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Substance and Function
Contents
Part I
The concept of thing and the concept of relation
Chapter I
on the theory of the formation of concepts
I. New Developments in logic.--The concept in Aristotelian logic.--Purpose and nature of the generic concept.--The problem of abstraction.--The metaphysical presuppositions of Aristotelian logic.--The concept of substance in logic and metaphysics...
II. The psychological criticism of the concept(Berkeley).--The psychology of abstraction.--Mill's analysis of mathematical concepts.--The defect of the psychological theory of abstraction.--The forms of series.--The place of the thing-concept in the system of logical relations...
III. The negative process of ``abstraction.''---The mathematical concept and its ``concrete universality.''--The criticism of the theories of abstraction.--Objects of the ``first'' and ``second'' orders.---The variety of objective ``intentions.''--The serial form and the members of a series...
Chapter II
The concept of number
I. The sensationalistic deduction of number.--Frege's foundations of arithmetic.--The system of arithmetic.--Number and presentation.--The content of presentation and the act of presentation.
II. The logical foundation of the pure concept of number (Dedekind).---The logic of relations.--The concept of progression.--Number as ordinal number.--The theories of Helmholtz and Kronecker.--Criticism of the nominalistic deduction.
III. Number and the concept of class.--Russell's theory of cardinal number.--Criticism of ``class theories.''---The logical definition of the zero and unity.--The presuppositions of the class concept.--The generic concept and the relational concept.
IV. Extension of the concept of number.--Gauss' theory of the negative and imaginary numbers.--The irrational numbers.--Dedekind's explanation of the irrational numbers.--The problem of the transfinite numbers.--The concept of ``power.''--The production of transfinite numbers.--The second ``principle of generation'' of numbers (Cantor)
Chapter III
The concept of space and geometry
I. Concept and form.--The method of ancient geometry.--The concept of space and the concept of number.--The fundamental principle of analytic geometry.--The infinitismal geometry.--Magnitude and functions.
II. Intuition and thought in the principles of the geometry of position...
III. Characteristic (Kombinatorik), as ``doctrine of forms''(Leibniz).--Geometry as pure ``doctrine of relations'' (Hilbert).--The syntheses of generating relations....
IV. The problem of metageometry.--The attempt at an empirical ground of geometry (Pasch).--Ideal objects in empirical geometry.--Veronese's modification of empiricism.--Rationalism and empiricism.--Mathematical space and sensuous space.--Objections to the Kantian theory of geometry.--Real space and experiment.--The conceptual principles of pure space.--Euclidean space and other forms of mathematical space.--Geometry and reality.
Chapter IV
The concepts of natural science
I. The constructive concepts and the concepts of nature.--The concept of traditional logic and the scientific ideal of pure description.--The apparent logical ideal of physics.--Is this the true ideal of physics?.
II. Numbering and measuring as presuppositions.--Mechanism and the concept of motion.--The ``subject'' of motion.--The ``limiting concept'' and its significances for natural science (Karl Pearson).--P. du Bois-Reymond's theory of the limiting concept.--The problem of existence.--The existence of the limiting point.--Logical idealism on the problem of existence.--Consequences of the confusion of truth and reality.--The ``idealization'' of presentations.--The relation of the ideal and reality.
III. The problem of the physical method and its history.--The problem of knowledge (Plato).--The sceptical theory of knowledge (Protagoras, etc.)--The concept of nature and purpose (Plato).--Mathematics and teleology (Plato, Aristotle, Kepler).--The concept of hypothesis (Kepler and Newton).--The logical and ontological ``hypothesis.''
IV. Robert Mayer's methodology of natural science.--Hypotheses and natural laws.--The presuppositions of physical ``measurement.''--The physical ``fact'' and the physical ``theory.''--Units of measurement.--The verification of physical hypothesis.--The motive of serial construction.--The physical concepts of series.
Here I am going to stop because it is getting too long. Below is a paragraph on his analysis of the Aristotelian concept:
``...Nothing is presupposed save the existence of things in their inexhaustible multiplicity, and the power of the mind to select from this wealth of particular existences those features that are common to several of them. When we thus collect objects characterized by possession of some common property into classes, and when we repeat this process upon higher levels, there gradually arises an ever firmer order and division of being, according to the series of factual similarities running through the particular things. The essential functions of thought, in this connection, are merely those of comparing and differentiating a sensuously given manifold. Reflection, which passes hither and thither among the particular objects in order to determine the essential features in which they agree, leads of itself to abstraction. Abstraction lays hold upon and raises to clear consciousness these related features,---pure, by themselves, freed from all admixture of dissimilar elements. Thus the peculiar merit of this interpretation seems to be that it never destroys or imperils the unity of the ordinary view of the world. The concept does not appear as something foreign to sensuous reality, but forms a part of this reality; it is a selection from what is immediately contained in it....'' (from The theory of the formation of concepts, p4-5)
The import of the above is that it forms the philosophical basis for the axiom of comprehension, under the map: concept -> set.
To paraphrase the axiom, a set is a collection of elements with a shared characteristic. The shared characteristic is given the mathematicaly expression of a relation between the elements. An equivalence relation for example is such that all elements of the set are said to be equal under the given relation. For instance, every object that can be labeled as blue in color forms the set of blue objects in the universe of discourse.
So what is Cassirer up to in the above? He is prefacing the most elementary of all concepts, the concept of number.
On a deeper level Cassirer is making his opening moves to destroy the primacy of place for the Aristotelian object and the Aristotelian substance as the primordial stuff of reality. And, under it all is the development of the idea that perhaps more important to understanding in the theory of knowledge is not objects and substance, but the transcendent symbolic realm of concepts like relations, operations, and transformations, or mappings, and that a more thorough going understanding and examination of these may help to unravel the mystery of thought and language.
How can I sum this up? Cassirer apprehended what he thought to be the most radical intellectual revolution of the 20thC, by understanding that the most widespread of implications to Einstein's theory of relativity was found in its potential generalization across many realms of human activity, such that the multiplicity of these human worlds could be made related endeavors, that is to say both understandable and intelligible, if only we could find the appropriate mappings between them. Cassirer theorized the direction such a scheme of transformative relations might follow in his Philosophy of Symbolic Forms.
In contrast to what I think of as a grand scheme for progress in thought and the material reality of society, instead we have been made to live under generation after generation of reactionary power elites and their intellectual hacks who maintain almost the exact converse. That is to say, that our own limited and bounded frame of reference (our culture, our state, our science, our thought, our value system) is utterly unique and god given and that there is nothing out there that shares the slightest similarity or humanity to our own special point of origin and point of view. And further, that there is an inhumanity and godlessness to any thought that claims a mapping (or approximating) scheme for human existence that embraces its multitudes and the multitude of cultural expressions, that is to say their states of being.
So it follows almost without saying, since we (or rather our power elite and their hacks) are hegemon, it's our personal, political, social, economic, military, and cultural teleos and duty to conquer and erase the Other....
CG