The partial piece quoted below is written by Michael Friedman at Stanford. Friedman also wrote on Davos, The Parting of Ways, Open Court, 2000, and Foundations of Space-Time Theories, Princeton Uni P, 1983. The latter is highly technical, (i.e. I couldn't understand a lot of it). In Foundations, Friedman is attempting a mathematical and philosophical critique of standard versions of Special and General Relativity, and proposes mathematical additions, motifications, and recombinations to the theory, essentially a mathematical tunning to answer his own philosophical critiques. It is very much in the Cassirer spirit. As mathematicians might say, it seemed quite elegant to me.
This work must have prepared him very well for Cassirer whom he must have encountered during his research on period works contemporary with the development of special and general relativity. Friedman obviously became a convert. His rendition given below of Substance and Function is the best I've read.
Now it may seem completely irrelevant to the recent Butler threads, the several different tracks Charles, Carrol, Shag, Jerry, Ted, Dennis, and others have taken. Trust me here. The relevance will build up over the next few posts.
I am going to post this under the subject Cassirer 1. The next post will be Cassirer 2 and will have a section from the table of contents from Substance and Function. The next post will be Cassirer 3 and have Friedman's sketch on The Philosophy of Symbolic Forms. The post after that sometime tomorrow or the next day will be long and be a general rambling on constructionism and try to tie all this together.
Just so we are clear, in other words so to center and align the categories and involved. Constructionism is a school of epistemology. and Cassirer's work bares heavily on the general concept of epistemology, that is on most theories of knowledge.
CG
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3. Philosophy of Mathematics and Natural Science
It was noted above that Cassirer's early historical works interpret the development of modern thought as a whole (embracing both philosophy and the sciences) from the perspective of the philosophical principles of Marburg neo-Kantianism, as initially articulated in [Cohen 1871]. On the `genetic' conception of scientific knowledge, in particular, the a priori synthetic activity of thought, the activity Kant himself had called `productive synthesis', is understood as a temporal and historical developmental process in which the object of science is gradually and successively constituted as a never completed `X' towards which the developmental process is converging. For Cohen, this process is modelled on the methods of the infinitesimal calculus (in this connection, especially, see [Cohen 1883]). Beginning with the idea of a continuous series or function, our problem is to see how such a series can be a priori generated step-by-step. The mathematical concept of a differential shows us how this can be done, for the differential at a point in the domain of a given function indicates how it is to be continued on succeeding points. The differential therefore infinitesimally captures the rule of the series as a whole, and thus expresses, at any given point or moment of time, the general form of the series valid for all times.
Cassirer's first `systematic' work, Substance and Function [Cassirer 1910], takes an essential philosophical step beyond Cohen by explicitly engaging with the late nineteenth-century developments in the foundations of mathematics and mathematical logic that exerted a profound influence on twentieth-century philosophy of mathematics and natural science. Cassirer begins by discussing the problem of concept formation, and by criticizing, in particular, the `abstractionist' theory characteristic of philosophical empiricism, according to which general concepts are arrived at by ascending inductively from sensory particulars. This theory, for Cassirer, is an artifact of traditional Aristotelian logic; and his main idea, accordingly, is that developments in modern formal logic (the mathematical theory of relations) allows us definitively to reject such abstractionism (and thus philosophical empiricism) on behalf of the genetic conception of knowledge. In particular, the modern axiomatic conception of mathematics, as exemplified especially in Richard Dedekind's work on the foundations of arithmetic and David Hilbert's work on the foundations of geometry, has shown that mathematics itself has a purely formal and ideal, entirely non-sensory and thus non-intuitive meaning. Pure mathematics describes abstract `systems of order' , what we would now call relational structures , whose concepts can in no way be accommodated within abstractionist or inductivist philosophical empiricism. Cassirer then employs this `formalist' conception of mathematics characteristic of the late nineteenth century to craft a new, and more abstract, version of the genetic conception of knowledge. We conceive the developmental process in question as a series or sequence of abstract formal structures (systems of order), which is itself ordered by the abstract mathematical relation of approximate backwards-directed inclusion (as, for example, the new non-Euclidean geometries contain the older geometry of Euclid as a continuously approximated limiting case). In this way, we can conceive all the structures in our sequence as continuously converging, as it were, on a final or limit structure, such that all previous structures in the sequence are approximate special or limiting cases of this final structure. The idea of such an endpoint of the sequence is only a regulative ideal in the Kantian sense , it is only progressively approximated but never in fact actually realized. Nevertheless, it still constitutes the a priori `general serial form' of our properly empirical mathematical theorizing, and, at the same time, it bestows on this theorizing its characteristic form of objectivity.
In explicitly embracing late nineteenth-century work on the foundations of mathematics, Cassirer comes into very close proximity with early twentieth-century analytic philosophy. Indeed, Cassirer takes the modern mathematical logic implicit in the work of Dedekind and Hilbert, and explicit in the work of Gottlob Frege and the early Bertrand Russell, as providing us with our primary tool for moving beyond the empiricist abstractionism due ultimately to Aristotelian syllogistic. The modern `theory of the concept,' accordingly, is based on the fundamental notions of function, series, and order (relational structure) , where these notions, from the point of view of pure mathematics and pure logic, are entirely formal and abstract, having no intuitive relation, in particular, to either space or time. Nevertheless, and here is where Cassirer diverges from most of the analytic tradition, this modern theory of the concept only provides us with a genuine and complete alternative to Aristotelian abstractionism and philosophical empiricism when it is embedded within the genetic conception of knowledge. What is primary is the generative historical process by which modern mathematical natural science successively develops or evolves, and pure mathematics and pure logic only have philosophical significance as elements of or abstractions from this more fundamental developmental process of `productive synthesis' aimed at the application of such pure formal structures in empirical knowledge (see especially [Cassirer 1907b]).
Cassirer's next important contribution to scientific epistemology [Cassirer 1921] explores the relationship between Einstein's general theory of relativity and the `critical' (Marburg neo-Kantian) conception of knowledge. Cassirer argues that Einstein's theory in fact stands as a brilliant confirmation of this conception. On the one hand, the increasing use of abstract mathematical representations in Einstein's theory entirely supports the attack on Aristotelian abstractionism and philosophical empiricism. On the other hand, however, Einstein's use of non-Euclidean geometry presents no obstacle at all to our purified and generalized form of (neo-)Kantianism. For we no longer require that any particular mathematical structure be fixed for all time, but only that the historical-developmental sequence of such structures continuously converge. Einstein's theory satisfies this requirement perfectly well, since the Euclidean geometry fundamental to Newtonian physics is indeed contained in the more general geometry (of variable curvature) employed by Einstein as an approximate special case (as the regions considered become infinitely small, for example). Moritz Schlick published a review of Cassirer's book immediately after its first appearance [Schlick 1921], taking the occasion to argue (what later became a prominent theme in the philosophy of logical empiricism) that Einstein's theory of relativity provides us with a decisive refutation of Kantianism in all of its forms. This review marked the beginnings of a respectful philosophical exchange between the two, as noted above, and it was continued, in the context of Cassirer's later work on the philosophy of symbolic forms, in [Cassirer 1927b] (see [Friedman 2000, chap. 7]).
Cassirer's assimilation of Einstein's general theory of relativity marked a watershed in the development of his thought. It not only gave him an opportunity, as we have just seen, to reinterpret the Kantian theory of the a priori conditions of objective experience (especially as involving space and time) in terms of Cassirer's own version of the genetic conception of knowledge, but it also provided him with an impetus to generalize and extend the original Marburg view in such a way that modern mathematical scientific knowledge in general is now seen as just one possible `symbolic form' among other equally valid and legitimate such forms. Indeed, [Cassirer 1921] first officially announces the project of a general `philosophy of symbolic forms,' conceived, in this context, as a philosophical extension of `the general postulate of relativity.' Just as, according to the general postulate of relativity, all possible reference frames and coordinate systems are viewed as equally good representations of physical reality, and, as a totality, are together interrelated and embraced by precisely this postulate, similarly the totality of `symbolic forms', aesthetic, ethical, religious, scientific, are here envisioned by Cassirer as standing in a closely analogous relationship. So it is no wonder that, subsequent to taking up the professorship at Hamburg in 1919, Cassirer devotes the rest of his career to this new philosophy of symbolic forms. (Cassirer's work in the philosophy of natural science in particular also continued, notably in [Cassirer 1936].)
From:
http://plato.stanford.edu/entries/cassirer/
Note. In this last paragraph I think you can already hear the reactionary howls of derision and hysteria about how liberalism and relativistic thought leads to a break down of tradtional values, etc, etc, etc.
The above noted imaginary world of the rightwing is quite literally a great example of mythological thinking, that to think a thought or express it, or even worse to enact such an idea into law in the name of human rights, is tantamount to making it empirically manifest. In effect passing same-sex marriage will turn the whole universe queer. The horror, the horror.