Kantian idea of constructing concepts.

It stands as fundamental and key one for his conception (understanding) of mathematics. Although Kant's constructing is present, as we explained above, at the level of definitions and axioms, but only in mathematical activity constructing finds its essential meaning. In this case mathematics is conceived by Kant as a complex two-level (two-component) way of cognition. It begins with the creation by using definitions “pure sensible concept”.

Their specificity is that they are formed by “arbitrary synthesis”, i.e., contain some mathematical [mental/mind] action. Further, when we constructing concepts we implement descent to the level of (quasi)sensuality (imagination) and relating of the concept to universally valid contemplation — scheme. Here, as if upon the reverse reading (from left to right) of the Hume’s principle, happens the decoding (or construction) of the concept, i.e., the transition to a deep information level: from rational (declarative) level of the concept on contemplative (procedural pragmatic) level of schemes. This can be represented as an expansion of the original abstract concept to lower-level objects that are in some [space-time] environment and with which (therefore) we can perform certain mathematical operations. Actually exactly here the creative mathematical activity of the corresponding type is performed: geometric constructions, algebraic calculations or logical-mathematical proofs, each of which, in turn, represents a certain set of permissible in this environment local action – operations (like drawing the line, division of numbers, etc.). We can say that in this “descent” through sensory intuition the egress beyond the original concepts and the [synthetic] increment of knowledge occurs, as any [dynamic] action (as opposed to static concepts) is a synthesis of at least two views42. The result of this synthesis by reverse return (rise) on the cerebral (conceptual) level is fixed as a formal result of the construction, calculation or the proved theorem.
Schematically, mathematical acts can be represented as follows:

(Understanding) formalized approach
Concept A -------------------> Concept B
^
|
| "Constructing a
concept to descent" |
|
V "Return rise backwards"
(Imagination) |

Scheme A --------------------> Intuition B
mathematical act

Here, following by I. Lakatos (Lakatos, 1976), we should distinguish the actual mathematical activity as a certain sequence of mathematical actions in contemplative environments (space and time) in the lower part of the scheme and its logical design, which can be represented in the upper block of the scheme as a formal-logical transition (“conclusion”) from the concept (formula) A to the concept (formula) B. And the first can not be completely reduced to the second, as the task of the formal proof is not in the modeling of real mathematical activity (for example, as a process of mathematical constructions in the proof of the theorem on the sum of the angles of triangle), but in ensuring the logical rightness (correctness) of its implementation. Therefore, the structure of the real mathematical process differs from its logical design in some formal meta-language.

It is also clear that the situation in modern mathematics is much more difficult, since the above principle of abstraction can be applied iteratively, generating abstractions of increasingly higher levels. Accordingly, the specifying these abstractions “descents” will also be multi-level ones, and intermediate “descents” will likely be not “descents” to the level of sensuality (imagination), but on a preceding more specific rational level. However, the Kantian thesis about the need to “to make an abstract concept sensible” should be the principal one here, it suggests a final “descent” to the level of sensible intuition, for example, geometric drawing.

In his analysis of mathematical activity, Kant distinguishes two types of constructing: geometric and algebraic. Along with the ostensive (from Lat. Ostentus — showing) constructing, based on the spatial intuition, or the intuition in space, Kant also distinguishes founded by time intuition — symbolic construction, underlying the algebraic operations.

But mathematics does not merely construct magnitudes (quanta), as in geometry, but also mere magnitude (quantitatem), as in algebra, where it entirely abstracts from the constitution of the object that is to be thought in accordance with such a concept of magnitude. In this case it chooses a certain notation for all construction of magnitudes in general (numbers), as well as addition, subtraction, extraction of roots, etc. and, after it has also designated the general concept of quantities in accordance with their different relations, it then exhibits all the procedures through which magnitude is generated and altered in accordance with certain rules in intuition; where one magnitude is to be divided by another, it places their symbols together in accordance with the form of notation for division, and thereby achieves by a symbolic construction equally well what geometry does by an ostensive or geometrical construction (of the objects themselves), which discursive cognition could never achieve by means of mere concepts.

Turning to the analysis of more abstract algebra as a kind of mathematical activity we would like to draw attention to two things. Firstly, the use of “language of X-s and Y-s” or transition compared to the arithmetic meta-language of variables, which allows us to “work” not only with certain values as in arithmetic (e.g., distinguish even and odd numbers), but also with abstract values whose validity is spontaneous, i.e. with variables, is one of the major constituents of algebra. Secondly, the language of algebra allows expressing not only abstract symbols, but also [arithmetic] operations (“actions”), which can be done with these symbols, is no less important factor, although it almost falls out of the scope of the analysis.

Thus the algebraic language is, unlike declarative language that is used, for example, in metaphysics (“philosophy vs. mathematics”), a specific procedural language in which the possible ways to work with mathematical objects is fixed, i.e. how we need to make certain mathematical operations. Moreover, if earlier such procedural language was the prerogative of algebra only, now it applies to all branches of the modern, which has become super-abstract, mathematics: no logical-mathematical language without special characters for expressing operations on mathematical objects is possible. It is important to note that the symbolic constructing is much more transparent than ostensive, since in the latter actions are is not expressly “affect” but only “shown” (Wittgenstein) through their real implementation in geometric constructions, although they may be explicated by describing the methods of constructing, in meta-languages. Such codification of possible mathematical actions makes mathematics more rigorous, although it restricts its heuristic potential, as it becomes impossible to introduce new mathematical structures.

However, the symbolic constructing, as indicated by its title, is an abstract one in another respect too. In essence, by mathematical symbols of [algebraic] operations the latter are only coded, i.e. presented in abbreviated [symbolic] form, what suggests their real existence already beyond formulaic expressions. So, formulaic (symbolic) record of multiplying of two numbers “a × b” supposes the real action of multiplying of a and b, for example, by multiplying in column, which serves a la geometric construction.

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