauung in einer möglichen Erfahrung.» (KrV, B197).

This principle contains the very short answer to the question about the possibility of a priori synthetic judgments. It is also a specimen of transcendental knowledge and it should be dealt with in transcendental logic, as Kant emphasised (KrV, B193). The principle of contradiction, on the other hand, «gehört aber... bloß in die Logik» (KrV, B190), thus stressing once more the disregard of content in this type of logic.

In the synthetic a priori, however, content is certainly taken into account, which is indicated in 8 by the occurrence of «Gegenstand» (object). The only objects given are objects of experience, in which case they should be taken as real objects. The principle of the synthetic a priori as stated in 8 specifies that the content of knowledge corresponds to an objective reality if the objects can be given in possible experience, i.e. if they comply with the synthetic requirements of sensibility (space and time), understanding (categories) and unity of apperception. Conformity to these formal, synthetic conditions of experience constitutes the objective reality or validity of knowledge. Synthetic principle 8 concerns content in so far as it specifies the conditions of what is to count as objectively real.

Kant's reference to «objective reality» (KrV, B194-196) in relation to the principle represents a major indication of how he conceived of the function of the three types of judgment in transcendental logic. In transcendental logic, affirmative judgments are true judgments, representing transcendental knowledge about what is to be considered as objectively real knowledge. This also makes it clear that an affirmative judgment is directly related to the category of «reality»: it affirms at a transcendental level what is valid and real at object level.

If this analysis is basically right, we may draw some tentative conclusions in respect to negative and infinite judgments in transcendental logic. Presumably, at a transcendental level negative judgment determines what should not be counted as a valid or objectively real claim of knowledge. In addition, infinite judgment is an affirmation, which is neither valid, nor invalid. Infinite judgment is non-valid. In the next section these conclusions will be further substantiated. Once again, we will be confronted with the limited applicability of the laws of contradiction and double negation as far as infinite judgment is concerned.

5. Infinite judgment, antinomy and “omnimoda determinatio”

If the proper function of negative judgment is to ward off error, its position in transcendental logic is implied in our analysis of affirmative judgment in the preceding section. At the level of the Transcendental Dialectic negative judgment is a true judgment about a false affirmative judgment regarding objects which cannot be given in possible experience (the soul, the world, God). This makes the Transcendental Dialectic the negative counterpart of the Transcendental Analytic. In this sense Kant referred several times to the important negative function of critique (KrV, Bxxivf., 25, 740, 823). Infinite judgment, the main focus of attention here, is also relevant in Transcendental Dialectic.

The clearest example of the presence of infinite judgment in the Transcendental Dialectic is Kant's solution of the antinomy, especially the first antinomy about the (in)finity of the world (KrV, B454-461). Reason gets entangled in the antinomy as soon as it tries to extend knowledge (cosmological ideas) beyond the limits of possible experience. The result is a deadlock; opposing rules seem to apply in the case of transcendent, cosmological knowledge, although they cannot both apply, because that would contradict their rule-like character. Furthermore, there does not seem to be a way out of this situation: antinomy is a fundamental conflict of reason. Kant's construction of the antinomy in transcendental logic shows the insufficiency of the logical principles involved. We will reconstruct Kant's analysis of the antinomy taking into account Ishikawa's study on infinite judgment and antinomy.

As in any other case of antinomy, the first antinomy consists of a thesis, supported by an argument to prove it, and an antithesis claiming the opposite of the thesis and also supported by proof. The thesis is, «The world is finite», whereas the antithesis runs, «The world is infinite». In support of the thesis, Kant constructed the following proof structure. Assume the opposite of the thesis (sc. the infinity of the world) to be true. According to Kant, this assumption would force us to accept contradictions which follow from it. Hence, we must reject this opposite claim because it is false, thus ipso facto accepting the thesis. Elsewhere (KrV, B817-822) this proof structure was called apagogical or indirect: justifying a claim by refutation of the opposite. Kant adopted the same strategy in case of the antithesis.The result, however, is a conflict of reason to which there is no apparent solution (for if there were, it would not be an antinomy).

The antinomy arises due to strict application of the logical principles mentioned above. The false statement about the infinity of the world entails a true statement about the finity (and vice versa in the case of the antithesis), given the principle of double negation (¬¬finity ? finity) and the principle of excluded middle (finity ? infinity). The result is a contradiction (antinomy, conflict of reason), which poses a serious problem because thesis and antithesis cannot both be true, given the principle of contradiction (¬(finity ? infinity)). In Kant's solution to the latter problem, infinite judgment plays a decisive role.

In the critical solution to the antinomy (section 7 of the antinomy chapter) Kant initially described the first antinomy as an analytical opposition. To solve it, however, this antinomy should be described in terms of a dialectical opposition, which is different from a contradictory opposition (KrV, B532). To do so, Kant employed predicate negation (sc. «nichtunendlich»), which is characteristic of infinite judgment, instead of a negative copula (sc. «non est infinitus») (KrV, B531f.). Then, the antinomy was rephrased as follows: «die Welt ist entweder unendlich, oder endlich (nichtunendlich)» (KrV, B532). Thus, double predicate negation may also occur in an infinite judgment.

In what way does infinite judgment help to solve the antinomy? Contrary to the antithesis in an analytical opposition, infinite judgment in a dialectical opposition attaches a determination to the world, regarded as a thing actually existing in itself. If the world does not exist as a thing in itself, both thesis and antithesis are false since there is no such thing that is either finite or infinite. Indeed, we cannot regard the world as something existing in itself; it exists only in the empirical regress of the series of appearances (according to the results of the transcendental analytic) (KrV, B532f.). Thus, the solution consists of the introduction of infinite judgment, which enables us to «unmask» the antinomy as an analytical opposition based on the assumption of the world existing in itself.

Although Kant only presents the solution in the case of the first antinomy, it also holds for the other three: «Was hier von der ersten kosmologischen Idee... gesagt worden, gilt auch von allen übrigen.» (KrV, B533; cf. Ishikawa, o.c., 93). As to the mathematical antinomy, thesis and antithesis are both wrong; in the dynamical antinomy, they may both be true (KrV, B556, 559f., 590).

According to Ishikawa there is yet another instance of infinite judgment present in Kant's solution. Kant's formulation of the solution has the form of infinite judgment (Ishikawa, o.c., p. 87, 96, 117f.). At this point it is clear that infinite judgment is proper to the perspective of transcendental knowledge, because the solution (or rather: «decision», «Entscheidung», KrV, B525) is part of the critical, transcendental level of knowledge.

In addition to Menne's historical account there is more to be said about infinite judgment if we look at the beginning of the third chapter of the Transcendental Dialectic. In KrV, B600f., Kant stated that every thing, as regards its possibility, is subject to the ontological principle of complete determination (omnimoda determinatio), «nach welchem ihm von allen möglichen Prädikaten der Dinge, so fern sie mit ihren Gegenteilen verglichen werden, eines zukommen muß.» (cf. also KrV, B596). This principle concerns the content, rather than the mere logical form of knowledge. This feature distinguishes it from the logical determinability of concepts, which is in fact the combination of the principle of contradiction and excluded middle (KrV, B600n.). As we have pointed out above, this distinction is exactly what makes the difference between infinite judgment, on the one hand, and affirmative and negative judgment, on the other.

There is a close link between infinite judgment and the omnimoda determinatio. This principle represents the (historical) origin of infinite judgement (Ishikawa, o.c., 57-69). It also confirms systematic points that were neglected by Menne, which make Kant's use of infinite judgment meaningful. The main point of agreement between infinite judgment and the principle is that they relate to a content, i.e. a thing. Every thing that exists is completely determined (KrV, B601). In accordance with this proposition specific pairs of contradictory predicates are not only compared with one another logically, but the thing itself is transcendentally compared with the sum total of all possible predicates (ib.). In its turn, the sum total itself is also completely determined as the concept of a thing called the ideal of pure reason (KrV, B601f.). This ideal represents the transcendental content, the transcendental substrate, the matter, the whole of reality or the omnitudo realitatis of the possibility and complete determination of all things.

The complete determination of a thing rests on the limitation of this omnitudo; it is completely determined in so far as the omnitudo is limited to a certain extent or, as Kant puts it «die durchgängige Bestimmung eines jeden Dinges beruht auf der Einschränkung dieses All der Realität» (KrV, B605). It is exactly at this point that infinite judgment is a meaningful means to express a partial step in the process of complete determination. The negation contained in the judgment does not affect the copula nor does it express the non-existence of a thing, since the existence is presupposed as soon as we take up the determination of a thing. Kant referred to this «presupposition» by calling it «transzendentale Bejahung» in contradistinction to a «transzendentale Verneinung», which means «das Nichtsein an sich selbst» (KrV, B602). Infinite judgment refers to a thing while expressing transcendental affirmation, but it also expresses that only one of two contradictory predicates is to be assigned to a thing, thereby limiting the infinite sphere of all possible predicates represented by the omnitudo. In this context it is clear why Kant's example of the non-mortal soul in KrV, B97f. is not identical to a negative judgment. In «The soul is non-mortal», “the soul” refers to something and this something is determined to the extent that «mortal» is not an appropriate predicate belonging to that thing.

6. Infinite judgment and the polemical employment of reason

Saying something does not entail the existence of what is said. In fact, in the transcendental analytic existing things are limited to whatever can be given in possible experience. In the Transcendental Dialectic, judgments about illusory objects and their presumed existence are unmasked. Still, Kant allowed for some kind of judgment about these objects, although not in theoretical perspective. In order to say something relevant about these «objects» we see them in a practical perspective, as if they were practically real, not objectively real. In this sense infinite judgment plays its major role.

We have to be brief about this «as if» character of objects in a practical perspective. As early as his critical examination of dialectical illusion Kant alluded to the practical use of objects. As to the substantiality and permanence (immortality) of the soul, for example, we cannot claim knowledge because any such claim would transcend the limits of possible experience, «Gleichwohl wird hiedurch für die Befugnis, ja gar die Notwendigkeit, der Annehmung eines künftigen Lebens, nach Grundsätzen des mit dem spekulativen verbundenen praktischen Vernunftgebrauchs, hiebei nicht das mindeste verloren» (KrV, B424). But although we cannot claim knowledge, we are entitled to postulate a future life. Strictly speaking, this postulate is not legitimate because there is no possible proof to support it; it transcends the possible limits of knowledge. It is not illegitimate, either, for the very same reason. The only way left open is to claim some right to postulate a future life as if the soul were an object, as long as it serves a practical interest. The proper judgment to express this claim is infinite: it is not illegitimate to claim a future life, since this illegitimacy, expressed in a negative judgment, cannot be proven.

Apparently, an interpretation of the critical result in terms of infinite judgment must also allow the occurrence of double negation (e.g. «not illegitimate») in infinite judgment. In the previous section this was already pointed out in the case of Kant's solution to the antinomy. There are also examples of Kant's employment of double negation concerning the critical results in view of a practical perspective: the severity of criticism has rendered reason a not unimportant service (KrV, B424); transcendental employment of pure reason prepares the site for building our moral edifices, a task that is not unmeritorious (KrV, B375); we limit the law of the empirical use of understanding, but not to declare the intelligible impossible (KrV, B590) (all italics mine). These are telling examples, but Kant's systematic employment of infinite judgment and his arguments are found in the Transcendental Doctrine of Methods.

In the chapter on the discipline of pure reason, Kant supplied a negative legislation (discipline) in addition to the positive legislation of the transcendental analytic. In so far as the speculative use of reason is dialectical, transcendental logic is nothing but a discipline (KrV, B824). Hence, a discipline of pure reason supplies a systematic account of the negative function of critique, which was mentioned at the end of section 4 and the beginning of section 5 above: it is «admonitory negative teaching» (KrV, B740).

The section on the discipline of the polemical employment of reason seems to be an exception, since it seems to represent a relapse into a purely polemical situation, i.e. a situation in which opposing claims constitute a conflict of reason. However, conflicts of this kind have already been solved in the transcendental dialectic on the basis of the formal conditions of knowledge. Hence, conflicts and polemics have basically been prevented: «So gibt's demnach keine eigentliche Polemik im Felde der reinen Vernunft.» (KrV, B784; cf. KrV, B771, 778). Nevertheless, reason may be employed polemically, but only to defend certain propositions against opposite dogmatic denials (KrV, B767). These propositions themselves are also dogmatic, but they are affirmative and they are made in view of the practical interest of reason (KrV, B769f.; cf. KrV, B777). Here we can see the same strategy that Kant employed in his solution to the antinomy. Neither the assertion nor the denial can be demonstrated conclusively. If the assertion is made in respect of the practical interest, however, there is no conclusive proof to support it, but, what is more important, there is also no valid proof of the opposite dogmatic denial. This lack of proof leaves open the very possibility of practical dogmatic assertions and, consequently, these assertions are not illegitimate. In this sense the position of polemical employment of reason is expressed by the double negation of «not illegitmate» in an infinite judgment, which should not be reduced to the simple affirmation of the legitimacy of that claim. An infinite judgment declaring certain claims «not illegitimate» secures the provisional competence or authority in preparation for the practical employment of reason. Its function is exactly to formulate one of Kant's main objectives: to deny knowledge in order to make room for faith (KrV, Bxxx).

Finally, even this denial of knowledge is expressed in terms of an infinite judgment. In the section on the impossibility of a sceptical satisfaction of reason, following the section on polemical employment, Kant presented his version of the docta ignorantia in the form of critical ignorance expressing the scope and limits of our knowledge. Ignorance in this sense is not a lack of knowledge, but it is affirmative knowledge about what cannot be known. To be critically ignorant one needs fundamental knowledge of what can and what cannot be known. This ignorance delimits the room available for practical assertions and faith.

7. Conclusion

Infinite judgment is not insignificant. In fact, it is the most important kind of judgment in KrV, since it has the particular properties that are necessary to express the results of Kant's critical project. Apart from the fact that the distinction between affirmative, negative and infinite judgments corresponds to the structure of KrV (viz. Transcendental Analytic, Dialectic and Doctrine of Methods), Kant employed infinite judgment in his solution to antinomy and paralogisms and in the critical limitation of knowledge which prepares for the practical employment of reason. Furthermore, infinite judgment is used in polemical and defensive strategies to ensure realisation of the practical interest of reason. Neglect of the structural and systematic importance of infinite judgment would pose insurmountable problems to any interpretation of Kant's critical work.

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«ÇÍÀÊÈ IN CONCRETO»* - Ê ÊÀÍÒÎÂÑÊÎÌÓ «ÔÎÐÌÀËÈÑÒÈ×ÅÑÊÎÌÓ» ÎÁÎÑÍÎÂÀÍÈÞ ÌÀÒÅÌÀÒÈÊÈ

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 XIX âåêå ñòðåìëåíèÿ ðàçëè÷íûõ ìàòåìàòèêîâ áûëè íàïðàâëåíû ê òîìó, ÷òîáû ñâåñòè âñå ïîíÿòèÿ ðåàëüíîé è êîìïëåêñíîé òåîðèè ÷èñåë ê îäíîìó îñíîâíîìó ïîíÿòèþ, à èìåííî ìíîæåñòâó. Íà ðóáåæå XIX è XX âåêîâ óñèëèÿ ïî ïîñòðîåíèþ ìàòåìàòèêè íà ôóíäàìåíòå òåîðèè ìíîæåñòâ ïîíà÷àëó ïðèíåñëè ñèëüíîå ðàçî÷àðîâàíèå. Ïîÿâëåíèå ïàðàäîêñîâ â òåîðèè ìíîæåñòâ ïðèâåëî ìàòåìàòèêó ê êðèçèñó, íàçâàííîìó Ãåðìàíîì Âåéëåì «êðèçèñîì îñíîâàíèé». Âîçìîæíî, èìåííî ýòè íàäåæäû è ñòåïåíü ðàçî÷àðîâàíèÿ â íèõ è âûçâàëè óïîðíûå äèñêóññèè ïî ïðîáëåìå ïàðàäîêñîâ, õîòÿ óæå â 1908 ã. Öåðìåëî ïðåäúÿâèë äëÿ íåå àêñèîìàòè÷åñêîå ðåøåíèå.

Êðèçèñ îñíîâàíèé ïîðîäèë ðàçëè÷íûå ïîäõîäû ê íîâîìó îáîñíîâàíèþ ìàòåìàòè÷åñêèõ íàóê - â ÷àñòíîñòè, èíòóèöèîíèçì è ôîðìàëèçì. Ïðè ýòîì èíòóèöèîíèñòû óäåëÿëè áîëüøîå âíèìàíèå ðàçëè÷íûì àñïåêòàì «Êðèòèêè ÷èñòîãî ðàçóìà» (äàëåå Ê×Ð). Òàê, èíòóèöèîíèñòû ïîëàãàëè, ÷òî èìåííî Êàíò ãíîñåîëîãè÷åñêè îáîñíîâàë èäåþ î êîíñòðóèðîâàíèè ìàòåìàòè÷åñêèõ îáúåêòîâ äåÿòåëüíûì ñóáúåêòîì è, â ÷àñòíîñòè, òîò ôàêò, ÷òî òàêàÿ äåÿòåëüíîñòü îñóùåñòâëÿåòñÿ ê òîìó æå âî âðåìåíè.

Äàâèä Ãèëüáåðò, ê êîòîðîìó âîñõîäèò ôîðìàëèçì â îñíîâàíèÿõ ìàòåìàòèêè, òàêæå íåîäíîêðàòíî ïîä÷åðêèâàë ñâÿçè ìåæäó ñâîèì îáîñíîâàíèåì ìàòåìàòèêè è êàíòîâñêîé ôèëîñîôèåé. Òàêèì îáðàçîì, íàðÿäó ñ äèñêóññèåé îá îòíîøåíèè èíòóèöèîíèçìà ê êàíòîâñêîé ôèëîñîôèè âîçíèêàåò âîñõîäÿùèé ê íåîêàíòèàíñòâó ïîäõîä, ðàññìàòðèâàþùèé Êàíòà â ñâÿçè ñ ãèëüáåðòîâñêèì ôîðìàëèçìîì.  íîâåéøåå âðåìÿ ýòà ñâÿçü, êîòîðàÿ äîëãî íàõîäèëàñü â òåíè äèñêóññèè îá èíòóèöèîíèçìå, ñíîâà òùàòåëüíî èçó÷àåòñÿ ñ ðàçëè÷íûõ ñòîðîí.

 äàëüíåéøåì ÿ ññûëàþñü íà ïðåäøåñòâóþùèå ìîåé ðàáîòå ïî ýòîé òåìå èññëåäîâàíèÿ Óëüðèõà Ìàéåðà èç Ãåòòèíãåíà.  íèõ îí óáåäèòåëüíî ïîêàçûâàåò, ÷òî âîïðåêè øèðîêî ðàñïðîñòðàíåííîìó ìíåíèþ Ãèëüáåðò ïî ïðàâó ìîæåò ññûëàòüñÿ íà Êàíòà, êîãäà óòâåðæäàåò, ÷òî áåñêîíå÷íîå åñòü òîëüêî «èäåÿ» (â êàíòîâñêîì ñìûñëå). Ìàéåð óáåäèòåëüíî äîêàçàë, ÷òî òàêàÿ ññûëêà íà êàíòîâñêîå ó÷åíèå îá èäåÿõ ïðÿìî ñëåäóåò èç âçãëÿäîâ Ãèëüáåðòà, à èìåííî, ÷òî áåñêîíå÷íîå íå ìîæåò áûòü äàííûì íè êàê «àêòóàëüíî áåñêîíå÷íîå», íè êàê ñóùåñòâóþùåå â ýìïèðè÷åñêîì èëè ÷èñòîì ñîçåðöàíèè. Èñõîäÿ èç ýòîãî ïîíèìàíèÿ Ãèëüáåðò âïîñëåäñòâèè ñäåëàë âûâîäû, îïðîâåðãàþùèå ðàçâèòîå â Ê×Ð ó÷åíèå î ôîðìàõ ñîçåðöàíèÿ, â êîòîðûõ â ìàòåìàòèêå äàíî áåñêîíå÷íîå, âçÿòîå â íåêîòîðîì, åùå áîëåå îïðåäåëåííîì ñìûñëå. Òåì ñàìûì äëÿ áåñêîíå÷íîãî îñòàåòñÿ ëèøü ðîëü «÷èñòîé» èäåè â êàíòîâñêîì ñìûñëå.  îòíîøåíèè îáîñíîâàíèÿ ìàòåìàòèêè ãèëüáåðòîâñêàÿ ïîçèöèÿ, â ñîîòâåòñòâèè ñ òî÷êîé çðåíèÿ Ìàéåðà, íåñîìíåííî, áëèçêà äîêðèòè÷åñêîé ôèëîñîôèè, â êîòîðîé Êàíò îáîñíîâûâàåò ìàòåìàòèêó ïðè ïîìîùè «çíàêîâ in concreto».

Î÷åâèäíî, ÷òî ïîïûòêà îáîñíîâàòü ìàòåìàòèêó, îïèðàÿñü íà Ãèëüáåðòà è Êàíòà, âåñüìà ìíîãîîáåùàþùà, ïîñêîëüêó îáúåäèíÿåò ðàçëè÷íûå àñïåêòû êîíñòðóêòèâíîé òåîðèè ìàòåìàòèêè. Êðàòêî îáðèñîâàííûå òåçèñû Ìàéåðà ñ ýòîé òî÷êè çðåíèÿ âñå-òàêè âûãëÿäÿò îäíîñòîðîííèìè, èáî îí âî ìíîãîì ïðèñîåäèíÿåòñÿ ê ãèëüáåðòîâñêîé èíòåðïðåòàöèè Êàíòà. Ïîñêîëüêó Ãèëüáåðò èç ñèñòåìàòè÷åñêèõ ñîîáðàæåíèé íå ïðèíèìàåò ðÿä âàæíûõ ïîëîæåíèé, â ÷àñòíîñòè, îòíîñÿùèõñÿ ê òðàíñöåíäåíòàëüíîé ýñòåòèêå, èçëîæåíèå Ìàéåðà òàêæå îáðåìåíåíî òðóäíîñòÿìè (êîòîðûå áóäóò èçëîæåíû â äàëüíåéøåì) ôîðìàëèñòè÷åñêîãî îáîñíîâàíèÿ ìàòåìàòèêè â ñìûñëå Ãèëüáåðòà. Ìàéåð íå ïðåäëàãàåò íèêàêîãî ðàçðåøåíèÿ ýòèõ òðóäíîñòåé. Áîëåå ñèëüíîå âîçäåéñòâèå êàíòîâñêîé ôèëîñîôèè ìàòåìàòèêè â íåêîòîðûõ ñëó÷àÿõ ìîæåò ïîìî÷ü âûáðàòüñÿ èç òóïèêà.

 äàííîé ñòàòüå áåç ïðåòåíçèè íà èñ÷åðïûâàþùåå èçëîæåíèå èëè òðàêòîâêó ïðîáëåìû â öåëîì çàòðàãèâàþòñÿ òîëüêî äâà ïðîáëåìàòè÷åñêèõ ïóíêòà òåîðèè ìàòåìàòèêè ïî Ãèëüáåðòó:

1) âîïðîñ î ïðîèñõîæäåíèè ïîíÿòèÿ ìàòåìàòè÷åñêîé áåñêîíå÷íîñòè: â ñëó÷àå, êîãäà áåñêîíå÷íîå â ìàòåìàòèêå è ëîãèêå, äàæå åñëè îíî îáîçíà÷àåòñÿ êàê «îòðèöàíèå êîíå÷íîãî» èëè êàê ÷èñòàÿ èäåÿ, «íå ñîòâîðåíî ëþáèìûì Áîãîì» (çäåñü Ãèëüáåðò èðîíè÷åñêè èñïîëüçóåò âûñêàçàííûé â äðóãîé ñâÿçè Êðîíåêåðîì îáîðîò), ìû ñíîâà îêàçûâàåìñÿ ïåðåä çàäà÷åé ïîëíîãî ïðîÿñíåíèÿ ãåíåçèñà ýòîé «èäåè»;

2) ôîðìàëèñòè÷åñêîå îáîñíîâàíèå ìàòåìàòèêè ïðåæäå âñåãî îñíîâûâàåòñÿ - è ýòè ñîîáðàæåíèÿ äîëæíû áûòü ïðåäïîñëàíû òåîðèè áåñêîíå÷íîãî - íà òîé ïðåäïîñûëêå, ñîãëàñíî êîòîðîé ÷óâñòâåííî äàííûå è àðòèêóëèðîâàííûå (ò. å. âîñïðèíèìàåìûå êàê òàêîâûå è êîëè÷åñòâåííî îïðåäåëåííûå) êîíôèãóðàöèè âñåãäà ìîãóò áûòü òîëüêî êîíå÷íûìè (áåñêîíå÷íîå íå ìîæåò áûòü äàííûì â ñîçåðöàíèè). Òî÷íåå ãîâîðÿ, ýòî ïîëîæåíèå îçíà÷àåò (ïðèíèìàÿ âî âíèìàíèå ãèëüáåðòîâñêîå íàìåðåíèå îáîñíîâàòü íàòóðàëüíûå ÷èñëà òîëüêî ïðè ïîìîùè òàêèõ êîíôèãóðàöèé), ÷òî óæå íàòóðàëüíûå ÷èñëà, áîëüøèå 10 èëè 12 íå ìîãóò áîëåå áûòü ïðåäñòàâëåííûìè â ñîáñòâåííîì ñìûñëå (ò. å. ïîñðåäñòâîì àðòèêóëèðîâàííîãî öåëîñòíîãî âîñïðèÿòèÿ, íàïðèìåð, êàê ìíîæåñòâà ìåëîâûõ øòðèõîâ).

Ïîëîæèâ â îñíîâó îïðåäåëåííîå òîëêîâàíèå Ê×Ð, â êàíòîâñêîì ó÷åíèè î ÷èñòîì ñîçåðöàíèè ìîæíî íàéòè íåêîòîðûå ïîëîæåíèÿ òåîðèè ñîâðåìåííîé ìàòåìàòèêè, êîòîðûå ïî äîñòîèíñòâó îöåíèâàþò êàê ãèëüáåðòîâñêèå ðàçäóìüÿ îòíîñèòåëüíî ýìïèðè÷åñêèõ èëè ÷èñòûõ ñîçåðöàíèé â êà÷åñòâå èñòî÷íèêà áåñêîíå÷íîãî, òàê è îáå íàçâàííûå çäåñü ñèñòåìàòè÷åñêèå òî÷êè çðåíèÿ. Òî, ÷òî äëÿ òàêîé òåîðèè «çíàêàì in concreto» ïðèíàäëåæèò ðåøàþùàÿ ðîëü, áûëî çàìå÷åíî Êàíòîì óæå â äîêðèòè÷åñêèé ïåðèîä. Îäíàêî ïðåäëîæåííóþ Êàíòîì òåîðèþ ñëåäóåò âñå-òàêè äîïîëíèòü, ïðèíèìàÿ âî âíèìàíèå ñîâðåìåííóþ àáñòðàêòíóþ (ñôîðìóëèðîâàííóþ Ãèëüáåðòîì) ìàòåìàòèêó. Èìåííî êîíñòðóêöèè òàêîé ìàòåìàòèêè, ïðåæäå âñåãî íåýâêëèäîâà ãåîìåòðèÿ, áëèæå âñåãî ïîäõîäÿò ê ÷èñòî «ôîðìàëèñòè÷åñêîìó» âèäåíèþ ìàòåìàòèêè. Çäåñü ñëåäóåò îáðàòèòü âíèìàíèå íà òåîðåòèêî-ïîçíàâàòåëüíóþ ìîòèâàöèþ ôîðìàëèñòè÷åñêîãî ïîíèìàíèÿ ìàòåìàòèêè.

Ïîëó÷àþùàÿñÿ â ðåçóëüòàòå êàíòîâñêî-ãèëüáåðòîâñêàÿ ìîäåëü îáîñíîâàíèÿ ìàòåìàòèêè, çà êîòîðóþ ìû çäåñü âûñêàçûâàåìñÿ, äåìîíñòðèðóåò ñëåäóþùèé ôàêò: îáðàùåíèå ê Êàíòó ñòîðîííèêîâ íåñîâìåñòèìûõ ïîäõîäîâ ê íîâîìó îáîñíîâàíèþ ìàòåìàòèêè ìîæíî ðàññìàòðèâàòü êàê äâå ñòîðîíû îäíîé («êîíñòðóêòèâíîé») ìîíåòû.

Ïðåæäå âñåãî ñëåäóåò êðàòêî îáðèñîâàòü âàæíåéøèå ïîëîæåíèÿ òàê íàçûâàåìîé ïðîãðàììû Ãèëüáåðòà. Çàòåì ìû îïèøåì íåêîòîðûå àñïåêòû ãèëüáåðòîâñêîãî îáðàùåíèÿ ê Êàíòó è, íàêîíåö, â-òðåòüèõ, ðàññìîòðèì ãèëüáåðòîâñêóþ êðèòèêó Êàíòà, ïðè÷åì ïðèíèìàþòñÿ âî âíèìàíèå îáå ðàññìîòðåííûå ðàíåå ñèñòåìàòè÷åñêèå òî÷êè çðåíèÿ.

1. Ïðîãðàììà Ãèëüáåðòà

Ãèëüáåðòîâñêàÿ ïîïûòêà ôîðìàëèñòè÷åñêîãî îáîñíîâàíèÿ ìàòåìàòèêè - è â ýòîì îòíîøåíèè îí ðàñõîäèòñÿ ñ èíòóèöèîíèñòàìè - äîëæíà áûëà ñîõðàíèòü ïîñëåäíþþ â åå ñóùåñòâóþùèõ ôîðìàõ.  1922 ã. Ãèëüáåðò îáðèñîâàë ñâîè íàìåðåíèÿ â ñëåäóþùèõ ñëîâàõ: «Ìîåé öåëüþ ÿâëÿåòñÿ íàäåæíî îáîñíîâàòü ìàòåìàòèêó; ÿ õîòåë áû âîññòàíîâèòü ðåïóòàöèþ íåóÿçâèìîé èñòèííîñòè ìàòåìàòèêè, êîòîðàÿ, êàçàëîñü, áûëà ïîòåðÿíà áëàãîäàðÿ ïàðàäîêñàì òåîðèè ìíîæåñòâ; íî ÿ äóìàþ, ÷òî ýòî âîçìîæíî ïðè ïîëíîñòüþ ñîõðàíåííûõ åå ïîçèöèÿõ». Åãî óñèëèÿ ïðè ýòîì áûëè ñîñðåäîòî÷åíû íà ïðîáëåìå áåñêîíå÷íîãî è íàöåëåíû íà ðåøåíèå ýòîé ïðîáëåìû, êîòîðàÿ, êàê åìó ïðåäñòàâëÿëîñü, âûõîäèò çà ïðåäåëû îòäåëüíî âçÿòîé íàóêè.  ðàáîòå «Î áåñêîíå÷íîì» Ãèëüáåðò óòâåðæäàåò, «... ÷òî îêîí÷àòåëüíîå âûÿñíåíèå ñóùíîñòè áåñêîíå÷íîãî âûõîäèò çà ïðåäåëû óçêèõ èíòåðåñîâ ñïåöèàëüíûõ íàóê è, áîëåå òîãî, ÷òî îíî ñòàëî íåîáõîäèìûì äëÿ ÷åñòè ñàìîãî ÷åëîâå÷åñêîãî ðàçóìà».

Ãèëüáåðò äîñòèã áû ñâîèõ öåëåé, åñëè áû ìàòåìàòè÷åñêèå äåéñòâèÿ ñ ïîíÿòèåì áåñêîíå÷íîñòè è âñåìè ïîíÿòèÿìè, êîòîðûå íà íåå ïðÿìî èëè êîñâåííî ññûëàþòñÿ, äåéñòâèòåëüíî îêàçàëèñü áû áåññìûñëåííûìè. Äîêàçàòåëüñòâî ýòîãî, êàê íàäåÿëñÿ Ãèëüáåðò, äîëæíî áûëî áû íàõîäèòüñÿ â ðàìêàõ åãî «ïðîãðàììû êîíñåðâàòèâíîñòè»: ñàìûìè ïðîñòûìè èç âîçìîæíûõ ñðåäñòâ íóæíî äîêàçàòü, ÷òî èñïîëüçîâàíèå àáñòðàêòíûõ ìåòîäîâ â ìàòåìàòèêå, ò. å. ïðèìåíåíèå àáñòðàêòíûõ îïåðàöèé ïðè ïîñòðîåíèè äîêàçàòåëüñòâ, ÿâëÿåòñÿ êîíñåðâàòèâíûì. Ýòî îçíà÷àåò, ÷òî êàæäîå êîíêðåòíîå óòâåðæäåíèå, ïîëó÷åííîå ñ ïîìîùüþ àáñòðàêòíûõ ìåòîäîâ, ìîæåò áûòü ïîëó÷åíî è áåç íèõ. Äëÿ óòî÷íåíèÿ òîãî, ÷òî Ãèëüáåðò çäåñü èìååò â âèäó, ñëåäóåò ïîäðîáíî ïðîàíàëèçèðîâàòü ïîíÿòèå «ôèíèòèçì», êîòîðîå ñàì Ãèëüáåðò íå ñôîðìóëèðîâàë ÿñíî.

 îáëàñòè êîíêðåòíîé ìàòåìàòèêè ñóùåñòâóþò ôèíèòíûå îñìûñëåííûå ïðåäëîæåíèÿ è ôèíèòíûå ñðåäñòâà äîêàçàòåëüñòâà. Ôèíèòíûå îñìûñëåííûå âûñêàçûâàíèÿ ñóòü òàê íàçûâàåìûå «ðåàëüíûå âûñêàçûâàíèÿ», òî åñòü óðàâíåíèÿ òèïà , ãäå f è g ÿâëÿþòñÿ â íåêîòîðîì â ðàçóìíîì ñìûñëå ïðîñòûìè (íàïðèìåð, ïðîñòåéøèìè ðåêóðñèâíûìè) ôóíêöèÿìè. Êîíå÷íûå äîêàçàòåëüñòâà ñîîòâåòñòâóþò, íàïðèìåð, âû÷èñëåíèÿì èëè êîìáèíàòîðíûì ïåðåñòàíîâêàì. Ñëîæíûå âûñêàçûâàíèÿ, ê ïðèìåðó, òàêèå âûñêàçûâàíèÿ, êîòîðûå ññûëàþòñÿ íà áåñêîíå÷íûå ìíîæåñòâà, à òàêæå è òàêèå âûñêàçûâàíèÿ, êîòîðûå óïîòðåáëÿþò , ñóòü òîëüêî «èäåàëüíûå âûñêàçûâàíèÿ». Êàê òàêîâûå îíè íå èìåþò íèêàêîãî ñìûñëà, íî òåì íå ìåíåå ñ íèìè ìîæíî ðàáîòàòü ïðè ïîìîùè àáñòðàêòíûõ ìåòîäîâ. Ýòè èäåàëüíûå âûñêàçûâàíèÿ, õîòÿ è íå ïîçâîëÿþò âûâåñòè íîâûå ðåàëüíûå âûñêàçûâàíèÿ, óïðîùàþò ñîîòâåòñòâóþùèå òåîðèè. Òàê, äëÿ îïðåäåëåííûõ òåîðèé ìîæåò ñòàòü íåîáõîäèìûì «... ê êîíå÷íûì âûñêàçûâàíèÿì ïðèñîåäèíèòü èäåàëüíûå âûñêàçûâàíèÿ äëÿ òîãî, ÷òîáû óäåðæàòü ôîðìàëüíî ïðîñòûå çàêîíû îáû÷íîé àðèñòîòåëåâîé ëîãèêè».

×òîáû ëó÷øå ïîíÿòü õîä ìûñëåé Ãèëüáåðòà, íóæíî ïðåäñòàâèòü ñåáå, ÷òî îí òâåðäî ïðèäåðæèâàåòñÿ ñëåäóþùèõ óáåæäåíèé.

1. Ìîæíî îáîéòèñü áåç áåñêîíå÷íîãî ôàêòè÷åñêè â êàæäîì îòäåëüíîì ñëó÷àå. Ýòî è áûë óïîìÿíóòûé «çàìûñåë» åãî òåîðèè, ñîñòîÿùèé â òîì, ÷òîáû â êîíöå êîíöîâ ñâåñòè ìàòåìàòèêó ê êîíå÷íûì ïðîöåññàì: «Ìû äîëæíû áåñêîíå÷íîå, â ñìûñëå áåñêîíå÷íîé ñîâîêóïíîñòè, â òåõ ñëó÷àÿõ, ãäå îíî âñòðå÷àåòñÿ â âûâîäàõ åùå è òåïåðü, ïîíèìàòü êàê íå÷òî êàæóùååñÿ, ïîäîáíî òîìó, êàê â ïðåäåëüíûõ ïðîöåññàõ èñ÷èñëåíèÿ áåñêîíå÷íî ìàëûõ îêàçàëîñü âîçìîæíûì ïîêàçàòü, ÷òî áåñêîíå÷íîå, â ñìûñëå áåñêîíå÷íî ìàëîãî è áåñêîíå÷íî áîëüøîãî, åñòü ïðîñòî îáîðîò ðå÷è. È ïîäîáíî òîìó, êàê äåéñòâèÿ ñ áåñêîíå÷íî ìàëûìè áûëè çàìåíåíû ïðîöåññàìè â êîíå÷íîì, êîòîðûå äàþò òå æå ðåçóëüòàòû è ïðèâîäÿò ê òåì æå èçÿùíûì ôîðìàëüíûì ñîîòíîøåíèÿì, âûâîäû, ñîäåðæàùèå áåñêîíå÷íîå, äîëæíû áûòü âîîáùå çàìåíåíû êîíå÷íûìè ïðîöåññàìè, äàþùèìè â òî÷íîñòè òå æå ðåçóëüòàòû, ò. å. ïîçâîëÿþùèìè ïðîâîäèòü òîò æå õîä äîêàçàòåëüñòâà è ïðèìåíÿòü òå æå ìåòîäû äëÿ ïîëó÷åíèÿ ôîðìóë è òåîðåì».

2. Ïðè ýòîì ñëåäóåò èñõîäèòü èç òîãî, «... ÷òî êàæäàÿ ìàòåìàòè÷åñêàÿ ïðîáëåìà ðàçðåøèìà». Ëèøü áëàãîäàðÿ ãåäåëåâñêîé òåîðåìå î íåïîëíîòå áûëî ðàçðóøåíî ðàçäåëÿåìîå Ãèëüáåðòîì óáåæäåíèå â òîì, ÷òî âñå ìàòåìàòè÷åñêèå èñòèíû ÿâëÿþòñÿ òàêæå äîêàçóåìûìè.

 ñâîåé àðãóìåíòàöèè Ãèëüáåðò â ïóíêòå 1 èñõîäèò ïðåæäå âñåãî èç áåñêîíå÷íîñòåé â ïðèðîäå. «Áåñêîíå÷íàÿ äåëèìîñòü êîíòèíóóìà - ýòî îïåðàöèÿ, ñóùåñòâóþùàÿ òîëüêî â ÷åëîâå÷åñêîì ïðåäñòàâëåíèè, ýòî òîëüêî èäåÿ, êîòîðàÿ îïðîâåðãàåòñÿ íàøèìè íàáëþäåíèÿìè íàä ïðèðîäîé è îïûòàìè ôèçèêè è õèìèè». Òî æå ñàìîå ìîæíî ñêàçàòü è î áåñêîíå÷íî áîëüøèõ âåëè÷èíàõ: «Èç òîãî ôàêòà, ÷òî âíå êàêîãî-ëèáî êóñêà ïðîñòðàíñòâà âñåãäà ñíîâà èìååòñÿ ïðîñòðàíñòâî, ñëåäóåò òîëüêî íåîãðàíè÷åííîñòü ïðîñòðàíñòâà, à íå åãî áåñêîíå÷íîñòü. Íî ïîíÿòèÿ íåîãðàíè÷åííîñòü è êîíå÷íîñòü íå èñêëþ÷àþò äðóã äðóãà». Êàê ìàëî áåñêîíå÷íîå âñòðå÷àåòñÿ â ïðèðîäå, ñòîëü æå ìàëî îíî ìîæåò îñìûñëåííûì îáðàçîì ñîñòàâëÿòü îñíîâó ìûøëåíèÿ.

Òåì ñàìûì ïîëó÷àåòñÿ, ÷òî íè ñòàíîâÿùååñÿ «ïîòåíöèàëüíîå áåñêîíå÷íîå», íè áåñêîíå÷íîå, ìûñëèìîå êàê àêòóàëüíî çàâåðøåííîå, íèêîèì îáðàçîì íå ìîæåò áûòü âîñïðèíèìàåìûì: «... áåñêîíå÷íîå íèãäå íå ðåàëèçóåòñÿ. Åãî íåò â ïðèðîäå, è îíî íåäîïóñòèìî êàê îñíîâà íàøåãî ðàçóìíîãî ìûøëåíèÿ - çäåñü ìû èìååì çàìå÷àòåëüíóþ ãàðìîíèþ ìåæäó áûòèåì è ìûøëåíèåì».

Ãèëüáåðò ñ÷èòàë ðåàëüíûå âûñêàçûâàíèÿ, íàïðèìåð, ïðåäëîæåíèÿ ýëåìåíòàðíîé òåîðèè ÷èñåë, îñìûñëåííûìè, òàê êàê ïîëàãàë, ÷òî ýòîò ðàçäåë êîíêðåòíîé ìàòåìàòèêè ìîæíî âîçâåñòè ê ñîçåðöàíèþ, ïðè÷åì îí èñòîëêîâûâàë íàòóðàëüíûå ÷èñëà êàê ïîñëåäîâàòåëüíîñòü øòðèõîâ, êîòîðûå îáúåäèíÿþòñÿ ñ ïîìîùüþ çíàêà «+», êàê, íàïðèìåð,

«? + ? + ?». «Ýòè ÷èñëîâûå ñèìâîëû - îíè è ÿâëÿþòñÿ îáúåêòàìè íàøåãî ðàññìîòðåíèÿ - ñàìè ïî ñåáå íå èìåþò íèêàêîãî çíà÷åíèÿ». Ïðè ïîìîùè «÷èñåë» ìîæíî ïðèäàòü ñîçåðöàòåëüíûé õàðàêòåð îñíîâîïîëàãàþùåé îïåðàöèè òåîðèè íàòóðàëüíûõ ÷èñåë «ïðèáàâëåíèå åäèíèöû». Íàøå ïîçèöèîííîå äåñÿòè÷íîå èçîáðàæåíèå íàòóðàëüíûõ ÷èñåë, íàïðèìåð, ñ ïîìîùüþ àðàáñêèõ öèôð, ÿâëÿåòñÿ ñèìâîëè÷åñêèì, â òîì ñìûñëå, ÷òî àðàáñêèå çíàêè «1», «2»... îáîçíà÷àþò è òàêèå êîíôèãóðàöèè øòðèõîâ. Ñàìè ÷èñëà - íàïðèìåð, «? + ? + ?» - î÷åâèäíî, íå èìåþò íèêàêîãî äîïîëíèòåëüíîãî ñìûñëà, àðàáñêèå æå çíàêè, íàïðîòèâ, èìåþò ñâîèì çíà÷åíèåì èìåííî ÷èñëà.  ðàìêàõ òàêîãî îáîñíîâàíèÿ ìàòåìàòèêè íå òîëüêî áåñêîíå÷íûå îáúåêòû ÿâëÿþòñÿ èäåàëüíûìè ýëåìåíòàìè, íî óæå è îòðèöàòåëüíûå ÷èñëà íå ìîãóò ïîëó÷èòü íèêàêîãî íàãëÿäíîãî çíà÷åíèÿ.

 èíòåðåñàõ äàëüíåéøåãî èçëîæåíèÿ âàæíî ïîä÷åðêíóòü, ÷òî Ãèëüáåðò â ñâÿçè ñ ýòèì ÿâíî íàñòàèâàåò íà «ñîäåðæàòåëüíîñòè» ýëåìåíòàðíîãî ìàòåìàòè÷åñêîãî ìûøëåíèÿ: ïðåæäå âñåãî äîëæíû áûòü äàíû «...îïðåäåëåííûå âíåëîãè÷åñêèå êîíêðåòíûå îáúåêòû, êîòîðûå ñóùåñòâóþò íàãëÿäíî, â êà÷åñòâå íåïîñðåäñòâåííûõ ïåðåæèâàíèé äî êàêîãî áû òî íè áûëî ìûøëåíèÿ».  ïðèìåíåíèè ê íèì ñòàíîâÿòñÿ «íàäåæíûìè» ëîãè÷åñêèå âûâîäû, ò. å. àáñòðàêòíîå îïåðèðîâàíèå ñ îáùèìè îáúåìàìè è ñîäåðæàíèÿìè ïîíÿòèé. Äëÿ îáåñïå÷åíèÿ òàêîé íàäåæíîñòè «...ýòè îáúåêòû äîëæíû áûòü ïîëíîñòüþ âî âñåõ ñâîèõ ÷àñòÿõ îáîçðèìûìè; èõ ïîêàç, èõ ðàçëè÷èå, èõ ñëåäîâàíèå äðóã çà äðóãîì è ñóùåñòâîâàíèå îäíîãî èç íèõ íàðÿäó ñ äðóãèìè äàþòñÿ íåïîñðåäñòâåííî, íàãëÿäíî, âìåñòå ñ îáúåêòàìè êàê íå÷òî, íå ìîãóùåå áûòü ñâåäåííûì íè ê ÷åìó äðóãîìó è íå íóæäàþùååñÿ â òàêîì ñâåäåíèè...  ÷àñòíîñòè, â ìàòåìàòèêå ïðåäìåòîì íàøåãî ðàññìîòðåíèÿ ÿâëÿþòñÿ ñàìè êîíêðåòíûå çíàêè, âèä êîòîðûõ, ñîãëàñíî íàøåé óñòàíîâêå, ìîæåò áûòü íåïîñðåäñòâåííî îò÷åòëèâî è ìíîãîêðàòíî îïîçíàí».

Äàëåå Ãèëüáåðò äîáàâëÿåò, ÷òî îí êîðîòêî íàçûâàåò çíàêè òàêèõ îáðàçîâ òàêèìè æå çíàêàìè.

 ýòîì ïóíêòå ôîðìàëèçì ÷àñòî ñòàëêèâàåòñÿ ñ âîïðîñîì î òîì, íà êàêîì îñíîâàíèè â òàêîì ñëó÷àå êîíôèãóðàöèè, íàïðèìåð, ïÿòè ìåëîâûõ øòðèõîâ (êîòîðûå, åñòåñòâåííî, îòëè÷àþòñÿ äðóã îò äðóãà), ìîãóò áûòü ïðèçíàíû «îäíèì è òåì æå» çíàêîì, íà îñíîâàíèè ïîñòîÿíñòâà èëè æå ñïîñîáà ïîðîæäåíèÿ. Ýòîò âîïðîñ, êàê ïîêàçàë Ìàéåð, ìîæåò áûòü ïîñòàâëåí óæå èñõîäÿ èç òåêñòîâ ñàìîãî Ãèëüáåðòà. Ãèëüáåðò, õîòÿ è ãîâîðèò, ÷òî äëÿ ïîñòðîåíèÿ òåîðèè ÷èñåë íåîáõîäèìà èçâåñòíàÿ íàãëÿäíàÿ óñòàíîâêà a priori, îòíîñèòåëüíî Êàíòà âñå-òàêè ñ÷èòàåò, ÷òî ïîñëåäíèé «äàëåêî ïåðåîöåíèë ðîëü è îáúåì àïðèîðíîãî». Ìàéåð çäåñü ññûëàåòñÿ íà òî, ÷òî Ãèëüáåðò îòêëîíÿåò êàê ëîæíóþ àëüòåðíàòèâó ñòðîãîå ðàçëè÷èå ìåæäó àïðèîðíûì (â êàíòîâñêîì ñìûñëå, à çíà÷èò, ïî ìåíüøåé ìåðå ïîòåíöèàëüíîé áåñêîíå÷íîñòüþ) è ÷èñòûì ýìïèðè÷åñêèì ñîçåðöàíèåì. Ðàçðàáîòàííîå Ãèëüáåðòîì ïîíÿòèå ñîçåðöàíèÿ Ìàéåð îïèñûâàåò êàê «ñîçåðöàíèå a priori ôèíèòíîãî õàðàêòåðà». Îòëè÷èå ýòîãî ïîíÿòèÿ ñîçåðöàíèÿ îò êàíòîâñêîãî ÷åòêî ïðîÿâëÿåòñÿ â òîì, ÷òî äëÿ Ãèëüáåðòà áåñêîíå÷íîå íå ìîæåò áûòü äàíî â ñîçåðöàíèè (÷èñòîì, èëè ýìïèðè÷åñêîì, â ñìûñëå Êàíòà). «Èìåííî «ôèíèòíûé» ÿâëÿåòñÿ äëÿ Ãèëüáåðòà ïðèçíàêîì ñîçåðöàíèÿ, è íàîáîðîò, ñîçåðöàíèå ÿâëÿåòñÿ ãàðàíòîì ôèíèòíîñòè».

Äëÿ Ãèëüáåðòà ìàòåìàòèêà, òåì ñàìûì, íà÷èíàåòñÿ ñ ñîçåðöàíèÿ: «... â íà÷àëå áûë çíàê». Ýòè óñèëèÿ Ãèëüáåðòà ïî íîâîìó îáîñíîâàíèþ ìàòåìàòèêè ïîñðåäñòâîì âîçâðàùåíèÿ îñíîâíûõ ìàòåìàòè÷åñêèõ ïîíÿòèé ê ðåöåïòèâíîñòè, ñ êîòîðîé è â êîòîðîé ìîæåò áûòü äàíî òîëüêî êîíå÷íîå, î÷åâèäíûì îáðàçîì íàõîäÿòñÿ â ïîëíîì ïðîòèâîðå÷èè ñ òîé ðîëüþ, êîòîðóþ Êàíò â «Êðèòèêå ÷èñòîãî ðàçóìà» ïðèïèñûâàë â ìàòåìàòè÷åñêîì ïîçíàíèè ïðîñòðàíñòâó è âðåìåíè, ïîñêîëüêó ñ ââåäåíèåì ôîðì ñîçåðöàíèÿ â ìàòåìàòèêó ââîäèòñÿ áåñêîíå÷íîå.

Ê ñîæàëåíèþ, èç-çà îãðàíè÷åííîãî îáúåìà ñòàòüè ìû íå áóäåì çäåñü çàòðàãèâàòü ãèëüáåðòîâñêóþ ïðîãðàììó äîêàçàòåëüñòâà íåïðîòèâîðå÷èâîñòè (âòîðóþ ÷àñòü òàê íàçûâàåìîé ïðîãðàììû Ãèëüáåðòà) è åå ñâÿçü ñ êàíòîâcêèì ó÷åíèåì î ðåãóëÿòèâíîì óïîòðåáëåíèè èäåé. Óêàæåì ëèøü íà òî, ÷òî îïðîâåðãíóòàÿ òåîðåìîé Ãåäåëÿ ïðîãðàììà äîêàçàòåëüñòâà íåïðîòèâîðå÷èâîñòè íàðÿäó ñ äðóãèìè îñìûñëåííûìè ïðåäëîæåíèÿìè âêëþ÷àåò òàêæå è ãèëüáåðòîâñêóþ ïðîãðàììó äîêàçàòåëüñòâà êîñåðâàòèâíîñòè

2. Ãèëüáåðò è Êàíò

Êàê ðàç, â ñâÿçè ñ òîëüêî ÷òî óêàçàííûì íàìè ïîíèìàíèåì ïîíÿòèÿ áåñêîíå÷íîñòè, Ãèëüáåðò ññûëàåòñÿ íà Êàíòà: «Ðîëü, êîòîðàÿ îñòàåòñÿ áåñêîíå÷íîìó, ýòî òîëüêî ðîëü èäåè, - åñëè, ñîãëàñíî Êàíòó, ïîä èäååé ïîäðàçóìåâàòü ïîíÿòèå, îáðàçîâàííîå ðàçóìîì, êîòîðîå âûõîäèò çà ïðåäåëû âñÿêîãî îïûòà è ïîñðåäñòâîì êîòîðîãî êîíêðåòíîå äîïîëíÿåòñÿ â ñìûñëå öåëüíîñòè, - áîëåå òîãî, èäåè, êîòîðîé ìû ìîæåì âïîëíå äîâåðÿòü â ðàìêàõ, ïîñòàâëåííûõ òåîðèåé, íàìå÷åííîé è çàùèùàåìîé ìíîþ çäåñü».

Ïî ðàçëè÷íûì ïðè÷èíàì ýòà ññûëêà íà Êàíòà çàãàäàëà çàãàäêó ìíîãî÷èñëåííûì èíòåðïðåòàòîðàì Ãèëüáåðòà.  ÷àñòíîñòè, åñëè ó÷åñòü ðàçâèòèå íåýâêëèäîâîé ãåîìåòðèè, Ãèëüáåðò äîëæåí áûë áû îòêëîíèòü êàíòîâñêóþ òðàíñöåíäåíòàëüíóþ ýñòåòèêó è òåì ñàìûì òåîðèþ ñèíòåòè÷åñêèõ ñóæäåíèé a priori, à âìåñòå ñ íåé, êàê ÷àñòî íåîáîñíîâàííî óòâåðæäàþò, è êàíòîâñêóþ ôèëîñîôèþ âîîáùå. Åãî îáðàùåíèå ê êàíòîâñêèì èäåÿì ðàçóìà áûëî ïðåäïðèíÿòî êàê áû äëÿ êðàñíîãî ñëîâöà, è íà íåãî ìîæíî ñìîòðåòü êàê íà ñîâåðøåííûé ìàòåìàòèêîì ïðîñòèòåëüíûé ïðîìàõ.

Ìàéåð óáåäèòåëüíî äîêàçûâàåò ñ ïîìîùüþ ðàçëè÷íûõ, ÷àñòüþ íåîïóáëèêîâàííûõ, òåêñòîâ Ãèëüáåðòà, ÷òî îáðàùåíèå ïîñëåäíåãî ê êàíòîâñêîìó ó÷åíèþ îá èäåÿõ ðàçóìà è èõ ÷èñòî ðåãóëÿòèâíîì óïîòðåáëåíèè ñîâåðøåííî îïðàâäàíî. Ñîãëàñíî Ìàéåðó, èç ãèëüáåðòîâñêîé êðèòèêè êàíòîâñêîãî îáúÿñíåíèÿ ïðîñòðàíñòâà è âðåìåíè ïðÿìî ñëåäóåò, ÷òî, âîïðåêè îïèñàííîìó âûøå ïîñïåøíîìó èãíîðèðîâàíèþ ãèëüáåðòîâñêîãî îáðàùåíèÿ ê Êàíòó, Ãèëüáåðò îáîçíà÷àåò áåñêîíå÷íîå, êîòîðîå äëÿ íåãî íå ìîæåò áûòü äàíî íè â ýìïèðè÷åñêîì, íè â ÷èñòîì ñîçåðöàíèè, òîëüêî êàê èäåþ â êàíòîâñêîì ñìûñëå.

 «Êðèòèêå ÷èñòîãî ðàçóìà» åñòü äâà î÷åíü ïîïóëÿðíûõ âûñêàçûâàíèÿ î ïðîñòðàíñòâå è âðåìåíè:

1. «Ïðîñòðàíñòâî ïðåäñòàâëÿåòñÿ êàê áåñêîíå÷íàÿ äàííàÿ âåëè÷èíà» ( 39, à òàêæå À 25). Àíàëîãè÷íî äëÿ âðåìåíè, êîòîðîå òàêæå äîëæíî âîñïðèíèìàòüñÿ «êàê íåîãðàíè÷åííî äàííîå» (Â48 / À32).

2, à. «Èñòèííîå (òðàíñöåíäåíòàëüíîå) ïîíÿòèå áåñêîíå÷íîñòè çàêëþ÷àåòñÿ â òîì, ÷òî ïîñëåäîâàòåëüíûé ñèíòåç åäèíèöû èçìåðåíèÿ êîëè÷åñòâà íèêîãäà íå ìîæåò áûòü çàêîí÷åí» (Â460). Ê ýòîìó ìîæíî ïðèáàâèòü:

2, á. Ïðîñòðàíñòâî «äåëèìî äî áåñêîíå÷íîñòè» ( 553).

 ñîîòâåòñòâèè ñ âûøåèçëîæåííîé ãèëüáåðòîâñêîé àðãóìåíòàöèåé, êàñàþùåéñÿ ïîíÿòèÿ áåñêîíå÷íîñòè, è åãî íàìåðåíèåì îòîæäåñòâèòü ÷èñëà â ñîáñòâåííîì ñìûñëå ñ ðåöåïòèâíî äàííûìè è ïîòîìó ïî íåîáõîäèìîñòè êîíå÷íûìè êîíôèãóðàöèÿìè ìîæåò ïîêàçàòüñÿ, ÷òî Ãèëüáåðò äîëæåí ðåøèòåëüíî è âïîëíå ñïðàâåäëèâî îòêàçàòüñÿ êàê îò âûñêàçûâàíèé (1), ñîãëàñíî êîòîðûì è ïðîñòðàíñòâî è âðåìÿ (êîòîðûå äîëæíû áûòü áåñêîíå÷íûìè âåëè÷èíàìè) äàíû (â ñìûñëå èíòóèòèâíî äàííûõ), òàê è îò (2) - ïðîñòðàíñòâî è âðåìÿ ñóòü áåñêîíå÷íûå â ñìûñëå áåñêîíå÷íûõ äåëèìîñòè è ïðîäîëæàåìîñòè. Ýòîò îòêàç áûë âûðàæåí â ñëåäóþùåì âåñüìà òèïè÷íîì äëÿ Ãèëüáåðòà âûñêàçûâàíèè, êîòîðîå ïî ñîäåðæàíèþ ïîëíîñòüþ àíàëîãè÷íî óæå ïðèâåäåííîé âûøå öèòàòå èç ñòàòüè «Î áåñêîíå÷íîì»: «Âòîðîé ðàç ìû íàòàëêèâàåìñÿ â ïðèðîäå íà âîïðîñ î áåñêîíå÷íîñòè ïðè ðàññìîòðåíèè âñåëåííîé â öåëîì. Ìû äîëæíû òåïåðü èññëåäîâàòü ïðîòÿæåííîñòü âñåëåííîé, ÷òîáû óçíàòü, íåò ëè çäåñü áåñêîíå÷íî áîëüøîé âåëè÷èíû... Ïðè ïîïûòêàõ óìîçðèòåëüíî ïîêàçàòü áåñêîíå÷íîñòü ïðîñòðàíñòâà âêðàäûâàëèñü òàêæå è î÷åâèäíûå îøèáêè. Èç òîãî ôàêòà, ÷òî âíå êàêîãî-ëèáî êóñêà ïðîñòðàíñòâà âñåãäà ñíîâà èìååòñÿ ïðîñòðàíñòâî, ñëåäóåò òîëüêî íåîãðàíè÷åííîñòü ïðîñòðàíñòâà, à íå åãî áåñêîíå÷íîñòü. Íî ïîíÿòèÿ íåîãðàíè÷åííîñòü è êîíå÷íîñòü íå èñêëþ÷àþò äðóã äðóãà. Ìàòåìàòè÷åñêèå èññëåäîâàíèÿ äàþò íàì òàê íàçûâàåìóþ ýëëèïòè÷åñêóþ ãåîìåòðèþ - åñòåñòâåííóþ ìîäåëü êîíå÷íîãî ìèðà».

Ïðèíèìàÿ âî âíèìàíèå ýòó ñèòóàöèþ, Ìàéåð ññûëàåòñÿ òåïåðü íà prima facie (ñ ïåðâîãî âçãëÿäà, ïî âèäèìîñòè (ëàò.). - Ïðèì. ðåä.) ñóùåñòâåííîå ñõîäñòâî ìíåíèÿ Ãèëüáåðòà ñ îáîñíîâàíèåì ìàòåìàòèêè ïðè ïîìîùè «çíàêîâ in concreto», êîòîðîå Êàíò ïðåäïîëîæèòåëüíî ïðåäëàãàåò â 1764 ãîäó â ñâîåé ñòàòüå «Èññëåäîâàíèå î ÿñíîñòè ïîëîæåíèé íàòóðàëüíîé òåîëîãèè è ìîðàëè».

 ýòîé ñòàòüå Êàíò êàæåòñÿ èññëåäîâàòåëåì èç ÕÕ âåêà, îïèñûâàþùèì ñîâðåìåííóþ ìàòåìàòèêó ñ ïîçèöèé ôîðìàëèçìà. Îí ãîâîðèò çäåñü î ÷èñòîé «...ïðîèçâîäèìîé ïî ïðàâèëàì ïåðåñòàíîâêå çíàêîâ...», ÷åì äåëàåò âîçìîæíûì òî, ÷òî â õîäå äîêàçàòåëüñòâà «ñàìè îáîçíà÷àåìûå âåùè îñòàþòñÿ ïðè ýòîì ñîâåðøåííî âíå ñôåðû ìûñëè...». Êàê âñåãäà ñëó÷àåòñÿ ïðè ÷òåíèè ïîäîáíûõ ýòîìó ìåñò, Êàíò çäåñü íå èìåë â âèäó ôîðìàëèñòè÷åñêîå îáîñíîâàíèå ìàòåìàòèêè, ò. ê. îí ïîëàãàë, ÷òî «...â êîíöå êîíöîâ íå ðàñøèôðîâûâàåòñÿ çíà÷åíèå ñèìâîëè÷åñêîãî âûâîäà». Ýòè ìàòåìàòè÷åñêèå îáúåêòû óæå â äîêðèòè÷åñêèé ïåðèîä èìåþò ñèíòåòè÷åñêóþ ïðèðîäó: «Äåëî â òîì, ÷òî ìàòåìàòèêà íèêîãäà íå îáúÿñíÿåò êàêîãî-ëèáî äàííîãî ïîíÿòèÿ ïîñðåäñòâîì ðàñ÷ëåíåíèÿ, à âñåãäà ðàçúÿñíÿåò îáúåêò, ïðîèçâîëüíî ñîåäèíÿÿ [ïðèçíàêè], áëàãîäàðÿ ÷åìó òîëüêî è ñòàíîâèòñÿ âîçìîæíîé ñàìà ìûñëü îá ýòîì îáúåêòå».

Êàíò ãîâîðèò â ïðèâåäåííûõ öèòàòàõ îá ýòèõ ñèíòåòè÷åñêèõ îáúåêòàõ êàê î «çíà÷åíèÿõ» çíàêîâ. Íà ýòè ïîíÿòèÿ óêàçûâàþò ëèòåðû, êîòîðûìè ìàíèïóëèðóþò, íå ïðèíèìàÿ âî âíèìàíèå èõ çíà÷åíèé. Íî êàê âûãëÿäèò ýòî «óêàçàíèå» ó Êàíòà? Äëÿ îòâåòà íà ýòîò âîïðîñ äîëæíî õâàòèòü êîðîòêîé öèòàòû èç Ê×Ð. Ïîñëå òîãî êàê Êàíò ðàçäåëèë «èíòóèòèâíûé ñïîñîá ïðåäñòàâëåíèÿ» íà «ñõåìàòè÷åñêèé» è «ñèìâîëè÷åñêèé», îí èìååò â âèäó ïîä ýòèì òîëüêî ÷èñòûå «õàðàêòåðèñòèêè», «... ò.å. îáîçíà÷åíèÿ ïîíÿòèé ÷åðåç ñîïóòñòâóþùèå ÷óâñòâåííûå çíàêè, êîòîðûå âîâñå íå ñîäåðæàòñÿ â ñîçåðöàíèè îáúåêòà êàê íåîáõîäèìûå, à òîëüêî ïî çàêîíó àññîöèàöèè ñèëû âîîáðàæåíèÿ, ïîýòîìó â ñóáúåêòèâíîì íàìåðåíèè ñëóæàò ñðåäñòâîì âîñïðîèçâåäåíèÿ». Ê ïîñëåäíèì îí ïðè÷èñëÿåò «âèäèìûå» - äàííûå çðèòåëüíî - «àëãåáðàè÷åñêèå çíàêè». Çäåñü âàæíî, îäíàêî, ÷òî îíè ñóòü òîëüêî «âûðàæåíèÿ äëÿ ïîíÿòèÿ». Êàíò ïðèçíàåò â ýòîì ñìûñëå, ÷òî ìàòåìàòè÷åñêèå îáúåêòû - «ïîíÿòèÿ» - îòëè÷èìû îò «õàðàêòåðèñòèê». Îí îáñóæäàåò îáúåêòû ìàòåìàòèêè - è ñäåëàë áû ýòî òàê æå è ñ «ïîíÿòèÿìè» ñîâðåìåííîé ëîãèêè - ñ òî÷êè çðåíèÿ çíà÷åíèé.

Ïðåäìåòû, íà êîòîðûå óêàçûâàþò ñèãíèòèâíî èñïîëüçóåìûå çíàêè, ïî Êàíòó ÿâëÿþòñÿ òåì ñàìûì çíà÷åíèÿìè, à ñëåäîâàòåëüíî, ìåíòàëüíûìè ïðåäìåòàìè. Ïî Ãèëüáåðòó, íàïðîòèâ, ÷èñëîâûå çíàêè - ýòî äàííûå â âîñïðèÿòèè ïðåäìåòû. Ñ ýòîé òî÷êè çðåíèÿ ñõîäñòâî ìåæäó ãèëüáåðòîâûì ôîðìàëèçìîì è Êàíòîâûì èçëîæåíèåì 1764 ãîäà - âîïðåêè òåçèñó Ìàéåðà - ïðåäñòàâëÿåòñÿ ñïîðíûì.  äàëüíåéøåì ìû åùå âîçâðàòèìñÿ ê ýòîìó ïðåäïîëàãàåìîìó îòíîøåíèþ.

3. «Çíàêè in concreto» è ôîðìû ñîçåðöàíè

Íà ôîíå ãèëüáåðòîâñêîé êðèòèêè ïðîèçâåäåííîãî â òðàíñöåíäåíòàëüíîé ýñòåòèêå âûâîäà ïîíÿòèÿ áåñêîíå÷íîñòè èç ôîðì ñîçåðöàíèÿ ñàìîå âðåìÿ çàíÿòüñÿ âîïðîñîì, êàêóþ öåííîñòü ìîæåò èìåòü êàíòîâñêàÿ ôèëîñîôèÿ ìàòåìàòèêè äëÿ îáîñíîâàíèÿ ìàòåìàòèêè íà «çíàêàõ in concreto». Êàê ïîêàçàíî â ïðåäûäóùåì ðàçäåëå, Êàíò â ñâîåì «Èññëåäîâàíèè ÿñíîñòè îñíîâíûõ ïîëîæåíèé...» äàæå íå îñóùåñòâëÿåò òàêóþ ïðîãðàììó.

Ìû íå ìîæåì óäåëèòü äîëæíîãî âíèìàíèÿ îáñòîÿòåëüíîìó èçëîæåíèþ òîé ðîëè, êîòîðàÿ â «Êðèòèêå ÷èñòîãî ðàçóìà» ïðèïèñûâàåò â ìàòåìàòèêå ïðîñòðàíñòâó è âðåìåíè. Ñëåäóåò óïîìÿíóòü òîëüêî î òîì, ÷òî ôîðìû ñîçåðöàíèé êîíñòèòóèðóþò îáúåêòû ìàòåìàòèêè íå per se (ñàìè ïî ñåáå (ëàò.). - Ïðèì. ðåä.) - ò. å. íå ïîñðåäñòâîì «÷èñòîãî ñîçåðöàíèÿ» íåêîòîðîãî ðîäà. Ñèíòåçû, â êîòîðûõ ïîëó÷àþòñÿ «ôîðìàëüíûå ñîçåðöàíèÿ», ò. å. - â ïðåäìåòíîì àñïåêòå - ñòðóêòóðû îïðåäåëåííûõ îáúåêòîâ, íàïðèìåð, îïðåäåëåííûõ ïðîñòðàíñòâ â åâêëèäîâîé ãåîìåòðèè, ïðîèçâîäÿòñÿ íå òîëüêî ïðè ïîìîùè ïðîñòðàíñòâà è âðåìåíè, íî ïðåäñòàâëÿþò ñîáîé «ôèãóðíûå ñèíòåçû» [Â151-4, 204, 752, 196] ïðîäóêòèâíîãî âîîáðàæåíèÿ, â êîòîðûå âñòðîåíû (ìàòåìàòè÷åñêèå) êàòåãîðèè.  ýòîì ñìûñëå èìåþòñÿ îïåðàöèè, êîòîðûå îäíîâðåìåííî ÿâëÿþòñÿ ñîçåðöàíèåì è ìûøëåíèåì.

 ðàìêàõ ìàòåìàòèêè îïåðàöèè â ïðîäóêòèâíîì àðèôìåòè÷åñêîì è ãåîìåòðè÷åñêîì êîíñòðóèðîâàíèè òàêæå ïðîèñõîäÿò âñåãäà è íåîáõîäèìî â ôîðìàõ ñîçåðöàíèÿ. Åñëè ïðèíÿòü âî âíèìàíèå îäíó âàæíóþ äåôèíèöèþ, äàííóþ â ðàìêàõ òðàíñöåíäåíòàëüíîé ýñòåòèêè, à èìåííî îïðåäåëåíèå «ÿâëåíèÿ» êàê «ïðåäìåòà íàøåãî ÷óâñòâà» [À34/Â51], òî îòñþäà åñòåñòâåííî âûòåêàåò, ÷òî «â» ýòîì «ïðîñòðàíñòâå» è ýòîì «âðåìåíè» è, ñ äðóãîé ñòîðîíû, ñðåäè òîãî, ÷òî çäåñü ïîðîæäàåòñÿ, - ÷èñåë, ãåîìåòðè÷åñêèõ ôèãóð è ò. ä. - íå ìîæåò áûòü äàíî íèêàêîå ÿâëåíèå â ñìûñëå «ïðåäìåòà ÷óâñòâà». Èíà÷å ãîâîðÿ, ïî Êàíòó, ïðîñòðàíñòâî è âðåìÿ âïîëíå ìîæíî ìûñëèòü òàêèì îáðàçîì, ÷òî «íåòðóäíî ïðåäñòàâèòü ñåáå îòñóòñòâèå ïðåäìåòîâ â íåì» [Â38-9]. Íàïðèìåð, â îòíîøåíèè âðåìåíè: «ÿâëåíèå ïðåêðàñíî ìîæíî îòäåëèòü îò âðåìåíè» [À31/Â46]. Êàíò ïðàâ çäåñü ëèøü â îòíîøåíèè òîãî ôàêòà, ÷òî òàêæå òî÷êè è ëèíèè êîíêðåòíîé, ò. å. åâêëèäîâîé ãåîìåòðèè, íå ÿâëÿþòñÿ ïðåäìåòàìè íàøåãî ÷óâñòâà, äðóãèìè ñëîâàìè, íèêàêîé òî÷íûé ìàòåìàòè÷åñêèé îáúåêò íå ÿâëÿåòñÿ èäåàëèçèðîâàííûì ïðåäìåòîì «ÿâëåíèå» â ñìûñëå åãî îïðåäåëåíèÿ êàê «ïðåäìåòà íàøåãî ÷óâñòâà».

(i)

Ïîñëå ýòèõ ïðåäâàðèòåëüíûõ çàìå÷àíèé ñëåäóåò íà÷àòü ðàçúÿñíåíèå óñëîâèé (1) è (2). Ïðè ýòîì âàæíî èìåòü ïåðåä ãëàçàìè òî ïðîòèâîðå÷èå, êîòîðîå prima facie ñóùåñòâóåò ìåæäó ýòèìè ïóíêòàìè òðàíñöåíäåíòàëüíîé ýñòåòèêè. Ïåðâîå âûñêàçûâàíèå (ïðîñòðàíñòâî ÿâëÿåòñÿ ïðåäñòàâëåííûì êàê áåñêîíå÷íàÿ «äàííàÿ» âåëè÷èíà) êàæåòñÿ íåñîâìåñòèìûì ñî âòîðûì âûñêàçûâàíèåì (÷òî ñóùåñòâóåò íåêàÿ áåçãðàíè÷íîñòü â âîçìîæíîì ïðîäîëæåíèè ñîçåðöàíèÿ, íàïðèìåð, ïðîñòðàíñòâî áåñêîíå÷íî äåëèìî).

Ïðèíèìàÿ âî âíèìàíèå (2), ñíà÷àëà ñêàæåì, ÷òî ïîäðàçóìåâàåìûå ïîä ïðîñòðàíñòâîì è âðåìåíåì îïåðàöèè íèêîèì îáðàçîì íå ñîäåðæàò per se îãðàíè÷åíèÿ.  ýòîì ñìûñëå ñîâåðøåííî íîðìàëüíî ãîâîðèòü î «áåçãðàíè÷íîñòè âîçìîæíîãî ïðîäîëæåíèÿ ñîçåðöàíèÿ». Íà ýòîì ôîíå ïîíèìàåòñÿ òàêæå è ñìûñë âàæíîãî äëÿ ïðåäïîëàãàåìîãî ïðîòèâîðå÷èÿ ñëîâà «äàííûé» â (1). Êðàéçåëü â ñâÿçè ñ ýòèì ññûëàåòñÿ íà òî, ÷òî Êàíò çäåñü äîëæåí áûë ñêàçàòü «ïðîñòðàíñòâî» è «âðåìÿ» «äàíû» êàê áåñêîíå÷íûå = áåçãðàíè÷íûå, áåç óñòàíîâëåííûõ ãðàíèö, ïðè÷åì ñëîâî «äàíû» â ñîîòâåòñòâèè ñ çàìûñëîì Êðèòèêè ñëåäóåò åñòåñòâåííûì îáðàçîì ïîíèìàòü â òîì ñìûñëå, ÷òî ïðîñòðàíñòâî è âðåìÿ ñóòü ôîðìû ñîçåðöàíèÿ, îò êîòîðûõ ìû íå ìîæåì íè îñâîáîäèòüñÿ, íè çàìåíèòü èõ íà äðóãèå ïðàâèëà ñîçåðöàíèÿ.

Ýòî òîëêîâàíèå íå òîëüêî «ñïàñàåò» òðàíñöåíäåíòàëüíóþ ýñòåòèêó îò íåïðåîäîëèìûõ ôåíîìåíîëîãè÷åñêèõ íåîïðåäåëåííîñòåé, êîòîðûå íàäåÿëñÿ îáíàðóæèòü çäåñü Ãèëüáåðò, íàïðèìåð, îò ãðóáîãî ïðîòèâîðå÷èÿ, íî ê òîìó æå è â áîëüøåé ìåðå ñîîòâåòñòâóåò îñíîâíîé ìûñëè Êðèòèêè, êîòîðàÿ äîëæíà èìåòü ïðèîðèòåò äàæå òîãäà, êîãäà îíà âñòóïàåò â íåóñòðàíèìîå ïðîòèâîðå÷èå ñ îòäåëüíûìè ìåñòàìè òåêñòà. Ýòà îñíîâíàÿ ìûñëü, êîòîðàÿ a fortiori (òåì ñàìûì (ëàò.). - Ïðèì. ðåä.) âåðíà òàêæå è äëÿ òðàíñöåíäåíòàëüíîé ýñòåòèêè (è, òàêèì îáðàçîì, çàñòàâëÿåò êðèòèêó Ãèëüáåðòà áèòü ìèìî öåëè), ÿâëÿåòñÿ âàæíîé ïðåäïîñûëêîé «Êðèòèêè ÷èñòîãî ðàçóìà», ñîãëàñíî êîòîðîé ÷åëîâå÷åñêèé äóõ è âñå åãî äåéñòâèòåëüíûå àêòû èëè äåéñòâèÿ êîíå÷íû è âðåìåííû, ò. å. ñóêöåññèâíû. Ïî Êàíòó, î íå÷åëîâå÷åñêèõ ñïîñîáàõ ñîçåðöàíèÿ íåëüçÿ ñêàçàòü íè÷åãî îñìûñëåííîãî: ìû íå ìîæåì «... ñóäèòü î ñîçåðöàíèÿõ èíûõ ìûñëÿùèõ ñóùåñòâ, ñâÿçàíû ëè îíè ñóáúåêòèâíûìè óñëîâèÿìè, êîòîðûå îãðàíè÷èâàþò íàøè ñîçåðöàíèÿ...» (ïîä÷åðêíóòî ìíîé. - Î. Â.) è îò êîòîðûõ ìû íå ìîæåì îñâîáîäèòüñÿ, ò. å. êîòîðûå «ÿâëÿþòñÿ äëÿ íàñ îáùåçàêîííûìè» [À27/Â43].

(ii)

Èç ýòèõ ñîîáðàæåíèé ñëåäóåò òî, ÷òî êàæäàÿ ñêîíñòðóèðîâàííàÿ ôèãóðà, ÷èñëî è àðèôìåòè÷åñêàÿ èëè àëãåáðàè÷åñêàÿ ôîðìóëà è âñå ñêîíñòðóèðîâàííûå âåùè è ïðîöåññû äîëæíû áûòü ïîñòðîåíû ïðè ïîìîùè «ïðîñòðàíñòâà», «âðåìåíè» è êàòåãîðèé â âèäå áëîêà âðåìåííûõ ñóêöåññèâíûõ îïåðàöèé â ðàìêàõ íåêîòîðîãî ïðîñòðàíñòâà êàê «ôèãóðíî îïðåäåëåííûå ïðîñòðàíñòâà», ò. å. îïðåäåëåííûå ïðîñòðàíñòâåííûå ôèãóðû. Ïîñëå òîãî êàê áûëà âûÿâëåíà êîíå÷íîñòü òàêæå è êàíòîâñêîé êîíöåïöèè ñîçåðöàíèÿ (÷èñòîãî èëè ýìïèðè÷åñêîãî), âòîðîé øàã ê ïðîÿñíåíèþ ïîíÿòèÿ áåñêîíå÷íîñòè ó Êàíòà òåïåðü îñíîâûâàåòñÿ íà äàëüíåéøåì ðàçëè÷åíèè, êîòîðîå òîëüêî ÷òî îïèñàííîé, ñóêöåññèâíî êîíñòðóèðóåìîé ñóáúåêòèâíîñòè ñîïîñòàâëÿåò â êà÷åñòâå êîððåëÿòà îáúåêòèâíóþ ñòîðîíó