Bernard Bolzano: Study of science – Collections of representations

Bolzano's journey through characteristics of representations has reached the concept of characteristic itself. He really has no definition of the concept, except that a characteristic is something that an object has, no matter how temporary it is. This having, Bolzano thinks, has then no better explanation than being a relation of a thing to its characteristic (and not, say, possessing another thing, like when we say that I have money). It is then completely arbitrary, which of the two – characteristic or having – is taken as primary notion and which as defined, although Bolzano prefers having as the primary one. He also notes that spatial and temporal determinations are to not to be taken as characteristics, since in propositions they more conveniently characterise the subject position, while characteristics always fall to the predicate position.

Bolzano divides characteristics into internal and external or properties and relations. Of these, he insists, relation between things is actually a characteristic of the collection or the whole composed of the related things, for instance, the relation of three points forming an equilateral triangle is the characteristic of the collection of the three points, not of the points themselves, although we can say that the points have the (external) characteristic of being a part in a collection with such characteristic. Bolzano adds to his definition of relation that both the parts and the characteristic must be variable, because otherwise we would have to say that the primeness of number 13 is a relation, because it refers to the whole system of numbers.

Having defined relations and external characteristics, Bolzano can easily explain internal characteristics or properties as characteristics that are not external. From these definitions it easily follows that a relation between things is a property of their collection. Furthermore, Bolzano adds, an internal characteristic of even a simple thing can be seen as a relation, that is, between the thing and its characteristic (the having described earlier). He then briefly defines similarity or equality as a reciprocal relation, where objects have a same part in a characteristic belonging to the collection they form. A dissimilar or unilateral relation is then a relation that is not reciprocal.

Bolzano notes that with complex objects we can distinguish between their matter – the parts from which they are constituted – and their form – the manner in which the parts are combined. He notes that similar definitions can then be applied to representations of things, so that representation of their matter means representations showing constituent characteristics of the things and representation of their form means type of combination for these parts. Bolzano notes that the distinction between matter and form can be related to the distinction of internal and external characteristics. Matter of an object, he says, can always be determined as consisting just of internal characteristics, when the parts are thought as simple representations, but they can also be seen as external characteristics, if described by representations referring to other things: for instance, “not-human” consists of “not” and “human”, but also of “not” and “the most perfect type of living beings on Earth”. Then again, Bolzano thinks, some types of combinations in form can be represented only through relations.

We have already mentioned a couple of times the notion of a collection (Inbegriff). By this, Bolzano means simply a representation composed of other representations A, B, C…, where the order of these collected representations or parts of the collection is not yet considered: it might be relevant or irrelevant, depending on the type of collection. He also notes that we can either discuss this collection collectively – the whole formed of these parts – or distributively – each part of this collection (for instance, all players are a team, but each player is not). Bolzano also points out that sometimes we leave implicit whether the collection includes parts beyond those explicitly mentioned.

We just mentioned that the order of the parts might be irrelevant to the collection. In case it is explicitly so, Bolzano speaks of sets (Menge). Now, both in collections in general and in sets, the parts of parts are not usually parts of the collection or set – a set of people does not include their hands. Then again, Bolzano notes, there is an important type of sets – he calls them sums – where parts of parts are parts: lines are of this kind.

If the order of the parts does not matter in a set, a series is a collection where it does. Bolzano defines series through a law or rule, so that for any part or member M of the series can be found another member N, where either N is determined by the law from M or conversely. The N and M are then said to follow immediately one another, the determined member being later and the determining member earlier. Bolzano also defines internal members of a series as such that have both earlier and later members, while for external or limit members, either of them cannot be found. Limit members include the first or starting member that has no earlier member and the final or end member that has no later member.

Another important concept Bolzano mentions is that of unity or unit (Einheit can be translated in both ways), which is simply something that has a certain characteristic A (in the concrete sense) or then a property that makes something a unit of type A (in the abstract sense). Plurality in the concrete sense is then, for him, a collection of concrete units of some type A, while plurality in the abstract sense is the property, by which something is a concrete plurality. Bolzano notes that we could go on defining twos and threes and so forth by determining that the plurality is to have a unit and then a unit etc., but turns then to define a concrete whole or all as a collection that has each object belonging to some representation A and nothing else; similarly allness in abstract sense is a property that makes a concrete whole into such. He also points out that when the expression “all A” is used to mean A in general, it is used distributively, but here it is used collectively.

Bolzano’s account of the representations of set and series is quite close to later logicist ideas of defining basic mathematical concepts, so it is no surprise that he next tackles magnitudes. Types of magnitudes, he says, are characterised by the property that no matter which two of them are taken, they are either equal or have the relation of one of them being greater, that is, a whole with a part equal to the other. With this definition, Bolzano notes, pluralities, wholes and units can be seen as magnitudes.

Bolzano has something to say even about the notions of finite and infinite. A plurality of finite magnitude, he defines, is any plurality of type A that appears as a member in a series, where two of As is the first member and next member is reached from the previous by adding a new A. Plurality of infinite magnitude, on the other hand, is a plurality of type A, where each finite plurality of A appears only as a part. Furthermore, Bolzano defines number as a member of a series, the first member of which is a unit of any type A and each next member is a sum of previous with a new unit. Every finite plurality, he points out, is a number, while infinite pluralities are innumerable.

From these novel considerations, Bolzano moves on to more traditional logical notions, first of which is what he calls an exceptive representation, that is, a collection from which certain objects are excluded, either by individually naming the excluded objects or through general characteristic, for instance, by discussing a collection of all A that do not have the characteristic b, although Bolzano is not certain whether to really call this latter an exceptive representation. The second traditional notion Bolzano discusses is the concept of negation or “no”, which he considers undefinable. He defines all representations having “no” as constituent to be negative, although since two negatives cancel one another, he delineates a stricter sense of negative representations, which do not have an even series of negations – other representations are then affirmative.

Bolzano divides quite traditionally negative representations into two kinds. First of these he calls purely or completely negative representations. These deny a certain representation A without requiring any other representation in its stead – not even the very indeterminate representation of something. Such purely negative representations include, Bolzano notes, at least the concept of nothing. The other kind is, then, those of partially negative representations. In these, Bolzano explains, negation is only one of its constituents, like in the representation of A that is not B.

Bolzano closes off this section with the notion of symbolic representation. Quite appropriately, this notion is defined by the form “representation that has a characteristic b”. Thus, all the various concepts described in this section are of such a kind. Furthermore, all the concepts described are what Bolzano calls real or objective symbolic representations, since there have been representations having the described characteristics. He notes that we could also delineate the notion of mere symbolic representation, which are completely imaginary in the sense of having no objects: a notion of a representation that is also a judgement would be of such kind.

Immanuel Hermann Fichte: Outline of a system of philosophy. Second division: Ontology – Number

Last time, Fichte had ended with the concept of this (Dieses) that was empty or lacked all positive content and that was related to an equally empty this, and because they both lacked any content, they were not internally distinguished from one another. This concept, he continues, opens up a new field of thinking, where we have an infinite series of such “these” that have no internal, but only formal distinction and are thus unlimited in their similarity. Here, it is left expressly undetermined what internal relations the various “these” have to each other and whether they correspond or differ in some respect: they are just formally not same and are thus vanish in an infinity that is not separated by any content or qualitative difference. This being formally distinguished and qualitatively not distinguished, Fichte suggests, is the fundamental concept of quantity. Quantity, he adds, is the most formal or most abstract determination that still also leaves everything undetermined, since it does not yet point to any internally distinguishing quality.

Quantity is thus, for Fichte, the first of all proper categories or the most abstract manner of distinguishing beings. The basic characteristic of the quantity, he suggests, is the highest contradiction of abstraction where, on the one hand, we expressly affirm difference, but on the other hand, also expressly deny it again and resolve it into an unbroken, abstract similarity. This is the most formal level of thinking that does not yet determine anything, but still already tends toward determining and prepares for it the form of distinction. Fichte goes even so far as to suggest defining quantity as the unification or synthesis of formal distinguishing – indeed, of a possibility of infinite distinction – and of non-distinguishing, where internal distinctions are not permitted. To put it shortly, in quantity we can posit a limit, which is then immediately cancelled.

Fichte clarifies that what he has described here is especially the concept of pure quantity, which can then be regarded from different viewpoints, according to whether manifoldness or similarity is highlighted. This pure quantity, he suggests, should then be differentiated into determined quantities or magnitudes, which differ from pure quantity only insofar as they are results of determining quantity. In other words, magnitude still shares the general character of quantity that it is indifferent toward how it is determined. Therefore, Fichte explains, magnitude precisely negates the manner in which it is determined, as he thinks can be seen in the mathematical definition of magnitude: it is something that can be infinitely increased or diminished. Determined and pure quantities should thus contain the same contradiction that they are both determined and left undetermined.

Fichte reminds us that the universal meaning of all categories is to serve as fundamental determinations of the absolute. Thus, he notes, absolute could be defined in a very undeveloped manner as the pure quantity that infinitely comprehends or determines everything quantitative. While the pure quantity thus gives everything its quantity, it is not itself determined according to magnitudes or relations of quantity, but contains all of these as moments in itself. In other words, Fichte clarifies, the absolute is the unlimited that limits or measures everything else quantitatively, while itself it is measureless or infinite in the crudest sense of the word. He also suggests that we could give a more detailed meaning to this definition of absolute by relating it to the quantitative forms of intuition or space and time. Thus, Fichte continues, while absolute or God is thought as positing and filling space and time, as the universal essence it is not itself in space and time.

Fichte also suggests that we can  anticipate here the notion of indifference that is the one of the emptiest determinations of the absolute. Thus, the absolute as pure quantity should already be the abstraction from everything finite, including both quantitative and qualitative distinctions. In other words, absolute in this sense is the universal sphere and internal limit for distinctions, but is itself indifferent to these distinctions. Yet, Fichte notes, this notion of indifference is still just an anticipation, since we have not yet consciously discovered anything qualitative.

After these preliminary considerations, Fichte goes on to study in more detail magnitude as the first and thus the most universal form of all thinking of quantity: everything that is quantitative is at first to be determined by a magnitude, but abstractly and not as magnitude of any kind (e.g. not as magnitude of space or time). Now, Fichte continues, there is no internal distinction that would form any proper limits to the magnitude, or if there is such, it is expressly ignored. Thus, this immediate form of magnitude grasped is an unbroken, inseparable series or a continuous magnitude. What this continuity means, Fichte explains, is internal similarity or negation of any distinction posited in the magnitude formally: it is an infinite manifoldness that is immediately again united and resolved into similarity. Hence, continuous magnitude contains manifoldness only as a possibility, which could still be actually distinguished and separated into further magnitudes.

Continuous magnitude contains no distinction, but everything in it melts into a similar togetherness of what is distinguished, although still distinguishable. Fichte thinks that all external limitations must therefore also seem indifferent to the continuous magnitude: within and without the continuous magnitude, there is nothing qualitative that could limit its continuity that always remains the same. Thus, he notes, continuity is outwardly unlimited or infinite in the sense that it is a series that could be indifferently lengthened, while inwardly it is endlessly distinguishable or divisible, since its internal similarity allows an infinite possibility of distinctions.

Fichte notes that we can also emphasise the other side or multiplicity in the determination of magnitude or make distinguishability its prevalent characteristic. This leads us to the concept of a discrete magnitude. Discreteness, Fichte explains, is the formal separation of one (Eins) from another one, while this separation had vanished in the continuity. Fichte thus takes one as the fundamental element of concrete magnitudes. He also points out that discrete and continuous magnitudes are not two different kinds of magnitudes, but only complementing viewpoints on the same quantity. In other words, in magnitude taken as continuous, every distinction is extinguished into a similar, unbroken togetherness, while the same magnitude as discrete highlights multiplicity or infinitely distinguishable ones.

We have thus formally separated many ones, Fichte points out, but they also appear internally undistinguishable or similar. Thus, the moment of continuity is reproduced in discrete quantities, he thinks. Series of ones can be arbitrarily limited, but every limitation is again cancelled, just like in continuous magnitude. Difference between both notions of quantity is just that the one emphasises the internal multiplicity, while the other asserts the similarity, because multiplicity is just formal and not qualitative manifoldness.

Continuous and discrete magnitudes are, according to Fichte, only different viewpoints on quantity in general and on quantitative magnitudes in particular. Every quantity can thus be determined in this dual manner: on the one hand, we can grasp only its internal indistinguishability and make it a continuous quantity, on the other hand, we can highlight the possibility of infinite distinguishing and grasp it as discrete quantity splintered into an infinity of ones.

Fichte thinks that we can find a common expression for such a quantity that is from one respect similarity, from another respect distinguishability. This common expression, he suggests, is the number that flows into a continuity of endless multiplicity or an infinite series of similar ones, but also again collects these ones into discrete units in individual sets of ones. Number, Fichte describes, is the most comprehensible abstraction that formally emphasises distinction that at once becomes fully indifferent. Such a speculative category can exist, according to Fichte, only in the world of pure, abstract thinking. It is still the most effortless and easiest to handle, he adds, since it overlooks everything difficult and deep in thinking. Number encloses and governs every determination of thought and being, because it is the universal form of all determining and distinguishing. While we have not yet developed any internal distinction of things, Fichte emphasises, we can at least distinguish them according to numbers, and this opens up the road to the sphere of qualitative fundamental distinctions.

Fichte suggests that the principle of dialectics for numbers, leading to all numeric relations, is the opposition of continuity and discreteness that goes through everything quantitative and finds its most immediate reconciliation in numbers. Thus, all numeric relations appear from the double viewpoint, where numbers are regarded, on the one hand, continuous, on the other hand, discrete: either the multiplicity of internally similar ones can be again collected together and raised to a higher unity, or the ones are expressly fixated in their separation and enumerated as distinguishable.

In the further chapters, Fichte reveals, numbers will develop into more qualitative forms. As the common expression for all these further determinations of quantity, he explains, number can, despite its abstract position in the whole series of categories, still be a paradigmatic expression for properly qualitative relations or any determinations where it is not the question about merely quantitative. Thus, Fichte muses, if a symbol should be chosen to designate the eternal form of all determinations, the most fitting would certainly be the number, because it posits every distinction expressly as indifferent and can designate what is externally most formal in it.

Fichte returns from these general considerations of the nature of numbers to its further development. He points out that what was formerly designated as this (Dieses) is in its quantitative relation to others a mere one (Eins) related to other ones. This one, Fichte explains, is an empty abstraction that is still opposed to another, equally empty one. “One” is the simplest determination under the categories of quantity, he thinks, since everything in general can at least be called one, which can be thought only in relation to another one. Just like something (Etwas) was the fundamental concept of all determining, Fichte states, one as the quantitative expression of something is the fundamental element of numbers, which is determined only in relation to other ones.

With one, its relation to another one is already posited, Fichte continues: just like earlier developing the concept of something posited another, developing one leads to other ones. These other ones, Fichte explains, do not just appear contingently and form a plurality from an aggregate of externally collected units, but one is by its own nature comprehended in and beside other ones. Still, we at first see only the loosest, most external relation of ones to one another or an indeterminate, numeric multiplicity in general that can be increased and decreased. This plurality or “many”, Fichte emphasises, is the result of thoughtless not-counting or of unlimited and undetermined quantifying in general. Thus, it is the vaguest category of quantity.

Next, Fichte states, thinking proceeds to comprehend this empty manifold in allness: many are combined into a unity. One is thus comprehended not just in many ones, but in totality of ones that forms a synthesis to thesis of one and antithesis of many. Allness in in this sense also completely relative, Fichte thinks, since it is all just in relation to this particular set of combined similar ones. Therefore this relative allness is to be distinguished from absolute or conceptual allness, which, according to Fichte, we haven’t yet reached in relations of quantity.