Isn't Lukács usually read through the lens of the discovery of Marx's early writings and the debate between 'early' and 'later' Marx? — John Doe
Interesting! My reason for thinking that Lukács is dealing at bottom with what he takes to be a moral concern is that Kant explicitly states (if I read him correctly?) that his metaphysics is ultimately justified on moral grounds (in the First Critique). I guess it's probably impossible to come to any definite conclusion about this underdetermined aspect of Lukács's paper but I would love to pursue the point a bit.
Basically, my reasoning for thinking that Lukács finds an inherent connection between idealism and bourgeois morality (which perpetuates a politically retrogressive ideology) was that he's playing off of this connection that Kant draws between his morality and his metaphysics.
For example, from the closing sections of the first Critique: "Hence theology and morality were the two incentives, or better, the points of reference for all the abstract inquiries of reason to which we have always been devoted" (A853/B881).
So I think you're right that since I'm not a Marxist I may be missing the fact that thinking of metaphysics and morality as in this sense more fundamental than politics and ideology is a distinctly bad way of reading a Marxist thinker.
I like your way of looking at this but wonder if you might expand a tad? Definitely curious about how to work within the Marxist framework here to think through the difference between morality and the political order, as well as whatever personal views you might have about tensions in Marx's nihilism. — John Doe
It is known that geometry assumes, as things given, both the notion of space and the first principles of constructions in space. She gives definitions of them which are merely nominal, while the true determinations appear in the form of axioms.
The relation of these assumptions remains consequently in darkness; we neither perceive whether and how far their connection is necessary, nor a priori, whether it is possible.
The basic idea is this: Riemann is saying that space as we know it - the kind of space in which me move around and live - is but a particular case of a more general notion of space which can be constructed from 'general notions of magnitude'. Or, put the other way, one can construct more kinds of spaces out of 'general notions of magnitude' than only the kind of the space in which we live in. — StreetlightX
From Euclid to Legendre (to name the most famous of modern reforming geometers) this darkness was cleared up neither by mathematicians nor by such philosophers as concerned themselves with it. The reason of this is doubtless that the general notion of multiply extended magnitudes (in which space-magnitudes are included) remained entirely unworked. I have in the first place, therefore, set myself the task of constructing the notion of a multiply extended magnitude out of general notions of magnitude. It will follow from this that a multiply extended magnitude is capable of different measure-relations, and consequently that space is only a particular case of a triply extended magnitude. But hence flows as a necessary consequence that the propositions of geometry cannot be derived from general notions of magnitude, but that the properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience. Thus arises the problem, to discover the simplest matters of fact from which the measure-relations of space may be determined; a problem which from the nature of the case is not completely determinate, since there may be several systems of matters of fact which suffice to determine the measure-relations of space - the most important system for our present purpose being that which Euclid has laid down as a foundation. These matters of fact are - like all matters of fact - not necessary, but only of empirical certainty; they are hypotheses. We may therefore investigate their probability, which within the limits of observation is of course very great, and inquire about the justice of their extension beyond the limits of observation, on the side both of the infinitely great and of the infinitely small.
From Euclid to Legendre (to name the most famous of modern reforming geometers) this darkness was cleared up neither by mathematicians nor by such philosophers as concerned themselves with it.
The reason of this is doubtless that the general notion of multiply extended magnitudes (in which space-magnitudes are included) remained entirely unworked.
I have in the first place, therefore, set myself the task of constructing the notion of a multiply extended magnitude out of general notions of magnitude.
It will follow from this that a multiply extended magnitude is capable of different measure-relations, and consequently that space is only a particular case of a triply extended magnitude.
hence as a necessary consequence that the propositions of geometry cannot be derived from general notions of magnitude
the properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience.
Thus arises the problem, to discover the simplest matters of fact from which the measure-relations of space may be determined; a problem which from the nature of the case is not completely determinate, since there may be several systems of matters of fact which suffice to determine the measure-relations of space - the most important system for our present purpose being that which Euclid has laid down as a foundation.
These matters of fact are - like all matters of fact - not necessary, but only of empirical certainty; they are hypotheses. We may therefore investigate their probability, which within the limits of observation is of course very great, and inquire about the justice of their extension beyond the limits of observation, on the side both of the infinitely great and of the infinitely small.
It is known that geometry assumes, as things given, both the notion of space and the first principles of constructions in space. She gives definitions of them which are merely nominal, while the true determinations appear in the form of axioms. The relation of these assumptions remains consequently in darkness; we neither perceive whether and how far their connection is necessary, nor a priori, whether it is possible.
From Euclid to Legendre (to name the most famous of modern reforming geometers) this darkness was cleared up neither by mathematicians nor by such philosophers as concerned themselves with it.
it. The reason of this is doubtless that the general notion of multiply extended magnitudes (in which space-magnitudes are included) remained entirely unworked
it. The reason of this is doubtless that the general notion of multiply extended magnitudes (in which space-magnitudes are included) remained entirely unworked — Moliere
I think that the "darkness" referred to above is the relations between the assumptions geometry has -- I take it he means the 5 postulates of Euclids system as the primary example, though he does allude to the thought that there can be other axioms. As I read him here it seems that Reimann is motivated to understand the possible justification for just these axioms, and wants to understand the relationship they have to one another -- whether they are necessary, whether they are universal, and whether they are even possible. — Moliere
I was reading another paper about Reimann online, and from it I gleaned that I was sort of misreading "manifoldness" -- whereas I was sort of analogizing it with a vector before -- which I really only understand to be a magnitude with a direction -- it seemed to make more sense that Reimann is actually talking about the coordinate system itself. So Euclid or Cartesian coordinates are one example of a manifold, but the manifold could differ from these. — Moliere
Magnitude-notions are only possible where there is an antecedent general notion which admits of different specialisations. According as there exists among these specialisations a continuous path from one to another or not, they form a continuous or discrete manifoldness; the individual specialisations are called in the first case points, in the second case elements, of the manifoldness.
Notions whose specialisations form a discrete manifoldness are so common that at least in the cultivated languages any things being given it is always possible to find a notion in which they are included.
(Hence mathematicians might unhesitatingly found the theory of discrete magnitudes upon the postulate that certain given things are to be regarded as equivalent.)
On the other hand, so few and far between are the occasions for forming notions whose specialisations make up a continuous manifoldness, that the only simple notions whose specialisations form a multiply extended manifoldness are the positions of perceived objects and colours. More frequent occasions for the creation and development of these notions occur first in the higher mathematic.
Definite portions of a manifoldness, distinguished by a mark or by a boundary, are called Quanta.
Their comparison with regard to quantity is accomplished in the case of discrete magnitudes by counting, in the case of continuous magnitudes by measuring.
Measure consists in the superposition of the magnitudes to be compared; it therefore requires a means of using one magnitude as the standard for another. In the absence of this, two magnitudes can only be compared when one is a part of the other; in which case also we can only determine the more or less and not the how much.
The researches which can in this case be instituted about them form a general division of the science of magnitude in which magnitudes are regarded not as existing independently of position and not as expressible in terms of a unit, but as regions in a manifoldness.
Such researches have become a necessity for many parts of mathematics, e.g., for the treatment of many-valued analytical functions; and the want of them is no doubt a chief cause why the celebrated theorem of Abel and the achievements of Lagrange, Pfaff, Jacobi for the general theory of differential equations, have so long remained unfruitful.
Out of this general part of the science of extended magnitude in which nothing is assumed but what is contained in the notion of it, it will suffice for the present purpose to bring into prominence two points; the first of which relates to the construction of the notion of a multiply extended manifoldness, the second relates to the reduction of determinations of place in a given manifoldness to determinations of quantity, and will make clear the true character of an n-fold extent.
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