Smart Remarks 2: Einstein, Noether, Russell, Lockwood, Churchland, Atiyah, Weyl


Emmy Noether

In the judgment of the most competent living mathematicians, Fraulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.


Similar light produces, under like conditions, a like sensation of color.


Thus “this is red,” “this is earlier than that,” are atomic propositions.

Russell & Whitehead

Bertrand Russell

Take some range of phenomenal qualities. Assume that these qualities can be arranged according to some abstract n-dimensional space, in a way that is faithful to their perceived similarities and degrees of similarity -- just as, according to Land, it is possible to arrange the phenomenal colors in his three-dimensional color solid. Then my Russellian proposal is that there exists, within the brain, some physical system, the states of which can be arranged in some n-dimensional state space ... And the two states are to be equated with each other: the phenomenal qualities are identical with the states of the corresponding physical system.


Thus the colors with their various qualities and intensities fulfill the axioms of vector geometry if addition is interpreted as mixing; consequently, projective geometry applies to the color qualities.


color sphere

Complete Intersection Calabi-Yau Manifolds
The class of complete intersection manifolds embedded
in products of ordinary projective spaces ...

We said at an earlier place, that every difference in experience must be founded on a difference of the objective conditions; we can now add: in such a difference of the objective conditions as is invariant with regard to coordinate transformations, a difference that cannot be made to vanish by a mere change of the coordinate system used.



How far does the laser move in color space?

Where does the yellow come in?

The actual sense-data are neither true nor false. A particular patch of colour, which I see, for example, simply exists: it is not the sort of thing which is true or false. It is true that there is such a patch, true that it has a certain shape and a certain degree of brightness, true that it is surrounded by certain other colours. But the patch itself is of a radically different kind from the things that are true or false, and therefore cannot properly be said to be true [...]

We shall say that we have acquaintance with anything of which we are directly aware without the intermediary of any process of inference or any knowledge of truths. Thus in the presence of my table I am acquainted with the sense-data that make up the appearance of the table -- its colour, shape, hardness, smoothness, etc. [...]

The particular shade of colour that I am seeing may have many things said about it -- I may say that it is brown, that it is rather dark, and so on. But such statements, though they make known truths about the colour, do not make me know the colour itself any better than I did before: so far as concerns knowledge of the colour itself, as opposed to knowledge of truths about it, I know the colour perfectly and completely when I see it and no further knowledge of it itself is even theoretically possible.


Discussion so far has been concentrated on the impressive computational power of coordinate transformations of state spaces and on the possible neural implementations of such activity. But it is important to appreciate fully the equally powerful representational capacity of neural state spaces. The global state of a complex system of n distinct variables can be represented by a single point in an abstract n-dimensional state space.

Churchland, PM

We shall now recall the data of a classical theory as understood by physicists and then reinterpret them in geometrical form. Geometrically or mechanically we can interpret this data as follows. Imagine a structured particle, that is a particle which has a location at a point x of R4 and an internal structure, or set of states, labeled by elements g of G.



Calabi-Yau space

The internal space defined at each space-time point 
is called a fibre, and the union of this internal space with 
space-time is called fibre-bundle space.

Epistemologically it is not without interest that in addition to ordinary space there exists quite an- other domain of intuitively given entities, namely the colors, which forms a continuum capable of geometric treatment.


It seems useful to me to develop a little more precisely the "geometry" valid in the two-dimensional manifold of perceived colors. For one can do mathematics also in the domain of these colors. The fundamental operation which can be performed upon them is mixing: one lets colored lights combine with one another in space [...]














More smarts 3: Leibniz, Maxwell, Turing, Churchland, Lockwood, Salam, Wigner




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