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- mathematics and statistics online
- Kronecker in Contemporary Mathematics, General Arithmetic as a Foundational Programme
- Foundations of Mathematics
One makes sometimes a distinction between analogy and abstraction. A good example is given by J. Marquis, namely Dedekind's and Weber's work on algebraic number theory and algebraic functions. Another example is the transfer of algebraic laws and tools to logic in the works of G. Boole, A. Schroder, etc. Abstraction comes in play when several, and not only two, domains of entities or several classes of structures are a priori in question. Indeed, at a first step a theory is abstract when it has a priori a plurality of models.
The plurality criterion is indeed commonly used to distinguish between concrete or material axiomatics and abstract axiomatics, e. At a second step, domains of entities are neglected, while one considers a priori a plurality of structures along with their specific structure preserving rnorphisrns. Thus, searching after analogies involves an abstracting mind, if not yet a systematic use of the abstract method. Although I have taken examples mainly from modern mathematics, it must be stressed that abstraction is there from the very first beginning. It is only through a long habit that we consider positive integers as given intuitive concrete objects and geometrical figures as concrete spatial visualizations supporting the proof process.
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As Ch. Peirce, E. Husserl and J. The more advanced the abstraction process, the more concrete the abstract objects become—classes, structures, operations as such, functions as such, morphisms, categories, etc. Notably Frege rejected the Aristotelian dfaipeaig as being not the only sort of logical abstraction 24 and he dissented from the traditional view on concepts; he used mathematical tools, namely a functional relation and an equation for stating a putative logical definition of the concept of number.
In most elementary cases indeed a mathematical concept encompasses more thought-processes than only the logical subsumption, to which corresponds the set-theoretic operation of inclusion. I am just saying that, in mathematical practice, at any step, genuine mathematical stuff fills the logical move. This is why I have stressed hereinabove that the ascent is at once semantic and syntactic. Mathematical abstraction is a many-faceted and multi-leveled process and it leads to a sophisticated and branched hierarchy of mathematical concepts and operations.
Moreover it is not always the case that the more abstract a concept is the more undetermined it is. For instance, with just a general concept of set as a collection of any things one does not go far. It happens often that the more abstract is a structure the more overdetermined and stratified it is: axiomatics and category theory give many examples.
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Anyway, creative manipulation of symbols and diagrams does not dwell only on their drawings; it pertains their meanings and meaningful connections with other symbols and diagrams. There are really different levels of abstraction, even if there are connecting paths between levels. One must distinguish between axiomatics as a fruitful mathematical method and axiomatics as a putative foundation or useless mathematical ideology, which is an epiphenomenon harmful in teaching. In practice, it would be absurd to go without the axiomatic contributions: for instance Galois' theory has been deeply and effectively understood only after Dedekind's, Weber's and Artin's axiomatic presentations.
A systematic study of category theory then allows us to prove general results about any of these types of mathematical structures directly from the axioms of a category. Mathematics is always aiming at more and more general results about more and more complicated structures. I will now give a non-exhaustive list of other mathematical abstraction processes that interplay in mathematical thinking and actually illustrate the unceasing iteration of intertwining processes of setting up invariants, idealizing entities and procedures, transforming operations into objects thematizing , bringing to light analogies between sets, structures, categories, etc.
Representing an infinite numerical sequence by its law of recurrence. One gets the law by discarding concrete calculation and retaining only how one passes from any element n to its successor. One does not actually know all the elements of the sequence but one knows how to generate the sequence. Here it matters of finding out a rule of calculation, not a concept, but the rule dispenses with enumerating all the elements of the sequence like the concept of even integers dispenses with enumerating all the multiples of 2.
Discarding the specific nature of the elements forming a sequence, e. For that Dedekind invented the concept of chain. The level of abstraction is higher than in the example 1, because we are not concerned with a particular calculation law valid for one particular sequence but with a law type generating an order structure suitable for integers and for sequences of unspecified elements as well. Dedekind's definition shows that integers are a particular instantiation of a general structure; it indicates one way of linking abstraction and generalization.
Heinrich Weber gave, in , axiom systems for groups, and later on these axiom systems have been formalized and investigated in their own right.
Kronecker in Contemporary Mathematics, General Arithmetic as a Foundational Programme
We owe the abstract axiomatic characterization of the sequence of positive integers to Dedekind through the definition 73 of [Dedekind ]:. In set theory, morphisms are functions; in linear algebra they are linear transformations; in group theory, they are group homomorphisms; in topology, they are continuous functions, in manifold theory they are smooth functions functions having derivatives of all orders , and so on. Equivalent Cauchy sequences of rational numbers are identified for defining the concept of real number.
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Similarly, Prege used the process of forming equivalent, namely equinumerical classes for defining positive cardinal numbers. In particular, this way must not be confused with those listed in 2. Such structural definitions were not welcome in Prege's conception. Emil Artin later generalized this result to the case of Artinian rings rings satisfy the descending chain on ideals. Several levels of abstraction are crossed from the abstract concept of ring to Artin's theorem.
Another famous example is the classification of finite simple groups: every finite simple group belongs to one of four classes cyclic groups, alternating groups, classical Lie groups, sporadic simple groups. Classification is a down top process.
Going top down, the converse action is also a way to show the structure of an entity or a procedure by breaking it up into simple pieces: e. That shows deep connections between arithmetic and algebra: historically that was a result of the project, shared by Kronecker, Dedekind and Weber, to arithmetize algebra, i. When f is a bijection , a and f a are distinct but behave in the same way in the structure A and the structure B respectively , A and B being isomorphic.
This process turned out to be essential in category theory. As spotted by J. Each sort X is equipped with an internal relation of identity but there is no identity relation that would apply to all sorts. Probably the most fundamental action is thinking in terms of invariance; it operates in any mathematical area and corresponds to the task of isolating intrinsic or stable properties of the object under study.
Foundations of Mathematics
One wants indeed to study not only the structure of some entity but also how it behaves under transformations. A few examples are below, taken from arithmetic, geometry, algebra, topology, algebraic topology, and category theory. Felix Klein characterized a geometry by a set of geometric invariants under a given group of symmetries; e. Sylvester law of inertia: certain properties of the coefficient matrix of a real quadratic form homogeneous polynomial of degree 2 in a number n of variables remain invariant under a change of coordinates.
Translated into algebraic geometry that means that every algebraic set over a field can be described as the set of common roots to a finite number of polynomial equations. The normal subgroups of a certain group G are the subgroups of G invariant stable under the inner automorphisms of G. The dimension of a topological space is invariant under homeomorphism. Going further, one defines the homotopy category as the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps.
Two topological spaces X and Y are isomorphic in this category if and only if they are homotopy-equivalent. Then a functor on the category of topological spaces is homotopy-invariant if it can be expressed as a functor on the homotopy category. The question whether the abstraction process is logical or psychological gives rise to argument. I think the process has evidently a logical side and a psychological side, the latter being by now very much investigated by cognitive scientists and neuroscientists.
Prom the point of view of mathematical practice, abstraction is an indispensable tool of work and production. I have been interested here by the multiple ways of constructing and developing mathematical abstract objects, and I have tried to show which permanent actions are involved in all those ways. I had personally no example of a general procedure or entity, which would not involve abstraction at some level.
That leads me to think that generalization can sometimes be made without using abstracting processes, while any process of abstraction involves generalization. Moreover, different ways are simultaneously used in constructions of higher and higher levels.
Other ways are more specific of modern mathematics:. The abstraction process is open: new steps towards higher levels yielding more abstract, more sophisticated, and more encompassing concepts are to be expected. Jacobson, Springer-Verlag, , Tate, Reading, Mass.
Awodey , Steve , Structure in mathematics and logic: A categorical perspective, Philosophic Mathematica, 4 3 , Selected Readings, 2nd edn, Cambridge University Press, , Brouwer , Luitzen E. Burgess , John P. Cleary , John J. III, , Mathematische Werke, edited by R. Fricke, E. Ore, Braunschweig: Vieweg, vol. III, , trad. Frege , Gottlob , Die Grundlagen der Arithmetik.
Gentner , Dedre , Structure-mapping: A theoretical framework for analogy, Cognitive Science, 7, Lasson, Leipzig: Felix Meiner Verlag, Kreisel , Georg , Mathematical logic: Tool and object lesson for science, Synthese, 62 2 , Marquis , Jean-Pierre , Categorical foundations of mathematics, or how to provide foundations for abstract mathematics, The Review of Symbolic Logic, 6 1 , French translation by M.