Philosophical Investigations on Time, Space and the Continuum (Routledge Revivals)

Franz Brentano, Barry Smith

• Publisher: Routledge
• Year: 2009
• Print ISBN: 9780415563789
• eBook ISBN: 9781135175856
• Edition: 1
• Topic: Philosophy
• Subtopic: Philosophy History & Theory

Franz Brentano is recognised as one of the most important philosophers of the late nineteenth and early twentieth centuries. This work, first published in English in 1988, besides being an important contribution to metaphysics in its own right, has considerable historical importance through its influence on Husserl's views on internal time consciousness. The work is preceded by a long introduction by Stephan Körner in collaboration with Brentano's literary executor.

Part One
THE CONTINUUM

I.
On what is continuous

Dictated 22 November 1914 [Meg 24]

This item was transcribed for publication by Kastil on two occasions (in 1943 and 1944). The texts of the two versions differ slightly, and the version that is reproduced here is based on Kastil’s transcription from the year 1943. [Editors’ note.]

1. The question concerning the concept of continuity cannot be framed in such a way that one would call into doubt whether we do in fact possess such a concept. For otherwise we would not be able to understand ourselves when arguing about other aspects of this concept. This implies also that we have attained this concept in the same way in which we are able to attain other concepts. Thus it cannot be inherited and given from the start, since this holds of none of our other concepts.¹ Yet we have been in possession of it long before we began to philosophise, and our memory can no longer tell us about the way it first developed. Fortunately however it is true of all our concepts that they are actually not given to us without interruption, so that the process by which they develop is in fact repeated even now. We can perceive this process and analyse it while it is still fresh in our memory.²

2. All our concepts are either taken immediately from an intuition or combined out of marks that are taken from intuition.³ It has lately been asserted that it is in this second way that we arrive at the concept of what is continuous. One has, for example, pointed to the cases in which we insert fractions between two numbers, between 1 and 2, or indeed between 0 and 1. Thus for example we insert 1/2 between 0 and 1, and then continue halving by inserting 1/4 and 3/4, 1/8, 3/8, 5/8, 7/8, 1/16, 3/16 and so on. If one imagines this process of halving to be continued to infinity, then nowhere would there be manifested any gap of finite magnitude, and the result would already be something that one could accept as an example of continuity. Yet this continuity would be incomplete, for there would still remain certain relations of magnitude which, even after a completely executed process of halving, would be represented by no inserted fraction, as for example that relation which is signified by the fraction 1/3. This then leads to the further requirement that we should proceed as with 2 so also with 3, 5, and all other natural numbers not yet implicitly used. We seem thereby to have achieved something that approximates much more to an example of something completely continuous. But still, the algebraic relations of irrationals would remain unconsidered, and thus in order to fill the still unoccupied positions one would have to carry out further insertions of the algebraic irrational numbers. Once this has been done, it is still of course not excluded that there would be more to be inserted, but then one would already be so close to the idea of continuity, of that which would leave no room for any sort of intercalation, that one could surely ignore this latter incompleteness. If, however, one did not want to do this, then one could construct the concept of the completely continuous by conceiving of something which—in contrast to all our examples of incomplete continuity—was such as to allow no further insertions. One would in fact need for this purpose simply to carry out a process wholly analogous to the one already considered. For even the halvings were not executed in toto; rather, the idea was formed of an execution which would differ from the finite execution actually achieved in being infinite. Thus one now forms,—in opposition to the groups of intercalations carried out in infinitum of all halvings, rational fractions, algebraic irrational fractions, transcendental fractions, etc., where there always remain positions open for new insertions—the conception of a complete realisation of this process, in relation to every possible relation of magnitude. There would then be given something which would present itself as a perfect example of continuity.

The attempted construction here described is similar to one that is to be found for example in the work of Poincaré,⁴ though it is not identical therewith. Poincaré begins, certainly, with the insertion of all rational fractions between two whole numbers such as 0 and 1. He then goes on in the manner of Dedekind (see the Addendum on pp.39 ff. below) to the intercalation of all irrationals, calling the series so attained a continuous series of the second order. He then appeals to the fact that one can distinguish infinitely small magnitudes, magnitudes which are infinitely smaller than other infinitely small magnitudes. (A point is related to a finite line as the line itself to a finite plane and this to a finite body, and in this way relations of magnitude can be conceived in infinitum with regard to ever infinitely greater magnitudes.) Thus, he believes, one can speak of a third, fourth, etc. order of continuity. This last idea seems to be peculiar to Poincaré.

It is also worthy of note that, although he credits some sort of validity to this construction, Poincaré nonetheless admits that it gives rise to a number of problems. Already, he suggests, the very possibility of rational fractions becoming infinitely smaller would presuppose that continua exist. And a further reservation seems to attach itself to the transition from the rational to the irrational numbers since one would be allowing oneself in the latter case to insert mere ‘symbols’. Thus after dividing up the series of rational fractions which contain squares into those which are smaller than 1/2 and those which are larger than 1/2, one discovers in relation to the series of their roots, that the one part has no largest member, the other has no smallest. As a consequence of this, the two series of roots are supposed to border not on each other but rather on a cut, which is symbolised by

l/2. (In fact it seems unjustified to assert that a series without a last member can border on a series with a first member but not on a series with no first member. It is only through such an assertion however that Dedekind arrives at the insertion of the irrational fractions. If one takes the whole series of rational and irrational fractions, then one finds that no two follow each other immediately, so that the series differs in this respect not at all from the series made up exclusively of the rationals, and I do not see why it should be more unacceptable in the one case than in the other to say that one could divide a series in two parts of which the one has no initial, the other no final member. According to Dedekind the rational fractions would certainly admit of being dissected in this way, not however the irrationals.) Finally, Poincaré makes the striking remark that it can be established in no other way than through convention what magnitude should be ascribed to the distance between one fraction and another. In the case of a true continuum, however, the magnitude would be given independently of convention.

The idea of continua of various degrees of completeness seems also to be incompatible with the true solution to the problem of constructing the continuum. If one raises all the rational and irrational fractions between 0 and 1 to some power, then one obtains precisely the series with which one started but with a certain displacement, and the same holds where all members which are either rational or irrational are themselves raised to a certain power. In this way the magnitudes of the distances between the fractions appear not to be determined by the magnitudes of the fractions.

(Poincaré’s two orders of continuity recall the two powers of Cantor. Yet the fact that there exist infinitely small magnitudes of a higher order cannot be regarded as a demonstration of a higher power, indeed the points of a surface for example are supposed to be of the same power as the points of a line, etc.)

(How, according to Cantor, is one to relate univocally the totality of the irrational points on a line to the totality of all its points, and how is one to relate the totality of transcendental irrational points to the totality of algebraic irrationals?)

3. Proceeding in this way, we should have to ascribe to the concept of continuity an origin in operations of thought both artificial and involved. This seems unacceptable from the very start, for how could this concept then be found in the possession of the simple man or even of the immature child? And further, how dubious it appears to suppose that the halvings and other divisions have been executed to an actual infinity, that they have been brought to completion, just because one can assume without absurdity that they have been executed beyond any arbitrarily determined limit. It is not to be denied that one is here accepting something simply impossible.

That one has indeed here posited something completely absurd is seen immediately if one splits the supposedly continuous series of all fractions between 0 and 1 into two parts at some arbitrary position. One of the two parts will then end with some fraction f, the second however could now start only if there were some fraction in the series which was the immediate neighbour of f, which is however not the case. With what, then, does the second series begin? With a multiplicity of fractions rather than just one? But this, too, is impossible since every fraction is distinguished from every other by a before or after in the series. But if not with a single fraction and not with a multiplicity of fractions then with what, since there is nothing to be found in the series other than fractions taken either singly or in groups? We should apparently have something that began but without having any beginning.

One sees that in this entire putative construction of the concept of what is continuous the goal has been entirely missed; for that which is above all else characteristic of a continuum, namely the idea of a boundary in the strict sense (to which belongs the possibility of a coincidence of boundaries), will be sought after entirely in vain. Thus also the attempt to have the concept of what is continuous spring forth out of the combination of individual marks distilled from intuition is to be rejected as entirely mistaken, and this implies further that what is continuous must be given to us in individual intuition and must therefore have been abstracted therefrom.

Someone might however object at this point that the failure of the attempt here presented would not rule out the possibility that some alternative might have greater success, might even possess the advantage of greater simplicity. One could say that something continuous was present where a whole was given that could be thought of as divided not, certainly, into infinitely many parts, but still in infinitum into parts. The first-mentioned view would be absurd, and would be guilty of absurdity already in the idea of the completion of the totality of possible halvings; it would be possible to show further that with the completion of infinite halving the magnitude 1/3 would have to be arrived at, when, as has been correctly emphasised, this magnitude cannot ever be reached. Not quite so absurd, however, would be the idea of a magnitude which can be divided in infinitum. And nothing would be easier than to show how one arrives at this idea, since it forms the contradictory opposite of that which can only be divided into a finite number of not further divisible parts. One could easily see also how, having been taught by experience that in the division of bodies one never reaches any boundary that is not in all probability capable of being breeched, one is led by habit to take for granted the idea that every body is divisible. And since certainly every division of a body leads to parts which are themselves bodies, one is led further to believe in the possibility of a division in infinitum, even if not by any power of one’s own. Of course this habitual assumption would as such not yet be justified from a scientific point of view. Yet science in no way contradicts it; rather it lends support to it in its own fashion by showing how the phenomena of experience can be reduced to quite simple general laws only on the basis of the hypothesis of the continuity of what is spatial. Without the assumption of continuity the assumption of a law of inertia would become untenable and the entire remarkable power of our mechanics would thereby be destroyed. Both the origin of the concept of that which is continuous and also its significance for the actual world would in this way be explained.

4. Even should one accept, however, that this theory avoids some of the cruder errors of the earlier view, it is still far from satisfying all requirements. Democritus believed in the extendedness of his atoms, not however in their physical divisibility, and it is a long familiar fact that when we distinguish parts in our thoughts we soon arrive at a certain limit. Moreover, everyone knows that in virtue of their small size corpuscles are far from being capable of being separately noticed, even if they are still very much capable of being divided physically. Yet we still award them continuity, and it seems therefore indubitable that being continuous and being divisible in infinitum are concepts that do not coincide in their content. We can remind ourselves also of what was said earlier about the peculiarity of boundaries and the possibility of their coincidence. He who does not show how we arrive at these ideas is not, either, allowed to flatter himself with having sufficiently clarified the idea of the continuous.

5. Thus I affirm once more, and with still less contestability, that the concept of the continuous is acquired not through combinations of marks taken from different intuitions and experiences, but through abstraction from unitary intuitions.

One does not need to search long, either, for the intuitions at issue, since, as I dare to assert and shall attempt to prove, it is much rather the case that every single one of our intuitions—both those of outer perception as also their accompaniments in inner perception, and therefore also those of memory—bring to appearance what is continuous. Thus in seeing we have as object something that is extended in length and breadth which at the same time shows itself clearly as allowing us to distinguish a front and rear side and thus as characterised as the two-dimensional boundary of something extended in three dimensions. And since this continuous something presents itself to us who see as being our primary object, we see also at the same time and as it were incidentally, our seeing itself, that is, we are conscious of ourselves as ones who see, and we find that to every part of the seen corporeal surface there corresponds a part of our seeing, so that we also, as seeing subjects, appear to ourselves as something continuously manifold. And still more, what appears to us first and foremost is rest and motion; so also persistence and gradual change appear to us as primary qualitative objects. This happens in that, whilst certainly in our perceptual presentation of the primary object we are never able to present the same place filled with two qualities simultaneously,1* still we are able to present it as filled with one quality as present, with another as most recently past, and with yet another as further past, whereby the transition from present to further past takes place in an entirely continuous manner. Thus once more we appear to ourselves, in seeing phenomenal qualities following each other in a temporally continuous way or in seeing them persisting continuously in time, as something that is continuously manifold.

Given, therefore, that what is continuous is present in every intuition, the whole question as to the origin of the concept seems to have been dealt with in the simplest way. We have after all seen that this concept is gained not through any intricate process of combination but rather in immediate fashion through simple abstraction from our intuition.

6. Against this, however, objections will be raised from certain venerable quarters which we should not leave unconsidered. Thus it is said that according to general consensus it belongs to the nature of the continuous that it can be divided in thought over and over again in infinitum. Now it is certain, and we ourselves have conceded this earlier, that we are no longer able to distinguish with our senses particles which do not attain a certain size. It is therefore only in our thoughts that we are supposed to be able to undertake halvings and any other sorts of divisions. But who is to guarantee that in thinking this we do not go off entirely into the transcendent and do not lose ourselves in nothing but chimeras? Only in so far as we can distinguish with our senses does experience act as guarantor. Beyond that, we seem to be lacking in surety altogether. Our thought-apperceptions may not correspond phenomenally to anything at all, in which case the divisibility would not be a genuine divisibility but would be entirely imaginary.

Further, we have said earlier that the concept of a boundary and

1* In the intuition of a body I present the body primarily in recto. I can also however, by presenting to myself primarily a presenter of this body, present the body in obliquo. And it is in this way that I present it when I have a memory of an earlier experience which consisted in my presenting it in perception. It is not my intention to deny, by what has been said in the text, that I can in the same way, while presenting a given place in recto filled with blue, present this same place in obliquo filled with red. Also it seems as though I am able simultaneously to present the same place in obliquo filled in multiple fashion with colour-qualities. Similarly, when I judge something, I am also able to think more than one other judger judging about the same question in ways diverging variously from my own.⁵

the possibility of a coincidence of boundaries is essential to the concept of what is continuous. It was precisely the fact that nothing of such a boundary or coincidence of boundaries resulted from the attempts at combination discussed above which was to serve as proof of their total failure. But if this is so, then it must be demanded that the attempt to clarify the origin of the concept of the continuous via abstraction from what is given intuitively does not suffer from the same defect. And yet it seems obvious that it does, for if it is conceded that parts of a certain smallness are no longer noticeable, then this is to concede also that points and also other boundaries, all of which do after all dispense with all magnitude in certain directions—as for example a line should have no breadth—could least of all be capable of being distinguished in intuition. But then it seems also incapable of being established via intuition that boundaries and a coincidence of boundaries should exist at all in what is intuited.

7. These objections seem at first to be very plausible; yet the means to refute them are found if one inquires more closely into the peculiarities which, according to experience, our noticing has, and takes these into account. If we imagine a chess-board with alternative red and blue squares, then this is something in which the individual red and blue areas allow themselves to be distinguished from each other in juxtaposition, and something similar holds also if we imagine each of the squares divided into four smaller squares also alternating between these two colours. If, however, we were to continue with such divisions until we had exceeded the boundary of noticeability for the indiv...

Table of contents

• CONTENTS
• Editors’ Introduction to the English Edition
• A Note on Brentano’s Terminology
• Part One THE CONTINUUM
• Part Two TIME AND TIME-CONSCIOUSNESS
• Part Three SPACE AND TIME
• Notes
• Index