Carnap's Objection 25

In "Eigentliche und Uneigentliche Begriffe" ("Proper and Improper Concepts”) (1927), Carnap distinguishes between concepts that are defined concretely and those that are defined implicitly. The former he calls "proper", and the latter "improper". Distinguishing them, he writes:

Empirical concepts are constituted step by step in the systematic construction (Aufbau) of our world-knowledge. Each empirical concept, as a component of this structure, has a direct connection to reality. In contrast, the improper concepts hang in the air, so to speak, awaiting instructions. They are introduced by an axiom system, but that system doesn't relate directly to anything real. The axioms of this system and the theorems deduced from it don't properly form a theory (as they aren't actually about anything in

- 26 -

particular), but rather just a theory-schema, an empty framework for possible theories,

Carnap, R., 1927, translated and referenced in Awodey & Carus, 2001, p.149

Carnap’s principle objection here is that a network of implicit definitions

doesn’t say anything about the world. Just as the Peano axioms define the same structure in this universe or an empty one, the axioms of QED define the same system whether or not there are any such things as electrons. The consequences of this are that, although we can use QED to make predictions about electrons when electrons have been determined to have a suitable structure, this determination itself needs to be made in experience. We can define any number of axiom systems

in the abstract, but none of them can be said to define “electron” until we have

actually investigated electrons and found out what properties they have. The problem here is related to, but different from, the objection of Frege’s that we discussed above, made against Hilbert. Frege had objected that points and lines might have some properties that are not captured in the axioms that are supposed to define them, but we’ve argued that Benacerraf was right to say that mathematical terms simply have no properties other than those that relate them to other mathematical terms. Carnap’s objection here is not that electrons have some existence independent from their relational properties – as we will see, he thinks that structure is sufficient for the definition of any scientific term – but he does worry about whether or not anything in reality instantiates the relational properties stipulated in an implicit definition, and thus argues that for a structure to adequately describe a concept we must understand what it would be for that structure to be found in the world.

Carnap illustrates his point by considering the ways we can apply the Peano axioms in the world. The axioms are relevant any time we have a sequence with a start but no end, no repetitions and no loops. We can define such a sequence along the edge of a desk. We can take one corner as 0, the halfway point to the other corner as 1, halfway again as 2, and so on, getting half way closer to the edge of the table each time but allowing for an infinite number of points to be picked out in sequence.

- 27 -

With this definition in place we can now use the predetermined results of arithmetic to calculate things about the edge of the desk. If a snail is crawling along the desk at a rate of half-way-to-the-end per minute, we know that it will reach the point

designated by “10” at the end of the tenth minute. But creating this mapping from table edge to integers doesn’t mean that points on the table edge are integers. The

Peano axioms don’t now come to define the table. The concepts of Peano arithmetic

are useful when such a sequence is found, but on their own they don’t denote

anything in the real world. Nothing intrinsically is a number because the properties of numbers are not the kinds of thing that objects in reality instantiate, and likewise, Carnap says, no real thing can be defined by an abstract structure from implicit definitions. In contrast, for proper concepts we don’t need to say “such-and-such falls under this concept if we interpret it in a certain way”. We don’t say “Fido is a

dog if we can find a way to map his DNA onto our Dog Concept DNA” –we say “we

investigated Fido, found that he possessed the characteristics of a dog, and therefore

have determined that he is a dog”.

Carnap’s position was that the only way a concept could really be defined was by some link to empirical experience. If an object can be shown to have the properties attributed to a variable in the axiom system, suitably interpreted, then we have a realisation of that system and can use the axioms to make predictions. If we can show that electrons have the right properties, we can use QED to predict their behaviour. But it is essential that showing that they have the right properties occurs, and this can only be done by experience. However, Schlick’s arguments against the use of intuition to ground scientific concepts still stand, and so the role of experience can’t depend on the intuitive content of that experience.

Carnap did suggest an alternative to implicit definitions that would not result in Intuition re-entering the picture as a component of knowledge. In Der Logische Aufbau Der Welt (1928, [1967, §13-15]; henceforth “the Aufbau”) (which was in development and known to Schlick several years before publication), Carnap develops the notion of structure descriptions. Where implicit definitions serve to define properties in terms of networks of relations between one another, structure

- 28 -

descriptions pick out individual objects by the relations they hold to one another. Instead of definition by axioms, relations are described in totality by listing the objects between which they hold. This gives an extensionally-adequate way of picking out objects uniquely unless they are equivalent in all respects that science can distinguish, and if that is the case then science must treat those objects as identical. This picking out was to be done in experience by looking at the relations that actually hold between objects and determining how the network of relations in the world could be made isomorphic with the network of relations in the structure description.

The advance that structure descriptions introduce which takes them beyond implicit definitions is a way of bringing experience back into the picture without thereby introducing the imprecision that so bothered Schlick. Carnap’s complaint was that implicit definitions don’t have an appropriate link to empirical facts, but

Schlick’s arguments about the imprecision of intuitive content still carry the same

force that they did before. The response, therefore, is not to reintroduce intuitive content to the picture of scientific knowledge, but rather to purge what little of it remains in our account of meaning. After Carnap’s criticisms are taken on board, even terms that we might think could be defined straightforwardly by ostension, such as “blue”, will in fact be picked out by structure.

Schlick appears to endorse Carnap’s account shortly after publishing the second edition of AE, writing “Cf the acute and irrefutable remarks of Rudolf Carnap in his forthcoming work, Der Logische Aufbau Der Welt, where he shows that all

scientific judgements must confine themselves to purely structural assertions,”

(1926, [1979b, p.111]; Uebel, 2007, pp.75-76), and, as we noted before, the term

“implicit definitions” does not appear in his published work after Carnap’s criticisms

emerged. Nevertheless, many of the commitments which drew Schlick towards implicit definitions remain in place through his later philosophy. In particular, the distinction between knowledge and acquaintance, along with the resistance to the idea that perceptual images themselves enter into knowledge, remains in Schlick’s

- 29 -

Carnap’s objection convinced Schlick of the need to more closely tie experience to the system of scientific concepts, but the problem wasn’t that

formalisation had removed intuitive experience from the picture – it was that effecting a separation between the everyday language and the language of science made concepts in the language of science only ambiguously related to the things we find in everyday life. The solution to this was not to be found by relating the concepts of science back to intuitive experience, but rather by unifying the languages and ensuring that experience could be understood in a structural way rather than by ostensive definition.