Ueber die Hypothesen, welche der Geometrie zu Grunde liegen.

As I pounder the very basis of my thoughts about geometry based on the very fabric of our thinking minds, it has alway been a reductionist one in my mind, that the truth of the reality would a geometrical one.

The emergence of Maxwell’s equations had to be included in the development of GR? Any Gaussian interpretation necessary, so that the the UV coordinates were well understood from that perspective as well. This would be inclusive in the approach to the developments of GR. As a hobbyist myself of the history of science, along with the developments of today, I might seem less then adequate in the adventure, I persevere.

On the Hypotheses which lie at the Bases of Geometry.
Bernhard Riemann
Translated by William Kingdon Clifford

[Nature, Vol. VIII. Nos. 183, 184, pp. 14–17, 36, 37.]

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. 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.

For me the education comes, when I myself am lured by interest into a history spoken to by Stefan and Bee of Backreaction. The “way of thought” that preceded the advent of General Relativity.


Einstein urged astronomers to measure the effect of gravity on starlight, as in this 1913 letter to the American G.E. Hale. They could not respond until the First World War ended.

Translation of letter from Einstein’s to the American G.E. Hale by Stefan of BACKREACTION

Zurich, 14 October 1913

Highly esteemed colleague,

a simple theoretical consideration makes it plausible to assume that light rays will experience a deviation in a gravitational field.

[Grav. field] [Light ray]

At the rim of the Sun, this deflection should amount to 0.84″ and decrease as 1/R (R = [strike]Sonnenradius[/strike] distance from the centre of the Sun).

[Earth] [Sun]

Thus, it would be of utter interest to know up to which proximity to the Sun bright fixed stars can be seen using the strongest magnification in plain daylight (without eclipse).

Fast Forward to an Effect

Bending light around a massive object from a distant source. The orange arrows show the apparent position of the background source. The white arrows show the path of the light from the true position of the source.

The fact that this does not happen when gravitational lensing applies is due to the distinction between the straight lines imagined by Euclidean intuition and the geodesics of space-time. In fact, just as distances and lengths in special relativity can be defined in terms of the motion of electromagnetic radiation in a vacuum, so can the notion of a straight geodesic in general relativity.

To me, gravitational lensing is a cumulative affair that such a geometry borne into mind, could have passed the postulates of Euclid, and found their way to leaving a “indelible impression” that the resources of the mind in a simple system intuits.

Einstein, in the paragraph below makes this clear as he ponders his relationship with Newton and the move to thinking about Poincaré.

The move to non-euclidean geometries assumes where Euclid leaves off, the basis of Spacetime begins. So such a statement as, where there is no gravitational field, the spacetime is flat should be followed by, an euclidean, physical constant of a straight line=C?

Einstein:

I attach special importance to the view of geometry which I have just set forth, because without it I should have been unable to formulate the theory of relativity. … In a system of reference rotating relatively to an inert system, the laws of disposition of rigid bodies do not correspond to the rules of Euclidean geometry on account of the Lorentz contraction; thus if we admit non-inert systems we must abandon Euclidean geometry. … If we deny the relation between the body of axiomatic Euclidean geometry and the practically-rigid body of reality, we readily arrive at the following view, which was entertained by that acute and profound thinker, H. Poincare:–Euclidean geometry is distinguished above all other imaginable axiomatic geometries by its simplicity. Now since axiomatic geometry by itself contains no assertions as to the reality which can be experienced, but can do so only in combination with physical laws, it should be possible and reasonable … to retain Euclidean geometry. For if contradictions between theory and experience manifest themselves, we should rather decide to change physical laws than to change axiomatic Euclidean geometry. If we deny the relation between the practically-rigid body and geometry, we shall indeed not easily free ourselves from the convention that Euclidean geometry is to be retained as the simplest. (33-4)

It is never easy for me to see how I could have moved from what was Euclid’s postulates, to have graduated to my “sense of things” to have adopted this, “new way of seeing” that is also accumulative to the inclusion of gravity as a concept relevant to all aspects of the way in which one can see reality.

See:

  • On the Hypothese at the foundations of Geometry
  • Gravity and Electromagnetism?
  • “The Confrontation between General Relativity and Experiment” by Clifford M. Will
  • This entry was posted in Einstein, Emergence, Euclid, General Relativity, HENRI POINCARE and tagged , , , , . Bookmark the permalink.

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